Physical Optics: Concepts, Optical Elements, and Techniques

  • 94 168 2
  • Like this paper and download? You can publish your own PDF file online for free in a few minutes! Sign Up

Physical Optics: Concepts, Optical Elements, and Techniques

UNITEXT for Physics Giovanni Giusfredi Physical Optics Concepts, Optical Elements, and Techniques UNITEXT for Physic

2,219 551 10MB

Pages 946 Page size 453.544 x 683.151 pts Year 2019

Report DMCA / Copyright

DOWNLOAD FILE

Recommend Papers

File loading please wait...
Citation preview

UNITEXT for Physics

Giovanni Giusfredi

Physical Optics Concepts, Optical Elements, and Techniques

UNITEXT for Physics Series Editors Michele Cini, University of Rome Tor Vergata, Roma, Italy Attilio Ferrari, University of Turin, Turin, Italy Stefano Forte, University of Milan, Milan, Italy Guido Montagna, University of Pavia, Pavia, Italy Oreste Nicrosini, University of Pavia, Pavia, Italy Luca Peliti, University of Napoli, Naples, Italy Alberto Rotondi, Pavia, Italy Paolo Biscari, Politecnico di Milano, Milan, Italy Nicola Manini, University of Milan, Milan, Italy Morten Hjorth-Jensen, University of Oslo, Oslo, Norway

UNITEXT for Physics series, formerly UNITEXT Collana di Fisica e Astronomia, publishes textbooks and monographs in Physics and Astronomy, mainly in English language, characterized of a didactic style and comprehensiveness. The books published in UNITEXT for Physics series are addressed to graduate and advanced graduate students, but also to scientists and researchers as important resources for their education, knowledge and teaching.

More information about this series at http://www.springer.com/series/13351

Giovanni Giusfredi

Physical Optics Concepts, Optical Elements, and Techniques

123

Giovanni Giusfredi European Laboratory for Non-Linear Spectroscopy (LENS) Istituto Nazionale di Ottica—Consiglio Nazionale delle Ricerche (INO-CNR) Sesto Fiorentino, Italy

ISSN 2198-7882 ISSN 2198-7890 (electronic) UNITEXT for Physics ISBN 978-3-030-25278-6 ISBN 978-3-030-25279-3 (eBook) https://doi.org/10.1007/978-3-030-25279-3 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland



,QWKHEHJLQQLQJZDVWKH/zJRV WKH/zJRVZDVZLWK*RG DQGWKH/zJRVZDV*RG  +HZDVZLWK*RGLQWKHEHJLQQLQJ  7KURXJKKLPDOOWKLQJVFDPHLQWREHLQJ  QRWRQHWKLQJFDPHLQWREHLQJ  H[FHSWWKURXJKKLP  :KDWKDVFRPHLQWREHLQJLQKLPZDVOLIH  OLIHWKDWZDVWKHOLJKWRIPHQ  DQGOLJKWVKLQHVLQGDUNQHVV  DQGGDUNQHVVFRXOGQRWRYHUSRZHULW  «  7KH/zJRVZDVWKHUHDOOLJKW  WKDWJLYHVOLJKWWRHYHU\RQH  KHZDVFRPLQJLQWRWKHZRUOG  «  7KH/zJRVEHFDPHIOHVK  KHOLYHGDPRQJXV  DQGZHVDZKLVJORU\  WKHJORU\WKDWKHKDVIURPWKH)DWKHU  DVRQO\6RQRIWKH)DWKHU  IXOORIJUDFHDQGWUXWK  «  1RRQHKDVHYHUVHHQ*RG  7KHRQO\6RQ*RG  ZKRLVFORVHWRWKH)DWKHU¶VKHDUW  KDVUHYHDOHGKLP  )URPWKHPrologue, Gospel of John -RKQ  

v



)RUHZRUG 2SWLFVLVHYHU\ZKHUHLQRXUGDLO\OLYHV-XVWLPDJLQHZKDWWKH(DUWKZRXOGEH ZLWKRXW OLJKW /LJKW LV WKH SULPDU\ VRXUFH RI HQHUJ\ IRU OLIH RQ RXU SODQHW ,W LV DOVRRXUSULPDU\FKDQQHORIFRPPXQLFDWLRQQRWRQO\WKURXJKYLVLRQEXWDOVRLQ WUDQVPLVVLRQVRIRSWLFDOGDWDWKURXJKVXEWHUUHVWULDORUVXEPDULQHILEHURSWLFQHW ZRUNVDQG³LQYLVLEOHOLJKW´UDGLRFRPPXQLFDWLRQVLQWKHDWPRVSKHUH/LJKWLVDOVR RQH RI WKH SULPDU\ LQYHVWLJDWLRQ WRROV LQ VFLHQFH IURP PLFURVFRS\ LQ ELRORJLFDO VFLHQFHV WR 8QLYHUVHVFDOH DVWURQRPLFDO REVHUYDWLRQV LQ DVWURSK\VLFV 0DVWHULQJ WKHVWXG\RIOLJKWDQGLWVLQWHUDFWLRQZLWKPDWWHUHQDEOHGWKHVFLHQWLVWVRIWKH;; FHQWXU\WRXQYHLOWKHIXQGDPHQWDOODZVRIRXUSK\VLFDOZRUOGDQGSXWXOWLPDWHOLP LWVRQWKHLUYDOLGLW\ 7KHRGRU:+lQVFKUHFLSLHQWRIWKH1REHO3UL]HLQ3K\VLFVIRUWKHLQYHQ WLRQRIWKH³RSWLFDOIUHTXHQF\FRPE´DUHYROXWLRQDU\WHFKQLTXHIRUWKHPHDVXUH PHQW RI RSWLFDO IUHTXHQFLHVZLWKXQSUHFHGHQWHG DFFXUDF\ HQWLWOHGKLV1REHO OHF WXUH³$SDVVLRQIRUSUHFLVLRQ´7KHPRWLYDWLRQIRUWKDWWLWOHUHVLGHGLQWKHWKULOOLQJ SUHFLVLRQWKDWWKHH[WUHPHGHYHORSPHQWVLQWKHPDQLSXODWLRQRIOLJKWKDGEURXJKW LQWR WKH ILHOG RI DWRPLF DQG PROHFXODU VSHFWURVFRS\ DQG XOWLPDWHO\ LQ RXU NQRZOHGJHRIWKHTXDQWXPZRUOG *LRYDQQL *LXVIUHGL LV D WUXH PDVWHU RI RSWLFV , KDG WKH SULYLOHJH RI NQRZLQJ KLP VLQFH  \HDUV LQ  ZH PHW LQ 3LVD DW WKH 6FXROD 1RUPDOH 6XSHULRUH 7KHQ,VKDUHGZLWKKLPPDQ\H[FLWLQJVFLHQWLILFDGYHQWXUHVPRVWO\LQWKHFRQWH[W RI WKRVH KLJKSUHFLVLRQ VSHFWURVFRSLF VWXGLHV WKDW +lQVFK KDG SLRQHHUHG 'XULQJ WKRVH \HDUV , ZLWQHVVHG WKH VLQFHUH SHUVRQDO ³SDVVLRQ IRU SUHFLVLRQ´ WKDW PRYHV *LRYDQQL*LXVIUHGLLQDOOWKHDVSHFWVRIKLVZRUNIURPWKHRU\WRH[SHULPHQWV,Q KLVSHUVRQWKHDFXWHLQWHOOHFWRIWKHVFLHQWLVWDOZD\VVWULYLQJWRJUDVSWKHILQHVWGH WDLOVRIWKHSUREOHPVFRH[LVWVZLWKDQH[FHSWLRQDOWDOHQWDV³FUDIWVPDQ´RIUHILQHG H[SHULPHQWDOGHYLFHVWKDWKHNHSWEXLOGLQJSHUVRQDOO\GXULQJKLVZKROHVFLHQWLILF OLIH 7KLVERRNLVDWUXHJHP,WLVDQHVVHQWLDOUHVRXUFHIRUDQ\UHVHDUFKHUDSSURDFK LQJWKHVWXG\RIRSWLFVRUZLVKLQJWRFRQVROLGDWHKHUKLVNQRZOHGJH+HUHVWXGHQWV VFKRODUVDQGSURIHVVRUVFDQILQGEDVLFDOO\HYHU\WKLQJWKH\QHHGWRNQRZRIFODVVL FDORSWLFV*HRPHWULFDORSWLFVDQGZDYHRSWLFVDUHFRYHUHGLQDFRPSUHKHQVLYHDQG RULJLQDOZD\QRWRQO\ULJRURXVGHPRQVWUDWLRQVRIODZVDQGHTXDWLRQVEXWDOVRWKH SHUVSHFWLYHDQGWKH³WULFNV´RIDQH[SHULHQFHGWDOHQWHGH[SHULPHQWDOLVWZLWKYHU\ LQWHUHVWLQJKLVWRULFDOQRWHVFRPSOHPHQWLQJWKHVFLHQWLILFFRQWHQW 2SWLFVLVHYHU\ZKHUHLQRXUGDLO\OLYHV$QGWKLVWUXO\³HQOLJKWHQHG´ERRNPHU LWVWREHRQWKHVKHOIRIHYHU\VFLHQWLVWWKDWPDNHVXVHRIRSWLFVLQKHUKLVUHVHDUFK  )ORUHQFH-XO\ 0DVVLPR,QJXVFLR

vii

 

3UHIDFHWRWKH,WDOLDQHGLWLRQ ,Q,ZDVDVNHGWRWHDFKWKHFRXUVHLQ*HQHUDO2SWLFVRIWKHSRVWJUDGXDWH 6FKRRORI2SWLFVDWWKH8QLYHUVLW\RI)ORUHQFH,JODGO\DFFHSWHGWKLVDVVLJQPHQW ZLWKDOLWWOHIHDU,QGHHG,NQHZ,ZDVRQO\MXVWDEHJLQQHUDQGWKXV,VHWDUXOHIRU P\VHOIWKDWEHIRUHHDFKOHVVRQ,ZRXOGVWXG\WKHVXEMHFWWREHWUHDWHGLQGHSWK 7R HVWDEOLVK RUGHU , VWDUWHG ZULWLQJ WKHVH QRWHV FKRRVLQJ HYHU\ ZRUG FDUHIXOO\ DQGWKURXJKXVHRIWKHYDULRXVERRNVDYDLODEOHDWWKHVSOHQGLG/LEUDU\RIWKHIstituto Nazionale di Ottica GHWHUPLQLQJ WKH EHVW WHDFKLQJ PHWKRG IRU H[SRVLQJ WKH DUJXPHQWVDQGGHPRQVWUDWLRQVRI3K\VLFDO2SWLFV7KLVWH[WKDVWKXVJURZQOLWWOH E\ OLWWOH OHQJWKHQLQJ DQG WKLFNHQLQJ JUDGXDOO\ HYHU\ WLPH , OHDUQHG VRPHWKLQJ QHZ7KHJRDOWKDW,VHWIRUP\VHOIZDVWRJLYHP\VWXGHQWVDVROLGIRXQGDWLRQLQ WKHVWXG\RIRSWLFVOHDYLQJWKHGHHSHUVWXG\RIVSHFLILFWRSLFVVXFKDVRSWLFDOGH VLJQRUKRORJUDSK\WRRWKHUFRXUVHVDWWKH6FKRRO $VDVWDUWLQJSRLQW,FKRVH0D[ZHOO¶VHTXDWLRQVRIFODVVLFDOHOHFWURPDJQHWLVP WKLVLVWKHVXEMHFWRI3DUW,&KDSWHUZKLFKVHUYHVDVERWKDUHIUHVKHUDQGDVD VXSSRUWIRUWKHUHVWRIWKHERRN,QSDUWLFXODUSODQHZDYHVDUHGHVFULEHGLQFOXGLQJ WKHLU SRODUL]DWLRQ WKH ODZVRI UHIUDFWLRQ DQG UHIOHFWLRQ E\ GLHOHFWULF DQG PHWDOOLF PHGLDDEVRUSWLRQDQGGLVSHUVLRQ ,Q 3DUW ,, &KDSWHU  WKH PDLQ FRQFHSWV RI *HRPHWULFDO 2SWLFV DUH H[SRVHG 7KLVDSSURDFKUHGXFHVWKHZDYHVWRrays of lightE\QHJOHFWLQJWKHWUDQVYHUVHDQG RVFLOODWLQJQDWXUHRIWKHILHOGVGHYRWLQJLWVHOIWRWKHLQWHQVLW\DQGWKHQWRWKHincoherent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ast Fourier Transform7KHVH PHWKRGV DUH WKHQXVHG IRU ix

[

3UHIDFH

WKH DQDO\VLV RI WKH TXDOLW\ RI WKH RSWLFDO V\VWHPV ,Q DGGLWLRQ ZLWK WKH VWXG\ RI VSDWLDO DQG WHPSRUDO FRKHUHQFH ZH OHDYH WKH UHVWULFWHG DPELW RI WKH SODQH DQG PRQRFKURPDWLFZDYHV)LQDOO\,SUHVHQWDUHYLHZRIVSDWLDOILOWHULQJPHWKRGVDQG GHVFULEHWKHGLIIUDFWLRQJUDWLQJV ,Q3DUW9,UHWXUQWRGLVFXVVLQJZDYHSURSDJDWLRQLQLVRWURSLFDQGDQLVRWURSLF OLQHDU PHGLD ,Q SDUWLFXODU LQ &KDSWHU  , GHVFULEH WKH PRQRFKURPDWLF ZDYHV WKRVHRIODVHUEHDPVZDYHVQRORQJHUSODQDUEXWRIOLPLWHGWUDQVYHUVHH[WHQVLRQ ZKLFKDUHFDOOHGGaussian beams$OVRWKHWHFKQLTXHHPSOR\HGLQWKHLUGHVFULS WLRQ FRQWDLQV D VLPSOLILFDWLRQ WKDW RI FRQVLGHULQJ ZDYHV WKDW PDLQO\ SURSDJDWH DURXQG DQ D[LV ZKLFK LV FDOOHG Gaussian RU paraxial approximation 7KLV DS SUR[LPDWLRQ LV LQ WXUQ ERUURZHG IURP 3DUD[LDO 2SWLFV ZKRVH ODZV FDQ WKXV EH UHLQWHUSUHWHGDOORZLQJ XV WR HDVLO\ PDQLSXODWH HYHQ LQ D FRQFHSWXDO VHQVH ODVHU EHDPV ZLWK OHQV V\VWHPV , DOVR GLVFXVV WKH Bessel waves DQG WKH Bessel-Gauss beamsZKLFKDUHPRGHUQWRSLFVWKDW,SLFNHGXSGLUHFWO\IURPPDJD]LQHDUWLFOHV )LQDOO\ , SUHVHQW D EULHI GLVFXVVLRQRI WKH UHVRQDQW FDYLW\ ,Q &KDSWHU ,GHWDLO WKH SURSDJDWLRQ LQ DQLVRWURSLF PHGLD EHJLQQLQJ ZLWK D VXPPDU\ RQ FU\VWDOORJ UDSK\DQGWKHSXQFWXDODQGVSDWLDOV\PPHWU\JURXSV,QDGGLWLRQRQWKHEDVLVRI UHODWLYLW\,GLVFXVVWKHSURSDJDWLRQLQbianisotropic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

 &DPSL%LVHQ]LR'HFHPEHU

*LRYDQQL*LXVIUHGL

 

,QGH[  ,QWURGXFWLRQWRRSWLFV



3DUW, (OHFWURPDJQHWLVP  +LVWRULFDOQRWHV7KHILUVWGLVFRYHULHVDERXWPDJQHWLVPDQGHOHFWULFLW\ 7KHGDZQLQJ 7KHODZVRIHOHFWURPDJQHWLVP 7KHGLVFRYHU\RIHOHFWURPDJQHWLFZDYHV 6XJJHVWHGUHDGLQJ   5HFROOHFWLRQVRIHOHFWURPDJQHWLVP  ,QWURGXFWLRQ     7KHIXQGDPHQWDOV    &KDUJHVDQGILHOGV    7KHVFDODUDQGYHFWRUSRWHQWLDOV    5HODWLYLVWLFIRUPXODWLRQRIILHOGVDQG    /RUHQW]WUDQVIRUPDWLRQV   0D[ZHOO¶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



                                   xi

[LL

                    

,QGH[

  1DWXUDOOLJKW   &RKHUHQF\PDWUL[DQG6WRNHV¶SDUDPHWHUV  5HIOHFWLRQDQGUHIUDFWLRQRQSODQHLQWHUIDFHV   /DZVRIUHIOHFWLRQDQGUHIUDFWLRQIRUKRPRJHQHRXV   GLHOHFWULFPHGLD   )UHVQHO¶VIRUPXODV   5HIOHFWLYLW\DQGWUDQVPLVVLYLW\   %UHZVWHU¶VDQJOH   7RWDOUHIOHFWLRQ   (YDQHVFHQWZDYH   $SSOLFDWLRQRIWRWDOLQWHUQDOUHIOHFWLRQ  'LVSHUVLRQWKHRU\   'LVSHUVLRQDQGH[WLQFWLRQLQGLOXWHPHGLD   'LVSHUVLRQDQGH[WLQFWLRQLQGHQVHPHGLD   5HODWLRQVKLSEHWZHHQWKHUHDODQGLPDJLQDU\SDUWV   RIWKHSHUPLWWLYLW\   'LVSHUVLRQLQFRQGXFWLQJPHGLD   3KDVHJURXSDQGVLJQDOYHORFLWLHV  2SWLFVRIPHWDOVDQGDEVRUEHQWPDWHULDOV   5HIOHFWLRQDQGUHIUDFWLRQE\DEVRUEHQWPDWHULDOV %LEOLRJUDSKLFDOUHIHUHQFHV

3DUW,,*HRPHWULFDO2SWLFV  +LVWRULFDOQRWHV)URP(PSHGRNOƝVWR+X\JHQV   7KHGDZQLQJ    7KH$UDESHULRG    7KH0LGGOH$JHV   7KH5HQDLVVDQFH   7KH1HZ6FLHQFH   7KHODZRIUHIUDFWLRQDQGWKHVSHHGRIOLJKW   7KHGHEDWHRYHUWKHQDWXUHRIOLJKW   6XJJHVWHGUHDGLQJ   *HRPHWULFDO2SWLFV   ,QWURGXFWLRQ     'HULYDWLRQRI*HRPHWULFDO2SWLFVIRUOo   3URSDJDWLRQSURSHUWLHVLQ*HRPHWULFDO2SWLFV    /LJKWUD\VDQGWKHUDGLDQWHQHUJ\SURSDJDWLRQ    3URSDJDWLRQRISRODUL]DWLRQ   *HQHUDOSURSHUWLHVRIUD\V    7UDFLQJODZRIUD\VLQLQKRPRJHQHRXVPHGLD    &XUYDWXUHDQGWRUVLRQRIWKHUD\V

                  

                 

[LLL

,QGH[

                 

              

                    

                    



 3URSDJDWLRQLQPHGLDZLWKVSKHULFDOV\PPHWU\  $WPRVSKHULFUHIUDFWLRQ  7KHRSWLFDOSDWK  &ROOHFWLYHSURSHUWLHVRIUD\V  7KHODZVRIUHIUDFWLRQDQGUHIOHFWLRQLQ*HRPHWULFDO2SWLFV  7KH0DOXV'XSLQWKHRUHP  )HUPDW¶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

                                         

[LY

   &XUYDWXUHRIDTXDGULFVXUIDFH    7UDQVIRUPDWLRQRIWKHZDYHIURQW    6\PPHWU\UHODWLRQVDQGRSWLFDOSDWK   $SHUWXUHVRIDQRSWLFDOV\VWHP    $SHUWXUHGLDSKUDJPDQGWKHHQWUDQFHDQGH[LWSXSLOV    5HODWLYHDSHUWXUHRUfQXPEHUDQGQXPHULFDODSHUWXUH    3ULQFLSDOUD\V    )LHOGGLDSKUDJPDQGHQWUDQFHDQGH[LWZLQGRZV    9LJQHWWLQJ   $EHUUDWLRQV     7KHDEHUUDWLRQIXQFWLRQVIRUDQD[LDORSWLFDOV\VWHP    7KHFRVLQHWKHRUHP$EEH¶VVLQHFRQGLWLRQ    DQG+HUVFKHO¶VFRQGLWLRQ    6SKHULFDODEHUUDWLRQ    &RPD     $VWLJPDWLVPDQGFXUYDWXUHRIILHOG    'LVWRUWLRQ    $EHUUDWLRQRIDWKLQOHQV    &KURPDWLFDEHUUDWLRQV   3ODQHPLUURUVDQGSULVPV    3ODQHPLUURUV    5HIOHFWLQJSULVPV    'LVSHUVLYHSULVPV  %LEOLRJUDSKLFDOUHIHUHQFHV   3DUW,,,3K\VLFDO2SWLFV  +LVWRULFDOQRWHV)URP1HZWRQWR)UHVQHO   1HZWRQ¶V2SWLFV   7KHSURJUHVVRIWKHWKFHQWXU\   7KHHPLVVLRQWKHRU\DQGWKHVSHHGRIOLJKW   %HUQDO@7KH\KDGIHGWKHWKHRULHVRIDIILQLW\DQGDWWUDFWLRQDWDGLV WDQFHDQGWKHLGHDRID³YLUWXH´ DVLWZDVFDOOHGE\$ULVWRWpOƝV KHOGE\FHUWDLQ VXEVWDQFHV WKDW FRXOG EH WUDQVIHUUHG WR RWKHUV E\ VLPSOH FRQWDFW 7KLQN RI WKH P\WKRI.LQJ0LGDVDQGWKHSKLORVRSKHU¶VVWRQH,QWXUQWKHFRQFHSWRIWKHILHOG FRPHVIURPDORQJWUDGLWLRQWKDWDWWULEXWHVG\QDPLFRUHQHUJ\SURSHUWLHVWRVSDFH DVLQWKH$ULVWRWHOLDQLGHDRIDHWKHU+RZHYHUWKHVWXG\RIPDJQHWLVPEHFDPHD VFLHQFHRQO\DIWHUWKHPDJQHWLFYLUWXHZDVDVVRFLDWHGZLWKDSUDFWLFDOSXUSRVHDV ZLWK WKH FRPSDVV LW ZDV GLVFRYHUHG LQ &KLQD DURXQG WKH ILUVW FHQWXU\ %& DQG SHUKDSVLQWKHWZHOIWKFHQWXU\WUDYHOHGWRWKH:HVWLQDPDQQHUWKDWUHPDLQVXQ NQRZQ7KHILUVW:HVWHUQVFLHQWLILFVWXG\RQPDJQHWLVPZDVFDUULHGRXWE\3LHUUH OH 3qOHULQ GH 0DULFRXUW 3HWUXV 3HUHJULQXV  ZKR LQ KLV Epistola de magnete  JLYHVLQVWUXFWLRQVRQKRZWREXLOGDPDJQHWLFQHHGOHDQGGHVFULEHVIRUWKH ILUVWWLPHWKHVLPSOHSURSHUWLHVRIPDJQHWVLQFOXGLQJWKHODZWKDWRSSRVLWHSROHV DWWUDFWHDFKRWKHU7KLVVWXG\ZDVIROORZHGDIWHUDFRQVLGHUDEOHWLPHE\ The new attractive  ZULWWHQE\WKHQDYLJDWRUDQGPDQXIDFWXUHURIFRPSDVVHV5REHUW 5



+LVWRULFDOQRWHVWKHILUVWGLVFRYHULHVDERXWPDJQHWLVPDQGHOHFWULFLW\

1RUPDQ Pa ZKRGLVFRYHUHGWKHPDJQHWLFGHFOLQDWLRQ RQDYHUWLFDOSODQH  RIWHUUHVWULDOPDJQHWLVPDVVRFLDWHGZLWKWKHODWLWXGHDQGE\De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure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attraction WKDW NHHSV WKH SODQHWV WLHG WR WKHLUSDWWHUQRIPRWLRQ+LVERRNDOVREHFDPHDFODVVLFRIVFLHQWLILFOLWHUDWXUHIRU LWV PHWKRG RI LQYHVWLJDWLRQ LQGHHG LQ LW *LOEHUW VXSSRUWV WKH QHHG WR YHULI\ WKH WUDGLWLRQV SDVVHG GRZQ RYHU WLPH WKURXJK WKH H[HFXWLRQ RI DSSURSULDWH DQG GH WDLOHG H[SHULPHQWV +LV ZRUN ZDV DSSUHFLDWHG E\ *DOLOHR ZKR GHGLFDWHG VHYHUDO SDJHV WR LW LQ WKH WKLUG GD\ RI WKH Dialoghi DQG LW SDYHG WKH ZD\ IRU 1HZWRQ¶V JHQHUDOWKHRU\RIJUDYLWDWLRQ*LOEHUWDGGUHVVHGQRWRQO\PDJQHWVEXWDOVRHOHFWUL FDOSKHQRPHQD+HIRXQGWKDWYDULRXVPDWHULDOVEHVLGHVDPEHUFRXOGEHHOHFWUL ILHGWKURXJKUXEELQJDQGWKHRUL]HGWKDWWKLVZRXOGFDXVHWKHHPDQDWLRQRIFHUWDLQ effluviaWKDWUHWXUQLQJWRWKHUXEEHGERG\ZRXOGGUDJDORQJWKHVXUURXQGLQJOLJKW ERGLHV)XUWKHUPRUHKHLQYHQWHGWKHILUVWHOHFWULFLQVWUXPHQWWKHEDODQFHGQHHGOH RUversoriumDQGZDVWKHILUVWWRXVHWKHWHUPVelectric attractionelectric force DQGmagnetic pole )URP  RQZDUGV WKHUH KDYH EHHQ PDMRU QHZ GLVFRYHULHV DERXW HOHFWULFLW\ DQGPDJQHWLVPIURPZKLFKEHJDQWKHHOHFWURPDJQHWLFUHYROXWLRQRIKXPDQVRFL HW\WKDWLVVWLOOGUDPDWLFDOO\FKDQJLQJWKHFRQGLWLRQVRIOLIHRQ(DUWK$WILUVWHOHF WULFLW\ ZDV FRQVLGHUHG D PHUH LQWHOOHFWXDO SDVWLPH ZLWKRXW VHULRXV DSSOLFDWLRQV VXFKDVWKHOLWWOHFORXGVLJKWHGE\1RDKEHIRUHWKH*UHDW)ORRG 7KXV1LFFROz&DEHR  GHVFULEHGLQKLVPhilosophia MagneticaRI WKHGLVFRYHU\RIHOHFWURVWDWLFUHSXOVLRQ)UDQFHVFR0DULD*ULPDOGL    FODLPV LQ WKH IURQWLVSLHFH RI KLV De Lumine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± ZKRVKRZHG WKDWWKURXJKUXEELQJZHFRXOGFUHDWHOLJKWHIIHFWVLQYDFXXP,QSDUWLFXODULIWKH JODVVJOREHRIWKHHOHFWULFPDFKLQHZDVSDUWLDOO\HPSWLHGRIDLUWKHUXEELQJFDXVHG LQWHQVHOXPLQRXVSKHQRPHQDLQLWVLQWHULRU $WWKDWWLPHERGLHVZHUHGLVWLQJXLVKHGDVEHLQJHLWKHUelectricRUnon-electric

7KHGDZQLQJ



7RWKHILUVWJURXSEHORQJHGRQO\DPEHUJODVVDQGVRPHSUHFLRXVVWRQHVWKDWFRXOG EHHOHFWULILHGE\UXEELQJ,WZDVEHOLHYHGWKDWWKHVHZHUHDEXQGDQWO\VXSSOLHGZLWK D SDUWLFXODU HOHFWULFDO VXEVWDQFH WKDW HPDQDWHG LQ UHVSRQVH WR UXEELQJ 7KLV SKH QRPHQRQ LV FDOOHG triboelectricity 7KH RWKHU QRQHOHFWULF ERGLHV LQGLIIHUHQW WR UXEELQJZHUHGHHPHGWREHZLWKRXWWKLVVXEVWDQFH ,Q6WHSKHQ*UD\ a GLVFRYHUHGWKDWWKHHOHFWULFDOYLUWXHFDQ EHWUDQVPLWWHGIURPDQH[FLWHGHOHFWULFERG\WRDQRQHOHFWULFDORQHE\FRQWDFWDQG FRQGXFWLRQZLWKRXWDQ\DSSDUHQWPDWHULDOPRYHPHQWOLNHDQLPSRQGHUDEOHIOXLG DSLHFHRIPHWDOWRXFKHGE\DQHOHFWULILHGJODVVWXEHDFTXLUHGWKHSURSHUW\RIDW WUDFWLQJ WKH WHVW FRUSXVFOHV 0RUHRYHU KH IRXQG WKDW LQ VRPH FDVHV RQH FRXOG KDYHDQHOHFWULILFDWLRQIRULQGXFHGFKDUJHV,QSDUWLFXODUKHQRWHGWKDWHOHFWULFLW\ FDQ EH JHQHUDWHG DQG VWRUHG LQ FHUWDLQ PDWHULDOV WKDW KH FDOOHG electris ZKLOH LW IORZVLQPHWDOVDQGZHWERGLHVZKLFKKHFDOOHGnon electrisDIHZ\HDUVODWHUWKH ILUVW ZRXOG FRPH WR EH FDOOHG LQVXODWRUV DQG WKH VHFRQGV FRQGXFWRUV ,QGHHG D PHWDOSLHFHZDVHOHFWULILHGRQO\ZKHQLWZDVVXSSRUWHGE\VSHFLDOPDWHULDOVVXFK DVVLONWKUHDGVEXWQRWE\PHWDOOLFZLUHV,QDGGLWLRQE\VXVSHQGLQJDFRQGXFWLYH VWULSZLWKVLONWKUHDGVKHIRXQGWKDWKHFRXOGWUDQVPLWWKHHOHFWULFLW\WRPRUHWKDQ IHHW 7KHVHVWXGLHVZHUHH[WHQGHGE\WKHVXSHULQWHQGHQWRIWKHUR\DOJDUGHQVRIWKH .LQJRI)UDQFH&KDUOHV)UDQoRLV'XID\  ZKRHPSKDVL]HGWKHQHHG WR SODFH WKH ERGLHV WR EH HOHFWULILHG RQ LQVXODWLQJ VXSSRUWV WKLV SUHFDXWLRQ ZDV FDOOHGDufay’s rule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prime conductorVXVSHQGHGZLWKLQVXODWLQJFRUGV'XID\¶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

 ,WVKRXOGEHQRWHGWKDWIRUWKHVPDOODPRXQWRIFKDUJHDQGWKHKLJKYROWDJHVJHQHUDWHGERWK

WKHYROXPHDQGVXUIDFHUHVLVWDQFHRIWKHLQVXODWRUVPXVWEHSDUWLFXODUO\KLJKLQGHHGLWMXVWQHHGV DELWRIPRLVWXUHWRGLVSHUVHWKHFKDUJHVDQGFRQYHUWWKHPLQWRFRQGXFWRUV



+LVWRULFDOQRWHVWKHILUVWGLVFRYHULHVDERXWPDJQHWLVPDQGHOHFWULFLW\

VKRFNLQJYLRODWLRQRI'XID\¶VUXOH7RWKHVHUHVHDUFKHUVLVFUHGLWHGWKHLQYHQWLRQ RIWKHLeyden jarZKLFKZDVDSULPLWLYHIRUPRIDQHOHFWULFDOFDSDFLWRU,Q -UJHQYRQ.OHLVWGHDQRIWKH&DWKHGUDORI.DPPLQLQ3RPHUDQLDWRRNDQ³HOHF WULFVKRFN´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personally YHULI\ WKH SDLQIXO HIIHFWV RI WKH HOHFWULF VKRFN&RQWUDU\WRH[SHFWDWLRQVWKHFKDUJLQJRIWKHERWWOHZDVPXFKVWURQJHUWKDQ ZKDWZDVREWDLQHGE\IROORZLQJ'XID\¶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eclaration of IndependenceDQGZDVWKHILUVWIDPRXVSK\VLFLVWRIWKH1HZ:RUOG+HLQWHU SUHWHGWKH©HOHFWULFILUHªDVDQLPPDWHULDOIOXLGWKDWH[LVWHGLQHYHU\ERG\LPSHU FHSWLEOHXQGHUQRUPDOFRQGLWLRQVEXWWKLVIOXLGFRXOGEHSXPSHGIURPRQHERG\WR DQRWKHU WKURXJK UXEELQJ 7KH ERG\ WKDW UHFHLYHV WKH HOHFWULFLW\ LV FKDUJHG LQ WKH positiveVHQVHWKHRWKHULQWKHnegativeZLWKWKLV)UDQNOLQHQXQFLDWHGWKHODZRI FRQVHUYDWLRQRIHOHFWULFFKDUJHDQGSURGXFHGDQHZLQWHUSUHWDWLRQRIWKHWZRW\SHV RIHOHFWULFLW\GLVFRYHUHGE\'XID\XVLQJDSQHXPDWLFDWWUDFWLYHPRGHO7KHHOHF WULFDODWWUDFWLRQZDVWKHUHIRUHGXHWRWKHWHQGHQF\WRZDUGVWKHHTXLOLEULXPFRQGL WLRQZKLFK FRXOGEHUHHVWDEOLVKHGE\ D VSDUN 7KURXJKYDULRXV H[SHULPHQWV KH H[SODLQHGWKHIXQFWLRQLQJRIWKH/H\GHQMDUFRQFOXGLQJWKDWDsuperficialH[FHVV RIHOHFWULFDOVXEVWDQFHLVIRUPHGRQRQHIDFHRIWKHJODVVZDOODQGDQHTXLYDOHQW GHSOHWLRQRQWKHRSSRVLWHIDFH)UDQNOLQGLVFRYHUHGWKDWWKHERWWOHZRUNVEHWWHUE\ FRDWLQJWKHRXWHUDQGLQQHUVXUIDFHVZLWKWKLQPHWDOVKHHWV,QDGGLWLRQWKHVKDSH RIWKHERWWOHZDVLQHVVHQWLDODQGWKHUHIRUHKHXVHGVLPSOHJODVVVODEVDUPHGZLWK PHWDOOLFVKHHWVRQERWKVXUIDFHVWKLVGHYLFHZDVFDOOHGFranklin’s square,IDWKLQ VXUIDFHOD\HULVVXIILFLHQWWRSURGXFHVXFKJUHDWHIIHFWVWKHelectrisPXVWDOVREH ULFK LQ HOHFWULFDO VXEVWDQFH ZLWKLQ WKH H[SDQVLYH SRZHU RI ZKLFK LV FRXQ WHUEDODQFHG E\ WKH ³SRZHU RI DWWUDFWLRQ DQG FRQGHQVDWLRQ´ WKDWWKH PDWWHU RI WKH VXEVWDQFHH[HUWVRQWKHHOHFWULFILUH

7KHGDZQLQJ



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rZKLOHWKHDWWUDFWLRQaLVEHWZHHQWKHILUHRI$RQ WKHPDWWHURI%DQGWKHDWWUDFWLRQa'LVEHWZHHQWKHILUHRI%RQWKHPDWWHURI$ 6LQFH WKHUH LV QR DWWUDFWLRQ RU HOHFWULF UHSXOVLRQ EHWZHHQ QHXWUDO ERGLHV WKHVH IRUFHV VKRXOG EDODQFH HDFK RWKHU IRU ZKLFK ZH KDYH raa'   2Q WKH RWKHU KDQGWKHUHLVQRDWWUDFWLRQRUUHSXOVLRQQRWHYHQEHWZHHQWKHHOHFWULFILUHRIWKH ERG\$DQGWKHQHXWUDOERG\%VRLWPXVWDOVREHYDOLGWKDWra IURPZKLFK ZH REWDLQ WKH DEVXUG UHVXOW RI a'   $HSLQXV FRQFOXGHG WKDW WKLV FRQWUDGLFWLRQ FRXOG EH UHVROYHG DVVXPLQJ D IRXUWK IRUFH repulsive EHWZHHQ WKH PDWWHU RI WKH WZRERGLHV+HFDPHWRWKLVUHDOL]DWLRQWRKLVRZQKRUURUVLQFHWKLVIDFWFRQWUD GLFWHG WKH LGHD RI DQ DWWUDFWLYH IRUFH JUDYLWDWLRQDO  DPRQJ WKH PDWWHU EXW UH PDLQHGFRQYLQFHGRIWKHYDOLGLW\RIKLVSURRI$HSLQXVDOVRZRUNHGRQPDJQHWLVP DQG LQ  SXEOLVKHG KLV VWXGLHV LQ Tentamen theoriae electricitatis et magnetismiZKLFKZDVFRQVLGHUHGKDUVKRZLQJWRLWVVWURQJPDWKHPDWLFDOFRQWHQW DQGZDVWKXVQHJOHFWHGE\HOHFWURORJLVWVRIWKHWLPH

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



+LVWRULFDOQRWHVWKHILUVWGLVFRYHULHVDERXWPDJQHWLVPDQGHOHFWULFLW\

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heoria philosophiae naturalis redacta ad unicam legem virium in natura existentium'LVVDWLVILHGZLWK1HZWRQ¶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oyal Society RI /RQGRQ LQ  7KHUH KH DOVR GHVFULEHV DQ HOHFWURPHWHU FRQVLVWLQJRIWZRVLOYHUZLUHVOHIWIUHHWRGLYHUJHIRUHOHFWULFUHSXOVLRQZKHQWKH\ DUHSXWLQWRFRQWDFWZLWKDFKDUJHGF\OLQGULFDOFRQGXFWRU)ROORZLQJWKHSQHXPDWLF DQDORJ\%HFFDULDDWWULEXWHGDSUHVVXUHWRWKHFKDUJHSRVVHVVHGE\WKHFRQGXFWRU ERG\ DQG ZLWK KLV HOHFWURPHWHU YHULILHG WKDW WKLV SUHVVXUH ZDV KDOYHG ZKHQ WKH F\OLQGHUZDVSODFHGLQFRQWDFWZLWKDQRWKHUHTXDOEXWGLVFKDUJHGF\OLQGHU,QWKH VHFRQG PHPRU\ %HFFDULD GHVFULEHG VRPH REVHUYDEOH HIIHFWV ZLWK WKH )UDQNOLQ¶V VTXDUHKRZHYHUIRUWKHVHHIIHFWVDQGIRUWKHWULERHOHFWULFLW\KHSRVWSRQHGWKHH[ SODQDWLRQUHIHUULQJWRYDJXHSULQFLSOHVWKDWVWLOOKDG\HWWREHGLVFRYHUHG 6WLPXODWHG E\ WKLV LQFRPSOHWHQHVV 9ROWD WKHQ SURSRVHG DQ HOHFWULF WKHRU\ EDVHGRQWZRSULQFLSOHV7KHILUVWZDVVWLOOWKHSQHXPDWLFUHSXOVLRQRIWKHHOHFWULF

7KHODZVRIHOHFWURPDJQHWLVP



ILUH ZKLOH WKH VHFRQG ZDV WKH H[LVWHQFH RI D QRQPHFKDQLFDO IRUFH RI DWWUDFWLRQ EHWZHHQ WKH PDWWHU DQG WKH HOHFWULF ILUH 9ROWD ZDV LQIOXHQFHG E\ WKH ZRUN RI %RVFRYLFK DQG UHVXPHG WKH VDWXUDWLRQ FRQFHSW WKDW KH KDG XVHG LQ KLV WKHRU\ RI KHDW WKH FRQGLWLRQ RI HTXLOLEULXP EHWZHHQ WZR ERGLHV RI GLIIHUHQW VXEVWDQFHV LV UHDFKHGZKHQWKHIRUFHRIDWWUDFWLRQWKDWWKHVHH[HUWRQWKHVXEVWDQFHRIWKHKHDW EDODQFHRXWWKHH[SDQVLYHSRZHURIWKHSQHXPDWLFUHSXOVLRQ6RWKHGLIIHUHQWVXE VWDQFHV EHFRPH ³VDWHG´ ZLWK DQ DPRXQW RI KHDW SURSRUWLRQDO WR WKHLU GLIIHUHQW IRUFHVRIDWWUDFWLRQ,QDQDQDORJRXVZD\9ROWDFRXOGFRKHUHQWO\H[SODLQERWKWKH HOHFWULFDODWWUDFWLRQRUUHSXOVLRQPRWLRQVEHWZHHQHOHFWULILHGERGLHVDQGWKHHIIHFWV RIHOHFWURVWDWLFLQGXFWLRQ,QGHHGWZRSODWHVFKDUJHGLQWKHRSSRVLWHZD\DQGDS SURDFKHG DW D VPDOO GLVWDQFH ZLWKRXW WRXFKLQJ ORVH WKHLU HOHFWULF VLJQV ZKLOH ZKHQWKHLUGLVWDQFHLVLQFUHDVHGWKHSODWHVUHJDLQVXFKVLJQVDVKDSSHQVIRUWKH DUPDWXUHVRI)UDQNOLQ¶VVTXDUH,QDGGLWLRQKHFRXOGH[SODLQWKHWULERHOHFWULFLW\E\ DSSHDOLQJLQWKLVFDVHWRWKHPLFURVFRSLFOHYHORI%RVFRYLFK¶VWKHRU\RQPDWWHU ,Q  RQO\  \HDUV ROG 9ROWD ZURWH D SDSHU HQWLWOHG De vi attractiva ignis electrici ac phaenomenis independentibusZKLFKKHVHQWWRWKHPDMRUHOHFWURORJLVW RIKLVWLPHDOWKRXJKKHJRWDUDWKHUFROGDQVZHU 9ROWDGLGQRWORVHKHDUWDQGRQWKHVXJJHVWLRQRI/D]]DUR6SDOODQ]DQLGHYRWHG KLPVHOI WR QXPHURXV H[SHULPHQWDO WHVWV ,Q  KH DQQRXQFHG WKH LQYHQWLRQ RI WKH³perpetual electrophorus´E\LWVHOIVLPLODUWR)UDQNOLQ¶VVTXDUHEXWFRQVLVWLQJ RI D PHWDO SODWH RQZKLFK ZDV GHSRVLWHG DOD\HURIUHVLQPXFK PRUHFDSDEOHRI UHWDLQLQJHOHFWULFLW\WKDQWKHJODVVDQGDPHWDOOLFVKLHOGZLWKURXQGHGHGJHVDQG DQ LQVXODWLQJ KDQGOH 7KH UHVLQ ZDV LQLWLDOO\ HOHFWULILHG 7KHQ WKH VKLHOG ZDV SODFHGRYHUWKHUHVLQVKRUWFLUFXLWHGZLWKWKHGLVKDQGILQDOO\UDLVHGSURGXFLQJD VWURQJVSDUN7KHVHTXHQFHFRXOGEHUHSHDWHGPDQ\WLPHVFDXVLQJWKHLQVWUXPHQW WRVHHPWRJHQHUDWHDQHDUO\XQOLPLWHGTXDQWLW\RIHOHFWULFLW\7KLVSURGXFHGDVHQ VDWLRQ LQ (XURSH VLPLODU WR WKDW RI WKH LQYHQWLRQ RI WKH /H\GHQ MDU KRZHYHU DO PRVWQRRQHXQGHUVWRRGWKHRSHUDWLRQDQG9ROWD¶VDSSHDOWRUHFRQVLGHUKLVDe vi actractiva ZDV QRW \HW DFFHSWHG $V D UHVXOW RI WKH H[SHULPHQWV FDUULHG RXW ZLWK ORQJFRQGXFWLQJZLUHVKHUHVXPHGWKHSQHXPDWLFFRQFHSWRItensionXVHGE\%HF FDULDWRLQGLFDWHWKH³SUHVVXUH´RIWKHHOHFWULFIOXLGLQDFKDUJHGERG\DQGLQWUR GXFHGWKHFRQFHSWRIcapacityWRLQGLFDWHWKHZLUH¶VDWWLWXGHWRZDUGFRQWDLQLQJWKH HOHFWULF IOXLG 8VLQJ DQ HOHFWURPHWHU IRU YROWDJH PHDVXUHPHQWV DQG FRXQWLQJ WKH QXPEHURIUHYROXWLRQVRIWKHHOHFWULFPDFKLQHDVDQHVWLPDWHRIWKHFKDUJHQWKDW ZDV QHFHVVDU\ WR ORDG D FRQGXFWRU ZLWK D FDSDFLW\ C XS WR WKH YROWDJH V 9ROWD ZDV WKXV DEOH WR IRUPXODWH WKH ODZ RI SURSRUWLRQDOLW\ Q v VC ,Q  9ROWD UHDOL]HG WKH condenser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



+LVWRULFDOQRWHVWKHILUVWGLVFRYHULHVDERXWPDJQHWLVPDQGHOHFWULFLW\

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volt $IWHUZDUGVWKHJUHDW)UHQFKSK\VLFLVWVDQGPDWKHPDWLFLDQVGHDOWZLWKWKHWKHR U\RIHOHFWULFLW\GHYHORSLQJDJHQHUDOIRUPXODWLRQRIWKHSKHQRPHQDJRYHUQHGE\ IRUFHODZVRIWKHW\SHrLQWKHFRQWH[WRIWKHGRFWULQHRIDFWLRQDWDGLVWDQFH,Q SDUWLFXODUWKH\FRQVLGHUHGWKHHIIHFWVRIDPDVVRUDFKDUJHGLVWULEXWHGLQWKHERG LHV&RXORPEKDGDOUHDG\H[DPLQHGWKHFDVHRIWKHVSKHUH/DSODFHLQKLVVWXGLHV RIFHOHVWLDOPHFKDQLFVZDVFRQFHUQHGZLWKHVWDEOLVKLQJWKHJUDYLWDWLRQDOIRUFHDFW LQJRQDSRLQW3RIPDVVmE\DQH[WHQGHGERG\RIDJLYHQJHRPHWULFVKDSH,Q SDUWLFXODUKHFRQVLGHUHGDVSKHULFDORUHOOLSVRLGDOERG\LQZKLFKLWVPDVVMZDV GLVWULEXWHGLQDFRQWLQXRXVPDQQHUDQGLQWURGXFHGDIXQFWLRQRIWKHVSDWLDOFRRU GLQDWHVRI3ZKLFKLVNQRZQWRGD\DVWKHpotential functionZLWKZKLFKWRH[SUHVV WKLVIRUFHLQWKHGLIIHUHQWLDOHTXDWLRQVRIPRWLRQ )URP  6LPpRQ 'HQLV 3RLVVRQ   DVVXPLQJ WKH H[LVWHQFH RI ERWK SRVLWLYH DQG QHJDWLYH HOHFWULF IOXLGV GHYHORSHG WKH WKHRU\ RI HOHFWURVWDWLFV ZKLFKKHSXEOLVKHGLQLQKLVMémoire sur la distribution de l’électricité à la surface des corps conducteurs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

7KHODZVRIHOHFWURPDJQHWLVP



IURJOHJZDVRQO\DGHOLFDWHHOHFWURPHWHUIRUGHWHFWLQJWKHSDVVDJHRIWKHFXUUHQW +RZHYHU*DOYDQLKDGVKRZQWKDWHYHQWKHWUDQVPLVVLRQRIVLJQDOVLQDOLYLQJEH LQJLVGRQHWKURXJKHOHFWULFLW\ ,Q9ROWDEXLOWWKHILUVWHOHFWULFDOSRZHUVRXUFHQRWRIDQHOHFWURVWDWLFW\SH QDPHO\DpileFRQVWLWXWHGE\DVHULHVUHSHDWHGPDQ\WLPHVRYHURIDFRSSHUGLVN D ]LQF GLVF DQG D ZHW FORWK GLVN ZLWK ZKLFK 9ROWD ZDV DEOH WR GHPRQVWUDWH WKH VDPHHOHFWULFDOHIIHFWVWKDWZHUHREWDLQHGZLWKWKHUXEELQJPDFKLQHV6RRQFKHP LVWVEHJDQWRUHDOL]HWKHLPSRUWDQFHRIWKLVGLVFRYHU\DQGWKH\XVHGODUJHDQGH[ SHQVLYH EDWWHULHV IRU WKH VHSDUDWLRQ DQG LGHQWLILFDWLRQ RI WKH FKHPLFDO HOHPHQWV $PRQJ WKHP +XPSKU\ 'DY\   GLUHFWRU RI WKH Royal Institution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³FRQIOLFW´EHWZHHQWKHSULPRUGLDOIRUFHVRSSRVLQJHDFKRWKHU 7KXV+DQV&KULVWLDQ2HUVWHG  SURIHVVRURI3K\VLFVLQ&RSHQKDJHQ LQVSLUHGE\WKLVNaturphilosophiae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³HOHFWULF FRQIOLFW´ WKDW RSHUDWHV LQ D FLUFOH DURXQG D VWUDLJKW ZLUH FDUU\LQJ D FXU UHQW LH ZLWK WKH )DUDGD\ H[SUHVVLRQ WKDW WKHUH DUH FLUFXODU OLQHV RI PDJQHWLF IRUFHDURXQGWKHZLUH2HUVWHGVDLGDPRQJRWKHUWKLQJVWKDWWKLVHOHFWULFFRQIOLFW ZDVDEOHWRFURVVWKHQRQPDJQHWLFPDWHULDOVDQGGHGXFHGIURPWKLVWKDWLW³FDQDFW RQO\RQWKHPDJQHWLFSDUWLFOHVRIPDWWHU´2Q-XO\KHVHQWKLVUHVXOWVWR WKH 'DQLVK $FDGHP\ RI 6FLHQFHV ZKLFK SXEOLVKHG WKHP XQGHU WKH WLWOH Experimenta circa effectum conflictus electrici in acum magneticam,QDVXEVHTXHQWDU WLFOH2HUVWHGVKRZHGWKDWWKHHIIHFWZDVPXWXDOLIDPDJQHWLFQHHGOHLVKHOGQHDU DPRYDEOHZLUHWKLVZLUHURWDWHVLQWKHRSSRVLWHGLUHFWLRQZKHQDFXUUHQWSDVVHV WKURXJK LW 7KHUHIRUH LQ DGGLWLRQ WR ILQGLQJ DQ XQH[SHFWHG FRQQHFWLRQ EHWZHHQ



+LVWRULFDOQRWHVWKHILUVWGLVFRYHULHVDERXWPDJQHWLVPDQGHOHFWULFLW\

PDJQHWLVPDQGHOHFWULFLW\KHKDGGHPRQVWUDWHGWKHH[LVWHQFHRIDVKRFNLQJnoncentral forceDFWLQJDWDGLVWDQFH 7KHQHZVRIWKLVGLVFRYHU\ZDVLPPHGLDWHO\WDNHQWR3DULVE\-HDQ'RPLQLTXH )UDQoRLV $UDJR ZKR UHSHDWHG WKH H[SHULPHQW LQ IURQW RI WKH PHPEHUV RI WKH )UHQFK$FDGHP\ZKLFKLQFOXGHGKLVIULHQG$QGUp0DULH$PSqUH   $PSHUH UHDOL]HG WKDW LW ZDV DSSURSULDWH WR FDQFHO WKH (DUWK¶V PDJQHWLF ILHOG DURXQGWKHFRPSDVVWRGHWHUPLQHWKHGLUHFWLRQRIWKHPDJQHWLFILHOGJHQHUDWHGE\ FXUUHQWV7KDQNVWRWKLVPHDVXUHKHFUHDWHGDWRROFDOOHGDJDOYDQRPHWHUZLWKLW KHFRXOGGHILQHWKHFXUUHQWSDWKLQDFORVHGFLUFXLWDQGGLVFRYHUHGWKDWZLWKLQWKH SLOHLWIORZVWKHVDPHFXUUHQWWKDWJRHVWKURXJKWKHUHVWRIWKHFLUFXLW $PSqUH WKHQ SHUIRUPHG D VHULHV RI H[SHULPHQWV LQ ZKLFK KH IRXQG WKDW EH WZHHQWZRSDUDOOHOFRQGXFWRUVDIRUFHRIDWWUDFWLRQRUUHSXOVLRQLVGHYHORSHGGH SHQGLQJRQWKHPXWXDORULHQWDWLRQLQZKLFKWKHFXUUHQWVIORZDQGWKDWDVROHQRLG EHKDYHVLQDZD\VLPLODUWRDPDJQHW)URPKHUHKHFRQFOXGHGWKDWWKHVDPHPDJ QHWLVP RI WKH PDJQHWV ZDV JHQHUDWHG E\ LQWHUQDO FXUUHQWV ,Q WKLV ZD\ $PSqUH FRXOGEULQJWKHPDJQHWLFIRUFHEDFNWRDFHQWUDOIRUFHGXHWRFXUUHQWVXQGHUVWRRG DVDUHVXOWRIWKHLQWHJUDWLRQRQFORVHGFLUFXLWVRUH[WHQGHGLQGHILQLWHO\>-DFNVRQ &DS@)DUDGD\FKDOOHQJHGWKLVFRQFOXVLRQLQDQDQRQ\PRXVSXEOLFDWLRQRI FLWLQJWKHFRXQWHUH[DPSOHRIRQHRIKLVH[SHULPHQWVLQZKLFKKHQRWHGWKDW WKHPDJQHWLFILHOGRIDPDJQHWL]HGLURQWXEHUHYHUVHVLWVGLUHFWLRQE\JRLQJIURP RXWVLGHWRLQVLGHRIWKHWXEHDORQJLWVD[LVZKLOHWKLVGRHVQRWKDSSHQIRUWKHVROH QRLG2QWKHVXJJHVWLRQRIKLVIULHQG$XJXVWLQ)UHVQHO$PSqUHZDVDEOHWRVROYH WKHTXHVWLRQE\DVVXPLQJWKHQWKDWWKHFXUUHQWVRIDSHUPDQHQWPDJQHWFLUFXODWH DURXQGWKHLQGLYLGXDOPROHFXOHVWKDWFRPSRVHLWLQVWHDGRIXQLIRUPO\LQDFLUFOH DURXQGLWVPDJQHWLFD[LVLHWKHD[LVRIWKHWXEHFRQWUDU\WRZKDWRFFXUVLQWKH VROHQRLG $PSqUH ILQDOO\ VXFFHHGHG LQ JLYLQJ D PDWKHPDWLFDO H[SUHVVLRQ IRU WKH IRUFH WKDW DFWV EHWZHHQ DQ\ WZR FRQGXFWRUV E\ YHFWRULDOO\ LQWHJUDWLQJ WKH HOH PHQWDU\IRUFHVSURGXFHGE\WKHFXUUHQWLQWKHLQGLYLGXDOYROXPHHOHPHQWVRIWKH FRQGXFWRUV7KHVHIRUFHVUHVXOWHGGHSHQGHQWRQERWKWKHGLVWDQFHDVr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caloric2KPGHILQHGWKHFRQFHSWVRI UHVLVWDQFHDQGSRWHQWLDOGLIIHUHQFHRUHOHFWURPRWLYHIRUFHDQGOLQNHGWKHPWRWKH

7KHODZVRIHOHFWURPDJQHWLVP



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magneto electric induction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¶V PHPRU\ LQ  7KDW VDPH \HDU +HQU\ DOVR GLVFRYHUHGDQGFDPHWRXQGHUVWDQGWKHSKHQRPHQRQRIVHOILQGXFWLRQXOWLPDWHO\ SXEOLVKLQJKLVZRUNLQWKHAmerican Journal of Science )DUDGD\ FDPH WR WKH GLVFRYHU\ RI LQGXFWLRQ E\ SODFLQJ KLV DWWHQWLRQ WR WKH ³RYHUDOOSURFHVV´XQGHUVWRRGE\PHDQVRIFRQFHSWXDOJHRPHWULFWRROVUDWKHUWKDQ ZLWKWKHDQDO\WLFDOPHWKRGVRIWKH)UHQFKSK\VLFLVWV VXFKDV$PSHUH ZKLFKLQWH JUDWHGWKHFRQWULEXWLRQVRIHDFKHOHPHQWRIFRQGXFWRUVDQGPDJQHWV+HLQIDFW UHMHFWHG WKH DWRPLVWLF FRQFHSWLRQ ZKLFK GLYLGHV PDWWHU LQWR DWRPV DQG HPSW\ VSDFHVLQFHKHFRQVLGHUHGLWDEVXUGWRVHSDUDWHPDWWHUIURPWKHDFWLRQVLWFDUULHV ZKLFKH[WHQGHYHU\ZKHUHLQVSDFH,QVWHDGKHUHVXPHG%RVFRYLFK¶VDWRPLFWKHR U\ DVVXPLQJ WKDW PDWWHU ZDV PDGH XSRI FHQWHUVUDGLDWLQJ lines of force WKDW ILOO WKHHQWLUHVSDFHJHQHUDWLQJWKHYDULRXVHOHFWULFDOPDJQHWLFJUDYLWDWLRQDOSKHQRP HQD DQG WKH UDGLDWLRQ LWVHOI ,Q SDUWLFXODU LQ WKH FDVH RI D PDJQHW WKHVH OLQHV RI IRUFH©DUHLQGLFDWHGLQDJHQHUDOPDQQHUE\WKHGLVSRVLWLRQRILURQILOLQJVª)DUD GD\UHJDUGHGWKLVDVDQH[SHULPHQWDOSURRIRIWKHLUH[LVWHQFHZKLOHWKHUHZDVQRQH ZKDWVRHYHUIRUWKHDHWKHUWKDWKHFRQVLGHUHGWKHUHVXOWRIVSHFXODWLRQDQGQRWRE VHUYDEOH)DUDGD\EHOLHYHGWKDWPDWWHUZDVDFRQWLQXRXVHQWLW\GLVWULEXWHGHYHU\ ZKHUH DQG FRQFOXGHG WKDW ©WKH SRZHUV DURXQG WKH FHQWHUV JLYH WKHVH FHQWHUV WKH SURSHUWLHVRIDWRPVRIPDWWHUªDVKHZURWHLQWKHLVVXHRI3KLORVRSKLFDO0DJD]LQH IURPSDJH )ROORZLQJWKHVHLQVLJKWV)DUDGD\GHYRWHGKLPVHOIWRWKHVWXG\RIHOHFWULFDOLQ GXFWLRQ WKURXJK GLHOHFWULF PDWHULDOV ILQGLQJ WKDW WKH\ PRGLI\ LW DQG WKHUHIRUH WKHUHDUHFKDUJHV³VSUHDGLQVLGH´RIWKHVHVXEVWDQFHV,QDGGLWLRQGULYHQE\WKHDW WHPSWWRXQLI\WKHIRUFHVRIQDWXUHKHZHQWLQVHDUFKRIWKHHIIHFWVWKDWDQHOHFWULF RUPDJQHWLFILHOGPD\KDYHRQOLJKW,QKHIRXQGWKDWDEORFNRIJODVVVXE MHFWHGWRWKHDFWLRQRIDVWURQJPDJQHWLVDEOHWRURWDWHWKHSODQHRISRODUL]DWLRQRI DEHDPRIOLJKWWKDWSDVVHVWKURXJKLW7KHUHDUHQRWWKHUHIRUHRQO\DIHZPDWHUL DOVVXFKDVLURQWKDWDUHDIIHFWHGE\PDJQHWLVP7KHQDIHZZHHNVODWHUKHSHU



+LVWRULFDOQRWHVWKHILUVWGLVFRYHULHVDERXWPDJQHWLVPDQGHOHFWULFLW\

IRUPHGDVHULHVRIH[SHULPHQWVRQYDULRXVVXEVWDQFHVWUDGLWLRQDOO\FRQVLGHUHGLQHUW WRPDJQHWLVPDVKHH[SHFWHGKHIRXQGWKDWWKHVHKDGDQLQIOXHQFHRQWKHLQGXF WLRQ+HDOVRGLVFRYHUHGWKDWVRPHVXEVWDQFHVKLQGHUWKHIORZRIWKHOLQHVRIIRUFH ZKLOHRWKHUVIDFLOLWDWHLWDQGGLVWLQJXLVKHGWKHPDVdiamagneticDQGparamagnetic ,QFRQFOXVLRQLIWKHPHGLXPFKDQJHVWKHLQGXFWLRQWKHFRQFHSWRIaction at a distanceORVHVYDOLGLW\WKHUHIRUHHYHQWKHSRVVLELOLW\RIDQLQVWDQWDQHRXVSURSDJD WLRQRISK\VLFDODFWLRQVZDVEHLQJTXHVWLRQHGDV)DUDGD\QRWHGLQDOHWWHUHQWLWOHG Thoughts on Ray-Vibrations SXEOLVKHG LQ WKH LVVXH RI 3KLORVRSKLFDO 0DJD]LQH IURP,QVWHDGKHLQWURGXFHGWKHFRQFHSWRIDILHOGWKDWZDVHIIHFWLYHO\UHSUH VHQWHGZLWKelectric or magnetic lines of force,QSDUWLFXODUKHLQWHUSUHWHGWKHUD GLDWLYHSURFHVVHVVXFKDVWKHYLEUDWLRQRIWKHOLQHVRIIRUFHVRZHFRXOG©GLVPLVV WKHDHWKHUª)XUWKHUPRUHVLQFH©WKHSURSDJDWLRQRIOLJKWDQGWKHUHIRUHSUREDEO\ RIDOOUDGLDQWDFWLRQRFFXSLHVWLPHªKHFRQFOXGHG©WKDWDYLEUDWLRQRIWKHOLQHRI IRUFH>IURPRQHH[WUHPHWR[email protected]VKRXOGDFFRXQWIRUWKHSKDHQRPHQDRIUDGLD WLRQLWLVQHFHVVDU\WKDWVXFKYLEUDWLRQVKRXOGRFFXS\WLPHDOVRª 0HDQZKLOHDJURXSRI*HUPDQSK\VLFLVWVDQGPDWKHPDWLFLDQVZHUHFRQWLQXLQJ WKH UHVHDUFKHV LQ WKH WUDGLWLRQ RI DFWLRQ DW D GLVWDQFH :LOKHOP (GXDUG :HEHU   EHWZHHQ  DQG  GHYHORSHG WKH ILUVW WKHRU\ RI HOHFWULF DQG PDJQHWLFSKHQRPHQDEDVHGRQWKHLGHDRIGLVFUHWHHOHFWULFFKDUJHDQGRIHOHFWULF FXUUHQWDVWKHPRWLRQRIFKDUJHVVXFFHHGLQJLQJLYLQJDQDQDO\WLFDOH[SUHVVLRQIRU WKHIRUFHEHWZHHQWZRFKDUJHVLQUHODWLYHPRWLRQWRHDFKRWKHU,QDGGLWLRQWRWKH WHUP&RXORPE¶VIRUFHDWHUPDSSHDUHG RIUHODWLYLVWLFFRUUHFWLRQ SURSRUWLRQDOWR vc ZKHUHvLVWKHLUUHODWLYHVSHHGDVZHOODVD UDGLDWLYH WHUPSURSRUWLRQDOWR WKH DFFHOHUDWLRQ DQG LQYHUVHO\ SURSRUWLRQDO WR WKH GLVWDQFH ,Q ERWK WHUPV WKHUH ZDVWKHQDFRHIILFLHQWcZKLFKZRXOGODWHUSURYHHTXDOWRWKHVSHHGRIOLJKW%H VLGHVEHLQJDQRQUDGLDOIRUFHLWZDVQRORQJHUGHSHQGHQWRQWKHGLVWDQFHDORQHD SRVVLEOHVROXWLRQZDVWKDWWKLVIRUFHZDVQRWLQVWDQWDQHRXVEXWGHOD\HGDFFRUGLQJ WRDFRQMHFWXUHWKDW&DUO)ULHGULFK*DXVV  KDGSURSRVHGWR:HEHULQD OHWWHURI 7KH FROODERUDWLRQ EHWZHHQ *DXVV DQG :HEHU EHJDQ LQ  7KHLU ILUVW MRLQW VXFFHVVZDVWKHFRQVWUXFWLRQRIWKHILUVWHOHFWURPDJQHWLFWHOHJUDSKZKLFKZDVRS HUDWLYHIURPWREHWZHHQWKH2EVHUYDWRU\DQGWKHSK\VLFVODERUDWRU\DW WKH8QLYHUVLW\RI*|WWLQJHQ7RVWXG\WKH(DUWK¶VPDJQHWLVPDPDJQHWLFREVHUYD WRU\ZDVEXLOWLQLQ*|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

7KHODZVRIHOHFWURPDJQHWLVP



SURSULDWHO\HQRXJKWKHgaussZKLFKZDVLQVHUWHGEHWZHHQWKH&*6XQLWVE\WKH 3DULV,QWHUQDWLRQDO&RQJUHVVRI)ROORZLQJWKHVHVWXGLHV*DXVVGHPRQVWUDW HGKLVLPSRUWDQWWKHRUHPRQWKHIORZZKLFKLVDIXQGDPHQWDOSDUWRIWKHHOHFWUR PDJQHWLFODZV $PRQJWKHIHZZKRDSSUHFLDWHG)DUDGD\¶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¶HTXDWLRQVPDQLIHVWHGWKHVDPHSURSHUWLHVRIWKHHOHFWURVWDWLFPDJQHWRVWDW LFDQGHOHFWURG\QDPLFPDJQLWXGHV>3HUX]]L@)XUWKHUPRUHZLWKWKHDIILUPD WLRQRIWKHZDYHWKHRU\RIOLJKWYDULRXVUHVHDUFKHUVDPRQJWKHP*HRUJH*UHHQ  DQG6WRNHVWULHGWRIRUPXODWHPRGHOVRIHODVWLFOXPLQLIHURXVDHWKHU FRQVLVWHQW ZLWK REVHUYDWLRQV ,Q SDUWLFXODU WKH PDJQHWRRSWLFV HIIHFW GLVFRYHUHG E\ )DUDGD\ KDG VXJJHVWHG WR 7KRPVRQ LQ  WKH H[LVWHQFH RI D FRXSOLQJ EH WZHHQWKHOXPLQLIHURXVDHWKHUDQGWKHHOHFWURPDJQHWLFRQH

7KHGLVFRYHU\RIHOHFWURPDJQHWLFZDYHV ,QWKH\HDU:LOOLDP+HUVFKHO  H[DPLQLQJWKHZDUPLQJSUR GXFHGE\GLIIHUHQWSRUWLRQVRIWKHVRODUVSHFWUXPZLWKDWKHUPRPHWHUIRXQGWKDWLW H[WHQGVWR³LQIUDUHG´EH\RQGWKHYLVLEOHVSHFWUXP6WLOOWKHQDWXUHRIWKHVHinfrared raysZDVQRWFOHDUWKDWLVWRVD\LIWKH\ZHUHDSDUWLFXODUFDORULFUDGLDWLRQRU DQLQYLVLEOHH[WHQVLRQRIWKHOXPHQ7KLVVHFRQGK\SRWKHVLVEHFDPHPRUHFUHGLEOH ZKHQWKHSK\VLFDOFKHPLVWV-RKDQQ:LOKHOP5LWWHU  LQDQGLQ GHSHQGHQWO\ :LOOLDP +\GH :ROODVWRQ   GLVFRYHUHG D VLPLODU H[WHQ VLRQRQWKHRWKHUHQGRIWKHVSHFWUXPWKHVHultraviolet rays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ynamical Theory of the electromagnetic FieldGLYLGHGLQWRVHYHQSDUWVSXEOLVKHG LQLQWKH3KLORVRSKLFDO7UDQVDFWLRQVRIWKHRoyal SocietyDQGWKHQXOWLPDWHO\ LQKLVPRQXPHQWDOTreatise on Electricity and MagnetismRI,QKHDO VR SXEOLVKHG DQ H[SHULPHQWDO ZRUN >0D[ZHOO @ FDUULHG RXW ZLWK WKH KHOS RI &KDUOHV+RFNLQLQZKLFKWKH\UHSHDWHGWKHPHDVXUHVRI:HEHUDQG.RKOUDXVFKE\ XVLQJDUHVLVWRULQVWHDGRIDFDSDFLWRUDQGREWDLQLQJNPV ,Q Dynamical Theory 0D[ZHOO SHUIHFWV WKH HOHFWURPDJQHWLF ILHOG WKHRU\ LQ ZKLFKWKHILHOGEHFRPHVDG\QDPLFHQWLW\ GLVWULEXWHGLQVSDFHDWKHRU\IRXQGHG RQWKHHVVHQWLDOFRQFHSWVRIPHFKDQLFVVXFKDVHQHUJ\DQGPRPHQWXPDQGRQWKH XVH RI WKH /DJUDQJLDQ PDWKHPDWLFDO PHWKRGV $W WKH VDPH WLPH KH DYRLGV UH FRXUVH WR PHFKDQLFDO PRGHOV RI ZKLFK KHKDG PDGH XVHEHIRUH FRQVLGHULQJ WKH SURSRVLWLRQVUHODWHGWRWKHVHPRGHOV©DVLOOXVWUDWLYHDQGQRWDVH[SODQDWRU\ª%XW KH ZULWHV ©LQ VSHDNLQJ RI WKH (QHUJ\ RI WKH ILHOG KRZHYHU , ZLVK WR EH XQGHU VWRRG OLWHUDOO\ª 6WDUWLQJ IURP WKH SRWHQWLDO IXQFWLRQ 0D[ZHOO FDQ H[SUHVV WKH ODZVRILQGXFWLRQDQGIURPWKHHQHUJ\FRQVHUYDWLRQSULQFLSOHKHGHULYHVWKHH[ SUHVVLRQVIRUWKHHOHFWURPDJQHWLFIRUFHV,QWKHWKLUGSDUWRIWKHZRUNKHVXPVXS KLVWKHRU\ZLWKDVHWRIGLIIHUHQWLDOHTXDWLRQVIRUWKHYDULDEOHVXVHGWRGH



+LVWRULFDOQRWHVWKHILUVWGLVFRYHULHVDERXWPDJQHWLVPDQGHOHFWULFLW\

VFULEH WKH DFWLRQV RI WKH HOHFWURPDJQHWLF ILHOG $PRQJ WKHVH LQ DGGLWLRQ WR WKH YHFWRUSRWHQWLDOWKHUHLVHYHQSUHVHQWDWHUPWRWDNHDFFRXQWIRUPDJQHWLFPRQR SROHVLIWKH\H[LVWHG7KHVHHTXDWLRQVZHUHODWHUVLPSOLILHGLQWKHVE\+HLQ ULFK +HUW]   DQG 2OLYHU +HDYLVLGH   LQ WKH YHUVLRQ ZH NQRZWRGD\DVMaxwell’s equations)LQDOO\LQWKHVL[WKSDUWRIWKHZRUNZLWKWKH WLWOHRIElectromagnetic Theory of Light0D[ZHOOGHULYHVWKHILUVWZDYHHTXDWLRQV IRU WKH SURSDJDWLRQ RI WKH SRWHQWLDO IXQFWLRQ DQG WKH PDJQHWLF ILHOG ZKLFK LQGL FDWHVWKDWWKH\DUHWUDQVYHUVHZDYHVWUDYHOLQJDWWKHVSHHGRIOLJKW ,QTreatise0D[ZHOOFROOHFWVDQGPDWKHPDWLFDOO\UHILQHVWKHMREGRQHWU\LQJWR FODULI\ WKH PHDQLQJ RI WKH HOHFWURPDJQHWLF ILHOG WKHRU\ +LV HTXDWLRQV DOVR UH TXLUHGQHZPDWKHPDWLFDOWRROVVR0D[ZHOOHPSKDVL]HVWKHUROHRIWKHYHFWRUVWR JHRPHWULFDOO\UHSUHVHQWWKHSK\VLFDOTXDQWLWLHVDQGLQWURGXFHVWKHGLIIHUHQWLDORS HUDWRUV rotor DQG convergence ZKLFK ZH QRZ FDOO divergence  $OWKRXJK 0D[ ZHOOVWLOOLOOXVWUDWHVWKHVHHTXDWLRQVZLWKWKHLGHDRIHODVWLFDFWLRQVRQDHWKHUUH VSRQGLQJ WR WKH YLHZV RI KLV HUD WKH\ DUH LQ IDFW LQGHSHQGHQW RI DQ DHWKHU PRGHOLQJ DQG WKXV KDYH ZLWKVWRRG WKH WHVW RI WLPH ,QFLGHQWDOO\ 0D[ZHOO HYHQ GHULYHGWKHH[LVWHQFHRIDUDGLDWLRQSUHVVXUHZKLFKKDGDOUHDG\EHHQLQYRNHGE\ .HSOHUWRH[SODLQWKHVKDSHRIFRPHWVWKHH[SHULPHQWDOHYLGHQFHRIWKLVSUHVVXUH ZDVREWDLQHGLQE\3\RWU1/HEHGHY  >/DQGVEHUJ@ 0D[ZHOO¶VWKHRU\KRZHYHUFRQWDLQHGDSUREOHPDWLFDVSHFWKHFRQVLGHUHGWKH HOHFWULFFKDUJHDVDQ³HOHFWULFGLVSODFHPHQW´SURGXFHGE\WKHILHOGXQOLNHWKHFRQ WLQHQWDO SK\VLFLVWV VXFK DV :HEHU ZKR VXSSRVHGO\ FRQVLGHUHG WKH FKDUJH DV D SULPLWLYHQRWLRQ,ILQGHHGWKHLGHDRIGLVSODFHPHQWKDGIDYRUHGWKHFRPSOHWLRQ RI$PSHUH¶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¶V WKHRU\ WKH H[LVWHQFH RI HOHFWURPDJQHWLF ZDYHV DSSHDUHG HYLGHQW WR KLP )RUWKHZDYHVWKHIUHTXHQF\H[HUWVWKHGRPLQDQWUROH+HUW]VWURYHWKHQDVWRKRZ WR REWDLQKLJKIUHTXHQF\ RVFLOODWLRQV ,Q WKH 'DQLVK SK\VLFLVW %HUHQG:LO KHOP)HGGHUVHQ  ZLWKWKHXVHRIURWDWLQJPLUURUVKDGGLVFRYHUHGWKDW

7KHGLVFRYHU\RIHOHFWURPDJQHWLFZDYHV



WKH GLVFKDUJH RI DQ HOHFWULF VSDUN FXUUHQW KDV D YHU\ UDSLG RVFLOODWLRQ WKLV IDFW RSHQHG XS WKH SRVVLELOLW\ RI JHQHUDWLQJ HOHFWURPDJQHWLF ZDYHV ZLWK WKH PHDQV DYDLODEOHDWWKDWWLPH,Q+HUW]H[SHULPHQWDOO\YHULILHGWKHH[LVWHQFHRIWKHVH ZDYHVLQWKHEDQGRIFHQWLPHWHUZDYHVKHWULJJHUHGKLJKIUHTXHQF\GLVFKDUJHVLQ VLPSOH HOHFWULFDO FLUFXLWV Hertz oscillators  LQFOXGLQJ DQ LQGXFWDQFH DQG VXF FHHGHGLQSLFNLQJWKHPXSXVLQJUHVRQDQWFLUFXLWVZKHUHWKHZDYHUHFHSWLRQZDV DJDLQUHYHDOHGE\DGLVFKDUJH0RUHRYHUKHGHPRQVWUDWHGWKDWWKHVHZDYHVH[KLE LWWKHVDPHSKHQRPHQDDVOLJKWVXFKDVUHIOHFWLRQUHIUDFWLRQGLIIUDFWLRQLQWHUIHU HQFHDQGSRODUL]DWLRQ7KHVHVWXGLHVFRQILUPHGWKHHOHFWURPDJQHWLFWKHRU\RIOLJKW H[SRXVHGE\0D[ZHOO

6XJJHVWHGUHDGLQJ 7KHVHKLVWRULFDOQRWHVDUHGHULYHGIURPWKHIROORZLQJWH[WV 9$ Encyclopaedia BritannicaDWHQWULHV³(OHFWULFLW\´³*LOEHUW´ 9$ Enciclopedia della scienza e della tecnica0RQGDGRUL(GDWWKHHQWU\³(OHWWULFLWj´ %HUQDO-RKQ'Storia della Scienza(GLWRUL5LXQLWL5RPD,,,HG   7KLVERRNGH VFULEHV WKH PXWXDO UHODWLRQVKLS EHWZHHQ VFLHQFH DQG VRFLHW\ DQG DOORZV XV WR IUDPH WKHSURJUHVVRIRSWLFVDQGHOHFWURPDJQHWLVPLQWKHJHQHUDOFRQWH[WRIVFLHQWLILFGHYHO RSPHQWDQGFLYLOL]DWLRQ 'UDJRQL*LRUJLR Le origini del campo elettromagnetico/H6FLHQ]HQqLQVHUWI grandi esperimenti $SULO ,QDGGLWLRQWRGLVFXVVLQJWKHKLVWRU\RIWKHHOHFWURPDJQHWLF ILHOGLWJLYHVPDQ\GHWDLOVRIWKHH[SHULPHQWVRI2HUVWHG$UDJR$PSHUHDQG)DUD GD\ )UHJRQHVH/XFLRVolta: teorie ed esperimenti di un filosofo naturale,JUDQGLGHOODVFLHQ]D Qq  HG/H6FLHQ]H6S$0LODQR )XFKV :DOWHU 5 La fisica moderna illustrata 5L]]ROL HG 0LODQR   7KLV GHVFULEHV WKHSDWKRISK\VLFVRIWKHODVWWKUHHFHQWXULHVDQGDGGUHVVHVWKHGHWDLOVRILQGLYLGXDO GLVFRYHULHVDQDO\]LQJWKHFRQFHSWXDODQGSKLORVRSKLFDOYDOXHZLWKSDUWLFXODUHPSKD VLVRQWKHGLIILFXOWUHODWLRQVKLSEHWZHHQPRGHODQGUHDOLW\ +DOO$5H%RDV+DOO0Storia della Scienza(GLO0XOLQR%RORJQD  7KLVWHOOV WKH KLVWRU\ RI VFLHQFH WKURXJK WKH PHQ ZKR WKURXJK WKHLU WKLQNLQJ DQG WKHLU ZRUN KDYHOHIWDIXQGDPHQWDOLPSULQW -DFNVRQ-'Classical Electrodynamics-RKQ:LOH\ 6RQV1HZ (@

1 &

9 P

> %@

1 ˜ VHF &˜P

9 ˜ VHF P

:E P

7HVOD *DXVV 

,Q WXUQ WKHVH ILHOGV GHWHUPLQHG E\ WKH FKDUJHV DQG WKHLU PRWLRQ HYROYH DF FRUGLQJWRWKHVSOHQGLGHTXDWLRQVRI0D[ZHOOKHUHWUDQVFULEHGLQWKHLUSXUHGLIIHU HQWLDOIRUPWKDWLVLQHPSW\VSDFH ­’[email protected]

$ P 

&RQFHUQLQJWKHGLPHQVLRQRI 0LWVKRXOGEHQRWHGWKDWLQWKHOLWHUDWXUHWKHUH DUHGLIIHUHQWGHILQLWLRQVRILW7KHRQHXVHGKHUHFRXOGEHWKHPRVWVXLWDEOH )RUDPHGLXPLQPRWLRQZLWKUHVSHFWWRDQREVHUYHUZHKDYHIXUWKHUFRQWULEX WLRQV WR WKH DYHUDJH GHQVLWLHV RI FKDUJH DQG FXUUHQW )RU VPDOO YHORFLWLHV LQ WKH QRQUHODWLYLVWLFOLPLW(TV  EHFRPH 

U U P  ’-DFNVRQ @ WKH ILUVW RQO\ IROORZV IURP UHODWLYLVWLF FRQVLGHUDWLRQV >%HFNHUDQG6DXWHU@+HUH -PLVWKHPDFURVFRSLFFXUUHQWGHQVLW\LQWKHRE VHUYHU UHIHUHQFH ZKLFK WKHUHIRUH DOVR LQFOXGHV WKH SURGXFW RI WKH PDFURVFRSLF FKDUJHGHQVLW\IRUWKHYHORFLW\RIWKHPHGLXP )LQDOO\WKHPDFURVFRSLF0D[ZHOO¶VHTXDWLRQVDUH

0D[ZHOO¶VHTXDWLRQVLQPDFURVFRSLFPHGLD









­ °’Qu ( ± ( @ l ZKHUHQLVSHUSHQGLFXODUWRWKHUHFWDQJOH¶VVXUIDFH+RZHYHUWKHGLUHFWLRQRIQLV DUELWUDU\ WKHUHIRUH WKH DUJXPHQW LQ WKH VTXDUH EUDFNHW PXVW EH ]HUR 7KH UHVXOW IRUWKH+ILHOGLVREWDLQHGZLWKDVLPLODUSURFHGXUH,QFRQFOXVLRQZHILQGWKDW 

Q u (  (

 Q u +   +

-V  

 



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

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

&RPSOH[ILHOGUHSUHVHQWDWLRQ ,QRSWLFVZHDUHHVVHQWLDOO\LQWHUHVWHGLQRVFLOODWLQJILHOGVDQGWKHSURSDJDWLRQ RIwavesWKDWLVRSWLFDOsignalsIRUZKLFKDSSURSULDWHPDWKHPDWLFDOWRROVDUHUH TXLUHG $PRQJ WKHVH WKH FRQFHSW RI DQ analytical signal RU D complex analytic functionZDVLQWURGXFHGLQE\'*DERULQFRPPXQLFDWLRQVWKHRU\,WILQGV ZLGHDSSOLFDWLRQLQRSWLFVLQZKLFKWKHUHDOILHOGVDUHUHSODFHGZLWKFRPSOH[YDUL DEOHV ZLWK D IRUPDOLVP WKDW EHFRPHV HVVHQWLDO LQ GHDOLQJ ZLWK GLVSHUVLYH PHGLD DQGQRQPRQRFKURPDWLFZDYHV $FRPSOH[IXQFWLRQg z RIWKHFRPSOH[YDULDEOHz xiyLVFDOOHGholomorphicRUanalyticLQzRLILWVGHULYDWLYHH[LVWVLQDILQLWHQHLJKERUKRRGRIzRLQFOXG LQJ WKH VDPH SRLQW zR 7KH WZR WHUPV DQDO\WLFDO DQG KRORPRUSKLF DUH V\QRQ\ PRXV,IWKHIXQFWLRQLVKRORPRUSKLFLQWKHQHLJKERUKRRGRIzREXWQRWLQzRVXFK DSRLQWLVFDOOHGsingularRUDsingularityofg z ,IDJLYHQSRLQWzRLVQRWVLQJXODU WKHQLWFDQEHVKRZQWKDWDWWKLVSRLQWallGHULYDWLYHVRIg z H[LVW7KHDQDO\WLFLW\ RIg z LPSRVHVVWULQJHQWFRQGLWLRQVEHWZHHQLWVUHDODQGLPDJLQDU\SDUWV:ULWLQJ gDV g z u x y  iw x y 



 

ZKHUH u DQG w DUH UHDO IXQFWLRQV WKH QHFHVVDU\ DQG VXIILFLHQW FRQGLWLRQ VR WKDW g z LVDQDO\WLFLVWKDWLWIXOILOOVWKHUHODWLRQVRICauchy-Riemann



wg x  iy wx wu x y wx



wg x  iy  WKDWLV wy

ww x y wu x y  wy wy

ww x y   wx



 

&RPSOH[ILHOGUHSUHVHQWDWLRQ



  7KLVFRQGLWLRQDOVRHQVXUHVWKDWu xy DQGw xy DUHharmonicIXQFWLRQVLHWKDW WKH\VDWLVI\WKHWZRGLPHQVLRQDOHTXDWLRQVRI/DSODFH ’u x y ’ w x y  



 

,QPDQ\DSSOLFDWLRQVZHKDYHWRZRUNZLWKUHDOIXQFWLRQVWKDWRVFLOODWHLQWLPH DWDVLQJOHIUHTXHQF\QIRUH[DPSOHWKHUHDOILHOGVWKDWUHSUHVHQWPRQRFKURPDWLF ZDYHV ,W LV FRQYHQLHQW WR UHSUHVHQW WKHP PDWKHPDWLFDOO\ ZLWK WKH UHDO SDUW RI D FRPSOH[H[SRQHQWLDO 

J t

$ FRV SQt  M ƒe ^ A H[S Si Qt  iM ` 

 

ZKHUH $ LV D FRQVWDQW 0RUH RIWHQ ZH KDYH WR FRQVLGHU PRUH JHQHUDO IXQFWLRQV ZKHUH IRU H[DPSOH WKHLU DPSOLWXGHA LV D FRPSOH[ IXQFWLRQRI WLPH A t  7KHQ ZDQWLQJWRH[SUHVVJ t DVWKHUHDOSDUWRIDFRPSOH[IXQFWLRQg t LQSULQFLSOHZH FDQVLPSO\ZULWH g t



J t  i K t 

 

ZKHUH K t  LV D UHDO DUELWUDU\ IXQFWLRQ +HUH , XVH UHJXODU FKDUDFWHUV WR LQGLFDWH WKDW WKH IXQFWLRQ J DQG K WRR  LV UHDO VR DV WR GLVWLQJXLVK LW IURP WKH FRPSOH[ IXQFWLRQgVKRZQLQLWDOLFV7KHEHVWZD\WRFKRRVHK t LVWRPDNHg t DQDO\WLF ZKHQWKHWLPHtLVUHSODFHGZLWKWKHFRPSOH[YDULDEOHW tit'7KLVFKRLFHHOLP LQDWHVWKHDUELWUDULQHVV/HWXVVHHKRZ &RQVLGHUD³JRRG´UHDOIXQFWLRQJ t LWV)RXULHUWUDQVIRUPLVGHILQHGDV G Z



f

³f J t eiZt dt 

 

%HFDXVHJ t LVUHDOKHUHZHFDQHDVLO\SURYHWKDW G Z G Z 



 

7KHLQYHUVHWUDQVIRUPLV 

J t

 f G Z e iZt d Z  S ³f

 

  +HUH,DVVXPHWKHSURSHUW\DQGWKHFRQGLWLRQVRIWKHH[LVWHQFHRIWKH)RXULHUWUDQVIRUPVWREH

NQRZQ 7KH\ ZLOO EH UHFDOOHG PRUH IXOO\ LQ &KDS  ZKLFK LV GHGLFDWHG WR )RXULHU 2SWLFV 0RUHRYHU , RQO\ FRQVLGHU ³JRRG´ IXQFWLRQV KHUH WKDW LV IXQFWLRQV VXIILFLHQWO\ UHJXODU WR EH FRQWLQXRXV GLIIHUHQWLDEOH OLPLWHG DQG GHOLPLWHG )LQDOO\ FRPSDUHG WR WKH VWDQGDUG GHILQLWLRQ WKHH[SRQHQWLDODUJXPHQWLVWDNHQZLWKWKHRSSRVLWHVLJQ VHHEHORZ 



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 :HQRZGHILQHWZRQHZFRPSOH[IXQFWLRQV 

g t

 f G Z e iZt d Z DQG S ³

g t

  G Z e iZt d Z  S ³f

 

7KHLUVXPUHWXUQVWKHGRXEOHRIWKHUHDOIXQFWLRQJ5HSODFLQJ ZZLWK ZLQWKH VHFRQGLQWHJUDODQGDSSO\LQJ(T  ZHILQGWKDW g t



g t 

 

,IWKHUHIRUHLQg t DQGWKHLQWHJUDOVRI(TV  ZHUHSODFHtZLWKWKHFRPSOH[ YDULDEOH W  t  it' ZH FDQ HDVLO\ YHULI\ WKDW g W  VDWLVILHV WKH &DXFK\5LHPDQQ UHODWLRQV7KHUHIRUH gLVWKHVRXJKWanalytic functionFRUUHVSRQGLQJWRJ:HFDQ WKHQZULWH J t



  g t  g t   

 

DQGUHSODFHgZLWKgLQ(T  ZLWKWKLVFKRLFHWKHIXQFWLRQKWKDWUHSUHVHQWV WKHLPDJLQDU\SDUWRI gWXUQVRXWWREHWKHHilbert transformRILWVUHDOSDUWJ,Q GHHGXVLQJWKHVLJQIXQFWLRQVJQ(T D FDQEHUHZULWWHQDV 

g t

 f G Z ª¬  VJQ Z º¼ e iZt d Z  S ³f

 

7KHUHIRUH WKH )RXULHU WUDQVIRUP RI g LV JLYHQ E\ G Z  G Z VJQ Z  )RU WKH FRQYROXWLRQWKHRUHP VHH&KDS WKHVHFRQGWHUPLVWKH)RXULHUWUDQVIRUPRIWKH FRQYROXWLRQLQWHJUDOEHWZHHQJ t DQGWKHIXQFWLRQ i §·  P ¨ ¸  S ©t¹



ZKHUHWKHV\PERO PLQGLFDWHVWKHSULQFLSDOSDUW7KHWUDQVIRUPRIWKLVIXQFWLRQLV VJQ Z 7KXV 

g t

f J t c  J t  i P ³ dt c J t  iK t  S f t  t c

 

$UHDOIXQFWLRQKDVDQLPSRUWDQWSURSHUW\ 7KHDQDO\WLFIXQFWLRQRIWKHGHULYDWLYHRIDUHDOIXQFWLRQLVHTXDOWRWKHGH ULYDWLYHRIWKHDQDO\WLFIXQFWLRQRIWKHUHDOIXQFWLRQ

&RPSOH[ILHOGUHSUHVHQWDWLRQ



  ,QGHHGLQWHJUDWLQJE\SDUWVZHILQGWKDWWKH)RXULHUWUDQVIRUPRIWKHGHULYDWLYHRI D³JRRG´UHDOIXQFWLRQJ t LV 

f

³f

wJ t iZt e dt wt

iZ³

f

f

J t eiZt dt

iZ G Z 

7KHUHIRUHWKHDQDO\WLFDOIXQFWLRQFRUUHVSRQGLQJWRwJ wtLV 

 f G Z iZ e iZt d Z S ³

w  f G Z e iZt d Z wt S ³

wg t  wt

7KLV UXOH H[WHQGV WR DOO GHULYDWLRQ RUGHUV ,W IROORZV LQ SDUWLFXODU WKDW LI D UHDO IXQFWLRQJ rt VDWLVILHVWKHZDYHHTXDWLRQ ’ J r  t 



 w J r  t   v  wt 

 

ZKHUH’LVWKH/DSODFHRSHUDWRURQWKHFRRUGLQDWHVRIrDQG vLVDYHORFLW\WKHQ WKHVDPHZDYHHTXDWLRQLVYDOLGIRULWVDQDO\WLFDOIXQFWLRQ ’ g r  t 



 w g r  t   v  wt 

 

7KHVHHTXLYDOHQFHVOHDGXVWRDSDUWLFXODUO\XVHIXOUHSUHVHQWDWLRQRIWKHYDULRXV UHDO HOHFWURPDJQHWLFHQWLWLHVLQWHUPVRIFRPSOH[DPSOLWXGHV,QGHHGE\DSSO\ LQJ(T  WRWKHHOHFWULFILHOGZHJHWWKDWLWFDQEHGHFRPSRVHGDVWKHVXP RIWZRFRPSOH[FRQMXJDWHYHFWRUV 

( r t

  E r  t  E r  t ƒe ª¬E r  t º¼   

 

ZKHUHELVWKHDQDO\WLFIXQFWLRQDVVRFLDWHGZLWK(DQGWKDWLV 

E r t

 f Ö i Zt d Z  S ³ E r  Z e

 

$VLPLODUGHFRPSRVLWLRQLQWRWZRFRPSOH[YHFWRUV FRQMXJDWHGWRHDFKRWKHU  FDQEHGRQHIRUWKHRWKHUUHDOYDULDEOHV7KLVGHFRPSRVLWLRQDOORZVIRUVLPSOLI\LQJ WKHVROXWLRQRI0D[ZHOO¶VHTXDWLRQVDQGDZLGHXVHLVPDGHRILWWRWKHSRLQWRI WKHFRPSOH[YHFWRUVQRWEHLQJGLVWLQJXLVKHGIURPWKHUHDORQHV7RDYRLGFRQIX VLRQLQWKLVWH[WWKHYHFWRUVIRUUHDOHOHFWURPDJQHWLFYDULDEOHVDUHGHQRWHGE\UHJ XODU5RPDQOHWWHUVZKLOHWKHFRUUHVSRQGLQJFRPSOH[YHFWRUVDUHLQGLFDWHGE\italicOHWWHUV



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 7KH0D[ZHOO¶VHTXDWLRQV  FDQWKHQEHZULWWHQLQWHUPVRIDQDO\WLFIXQF WLRQVRIWKHILHOGVDQGWKHVRXUFHGHQVLWLHV



­’< D r ,t ° °’< B r ,t ® °’ u E r ,t °¯’ u H r ,t

U P r ,t 

  w B r ,t wt 



 

JP r ,t  w D r ,t wt

ZKHUH E B D H U P DQG JPGHQRWHWKHDQDO\WLFDOIXQFWLRQVRI ( % ' +UP DQG -PUHVSHFWLYHO\7KHVHHTXDWLRQVDUHJHQHUDODOWKRXJKWRUHVROYHWKHPZH DOVRQHHGWKHFRQVWLWXWLYHUHODWLRQVEHWZHHQ E BDQG D H U P   JPZKLFKDUH JHQHUDOO\JLYHQE\DGGLWLRQDOGLIIHUHQWLDORULQWHJUDOHTXDWLRQVLQVSDFHDQGWLPH 6LQFHWKH0D[ZHOOHTXDWLRQVDUHOLQHDURQHRIWKHLUDOWHUQDWLYHIRUPVLVWRH[ SUHVVWKHPLQWKHVSDFHRIWKHVSDWLDODQJXODUIUHTXHQF\kDQGWHPSRUDORQHVZ WKURXJKWKH)RXULHUWUDQVIRUPRIWKHILHOGVDQGRIWKHUHDOVRXUFHVGHILQHGKHUHDV f f f f



G k  Z

³ ³ ³ ³ J r t ei k r  Zt dr dt  x

 

f f f f

,QWKLVH[SUHVVLRQDFHUWDLQFRQYHQWLRQLVHVWDEOLVKHGLQWKHVLJQVRIWKHWHUPVLQ WKHH[SRQHQWLDOLWLVQRWHZRUWK\WKDWZKLOHWKHVLJQRIk ‡ rDJUHHVZLWKWKHVWDQG DUGGHILQLWLRQDVD)RXULHUWUDQVIRUPWKHVLJQRIZtLVLQVWHDGRSSRVLWH ,QIUHTXHQF\VSDFHWKHGLIIHUHQWLDORSHUDWRUVDUHUHSODFHGE\PXOWLSOLFDWLRQV 

’ < o ik <  ’ u o ik u 

w o i Z  wt

 

DQGWKH0D[ZHOO¶VHTXDWLRQVEHFRPH



­ik < DÖ k ,Z UÖ P k ,Z  ° °ik < BÖ k ,Z   ® °ik u EÖ k ,Z iZBÖ k ,Z  °ik u HÖ k ,Z JÖ k ,Z  iZDÖ k ,Z  P ¯

 

ZKHUHWKHAV\PEROLQGLFDWHVWKHWUDQVIRUP 9HU\ RIWHQ LQ WUHDWLQJ WKH ZDYHV ZH FRQVLGHU WKH FDVH RI monochromatic wavesZLWKDQJXODUIUHTXHQF\ZRDVVLJQHG,QWKLVFDVHLQSODFHRI(T     7KHFRQYHQLHQFHRIWKLVRSSRVLWLRQRIVLJQVZLOOEHDSSDUHQWKHUHLQDIWHU7KLVFKRLFHLVWKHRQH

PRVWFRPPRQO\IROORZHGLQRSWLFVEXWEHDZDUHWKDWVRPHDXWKRUVUHYHUVHERWKVLJQV

&RPSOH[ILHOGUHSUHVHQWDWLRQ



 ZHFDQZULWH



E r t



E r e i ZR t 

 

ZKHUH E r LVDFRPSOH[YHFWRUWKDWFDUULHVZLWKLWDOORIWKHVSDWLDOGHSHQGHQFH EXWLVFRQVWDQWLQWLPH,WUHSUHVHQWVWKHamplitudeRIWKHILHOGZKLOHWKHH[SRQHQ WLDOUHSUHVHQWVWKHtemporal carrier:KHQWKHZDYHKDVWKHGHILQLWHGLUHFWLRQkR RIDSODQHZDYHWKHDQDO\WLFDOIXQFWLRQRIWKHILHOGFDQEHIDFWRUHGDV

E r t



ER ei kR < r ZR t 

 

2Q WKH RWKHU KDQG D PRQRFKURPDWLF ZDYH FRQVWLWXWHV D VWURQJ LGHDOL]DWLRQ ZKLOH LQ JHQHUDO WKH ILHOG LV FRPSRVHG RI D EDQG RI IUHTXHQFLHV IRU ZKLFK LQ SULQFLSOHZHVKRXOGUHWXUQWRXVHLWVUHSUHVHQWDWLRQLQWKHIUHTXHQF\VSDFH+RZ HYHULIWKHVSHFWUXPRIWKHILHOG (LVVXIILFLHQWO\QDUURZDURXQGWKHDQJXODUIUH TXHQF\ZRWKHDQDO\WLFIXQFWLRQRIWKHILHOGLVXVXDOO\IDFWRUHGDV E r t



E r  t e iZR t 

 

ZKHUH E r t  LV D VORZO\ YDU\LQJ DPSOLWXGH LQ WLPH 7KLV IDFWRUL]DWLRQ FDQ EH JHQHUDOL]HGWRLQFOXGHWKHGHSHQGHQFHRQWKHDQJXODUVSDWLDOIUHTXHQF\ kRLQWKH FDUULHU DV ZHOO 7KH FRQGLWLRQV LQ ZKLFK ZH KDYH D TXDVLPRQRFKURPDWLF ZDYH ZLOOEHVSHFLILHGLQWKHQH[WVHFWLRQ 7KHVDPHW\SHRIH[SUHVVLRQVDSSOLHVWRDOORIWKHRWKHUYDULDEOHV

1.3.1 Generalization of susceptibility for dispersive media ,QWKHPRVWJHQHUDOFDVHWKHSRODUL]DWLRQ 3GRHVQRWLQVWDQWO\IROORZWKHILHOG (QRUWKHPDJQHWL]DWLRQ0WKHILHOG%DVZRXOGEHVXJJHVWHGE\(TV  ,Q SULQFLSOH DQ\ PHGLXP UHTXLUHV D FHUWDLQ UHVSRQVH WLPH WR DGDSW WR WKH ILHOG DQG UHWDLQVVRPHPHPRU\RIWKHSDVW7KHUHIRUHUHPDLQLQJLQWKHDPELWRIlinearPH GLDDJHQHUDOL]DWLRQRI(T D FDQEHJLYHQE\DQLQWHJUDWLRQRQSDVWWLPHV IRUH[DPSOH 

3 r t

HR ³

t  F f e

r t  t c ( r t c dt c 

 

  ZKHUH rLVWKHSRVLWLRQLQVSDFH+HUH Fe r  t Fe r  t LVDIXQFWLRQRIDWLPH GLIIHUHQFHLWOLWHUDOO\H[SUHVVHVWKHPHPRU\RIWKHPHGLXPDQGWKHXSSHUOLPLWRI WKHLQWHJUDOFRUUHVSRQGVWRWKHprinciple of causalityIRUZKLFKRQHcannotKDYHD PHPRU\RIWKHIXWXUH7KLVH[SUHVVLRQLVRQO\YDOLGLQWKHFDVHLQZKLFKWKHLQWHU



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 DFWLRQLVlocalZKLOHLQWKHFDVHLQZKLFK3LVDOVRLQIOXHQFHGE\WKHILHOGLQRWKHU SRVLWLRQVLQWKHVSDFHWKHLQWHJUDWLRQPXVWEHH[WHQGHGHYHQWRWKHYDULDEOH rRQ WKHwholeVSDFH  7KHHTXDWLRQIRU 3LVMXVWWKHWHPSRUDOcorrelationRI Fe r  t ZLWK ( r t DQG  LW LV DOVR WKH WHPSRUDO convolution RI Fe r  t  ZLWK ( r t  (T   FDQ EH WUHDWHG PRUH FRQYHQLHQWO\ E\ PHDQV RI WKH WHPSRUDO  )RXULHU WUDQVIRUP LQ WKH VSDFHRIWKHDQJXODUIUHTXHQFLHVZ LQSODFHRIt ,IWKHLQWHUDFWLRQLVnon-local WKH VSDWLDO WUDQVIRUP LV DOVR UHTXLUHG VR DV WR H[SUHVV WKH ILHOG LQ WKH VSDFH RI wave vectors k LQSODFH RI r  )RU H[DPSOH WKH RYHUDOO VSDFHWLPH WUDQVIRUP RI ( r t LVFRPPRQO\IRUPXODWHGE\ 

EÖ k Z

³³³ dr ³ dt ( rt ei k˜r Zt 

ZKHUHWKHAVLJQGLVWLQJXLVKHVWKHWUDQVIRUPIURPWKHRULJLQDOIXQFWLRQ2UGLQDU\ PHGLDDUHKRZHYHUJHQHUDOO\localDQGLWLVJRRGWROLPLWRXUVHOYHVKHUHWRFRQ VLGHULQJWKHVROHWHPSRUDOWUDQVIRUP LQSDUWLFXODUWKLVDOORZVXVHDVLO\WRWUHDWWKH RUGLQDU\LQKRPRJHQHRXVPHGLD )RUH[DPSOHIRU(WKLVLVH[SUHVVHGE\WKH)RX ULHULQWHJUDO 

EÖ r  Z

f

³ f ( r t ei Zt dt 

ZKHUHZHVKRXOGUHPHPEHUWKDW EÖ r  Z

 

EÖ r  Z VLQFH(LVUHDO

,IZHDSSO\WKLVWUDQVIRUPWR(T  IRUWKHFRQYROXWLRQWKHRUHPZHILQG WKDW 

PÖ r Z

HR Fe Z EÖ r Z 

 

 ZKHUHFe Z LVWKHWUDQVIRUPRI Fe t WKDWLVWKHLQWHJUDO



Fe Z

f



³ f Fe t ei Zt dt 

 

 ZKHUHZHWDNH Fe t  IRUt )LJ (T  KDVDYHU\VLPLODUIRUP WR(T D DQGLQIDFWWKHLQVWDQWDQHRXVUHVSRQVHLPSOLHGE\WKDWHTXDWLRQ FRUUHVSRQGVWRFe Z  FRQVWDQWDVLVHYLGHQWIURP(T  IRUZKLFKDFRQ VWDQWFeFRUUHVSRQGVWRD'LUDFGHOWDIXQFWLRQFHQWHUHGDWt  $FRPSOHWHO\DQDORJRXVUHDVRQLQJFDQDOVREHDSSOLHGWRWKHPDJQHWLFVXVFHSWL ELOLW\IRUZKLFKZHKDYH



Ö r Z P F Z HÖ r Z  M R m

 

DQG LQ SDUDOOHO ZLWK (TV   ZH FDQ QRZ GHILQH WKH HOHFWULF SHUPLWWLYLW\ H

&RPSOH[ILHOGUHSUHVHQWDWLRQ







c(w)

c(t)=c(-t)

Ám (c) w

t

 e (c) (b)

(a)

)LJ D PHPRU\IXQFWLRQRIWKHPHGLXPLQWKHSUHVHQFHRIDUHVRQDQFH E UHDODQG LPDJLQDU\SDUWVRIWKHFRUUHVSRQGLQJVXVFHSWLELOLW\

DQGWKHPDJQHWLFSHUPHDELOLW\PDVWKHFRPSOH[TXDQWLWLHV 

H Z

ª¬  Fe Z º¼ HR  P Z

ª¬  Fm Z º¼ P R 

 

IRUZKLFK 

DÖ r  Z

H Z EÖ r  Z  BÖ r  Z P Z HÖ r  Z 

 

7KHQLIWKHPHGLXPLVlinearDQGlocalEXWQRWQHFHVVDULO\QRQGLVSHUVLYHLQ WKHFDVHRIDmonochromaticZDYHZLWKDQJXODUIUHTXHQF\ZRUHSUHVHQWHGIRUWKH HOHFWULFILHOGE\(T  (TV  DQG  FDQEHUHYHUVHGQRZWDN LQJWKHFDVDVVLJQHGFRHIILFLHQWV FRQVWDQWRYHUWLPH ,QWKHVHFRQGLWLRQVWKHSR ODUL]DWLRQDQGPDJQHWL]DWLRQKDYHWKHFRPSOH[UHSUHVHQWDWLRQ



P r t

HR Fe r  ZR E r  t 

M r t

Fm r  Z

R

H r t

Fm r  Z   R B r t    Fm r  Z PR

 

R

LQDQDORJ\WR(TV  )RURXUmonochromaticZDYHLQDdielectricPHGLXP ZKLFKLVlinearDQGnot in motion(TV  WKXVEHFRPH



­’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twoGLVWLQFWHOHFWURPDJQHWLFPRPHQWDVKRXOGEHFRQVLGHUHGWKHkinetic momentumDQGWKHcanonical momentumDOUHDG\GLVFXVVHGE\-&*DUULVRQDQG 5*DUULVRQDQG&KLDR@)RUDSKRWRQWKHILUVWLVƫZ cnJ DQGWKH VHFRQGDVVRFLDWHGZLWKWKHZDYHOHQJWKLVƫZncZKHUH nJDQGnDUHUHVSHFWLYH O\WKHUHIUDFWLYHLQGLFHVIRUWKHJURXSYHORFLW\DQGWKHSKDVHYHORFLW\7KHNLQHWLF PRPHQWXP LV HTXDOWRWKDWRI$EUDKDPDQGWKHFDQRQLFDO PRPHQWXPWRWKDWRI 0LQNRZVNL:KLFKRIWKHVHWZRPRPHQWDLVUHYHDOHGGHSHQGVRQWKHH[SHULPHQW LWVHOI

1.5.2 Flow of energy in simple media 1.5.2.1,QWHQVLW\RIDPRQRFKURPDWLFZDYH /HWXVQRZVHHWKHFDVHRIDPRQRFKURPDWLFZDYHLQDVLPSOHGLHOHFWULFPHGL XP ZKLFK LV linear local isotropic homogeneous transparent DQG nondispersive0DNLQJH[SOLFLWWKHWHPSRUDOSDUWLQWKHFRPSOH[UHSUHVHQWDWLRQRIWKH ILHOGVZHKDYH

(QHUJ\IORZDQGGHQVLW\RIPRPHQWXPDQGHQHUJ\







( r  t ƒe ª¬ E r e i Zt º¼   + r  t ƒe ª¬ H r e i Zt º¼ 



 

*LYHQWKDW (DQG +DUHRVFLOODWLQJ 6  (u+RVFLOODWHVZLWKDGRXEOHIUHTXHQ F\,WLVWKHUHIRUHXVHIXOWRFRQVLGHULWVPHDQYDOXHFDOFXODWHGRYHUDSHULRG6XFK YDOXHDVLVNQRZQIURP(T  LVJLYHQE\ 

6

 ƒe 

E uH 

 

7RHYDOXDWHWKLVYHFWRULQWKHVLPSOHVWFDVHVZHFDQORFDOO\DSSUR[LPDWHRXU ZDYHZLWKDKRPRJHQHRXVSODQHZDYH E r



Er ei kr˜r  H r

H r ei kr˜r 

 

ZKHUHWKHDPSOLWXGHV ErDQG HrDQGWKHZDYHYHFWRU krDUHVORZO\YDU\LQJIXQF WLRQV RI r 7KDQNV WR WKLV DVVXPSWLRQ E\ DSSO\LQJ (TV   WR WKH 0D[ZHOO HTXDWLRQVDQGWRWKHZDYHHTXDWLRQWKHFRQWULEXWLRQRIWKHWHUPVFRQWDLQLQJVSD WLDOGHULYDWLYHVRI Er HrDQG krDUHQHJOLJLEOHFRPSDUHGWRWKRVHZLWKWKHGHULYD WLYH RI WKH FDUULHU 7KHUHIRUH LW LV VWLOO IRXQG WKDW kr LV VXEMHFW WR WKH FRQVWUDLQWV >VHH(T  @ 

kr

HP Z

n S  Z n c Oo

 

,Q DGGLWLRQ LW LV RULHQWHG SHUSHQGLFXODUO\ WR Er DQG Hr DV LV FOHDU IURP (TV  ,QDGGLWLRQIURP(TV  ZHJHW kr u Er



ZPH r 

 

VRWKDWIRUWKHPRGXOHVRIErDQGHrZHKDYH 

P Hr

H Er 

 

)LQDOO\ZHFDQFDOFXODWHWKHWLPHDYHUDJHRIWKH3R\QWLQJYHFWRUVXEVWLWXWLQJ (T  DQGWKHQ(T D LQ(T   

6

ª

 §k ·º Er u ¨ r u Er ¸» « P ©Z ¹»¼ ¬

ƒe «

7KHQIRU(T  ZHFDQZULWH

  kr Er  Z P



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 

 H  Er V   P

6

D 

WKDWLV

6



c  H Er V  n

n  Er V RUDOVR 6 cP

EF 

ZKHUHVLVWKHYHUVRURIkr,QSDUWLFXODUIRUDQRQPDJQHWLFPHGLXPZLWKP PR (T D EHFRPHV 

nc  HR Er V  

6

 

/HWXVQRZH[DPLQHWKHWLPHDYHUDJHRIWKHHQHUJ\GHQVLW\ w RIRXUZDYHLQ WKHPHGLXPXQGHUH[DPLQDWLRQ)URP(T  ZHKDYH w

we  wm

 ƒe 

Er < Dr  Hr < Br

 H Er 





 P Hr





 

IURPZKLFKZHVHHZLWK(T  WKDWWKHHOHFWURPDJQHWLFHQHUJ\LVHTXDOO\ GLYLGHGEHWZHHQHOHFWULFDQGPDJQHWLFILHOGV 

we

wm 

 

DQGWKHUHIRUH(T  FDQEHVLPSOLILHGWRWKHH[SUHVVLRQ 

w

  H Er  

 

7KHLQWHQVLW\ IRIWKHZDYHLVWKHPRGXOXVRI 6 DQGVRLWFDQILQDOO\EHSXWLQ UHODWLRQZLWKWKHHQHUJ\GHQVLW\ w  

I

c H Er n



c w  n

 

ZKLFKH[SUHVVHVWKHIDFWWKDWWKHHQHUJ\LVPRYLQJDWDVSHHGv cn $QLPSRUWDQWUHVXOWLVREWDLQHGE\DSSO\LQJWKHILHOGVJLYHQE\(TV  WR WKH ODZ RI HQHUJ\ FRQVHUYDWLRQ IRU D non-conductive PHGLXP V   o J    ZKHUHQRPHFKDQLFDOZRUNLVGRQH RQWKHFKDUJHV DQGWKHUHIRUHQRQDEVRUEHQW E\WDNLQJWKHWHPSRUDODYHUDJHRI(T  ZHILQGWKDW 

GLY 6

 

 

(QHUJ\IORZDQGGHQVLW\RIPRPHQWXPDQGHQHUJ\







%\ LQWHJUDWLQJ WKLV HTXDWLRQ RQ DQ DUELWUDU\ YROXPH ZKLFK GRHV QRW FRQWDLQ KRZHYHUVRXUFHVRUDEVRUEHUVDQGDSSO\LQJWKH*DXVVWKHRUHPLWUHVXOWVWKDW

v³ 6VHHDOVR0DQVXULSXU  S HW VHT@ 7KH\ DUH VWLOO WKH VXEMHFW RI VWXG\ DQG GHEDWH $ GLIIHUHQW DSSURDFK ZKLFK RULJLQDWHV ZLWK)UHVQHOLVJLYHQIRUH[DPSOHE\6LYRXNKLQH  †DQG

5HIOHFWLRQDQGUHIUDFWLRQRQSODQHLQWHUIDFHV



  HQFH H[FHSW LQ WKH H[SHFWHG GLUHFWLRQ IURP WKH PDFURVFRSLF HTXDWLRQV ZLWK WKH PDLQHIIHFWRIDOWHULQJWKHSURSDJDWLRQVSHHGFRPSDUHGWRWKDWLQYDFXXP6LPLODU O\WRWKHGLIIUDFWLRQSURGXFHGE\WKHHGJHRIDQREVWDFOHWKHUHIOHFWLRQLVFDXVHG E\DGLVFRQWLQXLW\WKDWSUHYHQWVWKLVFRPSOHWHDQQLKLODWLRQ,IWKHLQWHUIDFHGLVFRQ WLQXLW\ ZHUH LQVWHDG GLVWULEXWHG LQ D FRQWLQXRXV PDQQHU RQ D WKLFNQHVV RI D IHZ ZDYHOHQJWKVWKHUHIOHFWHGZDYHZRXOGQRWEHSHUFHSWLEOH

1.7.1 Laws of reflection and refraction for homogeneous dielectric media &RQVLGHUDKDUPRQLFSODQHZDYHLQFLGHQWRQDSODQHVXUIDFHRIVHSDUDWLRQEH WZHHQ WZR GLIIHUHQW GLHOHFWULF PHGLD )LJ   7KLV ZDYH LV SDUWO\ WUDQVPLWWHG DQGSDUWO\UHIOHFWHGIURPWKLVLQWHUIDFH6ROHWXVWU\DVHOIFRQVLVWHQWVROXWLRQWKDW FRQVLVWVRIWKUHHSODQHZDYHVGHVFULEHGE\FRPSOH[HTXDWLRQV



­ E j ° ® j °H ¯

j Eo ei k j ˜r  i Z j t 

  k j u E j  PZj

 

ZLWK j  , 5 7 LQGLFDWLQJ UHVSHFWLYHO\ WKH LQFLGHQW UHIOHFWHG DQG WUDQVPLWWHG ILHOGV7KHVHFRQGHTXDWLRQLVGHULYHGIURP(T D  7KHWKUHHZDYHVFDQEHSXWLQUHODWLRQWRHDFKRWKHURQFHWKH\DUHGHWHUPLQHG E\LPSRVLQJFHUWDLQERXQGDU\FRQGLWLRQVRQWKHVXUIDFHRIGLVFRQWLQXLW\EHWZHHQ WKHWZRPHGLD7KHILUVWRIWKHVHFRQGLWLRQVDOVRFDOOHGkinematicsUHODWHVWRWKH GLUHFWLRQ RI WKH SURSDJDWLRQ YHFWRUV %HVLGHV D FRQVWDQW DPSOLWXGH WHUP WKH GH SHQGHQFHRQWKHVSDFHWLPHRIWKHWKUHHZDYHVLVJLYHQE\WKHH[SRQHQWLDORI(T D ,QRUGHUWKDWWKHUHFDQEHDUHODWLRQVKLSEHWZHHQWKHWKUHHZDYHVLQGH SHQGHQWIURPWLPHDQGIURPWKHSRVLWLRQRQWKHVXUIDFHZKLFKLVVXSSRVHGO\VHSD UDWLQJ WZRhomogeneous constant PHGLD LW LV QHFHVVDU\ WKDW WKH WKUHH SKDVHV EH HTXDOWRHDFKRWKHUDWDQ\WLPHDQGDWDQ\LQWHUIDFHSRLQWIRUZKLFK 

Z,

Z5

Z7

Z

 

k 7 < r  rR 

 

DQG 

k, < r  rR

k 5 < r  rR

ZKHUH rDQG rRLQGLFDWHDQ\WZRSRLQWVRQWKHLQWHUIDFHLWVHOI)URPWKLVDUELWUDUL QHVV LW IROORZV WKDW WKH SURMHFWLRQVRI WKHWKUHHYHFWRUV kj RQ WKH LQWHUIDFH SODQH DUHDOOHTXDODQGWKHUHIRUH k, k5DQG k7DUHcoplanar7KHSODQHRQZKLFKWKHVH YHFWRUVOLHLVFDOOHGWKHplane of incidenceDQGLWLVQHFHVVDULO\SHUSHQGLFXODUWR WKHLQWHUIDFHSODQH,QSDUWLFXODUZHFDQFKRRVHD&DUWHVLDQFRRUGLQDWHIUDPH2xyz



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 kR

kI

kT

JI

n1

JR

x

n2

JT

(a)

(b)

z



)LJ5HIOHFWLRQDQGUHIUDFWLRQ

VXFK WKDW IRU H[DPSOH WKH xySODQH ZLOO FRLQFLGH ZLWK WKH LQWHUIDFH DQG WKHzx SODQHZLWKWKDWRILQFLGHQFH:LWKUHIHUHQFHWR)LJ E LQZKLFKWKHDQJOHV DUH WDNHQ WR EH SRVLWLYH LQ WKH GLUHFWLRQ LQGLFDWHG E\ WKH DUURZV (T   EH FRPHV 

k , VLQ -,

k 5 VLQ -5

k 7 VLQ -7 

 

ZKHUHWKHkjDUHWKHYDULRXVPRGXOHVRIWKHZDYHYHFWRUVDQGWKH-jDUHWKHDQJOHV RILQFLGHQFHWKDWWKHVHYHFWRUVPDNHZLWKWKHQRUPDOWRWKHVXUIDFH%RWKLQFLGHQW DQGUHIOHFWHGZDYHVWUDYHOLQWKHVDPHPHGLXP KHUHZHFRQVLGHULWLVRWURSLF IRU ZKLFKk, k57KHILUVWRI(TV  WKXVOHDGVWRWKHIDPLOLDUODZVRIUHIOHFWLRQ 

-,

-5 

 

,QVWHDGIRUWKHWUDQVPLWWHGZDYHE\DSSO\LQJ(TV    DQG   ZHILQGWKH6QHOO¶VODZ k, k7



n, n7

VLQ -7  VLQ -,

 

ZKHUH n, DQG n7 DUH WKH LQGLFHV RI UHIUDFWLRQ IRU WKH LQFLGHQW DQG WUDQVPLWWHG ZDYHVUHVSHFWLYHO\

1.7.2 Fresnel’s formulas 1RZFRQVLGHUWKHDPSOLWXGHVRIWKHZDYHVUHIOHFWHGDQGWUDQVPLWWHG$VXVXDO ZHGHQRWHWKHFRPSRQHQWVRIWKHILHOGVWKDWDUHQRUPDOWRWKHSODQHRILQFLGHQFH ZLWKADQGWKHSDUDOOHOFRPSRQHQWVWRWKHSODQHRILQFLGHQFHZLWK&IRUZKLFK 

ER 

EA  E&  H A

 k u E&  H& PZ

 k u EA   PZ

 

7KH FKRLFH RI WKH SRVLWLYH GLUHFWLRQV IRU WKH SDUDOOHO FRPSRQHQWV LV VKRZQ LQ

5HIOHFWLRQDQGUHIUDFWLRQRQSODQHLQWHUIDFHV





E I

)LJ &RQYHQWLRQVLQFKRRVLQJSRVLWLYHGLUHFWLRQV



JI

JR

ER x ET

JT

z



)LJZKLOHWKDWIRUWKHQRUPDOFRPSRQHQWVFRLQFLGHVZLWKWKHSRVLWLYHGLUHF WLRQRIWKHyD[LV7KLVRSWLRQKDVWKHDGYDQWDJHWKDWWKHSRVLWLYHGLUHFWLRQVIRUWKH DPSOLWXGHV EADQGE&RIWKHWKUHHZDYHVDUHWKHVDPHLQWKHFDVHRIQRUPDOLQFL GHQFHEXWQRWDOODXWKRUVXVHWKLVVDPHFRQYHQWLRQIRUH[DPSOH%RUQDQG:ROI  GRQRW7KHFRPSRQHQWVxyzRIILHOGIRUWKHWKUHHZDYHDUH

j Ex

j E& FRV- j 

j Hx

j Ey

j EA 

j Hy

j Ez

sj

nj

j E FRV- j P jc A s j

nj

j j s j E& VLQ - j  H z

P jc



nj P jc

j E&

ZKHUH s j

­ SHUj ,7  ® ¯ SHUj 5

j EA VLQ - j 

 

7KHERXQGDU\FRQGLWLRQVGLVFXVVHGLQ†UHTXLUHWKDWWKURXJKWKHLQWHUIDFH WKHWDQJHQWLDOFRPSRQHQWVRI(DQG+DUHFRQWLQXRXV



, 5 Ex  Ex

7 E x 

, 5 Ey  Ey

7 Ey 

, 5 Hx  Hx

7 H x 

, 5 Hy  Hy

7 Hy 



 

ZKLFKE\DSSO\LQJWRWKHP(TV  FRQVWLWXWHOLQHDUHTXDWLRQVIRUWKHIRXU XQNQRZQVE) 5 EA 5 E) 7 EA 7 ZKHQE) , DQGEA , DUHDVVLJQHG7KHERXQG DU\FRQGLWLRQVIRUWKHFRPSRQHQWV 'DQG %QRUPDOWRWKHLQWHUIDFHDUHDXWRPDWL FDOO\IXOILOOHGWKLVFDQEHYHULILHGUHPHPEHULQJWKDW ' H(DQG % P+E\DS SO\LQJ6QHOO¶VODZDQGWKDWRIUHIOHFWLRQ>*XHQWKHU@ 6XEVWLWXWLQJ(TV  LQ(TV  ZHKDYH

n E E P

, 5 FRV -, E&  E& 

,

,

,

&

5

&

7 E& FRV -7  7 n7  E& P7

, 5 EA  EA



n, , 5 FRV -, EA  EA P,



7 EA  n 7 EA FRV -7 7  P7





5HFROOHFWLRQVRIHOHFWURPDJQHWLVP





1.0

amplitude coefficients

)LJ D  5HIOHFWLRQ DQG WUDQV PLVVLRQDPSOLWXGHFRHI ILFLHQWV DV D IXQFWLRQ RI WKH DQJOH RI LQFLGHQFH IRU D ZDYH LQFLGHQW externally RQ WKH DLUJODVV LQWHUIDFH ZLWK  n7n,  

t

2nI nT + n I

0.5

t r

0.0

JB nT - nI n T+ nI

-0.5

r 56.3

-1.0 0

15

30 45 60 angle of incidence J

75

90



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

5 EA , EA

rA



n, n FRV -,  7 FRV -7 P, P7 n, n FRV -,  7 FRV -7 P, P7

 t A



7 EA , EA



n, P,

n  , FRV -, P,  n7 FRV -,  FRV -7 P7



 

70ZDYHV  r&

5 E& , E&

n7 n FRV -,  , FRV -7 P, P7  t  & n7 n, FRV -,  FRV -7 P P7 ,





7 E& , E&

n, FRV -, P,   n7 n, FRV -,  FRV -7 P7 P, 

 

 7KHRQHVVKRZQKHUHDUHWKHPRVWFRPPRQO\XVHGWHUPV7KHPRVWH[SOLFLWDUH7(DQG70 8QIRUWXQDWHO\IRUIXUWKHUFRQIXVLRQWKHUHDUHRWKHUV3 V 7(6 S 70

5HIOHFWLRQDQGUHIUDFWLRQRQSODQHLQWHUIDFHV



1.0

amplitude coefficients

)LJ E  5HIOHFWLRQ DPSOLWXGH FR HIILFLHQWV r DV D IXQFWLRQ RI WKH DQJOH RI LQFLGHQFH IRU D ZDYH internally LQ FLGHQW RQ WKH DLUJODVV LQWHUIDFH ZLWK n,n7    %H\RQG WKH FULWLFDO DQJOH -&  WKH r FRHIIL FLHQWV DUH FRPSOH[ DQG HYHQ LI WKH\ DUH RI XQL WDU\ PRGXOXV FDQQRW EH UHSUHVHQWHGLQWKLVJUDSK

 

r

0.5

0.0

¢ JB

JC

-0.5

r 33.7

-1.0

15

0

41.8

30 60 45 angle of incidence J

75

90



1RWHWKDWIRUWKH7(ZDYHRQHKDVrA tAUHJDUGOHVVRIWKHDQJOHRILQFL GHQFH HQVXULQJ WKH FRQWLQXLW\ RI WKH WDQJHQWLDO FRPSRQHQWV RI WKH ILHOG &RQ WUDVWLQJO\ WKLV UHODWLRQ LV QRW WUXH IRU WKH 70 ZDYH :LWK QRUPDO LQFLGHQFH WKH GLVWLQFWLRQEHWZHHQWKHWZRW\SHVRIZDYHIDOOVDQGLQWKLVFDVHZHKDYHr t $SSO\LQJWKH6QHOO¶VODZDQGOLPLWLQJRXUVHOYHVWRFRQVLGHUDWLRQRIWKHQRUPDO WUDQVSDUHQWGLHOHFWULFPHGLDIRUZKLFKP, P7 PR(TV  DQG  FDQ EHZULWWHQLQDPRUHFRPSDFWIRUP rA

 r&

VLQ -,  -7   tA VLQ -,  -7  WDQ -,  -7    t& WDQ -,  -7 



 FRV -, VLQ -7   VLQ -,  -7  FRV -, VLQ -7  VLQ -,  -7 FRV -,  -7



 

7KHVLJQVRIWKHVHH[SUHVVLRQVGHWHUPLQHWKHSKDVHRIWKHUHIOHFWHGDQGWUDQV PLWWHG ZDYHV KRZHYHU WKH\ DUH DUELWUDU\ LQ WKH VHQVH WKDW WKH\ GHSHQG RQ WKH FKRLFHPDGHIRUWKHSRVLWLYHGLUHFWLRQVRIWKHILHOGV )LJ DQGRWKHUDXWKRUV UHSRUWGLIIHUHQWVLJQV,UHPDUNWKDW ZLWKWKHFKRLFHPDGHWKHSRVLWLYHGLUHFWLRQVRIWKHILHOGVRIWKHWKUHHZDYHV DUHWKHVDPHIRU-o de gustibus  7KH WHQGHQF\ RI WKHVH FRHIILFLHQWV WR YDU\WKH DQJOH RI LQFLGHQFH LV VKRZQ LQ )LJ D DQG E LQWKHWZRFDVHVRIH[WHUQDOLQFLGHQFHZLWKn,n7DQG LQWHUQDOLQFLGHQFHZLWKn,!n7,QWKHFDVHRIQRUPDOLQFLGHQFH DQGZLWKP, P7 PR IURP(TV  DQG  ZHLPPHGLDWHO\REWDLQWKDW



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 

r&

rA

n,  n7 t & n,  n7

tA

 n,  n,  n7

 

WKHWZRUHIOHFWLRQFRHIILFLHQWVDUHDOVRHTXDOLQVLJQEHFDXVHRIWKHFKRLFHPDGH 

1.7.3 Reflectivity and transmissivity /HWXVEULHIO\H[DPLQHKRZWKHHQHUJ\RIWKHLQFLGHQWILHOGLVGLYLGHGEHWZHHQ WKHUHIOHFWHGDQGWUDQVPLWWHGZDYHV)URP(T  LWIROORZVWKDWWKHLQWHQVLW\ RIRXUZDYHVLV I j



§ j   c j  · H j ¨ E&  EA ¸   nj © ¹

 

DQGWKHHQHUJ\IORZWKURXJKDQDUHD$RIWKHLQWHUIDFH LHWKHLQFLGHQWSRZHU3  LVWKHQ P j



§ j   c j  · H j ¨ E&  EA ¸ A FRV - j   nj © ¹

 

ZKHUHLWLVWREHQRWHGWKDWWRJHWKHUZLWKWKHILHOGPRGXOHVWKHSURSHUWLHVRIWKH PHGLXPDQGWKHDQJOHVRILQFLGHQFHDUHDOVRLQYROYHG7KXVLWLVIRXQGWKDWWKH reflectivity R DQG transmissivity T ZKLFK DUH D GLIIHUHQW WKLQJ IURP WKH FRHIIL FLHQWVrDQGtREWDLQHGHDUOLHUIRUWKHILHOGVDUH VWLOOIRUP, P7 PR  7(ZDYHV  RA

5 PA , PA

VLQ  -,  -7  TA VLQ  -,  -7

7 PA , PA

WDQ  -,  -7  T& WDQ  -,  -7

7 P& , P&

VLQ -, VLQ -7   VLQ  -,  -7

 

70ZDYHV  R &

5 P& , P&



VLQ 

VLQ -, VLQ -7  -,  -7 FRV -,  -7

  

  7KHVH WHUPV DUH XVXDOO\ XVHG WR LQGLFDWH WKH UDWLR EHWZHHQ UHIUDFWHG RU UHIOHFWHG SRZHU DQG

LQFLGHQWSRZHUIRUWKHFDVHRIVSDWLDOO\FRKHUHQWPRQRFKURPDWLFZDYHV,QUDGLRPHWU\WKHWHUPV reflectance DQG transmittance DUH XVHG 7KHVH UHIHU WR WKH UDWLR EHWZHHQ WKH IORZ RI HQHUJ\ UHIOHFWHGRUWUDQVPLWWHGZLWKWKHLQFLGHQWIOX[IRUJHQHUDOO\LQFRKHUHQWUDGLDWLRQ

5HIOHFWLRQDQGUHIUDFWLRQRQSODQHLQWHUIDFHV



1.0

intensity coefficients

)LJ D  5HIOHFWLRQ LQWHQVLW\ FRHIILFLHQWV DV D IXQF WLRQRIWKHDQJOHRILQ FLGHQFH IRU D ZDYH LQ FLGHQWH[WHUQDOO\RQWKH DLUJODVVLQWHUIDFHZLWK n7n, 

 

0.8 0.6

R

0.4

R

0.2

JB

0.0 0

15

30 45 60 angle of incidence J

75

90



2QHFDQHDVLO\YHULI\WKDW5A7A 5&7& ,QSDUWLFXODULQWKHFDVH RIQRUPDOLQFLGHQFHZHREWDLQ VWLOOIRUP, P7 PR  

R&

RA

n,  n7  T& n,  n7 

TA

n, nT

n,  n7 



 

7KH WUHQG RI WKH UHIOHFWLYLW\ ZKHQ YDU\LQJ WKH DQJOH RI LQFLGHQFH LV GUDZQ LQ )LJIRUWKHFDVHVRIH[WHUQDODQGLQWHUQDOLQFLGHQFH,WVKRXOGEHQRWHGWKDW LQWKLVODVWFDVHWKHUHIOHFWLYLW\YDULHVYHU\VWHHSO\DWWKHFULWLFDODQJOH

1.7.4 Brewster’s angle $V VHHQ IURP )LJ  WKH UHIOHFWLRQ DQG WUDQVPLVVLRQ FRHIILFLHQWV RI (TV   DUH PRQRWRQLF IXQFWLRQV RI WKH DQJOH RI LQFLGHQFH KRZHYHU r& KDV WKH UHPDUNDEOHSURSHUW\RIFKDQJLQJLWVVLJQE\SDVVLQJWKURXJK]HUR,QRWKHUZRUGV WKH UHIOHFWLYLW\ IRU WKH 70 ZDYH YDQLVKHV DW D SDUWLFXODU DQJOH RI LQFLGHQFH -% FDOOHGBrewster’s angle )LJ ,QGHHGLQWKHHTXDWLRQIRUr&WKHGHQRPLQDWRU GLYHUJHVWRLQILQLW\ZKHQ 

-%  -7

S  

$SSO\LQJ6QHOO¶VODZRIZHILQGWKDW 

n, VLQ -%

§ · n7 VLQ ¨ S  -% ¸  © ¹

 



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP



1.0

intensity coefficients

)LJ E  5HIOHFWLRQ LQWHQVLW\ FRHIILFLHQWV DV D IXQF WLRQ RI WKH DQJOH RI LQFLGHQFHIRUDQLQWHU QDOO\LQFLGHQWZDYHRQ DQ DLUJODVV LQWHUIDFH ZLWKn,n7 

0.8 0.6 0.4

R

0.2

R

¢ JB

0.0 0

15

30 45 60 angle of incidence J

75

90



IURPZKLFK 

-%

§n · DUFWDQ ¨ 7 ¸  © n, ¹

 

7KLV H[SUHVVLRQ DSSOLHV WR ERWK LQWHUQDO DQG H[WHUQDO LQFLGHQFH MXVW VZDS WKH valueRIWKHUHIUDFWLYHLQGLFHVLQWKHDUJXPHQWRIWKHDUFWDQJHQW)RUH[DPSOHIRU DQRUPDOJODVVZLWKUHIUDFWLYHLQGH[n WKHH[WHUQDO%UHZVWHU¶VDQJOH LQWKH DLU LVqDQGWKHLQQHURQH LQJODVV LVq,WVKRXOGWKHQEHQRWHGWKDWWKHVH WZRDQJOHVDUHUHODWHGE\6QHOO¶VODZVRWKDWLIDEHDPRIOLJKWSDVVHVWKURXJKD JODVVSODWHZLWKSDUDOOHOIDFHVZLWKLQFLGHQFHDW%UHZVWHU¶VDQJOHWKH70FRPSR QHQWKDVQRUHIOHFWLRQRQHLWKHUWKHILUVWRUWKHVHFRQGIDFHRIWKHSODWH,WLVDOVRWR EHQRWHGWKDWZKHQWKHUDWLRRIWKHLQGLFHVWHQGVWRWKH%UHZVWHU¶VDQJOHWHQGV WR ƒ :KHQ P, z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¶ RVFLOODWLRQ LQ WKH GLUHFWLRQ SHUSHQGLFXODU WR WKDW RI LWV SURSDJDWLRQ:HOOWKLVSURMHFWLRQLVFDQFHOHGZKHQ-,-7 qIRUZKLFKWKH GLSROHVH[FLWHGE\WKH70ZDYHDUHQRWDEOHWRUHHPLW WKHUHIOHFWHGZDYH7KLV

5HIOHFWLRQDQGUHIUDFWLRQRQSODQHLQWHUIDFHV







)LJ %UHZVWHU¶VDQJOH

EI

JI

TM waves JT



naïveUHDVRQLQJEHFRPHVPRUHFXPEHUVRPHLQWKHFDVHRILQWHUQDOLQFLGHQFHDQG FDQHYHQOHDGWRDQLQFRUUHFWFRQFOXVLRQLIWKHULJKWSUHPLVHVDUHQRWWDNHQ,OHDYH WKHVHWRWKHLPDJLQDWLRQRIWKHUHDGHUV

Apology of the Brewster’s angle 7KHSURSHUW\RI%UHZVWHU¶VDQJOHRIFDQFHOLQJWKHUHIOHFWHG70ZDYHKDVPDQ\ DSSOLFDWLRQV7KLVLVWKDQNVSULPDULO\WRWKHIDFWWKDWLWKDVDZLGHVSHFWUDOEDQG WKHLQGH[nRIWKHWUDQVSDUHQWFRPPRQVXEVWDQFHVLQIDFWYDULHVYHU\VORZO\ZLWK WKHZDYHOHQJWK7KHUHIRUHDZLQGRZSODFHGDW%UHZVWHU¶VDQJOHRIWHQKDVDQHJ OLJLEOHUHIOHFWLYLW\ IRUWKH70ZDYH IRUWKHHQWLUHYLVLEOHEDQG0RUHRYHUVXFKD ZLQGRZLVFKHDSHUDQGPRUHUREXVWWKDQRQHPDGHZLWKDQDQWLUHIOHFWLYHFRDWLQJ )RU WKHVH UHDVRQV ZLQGRZV DQG VXUIDFHV DW WKH %UHZVWHU¶V DQJOH DUH RIWHQ XVHG ZKHQHYHUWKHILHOGLVOLQHDUO\ SRODUL]HG,QVLGHRIDODVHUWKHYDULRXVVXUIDFHVDW WKH%UHZVWHU¶VDQJOHDOODOLJQHGZLWKRQHDQRWKHUDUHWKHRQHVWKDWIL[WKHSRODUL ]DWLRQRIWKHODVHUEHDPLQGHHGIRUWKH70ZDYHWKHORVVHVDUHQHJOLJLEOHZKLOH WKH7(ZDYHKDVORVVHVRQWKHRUGHURIIRUHDFKVXUIDFHZLWKWKHLQGH[RIWKH FRPPRQRSWLFDOJODVVHV $ SLOH RI YDULRXV SDUDOOHO SODWHV DW WKH %UHZVWHU¶V DQJOH DUH JRRG SRODUL]HUV XVHIXOLQSDUWLFXODULQWKHLQIUDUHGZKHUHWKHUHZDVQRDOWHUQDWLYHEXWZHFDQ XVHVXEVWDQFHVZLWKDKLJKUHIUDFWLYHLQGH[IRUZKLFKWKH7(ZDYHUHIOHFWLRQEH FRPHVVWURQJHU )XUWKHUPRUHLQWHUPVRILQWHQVLW\%UHZVWHU¶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

KDYHEHHQSXWRQWKHPDUNHW7KH\KDYHJRRGHIILFLHQF\DQGDQH[FHOOHQWH[WLQFWLRQUDWLR



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 'HYLDWLRQV IURP WKH H[DFW FDQFHOODWLRQ RI WKH UHIOHFWLRQ FRXOG KDSSHQ GXH WR OD\HUVRISROOXWDQWRQWKHVXUIDFHVEXWDOVRIRUWKHPLFURVFRSLFSURSHUWLHVRIWKH GLHOHFWULFVXUIDFHVWKHPVHOYHV>6LYRXNKLQH†@

1.7.5 Total reflection :HUHWXUQQRZWRH[DPLQLQJ6QHOO¶VODZFRQVLGHULQJLQSDUWLFXODUWKHFDVHLQ ZKLFKWKHLQFLGHQWZDYHSURSDJDWHVLQWKHGHQVHUPHGLXPDQGWKXVLVn,!n7,W WKHUHIRUHKDV VLQ -7 VLQ -,



n, !  n7

IURPZKLFKZHVHHWKDWDV-,LQFUHDVHVZHDOZD\VKDYH-,-7XQWLO-7UHDFK HVWKHYDOXHRIS q DWDQDQJOH-, -FVDLGcritical angleJLYHQE\ VLQ -F



n7  n,

 

:KDW KDSSHQV WKHQ WR WKH WUDQVPLWWHG ZDYH ZKHQ -, H[FHHGV -F" 0RUHRYHU ZKDW KDSSHQV WR -7" :HOO WKH VROXWLRQ RI 6QHOO¶V HTXDWLRQ IRU-7 QRZ OHDGV WR FRPSOH[YDOXHVRIWKHW\SH 

-7

S  i -c7 

rm S ZLWKm LQWHJHU 

7KHJHQHUDOL]DWLRQRIWKHVLQHDQGFRVLQHIXQFWLRQVLVLQGHHGHDVLO\REWDLQHGXV LQJWKHLUGHFRPSRVLWLRQLQH[SRQHQWLDO,WWKXVRFFXUVWKDW VLQ -7

 FRV -7

§S · VLQ ¨  i -c7 ¸ FRVK -c7 © ¹  §S · FRV ¨  i -c7 ¸ i VLQK -c7 © ¹

 

7KHILUVWH[SUHVVLRQDOORZVXVWRFDOFXODWHWKHLPDJLQDU\SDUWRIWKHDQJOH-7 

§n · -c7 FRVK  ¨ , VLQ -, ¸  n © 7 ¹

 

DOWKRXJKZHVWLOOGRQRWUHDOO\QHHGLWQRZ(TV  PDNHXVUHPHPEHUWKDW

5HIOHFWLRQDQGUHIUDFWLRQRQSODQHLQWHUIDFHV



  WKHPRVWJHQHUDOVROXWLRQIRUDSODQHZDYHDGPLWVDFRPSOH[ZDYHYHFWRU kIRU ZKLFKWKHZDYHLVLQKRPRJHQHRXVWKLVFRQVLVWVRIWKHWUDQVPLWWHGZDYH+RZHY HUJLYHQWKDWLWGHVHUYHVVSHFLDODWWHQWLRQZHSRVWSRQHWKHGHWDLOHGGLVFXVVLRQWR WKHQH[WVHFWLRQ ,Q FRQFOXVLRQ WKH )UHVQHO IRUPXODV GHULYHG DERYH DUH VWLOO YDOLG IRU -, ! -F ZLWKWKHRQO\ZDUQLQJEHLQJWKDWFRV-7LVQRZLPDJLQDU\ 

i VLQK -c7

FRV -7



§n · i ¨ , VLQ -, ¸    n © 7 ¹

 

)LQDOO\IURP(TV  DQG  LWLVVHHQWKDWIRU-,!-FWKHUHIOHF WLRQFRHIILFLHQWVDUHFRPSOH[EXWZLWKPRGXOXVHTXDOWRIRUZKLFKall of the energyLVUHIOHFWHGIURPWKHLQWHUIDFH)RUPDOO\WKLVKDSSHQVEHFDXVHLQWKHVHHTXD WLRQVWKHGHQRPLQDWRUVDUHWKHFRPSOH[FRQMXJDWHRIWKHQXPHUDWRUV ,QSDUWLFXODUIRUP, P7 PRZHKDYH 7(ZDYHV



rA

70ZDYHV

r&



FRV -,  i VLQ  -,   n  FRV -,  i VLQ  -,   n 



FRV -,  i n  VLQ  -,   n  FRV -,  i n  VLQ  -,   n 

D 



E 

ZKHUHn n,n73XWWLQJ(TV  LQWKHIRUPr eiGZHFDQGHULYHWKHSKDVH VKLIWVGADQGG&IRUWKHWZRW\SHVRIZDYHMXVWPDNHWKHUDWLREHWZHHQWKHUHDO SDUWDQGWKHLPDJLQDU\SDUWIRUZKLFKKDYLQJWDNHQWKDWr  aib  aLb ZH KDYH 

WDQ

G 

b   a

IURPZKLFK § VLQ  -   n  · , ¸  GA   DUFWDQ ¨ ¨ ¸ FRV -, © ¹

G&

§ VLQ  -,   n  · ¸  S   DUFWDQ ¨ n  ¨ ¸ FRV -, © ¹

IRUWKH7(ZDYHV

D 

IRUWKH70ZDYHV E 



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP



n I /n T = 1.5

phase shift

p

nT/n I = 1.5

p

d

d (c) 0

0

15

(a) 30

45

60

75

90

phase shift

0

0 p

0

15

30

45

60

75

90

60

75

90

d

d (b)

(d) p

15

30

45

60

75

90

0

0

15

30

45

angle of incidence J

phase shift

p

0

d

d

p

60

75

(e) 0

0

15

30

45

angle of incidence J



90



)LJ3KDVHVKLIWVIRUWKH7(DQG70FRPSRQHQWVRIWKHHOHFWULFILHOGLQWKHFDVHVRI LQWHUQDO OHIW DQGH[WHUQDO ULJKW UHIOHFWLRQ

ZKHUHWKHSLQWKHVHFRQGIRUPXODUHVSRQGVWRWKHSUHVHQFHRIDIXUWKHUPLQXVVLJQ IRUr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

5HIOHFWLRQDQGUHIUDFWLRQRQSODQHLQWHUIDFHV



 

(a)

z

E E

(b)

(I)

(I)

(c)

E

(R)

E (R)



)LJ 7RWDOUHIOHFWLRQIRUDQLQFLGHQWILHOGZLWKWKHOLQHDUSRODUL]DWLRQD[LVLQFOLQHGLQ WKHORZHUSDQHVWKHILHOGLVUHSUHVHQWHGDVLWLVYLHZHGORRNLQJWRZDUGVWKHZDYHVRXUFH 7KHDUURZVIRUD[LVADQG&IROORZWKHFRQYHQWLRQDGRSWHGLQ)LJ+HUHWKHUHIOHFWLRQ ILHOGLVULJKWHOOLSWLFDO7KHRSSRVLWHKDSSHQVE\UHYHUVLQJWKHLQLWLDOLQFOLQDWLRQ

XFW rAr&  H[S>i GA G& @IURPZKLFKSURFHHGLQJDVEHIRUHZHKDYH 

G

§ FRV - VLQ  -   n  · , , ¸  S   DUFWDQ ¨ ¨ ¸ VLQ  -, © ¹

 

)RUWKLVW\SHRISUREOHPKRZHYHULWLVEHWWHUWRGHILQHDQHZYDULDEOHLIZH ORRNDWHDFKZDYHLQFLGHQWRUUHIOHFWHGWRZDUGWKHGLUHFWLRQIURPZKLFKLWFDPH )LJ WKHSKDVHVKLIWWKDWLVPHDVXUHGLV 

G‫ މ‬GS

WKHWHUP±SGLVDSSHDUV WKHWUHQGRIG‫މ‬ZLWK-,LVVKRZQLQ)LJ H ,QSDUWLF XODUWKHSKDVHVKLIWG‫މ‬YDQLVKHVIRU-, -FDQGIRU-, S,QDGGLWLRQZHQRWH WKDWIRU-,!-FWKHDUJXPHQWRIWKHDUFWDQJHQWRI(T  LVSRVLWLYHDQG WKHUHIRUHDOVRG‫މ‬LVSRVLWLYHLWIROORZVWKDWWKH70ZDYHLVadvancedFRPSDUHGWR WKH7(ZDYHDVVKRZQLQ)LJ F  7KHSKDVHVKLIWLVPD[LPXPIRU FRV -P ,



Gcm

n   n   DQGLV

 ·º ª §  DUFWDQ « ¨ n  ¸ »  n ¹¼ ¬ ©



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 )RUDJODVVDLULQWHUIDFHZLWKn LWOLHVDW-, -P, qDQGLVG‫މ‬m  q:KHQLQVWHDG-, qRU-, qZHKDYHG‫ މ‬q

1.7.6 Evanescent wave 6LQFH ZH DUH FRQVLGHULQJ UHDO YDOXHV RI n7 DQG n,  ZH NQRZ IURP (T E WKDWWKHUHDOSDUW k7ƒ DQGWKHLPDJLQDU\SDUW k7‚ RI k7DUHRUWKRJR QDOWRHDFKRWKHU*LYHQWKHQWKDW(T  PXVWVWLOOEHYDOLGDQGVLQFHLWVILUVW WZRWHUPVDUHUHDOWKXV k7‚‡r LWIROORZVWKDW k7‚LVQRUPDOWRWKHLQWHUIDFH DQGWKDWk7ƒPXVWEHSDUDOOHOWRWKHLQWHUIDFH7KHUHIRUHZHKDYH 

k 7ƒ k , VLQ -,

n, Z VLQ -,  c

 

ZKLOHIURP(TV D ZHKDYHWKHLPDJLQDU\SDUWRIk 

Z n,  VLQ  -,  n7   c

k7‚

 

1RZLIZHDSSO\6QHOO¶VODZDQG(TV  ZHILQDOO\KDYH 

k7

n7 Z c c c [ FRVK -7 i ] VLQK -7 

 

%\FRPSDULVRQZLWK(T  LWLVVHHQWKDW-‫މ‬7FRLQFLGHVZLWKWKHSDUDPH WHU-XVHGWKHUHZKLOHLQADQGA' RI(TV  ZHFDQLGHQWLI\WKHDPSOL WXGHVE)DQGEAUHVSHFWLYHO\FKRRVLQJDVYHUVRUVH  [H  ] H  \)RUWKH WUDQVPLWWHGZDYH(TV  FRLQFLGHZLWK(TV  DVVRRQDVZHDSSO\ WKHVXEVWLWXWLRQRI(TV  6RZHILQGWKDWWKHDPSOLWXGHRIWKHWUDQVPLWWHG ZDYHGHFD\VH[SRQHQWLDOO\PRYLQJDZD\IURPWKHLQWHUIDFHZLWKWKHODZ 

Ez

ER e  k7‚ z 

ZKHUHzLVWKHGLVWDQFHIURPWKHVXUIDFH)RUWKLVUHDVRQWKLVZDYHLVFDOOHGDQevanescent wave,WVH[WHQVLRQLQVSDFHLVZHOOUHSUHVHQWHGE\WKHGHFD\GLVWDQFHIRU WKHLQWHQVLW\JLYHQE\ 

dH

  k7 ‚

OR S

n , VLQ  -,  n7 

 

7KLVGLVWDQFHJURZVE\GHFUHDVLQJ-,DQGLVQRUPDOO\YHU\VPDOO DIUDFWLRQRI

5HIOHFWLRQDQGUHIUDFWLRQRQSODQHLQWHUIDFHV







)LJ 7RWDOUHIOHFWLRQLQDULJKWDQJOHSULVP

45 45



WKHZDYHOHQJWKORLQYDFXXP H[FHSWZKHQ-,DSSURDFKHVWKHFULWLFDODQJOHIRU ZKLFKdHLQGHHGGLYHUJHVWRLQILQLW\,WPXVWEHUHPHPEHUHGKRZHYHUWKDWWKHVR OXWLRQV IRXQG KHUH DSSO\ LQ WKH VWULFW VHQVH RQO\ IRU D SODQH ZDYH LQILQLWHO\ H[ WHQGHG7KHWUHDWPHQWLVPRUHFRPSOLFDWHGIRUDOLJKWEHDPRIOLPLWHGVL]H

1.7.7 Application of total internal reflection )DU IURP H[KDXVWLQJ WKH DSSOLFDWLRQV RI WKLV SKHQRPHQRQ KHUH OHW XV ORRN DW VRPHVLJQLILFDQWH[DPSOHVRILW7KHILUVWLVWKHULJKWDQJOHSULVPXVHGDVDPLUURU DWƒ )LJ ,QIDFWZLWKDFRPPRQRSWLFDOJODVVZLWKn WKHFULWLFDO DQJOHLVDOUHDG\ƒWKXVZLWKDƒLQFLGHQFHDQJOHRQHKDVWRWDOUHIOHFWLRQ &RPSDUHG WR PHWDOOLF RU GLHOHFWULF PLUURUV WR EH XVHG DW ƒ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n   LVZHOO q )RU DQ LQFRPLQJ OLJKW EHDP OLQHDUO\ SRODUL]HG LQ DQ\ GLUHF WLRQZHFDQWKHUHIRUHKDYHDVWURQJHOOLSWLFLW\LQWKHRXWERXQGEHDPZKLFKKRZ HYHUFDQEHFRPSHQVDWHGDSDUW $VHFRQGLQWHUHVWLQJDSSOLFDWLRQZKLFKH[SORLWVWKHSKDVHVKLIWSURGXFHGE\WR WDOUHIOHFWLRQLVGRQHZLWKWKHFresnel’s rhombus.,WLVXVHGWRSURGXFHFLUFXODUO\ SRODUL]HGOLJKWIURPDOLQHDUSRODUL]DWLRQE\PHDQVRIWZRWRWDOUHIOHFWLRQV )LJ  ,QIDFWWKHSKDVHVKLIWZLOOVXPFRQVWUXFWLYHO\LQERWKUHIOHFWLRQVDQGLV y

y

y 45

z

x

J

J

J = 54.6 )LJ)UHVQHO¶VUKRPEXV

y

x

n = 1.51 z





5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 

G)

§ FRV - VLQ  -   n  · , , ¸    DUFWDQ ¨ ¨ ¸ VLQ  -, © ¹

 

,IWKHLQFLGHQFHDQJOHLVVXLWDEO\FKRVHQWKLVSULVPFRQVWLWXWHVDOSODWHZLWK D ZLGH VSHFWUDO EDQG LQWURGXFLQJ DGLIIHUHQFH LQ RSWLFDO SDWKRI D TXDUWHU ZDYH OHQJWK EHWZHHQ WKH 7( DQG 70 FRPSRQHQWV &RQWUDU\ WR WKH ELUHIULQJHQW SODWHV ZKLFKZHZLOOVWXG\IXUWKHURQ LWVSKDVHVKLIWYDULHVOLWWOHZLWKWKHZDYHOHQJWK LHYDULHVRQO\EHFDXVHRIWKHGHSHQGHQFHRInRQO,QSUDFWLFHZLWKDFRPPRQ JODVV WKH SULVP FRQVWLWXWHV D JRRGO SODWH IRU WKH HQWLUH YLVLEOH VSHFWUXP 7KH ULJKWFLUFXODUSRODUL]HGOLJKWLVREWDLQHGZLWKDOLQHDUSRODUL]DWLRQLQSXWRULHQWHGDW  ƒ FORFNZLVH IURP WKH SODQH RI LQFLGHQFH ILUVW TXDGUDQW ORRNLQJ WRZDUG WKH VRXUFH LQGHHGLQWKLVZD\WKHDPSOLWXGHVRIWKH7(DQG70ZDYHVDUHEDODQFHG 9LFHYHUVDZHKDYHDOHIWFLUFXODUSRODUL]DWLRQLIWKHLQSXWDQJOHLVq :LWKUHVSHFWWRWKHVHQVLWLYLW\WRWKHDQJOHRILQFLGHQFHZHQRWHWKDWLQWKHH[ DPSOH RI )LJ  D YDULDWLRQ RI rq LQ WKH DQJOH SURGXFHV D YDULDWLRQ LQ WKH q SKDVHVKLIWRIDERXW Bq  RQHRIrqo  q  $WKLUGPDMRUDSSOLFDWLRQLVLQWKHPHDVXUHPHQWRIWKHUHIUDFWLYHLQGH[PDQ\ UHIUDFWRPHWHUV DUH LQ IDFW EDVHG LQ WKH FULWLFDO DQJOH GHWHUPLQDWLRQ )URP )LJ LWLVVHHQWKDWDURXQGWKHFULWLFDODQJOHWKHLQWHUQDOUHIOHFWLRQFRHIILFLHQWKDV DQDEUXSWFKDQJHZKLFKFDQEHYHU\ZHOODSSUHFLDWHGYLVXDOO\2QFHWKHDQJOHKDV EHHQ GHWHUPLQHG RQH FDQ GHULYH WKH UHIUDFWLYH LQGH[ ZLWK WKUHH RU IRXU GHFLPDO SODFHV 

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uantum Mechanics 7KH UHODWLRQVKLS EHWZHHQ UDGLDWLRQ DQG PDWWHU LV WKHUHIRUH QRWDVLPSOHSURSRUWLRQDOUHODWLRQVKLSRQZKLFKRQWKHRWKHUKDQGDODUJHSDUWRI RSWLFV LV IRXQGHG WKH ILHOG RI /LQHDU 2SWLFV EXW GHYHORSV LQ D FRPSOH[ LQWHU WZLQHPHQWJLYLQJULVHWRG\QDPLFSKHQRPHQDIURPWKHUHODWLYHO\VLPSOHVXFKDV optical bistabilityWRWKHHVVHQWLDOO\LQGHWHUPLQDEOHVXFKDVchaos,QWKHVLPSOHVW FDVH ZH FDQ OLPLW RXUVHOYHV WR LQFOXGLQJ LQ WKH UHVSRQVH RI WKH PHGLXP VRPH WHUPVZLWKLQFUHDVLQJSRZHUVRIWKHILHOGLQZKDWLVFDOOHGNon-linear Optics +HUHZHZLOOVWXG\RQO\OLQHDUEHKDYLRUVWKDWUHVXOWIURPFRQVLGHULQJWKHPDW WHU DV EHLQJ FRPSRVHG RI GDPSHG KDUPRQLF RVFLOODWRUV IRUFHG E\ WKH UDGLDWLRQ

'LVSHUVLRQWKHRU\



  &DXFK\ZDVWKHILUVWWRWU\WRH[SODLQWKHGLVSHUVLRQKHSURSRVHGDGHYHORSPHQW RI WKH LQGH[ RI UHIUDFWLRQ n LQ D VHULHV RI HYHQ SRZHUV RI OR ZKHUH OR LV WKH ZDYHOHQJWK LQ YDFXXP LQWHUSUHWLQJ WKH RSWLFDO SURSHUWLHV RI WKH PHGLXP DV WKH PHFKDQLFDORQHVRIDQHODVWLFVROLG6XEVHTXHQWO\LQ0D[ZHOOVXJJHVWHGLQ DQH[DPLQDWLRQVXEMHFWWKDWDWRPVDQGPROHFXOHVDFWDVRVFLOODWRUVDQGKDYHQDWX UDOIUHTXHQFLHV7ZR\HDUVODWHU6HOOPHLHULQGHSHQGHQWO\XVHGWKLVLGHDE\GHULY LQJKLVHSRQ\PRXVHTXDWLRQn AOR OROr ZKHUHOr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

1.8.1 Dispersion and extinction in dilute media $JDVJHQHUDOO\FRQVLVWVRIQHXWUDOPROHFXOHVLQZKLFKWKHQHJDWLYHFKDUJHRI WKHHOHFWURQVFRPSHQVDWHVWKHSRVLWLYHRQHRIWKHQXFOHL$OWKRXJKWKHVHFKDUJHV DUHLQFRQWLQXRXVPRYHPHQWDFFRUGLQJWRWKHUXOHVRI4XDQWXP0HFKDQLFVFODVVL FDOO\ZHFDQLPDJLQHWKHPROHFXOHVDVFRQVLVWLQJRIQXFOHLVSDFHGDSDUWIURPHDFK RWKHU LQ SRVLWLRQV RI HTXLOLEULXP DQG VXUURXQGHG E\ D FORXG RI HOHFWURQV 7KH FKDUJH FHQWHU RI WKLV FORXG GRHV QRW QHFHVVDULO\ FRLQFLGH ZLWK WKDW RI WKH QXFOHL DQGLQWKLVFDVHWKHPROHFXOHLVFDOOHG polarEHFDXVHLWKDVDSHUPDQHQWHOHFWULF GLSROHPRPHQW,QDQRQSRODUPROHFXOHWKHVHFKDUJHFHQWHUVFRLQFLGHEXWDQH[ WHUQDOHOHFWULFILHOGPRYHVWKHPDZD\IURPWKHLUHTXLOLEULXPSRVLWLRQE\LQGXFLQJ DGLSROHPRPHQW:HDVVXPHIRUVLPSOLFLW\WKDWWKLVPRPHQWLVGXHWRDVLQJOH HOHFWURQ SHU PROHFXOH :KHQ WKH DSSOLHG ILHOG LV PXFK VPDOOHU WKDQ WKH ORFDO ILHOGVZHFDQDVVXPHWKDWWKHHOHFWURQLVVXEMHFWWRDQLVRWURSLFKDUPRQLFSRWHQWLDO GHVFULEHGE\DQHTXDWLRQRIFODVVLFDOG\QDPLFV 





me  r  *r  ZR r  J r

e(ORF 

 

ZKHUH rLVWKHGLVSODFHPHQWIURPWKHHTXLOLEULXPSRLQW meDQG eDUHUHVSHFWLYHO\ WKHPDVVDQGWKHFKDUJHRIWKHHOHFWURQDQGmeZRLVWKHFRHIILFLHQWRIWKHUHVWRU  (PSLULFDOO\LWDSSHDUVWKDWWKHQXPEHURIHOHFWURQVWKDWFRQWULEXWHWRWKHGLVSHUVLRQRIDJDV LV LQ JRRG DSSUR[LPDWLRQ HTXDO WR WKH YDOHQFH QXPEHU RI WKH PROHFXOHV WKDW FRQVWLWXWH LW >6RPPHUIHOG@



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 LQJIRUFH*LVDGDPSLQJFRHIILFLHQWWKDWLVLQWURGXFHGDVDIULFWLRQDOIRUFHWRWDNH LQWRDFFRXQWWKHHQHUJ\ORVVGXHWRFROOLVLRQVRIWKHPROHFXOHVZLWKHDFKRWKHURU E\WKHLQWHUDFWLRQZLWKSKRQRQVLQDVROLGPHGLXP (ORFLVWKH localHOHFWULFILHOG DSSOLHGZKLFKGRHVQRWFRLQFLGHZLWKWKHZDYHILHOGVLQFHWKHPROHFXOHVDUHDOVR VXEMHFWWRWKHILHOGJHQHUDWHGE\WKHLUFORVHUFRPSDQLRQVLQUHVSRQVHWRWKHZDYH ILHOG/DVWO\ 

J

e  SHR mec

 

LVWKHFRQVWDQWRIUDGLDWLYHGDPSLQJIRUWKHUDGLDWLRQSURGXFHGE\WKHDFFHOHUDWHG FKDUJH ZKLFK LQ TXDQWXP WHUPV FRUUHVSRQGV WR WKH spontaneous emission RI D SKRWRQ E\ Rayleigh scattering ZKHQ WKH DWRP RU PROHFXOH UHWXUQV WR LWV LQLWLDO VWDWH RU E\ Raman scattering ZKHQ WKH GHFD\ KDSSHQV LQ D GLIIHUHQW VWDWH. 7KH IUHTXHQF\RIWKH5D\OHLJKVFDWWHULQJLVFHQWHUHGRQWKDWRIWKHLQFLGHQWZDYHZLWK DVSHFWUXPEURDGHQHGE\WKH'RSSOHUHIIHFW7KDWRIWKH5DPDQVFDWWHULQJLVDOVR VKLIWHGIRUWKHHQHUJ\GLIIHUHQFHEHWZHHQWKHLQLWLDODQGILQDOVWDWHV7KHHPLWWHG UDGLDWLRQLVDOVRFDOOHGfluorescence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eDQG me UHVSHFWLYHO\ ZLWK WKH QXFOHDU FKDUJH DQG WKH QXFOHDU PDVV +HUH WKH PDJQHWLF ILHOG LV QRW FRQVLGHUHG IRU VLPSOLFLW\ KRZHYHU LW LQWHUYHQHV ERWK LQ WKH WHUP e vu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ƒe>EoORFH[S LZt @LV r ƒe>rRH[S LZt @ ZKHUH 

rR

e  ERORF    me ZR  Z  i *Z  i JZ

 

'LVSHUVLRQWKHRU\







7KHLQGXFHGSRODUL]DWLRQLVWKHUHIRUH 

PR

N erR

N HR D ERORF 

 

ZKHUHNLVWKHdensityRIWKHFKDUJHVDQGDLVWKHpolarizabilityRIWKHPROHFXOHV GHVFULEHGE\WKHLorentz equation 

e   HR me ZR   Z  i*Z  i JZ

D

 

7KHJDVVWUXFWXUHLVVXFKKRZHYHUWKDWLWVDWRPVRUPROHFXOHVSRVVHVVFRXQW OHVV UHVRQDQFH IUHTXHQFLHV ZKRVH LQWHUSUHWDWLRQ LV SRVVLEOH RQO\ E\ PHDQV RI 4XDQWXP 0HFKDQLFV 7KH TXDQWXP WKHRU\ RI VXVFHSWLELOLW\ LV ZHOO H[SRVHG E\ %R\G  ZKRLQSDUWLFXODUIRUDUDUHILHGLVRWURSLFPHGLXPLQZKLFKWKHPRO HFXOHVOLHLQWKHLUJURXQGVWDWHgUHSRUWVWKDW 

D Z

º ª   P ug «  »  R= «¬ Zug  Z  i J ug Zug  Z  i J ug »¼



¦u H

 

ZKHUH IRU VLPSOLFLW\ WKH FDVH LQ ZKLFK WKH VWDWH g LV QRQGHJHQHUDWH ZLWK ]HUR DQJXODU PRPHQWXP ZDV WDNHQ 7KH VXPPDWLRQ LV FRQGXFWHG RQ DOO RI WKH KLJKHU Pug_LVWKHVTXDUHPRGXOXVRIWKHGLSROHHOHPHQWEHWZHHQWKHVWDWHV uDQG VWDWHV_P g=ZugLVWKHHQHUJ\RIWKHVWDWH uZLWKUHVSHFWWR g=LVWKH3ODQFN¶VFRQVWDQW DQG Jug LV WKH FRQVWDQW RI dephasing LH RI WKH FRKHUHQFH GHFD\ EHWZHHQ WKHVH VWDWHVHLWKHUGXHWRVSRQWDQHRXVGHFD\RUWRRWKHU homogeneousEURDGHQLQJSUR FHVVHVVXFKDVFROOLVLRQV+HUHWKHLQKRPRJHQHRXVEURDGHQLQJVXFKDVWKDWGXH WRWKH'RSSOHUHIIHFWLVQRW\HWFRQVLGHUHG7KHWZRWHUPVLQVTXDUHEUDFNHWVFRU UHVSRQGWRWKHWZR)H\QPDQGLDJUDPVIRUWKHOLQHDUVXVFHSWLELOLW\LQWKHLQWHUDF WLRQRIWKHILHOGZLWKWKHDWRPLFV\VWHPEHWZHHQWKH gDQG uOHYHOV>'HORQHDQG .UDLQRY@)RUSRVLWLYHIUHTXHQFLHVRQO\WKHILUVWWHUPFDQEHUHVRQDQWEH FDXVH VLQFH g LV WKH JURXQG VWDWH Zug LV SRVLWLYH IRU DOO u 7KH VHFRQG WHUP LV FDOOHGDQWLUHVRQDQWDQGLVRIWHQRYHUORRNHG,QSUR[LPLW\RIDUHVRQDQFHZLWKZ| ZugWKHSRODUL]DELOLW\FDQWKHQEHDSSUR[LPDWHGE\



D Z

DR 

P ug



Zug  Z  i J ug

HR = Z  Z ug









 

 J ug 

  +HUH , XVHG )H\QPDQ¶V GHILQLWLRQ >)H\QPDQ @ IRU ZKLFKF

e ND +RSI DQG6WHJHPDQ  UHIHUWRSRODUL]DELOLW\DVWKDWZKLFKZHKHUHFDOOWKHTXDQWLW\HRD



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

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



e HR me

¦u Z

ug

f ug 

 Z  iZJ ug



 

ZKHUH 

f ug

 me Zug P ug



=e



 

LVWKHoscillator strength EHWZHHQWKHOHYHOVgDQGu)RUWKHVHGLPHQVLRQOHVVFR HIILFLHQWVLWLVZRUWKWKHVXPUXOH

¦u fug





 

DQGgEHLQJWKHIXQGDPHQWDOVWDWHIRUZKLFKZug!WKHLUYDOXHLVERXQGHGEH WZHHQ DQG 7KHGHSKDVLQJ FRQVWDQWJugLV W\SLFDOO\ JLYHQ E\ WZRFRQWULEX WLRQV 

J ug

 VS FRO  J  J ug  &ug

 

ZKHUHWKHVHFRQGWHUPLVGXHWRFROOLVLRQVZKLOH



J &VS ug

Zug  Pug SHR =c



e Zug   me SHRc

f ug 

 

LVWKHFRHIILFLHQWARI(LQVWHLQIRUWKHVSRQWDQHRXVHPLVVLRQ:LWKWKHVHVXEVWLWX WLRQV WKH GHQRPLQDWRU RI WKH IUDFWLRQV LQ(T   LV IRUPDOO\ HTXDO WR WKDW RI WKH/RUHQW]FODVVLFDOHTXDWLRQIRUWKHSRODUL]DELOLW\ 7KHSRODUL]DELOLW\DQGWKXVDOVRWKHVXVFHSWLELOLW\Fe NDDQGWKHSHUPHDELO LW\  $QRWKHUFDXVHRISRSXODWLRQGHFD\LVJLYHQE\quenchingLQZKLFKWKHHQHUJ\RIWKHH[FLWHG

VWDWHLVVXEWUDFWHGIRUFROOLVLRQ

'LVSHUVLRQWKHRU\



 



H

HR   N D 

 

DUHQRZFRPSOH[YDULDEOHV,QDQDORJ\WR(T  ZHFDQGHILQHDFRPSOH[ UHIUDFWLYHLQGH[JLYHQE\ n



c HP 

HP HRP R

 

ZKLFKLVPDGHH[SOLFLWLQWKHUHDODQGLPDJLQDU\SDUWDV n



n  iN 

 

ZKHUHnLVWKHUHDOLQGH[RIUHIUDFWLRQDQGNLVFDOOHGWKHH[WLQFWLRQLQGH[ )RUDUDUHILHGJDVVWLOODVVXPLQJP PRZHFDQWDNHEORF EDQGIRUWKHFRP SOH[UHIUDFWLYHLQGH[ZHKDYH n    F 



 

,Q FRQGLWLRQV IRU ZKLFK F   ZH FDQ PDNH WKH DSSUR[LPDWLRQ n      n    n   # n   DQGIURP(T  ZHWKHQKDYH 

 n Z  

N e HR me

¦u Z

f ug

ug



 Z  iZJ ug



 

)RUWKHUHDODQGLPDJLQDU\SDUWVRI n ZHKDYH





n Z  

 N Z

N e HR me

N e HR me

¦u

¦u



f ug Zug   Z

Zug   Z





f ug Z J ug

Zug   Z





D 

 Z J ug 



E 

 Z J ug 

6LQFH J LV JHQHUDOO\ VPDOO FRPSDUHG WR Z (TV   DUH QRW LQFRQVLVWHQW ZLWK (T   VLQFH LQ WKH YLFLQLW\ RI D UHVRQDQFH ZLWK WKH OHYHO u ZH KDYH Zug Z#Z ZugZ 7KXVGHVSLWHWKHDEXQGDQFHRIVTXDUHVLQWRWKHVHH[ SUHVVLRQV WKH WUHQG RI N LV WKDW RI D VXP RI /RUHQW]LDQ HDFK ZLWK KDOIZLGWK DW KDOIPD[LPXP ):+0 HTXDOWRJug0HDQZKLOHWKHWUHQGRInLVWKDWRIDVXPRI GLVSHUVLYH FXUYHV ZLWK D W\SLFDO RVFLOODWRU\ EHKDYLRU ZLWK D PLQLPXP DW ZQ ZugJugDQGDPD[LPXPDWZ[ ZugJug%HWZHHQWKHVHWZRYDOXHVRYHUWKH UHVRQDQFH WKH UHIUDFWLYH LQGH[ GHFUHDVHV ZLWK WKH IUHTXHQF\ DQG WKHUHIRUH LQ



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 FUHDVHVZLWKWKHZDYHOHQJWK7KLVYDULDWLRQLVFDOOHGanomalous dispersionZKLOH RQWKHWDLOVRXWVLGHRIWKHVHYDOXHVWKHRSSRVLWHRFFXUV7KHUHIRUHEHWZHHQWZR VXFFHVVLYHUHVRQDQFHVWKHUHIUDFWLYHLQGH[GHFUHDVHVZLWKZDYHOHQJWKDQGLWVYDU LDWLRQLVFDOOHGnormal dispersion 5HFDOOLQJ WKDW WKH PHDQLQJ RI n  LV WR GHWHUPLQH WKH SURSDJDWLRQ RI D PRQR FKURPDWLFSODQHZDYHDFFRUGLQJWRWKHUXOH



EZ r  t

Eo eikz iZt

i

Zn

Eo e c

z iZt

i

Zn

Eo e c

z iZt 

ZN z c



ZKHUHIRUVLPSOLFLW\WKHGLUHFWLRQRIkLVWDNHQDVEHLQJSDUDOOHOWRWKHzD[LVWKH /RUHQW]LDQRIN Z FRUUHVSRQGVWRDEVRUSWLRQOLQHVRIWKHPHGLXP)RUPDQ\PRO HFXOHVWKHVSHFWUXPFRQVLVWVRIDVHULHVRIOLQHVLQWKHIDULQIUDUHGGXHWRWKHGH JUHHVRIIUHHGRPRIURWDWLRQDVHULHVLQWKHLQIUDUHGIRUWKRVHRIYLEUDWLRQDQGD VHULHVLQWKHYLVLEOHDQGXOWUDYLROHWIRUWKRVHWKDWDUHHOHFWURQLF7KHELJWUDQVSDU HQF\RIDLUDQGZDWHULQWKHYLVLEOHUHJLRQRIWKHVSHFWUXPLVSUHFLVHO\GXHWRWKH LQWHUYDO OHIW IUHH EHWZHHQ WKH URYLEUDWLRQDO WUDQVLWLRQV LQ WKH LQIUDUHG DQG WKRVH WKDWDUHHOHFWURQLFLQWKHXOWUDYLROHW ,QDJDVZHPXVWDOVRFRQVLGHUWKHinhomogeneous broadeningIRUWKH'RSSOHU HIIHFW FDXVHG E\ WKH PRWLRQ RI WUDQVODWLRQ RI WKH PROHFXOHV :KHUHE\ QHDU WKH UHVRQDQFHZLWKWKHOHYHOuLQVWHDGRI(TV  ZHKDYHWKHFRQYROXWLRQZLWK WKH*DXVVLDQGLVWULEXWLRQRIWKHYHORFLW\vzLQWKHzGLUHFWLRQRIWKHZDYH 



f J ug e vz V N e f ug   N Z # N R  d vz     ³  HR me Zug V S f ª Zug  Z   no vz c º  J ug  ¬ ¼

ZKHUHNRDQGnRDUHWKHEDFNJURXQGYDOXHVOHIWE\RWKHUUHVRQDQFHVDQGVLVDSD UDPHWHUIRUWKH'RSSOHUEURDGHQLQJ7KHOLQHSURILOHWKDWLVREWDLQHGE\WKHFRQ YROXWLRQ  EHWZHHQD/RUHQW]LDQDQGD*DXVVLDQLVFDOOHGDVoigt ,QVSHFWUDOUHJLRQVIDUIURPWKHUHVRQDQFHVNLVQHJOLJLEOHIRUZKLFKZHKDYH 



n OR  

¦jO

A j O R R



 O Rj 



 

ZKHUHWKHUHIUDFWLYHLQGH[LVH[SUHVVHGLQDIXQFWLRQRIWKHZDYHOHQJWKLQYDFXXP OR ScZDQGORjDUHWKHZDYHOHQJWKVRIWKHUHVRQDQFHV(T  LVFDOOHG Sellmeier’s equationDQGLWLVSDUWLFXODUO\HIIHFWLYHLQGHVFULELQJWKHWUHQGRIWKH LQGH[RIUHIUDFWLRQRIPDQ\VXEVWDQFHVLQFOXGLQJWKRVHWKDWDUHVROLG7KLVLVWUXH IRU WZR UHDVRQV WKH HTXDWLRQ LV EDVHG RQ UHDOLVWLF SK\VLFDO SULQFLSOHV DQG LWV Aj DQG Ooj FRHIILFLHQWV DUH GHWHUPLQHG H[SHULPHQWDOO\ $ WDEXODWLRQ RI WKHVH HTXD WLRQVLVJLYHQLQWKHHandbook of Optics>7URSIHWDO@IRUPDQ\VXEVWDQFHV

'LVSHUVLRQWKHRU\







1.8.2 Dispersion and extinction in dense media ,WLVQRZQHFHVVDU\WRVSHFLI\WKHlocal fieldWKDWLQWHUYHQHVLQWKHVHHTXDWLRQV >+RSIDQG6WHJHPDQ@8QOHVVZHFRQVLGHULVRODWHGPROHFXOHVLQYDFXXPLW LVGLIIHUHQWIURPWKHaverageZDYHILHOGGXHWRWKHQHLJKERULQJPROHFXOHVWRWKDW FRQVLGHUHG7KLVGLVWLQFWLRQEHFRPHVVLJQLILFDQWZKHQWKHUHIUDFWLYHLQGH[LVVLJ QLILFDQWO\GLIIHUHQWIURPHLWKHULQWKHYLFLQLW\RIUHVRQDQFHVRULQKLJKSUHVVXUH JDV OLTXLGV RU VROLGV 7KH FDOFXODWLRQ RI WKLV ORFDO ILHOG LV LQ JHQHUDO GLIILFXOW VLQFH LW LQYROYHV WKH PLFURVFRSLF GHWDLOV RI WKH DUUDQJHPHQW RI WKH DWRPV ,WV WUHDWPHQWZRXOGWKHUHIRUHUHTXLUHXVHRIWKHPLFURVFRSLF0D[ZHOO¶VHTXDWLRQVIRU WKHILHOGV EDQG BLQDYDFXXPLQWKHSUHVHQFHRIFKDUJHVFRQVLVWLQJRIHOHFWURQV DQGQXFOHL+RZHYHUIRULVRWURSLFPHGLDZHFDQHYDOXDWHLWZLWKDUHDVRQLQJWKDW PLFURVFRSLFDOO\ WUHDWV RQO\ D VSKHULFDO UHJLRQ RI VSDFH ZKLFK LQFOXGHV D VLQJOH PROHFXOHRIWKHPHGLXPZKLOHXVLQJWKHPDFURVFRSLFHTXDWLRQVIRUWKHPHGLXP LQWKHVXUURXQGLQJVSDFH)URPWKLVDSSUR[LPDWLRQLWIROORZVWKDW>%RUQDQG:ROI @ 

E

EORF

P  HR

 

)RUDQLVRWURSLFPHGLDWKLVUHODWLRQVKLSFDQEHFRUUHFWHGE\DGGLQJDIXUWKHUVWUXF WXUHIDFWRU>9DQ.UDQHQGRQNDQG6LSH@ $VVXPLQJWKHFRQVWLWXWLYHHTXDWLRQ 

HR N DEORF 

P

 

DQGVXEVWLWXWLQJEORFIURP(T  ZHJHW 

P

HR FE

HR

N D E   ND

 

7KHVXVFHSWLELOLW\GLYHUJHVIRUYDOXHVRIƒe ND o‚m ND oZKLOHIRU NDofLQDQ\GLUHFWLRQLQWKHFRPSOH[SODQHFo:LWKP PRWKHUHIUDF WLYHLQGH[LVJLYHQE\ 

n

 F



N D   ND

 

)LJGHSLFWVDVDQH[DPSOHWKHFDVHLQZKLFKWKHVXPPDWLRQLQ(T  LV OLPLWHGWRDVLQJOHUHVRQDQFHZUIRUZKLFK 

N D Z

ZS ZU   Z  iZJ



 





5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

10

wp w r = 0.5

9

n k

8 7 6

n, k

w p wr = 0.3

5 4 3

wp wr = 0.1

2 1 0 -0.10

-0.08

-0.06

-0.04 -0.02

0.00

0.02

(w - wr) w r

0.04

0.06

0.08

0.10



)LJ   &RPSOH[ UHIUDFWLYH LQGH[ UHDO SDUW LQ EODFN DQG LPDJLQDU\ SDUW LQ JUD\  IURP (TV  IRUVRPHYDOXHVRIZSZUZLWKJ ZU

ZKHUH ZS LV D FRHIILFLHQW FDOOHG D plasma frequency DQG LV SURSRUWLRQDO WR WKH VTXDUHURRWRIN7KHSHDNWRWKHOHIWIRUWKHWZRODUJHUYDOXHVRIZSZUFRLQFLGHV ZLWKƒe ND  LQWKHGHQRPLQDWRURI(T  ZKLOHLQUHVRQDQFHNo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

D

 n     N n   

 

ZKLFKLVFDOOHGWKHLorentz-Lorenz equation([SHULPHQWDOO\LWDSSHDUVWKDWIDU   ,Q DGGLWLRQ WR +$ /RUHQW] ZKR REWDLQHG LW IURP WKH HOHFWURPDJQHWLF WKHRU\ WKLV ODZ ZDV

DOVRIRXQGLQGHSHQGHQWO\DIHZPRQWKVHDUOLHUE\//RUHQ]LQ&RSHQKDJHQIURPWKHWKHRU\RI HODVWLFPHGLD

'LVSHUVLRQWKHRU\



 

5

Ge

4

Si 3

GaAs

Diamond

ZnSe

n

b ZnS

Sapphire

2 CaF2 1

LiF

0 0.1

NaCl NaF

0.2

0.5

1

2

5

lo (mm)

10

20



)LJ,QGH[RIUHIUDFWLRQIRUYDULRXVFU\VWDOOLQHVXEVWDQFHV7KHFXUYHVDUHFDOFXODWHG IURPWKH6HOOPHLHU¶VIRUPXODVJLYHQLQHandbook of Optics>7URSIHWDO@

IURP WKH UHVRQDQFH WKH SRODUL]DELOLW\ LV DOPRVW LQGHSHQGHQW RI WKH GHQVLW\ IRU ZKLFK WKLV ODZ DOORZV XV WR H[WHQG WKH HVWLPDWH RI WKH LQGH[RI UHIUDFWLRQ WR WKH KLJKSUHVVXUHJDVDQGWROLTXLGVZLWKJRRGDFFXUDF\RQFHZHNQRZWKHLQWULQVLF SRODUL]DELOLW\ RI WKH PROHFXOH 7KLV ODZ DOVR H[WHQGV WR WKH GHWHUPLQDWLRQ RI WKH LQGH[RIUHIUDFWLRQRIPL[WXUHVRIVHYHUDOVXEVWDQFHVHDFKZLWKGHQVLW\NNN DQGSRODUL]DELOLW\DDDIRUZKLFKLQWKHWUDQVSDUHQF\UHJLRQVZKHUHN LVQHJOLJLEOH 



n   n  

ND  N  D  N D  " 

 

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



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

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

1.8.3 Relationship between the real and imaginary parts of the permittivity )RUDOLQHDULVRWURSLFPHGLXPWKHILHOG'LVGHILQHGDVDIXQFWLRQRIWKHILHOG( VHH†DQG† DV 

' r t

H R ( r t  H R ³

t  F f e

r t  t c ( rt c dt c 

 

7KLVHTXDWLRQUHSUHVHQWVWKHPRVWJHQHUDOspatially locallinearDQGcausalUHOD WLRQVKLSWKDWFDQEHZULWWHQEHWZHHQWKHVHWZRUHDOILHOGVIRUDtemporally invariant medium 2Q WKH RWKHU KDQG WKH UHODWLRQVKLS EHWZHHQ WKHVH WZR ILHOGV LV not temporally local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homogeneousPHGLXPLVJLYHQE\ 

H Z

H R  H R F Z

f HR  HR ³ Fe t eiZt dt  

 

ZKHUHWKHGHSHQGHQFHRIWKHVXVFHSWLELOLW\RQWKHSRVLWLRQZDVUHPRYHG)URPWKLV H[SUHVVLRQZHFDQJHWDQLPSRUWDQWJHQHUDOUHODWLRQVKLSEHWZHHQWKHUHDOSDUWDQG WKHLPDJLQDU\SDUWRIHXVLQJWKHWKHRU\RIKRORPRUSKLFIXQFWLRQV/HWXVDVVXPH WKDWZLVDFRPSOH[YDULDEOHJLYHQE\

'LVSHUVLRQWKHRU\



 



 Zc  iZcc  Z

$VDUHVXOWRIWKHSULQFLSOHRIFDXVDOLW\WKHLQWHJUDORI(T  WKXVEHFRPHVD /DSODFHWUDQVIRUPWKDQNVWRWKHIDFWWKDWWKHLQWHJUDWLRQLQWHUYDOLVGHOLPLWHGEH WZHHQDQGf  )XUWKHUPRUHVLQFH(DQG'DUHUHDOILHOGV Fe t LVDOVRDUHDOIXQFWLRQ7KHUH IRUHVSOLWWLQJWKHH[SRQHQWLDODV 

eiZt

eiZct e Zcct

FRV Zct e Zcct  i VLQ Zct e Zcct 

ZHILQGDWRQFHWKDWZLWKH HciHs 

 H Z



  H Z

 

  i ZsWKDWLVIRUZc   LVUHDOIRU Z ,QSDUWLFXODU H Z 7KHFRQYHUJHQFHRIWKHLQWHJUDOLQ(T  GHILQHGDVD/DSODFHWUDQVIRUP UHTXLUHVWKDWZs!lZKHUHlLVDUHDOYDOXHFDOOHGWKHabscissa of convergence2Q WKHRWKHUKDQGIRUSK\VLFDOUHDVRQVLWFDQQRWEH>&LFRJQD@



 Fe t | eat 

 LVKROR ZLWKa!IRUODUJH SRVLWLYH tLWIROORZVWKDWldDQGWKHUHIRUH H Z PRUSKLFLQWKHXSSHUKDOISODQHZs!  +HUHZHIROORZWKHIXUWKHUDVVXPSWLRQWKDW Fe t / \ LHLWLVDIXQFWLRQ ZKRVH VTXDUH PRGXOXV LV LQWHJUDEOH RYHU WKH ZKROH UHDO D[LV \ ,Q WKLV FDVH LWV  Zc  H[LVWV DQG LV HTXDO WR WKH OLPLW RI WKH KRORPRUSKLF )RXULHU WUDQVIRUP F Z  IRUZso2QWKHRWKHUKDQGWKHZLGWKRIWKHUDQJHRItVWDUW IXQFWLRQ F Z  LQJ IURP  IRU ZKLFK Fe t  LV VLJQLILFDQWO\ GLIIHUHQW IURP  LV RQ WKH RUGHU RI PDJQLWXGHRIWKHSRODUL]DWLRQUHOD[DWLRQWLPHV7KXVIRUWKHGLHOHFWULFVLWKDSSHQV  WKDW Fe t oIRUtofDQGWKXVWKHDVVXPSWLRQLVYHULILHG )RUFRQGXFWLQJPHGLDZHPXVWDOVRFRQVLGHUWKHFRQWULEXWLRQRIWKHFRQGXFWLY LW\ZKLFKGHSHQGVOLNH3RQWKHILHOG(,WFDQEHLQFRUSRUDWHGLQWR'DQGDGGVD   :HZLOOH[DPLQHWKLVFDVHLQWKHIROORZLQJVHF SROHRIWKHILUVWRUGHULQ Z WLRQ /HWXVQRZFRQVLGHUWKHIXQFWLRQ 

  HR H Z    ZR Z

ZKHUH ZR LV DQ\ UHDO YDOXH 7KLV IXQFWLRQ WRR LV KRORPRUSKLF LQ WKH XSSHU KDOI   ZRDQGDSRVVLEOHSROHDW Z   LIWKHPHGLXPLVFRQ SODQHZLWKRQHSROHDW Z GXFWLYH:HFDQWKHQDSSO\&DXFK\¶VWKHRUHPWRWKLVIXQFWLRQLQLWVKRORPRUSKLF



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP



imaginary axis

)LJ ,QWHJUDWLRQFLUFXLW



S

s1

R

s2 wo

0

real axis

w¢ 

ILHOGIRUZKLFK

v³ C



  HR H Z    dZ   ZR Z

 

ZKHUHCLVDFORVHGSDWKLQWKHXSSHUKDOISODQHIUHHRIVLQJXODULW\:HFKRRVHLQ SDUWLFXODUWKHSDWKVKRZQLQ)LJZKHUHWKH ROLQHLVSDUDOOHOWRWKHUHDOD[LV DWDGLVWDQFHhoIURPWKLV SLVDVHPLFLUFOHFHQWHUHGDWRIUDGLXVRofDQG sDQG sDUHWZRVHPLFLUFOHVRIUDGLXVroFHQWHUHGRQWKHSROHVUHVSHFWLYHO\ DWDQGZRRQWKHUHDOD[LV)RUZR!WKHLQWHJUDO  FDQWKHQEHVSOLWLQWR



§  r Zo r  R ¨  ³  ³ ¨ ³  R  r Zo  r ©

· § c ¸ H Z  ih  HR d Zc  ¨   ¸ Zc  ih  ZR ¨³ ³ ³ © s s S ¹



·  ¸ H Z  HR d Z      ZR ¸ Z ¹  

DQGLQDVLPLODUZD\LIZo,IWKHPHGLXPLVQRQFRQGXFWLYHZHFDQHOLPLQDWH WKH LQWHJUDWLRQ RQ s DQG UHGXFH WKH QXPEHU RI LQWHJUDOV RQ R WR WZR :H PXVW QRZFRQGXFWWKHOLPLWIRURofWKHWUHQGRI F Z IRUODUJHZFDQEHSXWLQUHOD  WLRQZLWKWKDWRI Fe t IRUVPDOOt'HYHORSLQJLQDQDV\PSWRWLFVHULHVWKHLQWH JUDORI(T  IRUUHDODQGODUJHYDOXHVRIZLWUHVXOWVWKDW>-DFNVRQ@ 

ƒe ª¬H Z  HR º¼

 ·   ¸¸ ©Z ¹ §

2 ¨¨

‚m ª¬H Z HR º¼

 ·  ¸¸ ©Z ¹ §

2 ¨¨



  GHFD\V WR  IRU Zsof IRU WKH SUHVHQFH RI WKH H[SRQHQWLDO )XUWKHUPRUH F Z eZstLQWKHLQWHJUDODUJXPHQW7KHUHIRUHWKHLQWHJUDWLRQRQWKHVHPLFLUFOHSWHQGV WRIRURof7KHQDVGLVFXVVHGDERYHLQWKHOLPLWIRUhoWKHLQWHJUDWLRQRQ RFRLQFLGHVZLWKWKDWWDNHQRQWKHUHDOD[LV)LQDOO\ZLWKLQWKHOLPLWroZHKDYH

'LVSHUVLRQWKHRU\



 



OLP ³

r o

  HR H Z  i S ª¬ H ZR  HR º¼  dZ s Z   ZR

 

,QFRQFOXVLRQIRUDQRQFRQGXFWLQJPHGLXPIURP(TV  VHSDUDWLQJWKH UHDODQGLPDJLQDU\SDUWVZHREWDLQWKHGLVSHUVLRQUHODWLRQV Hc ZR  HR

f Hcc Z  P³ d Z S Z  ZR f f

 Hcc ZR

Hc Z  H R  dZ   P³ S Z  ZR



 

f

ZKHUH PLQGLFDWHVWKDWWKHLQWHJUDOVKRXOGEHFDOFXODWHGDFFRUGLQJWRWKHSULQFLSDO SDUW7KHVHHTXDWLRQVDUHFDOOHGKramers-Kronig relationsZKHUHWKHUHDODQGLP DJLQDU\SDUWVRIWKHSHUPLWWLYLW\HRDUHUHVSHFWLYHO\WKHGLUHFWDQGLQYHUVHHilbert transform RQH RI WKH RWKHU 7KHVH UHODWLRQV KDYH D YHU\ JHQHUDO YDOLGLW\ LQ WKDW WKH\ DUH REWDLQHG E\ WKH K\SRWKHVHV RI FDXVDOLW\ DQG OLQHDULW\ DQG E\ D IHZ RWKHU PDWKHPDWLFDO K\SRWKHVHV WKDW DUH ³SK\VLFDOO\ UHDVRQDEOH´ ,Q SDUWLFXODU  DQG  DOVRUHVXOWVWREHDKRORPRUSKLFIXQFWLRQLQWKHXSSHUKDOISODQHRI Z n Z WKHUHIRUHWKH.UDPHUV.URQLJUHODWLRQVDOVRDSSO\EHWZHHQnDQGN6LQFHWKH V\PPHWU\UXOH n Z



n Z 

 

LVDOVRYDOLGIRU n WKHVHUHODWLRQVDUHXVXDOO\ZULWWHQDV n ZR  

f ZN Z  P³  d Z S Z  ZR 

 N ZR

f n Z   Z dZ  R P³  S Z  ZR



 



7KHVHDUHXVHIXOLQSDUWLFXODUIRUFDOFXODWLQJWKHWUHQGRInZKHQNLVGHWHUPLQHG H[SHULPHQWDOO\E\PHDQVRIDEVRUSWLRQPHDVXUHPHQWVRUYLFHYHUVD

1.8.4 Dispersion in conducting media 5HWXUQLQJ WR WKH FODVVLFDO PRGHOLQ D FRQGXFWRU JDV WKH FKDUJHV DUH IUHH IRU ZKLFK WKH UHVWRULQJ IRUFH LV ]HUR ,Q WKHVH FRQGLWLRQV VWDUWLQJ DJDLQ IURP (T



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

   ZH FDQ WDNH Zo   DQG DOVR QHJOHFWLQJ WKH WHUP LQ Z ZH KDYH WKH Drude equation  

H Z

§ · N e   HR ¨     i *Z ¸ m H Z R e © ¹

HR   N D

 

ZKHUH*LVDGDPSLQJFRQVWDQWGXHWRFROOLVLRQVDQGNLVWKHGHQVLW\RIIUHHFKDUJ HV7KHWHUPLQZLVRPLWWHGVLQFHWKH'UXGHPRGHODVVXPHVWKDW*ZLVGRPLQDQW RQJ Z)RUWKHFRPSOH[UHIUDFWLYHLQGH[ZHKDYH n 





Z p Z  i *Z



 

ZKHUH 

N e  HR me

Zp

 

LVWKH DQJXODU plasma frequency)RUQHJOLJLEOH*ZKHQ Z ZpWKHUHIUDFWLYH LQGH[ LV LPDJLQDU\ DQG LQ WKHVH FRQGLWLRQV D ZDYH LQFLGHQW RQ WKH PHGLXP LV FRPSOHWHO\ UHIOHFWHG )RU H[DPSOH >*XHQWKHU  -DFNVRQ @ LQ WKH LRQR VSKHUH DW DOWLWXGHV DERYH  NP WKH GHQVLW\ RI WKH FKDUJHV LV RQ WKH RUGHU RI PIRUZKLFKZp|[UDGV7KLVVLWXDWLRQLVNQRZQWRUDGLRDPDWHXUV ZKRH[SORLWWKHUHIOHFWLRQRIUDGLRZDYHVIURPWKHLRQRVSKHUHDWIUHTXHQFLHVEH ORZ0+]IRUFRPPXQLFDWLRQRQDZRUOGZLGHVFDOH $QRWKHUDSSURDFKLVWRJREDFNWRWKH0D[ZHOO¶VHTXDWLRQV  DOVRFRQVLG HULQJ WKH FRQWULEXWLRQ RI FXUUHQWV ZKLOH WKDW RI WKH IUHH FKDUJH GHQVLW\ FDQ EH RYHUORRNHGDVZHZLOOVHHODWHU(YHQIRUWKHFXUUHQWVZHFDQLQWURGXFHDFRQVWL WXWLYHHTXDWLRQDQDORJRXVWR  IRUWKHSRODUL]DWLRQIRUZKLFK 

- r t

t



³ f V r t  t c ( rt c dt c 

 

  ZKHUH V UHSUHVHQWVWKHG\QDPLFPHPRU\IRUWKHFXUUHQWVVLPLODUWR F IRUSRODUL ]DWLRQ 7KLV H[SUHVVLRQ LV D WHPSRUDOO\ QRQORFDO JHQHUDOL]DWLRQ RI Ohm’s law ZKLOH WKH DSSUR[LPDWLRQ RI VSDWLDO ORFDOLW\ LV VWLOO LPSOLHG ZLWK WKH OLPLWDWLRQV PHQWLRQHGLQWKHSUHFHGLQJSDUDJUDSK)URPWKH)RXULHUWUDQVIRUPRI(T   ZHILQG

 ZKHUH

JÖ Z V Z EÖ Z 

 

'LVSHUVLRQWKHRU\







V Z



f

³  V t eiZt dt 

 

LV WKH FRPSOH[ FRQGXFWLYLW\ RI WKH PHGLXP 2Q WKH RWKHU KDQG LQ WKH $PSHUH 0D[ZHOOHTXDWLRQIRUWKHFXUORIWKHILHOG +WKHFXUUHQWLQWHUYHQHVWRJHWKHUZLWK WKHWHPSRUDOGHULYDWLYHRIWKHILHOG')RUDPRQRFKURPDWLFZDYHZHKDYH 

ik u H

iZt D  V E 

:HFDQWKHQUHGHILQHWKHSHUPLWWLYLW\DV 

V Z º ª HR «  Fep Z  i »  HR Z ¼ ¬

H Z

 

ZKHUHFepLVWKHVXVFHSWLELOLW\RIWKHERXQGFKDUJHV,QWKLVZD\H Z SUHVHQWVD SROH DW Z   ZKLFK FRUUHVSRQGV WR WKH ]HUR UHVRQDQFH IUHTXHQF\ RI WKH 'UXGH PRGHOIRUWKHFRQGXFWRUV&RPSDULQJ(TV  DQG  ZHKDYH 

V Z

N e   me *  iZ

 

)RUFRSSHUDWURRPWHPSHUDWXUH>-DFNVRQ@IURPWKHVWDWLFYDOXHRIVDQGWKH GHQVLW\RIWKHFKDUJHVZHFDQHYDOXDWHWKDW*|uVHF7KHUHIRUHWKHFRQ GXFWLYLW\UHPDLQVSUHGRPLQDQWO\UHDOXSWRIUHTXHQFLHVRI|u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e t WKHFRQWULEXWLRQRIWKHFRQGXFWLYLW\LW  UHVXOWVWKDW Fe t o V HR IRUtofZKHUH V V  LVWKHVWDWLFFRQGXFWLYLW\ RIWKHPHGLXPDWZ 7KLVVWDWLFYDOXHDGGVRQHSROHWRWKHSHUPLWWLYLW\DWZ  DQGUHVXPLQJWKHUHDVRQLQJVRIWKHSUHYLRXVSDUDJUDSKIRUDFRQGXFWLYHPHGLXP WKHLQWHJUDORQsLV



OLP ³

  HR H Z

  ZR r o s Z

 S dZ

V ZR



DQGWKHFRUUHVSRQGLQJ.UDPHUV.URQLJUHODWLRQVEHFRPH



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 f Hcc Z  P³ d Z S Z  ZR

Hc ZR  HR

f f

 Hcc ZR

Hc Z  H R  V dZ    P³ S Z  ZR ZR



 

f

1.8.5 Phase, group and signal velocities 6RIDUZHKDYHFRQVLGHUHGHVVHQWLDOO\RQO\ PRQRFKURPDWLFZDYHVZKLFKDUH DQ LGHDO FDVH DQG DUH RI FRQVWDQW DPSOLWXGH E\GHILQLWLRQ ,QJHQHUDOZH LQVWHDG KDYHVLWXDWLRQVLQZKLFKWKHUDGLDWLRQLVFRQVWLWXWHGE\DPRUHRUOHVVEURDGVSHF WUXP RI IUHTXHQFLHV $Q LPSRUWDQW FDVH LV WKDW RI SURSDJDWLRQ RI D ³SDFNHW´ RI ZDYHVZKLFKLVGHOLPLWHGERWKWHPSRUDOO\DQGVSDWLDOO\VXFKDVWKDWJHQHUDWHGE\ DSXOVHGODVHU7KHSURSDJDWLRQRIVXFKDSXOVHLVVXEMHFWWRWKHGLVSHUVLRQDQGWKH GLVVLSDWLRQRIWKHPHGLXPZKHUHE\WKHSXOVHFKDQJHVLWVVKDSHRYHUWLPH/LPLW LQJRXUVHOYHVWRWKHRQHGLPHQVLRQDOFDVHRIDPHGLXPWKDWLVlinearhomogeneous DQG ZLWK QHJOLJLEOH PDJQHWL]DWLRQ IRU (T   D ZDYH LV JHQHUDOO\ H[ SUHVVHGDVDIXQFWLRQRIVSDFHDQGWLPHVXFKWKDWWKHDPSOLWXGH(RIWKHILHOG DQ HOHFWULFILHOGFRPSRQHQW LVFRQVWUDLQHGE\WKHHTXDWLRQ 

w w  w z,t  z,t P ( ( 3 z,t  o wz  wt  c wt 

 

ZKHUH 

3 z t

HR ³

t



F f e

z t  t c ( z t c dt c 

 

 %HVLGHVDSRVVLEOHFRQWULEXWLRQRIFRQGXFWLYLW\FDQEHLQFRUSRUDWHGLQWR Fe 7KH VSDWLDODQGWHPSRUDO)RXULHUWUDQVIRUPRI(T  LV



k  EÖ k,Z 

 Ö Z E k,Z c

  Z F Z EÖ k,Z  c

 

IURPZKLFK 

ª¬ k Z º¼



  Z ª¬  F Z º¼  c

 

H[SUHVVLQJDFRQVWUDLQWEHWZHHQWKHPRGXOXVRIWKHZDYHYHFWRUDQGWKHDQJXODU

'LVSHUVLRQWKHRU\



  IUHTXHQF\7KHUHIRUHLIZHNQRZWKHVSHFWUXP EÖ k Z ,Z RIRXUZDYHSDFNHW FUHDWLQJWKHLQYHUVHWUDQVIRUPRQLWZHDJDLQILQGLWVVSDFHWLPHIRUPJLYHQE\  f Ö E Z eik Z z iZt d Z  S ³f

( z t



 

ZKHUH IRU WKH FRQVWUDLQW PHQWLRQHG DERYH WKH LQWHJUDWLRQ UHPDLQV RQO\ IRU WKH WHPSRUDOIUHTXHQFLHV,QYHUWLQJWKHUHODWLRQEHWZHHQkDQGZWKDWLVH[SUHVVLQJZ DVDIXQFWLRQRIk(T  FDQEHZULWWHQLQDVLPLODUIRUPLQWHJUDWLQJLQkLQ VWHDGRIZ6XSSRVHQRZWKDWWKHZDYHVSHFWUXPLVVLJQLILFDQWO\GLIIHUHQWIURP RQO\RQDQLQWHUYDO'ZFHQWHUHGDWDQDQJXODUIUHTXHQF\ Z VXFKWKDW'Z Z  :HFDQWKHQUHZULWH(T  DV 

( z t

eikz iZt S

'Z

c c ª º EÖ Zc  Z ei ¬k Z Z k ¼ z iZ t d Zc  cc 

³

 

'Z

ZKHUH k k Z 2QWKHRWKHUKDQGLIkYDULHVOLWWOHZLWKLQWKHUDQJH'ZZHFDQ GHYHORSLWLQVHULHVDURXQG Z  

k Zc  Z

k

wk  w k  Zc  Zc  "  wZ  wZ

 

/LPLWLQJ RXUVHOYHV WR WKH ILUVW RUGHU LQ Zc DQG UHDUUDQJLQJ WKH H[SRQHQWLDO DUJX PHQWXQGHUWKHVLJQRILQWHJUDWLRQDV 

§ wk i¨ ƒ © wZ

· wk z  t ¸ Zc  ‚ wZ Z ¹

zZc  Z

ZKHUHkKDVEHHQGLYLGHGKHUHLQWRLWVUHDODQGLPDJLQDU\SDUWV,IZHFDQQHJOHFW WKH GLVVLSDWLRQ LQ WKH PHGLXP WKH LQWHJUDO DVVXPHV D YDOXH SURSRUWLRQDO WR WKH DPSOLWXGHEoRIWKHILHOGDWWKHSRVLWLRQx DQGDWWKHWLPHt‫ މ‬txvJZKHUH 

vJ

wZ wkƒ

k

§ wkƒ ¨ © wZ

· ¸ Z¹





 

LVFDOOHGWKHgroup velocity7KHQ(T  EHFRPHV 

( z t

 ikz iZt e ER z  vJ t  cc  





 

ZKHUHWKHH[SRQHQWLDOUHSUHVHQWVDFDUULHUZDYHZLWKDphase velocityJLYHQE\



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 vS



Z  kƒ Z

 

(T  WKHQUHSUHVHQWVWKHSURSDJDWLRQRIDTXDVLPRQRFKURPDWLFSODQHZDYH LQ D ZHDNO\  GLVSHUVLYH PHGLXP 7KH VTXDUH PRGXOXV RI Eo LV RQ WKH RWKHU KDQGSURSRUWLRQDOWRWKHHQHUJ\GHQVLW\VRWKDWLQthis approximation vJLVWKH HQHUJ\WUDQVSRUWYHORFLW\LWVHOIDQGWKHUHIRUHFRLQFLGHVZLWK WKHVLJQDOWUDQVPLV VLRQYHORFLW\ZKLFKLQ5HODWLYLW\PXVWDOZD\VEHOHVVWKDQRUHTXDOWRWKHVSHHG RIOLJKW,IDVXVXDOZHH[SUHVVWKDWkƒ Sn Z O Zn Z c,ZHWKHQKDYH 

c  vJ n Z

vS

c n Z  Z wn Z wZ



 

Z

7KHSK\VLFDOVHQVHRIvelocityLQWKHVHH[SUHVVLRQVLVHVVHQWLDOO\OLPLWHGWRWKH UHJLRQV RI QRUPDO GLVSHUVLRQ ZKHUH WKH GLVVLSDWLRQ LV QHJOLJLEOH DQG n  FDQ EH FRQVLGHUHGUHDO7KHSKDVHYHORFLW\LVKLJKHURUORZHUWKDQWKHVSHHGRIOLJKWGH SHQGLQJRQZKHWKHUnLVOHVVWKDQRUJUHDWHUWKDQXQLW\ZKLOHIRUWKHJURXSYH ORFLW\LQWKHGHQRPLQDWRUDIXUWKHUSRVLWLYHWHUPLVDGGHGWRnEHFDXVHLQWKHQRU PDOUHJLRQnJURZVZLWKZ ,Q WKH DQRPDORXV GLVSHUVLRQ UHJLRQV vS DQG vJ EHFRPH FRPSOH[ ZLWK D UHDO SDUW WKDW FDQ H[FHHG WKH VSHHG RI OLJKW RU HYHQ EH QHJDWLYH 2Q WKH RWKHU KDQG ZKHQWKLVRFFXUVWKH vJFRUUHVSRQGHQFHZLWKWKHWUDQVSRUWVSHHGRIWKHHQHUJ\RU WKHVLJQDOVHYHQWXDOO\IDOOVRII ,WLVLPSRUWDQWWRQRWHWKDWWKHpulse shape remains unchangedZKHQWKHWHUPV KLJKHUWKDQWKHILUVWRIWKHGHYHORSPHQWRIkLQ(T  DUHQHJOLJLEOHDQGWKH GLVVLSDWLRQLVDEVHQWZKLFKLVZKHQLQJRRGDSSUR[LPDWLRQ 

k

Sa  bZ IURPZKLFK n

aO R  bc 

ZKHUHaDQGbDUHUHDOFRQVWDQWVDQGWKXVWKHUHIUDFWLYHLQGH[YDULHVlinearlyZLWK WKH ZDYHOHQJWK DORQJ WKH HQWLUH SXOVH VSHFWUXP ,Q WKLV FDVH WKH group velocity dispersion *9' ZKLFKLVGHILQHGDV 

/HWXVFRQVLGHUKHUHIRUVLPSOLFLW\WKHFDVHRI k  :KHQvS vJWKHDUJXPHQWRIWKH ‚

H[SRQHQWLDODOVREHFRPHVDIXQFWLRQRI]vJtDQG(T  LVUHGXFHGWRWKHFODVVLFDOIRUP RI D ZDYH SDFNHW WKDW SURSDJDWHV DW WKH JURXS YHORFLW\ ZLWKRXW FKDQJLQJ LWV VKDSH ,I WKH WZR VSHHGVDUHGLIIHUHQWZHFDQUHZULWH(T  LQWKHIRUP



( z t





 i  SQ t ikƒ z vJt FHR e e ER z vJt  cc  



ZKHUH Q FHR kƒ vS vJ  S LVWKHIUHTXHQF\RIDSKDVHPRGXODWLRQRIWKHZDYHSDFNHW FHR carrier-envelope offset 

'LVSHUVLRQWKHRU\



 

5

(a) 4 3 2

k

n 1 FWHM

0 40

50

45

pulse spectrum 60

55

wavenumber

65

(10 2cm-1)

70

75

80

6 z (mm) 4

Intensity (arb. un.)

4 3

3 2

2 1

t=z/c

5

5

1 0.5 0.25

0

(b) 0

10

5

15

t (ps)

20

25



)LJ   3URSDJDWLRQ RI D SXOVH LQ D GLVSHUVLYH PHGLXP D  3XOVH VSHFWUXP DVVXPHG *DXVVLDQUHIUDFWLYHLQGH[nDQGH[WLQFWLRQLQGH[NRIWKHPHGLXP7KHFHQWUDOZDYHOHQJWK RIWKHSXOVHLVWDNHQKHUHDVHTXDOWRPP E 3XOVHVKDSHYHUVXVWLPHIRUVRPHGLVWDQFHV WUDYHOHGIURPWKHHQWUDQFHLQWKHPHGLXP



*9' {

wvJ wO R

ScvJ  w  k  O R  wZ

 

LV]HUR,QPRVWFDVHVWKLVGRHVQRWKDSSHQDQGLWLVQHFHVVDU\WRFRQVLGHUDWOHDVW WKHVHFRQGWHUPRIWKHGHYHORSPHQWRIk:LWKVRPHSDVVDJHVLWLV



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 w k wZ



O R w  n  Sc wO R

IURPZKLFKZHJHW 

*9'

SvJ 

OR w n  c wO R

 

7KHSKDVHYHORFLW\GLVSHUVLRQ 39' LVGHILQHGLQDVLPLODUZD\ 

39' {

wvS wO R



c wn  n  wO R

 

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t zcWKDWFRUUHVSRQGVWRWKH VSHHGRIOLJKWLQYDFXXP7KLVHQODUJHPHQWFRUUHVSRQGVWRDYDULDWLRQRIWKHLQ VWDQWDQHRXVIUHTXHQF\ chirp DORQJWKHSXOVHGXUDWLRQZLWKWKHORZHVWWRQHVWKDW SUHFHGHWKHKLJKHURQHVRUYLFHYHUVDGHSHQGLQJRQWKHVLJQRIWKH³FXUYDWXUH´RI WKHWUHQGRIn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coreZLWKDSDUWLFXODUFKRLFHRILWVGL DPHWHUWKH*9'FDQEHFDQFHOHGDWMXVWPP )LJLQVWHDGVKRZVWKHFDVHRIDSXOVHFHQWHUHGRQDUHVRQDQWIUHTXHQF\RI WKHPHGLXPZLWKDVSHFWUXPDERXWWLPHVZLGHUWKDQWKHDEVRUSWLRQEDQG7KH SURSDJDWLRQTXLFNO\H[WLQJXLVKHVWKHFHQWUDOSDUWRIWKHVSHFWUXPDQGWKHSXOVHLV GHIRUPHGUDSLGO\WKHRVFLOODWLRQLVHVVHQWLDOO\GXHWRWKHEHDWRIWKHWZRUHPDLQ LQJSDUWVRIWKHVSHFWUXPRUHTXLYDOHQWO\WRWKHORFDOIHOOLQJRIWKHVSHFWUXP

'LVSHUVLRQWKHRU\







(a)

n

1.0

0

pulse spectrum 0.125 0.25

0.5

0.5 1 1.5

k

2

0.0 -150

-100

-50

0

50

150

100

1/lo - 1/l r (cm -1)

1.5

(b)

Intensity (arb. un.)

z 1.0 2 mm

t = z/c

1.5 mm

0.5

1 mm

0.5 mm 0.25 mm 0.125 mm

0.0

0

5

10

15

20

t (ps)

25

30



)LJ([DPSOHRISURSDJDWLRQRIDSXOVHZKRVHIUHTXHQF\FDUULHULVFHQWHUHGRQDUHVR QDQWIUHTXHQF\RIWKHPHGLXPZLWKOU PP D UHIUDFWLYHLQGH[nDQGH[WLQFWLRQLQGH[ NZLWKJ S  *+]DQGZSZU RI(T  7KHQRUPDOL]HGVSHFWUXPRIWKH SXOVH LV VKRZQDV D IXQFWLRQ RI WKH ZDYH QXPEHU7KHQXPEHUV LQ WKH ILJXUHLQGLFDWHWKH GLVWDQFHLQPPWUDYHOHGIURPWKHHQWUDQFHLQWKHPHGLXP E SXOVHVKDSHIRUWKHVDPHSHQ HWUDWLRQGLVWDQFHVLQWKHPHGLXP7KHSHDNVLQJUD\UHSUHVHQWWKHSURSDJDWLRQLQYDFXXP

$W WKH UHVRQDQFH IUHTXHQF\ WKH JURXS YHORFLW\ FDOFXODWHG DFFRUGLQJ WR (T E LVQHJDWLYH+RZHYHUZHFDQVHHKRZLQWKHVHFRQGLWLRQVWKLVFRQFHSWRI VSHHGORVHVSK\VLFDOVHQVH



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP



2SWLFVRIPHWDOVDQGDEVRUEHQWPDWHULDOV $V ORQJ DV WKH ZDYHOHQJWK LV PXFK ODUJHU WKDQ WKH DWRPLF GLPHQVLRQV WKH SURSDJDWLRQ RI ZDYHV LQ PHWDOV LV DOVR GHVFULEHG E\ WKH PDFURVFRSLF 0D[ZHOO¶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conduction currents DQG WKRVH LQ TXDGUDWXUH DV polarization currents >6LYRXNKLQH @ WKLV LV D IXUWKHU ZD\ WR IRUPDOO\ VXEGLYLGH WKH UHVSRQVH RI WKH PHGLXP WR WKH HOHFWULF ILHOG 7KH FXUUHQWVLQSKDVHRUDQWLSKDVHJHQHUDOO\FDXVHDZDYHDEVRUSWLRQZKLOHWKRVHLQ TXDGUDWXUHGRQRW7KHODWWHUFDQEHH[SUHVVHGE\WKHHTXDWLRQ-SRO w 3wtZKHUH 3LVWKHSRODUL]DWLRQ 7KHPHWDOVLQVROLGIRUPDUHJHQHUDOO\FRQVWLWXWHGE\DQDJJUHJDWHRIUDQGRPO\ RULHQWHGFU\VWDOVLQDOOGLUHFWLRQV2IWHQWKHVL]HRIWKHVHFU\VWDOVLVVPDOOHUWKDQ WKHZDYHOHQJWKWKHUHIRUHWKH\DUHRSWLFDOO\LVRWURSLFHYHQZKHQWKHVLQJOHFU\ VWDOVDUHQRW:LWKWKLVDVVXPSWLRQWKHFRQVWLWXWLYHHTXDWLRQVRIWKHPDWHULDODUH 

3

HR Fc(

-

V (



 

ZKHUH ZLWK WKH SUHYLRXV VFKHPH ZH DVVXPH WKDW WKH VXVFHSWLELOLW\ F‫ މ‬DQG WKH conductivity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¶V ODZ LQ D FRQGXFWRU FRPPRQO\ DSSOLHG LQ HOHFWURQLFV ZKHUH WKH UH

2SWLFVRIPHWDOVDQGDEVRUEHQWPDWHULDOV



  VLVWDQFHRIDZLUHZLWKOHQJWKlDQGVHFWLRQALVJLYHQE\R l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‫ ( މ‬0RUHRYHU HYHQ DVVXPLQJ WKHUHODWLRQVKLS% P +WKHPDFURVFRSLF(TV  EHFRPH 

­ ’ < ( U m Hc  ® ¯’ u ( P w+ wt

’ < +   ’ u + Hc w( wt  V (

 

2QWKHRWKHUKDQGLQDPHWDOWKHFKDUJHGHQVLW\UPYHU\TXLFNO\UHOD[HVWR]HUR LQWLPHVRQWKHRUGHURI VHF>%RUQDQG:ROI@,QIDFWWDNLQJWKHGLYHU JHQFHRIWKHIRXUWKHTXDWLRQDQGVXEVWLWXWLQJLQWRLWWKHILUVWZHILQGWKDW 

Hc U P  VU P

 

 

ZKLFK H[SUHVVHV DQH[SRQHQWLDO GHFD\ ZLWK D WLPH FRQVWDQW HTXDO WR H‫މ‬V 7KHUH IRUHZHFDQWDNHUP ZKHQFH’ ‡ ( )URP(TV  ZHWKHQJHWWKH ZDYHHTXDWLRQ 

’  (  PHc

w w (  PV (    wt wt

 

ZKHUHWKHWHUPLQw(wtLPSOLHVWKDWWKHZDYHLVDWWHQXDWHGRQLWVZD\ )RUDPRQRFKURPDWLFZDYHRIDQJXODUIUHTXHQF\ZLQWKHFRPSOH[UHSUHVHQWD WLRQWKHODVWWZR0D[ZHOO¶VHTXDWLRQVEHFRPH



­’ u E iZP H  ° V·  ® § ° ’ u H i Z ¨ H c  i Z ¸ E  © ¹ ¯

 

 )RUWKHFRQWLQXLW\HTXDWLRQWKLVDOVRPHDQVWKDW’ ‡ - EXWWKLVLVVWLOOLQDJUHHPHQWZLWK

(T G IRUH[DPSOHZLWKD*DXVVLDQODVHUEHDPWKHFXUUHQWVIROORZFORVHOLQHVLQVWDQWE\ LQVWDQWVXFKDVWKRVHRIWKHHOHFWULFILHOG



5HFROOHFWLRQVRIHOHFWURPDJQHWLVP

 ZKLFKDUHIRUPDOO\HTXLYDOHQWWRWKHHTXDWLRQVIRUDGLHOHFWULFWDNLQJDVSHUPLWWL YLW\WKHFRPSOH[YDULDEOH H



Hc  iHcc Hc  i V Z 

 

ZLWK ZKLFK ZH UHGHILQH WKH ILHOG D DV D H E  7KLV HTXDWLRQ LV DQ DSSUR[LPDWH UHZULWLQJRI(T  WDNLQJIRUVWKHYDOXHRIFRQGXFWLYLW\DWZ $VORQJ DVH‫މ‬DQGVFDQEHFRQVLGHUHGUHDOFRPSDULQJ(TV  DQG  ZHKDYH cPHc nN

n  N



c PV  Z

 

ZKHUHWKHUHDOUHIUDFWLYHLQGH[nDQGWKHH[WLQFWLRQFRHIILFLHQWNZHUHLQWURGXFHG E\(T  7KHVROXWLRQVIRUnDQGNDUH  n 

c ª «  ¬«

PHc  

º P  V  PHc»  N   Z ¼»

c ª «  ¬«

PHc  

º P  V  PHc»      Z ¼»

ZKHUHWKHURRWZDVWDNHQZLWKWKHSRVLWLYHVLJQKDYLQJDVVXPHGnDQGNWREHUH DO7KHWHUPabsorptionZRXOGEHLPSURSHUIRUNEHFDXVHWKHUHPD\EHH[WLQFWLRQ HYHQZLWKRXWDEVRUSWLRQIRUH[DPSOHZLWKDSODVPDLWPD\KDSSHQWKDW H LVUHDO DQGQHJDWLYHIRUZKLFKNzDQGn >6LYRXNKLQH@ 2QWKHRWKHUKDQGWKHZDYHHTXDWLRQWKDWLVREWDLQHGIURP(T  LV ’  E  ZPH E



 

 

7KHUHIRUH ZH FDQ IRUPDOO\ H[WHQG WR WKH DEVRUEHQW PDWHULDOV WKH VDPH ODZV RI SURSDJDWLRQ UHIOHFWLRQ DQG WUDQVPLVVLRQ WKDW ZH KDYH IRU GLHOHFWULF PHGLD 7KH HVVHQWLDOGLIIHUHQFHWKDWRFFXUVZLWKWKHVHLVWKDWWKHZDYHYHFWRULVQRZQHFHVVDUL O\DFRPSOH[YDULDEOHH[SUHVVHGE\(T  )RUH[DPSOHWKHLQKRPRJHQHRXV SODQHZDYHVROXWLRQWR(T  LVJLYHQE\ 

k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± DVLPSOHXVKHURIWKH6WDWHV*HQ HUDORIWKH1HWKHUODQGVZDVYHU\VXFFHVVIXOLQEXLOGLQJVLPSOHPLFURVFRSHVZLWK VPDOOOHQVHVRIVKRUWIRFDOOHQJWK aPP ZLWKZKLFKKHGHYRWHGKLPVHOIWRQX PHURXVREVHUYDWLRQV 7KHWXUQLQJSRLQWZKLFKHYHQWXDOO\UHYROXWLRQL]HGRSWLFVLQWKHPRGHUQVHQVH DUULYHGWKDQNVWR-RKDQQHV.HSOHU ± ZLWKWKHVROXWLRQRIWKHSUREOHP RIYLVLRQ+HZDVDQDVVLVWDQWRI7\FKR%UDKHDWWKHDOFKHPLFDODVWURORJLFDOLQVWL WXWH RI 3UDKD 3UDJXH  ,Q  KH SXEOLVKHG WKH ERRN Ad Vitellionem Paralipomena ³DGGLWLRQVWR:LWHOR´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double coneRIUD\VHDFKREMHFWSRLQWLVWKHYHUWH[RIDFRQHZLWKWKHSXSLODV WKH EDVH ZLWK WKH UHIUDFWLRQ WKURXJK WKH WUDQVSDUHQW WXQLFV RI WKH H\H D VHFRQG FRQH LV IRUPHG ZLWK LWV YHUWH[ RQ WKH UHWLQD 7KHUHIRUH WKH UHIUDFWLRQ FUHDWHV DQ LQYHUWHG LPDJH RI WKH REVHUYHG VFHQH RQ WKH UHWLQD DQG WKH RSWLF QHUYH ZLWK LWV QXPHURXV ILEHUV FDUULHV LWV VLJQDOV WR WKH EUDLQ 7KHSV\FKH WKHQ KDV WKH WDVN RI UHEXLOGLQJDQGORFDWLQJLQWKHDSSDUHQWVSDFHFRUUHFWO\RULHQWHGWKHJKRVWVRIWKH REMHFWVVHHQ .HSOHU¶V NH\ VWHS ZKLFK HQDEOHG KLP WR FRQVLGHU WKH IRUPDWLRQ RI VKDUS LP

 +LVWRULFDOQRWHV)URP(PSHGRNOƝVWR+X\JHQV  DJHVRQWKHUHWLQDZDVWKDWRIDWWULEXWLQJWRWKHSXSLOWKHWDVNRIUHGXFLQJWKHVXU IDFHVRIWKHFRUQHDDQGWKHFU\VWDOOLQHWRVSKHULFDOFDSV,QWKLVZD\KHFRXOGH[ SODLQKRZWRDYRLGWKHIRUPDWLRQRIDFDXVWLFZKLFKDWWKDWWLPHZDVZHOONQRZQ DQG KDG EHHQ REVHUYHG ZLWK JODVV VSKHUHV 7KHUH ZDV VWLOO WKH SUREOHP RI GHSWK SHUFHSWLRQ WKDW LV WKH GLVWDQFH DW ZKLFK WKLQJV DUH ORFDWHG .HSOHU VROYHG WKLV SUREOHP ZLWK WKH FRQFHSW RI WKH distance-measuring triangle ZKLFK KDG DV LWV YHUWH[ WKHREVHUYHGSRLQW DQG DV LWV EDVHWKH GLDPHWHU RI WKH SXSLO KH WKRXJKW WKDWWKHSV\FKHZDVDEOHWRSHUFHLYHWKHDQJXODURSHQLQJRIWKLVWULDQJOHDQGWKHQ LWVKHLJKW2QWKHJURXQGVRIWKHVHFRQFHSWV.HSOHUPDGHDQLPSRUWDQWGLVWLQFWLRQ EHWZHHQpicturaeZKLFKDUHLPDJHVVHHQRQDVFUHHQDQGimagines rerumZKLFK DUHUHYHDOHGGLUHFWO\E\WKHH\HVDQGLQFOXGHLQSDUWLFXODUYLUWXDOLPDJHVRIRE MHFWV WKDW DUH VHHQ WKURXJK PLUURUV DQG OHQVHV )LQDOO\ WKHVH FRXOG QR ORQJHU EH ORRNHGDWDVIDOODFLRXVGHYLFHVWKDWGHIRUPUHDOLW\7KHVLPLODULW\ZLWKWKHZRUNRI 0DXUROLFR ZKLFK ZDV SXEOLVKHG RQO\ DIWHU WKH Paralipomena LV UHPDUNDEOH 1RQHWKHOHVV.HSOHUGRHVQRWFLWHLWDQGWKHUHLVQRHYLGHQFHWKDWKHKDGDPDQX VFULSWRILWZKLOHKHGRHVLQGHHGSUDLVH'HOOD3RUWD ,Q Paralipomena .HSOHU GLG QRW JLYH PXFK VSDFH WR WKH WUHDWPHQW RI OHQVHV EXWKHZDVODWHUFDOOHGXSRQWRJLYHDQRSLQLRQRQ*DOLOHR¶VWHOHVFRSH,QLWLDOO\KH UHPDLQHGQHXWUDOXQWLOKHFDPHLQWRSRVVHVVLRQRIRQHRIVDLGWHOHVFRSHVRQHRI WKRVHWKDWKDGEHHQVHQWWRWKH(OHFWRURI.|OQ &RORJQH WKURXJKWKH*UDQ'XNH RI7RVFDQD+HH[DPLQHGLWDQGZDVLPSUHVVHGZLWKLWVXEVHTXHQWO\UHSHDWLQJWKH REVHUYDWLRQRI-XSLWHU¶VVDWHOOLWHV,Q6HSWHPEHUKHSXEOLVKHGDYHU\IDYRUD EOHUHSRUWRQWKHLQVWUXPHQW$VLPLODUVLWXDWLRQDOVRRFFXUUHGZLWKWKHIDWKHUVRI WKH&ROOHJLR5RPDQRZKHQWKH\WRRUHFHLYHGDJRRGWHOHVFRSHEXLOWDQGVHQWWKHP E\$QWRQLR6DQWLQLRI9HQH]LD .HSOHUVRRQUHVXPHGKLVVWXG\RIRSWLFVDQGLQKHSXEOLVKHGDioptrice LQ ZKLFK ZH KDYH WKH ILUVW PRGHUQ WKHRUHWLFDO WUHDWPHQW RI OHQVHV DQG WKH WHOH VFRSH,QWKLVERRN.HSOHUSUHVHQWVWKHSKHQRPHQRQRIWRWDOUHIOHFWLRQZKLOHIRU VPDOODQJOHVKHDVVXPHVDODZRISURSRUWLRQDOLW\EHWZHHQWKHDQJOHRILQFLGHQFH DQGWKHDQJOHRIUHIUDFWLRQDQGGHWHUPLQHVWKHIRFDOOHQJWKRIWKHOHQVHV7KHUHE\ KHGHYHORSVWKHODZVRIRSWLFVWRWKHILUVWRUGHUIRUWKLQOHQVHVDQGWKHLUFRPELQD WLRQVFRPLQJWRSURSRVHDQHZWHOHVFRSHPDGHRIWZRSRVLWLYHOHQVHVZKLFKEHDUV KLVQDPH :LWK.HSOHUKDYLQJILQDOO\VROYHGWKHSUREOHPRIYLVLRQluxDQGlumenZHUH LGHQWLILHGDQGWKHLUVWXG\ZDVVHSDUDWHGLQWRGLVWLQFWVFLHQFHV3V\FKRORJLFDO2S WLFVDQG3K\VLFDO2SWLFV2QWKHSK\VLFDOVLGHLQWHUHVWWXUQHGWRZDUGVWKHSK\VL FDO REMHFWLYH QDWXUH RI lumen ZKLFK RYHU WLPH LPSURSHUO\ FDPH WR EH FDOOHG light LQ VFLHQWLILF ODQJXDJH DV ZHOO DV LQ WKDW RI OD\SHUVRQV GHVSLWH FHQWXULHV RI GHEDWHRYHUGLVWLQJXLVKLQJWKHWZR/DWLQWHUPV,QRXUVWXGLRZHDUHGHDOLQJZLWK WKHlumenEXWZHPXVWQRWIRUJHWWKHSK\VLRORJLFDOSV\FKRORJLFDODQGVSLULWXDO LPSOLFDWLRQVRIWKHlux$IWHUDOORXUVHQVHRIVLJKWJLYHVXVWKHEHVWRILWVHOIEXWLW PD\QRWUHSUHVHQWWKHZKROHUHDOLW\RIWKHZRUOGWRXVDWOHDVWQRWIRUQRZ 

7KHODZRIUHIUDFWLRQDQGWKHVSHHGRIOLJKW





7KHODZRIUHIUDFWLRQDQGWKHVSHHGRIOLJKW 7KHUHVXPSWLRQRIH[SHULPHQWDWLRQDOORZHG7KRPDV+DUULRW a ± WR LPSURYH3WROHPkLRV¶VWDEOHVRIWKHDQJOHVRIUHIUDFWLRQDQGWRLGHQWLI\WKHODZLQ JHRPHWULF WHUPV EXW KH GLG QRW SXEOLVK WKH UHVXOWV $URXQG WKH VDPH WLPH :LO OHEURUG6QHO ± EHWWHUNQRZQDV6QHOOSURIHVVRURI3K\VLFVDW/HLGHQLQ WKH1HWKHUODQGVZURWHDERRNRQRSWLFVFRQWDLQLQJWKHODZRIUHIUDFWLRQZKLFKKH KDG IRXQG HPSLULFDOO\ DQG ZKLFK WKH GHYHORSPHQWV RI WULJRQRPHWU\ KDG HQDEOHG KLP WR H[SUHVV DV D UDWLR RI FRVHFDQWV +RZHYHU WKLV GLVFRYHU\ ZDV FRQILQHG WR /HLGHQ5HQp'HVFDUWHV ± PXVWEHUHJDUGHGDVRQHRIWKHGLVFRYHUHUV RIWKHODZRIUHIUDFWLRQWRR,QGHHGKHKDGVXJJHVWHGWKHXVHRISODQHK\SHUEROLF OHQVHV WR IRFXV OLJKW RI D FROOLPDWHG EHDP WKUHH \HDUV EHIRUH YLVLWLQJ /HLGHQ LQ LQDQ\FDVHKHDWWULEXWHGWKHGLVFRYHU\WRKLPVHOIDQGSXEOLVKHGWKHODZRI WKH UDWLR RI WKH VLQHV LQ KLV ERRN Dioptrique LQ  $V WKH UHIUDFWLRQ ODZ KDG EHHQGHWHUPLQHGH[SHULPHQWDOO\'HVFDUWHVFRQVLGHUHGWKLVVLWXDWLRQDVXQVDWLVIDF WRU\DQGXQVFLHQWLILFEHFDXVHWKHODZKDGQRWEHHQLQVHUWHGLQDWKHRUHWLFDOFRQ WH[WDQGVRLWZDVQRWGHGXFLEOHIURPD[LRPVWKDWDUHPRUHJHQHUDO+HVXFFHHGHG LQJLYLQJLWDGHULYDWLRQE\XVLQJDPHFKDQLFDODQDORJ\DQGDcausalitySULQFLSOHDV WKHFRQVHUYDWLRQRIPRPHQWXP/DWHU)HUPDWGHULYHGWKHVDPHODZIURPDGLIIHU HQWFRQFHSWXDOEDVLVDQGLQSDUWLFXODUDFFRUGLQJWRDSULQFLSOHRIfinalityVXFKDV WKHYDULDWLRQDOSULQFLSOHRIPLQLPXPWLPH 'HVFDUWHV GHVSLWH KDYLQJ PDGH LPSRUWDQW FRQWULEXWLRQV WR PDWKHPDWLFV VXFK DVWKHFRRUGLQDWHPHWKRGIRUGHVFULELQJWKHSRVLWLRQRIDJHRPHWULFDOSRLQWEULQJ LQJJHRPHWU\EDFNWRDOJHEUDPHUHO\GHVFULEHGWKHSK\VLFDOSKHQRPHQDTXDOLWD WLYHO\XVLQJanalogies+HEXLOWIRUKLPVHOIDQHQWLUHSK\VLFVDQGDZKROHFRVPRO RJ\WKDWZDVSXUHO\mechanisticLQWKHVHQVHWKDWKHZDVFRQVLGHULQJDFWLRQVRQO\ IRUFRQWDFWFRQFHLYLQJWKHPRWLRQRIWKHSODQHWVDVEHLQJGUDJJHGE\DYRUWH[RI PLFURVFRSLFSDUWLFOHVLQDVWDWHRISHUHQQLDOFROOLVLRQEHWZHHQWKHP+HGHVSLVHG WKHNLQHPDWLFVRI*DOLOHRVLQFHDFFRUGLQJWRKLP*DOLOHRKDGQRWXQGHUVWRRGWKH meaning RI JUDYLW\ DQG PRYHPHQW +H DOVR UHMHFWHG .HSOHU¶V FHOHVWLDO SK\VLFV VLQFHKHFRQVLGHUHGLWSKLORVRSKLFDOO\HUURQHRXV,QVWHDG'HVFDUWHVUHTXLUHGPHW DSK\VLFVWRHVWDEOLVKDQGMXVWLI\VFLHQFHZLWKWKHILUVWVWHSEHLQJWRVHDUFKIRULQ GXELWDEOH D[LRPV WKH FHUWDLQW\ RI RXU RZQ H[LVWHQFH DQG WKDW RI *RG DQG VLQFH *RGFUHDWHGERWKRXUUHDVRQDQGDOOWKLQJV'HVFDUWHVGHULYHGIURPLWDSULQFLSOH RIWUXWKLQZKLFKWKLQJVWKDWDUHFOHDUO\DQGGLVWLQFWO\XQGHUVWRRGDUHWUXHWKHGHI LQLWLRQRIPDWWHUDVDSXUHVSDWLDOH[WHQVLRQLQILQLWHO\GLYLVLEOHWKHQRQH[LVWHQFH RIWKHYDFXXPWKHFRQVWDQF\RIWKH³TXDQWLW\RIPRWLRQ´RIWKH8QLYHUVH$FFRUG LQJ WR 'HVFDUWHV H[WHQVLRQ DQG PRYHPHQW DUH WKH RQO\ SULPDU\ SK\VLFDO UHDOLW\ ZKLOH RWKHU IDFWRUV VXFK DV FRORU WDVWH DQG VPHOO DUH VHFRQGDU\ %H\RQG WKHVH WKHUHH[WHQGVWKHUHJLRQRISDVVLRQGHVLUHORYHDQGIDLWKZKLFKUHPDLQLQDFFHV VLEOHWRSK\VLFV+LVZDVWKHUHIRUHDQHFHVVDU\GHGXFWLYHGHPRQVWUDWLYHDQGUD WLRQDOVFLHQFHDFFRUGLQJWRWKH$ULVWRWHOLDQLGHDOSURFHHGLQJdeductivelyIURPa priori FRQVLGHUDWLRQV ZKLFK FRQVWLWXWHG IRU KLP WKH JHQHUDO SULQFLSOHV RI DOO SK\VLFDO H[SODQDWLRQV ZKLOH H[SHULPHQWV ZHUH UHOHJDWHG WR D VHFRQGDU\ UROH RI

 +LVWRULFDOQRWHV)URP(PSHGRNOƝVWR+X\JHQV  FRQILUPLQJWKHWKHRUHWLFDOFRQFOXVLRQV+HUHSODFHGWKHSUDJPDWLFUHDVRQRI$ULV WRWpOƝVZLWKPDWKHPDWLFDOUHDVRQ+RZHYHUWKHZRUOGWKDW'HVFDUWHVGHULYHGIURP LWVD[LRPVZDVVWLOODZRUOGRISDSHUDQGLWVVFLHQWLILFSURJUDPSXUHO\deductive ZDVMXVWWKHRSSRVLWHRIWKHinductiveRI)UDQFLV%DFRQDQGVWRRGLQFRQWUDVWZLWK WKRVH RI *DOLOHR 0HUVHQQH DQG 3DVFDO 5RQFKL@,QVWHDGKHGLVFRYHUHGD QHZSKHQRPHQRQZKLFKZDV QRW UHIHUDEOH WR D WUHDWPHQW RI WKHJHRPHWULF W\SH DQG JDYH LW WKH QDPH RI GLIIUDFWLRQ +LV VWXG\ ZDV XQIRUWXQDWHO\ RQO\ SXEOLVKHG SRVWKXPRXVO\LQXQGHUWKHWLWOHPhisico-Mathesis de lumine, coloribus et iride ,W FRQVLVWV RI D SURHP DQG WZR ERRNV DQG LWV  SDJHV FRQWDLQ DQ DFFXUDWH GHVFULSWLRQRIWKHDXWKRU¶VQXPHURXVH[SHULPHQWVDQGWKHFRQFOXVLRQVWKDWKHKDG JUDGXDOO\UHDFKHG7KHVW\OHLVWKDWRIDGLDORJXHLQZKLFKWKHK\SRWKHVLVWKDWWKH OXPHQLVDVXEVWDQFHLVGHIHQGHGDJDLQVWRWKHUK\SRWKHVHVLQFOXGLQJWKHXQGXODWR U\RQH +HLPPHGLDWHO\VWDUWVZLWKH[SHULPHQWVRQGLIIUDFWLRQXVLQJDGDUNURRPZLWKD WLQ\KROHWKURXJKZKLFKVXQOLJKWSHQHWUDWHV,QWKHFRQHRIOLJKWWKDWLWIRUPVKH LQVHUWV YDULRXV REMHFWV DQG REVHUYHV WKHLU VKDGRZ RQ D ZKLWH VFUHHQ QRWLQJ WKDW WKHHGJHLVQRWFRPSRVHGRQO\RIWKHSHQXPEUDWKDWZDVWREHH[SHFWHGIURPWKH ILQLVKHG GLPHQVLRQV RI WKH KROH 5DWKHU WKH VKDGRZ DSSHDUV GLODWHG DQG DURXQG WKLVWKHUHDUHWKUHHseries lucidaeWKDWLVWKUHHOXPLQRXVDQGFRORUHGIULQJHVEOX LVK WRZDUGV WKH VKDGH DQG UHGGLVK WRZDUGV WKH RXWVLGH WKDW IDLWKIXOO\ IROORZ WKH HGJHRIWKHREVWDFOHV+HQRWHVDVZHOOWKDWEHKLQGORQJQDUURZERGLHVLPPHUVHG LQWKHOLJKWFRQHXQGHUFHUWDLQFRQGLWLRQVWKHUHDUHLULGHVFHQWIULQJHVHYHQLQWKH VKDGHWKDWYDU\LQQXPEHUDQGFKDUDFWHULVWLFV WRGD\ZHNQRZWKDWWKHVHDUHHV VHQWLDOO\LQWHUIHUHQFHIULQJHVIRUWKHGLIIUDFWHGOLJKWIURPWKHWZRHGJHV ,QDQRWK HUVHULHVRIH[SHULPHQWVKHLQWHUSRVHVDQRWKHUGLDSKUDJPZLWKDVPDOOKROHLQWKH VRODUOLJKWFRQH1RZWKHVSRWRIWKHOLJKWEHDPSURGXFHGE\WKHWZRVXFFHVVLYH RSHQLQJVRQWKHZKLWHVFUHHQLVPXFKODUJHUWKDQH[SHFWHGIURP*HRPHWULFDO2S WLFV WKH FHQWUDO SDUW LV ZKLWH DQG WKH HGJHV DUH FRORUHG UHG DQG EOXH,Q WKH WZR FDVHV KH FRQGXFWHG PDQ\ WHVWV YDU\LQJ WKH SRVLWLRQ RI WKH ZKLWH VFUHHQ DQG FKDQJLQJWKHVKDSHFRORUVXEVWDQFHDQGDUUDQJHPHQWRIWKHREVWDFOHVRUWKHGLD SKUDJP $V WR WKH IULQJHV KH QRWHV ©HDUXP SURJUHVVLRQ ILHUL SHU OLQHDP TXDH QHTXH HVW LQ GLUHFWXP FXP SULRUL OLQHD TXDH j IRUDPLQH UHFWD H[WHQGLWXU DG H[WUHPXP RSDFL LQVHUWL LQ FRQR UDGLRVR QHTXH UHFWD D IRUDPLQH DG HDP SDUWHP OXFLGDHEDVLVFRQLVXSHUTXDLOODHUHSUHVHQWDWXUSLFWDHDFWHUPLQDWHLQWDEHOODª 0RUHRYHUWKH\UHPDLQXQFKDQJHGZKDWHYHU©VLWFRUSXVLOOXGRSDFXPTXRGOXFLGR FRQRLQVHULWXUVLYHGHQVXPVLYHUDUXP VLYHODHYHDFSROLWXPVLYHDVSHUXPDF LQDHTXDOHVLYHGHQLTXHGXUXPVLWVLYHPROOHª+HH[FOXGHVWKDWWKHIULQJHVDUH GXHWRGLUHFWVXQOLJKWDVWKH\SURJUHVVLQDVWUDQJHZD\E\YDU\LQJWKHSRVLWLRQRI WKH VFUHHQ KH DOVR H[FOXGHV WKDW WKH\ DUH GXH WR WKH UHIUDFWHG RU UHIOHFWHG OLJKW

  3URS , Q  ³>[email protected] SURJUHVVLRQ DORQJ D OLQH LV QHLWKHU VWUDLJKW DV WKH ILUVW OLQH SDVVLQJ

WKURXJKWKHKROHDQGWDQJHQWWRWKHREVWDFOHLQWKH OLJKWFRQHQRURQDVWUDLJKW OLQHMRLQLQJWKH KROHWRWKHSRLQWRIWKHVFUHHQRQZKLFKWKH\DUHREVHUYHG´ 3URS,Q³«WKHRSDTXHERG\LQVHUWHGLQWRWKHFRQHRIOLJKW ZKHWKHUGHQVHUDUHILHG

VPRRWKDQGVKLQ\URXJKDQGXQHYHQWRXJKVRIWPLJKWEH´

 +LVWRULFDOQRWHV)URP(PSHGRNOƝVWR+X\JHQV  VLQFHWKH\GRQRWGHSHQGRQWKHVXEVWDQFHRIWKHREVWDFOHQHLWKHUGRWKH\PRYH ZKLOHURWDWLQJWKHREVWDFOHDURXQGLWVHGJH7KHUHIRUHWKH\PXVWEHSURGXFHGE\D QHZSKHQRPHQRQdiffraction ,Q DQ DWWHPSW WR H[SODLQ WKLV EHKDYLRU KH FRPSDUHV WKH OLJKW WR D IOXLG WKDW PRYHVYHU\IDVWDQGWKHZD\LQZKLFKZDWHUIRUPVZDYHVDURXQGDURFNWKDWRE VWUXFWVLWVSDWKVRKHWKLQNVWKDWWKHIULQJHVVKRXOGGHYHORSLQWKHVKDGHRIDQRE VWDFOHLPPHUVHGLQWKHFRQHRIOLJKW*ULPDOGLILQDOO\DGPLWVWKDW³DWOHDVWVRPH WLPHV´ WKH OLJKW DOVR SURSDJDWHV LQ DQ ³XQGXODWRU\ ZD\´ LQ WUDQVSDUHQW ERGLHV /HRQDUGRUHIHUULQJ WR WKHZDYHVRQ WKH ZDWHUKDG DOUHDG\ ZULWWHQ WKDW ©LO PRWR GHOO¶LPSUHVVLRQH ILD VRODPHQWH DFFRPSDJQDWR GDOO¶LPSHWR H QRQ GDO PRWR GHOOD PHGHVLPDDFTXDª,QVWHDGIRU*ULPDOGLWKHXQGXODWRU\PRWLRQRIWKHOXPLQRXV IOXLG DFFRPSDQLHV WKH PRWLRQ LQ WKH GLUHFWLRQ RI SURSDJDWLRQ ZLWK QRQSHULRGLF ³VLQXRXVHFULVSDWDH´WUDMHFWRULHVDQGIRUWKLVIOXLGKHUHFRJQL]HVWKHQHHGIRUD ILQLWHVSHHGRISURSDJDWLRQ *ULPDOGLWKHQJRHVRQWRH[DPLQHWKHSURSHUWLHVRIWKHOXPHQDQGLWVLQWHUDF WLRQVZLWKPDWWHULQYDULRXVVLWXDWLRQVLQGLDSKDQRXVERGLHVDQGLQWKRVHWKDWDUH RSDTXHUHIUDFWLRQ UHIOHFWLRQ DQGGLIIUDFWLRQ LQYLVLRQDQG HYHQ PRUH LQWHUHVW LQJO\ LQ WKH FRORULQJ 7KH JHQHUDO RSLQLRQ DW WKH WLPH DSDUW IURP WKDW RI 'HV FDUWHVZDVWKDWOLJKWZDVSXUHDQGZKLWHZKLOHWKHFRORUVZHUHDSURSHUW\RIERG LHVDQGZHUHFDOOHGpermanent+RZHYHUWKHUHZHUHDOVRP\VWHULRXVFRORUVOLNH WKRVHVHHQLQWKH5DLQERZDQGLQVRDSEXEEOHVWKDWVLQFHWKH\ZHUHQRWWKHUHVXOW RI D FRORUHG PDWHULDO ZHUH FDOOHG apparent 6XSSRVLQJ WKH H[LVWHQFH RI D ZKLWH OXPHQ *ULPDOGL GHVFULEHV QXPHURXV H[SHULPHQWV ZLWK ZKLFK D FRORUHG OLJKW LV REWDLQHGDQGKHFRQFOXGHVWKDWWKHFRORUPXVWEHDSURSHUW\RIWKHOXPHQLWVHOI :HSDUWLFXODUO\KDYHKLPWRWKDQNIRUWKHUHDOL]DWLRQRIVSHFWUDREWDLQHGIRUGLI IUDFWLRQIURPILQHO\VWULDWHGVXUIDFHVDQGE\UHWLFXODUVWUXFWXUHVOLNHWLVVXHV,QKLV ERRNKHZURWH©/XPLQLVPRGLILFDWLRYLFXLXVLOOXGWDPSHUPDQHQWHUTXDP XW DLXQW  DSSDUHQWHU FRORUDWXU VHX SRWLXV VLW VHQVLELOH VXE UDWLRQH FRORULV QRQ LPSUREDELOLWHUGLFLSRWHVWHVVHGHWHUPLQDWDLSVXVXQGXODWLRPLQXWLVVLPHFULVSDWDHW TXLGHPYHOXWWUHPRUGLIIXVLRQLVFXPFHUWDIOXLWDWLRQHVXEWLOLVVLPDTXDILWXWLOOXW SURSULD DF GHWHUPLQDWD DSSOLFDWLRQH DIILFLDW VHQVRULXPª +H WKHQ GHFODUHV WKDW WKHFRORUVDUHGLIIHUHQWLDWHGE\WKHQDWXUHDQGVSHHGRIWKHXQGXODWRU\PRWLRQDQG LQWKHILJXUHZLWKZKLFKKHGHVFULEHVWKLVPRWLRQWKHRVFLOODWLRQLVUHSUHVHQWHGE\ D]LJ]DJOLQHLQWKHtransverseGLUHFWLRQ$VDQH[DPSOHKHPHQWLRQVVRXQGZKLFK JLYHVVXFKDYDULHW\RIWRQHVZLWKDVPDQ\GLIIHUHQWYLEUDWLRQV$PRQJWKHFRORUV KHGHILQHVDVPDOOJURXSRIWKHPDVVLPSOHUHG\HOORZDQGEOXHDQGFRQVLGHUV WKHRWKHUVDFRPELQDWLRQRIWKHVH:KLOHLQWKHILUVWERRNKHKDGYLJRURXVO\GH

  ³WKH XQGXODWLQJ PRWLRQ LV DFFRPSDQLHG RQO\ E\ WKH LPSHWXV DQG QRW E\ WKH PRWLRQ RI WKH

VDPHZDWHU´ 3URS;/,,,³7KHPRGLILFDWLRQRIWKHOXPHQVIRUZKLFKLWLVFRORUHGSHUPDQHQWO\DQGDSSDU HQWO\ DVWKH\VD\ RUUDWKHULWPDNHVLWVHOIVHQVLEOHZLWKWKHTXDOLILFDWLRQRIFRORUFDQQRWLP SUREDEO\EHVDLGWREHDFHUWDLQXQGXODWLRQPLQXWHO\IOXFWXDWLQJDVDWUHPRURIGLIIXVLRQZLWKD FHUWDLQILQHRVFLOODWLRQZKLFKZLWKDJLYHQDSSOLFDWLRQVWLPXODWHVWKHVHQVRU\´

7KHGHEDWHRYHUWKHQDWXUHRIOLJKW   IHQGHGWKHWKHVLVWKDWWKHOXPHQZDVDsubstanceLQWKHVHFRQGERRNKHUHWXUQVWR JLYLQJ VFLHQWLILF OHJLWLPDF\ WR WKH accidental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±  SKLORVRSKHURIWKHQHZVFLHQFHZKRVWURYHWRIUHHWKHH[SHULPHQWIURPWKHGRXEW IXO FRQWH[W WR ZKLFK LW KDG EHHQ UHOHJDWHG WKDW LW PLJKW EHFRPH WKH LQGLVSXWDEOH EDVLV RI VFLHQWLILF H[SODQDWLRQ %HLQJ WKH GHPRFUDWLF LQWHOOHFWXDO WKDW KH ZDV KH XQGHUVWRRGWKDWRQO\WKHFROOHFWLYHHIIRUWRIPDQ\FRXOGSHQHWUDWHWKHFRPSOH[LW\ RI QDWXUH DQG XUJHG WKH IRUPDWLRQ RI VFLHQWLILF LQVWLWXWLRQV $OUHDG\ WKH DEERW 0DULQ0HUVHQQH ± ZDVDWLUHOHVVFRUUHVSRQGHQWRIDOO(XURSHDQVFLHQ WLVWVRIKLVWLPHDPRQJZKRPKHFDUULHGRXWWKHWDVNRIFROOHFWLQJDQGH[FKDQJLQJ OHWWHUV DQG PDQXVFULSWV 7KH ILUVW VFLHQWLILF DVVRFLDWLRQV ZHUH WKH Accademia dei Lincei LQ 5RPD ±  DQG WKH Accademia del Cimento LQ )LUHQ]H ±   ZKLFK DPRQJ RWKHU WKLQJV UHSHDWHG *DOLOHR¶V H[SHULPHQW IRU WKH PHDV XUHPHQW RI WKH VSHHG RI OLJKW 7KHVH 6RFLHWLHV ZKLFK ZHUH FUHDWLRQV RI SULYDWH SDWURQDJHKDGDVKRUWOLIHEHFDXVHRIWKHDWDYLVWLFHWHUQDOLQVLSLHQFHRIWKH,WDOLF UXOHUV%HWWHUOXFNZDVKDGE\WKHRoyal SocietyLQ/RQGRQ  DQGWKHAcadémie Royale des SciencesLQ3DULV  ZKLFKEHFDPHWKHDSSRLQWHGSODFHWR PDNHSXEOLFWKHZRUNDQGGLVFRYHULHVRIUHVHDUFKHUV 7KHVHFUHWDU\DQGFXUDWRURIWKHH[SHULPHQWVRIWKHRoyal SocietyZDV5REHUW +RRNH ±  ZKR LQ  SXEOLVKHG D ERRN HQWLWOHG Micrographia LQ ZKLFKKHUHSRUWHGKLVPLFURVFRSLFREVHUYDWLRQVLQFOXGLQJWKHGLVFRYHU\RIFHOOV +HZDVPDLQO\LQWHUHVWHGLQILQGLQJDQH[SODQDWLRQIRUWKHSKHQRPHQRQRIFRORU ,QWKHFRXUVHRIWKHVHREVHUYDWLRQVKHSDUWLFXODUO\GHDOWZLWKWKHFRORULQJRIWKH WKLQOD\HUVWKDWIRUPRQVWHHODIWHULWKDVEHHQKHDWHGWKHLULGHVFHQFHRIVRDSEXE EOHVDQGRIJODVVEORZQLQWRILQHOHDYHVDQGRIWKHIULQJHVREVHUYHGRQWKLQVKHHWV RIPLFDRUEHWZHHQWZRJODVVHVSUHVVHGWRJHWKHUZLWKDQLQWHUSRVHGOLTXLGRIYDUL RXVNLQGV+HZDVWKHILUVWWRFRQGXFWDV\VWHPDWLFLQYHVWLJDWLRQRIWKDWZKLFKLV WRGD\ FDOOHG fringes of equal thicknessRUHYHQ ³1HZWRQ ULQJV´ )URP LW KHGH GXFHG YDULRXV FLUFXPVWDQFHV WKDW DUH VWLOO FRQVLGHUHG YDOLG WRGD\ LQFOXGLQJ WKDW WKHFRORULQJDOZD\VDSSHDUVZLWKDYHU\WKLQODPLQDRIWUDQVSDUHQWPDWHULDOGHOLP LWHGE\PHGLDRIGLIIHUHQWUHIUDFWLRQIURPWKDWRIWKHODPLQD0RUHRYHUWKHFRORUD WLRQLVXQLIRUPLIWKHWKLFNQHVVLVXQLIRUP,ILWLVQRWWKHFRORUFDQWDNHWKHIRUP

 7KLVLV,WDO\ZHKDYHVPDUWJX\VDQGZRQGHUIXOODQGZKLOHWKHRWKHUVKDYHJRRGDGPLQLVWUD

WRUV

 +LVWRULFDOQRWHV)URP(PSHGRNOƝ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¶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¶VTXDQWDRU1HZWRQ¶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¶PHFKDQLFDOFRQFHSWVRILQ RUGHUWRDWWULEXWHDGLIIHUHQWVWUHQJWKDWWKHWZRHQGVRIWKHIURQW7KHUHIRUHLIWKH ZHDNSDUWRIDQLPSXOVHRIOLJKWSUHFHGHVWKHVWURQJRQHKDVDEOXHOLJKWPHDQLQJ

7KHGHEDWHRYHUWKHQDWXUHRIOLJKW   RQHKDVDUHGOLJKWLQWKHRSSRVLWHFDVH)LQDOO\WKHRWKHUFRORUVSURGXFHGE\UH IUDFWLRQPXVWFRUUHVSRQGWRD³GLOXWLRQ´RIUHGRUEOXHRUDFRPELQDWLRQWKHUHRI ,QKLVLQWHUSUHWDWLRQRIWKHFRORUVRIWKHIULQJHVRIWKLQODPLQDH+RRNHDWWULE XWHVHTXDOLPSRUWDQFHWRWKHUHIOHFWLRQVIURPWKHILUVWDQGVHFRQGIDFHVRIWKHODP LQDWKHSXOVHVUHIOHFWHGE\ERWKVLGHVZRXOGEH³FRPELQHG´ZLWKDWLPHGHOD\GH SHQGLQJ RQ LWV WKLFNQHVV ,I WKH IRLO LV YHU\ WKLQ WKH WZR SXOVHV ZRXOG EH VXSHULPSRVHG DQG QR FRORU ZRXOG EH DSSDUHQW WR WKH REVHUYHU KRZHYHU IRU D JUHDWHUWKLFNQHVVWKHVHFRQGSXOVHEHLQJDELWPRUH³ZHDN´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regionDQG QRW PHUHO\ D SRLQW RI WDQJHQF\ ZKHUH WKH composition RI WKH ZDYHOHWV EHFRPHV HIIHFWLYH+HWKXVGHILQHVLQWKHPRGHUQVHQVHDSK\VLFDOEHDPRIOLJKWDVDWKLQ UHJLRQDURXQGWKHJHRPHWULFUD\HYHQLIKHGRHVQRWLQGLFDWHLWVDFWXDOWKLFNQHVV )XUWKHUPRUHLQDSXUHO\JHRPHWULFFRQVWUXFWLRQWKHUHLVDOVRDQHQYHORSHVXUIDFH RI ZDYHOHWV LQ WKH UHWURJUDGH GLUHFWLRQ 7R UHVROYH WKLV GLIILFXOW\ +X\JHQV DS SHDOHGWRDSK\VLFDOSULQFLSOHWKDWLVWRWKHPHFKDQLFDODQDORJ\RIWKHEDOOVVRLI WKHVHDUHDOOHTXDOWKHLPSXOVHFDQRQO\SURSDJDWHIRUZDUGZLWKRXWJHQHUDWLQJD VLJQDOGLVSHUVLRQLQWKHRWKHUGLUHFWLRQV7KHobliquity factorLQYRNHGE\)UHVQHO QHDUO\DFHQWXU\DQGDKDOIODWHUKDGLWVILUVWGUDIWKHUH ,W LV VWULNLQJ WKDW +X\JHQV GLG QRW GHDO ZLWK GLIIUDFWLRQ >5RQFKL  @ $OW KRXJKWKHFRQFHSWRILQWHUIHUHQFHDQGSKDVHEHJDQYDJXHO\WRHPHUJHERWKLQWKH accessesRI1HZWRQZKRKDGDOVRH[DPLQHGWKHLQWHUIHUHQFHRIWKHZDYHVRQWKH VXUIDFH RI ZDWHU DQG LQ +X\JHQV¶ FRQFHSW RI composition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©ELHQ TXH OHV RQGHVSDUWLFXOLqUHV« VH UpSDQGHQW DXVVL KRUVGH FHW HVSDFH >RXWVLGH WKH OLJKW[email protected]WRXWHIRLVHOOHVQHFRQFRXUHQWSRLQWHQPrPH LQVWDQWjFRPSRVHUHQ VHPEOHXQHRQGHª7KHUHIRUHKHVWLIIHQHGRQWKHFRQFHSWRIUHFWLOLQHDUSURSDJD WLRQZLWKRXWHYHUGHILQLQJZKHWKHUWKHHQYHORSHDUHDEHKLQGDQREVWDFOHHQGVLQ D FRQWLQXRXV RU GLVFRQWLQXRXV ZD\ 7R XQGHUVWDQG +X\JHQV¶ ZRUGV ZH PXVW OHDYH DVLGH WKDW VKRUWFXW HVWDEOLVKHG E\ WKH VSHFWUDO GHFRPSRVLWLRQ LQ KDUPRQLF ZDYHV ZLWK ZKLFK ZH XVXDOO\ WUHDW UDGLDWLRQ VR DV WR UHDVRQ LQVWHDG LQ WKH WLPH GRPDLQ+HGLGLQGHHGUHJDUGOLJKWDVEHLQJFRPSRVHGRIYHU\VKRUWSXOVHVWKXV WKHGLIIHUHQFHVLQDUULYDOWLPHVSOD\WKHVDPHUROHDVWKHSKDVHGLIIHUHQFHVEHWZHHQ KDUPRQLFZDYHVWKDWLQWHUIHUH>GH/DQJ@ +X\JHQV¶JUHDWHVWVXFFHVVZDVWREHDEOHWRH[SODLQLQDVLPSOHZD\WKHODZV RI UHIOHFWLRQ DQG UHIUDFWLRQ 7KH FRPSDULVRQ EHWZHHQ KLV PRGHODQG REVHUYDWLRQ OHG KLP WR FRQFOXGH LQ DFFRUGDQFH ZLWK )HUPDW WKDW SURSDJDWLRQ LV VORZHU LQ GHQVHUPHGLXPVEXWQRZWKHODZRIWKHVLQHVZDVH[SODLQHGLQDFDXVDOZD\DQG QRWDFFRUGLQJWRDSULQFLSOHRIILQDOLW\)XUWKHUPRUHZKLOH3DUGLHVassumedWKDW LQUHIUDFWLRQWKHZDYHIURQWLVPDLQWDLQHGSHUSHQGLFXODUWRWKHGLUHFWLRQRISURSD

 >+X\JHQVS@³DOWKRXJKWKHSDUWLFXODUZDYHV«DOVRVSUHDGRXWRIWKLVVSDFHKRZHY

HUWKH\GRQRWFRQWULEXWHDWWKHVDPHLQVWDQWWRFRPSRVHWRJHWKHUDZDYH´

 +LVWRULFDOQRWHV)URP(PSHGRNOƝVWR+X\JHQV  JDWLRQ WKLV QRZ DSSHDUHG WR EH D consequence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raité de la Lumière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³PHUYHLOOHXVHPHQW´FRQILUPHG'XULQJWKHVH LQYHVWLJDWLRQVKHREVHUYHGDQXQH[SHFWHGSKHQRPHQRQE\DUUDQJLQJWZRFU\VWDOV VHSDUDWHGLQVXFFHVVLRQLQWKHSDWKRIDEHDPRIOLJKW:KHQWKHLUIDFHVZHUHDOO SDUDOOHOLQWKHILUVWWKHUHZDVWKHXVXDOGLYLVLRQRIWKHEHDPLQWRWZREXWQHLWKHU RIWKHVHXQGHUZHQWIXUWKHUGLYLVLRQLQWKHVHFRQGFU\VWDO+RZHYHUE\URWDWLQJWKH WZRFU\VWDOVEHWZHHQWKHPDORQJDQD[LVSHUSHQGLFXODUWRWKHIDFHVFURVVHGE\WKH OLJKW EHDPV WKLV VXEGLYLVLRQ UHDSSHDUV JLYLQJ SODFH WR IRXU EXQGOHV RI OLJKW RI YDULDEOH LQWHQVLW\ RQO\ WR GLVDSSHDU DJDLQ HYHU\ q +X\JHQV WKRXJKW WKDW WKLV ZDVGXHWRVRPHSURSHUW\acquiredE\WKHOLJKWFURVVLQJWKHILUVWFU\VWDO ZKLFKLV QRZNQRZQDVWKHSRODUL]DWLRQRIWKHZDYHV :KHUHE\ZKHQWKHOLJKWPHHWVWKH VHFRQGFU\VWDOODWWLFHLQDFHUWDLQSRVLWLRQLWLVVWLOODEOHWRPRYHWKHWZRGLIIHUHQW W\SHVRIPDWWHUZKLFKJLYHULVHWRWKHWZRUHIUDFWLRQVZKLOHIRUDQRWKHUGLVSRVL WLRQLWPD\PRYHRQO\RQH +X\JHQV ZURWH ©0DLV SRXU GLUH FRPPHQW FHOD VH IDLW MH Q¶DL ULHQ WURXYp MXVTX¶LFLTXLPHVDWLVIDVVHª+RZHYHUKHDOVRZURWHWKDW©&DUELHQTXHMHQ¶HQ DLHSDVS€WURXYHUMXVTX¶LF\ODFDXVHMHQHYHX[SDVODLVVHUSRXUFHODGHO¶LQGLTXHU

 >+X\JHQVS@³%XWWRVD\KRZWKLVLVGRQH,KDYHIRXQGQRWKLQJVRIDUWKDWVDWLVILHV

PH´

6XJJHVWHGUHDGLQJV   DILQGHGRQQHURFFDVLRQjG¶DXWUHVGHODFKHUFKHU,OVHPEOHTX¶LOIDXGURLWIDLUHHQ FRUHG¶DXWUHV VXSSRVLWLRQV RXWUH FHOOHV TXH M¶D\ IDLWHV TXLQH ODLVVHURQWSDV SRXU FHOD GH JDUGHU WRXWH OHXU YUDLVHPEODQFH DSUqV DYRLU HVWp FRQILUPpHV SDU WDQW GH SUHXYHVª :H DUH WKH KHLUV RI WKLV ZLVK WR GLVFRYHU WKH P\VWHULHV RI 1DWXUH FRXUDJH LPDJLQDWLRQ DQG UDGLFDOO\ QHZ DSSURDFKHV ZLOO EH QHFHVVDU\ EXW DV JRRG VRQV DQG GDXJKWHUV ZH ZLOO QRW IRUJHW WKH SUHFLRXV LQKHULWDQFH RI RXU ID WKHUV

6XJJHVWHGUHDGLQJV 7KHVHKLVWRULFDOQRWHVDUHGHULYHGIURPWKHIROORZLQJWH[WV $UJHQWLHUL$L’ottica di LeonardoLQWKHERRNLeonardo da Vinci&XUDWHGHGLWLRQRIWKH H[KLELWLRQRQ/GD9LQFLS %HUQDO-RKQ'Storia della ScienzaRI(GLWRUL5LXQLWL5RPD ,,,HG  %HYLODFTXD ) DQG ,DQQLHOOR 0* L’ottica dalle origini all’inizio del ‘700 /RHFKHU HG 7RULQR   7KLV LV DQ DQQRWDWHG DQWKRORJ\ RI YDULRXV DXWKRUV IURP 3OiWǀQ WR 1HZWRQ ZLWK WKH LQWHQWLRQ RI KLVWRULFDOO\ XQLI\LQJ WKH RSWLFV RI WKH YDULRXV GLVFL SOLQHV SK\VLFDO SK\VLRORJLFDO DQG SV\FKRORJLFDO  LQWR ZKLFK ZH VXEGLYLGH LW WRGD\ DQGWRVKRZWKHOLQNEHWZHHQ3K\VLFVDQG0HWDSK\VLFVFRQFHSWXDOPRGHOVUHJXODWLYH SULQFLSOHVDQGH[SHULPHQWDOUHVXOWV GH/DQJ+Christiaan Huygens, originator of wave opticsLQ³Huygens’ principle 16901990, theory and applications´(GGD+%ORN+$)HUZHGD+..XLNHQ1RUWK +ROODQG(OVHYLHU6FLHQFH3XEOLVKHUV%9   'UDNH6WLOOPDQGalileo(G,O0XOLQR%RORJQD  'UDNHZDVWKHPRVWHPLQHQWKLVWR ULDQRI*DOLOHR+LVERRNLVDQDFFXUDWHFKURQRORJLFDOELRJUDSK\RI*DOLOHRLQZKLFK WKHLQWHOOHFWXDOLWLQHUDU\RIWKLVJUHDWVFLHQWLVWDQGWKHHYHQWVRIWKHSHRSOHWKDWZHUH FORVHWRKLP DUHUHFRQVWUXFWHG )HUPDW3LHUGHEuvres37DQQHU\DQG&+HQU\HG3DULV  ,,SS *RX\0Sur le mouvement lumineux-3K\V7KHRU$SSO   +DOO$5DQG%RDV+DOO0Storia della scienza(GLO0XOLQR%RORJQD   +HU]EHUJHU0D[Optics from Euclid to Huygens$SSOLHG2SWLFV  7KLVLVD VXPPDU\ RI WKH VDOLHQW FRQWULEXWLRQV RI DXWKRUV WR RSWLFV IURP DQFLHQW WR PRGHUQ WLPHVZLWKDQLQWHUHVWLQJELEOLRJUDSK\ +X\JHQV&Traité de la Lumiere3LHUUH9DQGHU$D(G   /HQREOH 5 Le origini del pensiero scientifico moderno 8QLYHUVDOH /DWHU]D HG %DUL  7KLVGHVFULEHVLQDYLYDFLRXVZD\WKHKLVWRU\RIWKHGHYHORSPHQWRIVFLHQ WLILFWKRXJKWDQGRIWKHSHUVRQDJHVZKRFRQWULEXWHGWRLWLQWKHFHQWXU\ZKHQWKHSKL ORVRSK\RIVFLHQFHDQGPRGHUQVFLHQFHILQDOO\EHJDQGHHSO\WRLQIOXHQFHWKHKLVWRU\ RIKXPDQLW\ 0DUWLQL&DUG&DUOR0DULDL’ira di Dio(G(XURFOXE/RQJDQHVL &0LODQR   0ROHVLQL*Telescope lens-making in the 17th century: The legacy of Vangelista Torricelli2SWLFV 3KRWRQLFV1HZV $SULO S

 >+X\JHQVS@³)RUDOWKRXJKXQWLOQRZ,KDYHQRWEHHQDEOHWRILQGLWVFDXVH,

GRQRWZDQWWRUHIUDLQIURPLQGLFDWLQJLWLQRUGHUWRJLYHRFFDVLRQWRRWKHUVWRVHHNLW,WVHHPV WKDW VWLOO RWKHU VXSSRVLWLRQV ZRXOG KDYH WR EH PDGH EHVLGHV WKH RQHV , KDYH PDGH ZKLFK QHYHUWKHOHVVZLOOQRWIDLOWRSUHVHUYHDOORIWKHLUSUREDELOLW\DIWHUKDYLQJEHHQFRQILUPHGE\VR PDQ\SURRIV´



+LVWRULFDOQRWHV)URP(PSHGRNOƝVWR+X\JHQV

 0ROHVLQL*DQG*UHFR9 Galileo Galilei: Research and development of the telescopeLQ Trends in Optics, Research, Development and ApplicationsHGE\$QQD&RQVRUWLQL ,&29ROS$FDGHPLF3UHVV/RQGRQ  %DVHGRQKLVWRULFDOGRFX PHQWVWKLVGLVFXVVHVWKHUHDOL]DWLRQRIWKHWHOHVFRSHVRI*DOLOHRDQGUHSRUWVWKHDQDO\ VLVPDGHRQRQHRI*DOLOHR¶VRULJLQDOREMHFWLYHVWRGHWHUPLQHLWVRSWLFDOTXDOLW\ 3ROYDQL*LRYDQQLStoria delle ricerche sulla natura della luce,VWLWXWRGHOOD(QFLFORSHGLD ,WDOLDQD5RPD  $IWHUDEULHIUHIHUHQFHWRWKHDQFLHQWDXWKRUVLWGHVFULEHVWKH KLVWRU\RIRSWLFVEHWZHHQDQGDQGLQSDUWLFXODUWKHUHVHDUFKRQDHWKHU 5RQFKL9DVFR Storia della luce, da Euclide a Einstein/DWHU]DHG%DUL  WKLVLVWKH ODVWH[WHQGHGHGLWLRQRIStoria della luceE\9DVFR5RQFKL1=DQLFKHOOLHG%ROR JQD  ZKLFKFRQWDLQVPDQ\EHDXWLIXOLOOXVWUDWLRQVWKDWZHUHXQIRUWXQDWHO\RPLW WHGLQWKHQHZHGLWLRQ,WLVDERRNGHQVHZLWKQHZVDQGSDVVLRQDWHFULWLFLVPRQWKH FRQFHSWVLQQRYDWLYHLGHDVDQGPLVWDNHVRIDQFLHQWDQGPRGHUQDXWKRUVEXW\RXZLOO UHDGLQRQHEUHDWKScritti di Ottica(GL]LRQLLO3ROLILOR0LODQR  7KLVLVDSRQ GHURXVDQQRWDWHGDQWKRORJ\RIYDULRXV,WDOLDQDXWKRUVIURP/XFUHWLXVWR$PLFLStoria del Cannocchiale3RQWLILFLD$FFDGHPLDGHOOH6FLHQ]H&LWWjGHO9DWLFDQR   5RVHQ(GZDUGThe Invention of Eyeglasses-RIWKH+LVWRU\RI0HGLFLQHDQG$OOLHG6FL HQFHV   6DEUD $, Theories of Light, from Descartes to Newton &DPEULGJH 8QLYHUVLW\ 3UHVV &DPEULGJH  7KLVLVDIXQGDPHQWDOERRNWKDWDQDO\]HVVWHSE\VWHSWKHZRUN RIVFLHQWLVWVRIWKHVZKRZLWKWKHLUOHJDF\IURPSUHYLRXVDXWKRUVWKHLUSKLOR VRSKLFDOLGHDVDQGWKHLUGLVSXWHVFRQWULEXWHGWRWKHWKHRULHVRQWKHQDWXUHRIOLJKWDQG LWVSURSHUWLHV 9DYLORY6,Galileo in the history of optics6RYLHW3K\V8VS  $VZHOODV GLVFXVVLQJ*DOLOHR¶VDFWLYLW\LQ2SWLFVLWDOVRUHODWHVWKHVWRU\RIKLVSUHFXUVRUV :ROI (PLO The life and work of Christiaan Huygens LQ ³Huygens’ principle 1690-1990, theory and applications´ (G GD + %ORN +$ )HUZHGD +. .XLNHQ 1RUWK +ROODQG(OVHYLHU6FLHQFH3XEOLVKHUV%9  



 

Chapter 2 Geometrical Optics All what we see, it is seen in a rectilinear direction. Pseudo-Eukleidēs, Catoptrics, 2nd postulate

Introduction Geometrical Optics is one of the oldest of the physical sciences, but still remains the most effective approach for explaining a good part of the most common optical phenomena. It is particularly useful for tracing the propagation of light in inhomogeneous media and for describing or designing optical instruments. The emphasis of this discipline is to find the path of light rays, imagined as geometric lines along which energy flows. It is based on a few simple observations: a) light propagates in a straight line in homogeneous media and, in particular, it is possible to produce thin beams of light, similar to geometrical rays within the physically unattainable limit of an infinite subtlety; b) the laws of reflection and refraction; c) different light beams propagate without disturbing each other; d) “natural” sources are generally uncorrelated between them, for which their light beams overlap without showing interference. On the other hand, the electromagnetic field associated with visible light is characterized by very small wavelengths, on the order of 106  107 m. Therefore, the phenomena that violate the first and the last of the above observations can be observed only with accurate experiments. Indeed, the effects of diffraction or interference are almost hidden using natural sources, for which the visibility of the fringes is reduced. The diffraction phenomena appear when there are rapid changes in the amplitude of the field, such as that produced by a sharp obstacle, particularly when some dimension of the optical system, such as the diameter of an aperture, is comparable to the wavelength; or in the neighborhood of a focal point; or over long distances compared to the transverse dimension of a wave, particularly when there is a delimitation imposed upon it. Lastly, point (c) follows from the linearity of the media at the ordinary beams’ intensity. In this chapter, we will explore the consequences of such observations taken as empirical data. However, we will derive the laws of Geometrical Optics by Maxwell’s equations within the limit at which the wavelength  tends to zero. We will also see that, within such a limit, the intensity can be deduced from the transverse dimension of a thin pencil of rays and that the polarization state can be associated with each ray. Therefore, in Geometrical Optics, the rays are associated with the Electronic supplementary material The online version of this chapter (https://doi.org/10.1007/ 978-3-030-25279-3_2) contains supplementary material, which is available to authorized users.

© Springer Nature Switzerland AG 2019 G. Giusfredi, Physical Optics, UNITEXT for Physics, https://doi.org/10.1007/978-3-030-25279-3_2

159

*HRPHWULFDO2SWLFV  WUDQVSRUWRIHQHUJ\DQGPRUHJHQHUDOO\RILQIRUPDWLRQ,QSDUWLFXODUZHZLOOIR FXVRQWKHVSHFLDOIRUPRILQIRUPDWLRQWKDWLVFRQWDLQHGLQRSWLFDOLPDJHV 

'HULYDWLRQRI*HRPHWULFDO2SWLFVIRUOo ,W FDQ UHDVRQDEO\ EH DVVXPHG WKDW WKH ODZV RI *HRPHWULFDO 2SWLFV FDQ EH GH ULYHG IURP 0D[ZHOO¶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¶VHTXD WLRQV /HWXVFRQVLGHUWKHFDVHRIPRQRFKURPDWLFZDYHWKDWSURSDJDWHVLQDlinearDQG isotropic PHGLXP EXW QRW QHFHVVDULO\ D KRPRJHQHRXV RQH ,Q WKLV FDVH ZH FDQ DSSO\WKH0D[ZHOO¶V(TV  DQGWKHZDYH(TV  ,QSDUWLFXODUZHLQ WHQG WR ILQG D VROXWLRQ WR WKHVH HTXDWLRQV ZLWK D IRUP WKDW LV OLWWOH PRUH JHQHUDO WKDQWKDWRI(T  



( r  t ƒe ª ER r ei I r t º + r  t ƒe ª H R r ei I r t º  ¬ ¼ ¬ ¼

 

ZKHUHERDQGHRDUHFRPSOH[IXQFWLRQVRIWKHSRVLWLRQDQG



I r t

kR L r  Z t 

 

7KHFRHIILFLHQWkR SORZKHUHORLVWKHZDYHOHQJWKLQYDFXXPLVLQWURGXFHGWR VLPSOLI\VXEVHTXHQWFDOFXODWLRQVDQGL r LVDVVXPHGWREHDrealcontinuousDQG univocalIXQFWLRQRIWKHFRRUGLQDWHV7KHFRPSOLFDWLRQVDULVLQJIURPDQRWXQLYR FDOLVXFKDVLQWKHSUHVHQFHRIUHIOHFWLRQVRUFDXVWLFVZLOOEHFRQVLGHUHGODWHU 7KHORFXVRISRLQWVIRUZKLFKI FRQVWUHSUHVHQWVDJHQHULFPRYLQJZDYHIURQW ZKRVHG\QDPLFVZHDUHJRLQJWRGHWHUPLQH 7KHQ LI ZH DSSO\ WKH GLIIHUHQWLDO RSHUDWRUV WR WKH DQDO\WLF IXQFWLRQ E  ERH[S>iI rt @RIWKHUHDOILHOG(ZHJHW

’uE

^’< ER  ikR’L< ER `eiI rt  ^’ u ER  ikR’L u ER `eiI rt 

’E

^’ E

’ %RUQDQG:ROI@ ,QWKHFDVHRIWKHWRUXVZHKDYHWKDWWKHIRFDOOLQHVDUHMXVWOLQHV$FWXDOO\LQ PRVWFDVHVWKLVGRHVQRWRFFXUWKHVDPHGHILQLWLRQRIWKHFHQWHURIFXUYDWXUHLV GHULYHGIURPWKHOLPLWIRUxoRUyoLQWKHQHLJKERUKRRGRIrRDQGLVWKHUH IRUHLQGHSHQGHQWRIWKHWUDQVYHUVHFRRUGLQDWHVRQO\IRUZDYHIURQWVZKRVHVHFWLRQV DORQJWKHSULQFLSDOGLUHFWLRQVDUHWUXO\FLUFXODU)RUH[DPSOHLQWKHGHYHORSPHQW RILDFRQYHUJHQWSDUDEROLFZDYHIURQWGLIIHUVIURPDVSKHULFDOZDYHIURQWZLWKWKH VDPHFXUYDWXUHQIRUWKHWHUPQrDQGKLJKHUZKHUHr xy7KHUHIRUH WKHUD\WKDWSDVVHVWKURXJKWKHZDYHIURQWDWWKHKHLJKWrLQWHUVHFWVWKH]D[LVDWD GLVWDQFHRIDERXWQrDIWHUWKHFHQWHURIFXUYDWXUHREWDLQHGZLWKLQWKHOLPLWIRUr o DVZHZLOOVHHVWXG\LQJVSKHULFDODEHUUDWLRQ WKLVGLIIHUHQFHFRUUHVSRQGVWR WKH³DUURZ´zRQWKHZDYHIURQW ,Q JHQHUDO IRU HDFK SRLQW RI WKH ZDYHIURQW WKHUH DUH WZR FHQWHUV RI FXUYDWXUH



*HQHUDOSURSHUWLHVRIUD\V 

r2

)LJ :DYHIURQWVLQWKHYLFLQLW\RID FDXVWLF

r1

caustic P Q

T 

FDOFXODWHGE\WKHDERYHOLPLWDQGIRUWKHHQWLUHIURQWWKHVHWRIWKHVHFHQWHUVFRQ VWLWXWHVWZRVXUIDFHVFDOOHGcaustics ZKLFKDUHWKH³EHWHQRLUH´RIGHVLJQHUV ,Q WKHFDVHLQZKLFKWKHZDYHIURQWKDVURWDWLRQDOV\PPHWU\DERXWDQD[LVRQHRIWKH WZRVXUIDFHVLVUHGXFHGWRDQD[LDOIRFDOOLQHDQGWKHRWKHUWRD SRUWLRQ RIVXUIDFH RIURWDWLRQDURXQGWKHD[LV,IWKHQWKHSULQFLSDOUD\OLHVRQWKHD[LVRIURWDWLRQ VXFKDVLQWKHFDVHRIDFRQLFRIUHYROXWLRQWKLVVXUIDFHUHVHPEOHVDFRQHZLWKD FXVSLQFRUUHVSRQGHQFHZLWKRQHRIWKHH[WUHPHVRIWKHD[LDOIRFDOOLQH )LJ  /HWXVQRZWDNHDSRLQW3QHDUWKHFRQYH[VLGHRIDFDXVWLFWZRWDQJHQWUD\V SDVV IRU WKLV SRLQW r ZKLFK KDV OHIW WKH FDXVWLF VXUIDFH DQG r ZKLFK DS SURDFKHV LW )LJ   (DFK RI WKHP FRUUHVSRQGV WR D GLIIHUHQW YDOXH RI WKH HL NRQDODQGLQ3WKH\LQWHUVHFWWZRZDYHIURQWVWKHILUVW QRUPDOWR r GLYHUJLQJ DQGWKHVHFRQG QRUPDOWR r FRQYHUJLQJRQWKHSODQHGHILQHGE\WKHVHUD\V,Q WKHDPSOLWXGHRIWKHILHOGWKHUHLVLQWHUIHUHQFHEHWZHHQWKHWZRZDYHVDOVRLWRI WHQ KDSSHQV WKDW RWKHU UD\V WDQJHQW WR RWKHU SDUWV RI WKH FDXVWLF SDVV IRU 3 IRU ZKLFKWKHLQWHUIHUHQFHSDWWHUQLVHYHQPRUHFRPSOH[&RQVLGHUWKHVHFWLRQRIWKH FDXVWLFZLWKWKHSODQHIRUPHGE\WKHUD\VrDQGr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

FDXVWLFVDUHRQO\WKHHQYHORSHRIWKHFRQJUXHQFHRIUD\V

*HRPHWULFDO2SWLFV  VXUHWKDWVXFKSODQHLVDOZD\VGHWHUPLQHGE\WKHFRQJUXHQFHRIUD\VZHFDQWKHQ UHFRQVWUXFW WKH IDPLO\ RI FRQYHUJLQJ DQG GLYHUJLQJ ZDYHIURQWV WKDW RULJLQDWH WKH FDXVWLF,QSDUWLFXODUZHILQGWKDW 

DW HDFK SRLQW7 RI WKH FDXVWLF WKH FRUUHVSRQGLQJ FRQYHUJHQW DQGGLYHUJHQW ZDYHIURQWVDUHWDQJHQWWRHDFKRWKHU ,QGHHGWKHUD\WKDWLVWDQJHQWWRWKHFDXVWLFLQ7PXVWEHORQJWRERWKZDYHIURQWV PRUHRYHUHDFKUD\DUULYLQJDWWKHFDXVWLFJRHVIURPRQHZDYHIURQWWRWKHRWKHU ,QWKHIROORZLQJE\WUHDWLQJWKHPHWKRGRIPDWULFHVZHVKDOOVHHKRZWKHZDYH IURQW RI D WKLQ SHQFLO LV WUDQVIRUPHG E\ WKH UHIUDFWLRQ RU UHIOHFWLRQ RQ D VPRRWK VXUIDFH

2SWLFDOLPDJHV 2.4.1 Introduction *HRPHWULFDO 2SWLFV LV SULPDULO\ WKH VFLHQFH RI LPDJH IRUPDWLRQ DQG IRU WKLV SXUSRVHLWKDVLWVRZQMDUJRQOHWXVQRZORRNDWDQXPEHURILPSRUWDQWFRQFHSWV DQGGHILQLWLRQV ,IDOLJKWEHDPRULJLQDWHGE\DSRLQW3oFRQYHUJHVDWDSRLQW3iEHFDXVHRIUH IOH[LRQV UHIUDFWLRQV RU IRU WKH GHIOHFWLRQV XQGHUJRQH E\ LWV UD\V LQ D QRQ KRPRJHQHRXVPHGLDWKHSRLQW3iLVFDOOHGWKHoptical imageRIWKHSRLQW3o,IWKH FRQYHUJHQFHLVJHRPHWULFDOO\SHUIHFWLWLVVDLGWKDWWKHSRLQW3iLVDstigmatic imageRI3o,IWKHFRQYHUJHQFHLVQRWSHUIHFWEXWWKHHUURULVQHJOLJLEOHLWLVVDLG WKDWWKHLPDJHLVsharpLQWKLVFDVHZHFDQQRWULJRURXVO\VSHDNRIDVLQJOHLPDJH SRLQW KRZHYHU IRU VLPSOLFLW\ RI ODQJXDJH ZH ZLOO DVVXPH WKH FHQWUDO SRLQW DURXQGZKLFKWKHUD\VDUHFRQYHUJLQJWREHDQLPDJHRI3o7KHSRLQW3iLVDOVR FDOOHGWKHfocusRIFRQYHUJHQFHRIWKHUD\V7KHLPDJH3iLVDOVRVDLGWREHrealLI WKHOLJKWUD\VDFWXDOO\FRQYHUJHWRZDUGVWKHSRLQW3iZKLOHLWLVVDLGWREHvirtualLI WKHEHDPRIUD\V H[WHQGHGUHFWLOLQHDUO\ DSSHDUVWREHFRPLQJIURPRUJRLQJWR3i 7KH SRLQW 3o LV LQ WXUQ WKH LPDJH RI 3i JLYHQ WKDW UHYHUVLQJ WKH GLUHFWLRQ RI SURSDJDWLRQWKHSDWKRIWKHUD\VGRHVQRWFKDQJH7KHSRLQWV3oDQG3iDUHWKHUH IRUHVDLGWREHconjugatesRIHDFKRWKHU)LQDOO\WKHEHDPRIUD\VWKDWMRLQV DOVR YLUWXDOO\ 3oDQG3iLVVDLGWREHhomocentric7KHVLPSOHVWH[DPSOHRIFRQMXJD WLRQEHWZHHQWZRSRLQWVLVJLYHQE\WKHIRFLRIDURWDWLRQDOHOOLSVRLGDOPLUURUQRWH WKDWIRUGHILQLWLRQRIHOOLSVHDOORIWKHSDWKVFRQQHFWLQJWKHIRFL ZLWKDUHIOHFWLRQ RQWKHHOOLSVRLG DUHHTXDO7KLVSURSHUW\RIHTXDOLW\RIWKHRSWLFDOSDWKVLVJHQHUDO IRUDOOZHOOGHVLJQHGRSWLFDOLQVWUXPHQWV2QHRIWKHSRLQWVREMHFWRULPDJHFDQ DOVREHDWinfinityDQGLQWKHDERYHH[DPSOHWKHHOOLSVRLGDOPLUURUEHFRPHVSD  )RUDQDQDO\WLFDOWUHDWPHQWRIWKHFDXVWLFVDQGWKHLUGLVFXVVLRQLQWHUPVRIWKHILHOGDPSOLWXGH

VHHIRUH[DPSOH6ROLPHQRHWDO  

2SWLFDOLPDJHV 



UDERORLGDO 7KHV\VWHPRIPLUURUVOHQVHVHWFHPSOR\HGWRGHIOHFWWKHUD\VLVVDLGWREHDQ optical system)RUDJRRGRSWLFDOLQVWUXPHQWHDFKSRLQW3oRIDWKUHHGLPHQVLRQDO UHJLRQRIVSDFHVDLGWREHDQobject spaceKDVDQLPDJH3iWKDWLVDWOHDVWVKDUS 7KH VHW RI LPDJH SRLQWV GHILQHV WKH image space *HQHUDOO\ QRW DOO RI WKH UD\V HPHUJLQJIURP3oUHDFKWKHLPDJHVSDFH6RPHIRUH[DPSOHDUHH[FOXGHGE\WKH GLDSKUDJPVRIWKHLQVWUXPHQW,WLVFXVWRPDU\WRVD\WKDWWKRVHUD\VZKLFKLQVWHDG UHDFKWKHLPDJHVSDFHDUHORFDWHGLQWKHfield of the instrument :KHQ 3o GHVFULEHV D FXUYH C LQ WKH REMHFW VSDFH 3i GHVFULEHV D FRQMXJDWHG FXUYH C' 7KH WZR FXUYHV DUH QRW QHFHVVDULO\ JHRPHWULFDOO\ VLPLODU EXW LI HDFK FXUYHCLVJHRPHWULFDOO\VLPLODUWRLWVLPDJHC'ZHFDQVD\WKDWDVSDFHLVWKHperfect LPDJH RI WKH RWKHU ,Q SUDFWLFH KRZHYHU SHUIHFW LPDJHV RU HYHQ VWLJPDWLF RQHV DUH QRW REWDLQHG H[FHSW LQ VSHFLDO FDVHV 7KH LPSHUIHFWLRQ RI WKH UHDO LQ VWUXPHQWVLVQRWRQO\GXHWRWKHLUILQLVKHGRSHQLQJQDPHO\WRGLIIUDFWLRQEXWLV DOVRGXHWRWKH³JHRPHWULF´OLPLWVRIVXFKGHYLFHV,WVRKDSSHQVWKDWWKHLPDJHV DUHDIIHFWHGE\geometrical aberrationsWKDWFRQVLVWHLWKHURILQFRUUHFWVLPLODULWLHV ZLWK WKHLU FRUUHVSRQGLQJ REMHFW RU WKHLU UHGXFHG VKDUSQHVV ,Q WKH ILUVW FDVH ZH VSHDNRIdistortionDQGfield curvatureZKLOHLQWKHVHFRQGFDVHWKHDEHUUDWLRQV DUH FODVVLILHG DV spherical aberration chromatic aberration coma astigmatism DQG LQ WKHLU FRPELQDWLRQV RI KLJKHU RUGHU :KHQ WKHVH RIIHQVHVWR WKH VKDUSQHVV DUHQHJOLJLEOHLWLVVDLGWKDWWKHLQVWUXPHQWLVdiffraction limited 7KHFRQGLWLRQRIVWULFWVLPLODULW\LVJHQHUDOO\DEDQGRQHGLQWKHGHVLJQRIRSWLFDO LQVWUXPHQWV+RZHYHUWKHLQVWUXPHQWVWKDWDUHSHUIHFWLQWKHVHQVHGHILQHGDERYH DUHRIUHPDUNDEOHFRQFHSWXDOLQWHUHVWQRZ,ZLOOJLYHDEULHIUHIHUHQFHWRVRPH JHQHUDOWKHRUHPVEXWZLWKRXWUHSRUWLQJWKHLUGHPRQVWUDWLRQ)RUDGHWDLOHGGLVFXV VLRQVHH%RUQDQG:ROI  DQG+HU]EHUJHU  

2.4.2 General theorems $QRSWLFDOV\VWHPWKDWIRUPVstigmatic imagesRIDWKUHHGLPHQVLRQDOGRPDLQLV NQRZQDVDQabsolute instrumentLWZDVGHPRQVWUDWHGWKDW>0D[ZHOO@ LQDQDEVROXWHLQVWUXPHQWWKHRSWLFDOOHQJWKRIHDFKFXUYHLQDQREMHFWVSDFH LVHTXDOWRWKHRSWLFDOOHQJWKRILWVLPDJH 7KLV WKHRUHP IRUPXODWHG E\ 0D[ZHOO DQG WKHQ SHUIHFWHG E\ + %UXQV ) .OHLQ+/LHEPDQQDQGILQDOO\E\&&DUDWKpRGRU\DOVRDSSOLHVWRLVRWURSLFKHW HURJHQHRXVPHGLD)URPLWZHFDQGUDZVRPHLQWHUHVWLQJFRQFOXVLRQV,ILQIDFW DJLYHQDEVROXWHLQVWUXPHQWKDVLWVRZQREMHFWDQGLPDJHVSDFHVZLWKKRPRJHQH RXVUHIUDFWLYHLQGLFHVnoDQGniDQ\ILJXUHSODFHGLQDQREMHFWVSDFHZLOOEHGH SLFWHGZLWKDILJXUHLQWKHLPDJHVSDFHLQZKLFKDOOGLVWDQFHVDUHLQWKHUDWLRnoni  7KHVHFRQFHSWVDUHJHQHUDOL]DEOHWRLQKRPRJHQHRXVPHGLD

*HRPHWULFDO2SWLFV





n2

n1 P

P¢ r1

r2

n3

r 23c

O

n



n r n n 23 n



nn





)LJ(DFKUHDO YLUWXDO SRLQW3SODFHGDWOHIW ULJKW RIWKHOHQVKDVDVWLJPDWLFYLUWXDO UHDO LPDJH3' SODFHGRQWKHVDPHVLGH

ZLWKWKHFRUUHVSRQGLQJGLVWDQFHVLQWKHRULJLQDOILJXUH7KHUHIRUHDOORIWKHDQJOHV DUHDOVRUHSURGXFHGHTXDOO\DQGWKHILJXUHGHSLFWHGLVVLPLODUWRWKHRULJLQDORQH 7KXVKDVWKHIROORZLQJWKHRUHPRI&DUDWKHRGRU\EHHQHVWDEOLVKHG WKHSURFHVVRIUHSURGXFWLRQRIDQDEVROXWHLQVWUXPHQWEHWZHHQKRPRJHQHRXV VSDFHVLVDSURMHFWLYHWUDQVIRUPDWLRQDQLQYHUVLRQRUDFRPELQDWLRQRIWKH WZR>&DUDWKpRGRU\@ 7KHUHIRUH LIno DQG ni DUH FRQVWDQW LQ WKHLU UHVSHFWLYH VSDFHV DQ DEVROXWH LQ VWUXPHQWLVDOVRSHUIHFWLQWKHVHQVHVSHFLILHGHDUOLHUDQGKDVDPDJQLILFDWLRQUDWLR HTXDOWRnoni,QSDUWLFXODULIni no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¶V³ILVKH\H´,WFRQVLVWVLQDPHGL XPZKRVHLQGH[YDULHVZLWKDVSKHULFDOV\PPHWU\DFFRUGLQJWRWKHUDGLDOODZ 

n r

nR  r a





 

ZKHUHrLVWKHGLVWDQFHIURPDIL[HGFHQWHU2ZKLOHnRDQGaDUHFRQVWDQWV,QWKLV PHGLXP WKH UD\V DUH FLUFXODU OLQHV DQG WKH UD\V WKDW SDVV IURP D JLYHQ SRLQW 3 IRUPDV\VWHPRIFLUFOHVWKDWDOOLQWHUVHFWDWDVHFRQGSRLQW4DOLJQHGZLWK237KH GLVWDQFHVRIWKHVHWZRSRLQWVIURP2UHVSHFWWKHUHODWLRQVKLS



2SWLFDOLPDJHV 



rP rQ

a  

7KLV GHYLFH KDV WKH DGYDQWDJH RI UHSURGXFLQJ UHDO LPDJHV EXW LQ SUDFWLFH LW ³OLYHV´ RQO\ LQ ERRNV DERXW RSWLFV $QRWKHU LQWHUHVWLQJ RSWLFDO V\VWHP LV /XQH EXUJ¶VOHQVZKLFKLVFKDUDFWHUL]HGE\DQLQGH[RIUHIUDFWLRQ

n



§ r · no ¨    ¸ R ¹ © 

ZLWKLQDVSKHUHRIUDGLXVRDQGn noIRUrtRZKHUHrLVWKHGLVWDQFHIURPWKH FHQWHU$VRXUFHSODFHGRQLWVVXUIDFHJHQHUDWHVDQHOOLSWLFDOUD\V\VWHPLQVLGHRI WKHVSKHUHZKLFKDOOHPHUJHSDUDOOHOWRHDFKRWKHU ,QFRQFOXVLRQIURPWKHDERYHGHVFULSWLRQZHVHHWKDWWRJHWQRQWULYLDOUHSUH VHQWDWLRQVEHWZHHQWZRKRPRJHQHRXVVSDFHVRIHTXDOUHIUDFWLYHLQGH[ RUZLWKD GLIIHUHQWPDJQLILFDWLRQIURPnoniLIGLIIHUHQW WKHUHTXHVWIRUH[DFWDVWLJPDWLVP RUFORVHVLPLODULW\EHWZHHQWKHREMHFWDQGLPDJHPXVWEHDEDQGRQHG ,WLVQDWXUDOQRZWRDVNRXUVHOYHVZKHWKHUDWOHDVWVRPHsurfacesPD\LQDQRQ WULYLDOZD\EHUHSURGXFHGSHUIHFWO\RUDWOHDVWVKDUSO\E\DQRSWLFDOLQVWUXPHQW 7KLVDUJXPHQWKDVEHHQLQYHVWLJDWHGE\VHYHUDODXWKRUVZKRIRXQGWKDW ZKHQWKHREMHFWDQGLPDJHVSDFHVDUHKRPRJHQRXV DSDUWIURPVSHFLDOFDVHV LQ ZKLFK WKH HQWLUH YROXPH LV UHSURGXFHG VWLJPDWLFDOO\  QR PRUH WKDQ WZR REMHFW VXUIDFHV FDQ EH UHSURGXFHG VWLJPDWLFDOO\ E\ DQ RSWLFDO V\VWHP ZLWK URWDWLRQDOV\PPHWU\ 7KHFRQGLWLRQVIRUZKLFKWKLVLVSRVVLEOHDUHGHVFULEHGLQ+HU]EHUJHU  

2.4.3 Aplanatic surfaces of a refractive sphere $QLQWHUHVWLQJFDVHRFFXUVLQWKHUHSURGXFWLRQRIDVSKHULFDOVXUIDFHE\DKR PRJHQHRXVVSKHUHZLWKLQGH[nLPPHUVHGLQDKRPRJHQHRXVPHGLXPZLWKLQGH[ n':HFRQVLGHUKHUHWKHFDVHLQZKLFKn!n‫މ‬7DNHDQD[LV VD\z WKDWGLDPHWUL FDOO\LQWHUVHFWVWKHVSKHUHRIUDGLXVR$VVXPLQJWKHFHQWHU2RIWKHVSKHUHDVWKH RULJLQRIWKHD[HVFRQVLGHUWZRSRLQWV3oDQG3iSODFHGRQWKHVHPLD[LVz!ZLWK FRRUGLQDWHV zo n c n R DQG zi n n c R UHVSHFWLYHO\)LQDOO\ZHWDNHDSRLQW $RQWKHVXUIDFHRIWKHVSKHUHZLWKFRRUGLQDWHV z$ d zo DVVKRZQLQ)LJ D  7KHWZRWULDQJOHV$23oDQG$23iDUHVLPLODU,QIDFWWKH\KDYHDQDQJOHLQFRP PRQDQGWZRVLGHVZLWKWKHVDPHSURSRUWLRQ  $PRQJZKRPDUH+%RHJHKRGDQG0+HU]EHUJHU76PLWK6HHTXRWHVLQ%RUQDQG:ROI

 SDJHRU+HU]EHUJHU  

*HRPHWULFDO2SWLFV







J¢ A J

23o J¢

O Po n

J B

Pi

A

nc r n

J



z



(a)

O Pi

n r nc

23i



B J n

(b)

J¢ Po

z

n¢ 

)LJ$SODQDWLFSRLQWVRIDVSKHUH D n!n‫ މ‬E nn‫މ‬



23o  $2 $223i

nc  n 

Ö Ö Ö Ö 7KHQ 2$3 o 23i $ -  2$3i 23o $ -c )LQDOO\GUDZLQJIURP2WKHQRUPDO WRWKHVHJPHQW$3oZHVHHWKDW



VLQ -c VLQ -

$2 23o

n  nc

7KHUHIRUH WKH VLQHV RI DQJOHV - DQG -‫ މ‬DUH LQ WKH UDWLR SUHVFULEHG E\ WKH 6QHOO 'HVFDUWHVODZ,QRWKHUZRUGVDQ\UD\GLUHFWHGWRZDUGWKHSRLQW3iLQWHUVHFWLQJWKH VSKHUHDWDSRLQW$LVGHIOHFWHGWRZDUGVWKHSRLQW3o2QWKHRWKHUKDQGE\V\P PHWU\RIURWDWLRQDURXQGWKHFHQWHU2DZKROHVSKHULFDOVXUIDFHRIUDGLXVzoKDV DVYLUWXDOVWLJPDWLFLPDJHDFRQFHQWULFVSKHULFDOVXUIDFHZLWKUDGLXVzi7KHVHVXU IDFHVDUHFDOOHGDSODQDWLFVXUIDFHVRIDVSKHUH,QSDUWLFXODUWKHPDJQLILFDWLRQEH WZHHQWKHWZRVXUIDFHVLVJLYHQE\WKHsquareRIWKHUDWLREHWZHHQWKHUHIUDFWLYH LQGLFHV$VLPLODUUHDVRQLQJFDQEHFRQGXFWHGLQWKHRSSRVLWHFDVHLQZKLFKnn‫މ‬ DVVKRZQLQ)LJ E  ,W VKRXOGEH QRWHG WKDW LQ WKH FDVH FRQVLGHUHG DERYH ZKHQz$  zo - FRUUH VSRQGV WR WKH FULWLFDO DQJOH LQWHUQDO  EHWZHHQ WKH WZR PHGLD 7KH UD\V FRPLQJ IURP3oWKDWUHDFKWKHVXUIDFHRIWKHVSKHUHDWSRLQWV%IRUZKLFKz%!zoLQVWHDG HPHUJHDVUD\VWKDWGRQRWJLYHULVHWRDQ\VWLJPDWLFLPDJH 7KHSURSHUWLHVRIWKHVSKHULFDOUHIUDFWLYHVXUIDFHVVKRZQDERYHDUHXVHGLQWKH FRQVWUXFWLRQRILPPHUVLRQREMHFWLYHVRIKLJKPDJQLILFDWLRQPLFURVFRSHV

2.4.4 Aspherical surfaces /HQVHVDQGPLUURUVZLWKDVSKHULFDOVXUIDFHVRIVXLWDEOHVKDSHDUHXVHGWRFRO OHFWRUFRQFHQWUDWHUD\VRIOLJKWRQPXFKZLGHUDQJXODUILHOGVWKDQWKRVHJHQHUDOO\ DYDLODEOH ZLWK RQO\ VSKHULFDO VXUIDFHV ZLWK VLPLODU DEHUUDWLRQ HQWLWLHV ,Q UHIOHF WLRQWKHVHVXUIDFHVFDQKDYHWKHIDPLOLDOIRUPV SDUDERORLGVHOOLSVRLGDQGK\SHU



2SWLFDOLPDJHV 

0,55256

FRQFDYH

FRQYH[

paraboloid  

D

F

f

f

f

hyperboloid D    a  DC

F1

F2

g

g

C

F 

  D  D

C

F2 

g

g

a

F1

a

g

ellipsoid

  D   o SURODWH



F1

F2

a

g

g

g

a

 DC



  D  D

F1

F2

D  o VSKHUH 

C

g



g



a

)LJ   $VSKHULFDO VSHFXODU VXUIDFHV ZLWK V\PPHWU\ RI UHYROXWLRQ DURXQG DQ D[LV )RU VLPSOLFLW\WKHTXDQWLWLHVfgaDUHDOOWDNHQSRVLWLYH

ERORLGV OHDUQHGLQWKHVWXG\RIJHRPHWU\WKDWFDQEHGHVFULEHGE\WKHIRUPXOD 

z

 C x   y   Dz   





 

ZKHUHCLVWKHFXUYDWXUHDWWKHSRLQW  ZKHUHWKHVXUIDFHLQWHUVHFWVWKHz D[LVZKLOHDGHWHUPLQHVWKHDVSKHULFLW\6XUIDFHVRIWKLVW\SHZLWKWKHH[FOXVLRQ RIoblateHOOLSVRLGVIRUZKLFKD!SRVVHVVWZRfoci RQHFDQEHDWLQILQLW\ LH WZRSRLQWV))RQHDFKRIZKLFKWKHVXUIDFHFDQSURGXFHE\UHIOHFWLRQWKHVWLJ PDWLFLPDJHRIWKHRWKHU )LJ 7KLVSURSHUW\FDQDOVREHREWDLQHGLQUHIUDF WLRQ EHWZHHQ WZR KRPRJHQHRXV PHGLD ,Q ERWK FDVHV RQH ZD\ WR GHWHUPLQH WKH VXUIDFHWKDWSURGXFHVWKHVWLJPDWLFLPDJHRI)LQ)LVWRDSSO\WKHFRQVWDQWRSWL FDOSDWKUXOH

*HRPHWULFDO2SWLFV





DIOPTRES n1 < n 2

concave

convex

ellipsoid

n1 n 2

F2

a=

f =

f

0

n2 2 - n12

F2

n1 n 2

n2 2

n2

1

n2 - n1 C

f

0

hyperboloid

n1 n 2

F1

a=

f =

0

f

n12 - n2 2 n12

n1

F1

n1 n2

1

n2 - n1 C

f

0



)LJ$VSKHULFDOUHIUDFWLYHVXUIDFHV7KHVLJQRIfLVKHUHWDNHQDVSRVLWLYHRUQHJDWLYH GHSHQGLQJRQZKHWKHUWKH dioptreLVFRQYHUJHQWRUGLYHUJHQWWKHSRVLWLYHGLUHFWLRQLVLQGL FDWHGE\WKHDUURZ CLVWDNHQDVSRVLWLYHLIWKHVXUIDFHKDVLWVFHQWHURQWKHULJKW DVQHJD WLYHLQWKHRSSRVLWHFDVH 



n )2  n )2 FRVW 

 

ZKHUH2LVDSRLQWRIWKHVXUIDFHDQGnnDUHWKHUHIUDFWLYHLQGLFHVWKDWDUHDV VLJQHGWRWKHWZRVHJPHQWV)2DQG)2UHVSHFWLYHO\,QUHIOHFWLRQIRUZKLFKn nWKHGHYHORSPHQWRI(T  OHDGVWR(T  ,QVWHDGLQUHIUDFWLRQVR WKDW nz n WKH GHYHORSPHQW RI(T   OHDGV WR D YHU\ FRPSOLFDWHG H[SUHV VLRQZLWKWHUPVRIqJUDGHLQWKHFRRUGLQDWHVWKDWGHVFULEHVDVXUIDFHFDOOHGWKH oval of Descartes ,IRQHRIWKHSRLQWVLVDWLQILQLW\WKHRYDORI'HVFDUWHVLVUHGXFHGWRDQHOOLSVRLG RU D K\SHUERORLG )LJ   7KH UHIUDFWLYH VXUIDFHV RI JHQHULF IRUP ZLWK URWD WLRQDOV\PPHWU\DURXQGDQD[LVDUHFDOOHGdioptres7ZRRIWKHVHVXUIDFHVFDQ EHFRPELQHGWRJHWKHUWRIRUPDOHQVWKDWFDQSURGXFHDQLPDJHRIDVWLJPDWLFRE MHFWSRLQW )LJ $VSKHULFDOOHQVHVRIPRUHFRPSOLFDWHGVKDSHDUHDOVRVRPH WLPHVXVHGWRFRPSHQVDWH DQGWKXVWRFRUUHFW WKHDEHUUDWLRQVLQWURGXFHGE\WKH RWKHUHOHPHQWVRIDQRSWLFDOV\VWHP$IDPRXVH[DPSOHLVWKH6FKPLGWWHOHVFRSH LQZKLFKDFRUUHFWRUSODWHLVFRXSOHGZLWKDVSKHULFDOPLUURUFRPSHQVDWLQJIRUWKH VSKHULFDODEHUUDWLRQLQDQH[FHOOHQWZD\  DioptreLVD)UHQFKZRUGdiottroLQ,WDOLDQ)URP*UHHNGLµWKURXJK¶RSWRVµYLVLEOH¶



,GHDOLPDJHV 

F1

F2 double hyperboloid

F1 hyperboloid - plane

F2

F1 sphere - ellipsoid

plane - hyperboloid



)LJ(OHPHQWDU\DVSKHULFDOOHQVHV

$OWKRXJK WKH DVSKHULFDO VXUIDFHV PD\ FDUU\ RXW FHUWDLQ WDVNV LQ DQ H[FHOOHQW PDQQHUWKH\DUHGLIILFXOWWRPDQXIDFWXUHZLWKDTXDOLW\WKDWDSSURDFKHVWKDWRIWKH VSKHULFDO VXUIDFHV ZKRVH SURGXFWLRQ WHFKQLTXH LV LQKHUHQWO\ VLPSOH DQG ZHOO HV WDEOLVKHG +RZHYHUDQHZJHQHUDWLRQRIGLDPRQGFXWWLQJPDFKLQHVKDVGHYHORSHGFRQ WUROOHGE\FRPSXWHUZKLFKPDNHLWSRVVLEOHWRDFKLHYHWKHSUHFLVLRQDQGTXDOLW\ UHTXLUHG LQ RSWLFDO LQVWUXPHQWV ,Q SDUWLFXODU LW KDV EHFRPH FRPPRQ WR XVH WKH JRRGREMHFWLYHVRIFDPHUDVFRQWDLQLQJDVSKHULFDOOHQVHVZKRVHPDMRUFRVWLVRII VHW E\ WKH UHGXFWLRQ RI RSWLFDO HOHPHQWV WKDW DUH UHTXLUHG WR FRUUHFW DEHUUDWLRQV )XUWKHUPRUHWKHXVHRIDVSKHULFVXUIDFHVKDVEHHQDVVHUWHGHYHQIRUWKHRSKWKDO PLFOHQVHVZKLFKEHLQJFRQVWLWXWHGE\DVLQJOHRSWLFDOHOHPHQWPXVWFRUUHFWYLV XDOGHIHFWVZKLOHPLQLPL]LQJWKHLQWURGXFWLRQRIRWKHUDEHUUDWLRQV 6LQJOHDVSKHULFDOOHQVHVDUHQRZXVHGDVFROOLPDWRUVRIWKHUDGLDWLRQHPLWWHGE\ ODVHUGLRGHVDQGIRUFRXSOLQJWKHODVHUEHDPLQRSWLFDOILEHUV3UHYLRXVO\WKHXVH RIDVSKHULFDOOHQVHVRUPLUURUVZDVOLPLWHGWRFDVHVLQZKLFKWKHUHZDVQRFODVVLF DOWHUQDWLYHVROXWLRQDVLQDVWURQRPLFDOLQVWUXPHQWVRUWRV\VWHPVWKDWFROOHFWDQG FROOLPDWHWKHOLJKWRIODPSVRUILQDOO\ZKHQDPDVVSURGXFWLRQUHGXFHVWKHLUVLQ JOHFRVW2QHFDVHRIWKHODWWHULVSODVWLFOHQVHVSURGXFHGE\PROGLQJDQGXVHGLQ FHUWDLQORZFRVWFDPHUDV

,GHDOLPDJHV 2.5.1 The homography :H VDZ HDUOLHU WKDW D SHUIHFW UHSUHVHQWDWLRQ RI KRPRJHQHRXV VSDWLDO GRPDLQV PXVWQHFHVVDULO\EHDSURMHFWLYHWUDQVIRUPDWLRQRIRUGLQDU\W\SH+RZHYHUDEDQ

*HRPHWULFDO2SWLFV  GRQLQJ WKHUHTXHVW WR KDYH SHUIHFW LPDJHV LQ WKUHHGLPHQVLRQVRQH PD\ ZRQGHU ZKDWDQidealOLPLWUHDVRQDEO\DSSURDFKDEOHZLWKUHDOLQVWUXPHQWVFRXOGEH &RQVLGHU KHUH IRU VLPSOLFLW\ RQO\ WKH FDVH RI V\VWHPV ZLWK URWDWLRQDO V\P PHWU\DURXQGDQoptical axis VD\WKHzD[LV 7KLVLVDFDVHRIVSHFLDOLPSRUWDQFH LQ RSWLFV VLQFH WKH PDMRULW\ RI RSWLFDO V\VWHPV FRQVLVW RI VXUIDFHV RI UHYROXWLRQ DURXQG D FRPPRQ D[LV VR WKH\ DUH FDOOHG centered systems 7KLV V\PPHWU\ LP SOLHVWZRXVHIXOFRQVHTXHQFHV7KHILUVWLVWKDWWKHLPDJHRIDQ\SRLQW3oPXVWEH RQWKHmeridian planeZKLFKFRQWDLQVWKHRSWLFDOD[LVDQG3o7KHVHFRQGFRQVH TXHQFHLVWKDWZKDWKDSSHQVWRWKHSRLQWVRIDJLYHQPHULGLDQSODQH VD\WKHyz SODQH PXVWDOVRRFFXUIRUDOORIWKHRWKHUSRLQWV 0D[ZHOOSURSRVHGWKHFRQFHSWRIDQideal systemWKDWOHDYLQJDVLGHWKHSK\VL FDOPHFKDQLVPHPSOR\HGDFKLHYHVWKHIROORZLQJWUDQVIRUPDWLRQ>0D[ZHOO @ 

(DFKILJXUHSODFHGRQDQREMHFWSODQH xoyoQRUPDOWRWKHzD[LV ZLWKz zo IL[HG LVUHSURGXFHGVKDUSO\LQDJHRPHWULFDOO\VLPLODUILJXUHRQDQLPDJH SODQHxiyi ZLWKz ziIL[HG 7KHFRQVWDQWRISURSRUWLRQDOLW\LVLQVWHDGOHIW DVDIXQFWLRQRIWKHQDWXUHRIWKHV\VWHPDQGWKHSRVLWLRQRIWKHWZRSODQHV $VVXPHWKHQWKDWWKHREMHFWDQGLPDJHVSDFHV HYHQLIYLUWXDO DUHKRPRJHQH RXVVLQFHZHZDQWWRDFKLHYHWKHWUDQVIRUPDWLRQZLWKUD\VRIOLJKWRQHPXVWKDYH WKDW WKHSRLQWVRIHDFK VWUDLJKW UD\SODFHGLQWKHREMHFWVSDFHDUHUHSURGXFHGRQ DVWUDLJKWOLQHLQWKHLPDJHVSDFHDQGWKHQIRUPDQHZUD\ ,WKDSSHQVDQGZHZLOOVRRQFKHFNWKLVWKDWWKHVHFRQGLWLRQVDUHIXOILOOHGE\DQ D[LDOSURMHFWLYHWUDQVIRUPDWLRQH[SUHVVHGE\WKHHTXDWLRQV 

xi

a xo  bzo  c

yi

a yo  zi bzo  c

dzo  e   bzo  c

 

ZKLFKLVDOVRFDOOHGcollineationRUhomography>VHHWKHEHDXWLIXOERRNE\*LQR *LRWWL@7KHUHabcdeDUHFRQVWDQWVFKDUDFWHULVWLFRIWKHV\VWHPJLYHQ WKDWLQWKHFRHIILFLHQWVRQO\WKHUDWLRFRXQWVRQHFRXOGEHHOLPLQDWHGKRZHYHU WKLVQRWDWLRQZLOOEHFRQYHQLHQWODWHU (TV   DUH QRW OLQHDU LQ WKH z FRRUGLQDWHV DQG WKH LPDJH RI D WKUHH GLPHQVLRQDO REMHFW LV GHIRUPHG FRPSDUHG WR WKH RULJLQDO ,W UHPDLQV UHPDUNDEOH WKDWHDFKVWUDLJKWOLQHLVUHDOO\WUDQVIRUPHGLQWRDVWUDLJKWOLQHDVZHZLOOVRRQVHH EXWEHZDUHRIFHUWDLQVLQJXODULW\ ,WLVYHU\LPSRUWDQWWRVWXG\WKLV0D[ZHOOLDQ LGHDORIWKHIRUPDWLRQRILPDJHVVLQFHWKHDFWXDORSWLFDOV\VWHPVFDQDSSURDFKLW SUHWW\ ZHOO DW OHDVW IRU D QDUURZ VSDFH UHJLRQ DURXQG WKHRSWLF D[LV /HW XV WKXV H[DPLQHLQGHWDLOWKHSURSHUWLHVRI(TV  ZKLFKODWHUZLOOEHXVHIXOLQWKH VWXG\RIWKHVRFDOOHGparaxial approximation

,GHDOLPDJHV





2.5.2 Conventions on signs %HIRUHVWDUWLQJLWVKRXOGEHQRWHGWKDWWKHUHDUHHVVHQWLDOO\WZRRSSRVLWHFRQ YHQWLRQV ZLWK UHJDUG WR WKH zD[LV SRVLWLYH GLUHFWLRQ IRU WKH REMHFW VSDFH )DFHG ZLWKWKLVGLVDJUHHPHQWWKHYDULRXVDXWKRUVXVXDOO\WDNHDSRVLWLRQLQIDYRURIRQH RIWKHVHFKRLFHVERWKRIZKLFKKDYHVWUHQJWKVDQGZHDNQHVVHV 7KHILUVWLVWKH³FODVVLF´FRQYHQWLRQLQZKLFKWKHGLVWDQFHVDUHWDNHQDVSRVL WLYHPRYLQJDZD\IURPUHIUDFWLQJRUUHIOHFWLQJLQWHUIDFHV,WKDVWKHDGYDQWDJHRI FRQVLGHULQJ WKH REMHFW VSDFH DV specularly symmetric ZLWK UHVSHFW WR WKH LPDJH VSDFHPDNLQJWKHIRFDOGLVWDQFHVfoDQGfiUHVSHFWLYHO\DVVLJQHGWRWKHV\VWHPIRU WKHVHVSDFHVDJUHHLQVLJQ7KLVLVWKHFRQYHQWLRQWKDWLVIRXQGLQWKHPRVWEDVLF RSWLFDO PDQXDOV >+HFKW  -HQNLQV DQG :KLWH @ LQFOXGLQJ WKH H[FHOOHQW Optics Guide E\ 0HOOHV *ULRW ,Q IDFW WKH VSHFXODU V\PPHWU\ DVVXPHG EHWZHHQ REMHFWDQGLPDJHVSDFHVIDFLOLWDWHVWKHXQGHUVWDQGLQJRIWKHEHKDYLRURIWKHRSWLFDO V\VWHPV,WVIODZLVWKDWWKLVV\PPHWU\LVQRWWUXHIRUDOORIWKHYDULDEOHV,QIDFW WKHUDGLLRIFXUYDWXUHRIWKHRSWLFDOVXUIDFHVQRWEHLQJUHDVRQDEO\DWWULEXWDEOHWR RQHRUWKHRWKHURIWKHVSDFHVDUHWDNHQDVSRVLWLYHLIWKHFHQWHUVRIFXUYDWXUHDUH LQWKH³ULJKW´RIWKHUHVSHFWLYHVXUIDFH%\³ULJKW´ZHPHDQWKHGLUHFWLRQWRZDUG ZKLFKWKHOLJKWWUDYHOVDXVDJHERUQRIWKHIDFWWKDWZH:HVWHUQHUVZULWHIURPOHIW WR ULJKW,Q DGGLWLRQZKHQ DQDO\]LQJ LQVWUXPHQWV FRPSRVHG RIYDULRXVHOHPHQWV ZHULVNJHWWLQJORVWLQD%DEHORISDULW\WUDQVIRUPDWLRQV 7KHVHFRQGLVWKH³PRGHUQ´FRQYHQWLRQXVHGLQWKHGHVLJQRIRSWLFDOV\VWHPV DQGLQUD\WUDFLQJ7KLVFRQYHQWLRQKDVWKHDGYDQWDJHRIDVVLJQLQJWKHVDPHSRVL WLYHGLUHFWLRQIRUDOOD[HVZLQWKHVSDFHVWKDWIROORZHDFKRWKHULQDQRSWLFDOV\V WHPPDGHXSRQO\RIUHIUDFWLYHVXUIDFHVZLWKURWDWLRQDOV\PPHWU\DERXWDQD[LVz +RZHYHUWKLVXQLW\RIVLJQLVLOOXVRU\ZKHQGHDOLQJZLWKFRPSOH[RSWLFDOV\VWHPV LQFOXGLQJ WKH QRQSODQDU FRQWDLQLQJ SULVPV RU PLUURUV )XUWKHUPRUH WKH YDULRXV DXWKRUVZKRXVHWKLVFRQYHQWLRQDUHRIWHQUHWLFHQWWRGHILQHWKHYDULDEOHVZHOODQG GRQRWDJUHHRQHYHU\WKLQJ$GHIHFWRIWKLVFRQYHQWLRQLVWKDWIRUH[DPSOHIRUD converging OHQV LW DVVLJQV D negative YDOXH WR WKH IRFDO GLVWDQFH LQ WKH LPDJH VSDFH >%RUQ DQG:ROI  6LYRXNKLQH@ RU WR WKH IRFDO GLVWDQFH LQ REMHFW VSDFH>/DQGVEHUJ*XHQWKHU@$QRWKHUGHEDWDEOHSRLQWLVWKDWDOPRVWDOO RI WKH ³PRGHUQ´ DXWKRUV WUHDW UHIOHFWLRQ E\ DVVLJQLQJ Dnegative UHIUDFWLYH LQGH[ DVVRFLDWHG ZLWK WKH LPDJH VSDFH (YHQ LI D geometrical SUREOHP LV DSSDUHQWO\ VROYHG,EHOLHYHWKDWWKLVFKRLFHLVDQXQDFFHSWDEOHIRUFLQJRIDQHVVHQWLDOphysical SURSHUW\RIWKHPHGLXP +HUH,ZRXOGOLNHWROHDYHWKHFKRLFHWRWKHUHDGHUZULWLQJWKHYDULRXVH[SUHV VLRQVLQDIRUPWKDWUHPDLQVYDOLGLQERWKFDVHV)RUWKLVSXUSRVH,LQWURGXFHDQad hocV\PERO;ZKLFKKDVWKHYDOXHRIDRU±GHSHQGLQJRQWKHSUHIHUUHGFRQYHQ WLRQHYHQLIWKHQRWDWLRQLVDOLWWOHFRPSOLFDWHGDQGGRHVQRWVROYHDOORIWKHGLV FUHSDQFLHVEHWZHHQWKHYDULRXVDXWKRUV7KXVLWLV ;   IRU WKH FODVVLFDO FRQYHQWLRQ RI VSHFXODU V\PPHWU\ EHWZHHQ WKH REMHFW DQGLPDJHVSDFHV

*HRPHWULFDO2SWLFV  ; IRUWKHPRGHUQFRQYHQWLRQLQZKLFKWKDWRIDcentralRUprincipal ray RU chief ray LVWDNHQDVWKHzD[LVQRWQHFHVVDULO\FRLQFLGHQWZLWKDVWUDLJKWOLQHUH IUDFWHGDQGUHIOHFWHGIURPWKHVXUIDFHVRIWKHRSWLFDOV\VWHPJHQHUDOO\QHLWKHUD[L DOQRUSODQDU$VZLWKWKHSRVLWLYHGLUHFWLRQWKHRQHWKDWWKHOLJKWIROORZVLQHDFK VHFWLRQ DORQJ WKLV UD\ LV FKRVHQ )RU WKH IRFDO GLVWDQFHV , IROORZ WKH PLQRULW\ FKRLFHRI/DQGVEHUJ



2QWKHVLJQVRIYDULDEOHVZHZLOOVHHDVXPPDU\DWWKHHQGRI†$PRUHGH WDLOHGGLVFXVVLRQRIWKHVHFDVHVZLOOEHJLYHQODWHULQWKHWUHDWPHQWRIWKHPHWKRG RI[PDWULFHV

2.5.3 Non telescopic homography )RFDOSODQHV

7KHSODQHzo ± cbIRUZKLFKWKHUHLVDGLYHUJHQFHLQ(TV  LVFDOOHG WKHfocal planeRIWKHREMHFWVSDFHDQGDVLPLODUSODQHIRUZKLFKzi dbLVDOVR SUHVHQWIRUWKHLPDJHVSDFH,QWKHVSHFLDOFDVHWKDWLVb ERWKSODQHVDUHORFDW HGDWLQILQLW\WKHQWKHWUDQVIRUPDWLRQLVFDOOHGaffineRUtelescopic7KLVFDVHZLOO EHGLVFXVVHGODWHU7KHSRLQWV)oDQG)iZLWKFRRUGLQDWHV zo ± cb DQG  zi db UHVSHFWLYHO\DUHFDOOHGprincipal foci &RQMXJDWHSRLQWV

)RU QRQWHOHVFRSLF V\VWHPV ZLWK b z   LW VRRQ DSSHDUV HYLGHQW WKDW (TV  FDQEHFRQYHQLHQWO\VLPSOLILHGE\FKRRVLQJWZRGLVWLQFWUHIHUHQFHV\VWHPV IRU WKH REMHFW DQG LPDJH VSDFHV WR EH FHQWHUHG LQ WKHLU UHVSHFWLYH SULQFLSDO IRFL :HWKHQDSSO\WKHUHSODFHPHQWV 

Zo

c· § ; ¨ zo  ¸  Zi b¹ ©

zi 

d  b

7KHWZRQHZFRRUGLQDWHVIRUWKHzD[LVDUHDOVRLQGLFDWHGZLWKFDSLWDOOHWWHUZVLQ RUGHUWRKLJKOLJKWWKDWWKHLUD[HVLQDGGLWLRQWRKDYLQJWUDQVODWHGRULJLQVRIWHQGR QRW³SK\VLFDOO\´DJUHHDVWRGLUHFWLRQ,QDGGLWLRQ,LQWURGXFHWKHfocal lengths 

fo

;

a b

fi

cd  be  ab

 

7KHV\PERO;LVZRUWKRU±DFFRUGLQJWRWKHFKRVHQFRQYHQWLRQ,WVLQWURGXF WLRQLVHTXLYDOHQWWRFKDQJLQJWKHVLJQRIWKHYDULDEOHVZofoDQGso ZKLFKZLOOEH LQWURGXFHGODWHULQWKHWH[W 6XEVWLWXWLQJWKHQHZYDULDEOHV(TV  XOWLPDWHO\ EHFRPH



,GHDOLPDJHV 

y

O

y = yo

straight line X Zo

y

P¢i I Pi fi

Fo 0

straight line y =

Fi 0 yo Z fi i

Zi

 )LJ   7UDQVIRUPDWLRQ RI D VWUDLJKW OLQH SDUDOOHO WR WKH RSWLFDO D[LV 7KH WZR VWUDLJKW OLQHVODQGIDUHHDFKWKHLPDJHRIWKHRWKHUEXWWKHSRLQW)i ZLWKZi  KDVDVFRQMXJDW HGWKHSRLQWVDWZo rf6LPLODUO\IRU ZoorWKHSRLQWVDW Ziorf7KHHPSW\DUURZ LQGLFDWHVWKHSRVLWLYHGLUHFWLRQIRU Zo ZKHQ ; ZKLFKUHYHUVHV ZKHQ ; 





xi

fo xo  Zo

yi



fo yo  Zi Zo

fi fo  Zo

 

7RFODULI\WKHZD\LQZKLFKWKHWUDQVIRUPDWLRQ  DFWVZHFRQVLGHUWKHSRLQWV RIDVWUDLJKWOLQHREMHFWOSDUDOOHOWRWKHRSWLFDOD[LVZLWKFRRUGLQDWHVx y yo )LJ :LWK(TV  ZHKDYH 

yi



fo yo Zo



Zi yo  fi

7KHUHIRUHWKHVHSRLQWVDUHWUDQVIRUPHGLQWRDQHZVWUDLJKWOLQHIWKDWLQWHUVHFWV WKHRSWLFDOD[LVDW)i 7KHWZROLQHVDUHHDFKWKHLPDJHRIWKHRWKHUEXWDWWHQWLRQ PXVWEHSDLGWKHSRLQWVRIILQWKHQHLJKERUKRRGRI)iDUHFRQMXJDWHWRWKHSRLQWV RI O LQ WKH QHLJKERUKRRG RI Zo  rf 6LPLODUO\ WKH SRLQWV RI O ZLWK Zo LQ WKH QHLJKERUKRRG RI  WKDW LV DURXQG WKH REMHFW IRFDO SODQH DUH FRQMXJDWHG ZLWK SRLQWVRIILQWKHQHLJKERUKRRGRIZi rf ,Q DGGLWLRQ DQ\ VWUDLJKW OLQH LQ WKH SODQH yZ H[SUHVVHG E\ HTXDWLRQ yo  A  ;BZoLVWUDQVIRUPHGLQWRDVWUDLJKWOLQH,QGHHGLWVLPDJHKDVFRRUGLQDWHV 

yi



fo A  ;f o B Zo



A Z  ;fo B  fi i

,QSDUWLFXODUDWZi WKHVWUDLJKWOLQHLQWHUVHFWVWKHLPDJHIRFDOSODQHLPDJH LQyi ±;foBLUUHVSHFWLYHRIWKHYDOXHRIA)LQDOO\IRUWKHLQGHSHQGHQFHLQWKH WUDQVIRUPDWLRQ EHWZHHQ WKH FRRUGLQDWHV x DQG y LW DSSHDUV WKDW DQ\ VWUDLJKWOLQH REMHFWLVFRQMXJDWHGZLWKDVWUDLJKWOLQHLPDJH,QDGGLWLRQDOORIWKLVLVLQGLFDWLYH RIDQLPSRUWDQWIDFWDQ\REMHFWEHDPRISDUDOOHOUD\VDPRQJWKHPLVWUDQVIRUPHG GHIOHFWHG LQDFRQYHUJLQJEHDPRIUD\V³IRFXVHG´LQWRDVLQJOHSRLQWRQWKHIRFDO LPDJHSODQHDQGWKLVSRLQWFRLQFLGHVZLWK)iLIWKHLQFLGHQWEHDPLVSDUDOOHOWRWKH RSWLFDOD[LV

*HRPHWULFDO2SWLFV



 &RQMXJDWHGSODQHV 7KHWKLUG(T  FDQEHUHZULWWHQDV 

Zi Zo

fi fo 

 

ZKLFK LV WKH Huygens-Newton equation ,W HVWDEOLVKHV WKH UHODWLYH SRVLWLRQ DV VXPHGE\WZRFRQMXJDWHSODQHVREMHFWDQGLPDJH

/DWHUDODQGORQJLWXGLQDOPDJQLILFDWLRQ

7KHlateral magnificationEHWZHHQILJXUHVREMHFWDQGLPDJHHDFKSODFHGRQLWV FRQMXJDWHSODQH IRUZKLFKWKLVPDJQLILFDWLRQLVDOVRFDOOHGWKHconjugate ratio LV HDVLO\REWDLQHGIURP(TV   

mA

xi xo

yi yo



fo Zo



Zi  fi

 

7KLVPDJQLILFDWLRQYDULHVZLWKWKHSRVLWLRQRIWKHFRQMXJDWHGSODQHVLQSDUWLFXODU LWLVSURSRUWLRQDOWRZiDQGWKHGLYHUJHQFHDWZo FRUUHVSRQGVWRDQLPDJHLQZi  “f 7KHlongitudinal magnificationLVDPHDVXUHRIKRZDYDULDWLRQRIZoLVUHIOHFW HGLQZi'LIIHUHQWLDWLQJ(T F LWLVGHILQHGDV 

m&

wzi wzo

;

wZi wZo

f f ; i o Zo

;

Zi  Zo

 

ZKHUHWKHVLJQRIm)LVFKRVHQDFFRUGLQJWRWKHWUDQVIRUPDWLRQ  

3ULQFLSDOSODQHV

:KHQZo ± foLWLVVLPXOWDQHRXVO\WUXHWKDWZi ± fiDQGWKHODWHUDOPDJQLILFD WLRQ EHWZHHQ WKH FRQMXJDWHG SODQHV LV XQLWDU\ 7KHVH SDUWLFXODU SODQHV DUH FDOOHG principalRUunitary planesDQGDUHSODFHGDWDIRFDOGLVWDQFHIURPWKHFRUUHVSRQG LQJIRFDOSODQHV$ILJXUHSRVHGRQRQHRIWKHPLVUHSURGXFHGLGHQWLFDOO\RQWKH RWKHUSODQH7KHLQWHUVHFWLRQRIWKHSULQFLSDOSODQHVZLWKWKHRSWLFDOD[LVJLYHVULVH WR WZR SRLQWV 3o DQG 3i ZKLFK DUH FDOOHG principal points 7KH\ FRQVWLWXWH WKH RULJLQRIWKHRWKHUWZRD[LVV\VWHPVLQZKLFKWKHORQJLWXGLQDOGLVWDQFHVDUHPHDV XUHG VWDUWLQJ IURP WKH SULQFLSDO SODQHV UDWKHU WKDQ IURP WKH IRFDO SODQHV 7KLV PHDQVWKDWLQVWHDGRIWKHYDULDEOHVZoDQGZiRQHXVHVWKHYDULDEOHV



,GHDOLPDJHV 

P¢i

P¢o

Po

Fo

Fi

Pi

X fo

fi

 )LJ *UDSKLFSURFHGXUHIRUIL[LQJWKHUHODWLYHSRVLWLRQVRIIRFLDQGSULQFLSDOSODQHV 7KHDUURZVDVVRFLDWHGZLWKWKHIRFDOGLVWDQFHVfoDQGfiLQGLFDWHWKHGLUHFWLRQLQZKLFKWKH\ DUHFRQYHQWLRQDOO\WDNHQSRVLWLYH ZKHQ ; 



so

Zo  fo  si

Zi  fi 

 

,QSDUWLFXODUWKH+X\JHQV1HZWRQHTXDWLRQWDNHVWKHIRUP 

fo fi  so si

 

 

ZKLFKLVFDOOHGWKHGauss equation,QDGGLWLRQWKHFRQMXJDWHUDWLREHFRPHV 

mA



si fo  so fi

 

&RPSDUHWKLVH[SUHVVLRQZLWK(T   7KLVD[LVV\VWHP LVSUREDEO\PRUHIDPLOLDUWKDQWKDWRI+X\JHQVDQG1HZWRQ EDVHGRQIRFDOSODQHV+RZHYHU,FDQDVVXUH\RXIURPH[SHULHQFHWKDWWKH*DXVVL DQIRUPLQWKHFDOFXODWLRQVFDQOHDGWRFRQYROXWHGDQGLQFRPSUHKHQVLEOHH[SUHV VLRQVZKLOHWKH+X\JHQV1HZWRQIRUPOHDGVPRUHHDVLO\WRHOHJDQWH[SUHVVLRQV ULFKLQSK\VLFDOPHDQLQJ7KHUHDVRQIRUWKLVOLHVLQWKHIDFWWKDWWKHVLQJXODULWLHV H[LVWLQJRQIRFDOSODQHVDUHKLGGHQZKHQXVLQJWKHSULQFLSDOSODQHVDVDUHIHUHQFH

&DUGLQDOSRLQWVDQGJUDSKLFDOFRQVWUXFWLRQV

)RUDQLGHDOV\VWHPWKHSRVLWLRQRIWKHSULQFLSDOSRLQWV3oDQG3iFDQDOVREH GHILQHGZLWKDJUDSKLFDOO\HTXLYDOHQWPHWKRG&RQVLGHUDVLQ)LJDQREMHFW UD\SDUDOOHOWRWKHRSWLFDOD[LVODQGH[WHQGLWLQWKHLPDJHVSDFHXQWLOLWLQWHUVHFWV LWVRZQLPDJHVWUDLJKWOLQH I7KHSRLQW 3ic LVWKHRQO\SRLQWRI IZLWKWKHVDPH WUDQVYHUVHFRRUGLQDWHV xy RILWVFRQMXJDWH 3oc ZKHUHYHULWLVLQWKHREMHFWVSDFH $VLPLODUFRQVWUXFWLRQFDQDOVREHXVHGWRIL[WKHSRVLWLRQRIWKHREMHFWSULQFLSDO  %RUQDQG:ROI  ZULWHfosofisi VLQFHWKH\XVHVRSSRVLWHVLJQVIRUfoDQGfi

*HRPHWULFDO2SWLFV





Ao Fi Fo

Pi

Po

Ai

 )LJ*UDSKLFGHWHUPLQDWLRQRIDQLPDJHSRLQW

SODQH )LJ  %\GHILQLWLRQ)oDQG3oDUHDOZD\VSRLQWVLQWKHREMHFWVSDFHDQG)iDQG3iDUH DOZD\V SRLQWVRI WKH LPDJH VSDFH +RZHYHU LW PD\ KDSSHQIRU H[DPSOH WKDW)i DQG3iDUHSK\VLFDOO\RQWKHLQSXWVLGHRIWKHRSWLFDOV\VWHPWKHRQHIURPZKLFK WKHREMHFWUD\VRULJLQDWHDQGWKH\DUHWKHUHIRUHYLUWXDO 7KH IRXU SRLQWV )o 3o )i DQG 3i SODFHG DORQJ WKH RSWLFDO D[LV IXOO\ VHW WKH SURSHUWLHVRIDQLGHDORSWLFDOV\VWHP7KH\FDQEHXVHGWRGHWHUPLQHWKHLPDJHRI DQ\ REMHFW E\ PHDQV RI WKH IDPLOLDU JUDSKLF FRQVWUXFWLRQ XVHG WR VWXG\ WKH IRU PDWLRQRILPDJHVZLWKOHQVHV)LJJLYHVXVDQH[KDXVWLYHH[DPSOHRILWQRWH WKDWLWLVHQRXJKWRFRQVLGHURQO\UD\VSDUDOOHOWRWKHRSWLFDOD[LVDQGWKHLUWUDQV IRUPHGYHUVLRQVZKLFKLQWHUVHFWWKHD[LVLQ)oDQG)i,WVKRXOGEHQRWHGWKDWWKHUH LVDVPDOOLQFRQVLVWHQF\LQWKHQRWDWLRQXVHGKHUH:LWKWKHLQGLFHVoDQGiWKHRE MHFWSRLQWVoDUHJHQHUDOO\GLVWLQJXLVKHGE\WKHLUFRQMXJDWHLPDJHiEXW)oDQG)i DUHDQH[FHSWLRQVLQFHthey are notHDFKWKHFRQMXJDWHRIWKHRWKHU+RZHYHUD GLIIHUHQWZD\WRLQGLFDWHWKHIRFDOSRLQWVPD\EHLQDSSURSULDWH 2Q WKH RSWLFDO D[LV ZH FDQ XVHIXOO\ GHILQH D WKLUG SDLU RI SRLQWV 1o DQG 1i FDOOHGnodal points7KH\DUHVXFKWKDW $Q\UD\WKDWUHDFKHV1oHPHUJHVIURP1iSDUDOOHOWRWKHLQLWLDOGLUHFWLRQ )LJ VKRZV WKHLUJUDSKLFDO GHWHUPLQDWLRQ VWDUWLQJ IURP WKH RWKHU SRLQWV )o3o)iDQG3i$UD\rLVGUDZQIURP)o,WLVWUDQVIRUPHGLQWRDUD\SDUDOOHOWR WKHRSWLFDOD[LVDQGUHDFKHVWKHIRFDOSODQHLPDJHLQ )ic $VHFRQGUD\ rLVWKHQ ODXQFKHG IURP )ic  VR WKDW LW LV SDUDOOHO WR r ,Q WKH REMHFW VSDFH WKH UDGLXV r PXVWHPHUJHDVVWLOOSDUDOOHOWRrVLQFHrDQGrLQWHUVHFWWKHIRFDOSODQHLPDJH DWWKHVDPHSRLQW )ic 7KHLQWHUVHFWLRQVWKDWWKHWZRVWUDLJKWOLQHVRIrPDNHZLWK P¢o

)LJ *UDSKLFDO GHWHU PLQDWLRQ RI WKH QRGDOSRLQWV

P¢i

F¢i

r1 Po Fo

No

r2

Pi Ni

Fi





,GHDOLPDJHV 

WKHRSWLFDOD[LVDUHWKXVWKHSRLQWV1oDQG1iWKDWZHKDYHVHDUFKHG )RUWKHVLPLODULW\RIWKHWULDQJOHV)o3o 3oc DQG1i)i )ic ZHKDYHWKDW 

1 i )iC

C ; fo DQGVLPLODUO\1 o )o

 fi 

 

ZKHUHZLWKWKHVLJQX,LQGLFDWHWKDWWKHVHJPHQWKHUHLVWDNHQZLWKDGHILQHGGL UHFWLRQVHHWKHQRWHLQ7DEOH,IWKHWZRIRFDOGLVWDQFHVDUHHTXDOWKHQRGDO SRLQWV1RIDQLGHDOV\VWHPFRLQFLGHZLWKWKHSULQFLSDOSRLQWV3 (YHQWKHVHWRIIRXUSRLQWV)o1o)i1iLVHQRXJKWRGHILQHWKHSURSHUWLHVRI DQLGHDOV\VWHPDQGWRJHWKHUWKHVL[SRLQWV)o3o1o)i3i1iDUHFDOOHGcardinal points DQG FDQ VWD\ RQ WKH RSWLFDO D[LV LQ DQ\ RUGHU SURYLGHG WKDW WKH (TV  DUHUHVSHFWHG

2.5.4 The telescopic case $VZHVDZHDUOLHUDWHOHVFRSLFWUDQVIRUPDWLRQLVFKDUDFWHUL]HGE\KDYLQJERWK IRFDOSODQHVDWLQILQLW\7KHFRHIILFLHQWbLQ(TV  LV]HURDQGVRWKH\DUH UHGXFHGWR 

xi

a xo  yi c

a yo  zi c

dzo  e  c

 

(YHQ LQ WKLV FDVH LW LV FRQYHQLHQW WR FKRRVH WZRGLIIHUHQW FRRUGLQDWH V\VWHPV IRUWKHREMHFWDQGLPDJHVSDFHV$VWKHRULJLQRIWKHWZRV\VWHPVZHFDQFKRRVH DQ\SDLURIconjugateSRLQWV2oDQG2i$FFRUGLQJWRWKHFRQYHQWLRQRQWKHFKRLFH RIVLJQVIRU(TV  DQGZLWKUHIHUHQFHWRWKHQHZV\VWHPRID[HVWKHWUDQV IRUPDWLRQ  WDNHVWKHIRUP 

xi

mA xo  yi

mA yo  Zi

;m & Zo 

 

ZKHUHZoZiDUHWKHGLVWDQFHIURP2oDQG2iUHVSHFWLYHO\DQG 

mA

a d DQGm &  c c

 

DUH WKH ODWHUDO DQG ORQJLWXGLQDO PDJQLILFDWLRQV UHVSHFWLYHO\ WKH\ DUH WKHUHIRUH FRQVWDQWLQWKLVFDVH$QLPSRUWDQWSHFXOLDULW\RIWKHWHOHVFRSLFV\VWHPVLVWKDWD EHDPRISDUDOOHOUD\VEHWZHHQWKHPQDPHO\DFROOLPDWHGEHDPHPHUJHVIURPWKH V\VWHPVWLOOFROOLPDWHGLQSDUWLFXODUDUD\SDUDOOHOWRWKHRSWLFDOD[LVDWDGLVWDQFH hIURPLWHPHUJHVDVVKLIWHGDWDQHZGLVWDQFHmAh(YHQWKHDQJXODUPDJQLILFD WLRQEHWZHHQWZRFRQMXJDWHUD\VLVFRQVWDQWVXFKPDJQLILFDWLRQLVGHILQHGDVWKH UDWLRWDQ-iWDQ-oZKHUH-iDQG-oDUHWKHDQJOHVWKDWWKHWZRUD\VPDNHZLWKWKH

*HRPHWULFDO2SWLFV



 )LJ $QJXODUPDJQLILFDWLRQ

h

m h

Jo

l

Ji m l z

RSWLFDOzD[LV )LJ 7KHUHIRUHLWLV 



WDQ -i WDQ -o

mA  m&

 

:HOIRUG@ ,QWKHVXEVHTXHQWVHFWLRQVZHZLOOFRPSOHWHWKHGHPRQVWUDWLRQWKDWWKHLQVWUX



3DUD[LDO2SWLFV 

PHQWVZLWKD[LDOURWDWLRQDOV\PPHWU\UHSURGXFHLQWKH*DXVVUHJLRQWKHSURSHU WLHVRID[LDOSURMHFWLYHWUDQVIRUPDWLRQV:HZLOODOVRUHIHUWRWKHVHZLWKWKHQDPHV RI*DXVVKRPRJUDSKLHV

2.6.2 Refraction through a spherical surface /HWXVWDNHDFRQFUHWHORRNDWWKHSURSHUWLHVRIRQHRIWKHVLPSOHVWRSWLFDOHOH PHQWVWKHdioptreLQWKHSDUD[LDODSSUR[LPDWLRQ&RQVLGHUWKHQDVLQJOHUHIUDF WLYHVXUIDFHSZLWKFXUYDWXUHRZKLFKVHSDUDWHVWZRPHGLDRILQGH[noDQGniDV VKRZQLQ)LJ/HWFoDQG FiEHWZRZDYHIURQWVRIDEHDPRIOLJKWHPDQDWLQJ IURPDSRLQW2RQWKHD[LVDQGORFDWHGUHVSHFWLYHO\LPPHGLDWHO\EHIRUHDQGDIWHU WKH UHIUDFWLRQ LQ )LJ  LW KDSSHQV WKDW Fi LV YLUWXDO EXW WKLV GRHVQRW PDWWHU KHUH $UD\WKDWVWDUWVIURP2PHHWVWKHZDYHIURQWVLQ0oDQG0iDQGWKHUHIUDFW LQJVXUIDFHLQ16LQFHWKHRSWLFDOSDWKVEHWZHHQWKHWZRZDYHIURQWVRIWKHVDPH IDPLO\RIUD\VPXVWEHHTXDODQGIRUWKH0DOXV'XSLQWKHRUHPWKLVDSSOLHVHYHQ LQWKHSUHVHQFHRIDUHIUDFWLRQZHKDYH no 0 o 1C



ni 0 i 1C 

 

,QWKH*DXVVLDQUHJLRQWKLVHTXDWLRQLVFRUUHFWZLWKQRRWKHUUHVWULFWLRQVKRZHYHU ZLWKLQWKLVUHJLRQZHKDYH  h  h ;   R  so

0 o 1C



ZKHUHhLVWKHKHLJKWRILQFLGHQFHRIWKHEHDPDQGsoLVWKHGLVWDQFHEHWZHHQWKH UHIUDFWLYHVXUIDFHDQGWKHSRLQW27KHVLJQVDSSOLHGLQWKLVH[SUHVVLRQUHIOHFWWKH FRQYHQWLRQWUDQVFULEHGLQ7DEOH$VLPLODUHTXDWLRQWKHUHLVIRU 0 i 1C   Fo h

Mo

R

V

O

I

no X so

S

Fi Mi N

z

ni Fo

Fi

S

si



)LJ 5HIUDFWLRQRIDVSKHULFDOZDYHIURQWRQDGLRSWULFVXUIDFH+HUHWKHUDGLXV RRI WKHVXUIDFHLVWDNHQDVSRVLWLYHDQGni!noIRUZKLFKWKLVLVDFRQYHUJLQJdioptre

*HRPHWULFDO2SWLFV



 O¢

yo

Jo O

Ji

V no

I I¢

ni

yi

z

S

X so

0

si



)LJ5HIUDFWLRQRIDUD\DWWKHYHUWH[9RIDGLRSWULFVXUIDFH

0 i 1C



 h  h    R  si

6XEVWLWXWLQJWKHVHH[SUHVVLRQVLQ(T  DQGVLPSOLI\LQJZHJHW § · no ¨  ; ¸ so ¹ ©R



§ · ni ¨  ¸  © R si ¹

WKDWLVUHDUUDQJLQJWKHWHUPV n ; o so



n  i si

ni  no  R

 

ZKHUH\RXFDQUHFRJQL]HWKHIRUPRIWKH*DXVV(T  IRUFRQMXJDWHGGLVWDQF HV)URPWKHFRPSDULVRQLWLV 

fo

; no

R  fi ni  no

ni

R  ni  no

 

1RZ OHW XV VHH ZKDW KDSSHQV WR D SRLQW 2' SODFHG RIID[LV DERYH 2 $V GLV FXVVHGLQ†LWKDVDQLPDJH,'SODFHGXQGHU,:HFRXOGDSSO\WKHV\PPHWU\ SURSHUWLHVRIWKHVSKHULFDOVXUIDFHLQGHHGIRUVXFKDVXUIDFH2' ,'GHILQHVDQHZ D[LVHTXLYDOHQWWRWKDWRI2,7KHWZRD[HVQHFHVVDULO\LQWHUVHFWLQWKHFHQWHURIWKH VSKHUH DQG WKH SUREOHP FDQ EH VROYHG TXLFNO\ +RZHYHU , SUHIHU KHUH WR DSSO\ 6QHOO¶V ODZ WRD UD\ JRLQJ WKURXJK WKHYHUWH[ 9RI WKH LQWHUVHFWLRQRI S ZLWK 2, )LJ )RUVPDOODQJOHVZHKDYH 

; no

yo so

y ni i  si

 



3DUD[LDO2SWLFV 

7KHUHIRUHWKHWUDQVYHUVHPDJQLILFDWLRQLV 

mA

yi yo

s ; i so

no  ni

 

7KXV ZH KDYH ILQDOO\ FRQILUPHG WKDW D dioptre LQ WKH SDUD[LDO UHJLRQ UHSUR GXFHVWKHSURSHUWLHVRIWKHKRPRJUDSKLHVGLVFXVVHGLQ†,QSDUWLFXODUZLWKUH VSHFWWRVXFKDQRSWLFDOHOHPHQWWKHSULQFLSDOSODQHVFRLQFLGHDQG3o 3i 9,Q DGGLWLRQWKHQRGDOSRLQWVDUHERWKORFDWHGLQWKHFHQWHURIFXUYDWXUHRIWKHGLRSWULF VXUIDFH:HFDQWKHQDSSO\DOORIUXOHVDQGWULFNVRILGHDOLPDJHVWRLW7KHWHUP P



ni  no  R

 

LVFDOOHGWKHpower of the surfaceDQGLVPHDVXUHGLQdiopters GLRSWHU P  ,I P!DQGWKHUHIRUH;fofi!DVZHOOLWLVVDLGWKDWWKHVXUIDFHLVpositiveRU converging&RQYHUVHO\IRU PWKHVXUIDFHLVFDOOHGnegativeRUdiverging,Q 2SKWKDOPLF 2SWLFV WKH WHUPV ;noso DQG nisi RI (T   DUH FDOOHG reduced vergenceDQGWKHLUVXPLVWKHSRZHURIWKHGLRSWULFVXUIDFHE\vergenceZHPHDQ WKHFRQYHUJLQJRUGLYHUJLQJZDYHIURQWFXUYDWXUHDQGLWWRRLVPHDVXUHGLQGLRS WHUV

2.6.3 Reflection from a spherical mirror )RUWKHFDVHRIDPLUURUZHFDQSURFHHGLQDVLPLODUPDQQHUWRWKDWZKLFKZDV GRQH IRU WKH GLRSWULF VXUIDFH $GRSWLQJ WKH FRQYHQWLRQV VKRZQ LQ )LJ  LQ

R

0

yo I O

C

z

yi S

X so

si

0



)LJ 5HIOHFWLRQIURPDVSKHULFDOPLUURU,QWKLVFRQYHQWLRQWKHSRVLWLYHGLUHFWLRQIRU ; soDQGsiLVWKHRSSRVLWHWRWKDWIRURWKXVLQWKHFDVHRIWKLVILJXUH; sosi!DQGR

*HRPHWULFDO2SWLFV  ZKLFKWKHDUURZLQGLFDWHVWKHSRVLWLYHGLUHFWLRQVIRUDPLUURURIUDGLXVRZHKDYH 



;

   so si



  xi R

s ; i xo  so

yi

R  P 



s ; i yo  so

 

DQGLQSDUWLFXODU 

; fo

fi

f



  R

 

$OVR LQ WKLV FDVH WKH SULQFLSDO SRLQWV FRLQFLGH ZLWK WKH YHUWH[ RI WKH PLUURU DQGWKHQRGDOSRLQWVZLWKLWVFHQWHURIFXUYDWXUH :KLOHDSDUDEROLFPLUURUIRFXVHVDOORIWKHUD\VSDUDOOHOWRLWVD[LVLQLWVIRFDO SRLQWLQWKHVDPHVLWXDWLRQDVSKHULFDOPLUURUVXIIHUVIURPspherical aberration +RZHYHU WKH SDUDEROLF PLUURU VXIIHUV IURP coma DQG astigmatism IRU GLVWDQW VRXUFHVRIIRILWVD[LVZKLOHDQRSWLFDOV\VWHPFRQVLVWLQJRIDVSKHULFDOPLUURUDQG DVPDOODSHUWXUHGLDSKUDJPSODFHGRQLWVFHQWHURIFXUYDWXUHSURGXFHVDVKDUSLP DJHRIVXFKVRXUFHVRQDVSKHULFDOfocal surfaceFRQFHQWULFZLWKWKDWRIWKHPLUURU

2.6.4 Relationship between the two focal lengths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foDQGfiLVJLYHQE\ 

f ; i fo

ni  no

 

:HOOLWLVSRVVLEOHWRGHPRQVWUDWHWKDWWKLVUHODWLRQLVYDOLGLQJHQHUDOWKURXJK WKH0DOXV'XSLQWKHRUHPWKDWOLQNVWKHHTXDOLW\RIWKHRSWLFDOSDWKWRWKHZDYH IURQWRULI\RXOLNHWKURXJK)HUPDW¶VSULQFLSOH7RYHULI\WKLVZHDJDLQGHQRWH ZLWK)o3o)iDQG3iWKHIRFLDQGWKHSULQFLSDOSRLQWVRIDQRSWLFDOV\VWHP )LJ   DQG FRQVLGHU D SDUD[LDO UD\ r RI D OLJKW SHQFLO ZLWK IRFXV LQ )o 7KLV UD\ PHHWVWKHSULQFLSDOSODQHVLQ 3oc DQG 3ic *LYHQWKDW 3ic DQG3iDUHERWKSRLQWVRI WKHplaneZDYHIURQWRIWKHSHQFLORIUD\VFRPLQJIURP)oWKHHTXDOLW\RIRSWLFDO SDWKVUHTXLUHVWKDW



3DUD[LDO2SWLFV 

r1

P¢o

r2 Fo

r2

P¢i

r1

h no

Po

Pi

ni

Fi



)LJ'HWHUPLQDWLRQRIWKHUHODWLRQVKLSEHWZHHQWKHWZRIRFDOOHQJWKV



no )o 3oc  > 3oc [email protected] ; no fo  > 3o 3i @ 

ZKHUH WKH VTXDUH EUDFNHWV PHDQ WKH ZRUGV ³RSWLFDO SDWK OHQJWK RI´ 7KXV LI WKH GLVWDQFH3o 3oc  3i 3ic LVLQGLFDWHGZLWKhH[SDQGLQJWKHGLVWDQFH)o 3oc LQSRZ HUVRIhDQGOLPLWLQJWKHH[SDQVLRQXSWRWKHWHUPLQhZHKDYH 

n h ; o  fo

> 3o 3i @  > 3oc [email protected] 

6LPLODUO\WDNLQJDUD\rWKDWHQWHUVWKHV\VWHPSDUDOOHOWRWKHD[LVDQGZLWKWKH VDPHKHLJKWhZHKDYH 

ni h   fi

> 3o 3i @  > 3oc [email protected] 

)URPDFRPSDULVRQRIWKHVHWZRH[SUHVVLRQVZHJHW(T  ZKLFKLVWKHQ YDOLGLQJHQHUDO,WFDQEHUHH[SUHVVHGLQWKHIRUP 

P

n ; o fo

ni  fi

 

PLVFDOOHGWKH powerRIWKHLQVWUXPHQWDQGLWVLQYHUVHLVVDLGWREHWKHeffective focal distance RUequivalent ZKLFKLVLQIDFWWKHIRFDOOHQJWKRIDV\VWHPZLWK WKH VDPH YDOXH P EXW LPPHUVHG ³LQ WKH DLU´ WKDW LV ZLWK no  ni    DW ERWK VLGHV

2.6.5 The lateral, angular and longitudinal magnification, and the Smith-Helmholtz-Lagrange optical invariant /HW XV QRZ H[DPLQH DQ LPSRUWDQW UHODWLRQVKLS LQYROYLQJ WKH convergence anglesRIWKHUD\VWKDWLVWKHDQJOHWKDWWKH\PDNHZLWKWKHRSWLFDOD[LV/HWXVWDNH DVLQ)LJWZRFRQMXJDWHGSRLQWV2‫މ‬DQG,‫މ‬DQGWKHLUSURMHFWLRQRQWKHD[LV2

*HRPHWULFDO2SWLFV



 O¢ yo

O¢¢

O

y¢¢ o

y¢¢i

h

Jo Po

Pi

j

Ji

I yi

I¢¢

I¢ X so

si

0

0

Di

XDo



)LJ$QJXODUPDJQLILFDWLRQ

DQG , ZKLFK DUH FRQMXJDWHG DV ZHOO 7KH lateral magnification EHWZHHQ WKH GLV WDQFHV22 ‫މ‬DQG,, ‫މ‬DVZHNQRZIURP(TV  DQG  LV 

yi yo

mA

s f  i o so fi

s n ; i o  so ni

 

5HDUUDQJLQJWKHWHUPVZHJHW yi ni si



y ; o no  so

 

,IhLVWKHKHLJKWDWZKLFKDUD\SDVVLQJWKURXJK2PHHWVWKHSULQFLSDOSODQHV )LJ WKHDQJOHVVXEWHQGHGE\WKLVUD\ZLWKWKHRSWLFDOD[LVLQWKHWZRVSDF HVLQWKLVRUGHURISUHFLVLRQDUH 

-o

;

h  -i so



h  si

6RPXOWLSO\LQJERWKVLGHVRI(T  IRUhZHJHW 

-i yi ni

-o yo no

H 

 

(YHQWKHSODQHVGHILQHGE\2‫މމ‬DQG,‫މމ‬LQ)LJDUHFRQMXJDWHGDQGIURP(T  E\UHSODFLQJsoZLWK 'o h y o" DQGsiZLWK 'i h y "i  ZHILQGWKDW 

y "i yi ni 'i

y o" yo no  ;' o

6XSSRVHWKDWRXUV\VWHPIURP)LJLVIROORZHGE\DQRWKHUQRZWKHRXWSXW YDOXHV-iyiniRIWKHILUVWPXVWEHDVVLJQHGWRWKHYDULDEOHV-oyonoRIWKHVHF RQG VLQFH WKH LPDJH RI WKH ILUVW V\VWHP EHFRPHV WKH REMHFW IRU WKH VHFRQG DQG ERWKWKHUHIUDFWLYHLQGH[DQGWKHDQJOHRIFRQYHUJHQFHDUHWKHVDPH6RWKHTXDQ



3DUD[LDO2SWLFV 

WLW\ 

H

-yn 

 

LVLQYDULDQWWKURXJKWKHWZRV\VWHPVDQGWKHUHIRUHLWPXVWEHLQYDULDQWIURPWKH REMHFW WR WKH LPDJH VSDFH WKURXJK DOO RI WKH LQWHUPHGLDWH VSDFHV RI HDFK D[LDOO\ V\PPHWULFDO RSWLFDO V\VWHP 7KLV LQYDULDQW ZDV LQWURGXFHG E\ 5REHUW 6PLWK LQ LQKLVERRNCompleat System of OpticsDQGLWZDVVXEVHTXHQWO\WDNHQXSE\ /DJUDQJH DQG +HOPKROW] LW LV LQGLFDWHG E\ WKH QDPH RI WKH Smith-HelmholtzLagrangeoptical invariant RQH RU WZR QDPHV DUH XVXDOO\ RPLWWHG  $PRQJ WKH DSSOLFDWLRQVRIWKLVRSWLFDOLQYDULDQWWKHUHLVWKHGHULYDWLRQRIWKHUHODWLRQVKLSEH WZHHQODWHUDOPDJQLILFDWLRQDQGDQJXODUPDJQLILFDWLRQ)URP(T  ZHLP PHGLDWHO\ILQGWKDWWKHangular magnification 



-i -o

no yo ni yi

no   ni mA

 

DPRQJ WKH DQJOHV RI FRQYHUJHQFH LW LV LQYHUVHO\ SURSRUWLRQDO WR WKH ODWHUDO RQH DQGWKHFRQVWDQWRISURSRUWLRQDOLW\GRHVnotGHSHQGRQWKHSRVLWLRQRIWKHREMHFW LPDJHFRQMXJDWHSODQHV ,WPD\KDSSHQWKDWLQRQHRIWKHVSDFHVRIDQRSWLFDOV\VWHPWKHREMHFWRURQH LQWHUPHGLDWHLPDJHLVDWLQILQLW\$VGHVFULEHGE\(T  LQWKLVFDVHHEH FRPHVLQGHWHUPLQDWHWKHUHVLQFH-oDQGyof7KLVLQGHWHUPLQDWLRQFDQEH UHVROYHGE\DJDLQSODFLQJ h -  DQGy s



Ms 

ZKHUHMLVDILHOGDQJOHDVGHILQHGLQ)LJ7KHUHIRUHLQWKLVVSDFHZHFDQ ZULWH 

H

 nhM 

 

,WVKRXOGEHQRWHGWKDWIRUVPDOODQJOHVWKLVUHODWLRQVKLSLVYDOLGHYHQLIWKHTXDQ WLWLHVhsMDUHWDNHQZLWKUHVSHFWWRDQ\SODQH QRUPDOWRWKHD[LV LHQRWRQO\ ZLWKUHVSHFWWRDSULQFLSDOSODQHDVLQ)LJ /DVWO\OHWXVORRNDWDQLPSRUWDQWUHODWLRQEHWZHHQlongitudinal magnification DQGlateralmagnification)URP(T  DQGWKH*DXVV(T  ZHKDYH



m&

Z ; i Zo

s  fi ; i so  fo

f  i si si ; so   fo so

fo si so ; so fi si

DQGUHPHPEHULQJ(T  ZHXOWLPDWHO\KDYH

s ; i so 

fo  fi

*HRPHWULFDO2SWLFV



 

m&

si  no  so  ni

 

7KLVHTXDWLRQFDQEHFRPSDUHGWR(T  IRUWKHODWHUDOPDJQLILFDWLRQ,Q SDUWLFXODU 

m&

mA 

ni  no

 

DQGLQWKHIUHTXHQWFDVHLQZKLFKno niLWIROORZVWKDWWKHORQJLWXGLQDOPDJQLIL FDWLRQLVWKHVTXDUHRIWKHODWHUDORQH,WLVWKLVIDFWWKDWDOORZVXVWRVHHDQGWDNH SLFWXUHVZLWKDJRRGdepth of fieldPHWHUVRIWKLFNQHVV GLYHUJLQJWRLQILQLW\IRU PRUHGLVWDQWREMHFWV DUHFRPSUHVVHGRQWKHUHWLQDRURQWKHILOP

2.6.6 The lens 7KLV LV GHILQLWHO\ WKH PRVW LPSRUWDQW RI DOO RSWLFDO GHYLFHV 7KH OHQV LV RI XW PRVWXVHIXOQHVVWRRXURZQOLYHV/HQVHVRIPDUYHORXVSHUIHFWLRQFDQEHIRXQGLQ QDWXUH DQG HYHQ LQ RXUVHOYHV WKH\ DUH D SUHFLRXVJLIW IURP *RG ,Q WKH EURDGHVW VHQVHWKHOHQVLVDUHIUDFWLYHGHYLFHWKDWLVXVHGWRUHVKDSHZDYHIURQWVLQDFRQ WUROOHGPDQQHU7KHPHGLXPXVHGLVQRWQHFHVVDULO\KRPRJHQHRXV)RUH[DPSOHLW LVSRVVLEOHWRUHFRQILJXUHDZDYHIURQWE\SDVVLQJLWWKURXJKDPHGLXPLQZKLFKWKH UHIUDFWLYHLQGH[YDULHVDFFRUGLQJWRDFHUWDLQODZ7KHVHOHQVHVDUHFDOOHG*5,1 LH JUDGLHQWLQGH[ 2QH H[DPSOH LV WKH crystalline lens RI WKH H\H ,Q WKH PRUH WUDGLWLRQDOVHQVHDOHQVLVDQRSWLFDOV\VWHPFRQVLVWLQJRIWZRRUPRUHUHIUDFWLQJ LQWHUIDFHV EHWZHHQ KRPRJHQHRXV PHGLD RI ZKLFK DW OHDVW RQH LV FXUYHG $ OHQV WKDWFRQVLVWVRIDVLQJOHHOHPHQWRIWUDQVSDUHQWPDWHULDOQDPHO\WKDWRQO\KDVWZR UHIUDFWLYH VXUIDFHV LV FDOOHG Dsimple lens RU singlet LQVWHDG LI LW LV PDGH XS RI PXOWLSOH HOHPHQWV LW LV VDLG WR EH D compound lens 7KHVH OHQVHV DUH W\SLFDOO\ FRPSRVHGRIWZRRUWKUHHHOHPHQWV doubletsRUtriplets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convergingRUpositiveZKHQLWLVWKLFNHULQWKHFHQWHUWKDQDW WKHHGJHV DQGVLPXOWDQHRXVO\LWVUHIUDFWLYHLQGH[LVJUHDWHUWKDQWKDWRIWKHVXU URXQGLQJ PHGLXP DV XVXDOO\ KDSSHQV  ,QGHHG LW GHOD\V WKH EHDP¶V ZDYHIURQW ODUJHO\DWWKHFHQWHUUDWKHUWKDQDWWKHHGJHVDQGWKHQLQFUHDVHVWKHIRUZDUGEHQG



3DUD[LDO2SWLFV 

LQJ ZLWKWKHVLJQGHILQHGLQ7DEOH 7KHOHQVHVWKDWSURGXFHWKHRSSRVLWHHI IHFWDUHFDOOHGdivergingRUnegative /HWXVQRZORRNDWWKHFDVHRIDVLPSOHOHQVFRQVWLWXWHGE\WKHFRPELQDWLRQRI WZR dioptre ZLWK UDGLXV RI FXUYDWXUH R DQG R ZKRVH YHUWLFHV 9 DQG 9 DUH VSDFHGDSDUWE\DGLVWDQFHt0RUHRYHUZHGHQRWHZLWKnnnWKHLQGH[HVRI UHIUDFWLRQRI WKUHH PHGLD WKH VXUIDFHV RI ZKLFK VHSDUDWH LQ VHTXHQFH )LJ   )URP(TV  WKHIRFDOOHQJWKVRIWKHILUVWDQGVHFRQGVXUIDFHVDUHJLYHQUH VSHFWLYHO\E\ 

fo



fo 

n ;  P

fi

n  ZLWKP P

n  P

fi 

n  ZLWKP P

;

n  n  R n  n  R

D 

E 

7KHIRFDOOHQJWKVRIWKHFRPELQDWLRQDUHVWLOOJLYHQE\(TV  ZKHUHdLV QRZWKHGLVWDQFHEHWZHHQWKHWZRIRFL)iDQG)oRI)LJ)RUWKHXVXDOFRQ YHQWLRQRQWKHVLJQVVKRZQLQ)LJDQGIRU(TV  DQG  VXFKD GLVWDQFHLV 

d

§   · t  n ¨  ¸  ¨ P P ¸ © ¹

t  fi  ; fo 

ZKHUHLQWXUQWKHGLVWDQFHtEHWZHHQWKHSULQFLSDOSRLQWV3iDQG3oLVJLYHQE\ WKHWKLFNQHVVRIWKHOHQVRQLWVD[LV)URP(T  LQWURGXFLQJWKHSDUDPHWHU P





P P d

n



 

n  P

 

ZHXOWLPDWHO\ILQGWKDW 

fo

n ; 

P

fi

ZKHUHDVDIXQFWLRQRIWKHWKLFNQHVVt 

P

P  P 

t P P  n

 

UHSUHVHQWV WKH WRWDO SRZHU RI WKH WZR dioptres FRPELQHG WRJHWKHU QDPHO\ WKH SRZHURIWKHOHQV,WVKRXOGEHQRWHGWKDWIRUDWKLQOHQVZLWKtRRWKHYDO XHRIPLVVLPSO\WKHVXPRIWKHSRZHURIWKHWZRdioptres:HZLOODOVRHQFRXQWHU

*HRPHWULFDO2SWLFV



 )LJ 6LPSOHWKLFNOHQV

n1

R1

n2

V1

Po

R2

n3

V2

z

Pi

p2 0

R2 0 p1

R1

t



DUHODWLRQVKLSVLPLODUWR(T  LQWKHFRPELQDWLRQRIWZROHQVHV 7R VSHFLI\ IXOO\ WKH *DXVVLDQ SURSHUWLHV RI RXU OHQV ZH PXVW QRZ GHWHUPLQH WKHSRVLWLRQRIWKHSULQFLSDOSRLQWV3oDQG3iUHODWLYHWRWKHOHQVLWVHOI7KLVLVHDVL O\GRQHZLWKWKHKHOSRI(TV  :LWKUHVSHFWWRWKHYHUWH[9RIWKHOHQV , UHPHPEHUWKDWERWKRIWKHSULQFLSDOSRLQWVRIWKHILUVWVXUIDFHDUHORFDWHGWKHUH  WKH SRVLWLRQ RI WKH REMHFW SULQFLSDO SRLQW 3o RI WKH FRPELQDWLRQ )LJ   IRU (TV  LV 

; fo

p

t d

n P t  n P

D 

n P t  n P

E 

6LPLODUO\WKHGLVWDQFHRI3iIURP9LV 

p

fi 

t d



,QWKHVLPSOHVWFDVHRIDlens immersed in the airWKDWLVZLWKLQGLFHVn n  DQGn n(TV  DUHVLPSOLILHGLQWKHZHOONQRZQlens-maker’s equation 

f

ª §   n   t ·º  « n   ¨  ¸» n RR ¹ ¼ © R R ¬

 

 

ZLWK 

f

; fo

fi

  P

 

,Q RWKHU ZRUGV WKH WZR IRFDO OHQJWKV QRZ FRLQFLGH ZLWK HDFK RWKHU 7KLV KDV WKH LPSRUWDQW LPPHGLDWH FRQVHTXHQFH WKDW WKH QRGDO SRLQWV RI WKH OHQV FRLQFLGH ZLWKWKHLUSULQFLSDOSRLQWV1o 3o1i 3i2QWKHRWKHUKDQGZHFDQTXLFNO\ VHH IURP (T   WKDW WKLV FRLQFLGHQFH KDSSHQV ZKHQHYHU LQ D JHQHULF LQ

3DUD[LDO2SWLFV  si



Zi

Zi

(a)

si

(b) X Zo

fi X fo

X fo

X so

X so

fi

X Zo

 )LJ +\SHUERODVZKRVHFRRUGLQDWHV so siVDWLVI\WKH*DXVVHTXDWLRQ D FRQYHUJLQJ V\VWHP E GLYHUJLQJV\VWHP

VWUXPHQW WKH REMHFW VSDFH DQG WKH LPDJH VSDFH LQGLFHV DUH HTXDO ,Q WKH FDVH LQ ZKLFK WKH OHQV LV LPPHUVHG LQ D JHQHULF PHGLXP ZLWK LQGH[ nP  n  n (T  FDQEHIRUPDOO\UHFRYHUHGE\QRUPDOL]LQJWKHLQGH[HVZLWKnPLHSODF LQJn nnP 7RJHWKHUZLWK(T  WKHSRVLWLRQVRIWKHSULQFLSDOSODQHVRIDOHQVLP PHUVHGLQDLUDUH 

93oC



n  f t  9 3iC n R



n  f t  n R

 

ZKHUH,SUHIHUUHGKHUHWRPDNHH[SOLFLWWKHSRLQWVLQYROYHG $ IXUWKHU LPSRUWDQW VLPSOLILFDWLRQ FDQ EH PDGH LQ WKH FDVH RI D WKLQ OHQV IRU ZKLFK t  R  R  R  R 7KLVDSSUR[LPDWHLGHDRIOHQVLVYHU\XVHIXOIRU LWV VLPSOLFLW\ LQ WKH ILUVW SKDVH RI WKH GHVLJQ RI RSWLFDO V\VWHPV RU ZKHQ JUHDW SUHFLVLRQLVQRWQHFHVVDU\)RUDWKLQOHQV(TV  DQG  DUHUHVSHF WLYHO\UHGXFHGWR 



f

93oC



ª §   ·º « n   ¨  ¸» © R R ¹ ¼ ¬ R t  9 3iC R  R n







R t  R  R n

 

 

DQGWKHGLVWDQFHEHWZHHQWKHSULQFLSDOSODQHVLVDSSUR[LPDWHO\ 

3o 3iC

§ · t ¨  ¸  © n¹

 

*HRPHWULFDO2SWLFV  IRUWKHPRVWFRPPRQOHQVHVEXWLWLVH[DFWIRUR ±RDQGIRUR fDQGRUR f$VDIXUWKHUDSSUR[LPDWLRQLWLVQRUPDOO\DVVXPHGWKDWWKHIRXUSRLQWV9 93o3iFRLQFLGHZLWKWKHSRVLWLRQRIWKHFHQWHURIWKHOHQV+RZHYHUWKLVLVQRW SDUWLFXODUO\FRUUHFWIRUWKHmenisciZKLFKZHZLOOGLVFXVVODWHUIRUVRPHRIZKLFK (T  LWLVDOVRQRWDSSOLFDEOH 7KHOHQVHVDUHFRPPRQO\FKDUDFWHUL]HGE\LQGLFDWLQJWKHLUeffective focal distanceZKLFKLVWKHYDOXHRIfZKHQWKHOHQVLVLPPHUVHGLQDLU7KHUHDUHWZRRWKHU TXDQWLWLHVWKDWDUHDOVRFRPPRQO\XVHGFDOOHGWKHfrontDQGrear focal distance 



C  f

; )o 9

fI

S

9 )iC  

 

PHDVXUHGE\WKHYHUWLFHV9DQG9RIWKHOHQVLPPHUVHGLQDLUDQGWKHUHIRUHnot WREHFRQIXVHGZLWKfoDQGfi,QSDUWLFXODULQ2SKWKDOPLF2SWLFVXVHLVPDGHRI WKHrearRUback powerIRU(T E LWLVH[SOLFLWO\JLYHQE\ 

 fS

PS

P  P  t n P P   t n P



 

ZKHUHWKHGHQRPLQDWRUFRQVWLWXWHVDUHODWLYHO\VPDOOFKDQJHRI(T  )LQDO O\IRUDOHQVLQDLUWKH*DXVVHTXDWLRQDQGWKDWRI+X\JHQV1HZWRQEHFRPHUH VSHFWLYHO\ 

;

   so si

  Z o Zi f

f  

 

7KHJUDSKLFDOVROXWLRQWRWKHVHHTXDWLRQVLVVKRZQLQ)LJ

2.6.7 Variety of simple lenses 7KLFNOHQVLPPHUVHGLQDLU)URP(T  DQG  LWLVVHHQWKDWWKHUH DUHGHJUHHVRIIUHHGRPtnRDQGRWKDWDOORZIRUYDU\LQJWKHFKDUDFWHULVWLFV RIDOHQV$PRQJWKHVHtLVJHQHUDOO\UHGXFHGWRWKHPLQLPXPFRPSDWLEOHZLWKWKH GLDPHWHU DQGPHFKDQLFDO VWUHQJWKDQG WR LPSURYH WKHRSWLFDO TXDOLW\ RI WKH OHQV ,QGHHG WKH DEHUUDWLRQV RI WKH OHQV JHQHUDOO\ LQFUHDVH ZLWK LWV WKLFNQHVV 7KH UH IUDFWLYHLQGH[nKDVRQO\DOLPLWHGYDULDELOLW\LQWKHILHOGRIYLVLEOHUDGLDWLRQWKH YDOXHVnRIRSWLFDOJODVVHVYDU\EHWZHHQDERXW WKHPRVWFRPPRQJODVV DQG IRU VRPH VSHFLDO JODVV *ODVVHV RI KLJK LQGH[KDYH WKH DGYDQWDJH RI OHDGLQJ WR PLQRU JHRPHWULFDO DEHUUDWLRQV EXW WKH\ DUH DOVR PRUH GLVSHUVLYH DQG DEVRUEHQW )RU YLVLEOH UDGLDWLRQ WKH LGHDO ZRXOG EH D JODVV ZLWK LQGH[  DQG DQ DQWL UHIOHFWLYHFRDWLQJZLWKDVLQJOHOD\HURIPDJQHVLXPIOXRULGH2IWKHWZRUHPDLQLQJ GHJUHHVRIIUHHGRPRQHFDQEHLQGLFDWHGLQR±RZKLFKIRUWKHFDVHRID

3DUD[LDO2SWLFV 

)LJ 9DULHW\RIOHQVHV



plano-convex

biconvex

positive meniscus Po Pi

Po Pi

Po Pi

q > 1 and t < R 2

q=1 Po

Po Pi

Pi

biconcave

R1

Po Pi

plano-concave

negative meniscus



WKLQOHQVLVSURSRUWLRQDOWRWKHSRZHURIWKHOHQV7KXVE\VHWWLQJWKHIRFDOOHQJWK LWVWLOOUHPDLQVDIUHHSDUDPHWHUZKLFKLVFRQYHQWLRQDOO\LQGLFDWHGE\WKHUDWLR 

q

R  R  R  R

 

ZKLFKLVFDOOHGCoddington’s form factor 7KHUHLVDVSHFLILFUHDVRQWKDWMXVWLILHVWKHYDULHW\RIIRUPVLQZKLFKOHQVHVDUH PDGH )LJ   ,QGHHG WKH TXDOLW\ RI LPDJHV SURGXFHG E\ D OHQV GHSHQGV VWURQJO\RQKRZWKHOHQVLWVHOILVXVHGDQGE\LWVVKDSH7KHDELOLW\WRFKDQJHWKH VKDSHZLWKRXWFKDQJLQJWKH*DXVVLDQSURSHUWLHVRIWKHOHQV WKLVRSHUDWLRQLVFDOOHG bending LVWKHUHIRUHXVHIXOIRUEDODQFLQJWKHDEHUUDWLRQV,QRWKHUZRUGVIRUHDFK FRQMXJDWHUDWLRDVVLJQHGWRDOHQVZLWKIL[HGIRFDOOHQJWKIRUDSDUWLFXODUDSSOLFD WLRQZHFDQFKRRVHWKHYDOXHRIqWKDWDOORZV IRUH[DPSOH IRUWKHVPDOOHVWVSKHU LFDODEHUUDWLRQ$SUDFWLFDOFULWHULRQIRUWKHFKRLFHRIWKHEHVWIRUPLVWRHTXDOL]H WKHDYHUDJHDQJOHVWKDWWKHPDUJLQDOUD\VLHWKHUD\VWKDWLQWHUVHFWWKHOHQVQHDU LWVHGJHPDNHZLWKWKHWZRVXUIDFHVRIWKHOHQV 7KHVH FRQFHSWV ZLOO EH IXUWKHU HODERUDWHG LQ WKH VWXG\ RI DEHUUDWLRQV )RU WKH PRPHQWLWLVLQWHUHVWLQJWRQRWHKRZWKHSRVLWLRQRIWKHSULQFLSDOSRLQWVYDULHVE\ FKDQJLQJWKHVKDSHRIWKHOHQV(TV  H[SUHVVHGDVDQH[SOLFLWIXQFWLRQRI RDQGREHFRPH 

93oC



tR  9 3iC n R  R  t n  



tR     n R  R  t n  

ZKLFKIRUDWKLQOHQVDUHUHGXFHGWR(TV  JLYHQDERYH7KHGLVWDQFHEH WZHHQWKHSULQFLSDOSRLQWV3oDQG3iLVFDOOHGWKHhiatusRUgap6LQFH 

3o 3iC

DFFRUGLQJWR(T  ZHJHW

t  93oC  9 3iC 

*HRPHWULFDO2SWLFV



 R

R

  R  3 3C  n  o i

)LJ8VLQJV\PPHWULFOHQVHV )RUt  R f



3o 3iC

ª º R  R t «  » ¬ n R  R  t n   ¼

t

t

n    n

n   R  R  t  n R  R  t n  

 

6\PPHWULFOHQV,QWKLVFDVHZHKDYHR ±R R )LJ 7KHSRVLWLRQRI WKHSULQFLSDOSRLQWVFDQEHTXLFNO\REWDLQHGIURP(TV  EXWIRUDV\PPHW ULFthin lensZLWKtRRQHVLPSO\KDV 93oC



9 3iC

t  3o 3iC n

n  t  n

 

6LQFH WKH UHIUDFWLYH LQGH[ RI WKH PRVW FRPPRQ JODVVHV LV a  DV D rule of thumbZHFDQUHPHPEHUWKDW LQDV\PPHWULFDOOHQVWKHSULQFLSDOSODQHVDUHORFDWHGIURPWKHYHUWLFHVDW DERXWRIWKHFHQWUDOWKLFNQHVVRIWKHOHQVZLWKLQWKHOHQVLWVHOI 7KLVDSSOLHVWRERWKFRQYHUJLQJDQGGLYHUJLQJOHQVHV /DPLQDZLWKSODQHDQGSDUDOOHOIDFHV7KHVROXWLRQWRWKLVFDVHLVREWDLQHGE\ EULQJLQJRofLQWKHSUHYLRXVFDVHDQGWKHSRVLWLRQRISULQFLSDOSODQHVLVVWLOO JLYHQE\(TV  7KHSURSHUW\RIWKHVHODPLQDHLVWKDWHDFKUD\WKDWSDVVHV WKURXJKWKHPLVRQO\PRYHGSDUDOOHOWRLWVHOI,WPD\VHHPOLNHDSDUDGR[KRZHY HUDODPLQDLQWHUSRVHGRQDEHDPRIOLJKWPRYHVLWVIRFDOSRLQWforwardZKHWKHU LWLVUHDORUYLUWXDOE\DQDPRXQWHTXDOWRWKHKLDWXVEHWZHHQWKHSULQFLSDOSODQHV )LJ   $V D UXOH RI WKXPE LQ WKH FRQGLWLRQV GHVFULEHG DERYH LH IRU n a  ZHFDQUHPHPEHUWKDW )LJ (IIHFW RI D ODPLQD ZLWKSODQHIDFHV

t

n 1t n



3DUD[LDO2SWLFV 

)LJ

6SKHULFDOOHQV f

nR   n 

Po Fo

Pi

R Fi

 WKHGLVSODFHPHQWLQWURGXFHGE\DODPLQDZLWKSODQHIDFHVLVDERXWRILWV WKLFNQHVV 3OHDVHNHHSWKLVLQPLQGZKHQ\RXZLVKWRIRFXVDODVHUEHDPWKURXJKWKHZLQ GRZRIDFHOO 6SKHULFDOOHQV7KLVLVDQRWKHUVSHFLDOFDVHRIDWKLFNV\PPHWULFOHQVVRR ±R t )LJ ,QWKLVFDVHWKHSRLQWV3oDQG3iFRLQFLGHZLWKWKHFHQWHURIWKH VSKHUHDQGWKLVLPPHGLDWHO\OHDGVRQHWRWKLQNWKDWLQWKHFDVHVFRQVLGHUHGKHUH WKH QRGDO SRLQWV FRLQFLGH ZLWK WKH PDLQ RQHV 7KH IRFDO GLVWDQFH RI D VSKHULFDO OHQVLV



f

n R   n  

 

3ODQRFRQYH[RUSODQRFRQFDYHOHQV2QWKHEDVLVRIWKHFULWHULRQRI³EHVWIRUP´ PHQWLRQHGDWWKHEHJLQQLQJWKHV\PPHWULFDOOHQVHVPD\EHDSSURSULDWHIRUFRQMX JDWHGUDWLRVQHDU )LJ ,QVWHDGLQWKHFDVHRIDFRQMXJDWHUDWLRFORVHWR]H URRUfWKHPRVWDSSURSULDWHIRUPLVQHDUWKDWRIDplano-convexRUplano-concave OHQV DQG WKH FXUYHG VLGH VKRXOG EH RULHQWHG WRZDUG WKH GLUHFWLRQ LQ ZKLFK WKH EHDPLV PRUH FROOLPDWHG )LJ )RUWKHVHOHQVHV(TV  LPPHGLDWHO\ LQGLFDWHWKDWRQHRIWKHSULQFLSDOSRLQWVFRLQFLGHVZLWKWKHFRUUHVSRQGLQJYHUWH[RQ WKHFXUYHGVXUIDFHZKLOHWKHRWKHULVORFDWHGLQVLGHWKHOHQV7KHKLDWXVLV 3o 3iC



t

n   n

 

$JDLQIRUWKHPRVWFRPPRQJODVVHV 3o 3iC |   t  0HQLVFXV OHQVHV 7KHVH WDNH WKHLU QDPH IURP WKH *UHHN PKQLVNR9 ZKLFK LQGL FDWHVWKH0RRQVKRUWO\EHIRUHRUDIWHUWKHQHZPRRQDQGZHUHSDWHQWHGE\:RO

R

R

 )LJ3ODQRFRQYH[DQGSODQRFRQFDYHOHQV )RUt  R f

R  3 3C n  o i

n  t  n

*HRPHWULFDO2SWLFV



= 1

wi th R f th

em

en

isc i

quadrant of the menisci

eo lin

t/R2

n/t

P=

0

R2

es e ns f th le e o tric lin me sym

n n 1

separating line between negative and positive lenses

quadrant of the biconcave lenses

plano-concave



convex-plane

concave-plane

c oncentr i

quadrant of the biconvex lenses

line of the c meni sci

0 plano-convex

P=

t P = n/2

n n 1

n n 1

sphere

0

t/R1

quadrant of the menisci

line of the biconvex concentric lenses

n n 1 

)LJ3KDVHGLDJUDPRIWKHOHQVHV ZLWKn 

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tRDQG tRLQWRIRXUTXDGUDQWVWZRRIPHQLVFLRQHRIDELFRQFDYHOHQVDQGRQHRIDEL FRQYH[OHQVHV7KHVROLGOLQHVDUHDK\SHUEROHWKDWVROYHWKHHTXDWLRQ 

tP n 

n   t   t t   R R RR n

 

IRUWKUHHGLIIHUHQWYDOXHVRIP7KHFXUYHZLWKP FRLQFLGHVZLWKWKHERXQGDU\ EHWZHHQFRQYHUJLQJOHQVHVDWWKHERWWRPULJKWDQGGLYHUJLQJOHQVHVLQWKHXSSHU



3DUD[LDO2SWLFV 

)LJ &RQYHQWLRQVLQWKHPHDVXUHPHQWRIGLV WDQFHV SoSiIURPWKHYHUWLFHVRIDOHQV

V1

V2

t

0 X So

0 0

Si 

OHIW2QWKLVOLQHWKHSULQFLSDOSRLQWSRVLWLRQVGLYHUJHWRLQILQLW\>(TV  @DV C ZHOODVWKHGLVWDQFH 3o 3i  7KHPHQLVFLRIWKHGRWWHGGLDJRQDOOLQHZLWKR RIRU(TV  KDYH C 3o 3i  t 7KH WZR OLQHV RI WKH FRQFHQWULF PHQLVFL DQG WKH FRQFHQWULF ELFRQYH[ OHQVHVVROYHWKHHTXDWLRQR±R t)RUWKHP3oFRLQFLGHVZLWK3i,QWKHDUHD EHWZHHQWKHOLQHRIWKHFRQFHQWULFPHQLVFLDQGWKHOLQHZLWKP WKHSRLQW3oKDV RYHUWDNHQWKHSRLQW3i7KHVDPHLVWUXHLQWKHORZHUULJKWFRUQHUEHORZWKHOLQHRI WKHFRQFHQWULFELFRQYH[OHQVHV7KHGLYHUJHQFHVWKDWZHKDYHIRUPoLQWKHSR VLWLRQRIWKHSULQFLSDOSRLQWVVKRXOGQRWZRUU\XVWRRPXFK7KH\H[SUHVVRQO\WKH LQDGHTXDF\RIXVLQJWKHPDVWKHRULJLQRIWKHD[HVIRUPLQWKHQHLJKERUKRRGRI ]HUR,QIDFWLIZHLQVWHDGXVHWKHYHUWLFHV9DQG9RIWKHOHQVDVRULJLQVWKH GLVWDQFHIRUWKHREMHFWDQGWKHLPDJHIURPWKHVHSRLQWVDUH )LJ  )LJ$SODQDWLFPHQLVFL

(a) V1 Po Pi

D SRVLWLYHPHQLVFXV R

R §¨  n ·¸  t 

f

R § R · ¨ n   ¸  n  ¨© Rt ¸¹

©

93oC

t

R1

R  93iC Rt



t R  9 3iC n R t

B

nR1

0 t

R2 + t

R1



$UHDODSODQDWLFSRLQWIRUWKHLPDJH VSDFH %YLUWXDODSODQDWLFSRLQWIRUWKHREMHFW VSDFH

t R  n Rt



(b)

R1

V2 Po Pi

R § R ·   ¨¨ n   ¸¸  n  © R t ¹

9 3oC

A 1R n 1

¹

E QHJDWLYHPHQLVFXV R R §¨  n ·¸  t  © ¹ f

R2

t

R  R t

B R2

0 t

A

1R n 2 nR2 R2 + t

R1



*HRPHWULFDO2SWLFV



 )LJ D  PHQLVFXV ZLWK FRQFHQWULF VSKHUL FDOVXUIDFHV QHJDWLYH  R R  t  f



9 3oC

(a)

R2 V2

n R Rt  n  t

9 3iC

R1 0 t

R  

Pi Po

R1

E  PHQLVFXV ZLWK VSKHULFDO VXUIDFHV (b) RIHTXDOUDGLXV SRVLWLYH    R R R  f  n  R  Po Pi n  t C 9 3o



R n 

C  9 3i



t

R n 

; So

V1

R1



; so

R2

0 t

 93oC  Si

si  93iC 

R1 + t R1 

 

DQGZLWKDVPDOODPRXQWRIFDOFXODWLRQVWKDWZHDUHEHWWHURIIQRWUHSRUWLQJKHUH IURPWKH*DXVVHTXDWLRQZHFDQGHULYHWKHUHODWLRQEHWZHHQWKHVHQHZGLVWDQFHV 

; So

 Si 

t n

; P So Si

t

n   § ; So Si ·  ¨ ¸  n © R R ¹

 

IURPZKLFKLWLVVHHQWKDWIRUPoDQGRR!!SoSitWKHGLVWDQFHSiSo t EHWZHHQWKHREMHFWDQGWKHLPDJH )LJ LVt n± nZKLFKLVHTXDOWRWKDW DOUHDG\IRXQGIRUDODPLQDZLWKSODQDUIDFHV8QGHUWKHVDPHFRQGLWLRQVWKHORQ JLWXGLQDOPDJQLILFDWLRQWHQGVWRDQGWKHUHIRUHWKHODWHUDORQHGRHVWKHVDPH)LJ VKRZVVRPHW\SHVRIPHQLVFXVOHQVWKDWGHVHUYHVRPHDWWHQWLRQ7KHaplanatic menisciH[SORLWWKHSURSHUWLHVRIDVSKHUH VHH†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oDQG3iFRLQFLGHZLWKWKHLUFHQWHURIFXUYDWXUHWKH\DUHdiverging,QRWH ODVWO\ WKH PHQLVFL UHSUHVHQWHG LQ )LJ  E  ZKLFK DUH PDGH ZLWK spheres of equal radiusWKH\DUHconvergingDQGKDYHDKLDWXVHTXDOWRt



3DUD[LDO2SWLFV 

2.6.8 Optical center of a lens /HWXVQRZORRNDWDSDUWLFXODUSURSHUW\RIVLPSOHOHQVHV&RQVLGHUDWKLFNOHQV FRQVLVWLQJ RI WZR spherical dioptres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optical centerRI WKH OHQV 7KLV SRLQW GRHV QRW GHSHQG RQ WKH LQFOLQDWLRQ RI WKH WZR OLQHV SDVVLQJ WKURXJK & DQG & ,QGHHG WKH WULDQJOHV &26 DQG &26 DUH VLPLODU WR HDFK RWKHUVRWKDWWKHUDWLR 2&C 2& C





R  R

 

ZKHUHRDQGRDUHWKHUDGLLRIFXUYDWXUHRIWKHWZRGLRSWULFVXUIDFHVLVFRQVWDQW 7KLVGHPRQVWUDWLRQDOVRDSSOLHVXQGHUQRQSDUD[LDOFRQGLWLRQV2QWKHRWKHUKDQG WKHGLVWDQFHEHWZHHQWKHFHQWHUVRIFXUYDWXUHLV &&C



 R  R  t

2&C  2&C 

ZKHUH t LV WKH WKLFNQHVV RI WKH OHQV 7KH VLJQV RI WKH YDULDEOHV DUH WDNHQ LQ DF FRUGDQFHZLWKWKHSUHYLRXVFRQYHQWLRQV:LWKVRPHVWHSVZHILQGWKDW

C2

S1

e S1

i V1 O

V1 i

V2 C1 S2 e

S2 O V2

C2

C1

 )LJ   6WXG\ RI WKH RSWLFDO FHQWHU 2 RI D OHQV LPPHUVHG LQ DLU )RUD FRQYHUJLQJ PH QLVFXV2LVORFDWHGH[WHUQDOO\WRWKHFRQYH[VLGH,QSDUD[LDOFRQGLWLRQVDQGWKLVLVQRWWKH FDVHZLWKWKHUD\VWUDFHGKHUHWKHQRGDOSRLQWVDQGFRQVHTXHQWO\DOVRWKHPDLQSRLQWVDUH REWDLQHGIURPWKHLQWHUVHFWLRQZLWKWKHRSWLFDOD[LVRIWKHH[WHUQDOWUDLWVRIWKHUD\HLWKHU GLUHFWO\RUE\WKHLUH[WHQVLRQ LQJUD\

*HRPHWULFDO2SWLFV



 92C



Rt  92C R  R



Rt  R  R

 

2.6.9 Combination of two thin lenses /HW XV QRZ H[DPLQH WKH EDVLF EXW YHU\ LPSRUWDQW FDVH RI WKH FRPELQDWLRQ RI WZRWKLQOHQVHV)RUVLPSOLFLW\ZHDOVRDVVXPHWKDWWKH\DUHLPPHUVHGLQDLU7KH VROXWLRQWRWKLVSUREOHPLVTXLWHVLPLODUWRWKDWREWDLQHGIRUWKHFRPELQDWLRQRIWZR dioptres,QIDFWLIZHGHQRWHE\fDQGfWKHIRFDOOHQJWKVRIWKHWZROHQVHVDQG ZLWKlWKHGLVWDQFHEHWZHHQWKHPVWDUWLQJIURP(TV  DQGLQDQDORJ\ZLWK ZKDW ZDV GRQH WR GHULYH (T   ZH QRZ KDYH WKDW WKH IRFDO OHQJWK RI WKH FRPELQDWLRQLV  f



  l    f f  f f 

 

DQGIURP(TV  WKHSRVLWLRQRIWKHPDLQSODQHVZLWKUHVSHFWWRWKHSRVL WLRQ//RIWKHWZROHQVHV )LJ LV /3oC





f l f



f l  / 3iC f  f   l



f l f



f l f  f   l

 

(T  LQGLFDWHVDQLPSRUWDQWWKLQJQDPHO\WKDWE\YDU\LQJWKHGLVWDQFH EHWZHHQ WKH WZR OHQVHV ZH FDQ YDU\ WKH SRZHU RI WKHLU FRPELQDWLRQ 7KLV ILQGV DSSOLFDWLRQIRUH[DPSOHLQWKHWHOHVFRSLFREMHFWLYHVRIWKHFDPHUDVLQZKLFKWKH SRZHU RI WKH ILUVW FRQYHUJLQJ OHQV XVXDOO\ RI D ODUJH RSHQLQJ LV SDUWLDOO\ FRP SHQVDWHG E\ D VHFRQG GLYHUJLQJ OHQV WKXV REWDLQLQJ D VKRUWHU V\VWHP WKDQ LV UH TXLUHGZLWKDVLQJOHHTXDOSRZHUOHQV ,QSDUWLFXODUE\YDU\LQJlWKHUHDUHWZRLQWHUHVWLQJOLPLWFDVHVSOXVDWKLUGFDVH ZLWK LPSRUWDQW DSSOLFDWLRQV )RU WZR WKLQ OHQVHV LQ FRQWDFW l |  DQG IURP (T  LWLVVHHQWKDWWKHWRWDOSRZHU P f LVVLPSO\WKHVXPRIWKHLQGLYLGXDO SRZHUVWKLVLVDJRRGUHDVRQIRUWKHXVHRIWKHSRZHUVUDWKHUWKDQRIWKHIRFDOGLV WDQFHVDVDQLQGH[RIFRQYHUJLQJRUGLYHUJLQJSRZHUIRUH[DPSOHLQ2SKWKDOPLF 2SWLFV

l

)LJ &RPELQDWLRQRIWZRWKLQ OHQVHVLQDLU

P

P  P  l P P

Fo1

L1

L2

Fi2



3DUD[LDO2SWLFV 

7HOHVFRSHV

7KHRWKHULQWHUHVWLQJVLWXDWLRQLVIRUWHOHVFRSLFFRPELQDWLRQVIRUZKLFKl f fLQWKLVFDVHWKHWRWDOSRZHULV]HURDQGWKHSRVLWLRQRIWKHSULQFLSDODQGIRFDO SODQHVGLYHUJLQJWRLQILQLW\ORVHVLWVSUDFWLFDOVLJQLILFDQFH:HKDYHDOUHDG\VHHQ WKDWIRUWHOHVFRSLFV\VWHPV VHH† WKHKRPRJUDSK\WDNHVDVLPSOHDQDO\WL FDOIRUPWDNLQJDVWKHRULJLQRIFRRUGLQDWHVLQWKHREMHFWDQGLPDJHVSDFHVDQ\ WZRFRQMXJDWHSRLQWVIRUH[DPSOHWKHSRLQWV)oDQG)i )LJ )URP(TV  ZHVRRQJHWIRUWZROHQVHVLQDLU 

xi

f   xo  f

yi

f   yo  Zi f

f  ;  Zo   f

 

,DOVRUHFDOOWKDWWKHWHOHVFRSLFPDJQLILFDWLRQLVLQGHSHQGHQWRIWKHSRVLWLRQRI WKHFRQMXJDWHSRLQWV,QSDUWLFXODUDFROOLPDWHGEHDPWKDWUHDFKHVDWHOHVFRSLFV\V WHPHPHUJHVDVVWLOOFROOLPDWHG7KXVLIRXUOHQVHVDUHadjusted to infinityWKDWLV ZLWKl ffDEHDPRIGLDPHWHUDLQWHUFHSWHGE\WKHILUVWOHQVHPHUJHVIURP WKH VHFRQG OHQV ZLWK D GLDPHWHU D c mA D f  f D  DV FDQ LPPHGLDWHO\ EH VHHQIURP(TV D DQG   0RUHRYHUIRU(T  WKHDQJXODUPDJQLILFDWLRQLV 



f   f

  mA

 

DQGLVWKHUHIRUHWKHLQYHUVHRIWKDWODWHUDO7KLVUHVXOWFDQDOVREHUHDFKHGYLDWKH RSWLFDOLQYDULDQWLWLVQRWHZRUWK\WKDWIRUDWHOHVFRSLFV\VWHPVXFKDVWKHRQHGLV FXVVHGKHUHWKHSULQFLSDOSODQHVOLHDWLQILQLW\KRZHYHU(T  LVLQGHSHQG HQW RI WKHLU SRVLWLRQ DQG WKXV IRU HDFK SDLU RI FRQMXJDWH SRLQWV LW LV DSSOLFDEOH ZLWKRXW SUREOHPV $V DQ H[DPSOH RI FRQMXJDWH SRLQWV DW LQILQLW\ OHW XV FRQVLGHU WKHFDVHRIDWHOHVFRSHSRLQWHGDWDGRXEOHVWDUQRZZKDWPDWWHUVLVWKHDQJXODU VHSDUDWLRQDWZKLFKLWVWZRFRPSRQHQWV VD\$DQG% DUHREVHUYHGDQGLQSDUWLF XODUWKHDQJXODUPDJQLILFDWLRQRIWKHWHOHVFRSH)RUWKLVFDVHZHXVHWKHRSWLFDO LQYDULDQWZLWKLWVIRUPLQ(T  DQGDVUHIHUHQFHSODQHVIRULWVDSSOLFDWLRQ ZH WDNH WKRVHRQZKLFK WKH WZR OHQVHV RI WKH WHOHVFRSH OLH ,QSDUWLFXODU ZH DV VXPHWKDWWKHD[LVRIWKHWHOHVFRSHLVSRLQWHGWRZDUGWKHVWDU$,WVUD\VDUHWKHUH IRUHSDUDOOHOWRWKHD[LVDQGLIRQHRIWKHVHUHDFKHVWKHILUVWOHQVZLWKDKHLJKWh IURPWKHD[LVLWH[LWVWKHVHFRQGOHQVZLWKDKHLJKWh'   f  f h 7KXVIURP (T  WKHDQJXODUPDJQLILFDWLRQPDGHE\WKHWHOHVFRSHLVUHDGLO\REWDLQHG IRUDWHOHVFRSHLQDLU  



Ic I

h hc



f  f

 

*HRPHWULFDO2SWLFV



 A

j

D

C

E

F

B f1

G

f2

j'

D' D c

D

f  f



)LJ,QSXWDQGRXWSXWSXSLOVDDQGD ‫މ‬DQGDQJXODUPDJQLILFDWLRQRIDWHOHVFRSH7KH DQJXODUPDJQLILFDWLRQLVGLUHFWO\GHWHUPLQHGE\FRQVLGHULQJWKHWZRULJKWDQJOHGWULDQJOHV ()&DQG*)&LQZKLFKWKHFDWKHWXV)&LVFRPPRQDQGWKHFDWKHWV()DQG)&KDYHOHQJWKV HTXDOWRWKHIRFDOGLVWDQFHVfDQGfRIWKHWZROHQVHVUHVSHFWLYHO\7KHWH[WIROORZVDGLI IHUHQWUHDVRQLQJ

DQG WKLV H[SUHVVLRQ FRLQFLGHV ZLWK WKH DQJXODU PDJQLILFDWLRQ JLYHQ E\ (T  ,WLVLQWHUHVWLQJWRQRWHKHUHWKDWWKHDQJXODUPDJQLILFDWLRQRIWKHVHSDUD WLRQEHWZHHQWKHWZRVWDUVLVSUHFLVHO\WKHUHFLSURFDORIWKHODWHUDOPDJQLILFDWLRQ WKH ³VKULQNLQJ´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remote point of distinct visionRIWKHREVHUYHU¶VH\HZKLFKFRUUHVSRQGVWRWKHVWDWHRIUHOD[DWLRQ RIWKHPXVFOHVRIWKHFU\VWDOOLQHOHQV)RUDQH\HZLWKQRYLVXDOGHIHFWVWKLVSRLQW LVORFDWHGDWLQILQLW\VRWKHUDGLDWLRQFRPLQJIURPDGLVWDQWREMHFWHPHUJHVFROOL PDWHG E\ WKHH\HSLHFH DQG WKH LPDJH IRFDO SRLQWRI WKH REMHFWLYH FRLQFLGHVZLWK WKDWREMHFWRIWKHH\HSLHFH2WKHUZLVHIRUDmyopicH\HWKHOHQVHVDUHDSSURDFKHG SURGXFLQJDGLYHUJLQJHPHUJLQJZDYHZKLOHIRUDhyperopicH\HWKH\DUHPRYHG DZD\DQGWKHHPHUJLQJZDYHLVFRQYHUJLQJ7KHFRQILJXUDWLRQRIWKHWZROHQVHVRI DWHOHVFRSHFDQDOVREHXVHGWRREVHUYHHYHQQHDUE\REMHFWVDVPDJQLILHGLQDVLP LODUZD\WRWKDWZKLFKLVDOVRFUHDWHGZLWKDFRPSRXQGPLFURVFRSH,QWKLVFDVH IRUDKHDOWK\H\HWKHIRFXVFRUUHVSRQGVWRWKHSRVLWLRQIRUZKLFKWKHLPDJHSUR GXFHGE\WKHREMHFWLYHLVORFDWHGLQWKHREMHFWIRFDOSODQHRIWKHH\HSLHFHWKXVWKH DQJXODUPDJQLILFDWLRQLVQRZJLYHQE\ 



s  i  f

 

ZKHUHsiUHSODFHVfDQGFRUUHVSRQGVWRWKHGLVWDQFHRIWKHLPDJHSURGXFHGE\WKH REMHFWLYHIURPLWVLPDJHSULQFLSDOSODQH



3DUD[LDO2SWLFV 

0LFURVFRSHV

)LQDOO\DFDVHVLPLODUWRWKDWRIWKHWHOHVFRSHVLVWKDWRIWKHcompound microscopes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m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fRF WKHUHZRXOGEHDQDSSUR[LPDWHFRUUHVSRQGLQJPDJQLILFDWLRQRImRF FPfRF 7KH RYHUDOO PDJQLILFDWLRQ RI D PLFURVFRSH LV WKHQ JLYHQ E\ mWRW  mREîmRF 7KLVGRHVQRWPHDQWKDWWKHPDJQLILFDWLRQFDQEHLQFUHDVHGDUELWUDULO\E\VHQGLQJ WKHILQDOLPDJHSODQHWRLQILQLW\,QGHHGZHPXVWDOVRFRQVLGHUWKHH\H¶VRSWLFDO V\VWHPDQGWKHLPDJHWKDWLVIRUPHGRQWKHUHWLQDmWRWPXVWUDWKHUEHXQGHUVWRRG s h¢ = X s i1 h = mob h o1 Zi2 h¢= moc h¢ h¢¢= f2

Fo2

Fi2

h¢¢ J

h¢ h Zo1 fo1 so1

fi1

Zi1

Zo2

Zi2

f2

si1

f2

J @ mob h f2

)LJ 5HSUHVHQWDWLRQRIDPLFURVFRSHLQSDUD[LDODSSUR[LPDWLRQZLWKDOORIWKHOHQVHV LPPHUVHGLQDLU7KHREMHFWLYHPDJQLILFDWLRQLVREWDLQHGIURP(T  ZLWKno ni ZKLOHWKDWRIWKHH\HSLHFHLVJLYHQE\(T  6LQFHWKHH\HLVORFDWHGYHU\QHDUWRWKH LPDJHIRFDOSODQHRIWKHH\HSLHFHWKHREMHFWLVREVHUYHGDWDQDSSUR[LPDWHDQJOH-WKDWLV GHVFULEHGE\WKHH[SUHVVLRQVKRZQLQWKHILJXUH

*HRPHWULFDO2SWLFV  DVWKHDQJXODUPDJQLILFDWLRQWKDWRFFXUVZKHQFRPSDULQJWKHREMHFWREVHUYHGZLWK WKH QDNHG H\H XQGHU WKH EHVW FRQGLWLRQV LH FRQYHQWLRQDOO\ DW D GLVWDQFH RI  FPRUWKURXJKWKHPLFURVFRSH 

7KHPHWKRGRIPDWULFHV 2.7.1 2x2 matrices ,Q3DUD[LDO2SWLFVZHFDQXVHDVLPSOHDQGHOHJDQWPHWKRGIRULGHQWLI\LQJWKH EHKDYLRURIDQD[LDOO\FHQWHUHGV\VWHPRIOHQVHVDQGRUPLUURUV,WLVEDVHGRQWKH SURSDJDWLRQRIWKHLQGLYLGXDOUD\VDQGRQWKHLUGHIOHFWLRQDWWKHYDULRXVVXUIDFHV 7KLVPHWKRGLVHTXLYDOHQWWRWKHDSSUR[LPDWHWUHDWPHQWRI3DUD[LDO2SWLFVLQIDFW LWXVHVRQO\UD\VWKDWDUHSDUD[LDODQGFRSODQDUZLWKWKHRSWLFDOD[LV7KH\FDQEH GHVFULEHGZLWKD³KHLJKW´UDQGDQDQJOH-ZLWKZKLFKWKH\LQWHUVHFWDJLYHQSODQH QRUPDOWRWKHD[LVDQGWKHVHFRRUGLQDWHVFDQEHERWKSRVLWLYHDQGQHJDWLYH,QDG GLWLRQVLQ-LVVWLOODSSUR[LPDWHGZLWK-$UD\LVWKXVGHVFULEHGIRUDJLYHQzSR VLWLRQE\DFROXPQYHFWRU )LJ  §U· ¨ - ¸  © ¹



 

,WVSURSDJDWLRQFDQEHGHVFULEHGDVIROORZVWKURXJKPXOWLSOLFDWLRQE\DVXLWDEOH PDWUL[u)RUH[DPSOHLQWKHFDVHRIIUHHSURSDJDWLRQWRDGLVWDQFHdLQDKR PRJHQHRXVPHGLXPZHKDYH § Uc · § U  -d · §  d · § U · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸  © -c ¹ © - ¹ ©   ¹ © - ¹



 

§  d ·  LV FDOOHG WKHtranslation matrix. )RU WKH UHIUDFWLRQ ¨  ¸ © ¹ RQDSODQHVXUIDFHQRUPDOWRWKHD[LVZHKDYH

ZKHUH WKH PDWUL[ T

)LJ 3DUDPHWHUVRIDUD\



J1 r1 z1

z

7KHPHWKRGRIPDWULFHV



)LJ 'HIOHFWLRQ RI D EHDP WKURXJK D GLRSWULF VXU IDFH

n

n' r R

J

J¢ si

so

 § Uc · ¨ -c ¸ © ¹



§ U · ¨n ¸ ¨ -¸ © nc ¹

§  ·§U· ¨ ¸  ¨  n ¸ ¨© - ¸¹ c n © ¹

 

ZKHUHWKHYHFWRUVDUHDVVLJQHGWRWKHVDPHzSRVLWLRQ WKDWRIWKHLQWHUIDFH ³EH IRUH´ DQG ³DIWHU´ WKH UHIUDFWLRQ )RU D VSKHULFDO dioptre ZH FDQ XVH (T   ZLWKWKHVXEVWLWXWLRQV ; U so -  U si -c  )LJ REWDLQLQJ



§ Uc · ¨ ¸ © -c ¹

U § · ¨ n  nc ¸ n ¨ U  -¸ nc ¹ © n cR

§  ¨ n  nc ¨ © ncR

· §U· n ¸¸ ¨ ¸  nc ¹ © ¹

D 

6LPLODUO\IRUDVSKHULFDOPLUURUIURP(TV  DQGZLWKWKHVDPHVXEVWLWX WLRQVZHKDYH 

§ Uc · ¨ -c ¸ © ¹

§ U · ¨ ¸ ¨ U  -¸ ©R ¹

§ ¨ ¨ ©R

· ¸ § U ·  ¨ ¸  ¸ © -¹ ¹

E 

ZKHUH LQ DJUHHPHQW ZLWK (T   WKH DQJOHV DUH SRVLWLYH ZKHQ U LQFUHDVHV SURFHHGLQJLQWKHSRVLWLYHGLUHFWLRQRIWKHEHDPSURSDJDWLRQUHPHPEHULQJWKDWLQ WKHVSDFHVDLGGLUHFWLRQFKDQJHVDWUHIOHFWLRQ,QDFFRUGDQFHZLWKWKHFKRLFHPDGH LQ)LJRLVQHJDWLYHIRUDFRQFDYHPLUURUDQGSRVLWLYHIRUDFRQYH[RQH 7KXVLQJHQHUDODVORQJDVWKHFKDQJHVLQGXFHGE\WKHYDULRXVHOHPHQWVRIDQ RSWLFDOV\VWHPFDQEHDSSUR[LPDWHGE\OLQHDUHTXDWLRQVLQUDQG-HDFKHOHPHQW FDQEHGHVFULEHGE\DPDWUL[uNQRZQDVDtransfer matrixRUABCD matrix 7KHUHIRUHIRUWKHSURSHUWLHVRIPDWULFHVLQSDUD[LDODSSUR[LPDWLRQDQHQWLUHRSWL FDOV\VWHPFDQVWLOOEHGHVFULEHGE\DuPDWUL[WKDWLVWKHSURGXFWLQVXFFHVVLRQ RI WKH PDWULFHV RI WKH LQGLYLGXDO HOHPHQWV DQG WKH YDULRXV SURSDJDWLRQV 7KH UH VXOWLQJABCDPDWUL[H[SUHVVHVWKHWUDQVIRUPDWLRQXQGHUJRQHE\HDFKUD\EHWZHHQ DQ\WZRDVVLJQHGSODQHVnot necessarily conjugatedDPRQJZKLFKWKH PDWUL[LV FDOFXODWHGZKHUHE\ 

§ Uc · § A B · § U · ¨ -c ¸ ¨ C D ¸ ¨ - ¸  © ¹ © ¹© ¹

 

*HRPHWULFDO2SWLFV  7DEOHVKRZVWKHPDWULFHVFRUUHVSRQGLQJWRWKHFRQVWLWXHQWHOHPHQWVRIVLPSOH RSWLFDOV\VWHPV 6XSSRVH QRZ WKDW ZH VWDUW IURP DQ REMHFW SODQH IRU HDFK HOHPHQW GLV WDQFHVXUIDFH ZH PXOWLSO\ WKH UHODWLYH PDWULFHV XS XQWLO WKH ODVW VXUIDFH ZKRVH PDWUL[LVDOVRLQFOXGHGREWDLQLQJDQRYHUDOOABCDPDWUL[:HFDQQRZEHLQWHU HVWHGDVWRZKHUHWKHLPDJHSODQHLVORFDWHGIURPWKHODVWSRVLWLRQUHDFKHGZHLQ GLFDWHWKLVXQNQRZQGLVWDQFHZLWKd7KHSRODUFRRUGLQDWHVLQWKHLPDJHSODQHRID JLYHQLQLWLDOUD\H[SUHVVHGLQWKHREMHFWSODQHE\WKHYHFWRU Uo-o 7DUH 



§ Ui · ¨- ¸ © i¹

§  d · § A B · § Uo · § A  dC ¨  ¸¨C D¸¨- ¸ ¨ C © ¹© ¹© o ¹ ©

B  dD · § Uo · ¨ ¸  D ¸¹ © -o ¹

 

*LYHQWKDWLQWKHLPDJHSODQHUiPXVWQRWGHSHQGRQ-o DOORIWKHUD\VWKDW VWDUWZLWKFRRUGLQDWHGUoRQWKHREMHFWDUULYHZLWKFRRUGLQDWHUiRQWKHLPDJH LWLV QHFHVVDU\WKDWBdD 7KHUHIRUHWKH³PLVVLQJ´GLVWDQFHdIURPWKHLPDJHLV 



d

B  D

 

:HZLOOQRZVKRZWKDW WKHGHWHUPLQDQWRIDQ$%&'PDWUL[LVHTXDOWRWKHUDWLREHWZHHQWKHUHIUDF WLYHLQGLFHVnoDQGniRIWKHLQLWLDODQGILQDOVSDFH 

A

B

C

D

AD  CB

no  ni

 

,Q IDFW RQFH D PDWUL[ S WKDW GHVFULEHV WKH EHKDYLRU RI D V\VWHP EHWZHHQ DQ\ WZRSODQHVKDVEHHQDVVLJQHGWKHRYHUDOOPDWUL[SFEHWZHHQWZRFRQMXJDWHSODQHV LV 

Sc

§ Ac ¨C © c

Bc · §  di · § A B · §  do · Dc ¸¹ ¨©   ¸¹ ¨© C D ¸¹ ¨©   ¸¹

Ti S To 

 

ZKHUHdoDQGdiDUHWKHPLVVLQJGLVWDQFHVEHWZHHQWKHFRQMXJDWHGSODQHVDQGWKH LQLWLDODQGILQDOSODQHVRIWKHPDWUL[ SUHVSHFWLYHO\LQWKHREMHFWDQGLPDJHVSDF HV)URPDOJHEUDLWLVNQRZQWKDWWKHGHWHUPLQDQWRIWKHSURGXFWRIVHYHUDOPDWUL FHVLVHTXDOWRWKHSURGXFWRIWKHLQGLYLGXDOGHWHUPLQDQWVWKHUHIRUHWKHGHWHUPL QDQW RI SF LV HTXDO WR WKDW RI S 2Q WKH RWKHU KDQG DV ZH KDYH VHHQ EHIRUH EHWZHHQWZRFRQMXJDWHSODQHVWKHBcYDOXHLVQXOOIRUZKLFK 

S

Sc

Ac Dc 

,IZHQRZFRQVLGHUWZRUD\VFKDUDFWHUL]HGE\WKHYHFWRUV

 

7KHPHWKRGRIPDWULFHV



d

+RPRJHQRXV PHGLXP

2

1

§ d · ¨  ¸  © ¹

)UHH SDWK EH WZHHQ WKH VXU IDFHV 1DQG 2IRU WKHGLVWDQFH d z±z

§ ¨ ¨ ¨ ©

 · n ¸¸  n ¸¹

2QO\ WKH FURVV LQJ RI WKH VXU IDFH

§ ¨ ¨ ©

d· n¸ ¸ ¹

3DWKEHWZHHQWKH WZR VXUIDFHV 1 DQG 2 LQFOXGLQJ UHIUDFWLRQV

z

3ODQH VXUIDFH EHWZHHQ GLHOHF WULF PHGLD ZLWK LQGH[HV n DQG n

n1

n2 z



d

3ODQH SDUDOOHO 1 SODWH ZLWK LQGH[ n 1

n

1 2 z

'LRSWULF VXUIDFH EHWZHHQ PHGLD ZLWK LQGH[HV n DQG n ZLWK UD GLXV r ,I LW LV FRQYH[ DV LQ WKHILJXUHr! LI LW LV FRQFDYH r

z



/HQV ZLWK IRFDO OHQJWKfEHWZHHQ WZR LGHQWLFDO PHGLD DOVR DS SOLHVWRDPLUURU ZLWK UDGLXV r WDNLQJf r  2SWLFDO ILEHU RU *5,1 OHQV ZLWK D UDGLDOO\ JUDG HG LQGH[ ZLWK SDUDEROLFWUHQG

n2

n1

 § ¨ ¨  § n  · ¸ ¨ r ¨ n ¹ © ©

§  ¨ ¨  ¨ f ©

12

 · ¸ n ¸  n ¸¹

· ¸  ¸¸ ¹

2QO\ WKH FURVV LQJ RI WKH VXU IDFH

3DWKEHWZHHQWKH SULQFLSDO SODQHV 1 DQG 2 LQFOXG LQJUHIUDFWLRQV

z

1 1 n = n0

d

2

I d n n 

§ FRV I ¨ 1 ¨ 1 n r2 ¨  n n VLQ I 2 2  © 

VLQ I · 3DWKEHWZHHQWKH WZR VXUIDFHV 1 n n ¸¸ DQG 2 LQFOXGLQJ FRV I ¸¹ UHIUDFWLRQV

7DEOHABCD PDWUL[IRUVRPHHOHPHQWDU\RSWLFDOV\VWHPV

*HRPHWULFDO2SWLFV



 

§ · § Uo · ¨  ¸ WKHILUVWDQG ¨ - ¸ WKHVHFRQGUD\ © ¹ © o¹

ZHLPPHGLDWHO\UHDOL]HWKDWAcDQGDcDUHUHVSHFWLYHO\WKHODWHUDOPDJQLILFDWLRQ mA DQG WKH DQJXODU PDJQLILFDWLRQ m‘ DV WKH\ KDYH EHHQ GHILQHG LQ † DQG IURP(T  ZHKDYH 

S

mA m‘

no  ni

 

ZKLFKLVZKDWZHZDQWHGWRGHPRQVWUDWH,QSDUWLFXODU WKHPDWULFHVFDOFXODWHGEHWZHHQVSDFHVRIHTXDOUHIUDFWLYHLQGH[KDYHDXQL WDU\GHWHUPLQDQW ,QFRQFOXVLRQEHWZHHQWZRFRQMXJDWHSODQHVWKHFRUUHVSRQGLQJPDWUL[LV 

§ mA ¨  f i ©

 ·  m‘ ¸¹

 

ZKHUHWKHYDOXHRICcLVHDVLO\HYLQFHGIURPWKHWUDFNLQJRIWKHILUVWUD\PHQWLRQHG DERYH 7KH H[SUHVVLRQ   DOVR DSSOLHV LQ SDUWLFXODU EHWZHHQ WKH SULQFLSDO SODQHV RI DQ RSWLFDO V\VWHP IRU ZKLFK mA   DQG LQ WKLV FDVH WKH PDWUL[ EH FRPHV 

§  ¨  f i ©

 ·  no ni ¸¹

 

,QDGGLWLRQWRJHQHUDWLQJWKHPDWUL[EHWZHHQWZRFRQMXJDWHSODQHV(T   FDQDOVREHXVHGWRWUDQVIRUPWKHPDWUL[RIWKHV\VWHPLQWRDQRWKHUSDLURISODQHV WDNHQLQWRKRPRJHQHRXVREMHFWDQGLPDJHVSDFHV:HFDQHDVLO\QRWHWKDWWKHFR HIILFLHQWCUHPDLQVXQFKDQJHGE\WKLVWUDQVIRUPDWLRQLHC CcVRLWVYDOXHLV DOZD\VHTXDOWRfi PniZKHUHPLVWKHSRZHURIWKHLQVWUXPHQW,QSDUWLFX ODUCLV]HURLQWHOHVFRSLFV\VWHPV8OWLPDWHO\UHPDUNDEOHLVWKHPDWUL[EHWZHHQ WKHIRFDOSODQHVZKLFKLV 

§  ¨  f i ©

; fo ·

  ¸¹

 

ZKHUHE\WKHSRVLWLRQDQGLQFOLQDWLRQRIDUD\H[FKDQJHWKHLUUROHV,QIDFWWKHSR VLWLRQ LQFOLQDWLRQ  RQ WKH LPDJH IRFDO SODQH FDQ RQO\ GHSHQG RQ WKH LQFOLQDWLRQ SRVLWLRQ RIWKHUD\RQWKHREMHFWIRFDOSODQHIRUZKLFKWKHGLDJRQDOHOHPHQWVRI WKHPDWUL[EHWZHHQWKHVHWZRSODQHVDUHQXOO)RUDQABCDPDWUL[FDOFXODWHGEH WZHHQDQ\WZRSODQHVZLWKCzWKLVDOORZVXVWRFDOFXODWHWKHGLVWDQFHVd)oDQG

7KHPHWKRGRIPDWULFHV



7DEOH 5HODWLRQEHWZHHQWKHSDUDPHWHUVRI WKH PDWUL[ DQG WKH SRVLWLRQ RI WKH IRFDO DQG SULQFLSDO SRLQWV ZLWK UH VSHFWWRWKHSRLQWV/oDQG/iZKLFK DUH WKH LQWHUVHFWLRQ RI WKH RSWLFDO D[LVZLWKWKHLQLWLDODQGILQDOSODQHV UHVSHFWLYHO\ EHWZHHQ ZKLFK WKH PDWUL[LVFDOFXODWHG

)o /o



D  C

C



A  C

C

/i )i

C



3o /o

C

/i 3i

D no    C ni C



A    C C

d)iPLVVLQJIURPWKHREMHFWDQGLPDJHIRFDOSODQHVUHVSHFWLYHO\ 

d )o



D  d )i C



A  C

 

7DEOH  VXPPDUL]HV WKH UHODWLRQVKLS EHWZHHQ WKH KRPRJUDSK\ RI D QRQ WHOHVFRSLFRSWLFDOV\VWHPDQGWKHSDUDPHWHUVRILWVABCDPDWUL[ (T   FDQ EH LQYHUWHG WR REWDLQ WKH YHFWRU RI WKH LQLWLDO UDGLXV IURP WKH NQRZOHGJHRIWKHILQDORQHIRUZKLFK 

§U· ¨ -¸ © ¹

ni § D ¨ no © C

 B · § Uc ·  A ¸¹ ¨© -c ¸¹

 

ZKHUHWKHUDWLRnino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

ni § D B · ¨ ¸  no © C A ¹

 

2.7.2 Generalized tracing of rays by means of 4x4 matrices in the absence of axial symmetry 6RIDUZHKDYHRQO\WDNHQLQWRDFFRXQWD RSWLFDOV\VWHPVZLWKD[LDOURWDWLRQDO

*HRPHWULFDO2SWLFV  V\PPHWU\ E  UD\V VOLJKWO\ LQFOLQHG WR WKH RSWLFDO D[LV DQG F  VPDOO DQJOHV RI LQ FLGHQFHRQWKHRSWLFDOVXUIDFHV+RZHYHULWLVDOVRSRVVLEOHWRH[WHQGWKHSDUD[LDO WUHDWPHQW WR DV\PPHWULF RSWLFDO V\VWHPV VXFK DV WKRVH WKDW FRQWDLQ F\OLQGULFDO OHQVHV WKDW DUH RULHQWHG DQ\ZD\ DQG WR FDVHV ZKHQ WKH DQJOHV RI LQFLGHQFH DUH ³ODUJH´DVQRUPDOO\KDSSHQVZLWKWKHXVHRISULVPV6XFKDQH[WHQVLRQLVSRVVLEOH LQWKHFDVHRIWKLQpencilsRIUD\VDVKDSSHQVIRUH[DPSOHZLWKDODVHUEHDPLQDQ RSWLFDOULQJFDYLW\HYHQZKHQWKHV\VWHPVDUHQRWSODQDU,QWKHVHFDVHVLQVWHDGRI WKHRSWLFDOD[LVRIWKHV\VWHPZHWDNHDVWKHD[LVWKHcentral RUprincipal UD\RI WKH SHQFLO 6XFK DQ D[LV LV D EURNHQ OLQH FRPSRVHG RI VHYHUDO VHJPHQWV MRLQHG WRJHWKHULQFRUUHVSRQGHQFHZLWKWKHRSWLFDOVXUIDFHV7KHQFRQGLWLRQE EHFRPHV WKDWWKHUD\VRIWKHSHQFLOPDNHVPDOODQJOHVZLWKVXFKDQD[LVDQGFRQGLWLRQF  EHFRPHVWKDWWKHvariations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generalized ray tracingLQWURGXFHGE\$OOYDU*XOOVWUDQGLQWKHHDUO\V >*XOOVWUDQG@7KLVPHWKRGIDFLOLWDWHVLQDGGLWLRQWRWKHWUDFNLQJRIWKHFHQWUDO EHDPWKHFDOFXODWLRQRIWKHVKDSHRIWKHZDYHIURQWLQDVPDOOQHLJKERUKRRGRIWKH VDPH UD\ DQG LQ SDUWLFXODU LWV SULQFLSDO UDGLL RI FXUYDWXUH ,Q WKLV ZD\ ZH FDQ HDVLO\GHWHUPLQHWKHVKDSHRIWKHFDXVWLFQHDUWKHLPDJHSODQHDQGWKHSURSHUWLHV RID*DXVVLDQEHDPOHDYLQJWKHRSWLFDOV\VWHPDVZHVKDOOVHHODWHU 0HDQZKLOH LW LV QHFHVVDU\ WR GHILQH D YHFWRU WKDW H[KDXVWLYHO\ GHVFULEHV DQ\ UD\ RI WKH SHQFLO WKDW LQ DQ RSWLFDO V\VWHP GHYRLG RI V\PPHWU\ LV JHQHUDOO\ DV WLJPDWLF>%RUQDQG:ROI†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



U[  UK -[  -K 

 

ZKHUHWKHYDULDEOHVUDQG-RIWKHSUHYLRXVSDUDJUDSKDUHGHFRPSRVHGLQWRWKHLU

7KHPHWKRGRIPDWULFHV



FRRUGLQDWHVRQWKH][DQGK]SODQHVRIWKHUHIHUHQFHIUDPHFKRVHQIRUHDFKVHJ PHQWWKHSDLUVUjDQG-j ZLWKj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

2.7.3 Algorithm for the calculation of the matrix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laboratory frame L ZKLFK LV XQLTXH IRU ZKLFK ORZHUFDVH FKDUDFWHUV ZLOO KHUHLQ EH XVHG LL  WKH central ray frames Rm HDFK DVVRFLDWHG ZLWK LWV VHJPHQW m LQZKLFK WKH FRRUGLQDWHV DUH H[SUHVVHG E\*UHHN OHWWHUV DQG ZKLFKDUHIURPWLPHWRWLPHWKHUHDIWHUURWDWHGDURXQGWKHGLUHFWLRQRISURSDJD WLRQVRDVWRDJUHHZLWKWKHSODQHVRILQFLGHQFHDWWKHVXUIDFHV LLL WKHframes Sm of each surface m LQ ZKLFKWKH YDULDEOHV DUH H[SUHVVHG LQ XSSHUFDVH FKDUDFWHUV LY ILQDOO\WKHlocal frames lmFHQWHUHGRQWKHSRLQWRILQWHUVHFWLRQRIWKHFHQWUDO UD\ZLWKWKHVXUIDFHmZKLFKZLOOEHQHHGHGRQO\IRUWKHH[SODQDWLRQRIWKHSURE OHP7KHLQGH[mDIIL[HGWRWKHYHUVRUVZLOOLQGLFDWHWKDWWKH\DUHVSHFLILFDOO\DV VRFLDWHGZLWKWKHFHQWUDOUD\EXWLQVRPHYDULDEOHVVXFKDVWKHUDWLREHWZHHQWKH UHIUDFWLYHLQGLFHVDQGDQJOHVRILQFLGHQFHLWZLOOEHRPLWWHGIRUVLPSOLFLW\/DVWO\ WKH YDULRXVYHFWRUVRURSHUDWRUV LQGLFDWHGZLWKEROG FKDUDFWHUV DUH WREH XQGHU VWRRG LQ DQ ³DEVROXWH´ VHQVH ZKLOH WKHLU FRRUGLQDWHV DUH H[SUHVVHG DFFRUGLQJ WR WKHFKRVHQIUDPHRIUHIHUHQFH )LQDOO\,DGPLWWKDWLQWKHOLWHUDWXUHWKHUHDUHRWKHUZD\VWRUHIHUWRWKHYDULD EOHVDQGGHILQHWKHUHIHUHQFHV\VWHPVSHUKDSVWKHPRVWLPSRUWDQWLVWKDWRI21 6WDYURXGLV  

*HRPHWULFDO2SWLFV  D 6\VWHPGHILQLWLRQ)LUVWRIDOOLW LVQHFHVVDU\WRGHILQHWKHRSWLFDOV\VWHPE\ DVVLJQLQJWKHSDUDPHWHUVRIDOORIWKHMRSWLFDOVXUIDFHVDQGWKHUHIUDFWLYHLQGLFHV nmRIWKHLQWHUSRVHGPHGLD7KLVFDQEHGRQHE\H[SUHVVLQJLQ&DUWHVLDQFRRUGL QDWHVWKHYHUWLFHV YRIWKHVXUIDFHVDQGWKHYHUVRUV 1YQRUPDOWRWKHVXUIDFHVRQ WKHLUYHUWLFHVLQWKHV\VWHP L(DFKVXUIDFHFDQDOVREHGHILQHGLQLWVRZQUHIHU HQFHIUDPH SmZKLFKVLPSOLILHVLWVDQDO\WLFDOIRUPDQGLQSDUWLFXODUWKHDVVLJQ PHQWRIWKHSDUDPHWHUVRILWVFXUYDWXUH6XFKDIUDPHFDQEHGHILQHGE\DVVLJQLQJ LWV(XOHUDQJOHVZLWKUHVSHFWWRWKHIUDPHL ,Q SDUWLFXODU OHW XV FRQVLGHU RQO\ VXUIDFHV GHILQHG E\ D continuous IXQFWLRQ ZKLFKLVuniqueDQGcontinuously differentiableLQLWVXDQGYYDULDEOHV 



Z X  Y 

Z

 

ZKHUHXYZDUHWKHFRRUGLQDWHVLQWKHIUDPHSmRIRXUVXUIDFH E  &RRUGLQDWH WUDQVIRUPDWLRQV :H PXVW QRZ GHILQH WZR W\SHV RI WUDQVIRU PDWLRQ7KHILUVWFRQFHUQVWKHURWDWLRQDURXQGDFRPPRQRULJLQEHWZHHQWKHOD ERUDWRU\IUDPHLDQGWKHIUDPHRDVVRFLDWHGZLWKHDFKVHJPHQWRIWKHFHQWUDOUD\ ZKRVHYHUVRULV 57RUHVSRQGWRWKHFRQVWUDLQWVGHVFULEHGDERYHRQFHZHKDYH DVVLJQHGWKHFRPSRQHQWVrxryrzRI5LQWKHIUDPHLZHGHILQHWKHXQLWDU\WUDQV IRUPDWLRQ ZKHQ5GRHVQRWFRLQFLGHZLWKWKHyD[LV DV



Rr

§ r r ¨ z zx ¨ r y r x rzx ¨¨ rx ©

 rzx ry

 r x rzx · ¸ r y rz rzx ¸ ZKHUH rzx ¸¸ rz ¹

  r y  

 

IURPWKHODERUDWRU\IUDPH LZLWKFRRUGLQDWHVxyzWRWKHIUDPH RZLWKFRRUGL QDWHV [ K ] 7KH LQYHUVH WUDQVIRUPDWLRQ LV VLPSO\ LWV WUDQVSRVH 2QH FDQ HDVLO\ YHULI\WKDWWKHGLUHFWLRQRIWKHYHUVRU5FRPHVWRFRLQFLGHZLWKWKH]D[LVDQGWKDW WKH[D[LVOLHVRQWKHzxSODQHDQGLVWKXVSHUSHQGLFXODUWRWKHyD[LVUHVSRQGLQJ WRWKHDIRUHPHQWLRQHGFKRLFHLQWKHSUHYLRXV†IRUWKDWD[LV$IXUWKHUURWDWLRQ DURXQG 5LVWKHQQHFHVVDU\IRUWKHFDOFXODWLRQRIWKHUHIUDFWLRQDQGUHIOHFWLRQDV VWDWHG DW WKH HQG RI † 7KH FDVH ZKHQ 5 FRLQFLGHV RU LV QH[W WR WKH yD[LV VKRXOGEHWUHDWHGVHSDUDWHO\IRUH[DPSOHE\URWDWLQJWKHzDQGyD[HVDURXQGWKH xD[LVE\ƒDQGUHGHILQLQJWKHFRRUGLQDWHVRI 5,QWKLVFDVHWKHWUDQVIRUPDWLRQ LV



Rrc

§ r y ryx ¨ ¨ rz r x ryx ¨¨ © rx

 r x ryx r y rz ryx ry

 · ¸  ryx ¸ ZKHUH ryx ¸ rz ¸¹

  rz   ELV 

$VLPLODUURWDWLRQFDQDOVREHXVHGLQWKHWUDQVIRUPDWLRQVEHWZHHQWKHIUDPHVL

7KHPHWKRGRIPDWULFHV



DQGSmWDNLQJLQSODFHRI5WKHYHUVRU1YQRUPDOWRWKHVXUIDFHDWLWVYHUWH[7R FRPSOHWH WKH URWDWLRQ IURP L WR Sm LW ZLOO VXIILFH WR SHUIRUP D IXUWKHU URWDWLRQ DURXQGWKHZD[LVDORQJZKLFK1YOLHV ,QERWKFDVHVLWZLOOJHQHUDOO\EHQHFHVVDU\WRSHUIRUPWUDQVODWLRQVEHIRUHDQG DIWHUWKHURWDWLRQ F ,QLWLDOFRQGLWLRQV,WLVQHFHVVDU\WRFKRRVHWKHcentral rayDQGGUDZLWWKURXJK WKHRSWLFDOV\VWHP7KHQZHIL[WKHLQLWLDOSRVLWLRQ c DQGGLUHFWLRQ 5RIWKHUD\ LQWKHIUDPHL G 3URSDJDWLRQRIWKHFHQWUDOUD\)RUHDFKVXUIDFHmRIWKHV\VWHPWKHUHDUHD VWDUWLQJSRLQWcmDQGWKHYHUVRU,m 5mRIWKHLQFLGHQWUD\DQGZHKDYHWRFDO FXODWHWKHSRLQWRILQWHUVHFWLRQ cmDQGWKHYHUVRU 5mRIWKHUHIUDFWHGRUUHIOHFWHG UD\ 7R VLPSOLI\ FDOFXODWLRQV LW LV DSSURSULDWH WR XVH WKH WUDQVIRUPDWLRQ RI WKHVH LQLWLDOTXDQWLWLHVLQWKHIUDPH SmRIWKHVXUIDFHDQGFDOFXODWHWKHFRRUGLQDWHVXY ZRIWKHLQWHUVHFWLRQSRLQWLQWKLVIUDPH)RUWKHLUGHWHUPLQDWLRQDQLWHUDWLYHSUR FHGXUHLVJHQHUDOO\UHTXLUHGZKLOHWKHUHDUHDQDO\WLFDOVROXWLRQVIRUWKHFODVVLFDO VXUIDFHV VXFK DV WKRVH RI (T   ,Q DGGLWLRQ ZH PXVW YHULI\ WKDW WKH LQWHU VHFWLQJSRLQWRIWKHUD\ZLWKWKHVXUIDFHLVWKHILUVWDPRQJWKHSRVVLEOHRQHV:LWK DQLQYHUVHWUDQVIRUPDWLRQWRZDUGVWKHIUDPH LZHJHWWKHFRRUGLQDWHVRIWKHQHZ VWDUWLQJSRLQWcm H  ,QFLGHQFH YHUVRUV 'LIIHUHQWLDWLQJ (T   ZH FDQ GHWHUPLQH WKH YHUVRU 1mQRUPDOWRWKHVXUIDFHLQcm



dZ  Z X dX  ZY dY

 

 

ZKHUHWKHFRRUGLQDWHVDUHWKRVHLQWKHIUDPHSmDQGIRUEUHYLW\,LQGLFDWHLQVXE VFULSWV DIWHU WKH ³´ WR GLVWLQJXLVK WKHP IURP WKH QRUPDO LQGLFDWLRQ RI FRRUGL QDWHV WKHSDUWLDOGHULYDWLYHVRIWKHIXQFWLRQZ(T  FDQEHLQWHUSUHWHGDV WKHVFDODUSURGXFWRIDQLQILQLWHVLPDOYHFWRU dXdYdZ 7O\LQJRQWKHVXUIDFHDQG DQRUPDOYHFWRUWRWKLVODVWRQH7KHYHUVRUQRUPDOWRWKHVXUIDFHDWWKHFRRUGLQDWHV XDQGYLVWKHQH[SUHVVHGLQSE\



1

 r

§  Z X ¨ Z ¨ Y ¨  ©

· ¸  ZKHUH ¸ ¸ ¹S

r

Z X   ZY    

 

+LVSRVLWLYHGLUHFWLRQLVDUELWUDU\KHUHLWLVWDNHQDVFRQFRUGDQWZLWKWKHSRVLWLYH GLUHFWLRQRIZ7KHUHIRUHE\FDOFXODWLQJ 1DWWKHFRRUGLQDWHVXYRI cmZHJHW 1m 7KHUHDUHYDULRXVZD\VWRFDOFXODWHWKHUHIOHFWLRQDQGUHIUDFWLRQRQHSRVVLELOLW\ LVDVIROORZV7KHFRVLQHRIWKHDQJOHRILQFLGHQFHLVJLYHQE\  2QHGLIIHUHQWPHWKRGLVIRUH[DPSOHWKDWLQGLFDWHGE\'RQDOG3)HGHU>)HGHU@

*HRPHWULFDO2SWLFV



 

FRV -,

,m < 1m 

 

,WV VLJQ GHSHQGV RQ WKH FRQFRUGDQFH LQ WKH GLUHFWLRQ EHWZHHQ WKHVH WZR YHUVRUV )XUWKHUPRUHWKHSURMHFWLRQRI,mRQWKHWDQJHQWSODQHWRWKHVXUIDFHLQcmLV p ,m  FRV -I 1m 



 

ZKRVHPRGXOXVFRUUHVSRQGVWRWKHVLQHRIWKHDQJOHRILQFLGHQFH 

 VLQ -,

VJQ FRV -,   FRV -,

VJQ FRV -, p 

 

ZKHUH WKH VLJQ IXQFWLRQ VJQ LV LQWURGXFHG IRU FRQYHQLHQFH DV ZH ZLOO VHH ODWHU $SDUWIURPWKHGHJHQHUDWHFDVHRIQRUPDOLQFLGHQFHWKDWPXVWEHWUHDWHGSURSHUO\ WKHincidence planeImLVGHILQHGRQZKLFKWKHYHUVRUV,mDQG1mOLH1RUPDOL]LQJ WKHYHFWRUpZHJHWWKHYHUVRU3mWKDWLVSDUDOOHOWRWKHVXUIDFHDQGWRIm 

3m

p  VLQ -,

 

7KHUHIRUHWKHLQFLGHQFHYHUVRUFDQEHGHFRPSRVHGDV 

,m

VLQ -, 3m  FRV -, 1m 

 

DQGWKHIRUPRIWKLVHTXDWLRQGRHVQRWGHSHQGRQWKHSRVLWLYHGLUHFWLRQRI 1m)L QDOO\E\WKHSURGXFW 

1m u 3m

6m  IURPZKLFKLWIROORZVWKDW 1m

3m u 6m 

 

ZHJHWWKHsagittal versorZKLFKLVSHUSHQGLFXODUWRIm7KHGHILQLWLRQRIVLQ-,LQ (T  LVVXFKWKDWWKHSRVLWLYHGLUHFWLRQRI 6mLVLQGHSHQGHQWIURPWKDWRI 1mZKLOHWKDWRI 3mGHSHQGVRQLW,QWKLVZD\RQO\WKHORFDOIUDPH lLVDIIHFWHG E\WKHFKRLFHRIWKHVLJQRI1 I 9HUVRUVRIUHIUDFWLRQDQGUHIOHFWLRQ:HQRZGHWHUPLQHWKHYHUVRUV 5‫މ‬mDQG 5"m RI WKH UHIUDFWHG DQG UHIOHFWHG UD\V UHVSHFWLYHO\ ,Q WKH FDVH RI refraction IURP6QHOO¶VODZZHREWDLQWKHIROORZLQJSURFHGXUH



 , QRWH KHUH WKDW , KDYH FKRVHQ D GLIIHUHQW FRQYHQWLRQ WKDQ WKDW SUHVHQW LQ WKH OLWHUDWXUH LQ

ZKLFK 6 LV LQVWHDG REWDLQHG IURP WKH QRUPDOL]HG YHFWRU SURGXFW , u 1  _ , u 1 _ IRU ZKLFK WKH GLUHFWLRQRI6GHSHQGVRQWKDWRI1ZKLOH3GRHVQRWGHSHQGRQLW7KLVGHSHQGHQFHLVXVXDOO\ UHPHGLHGE\FKRRVLQJ1VXFKWKDWLWVDQJOHZLWK,LVDFXWH

7KHPHWKRGRIPDWULFHV









P

­°VLQ -5' P VLQ -,  ® °¯FRV -5' 5'm D" D'  C" B' Q  D" C'  C" A' @ > B" D'  A" B' Q  B" C'  A" A' @  

 

ZKHUHLQWKHODVWVWHSEHWZHHQWKHWZRVTXDUHEUDFNHWVWKHLGHQWLW\JLYHQE\WKH  SURGXFW B' Q  A' B' Q  A' ZDVLQVHUWHG 2QHFDVHRISDUWLFXODULQWHUHVWLVWKHRQHLQZKLFKWKHLQFLGHQWZDYHLVJHQHUDW HGE\DVRXUFHSRLQW2WDNHQRQWKHSULQFLSDOUD\VRWKDWLWVZDYHIURQWLVVSKHULFDO ZLWK Q  d  I  ZKHUH I LV WKH LGHQWLW\ DQG d LV WKH GLVWDQFH IURP 2 RI WKH ILUVWVXUIDFHFRQVLGHUHGE\WKHPDWUL[RIWKHV\VWHP,QWKLVFDVHDWWKHH[LWIURP WKHV\VWHPZHKDYH 

Q'

D  d C B  d A   

 

*HRPHWULFDO2SWLFV  DQGIURPWKHGLDJRQDOL]DWLRQRI Qү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үDUHUHVSHFWLYHO\QXOOZHKDYH





Q'

C A   RUQ

 D C 

 

7KHGLVWDQFHVRIWKHIRFDOSRLQWVIURPWKHLQLWLDOSRVLWLRQRIWKHPDWUL[RQWKHSULQ FLSDOUD\LQREMHFWVSDFHDQGIURPWKHILQDOSRVLWLRQLQWKHLPDJHVSDFHPD\EH REWDLQHGDVDERYHE\UHVSHFWLYHO\GLDJRQDOL]LQJQDQGQү2QWKHRWKHUKDQGWKH IRFDO OHQJWKV IRU WKH LPDJH VSDFH RI WKH V\VWHP DUH JLYHQ E\   CW DQG   CV  ZKHUHCDUHWKHSULQFLSDOFXUYDWXUHVFRUUHVSRQGLQJWRC ZLWKWIRUtangentialDQG VIRUsagittal DFKLHYDEOHZLWKDSSURSULDWHURWDWLRQVRQHLQWKHREMHFWVSDFHDQG RQHLQWKHLPDJHVSDFH7KHIRFDOOHQJWKVIRUWKHREMHFWVSDFHDUHREWDLQHGE\PXO WLSO\LQJWKRVHRIWKHLPDJHVSDFHIRUnoni

2.7.6 Symmetry relations and optical path ,QWZRLPSRUWDQWZRUNVZHUHSXEOLVKHGLQGHSHQGHQWO\MXVWWZRPRQWKV DSDUW WKH ILUVW E\ 6$ &ROOLQV DQG WKH VHFRQG E\ -$ $UQDXG HVWDEOLVKLQJ WKH SDUD[LDO SURSHUWLHV RI QRQRUWKRJRQDO RSWLFDO V\VWHPV >&ROOLQV  $UQDXG @ 7KHVHSURSHUWLHV FRQFHUQ LQSDUWLFXODU WKH V\PPHWU\ UHODWLRQVKLSVRI WKH [PDWULFHVGHVFULELQJWKHV\VWHPDQGWKHFDOFXODWLRQRIWKHRSWLFDOSDWKEHWZHHQ WZRJHQHULFSRLQWVRQWKHLQLWLDODQGILQDOSODQHV 

%RWK&ROOLQVDQG$UQDXGGHILQHDQRSWLFDOV\VWHPLQZKLFKWKHUHIUDFWLYHLQGH[KDVDJHQHULF

YDOXH no LPPHGLDWHO\ DIWHU WKH LQLWLDO SODQH  DQG ni LPPHGLDWHO\ EHIRUH WKH ILQDO SODQH 2 +RZHYHU WKHLU DQDO\VLV FDQ EH XQGHUVWRRG E\ DVVXPLQJ WKDW LQLWLDO DQG ILQDO VSDFHV KDYH DQ LQGH[ RI XQLWDU\UHIUDFWLRQ ,Q RWKHU ZRUGV WZR SODQH LQWHUIDFHVZKLFK FRLQFLGHZLWK SODQHV 1 DQG2DUH³DGGHG´WRWKHRSWLFDOV\VWHPZKLFKLPSO\DQLQGH[MXPSWRWKHXQLWDU\YDOXH7KHLU $%&' PDWULFHV LQFOXGH WKHVH HOHPHQWV DQG KDYH LQ IDFW D XQLWDU\ GHWHUPLQDQW +HUH , KDYH SUHIHUUHG WR DYRLG WKLV DGGLWLRQ VR LQ WKLV QRWDWLRQ WKH FRHIILFLHQWV A B C D RI $UQDXG¶V H[SUHVVLRQV DUH UHSODFHG UHVSHFWLYHO\ ZLWK A Bno Cni Dnino )RU WKRVH RI &ROOLQV LW LV

7KHPHWKRGRIPDWULFHV



)RUFRPSOHWHQHVV,PHQWLRQKHUHWKHV\PPHWU\UHODWLRQVKLSVIRUDJHQHULF[ PDWUL[ 7KH\ FDQ EH GHPRQVWUDWHGE\ LQGXFWLRQ E\QRWLQJ WKDW WKH\ DUHYDOLG IRU HDFKHOHPHQWDU\PDWUL[DQGWKDWLIWZR[PDWULFHVUHVSHFWWKHPWKHVHUHODWLRQV ZLOODOVREHSUHVHUYHGIRUWKHLUSURGXFW,QRXUQRWDWLRQWKHUHODWLRQVRI&ROOLQVDUH 

DC 7

CD7 

BA 7

AB7 

D7 B

B7 D C 7 A

A 7 C



 

ZKHUHWKHVXSHUVFULSW7LQGLFDWHVWKHWUDQVSRVH,QRWKHUZRUGV DC7LVDV\P PHWULFPDWUL[ EXWWKLVdoes notLPSO\WKDW D7CLVV\PPHWULFWRR DQGWKHVDPH JRHVIRUWKHRWKHUV)XUWKHUPRUH DA 7  CB7

no I, ni

D7 A

no I ni

 

B7 C



 

ZKHUHnoLVWKHUHIUDFWLYHLQGH[IRUWKHLQLWLDOVSDFHDQGniLVWKHRQHIRUWKHILQDO VSDFHZKLOH IUHSUHVHQWVWKH[LGHQWLW\PDWUL[7KH\FRUUHVSRQGWR(T   IRUWKHGHWHUPLQDQWRI[PDWULFHV &ROOLQVXVHVWKHVHH[SUHVVLRQVWRGHWHUPLQHWKHUHODWLRQVKLSEHWZHHQWKHFRRU GLQDWHVDQGVORSHVRQWKHLQLWLDODQGILQDOSODQHVRIDJHQHULFUD\WKDWFURVVHVWKH V\VWHP+HH[SUHVVHVWKHWZRSDLUVRIFRRUGLQDWHVDQGVORSHVE\WKH XDQGTYHF WRUVWKDWLQWKLVSUHVHQWQRWDWLRQDUHUHVSHFWLYHO\GHILQHGDV 

X7

U x  U y  Ucx  Ucy 

T7

no-x  no- y  ni -cx  ni -cy 

 

7KH PLQXV VLJQV LQ IURQW RI -x DQG -y H[SUHVV WKH VLJQ FKDQJH LQ WKH VORSH WKDW KDSSHQVZKLOHUHYHUVLQJWKHGLUHFWLRQRISURSDJDWLRQLQWKHV\VWHP:LWKWKLVH[ FKDQJHRIYDULDEOHV(T  EHFRPHV 

 D-  -c CU   BAU  Uc

ZKLFKZLWKWKHGHILQLWLRQ  FDQEHUHZULWWHQDV 

§ D no I ni · §C 0 ·   ¨ ¸T ¨ ¸ X  0 ¹ © A I ¹ © B no 

QHFHVVDU\WKDWZHDOVRGRWKHH[FKDQJHVAlDDQGBlC ,UHPHPEHUWKDWIRUHDFKSDLURIPDWULFHVabEHWZHHQZKLFKLWLVSRVVLEOHWRGRWKHSURGXFW

LWLV ab 7 b7a7,QDGGLWLRQIRUHDFKVFDODUcLWLVc c7

*HRPHWULFDO2SWLFV



 ZKHUH0UHSUHVHQWVWKH]HUR[PDWUL[7KHUHIRUH



T

LX  ZKHUH L

§ D no ¨ © B no

I ni · ¸ 0 ¹



§C ¨ ©A

0· ¸  I ¹

 

$IWHU VHYHUDO SDVVDJHV DSSO\LQJ WKH SUHYLRXV V\PPHWU\ UHODWLRQV DV ORQJ DV WKH GHWHUPLQDQWRIBLVGLIIHUHQWIURPZHILQDOO\JHW 

L

§ no B  A ¨ 7 ©  no B

 no B  · ¸  ni DB ¹

 

7KH[PDWUL[ LLVV\PPHWULFVLQFHIRU(T EF  BADQG DBDUH DOVRV\PPHWULF )LQDOO\ &ROOLQV VKRZV WKDW LQ WKH SDUD[LDO DSSUR[LPDWLRQ WKH RSWLFDO SDWK WKURXJKWKHV\VWHPEHWZHHQDSRLQWRIFRRUGLQDWHVU[UKRQWKHLQLWLDOSODQHDQG DSRLQWRIFRRUGLQDWHU‫[މ‬U‫މ‬KRQWKHILQDOSODQHLVJLYHQE\ 

LR 

L

 T X LX  

 

ZKHUH LR LV WKH RSWLFDO SDWK RI WKH SULQFLSDO UD\ EHWZHHQ WKH VDPH SODQHV RQ ZKLFK WKH PDWUL[ LV GHILQHG $V ZH ZLOO VHH ODWHU WKLV DOORZV IRU FDOFXODWLQJ WKH )UHVQHO¶VGLIIUDFWLRQLQWHJUDOVHYHQWKURXJKDQRSWLFDOV\VWHP$UQDXGLQWKHDUWL FOHPHQWLRQHGDERYHH[WHQGVWKLVDQDO\VLVWRWKHSURSDJDWLRQRIODVHUEHDPVDQG WKHVWXG\RIUHVRQDQWFDYLWLHV ,Q KLV GHPRQVWUDWLRQ &ROOLQV XVHV WKH FDOFXODWLRQ RI YDULDWLRQ WR ILQG WKH VWD WLRQDU\SDWKEHWZHHQWKHLQLWLDODQGILQDOSRLQWVWKURXJKWKHRSWLFDOV\VWHP+HUH ZH FDQ IROORZ D VLPSOHU GHPRQVWUDWLRQ XVLQJ WKH UHVXOWV DOUHDG\ REWDLQHG LQ †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r WKDW IURP 2 UHDFKHV D JHQHULF SRLQW 3 RQ WKH LQLWLDO SODQH 7KLV LV SRVVLEOH DV ORQJDVWKHSODQHV1DQG2DUHQRWFRQMXJDWHG.QRZLQJWKHV\VWHPPDWUL[ZHFDQ ZULWH 



AU  B- 

7KHPHWKRGRIPDWULFHV



z2

z1

)LJ 6WXG\RIWKHRSWLFDOSDWK

r2 r1

P O1

r2 r1

no 1

Q

O2 ni

S2 S1

2



ZKHUH-DQGUDUHUHVSHFWLYHO\WKHSRVLWLRQDQGVORSHRIrLQ35HVROYLQJIRU -LQIXQFWLRQRIUZHKDYH - B  AU ZKHUH B  A LVWKHFXUYDWXUHPD WUL[RIWKHVXUIDFHSZKLFKLVWKHUHIRUHGHILQHGE\ 

z

 7  U B AU  

ZKHUH WKH VLJQ RI z LV WDNHQ DV SRVLWLYH IRUZDUGV 7KH RSWLFDO SDWK EHWZHHQ 2 DQG3LVWKHUHIRUHDSSUR[LPDWHGE\L LRnoz/HWXVQRZFRQVLGHUDZDYH JHQHUDWHGLQ3DQGWKHFRUUHVSRQGLQJZDYHIURQW SFORVHWRWKHSODQH 2VXFKWKDW LWLQWHUVHFWVLWLQ2,QWKLVZD\WKHRSWLFDOSDWKEHWZHHQ3DQGSLVVWLOOL7DNH QRZDUD\rIURP3WRDJHQHULFSRLQW4RQSODQH2IRUZKLFKZHKDYH 

U

AU  B-c 

-

CU  D-c 



ZKHUH-DQGUDUHUHVSHFWLYHO\WKHSRVLWLRQDQGVORSHRI rLQ4ZKLOH-‫މ‬LV WKHVORSHLQ35HVROYLQJ-DVDIXQFWLRQRIUZHKDYH 

-





DBU  C  DB A U

DBU 

no 7 B U  ni

ZKHUHLQWKHODVWVWHSWKHV\PPHWU\RI DBDQG(T E ZHUHH[SORLWHG 7KHWHUPLQUUHSUHVHQWVWKHFXUYDWXUHRIWKHVXUIDFH SZKLOHWKHRQHLQUUHS UHVHQWVLWVLQFOLQDWLRQLQ27KLVVXUIDFHLVWKHQGHILQHGE\ 

z

n   U7 DB U  o U7 B7 U  ni 

ZKHUHWKHVLJQRIzLVDOVRSRVLWLYHIRUZDUGV7KHPL[HGWHUPLQUDQGUEHLQJ VFDODUKDVWKHVDPHYDOXHDVLWVWUDQVSRVHDQGFDQEHVXEGLYLGHGLQWRWZRWHUPVRI HTXDOYDOXH)LQDOO\WKHRSWLFDOSDWKEHWZHHQ3DQG4DOVRFDOOHGWKHpoint characteristicLVDSSUR[LPDWHGE\

*HRPHWULFDO2SWLFV



 L

L  ni z

     LR  no U7 B  AU  no U7 B U  no U7 B7 U  ni U7 DBU      ZKLFKUHSURGXFHV(T  ,IWKHWZRSODQHVZHUHFRQMXJDWHGHYHQRQO\DV WLJPDWLFDOO\IRUDIRFDOOLQHBZRXOGQRWEHLQYHUWLEOHDQG(T  ZRXOGQRW EHDSSOLFDEOH /DVWO\,UHFDOORQHRI$UQDXG¶VGHILQLWLRQFRQFHUQLQJWKH³VFDODUSURGXFW´EH WZHHQWZRUD\VR7{ U U7n-7 DQGR‫މ‬7{ U U‫މ‬7n-‫މ‬7 FURVVLQJWKHV\VWHP 

R  R c

nU7 -c  nUc7 - 

 

ZKHUHnLVWKHUHIUDFWLRQLQGH[$SSO\LQJWKHWUDQVIRUPDWLRQUXOHVIURPSODQH 1WR SODQH 2IRUWKHYDULDEOHVnU7-7U‫މ‬7-‫މ‬7ZHILQGWKDWWKLVTXDQWLW\LVFRQVWDQW ,WFRQVWLWXWHVDJHQHUDOL]DWLRQRIWKH6PLWK+HOPKROW]/DJUDQJHLQYDULDQWLQSDU D[LDODSSUR[LPDWLRQ

$SHUWXUHVRIDQRSWLFDOV\VWHP 2.8.1 Aperture diaphragm and the entrance and exit pupils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in focusDOWKRXJKDVZHKDYHVHHQWKHJUHDWHVWORQJLWXGLQDOVKULQNDJHFRPSDUHG WRWKHODWHUDORQHLVRIJUHDWKHOSLQFUHDWLQJSKRWRJUDSKV+HUHWKHQZHZLOOVHH WKDWDGLDSKUDJPFDQEHLQVHUWHGLQWRWKHRSWLFV\VWHPWRJUDGXDWHWKHVHOHFWLRQRI WKHUD\VXSWRWKHGHVLUHGVKDUSQHVVRUdepth of field ,QDQ\FDVHZLOOLQJO\RUQRWLWLVQHFHVVDU\WRWDNHQRWHRIWKHYDULRXVDSHUWXUHV SUHVHQWLQRSWLFDOV\VWHPV%\ZD\RIH[DPSOHZHFRQVLGHUWKHFDVHRIDV\VWHP FRQVLVWLQJRIWZROHQVHVDQGDGLDSKUDJPLQWHUSRVHGEHWZHHQWKHP )LJ ,Q JHQHUDOWKHEHDPUHVWULFWLRQRFFXUVGLIIHUHQWO\IRUWKHUD\VFRPLQJRXWRIWKHGLI

$SHUWXUHVRIDQRSWLFDOV\VWHP

entrance pupil C¢ B¢

O

A

Fo1 B¢





aperture diaphragm

Fo2

Fi1 marginal ray

L1 A

C

B

Fi2

B¢¢

O¢¢ A¢¢

L2 B

A¢¢

exit pupil

C

B¢¢



)LJ'LDSKUDJPVRIDQRSWLFDOV\VWHP

IHUHQWSRLQWVRIWKHREMHFW7KHGLDSKUDJPWKDWPRVWOLPLWVWKHEHDPIURPDJLYHQ SRLQW 2 RQ WKH D[LVRI WKH V\VWHP LV FDOOHG WKHaperture diaphragm DQGWKH UD\V JUD]LQJLWVHGJHDUHFDOOHGmarginal,QWKHH[DPSOHIURP)LJWKHUROHRIDS HUWXUHGLDSKUDJP WKHRQHWKDWPRVWOLPLWVWKHEHDPRIOLJKWIURPWKHSRLQW2 LV DVVXPHG E\ WKH GLDSKUDJP %% $ XVHIXO ZD\ WR VWXG\ WKH HIIHFWV RI WKH YDULRXV DSHUWXUHVLVWRORFDWHWKHLPDJHV UHDORUYLUWXDO RIHDFKHOHPHQWRIWKHLQVWUXPHQW LQWKHREMHFWVSDFHRUWKHLPDJHVSDFH7KXVLQWKHILJXUHWKHLPDJH%‫މ‬%‫މ‬RIWKH GLDSKUDJP%%DQGWKHLPDJHRIWKHOHQV/DUHGUDZQLQWKHREMHFWVSDFHZKLOH LQWKHLPDJHVSDFHWKHLPDJH%‫ މމ‬%‫ މމ‬RIWKHGLDSKUDJPDQGWKDWRIWKHOHQV/DUH VKRZQ7KHUHDODSHUWXUHRULWVLPDJHWKDWOLPLWVWKHHQWU\EHDPPRUHWKDQDOORI WKHRWKHURSHQLQJVDVWKH\DUHVHHQIURPWKHREMHFWSRLQW2LVFDOOHGWKHentrance pupil7KHDSHUWXUHRULWVLPDJHWKDWOLPLWVWKHRXWJRLQJEHDPIURPWKHRSWLFDOV\V WHPLVFDOOHGWKHexit pupil,WLVREYLRXVWKDWWKHHQWU\DQGH[LWSXSLOVDUHFRQMX JDWHGWRHDFKRWKHUE\WKHHQWLUHRSWLFDOV\VWHP ,QVRPHLPSRUWDQWFDVHVLQZKLFKWKHREMHFWWREHUHSURGXFHGLVDQLOOXPLQDWHG DSHUWXUHHYHQDVOLGHRUWUDQVSDUHQF\WKHDSHUWXUHGLDSKUDJPPD\EHWKHIURQWLHU RIWKHOLJKWVRXUFHRUWKHHOHPHQWV FRQGHQVHUOHQVDPRQJRWKHUV XVHGIRUOLJKW LQJDQGPXVWEHWDNHQLQWRDFFRXQWZKHQGHWHUPLQLQJWKHSXSLOV,QDQ\FDVHWKH EULJKWQHVVRIWKHLPDJHGHSHQGVRQWKHDUHDRIWKHSXSLOZKLFKLVSURSRUWLRQDOWR WKHDUHDRIWKHDSHUWXUHGLDSKUDJP

2.8.2 Relative aperture, or f-number, and numerical aperture $Q\RQHZKRKDVXVHGDQROGQRUPDOFDPHUD QRWRQHWKDWGRHVHYHU\WKLQJDOO E\LWVHOI PD\KDYHQRWHGWKDWWKHGLDSKUDJPSRVLWLRQV ZKLFKOLPLWWKHXVHIXOGL DPHWHU D RI WKH REMHFWLYH  DUH LQGLFDWHG E\ D FKDUDFWHULVWLF VHULHV RI QXPEHUV 7KH\JURZVWHSE\VWHSE\DIDFWRURIDERXW  FRUUHVSRQGLQJWRDGHFUHDVHRID IDFWRURILQWKHFROOHFWHGOLJKW,QGHHGHYHU\VWHSIRUZDUGRIWKHGLDSKUDJPFDQ EHFRXQWHUEDODQFHGE\DGRXEOLQJRIH[SRVXUHWLPH,I IÖo LVWKHLQWHQVLW\HPLWWHG IRUZDUGIRUWKHXQLWVROLGDQJOHIURPDVXUIDFHHOHPHQW AoWKHWRWDOLQWHQVLW\UH FHLYHGRQLWVLPDJHAiDFFRUGLQJWR(T  LV

*HRPHWULFDO2SWLFV





exit pupil

)LJ 'HWHUPLQLQJ WKH GHSWK RIILHOG

film ~f I

D I¢



S D  Ao I i # IÖo  so  Ai

D si

d



S D Ö S D IÖo  # Io  si   f

ZKHUH WKH ILQDO VWHS LV YDOLG LQ QRUPDO FRQGLWLRQV RI WKH XVH RI D SKRWR FDPHUD 7KH QXPEHUV LQGLFDWHG RQ WKH GLDSKUDJP DUH WKXV WKH GLPHQVLRQOHVV UDWLR fD FDOOHGWKHrelative apertureRUf-number 

f QXPEHU {

f  D

 

7KLVLVXVHGWRFKDUDFWHUL]HWKHOHQVHVLQSDUWLFXODUWRLQGLFDWHWKHLUDELOLW\WR FRQFHQWUDWHWKHOLJKW7KHV\PEROf LVDSHFXOLDUZD\WRLQGLFDWHWKHfQXPEHURI DOHQVIRUH[DPSOHDOHQVZLWKPPIRFDOOHQJWKDQGDPPDSHUWXUHLVLQGL FDWHGE\WKHV\PEROfZKHUHWKHVWDQGVIRULWVfQXPEHU,QDJUHHPHQWZLWKWKH SKRWRJUDSKLF REMHFWLYHV D OHQV LV FDOOHGfast LI LW KDV D VPDOOfQXPEHU EHFDXVH ZLWKLWWKHH[SRVXUHWLPHZRXOGEHVPDOOHUDQGslowLQWKHRSSRVLWHFDVH 7KH numerical aperture DEEUHYLDWHG WR 1$  LV E\ GHILQLWLRQ WKH sine RI WKH DQJOH -mWKDWDPDUJLQDOUD\SDVVLQJWKURXJKWKHIRFDOSRLQWPDNHVZLWKWKHRSWL FDOD[LV 

1$

VLQ -m |

   f QXPEHU

 

ZKHUHWKHODVWHTXDOLW\KROGVonlyLQWKHSDUD[LDOOLPLWIRUDVPDOODQJOH /DVWO\DQRGWRWKHdepth of field )LJ :LWKDJRRGREMHFWLYHWKHSRLQWV RQWKHREMHFWSODQH OFRQMXJDWHGWRWKHILOPDUHVKDUSO\UHSURGXFHGLQVWHDGWKH SRLQWVRQDQRWKHUSODQH O ‫މ‬SURGXFHDFLUFXODUVSRW LIWKHGLDSKUDJPLV FLUFXODU  ZLWKDGLDPHWHU GWKDWJURZVZLWKWKHGLVWDQFH 'soEHWZHHQWKHSODQHV ODQG O ‫މ‬ DQGSURSRUWLRQDOO\WRWKHGLDPHWHUDRIWKHGLDSKUDJP1RZIRUVLPSOLFLW\ZHDV VLPLODWHWKHREMHFWLYHWRDWKLQOHQVZLWKWKHGLDSKUDJPFRLQFLGLQJZLWKLW7DNLQJ WKH GLVWDQFHV IURP WKH IRFDO SODQHV DQG UHPHPEHULQJ WKH +X\JHQV1HZWRQ (T E IRUZKLFK'Zi f  '  Zo ZHKDYH 

G#

D 'si f

§  · §· Df ' ¨ ¸ # Df ' ¨ ¸  © Zo ¹ © so ¹

$SHUWXUHVRIDQRSWLFDOV\VWHP



A O¢¢

O

O¢ S

f



)LJ   7HOHFHQWULF V\VWHP WKH DSHUWXUH GLDSKUDJP $ LV ORFDWHG RQ WKH VHFRQG IRFDO SODQH RIWKH RSWLFDO HOHPHQWVWKDW SUHFHGH LW LQWKLV FDVH WKH ILUVW OHQV 7KH KHLJKW RIWKH SRLQW2‫މމ‬GRHVQRWGHSHQGRQWKHORQJLWXGLQDOSRVLWLRQRIWKHSRLQW2

$QDOWHUQDWLYHZD\WRGRWKLVZRXOGEHWRXVH(T  RIORQJLWXGLQDOPDJ QLILFDWLRQ ,I ZH QRZ IL[ D PD[LPXP YDOXH IRU G VD\  PP ZH FDQ JHW WKH DPSOLWXGHRIWKHLQWHUYDOV 'so 'soDURXQGsoLHWKHGHSWKRIILHOGVRWKHLP DJHZLOOVWLOOEHVXIILFLHQWO\VKDUS 

'so #

 G   so Df

 so  'so  #

 G   so Df

 so 

 

:LWKWKHYDOXHRI GFRQVLGHUHGDERYHDQGZLWKf FP f  f  so P WKHRYHUDOOGHSWKRIILHOGLV|±PlP

2.8.3 Principal rays 7KHUD\VWKDWSDVVWKURXJKWKHFHQWHURIWKHDSHUWXUHGLDSKUDJPDUHFDOOHGprincipal rays.,QWKHSDUD[LDODSSUR[LPDWLRQWKH\DOVRSDVVWKURXJKWKHFHQWHURIWKH HQWUDQFHDQGH[LWSXSLOV7KHVHFHQWHUVFDQEHWKRXJKWRIDVWKHSURMHFWLRQFHQWHUV EHWZHHQ REMHFW DQG LPDJH ZLWK DQ DQJXODU PDJQLILFDWLRQ UDWLR JLYHQ E\ (T  ZKHUHmALVWKHPDJQLILFDWLRQEHWZHHQWKHSXSLOV

2.8.4 Field diaphragm and entrance and exit windows 7KHSRVLWLRQRIWKHHQWUDQFHSXSLOZLWKUHVSHFWWRWKHSRVLWLRQRIWKHREMHFWKDV DFHUWDLQLPSRUWDQFH+HUH,ZDQWWRSRLQWRXWRQO\RQHSDUWLFXODUFDVHZKHQWKH DSHUWXUHGLDSKUDJPLVORFDWHGLQWKHIRFDOSODQHRIWKHVHWRIV\VWHPHOHPHQWVWKDW SUHFHGHGLWWKHQWKHHQWUDQFHSXSLOLVORFDWHGDWLQILQLW\&RQVHTXHQWO\DOORIWKH SULQFLSDOUD\VLQWKHREMHFWVSDFHDUHSDUDOOHOWRWKHRSWLFDOD[LV8QGHUWKHVHFRQ GLWLRQVWKHV\VWHPLVFDOOHGtelecentric7KLVW\SHRIFRQILJXUDWLRQILQGVDSSOLFD

*HRPHWULFDO2SWLFV





entrance windows

aperture diaphragm

field diaphragm

exit windows

O¢¢ O entrance pupil



)LJ)LHOGGLDSKUDJPDQGHQWUDQFHDQGH[LWZLQGRZV

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‫މ‬DQG2 ‫މމ‬PRYHDORQJWKH VDPH UD\ LQ WKH UHVSHFWLYH VSDFHV DQG WKH OLJKW VSRW RQ WKH VFUHHQ HQODUJHV EXW GRHVQRWWUDQVODWHDORQJWKHVFUHHQ,IZHKDYHWZRSRLQWV2WKHFHQWHUVRIWKHUH VSHFWLYHVSRWVRQWKHVFUHHQUHPDLQDWWKHVDPHGLVWDQFH :HKDYHXQGHUVWRRGWKDWWKHDSHUWXUHGLDSKUDJPFRUUHVSRQGVWRRQHRIWKHV\V WHPDSHUWXUHVWKDWLVWKHsmallerRQHDVLV³VHHQ´IURPWKHREMHFWSODQH7KHUH PDLQLQJDSHUWXUHVKDYHWKHHIIHFWRIOLPLWLQJWKHfield of viewRIWKHLQVWUXPHQW,Q SDUWLFXODUWKHGLDSKUDJPWKDWPRVWOLPLWVWKHDQJXODUDSHUWXUHRIWKHSULQFLSDOUD\V LVFDOOHGWKHfield diaphragmLWVLPDJHLQWKHREMHFWVSDFHLVFDOOHGWKHentrance windowDQGVLPLODUO\LQWKHLPDJHVSDFHWKHUHLVDQexit windowFRQMXJDWHGWRLW )LJ 

2.8.5 Vignetting 7KHSULQFLSDOUD\VFRQVWLWXWHRQO\DSDUWLFXODUVHWRIWKHPXOWLWXGHWKDWFRQWULE XWHVWRLPDJHIRUPDWLRQDQGWKHILQLWHVL]HRIWKHHQWUDQFHSXSLOJHQHUDOO\VKDGHV WKHHGJHRIWKHLQVWUXPHQW¶VILHOGRIYLHZ,QRWKHUZRUGVWKHYDULRXVSRLQWVRIWKH REMHFWLOOXPLQDWHWKHLPDJHSODQHZLWKDQHIILFLHQF\WKDWGHSHQGVRQWKHLUSRVLWLRQ DQGWKDWJHQHUDOO\GHFUHDVHVFRQWLQXRXVO\IURPWKHD[LVWRWKHHGJH7KLVLVGXHWR WKH VKDGRZ DQG SHQXPEUD HIIHFWV SURGXFHG E\ WKH VXFFHVVLRQ RI RSHQLQJV )LJ  7KLVYDULDWLRQRILOOXPLQDWLRQSURGXFHGE\WKHLQVWUXPHQWLVFDOOHGvignetting7RDYRLGWKLVHIIHFWWKHRXWSXWZLQGRZLVSODFHGRQWKHLPDJHSODQH DQGDW WKH VDPH WLPH WKH LQSXW ZLQGRZ LV SODFHG RQ WKH REMHFW SODQH  +RZHYHU WKLV GRHVQRWDOZD\VHQVXUHWKDWDWKLUGRSHQLQJZLOOQRWSURGXFHDQ\RWKHUYLJQHWWLQJ

$EHUUDWLRQV



image space

)LJ 9LJQHWWLQJ

shadow

penumbra light

pupil

window

image



7\SLFDOO\ LQ YLVXDO LQVWUXPHQWV ³DGMXVWHG IRU LQILQLW\´ WKH ILHOG GLDSKUDJP LV SODFHGLQWKHREMHFWIRFDOSODQHRIWKHH\HSLHFH

$EHUUDWLRQV 6RIDUZHKDYHOLPLWHGRXUVHOYHVWRFRQVLGHULQJRSWLFDOV\VWHPVLQWKHSDUD[LDO DSSUR[LPDWLRQIRUZKLFKHDFKSRLQWRIDQREMHFWLVUHSURGXFHGLQD³VKDUS´PDQ QHUWKDWLVZHKDYHFRQVLGHUHGLGHDORSWLFDOV\VWHPVLQZKLFKDOORIWKHUD\VWKDW VWDUWIURPDQREMHFWSRLQWDUHGHIOHFWHGWRDVLQJOHSRLQWLPDJH7KLVLVXVXDOO\RQ O\SRVVLEOHE\OLPLWLQJWKHILHOGRIWKHLQVWUXPHQWWKXVOLPLWLQJWKHFROOHFWHGOLJKW HQHUJ\RULQSDUWLFXODUFDVHVIRUZKLFK)HUPDW¶VSULQFLSOHRIWKHHTXDOLW\RIWKH RSWLFDOSDWKVRIDOORIWKHUD\VWKDWIRUPWKHLPDJHLVH[DFWO\DSSOLHG2IWKHVHFDV HVZHKDYHVHHQRUUHFDOOHGVRPHH[DPSOHVRIDVSKHULFVXUIDFHVDSODQDWLFOHQVHV DQGPHGLDZLWKYDULDEOHUHIUDFWLYHLQGH[HV ,QRUGHUWRKDYHOXPLQRXVRSWLFDOV\VWHPVLWLVRIWHQQHFHVVDU\WRDEDQGRQWKH SDUD[LDOWKHRU\+RZHYHULWLVQRWDOZD\VSRVVLEOHWRXVHSDUWLFXODURSWLFDOFRP SRQHQWVERWKIRUFRQYHQLHQFHDQGIRUHFRQRP\6\VWHPVZLWKDVSKHULFDOVXUIDFHV DUHQRWIUHHIURPGHIHFWVIRUREMHFWSRLQWVWKDWDUHQRWLQWKHVSHFLDOSRVLWLRQVIRU ZKLFKWKHVXUIDFHVDUHPDGHIRUH[DPSOHSRLQWVQRWQHDUWKHRSWLFDOD[LV0RUHR YHU DW OHDVW IRU QRZ DVSKHULFDO VXUIDFHVRU YDULDEOH LQGH[ PHGLD FDQQRW EH SUR GXFHG ZLWK JRRG RSWLFDO TXDOLW\ XQOHVV DW D KLJK FRVW VR WKH\ DUH RIWHQ MXVWLILHG RQO\ LQ VSHFLDO DSSOLFDWLRQV WKH PRVW UHPDUNDEOH H[DPSOH LV WKH FDVH RI DVWUR QRPLFDO WHOHVFRSHV &RQYHUVHO\ WKH SURGXFWLRQ RI KRPRJHQHRXV RSWLFDO JODVVHV ZLWK RSWLFDOO\ SROLVKHG VSKHULFDO VXUIDFHV KDV UHDFKHG D KLJK OHYHO RI H[FHOOHQFH DQGJRRGHFRQRPLFHIILFLHQF\ ,QIDFWLWLVQRWSRVVLEOHWRSURGXFHVXFKDVKDUSDQGGLVWRUWLRQIUHHLPDJHDV WKHRQHSUHGLFWHGE\WKHKRPRJUDSK\LQZKLFKDOORIWKHUD\VRIDQREMHFWSRLQW DUH GHIOHFWHG VR LGHDOO\ 7KH GLVFUHSDQFLHV EHWZHHQ WKH UHDO DQG WKH LGHDO LPDJH JHQHUDOO\WDNHWKHQDPHRIaberrations+RZHYHULWPXVWEHQRWHGWKDWGHIHFWVLQ UHDOLPDJHVFDQKDYHPXOWLSOHFDXVHV7KHILUVWGHIHFWVFRPHIURPWKHLUUHJXODULW\ RIWKHRSWLFDOVXUIDFHVDQGWKHLQKRPRJHQHLW\RIWKHRSWLFDO PDWHULDOV 7KHUHDUH DOVRerrors of positioningLQWKHPHFKDQLFDOPRXQWLQJLQWKHV\VWHP7KHVHGHIHFWV DUHLQGLYLGXDOLQWKHVHQVHWKDWWKH\YDU\IURPVDPSOHWRVDPSOH7KHVHFRQGW\SH RIGHIHFWLVFODVVLILHGDVaberrationsLQWKHVWULFWHVWVHQVHEHLQJWKDWWKH\DUHLQ

*HRPHWULFDO2SWLFV  KHUHQWWRWKHJLYHQW\SHRUGHVLJQRIWKHRSWLFDOV\VWHPDQGGRQRWGHSHQGRQLWV UHDOL]DWLRQ7KHVHLQWXUQDUHGLYLGHGLQWRWZRJURXSVWKHchromaticDEHUUDWLRQV ZKLFKGHSHQGRQWKHIDFWWKDWWKHUHIUDFWLYHLQGH[HVYDU\ZLWKWKHZDYHOHQJWKDQG WKHmonochromatic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pupilZKLFKZKHQ SODFHGLQIURQWRIDWUDQVSDUHQWVSKHUHDOORZHGWKHIRUPDWLRQRIVKDUSLPDJHV Dioptrice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third degree aberrationsWR ZKLFKZHZLOOVRRQVHHDEULHIQRG7KHQXSRQWKHGHYHORSPHQWRIWKHVHDQDO\WL FDO PHWKRGV ( $EEH FRQWULEXWHG E\ LGHQWLI\LQJ D SDUWLFXODU FRQGLWLRQ ZKLFK WDNHVKLVQDPHIRUWKHUHGXFWLRQRIRQHRIWKH6HLGHODEHUUDWLRQVWKHcoma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thePLQLPXPLIQRWWKHH[SHUL HQFHDQGRUWKHUHSHDWHGWHVWVLQRWKHUSDUDPHWHUUHJLRQV 7KHUHDUHEDVLFDOO\WZRZD\VWRFKDUDFWHUL]HDEHUUDWLRQV7KHILUVWLVWRUHFRQ 

$EHUUDWLRQV



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spot diagram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†  7KH FKRLFH WR FRQVLGHU WKH H[LW SXSLO DV D SODFH RQ ZKLFK WR GHWHUPLQH WKH ZDYHIURQWHPHUJLQJIURPWKHLQVWUXPHQWPD\VHHPDUELWUDU\EXWLWLVQRW,WUHSUH VHQWVWKHDSHUWXUHGLDSKUDJP RULWVLPDJH ZKLFKLVGHFLVLYHLQYDU\LQJWKHPDJ QLWXGHRIWKHDEHUUDWLRQVERWKE\LWVVL]HDQGLWVSRVLWLRQ

2.9.1 The aberration functions for an axial optical system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email protected]EHWZHHQWKHVHWZRSRLQWVUHSUHVHQWVDPHDV XUHRIUD\DEHUUDWLRQ,WFDQEHZULWWHQDV /  >[email protected]± >2)@  >[email protected]± >[email protected]ZKHUH ZH XVHG WKH IDFW WKDW WKH SRLQWV & DQG ) DUH RQ WKH VDPH ZDYHIURQW DQG LW LV D IXQFWLRQRIWKHFRRUGLQDWHVRIWKHSRLQWV2DQG6%\FKRRVLQJWKHRSWLFDOD[LVDV WKHzD[LVWKHDEHUUDWLRQIXQFWLRQRIWKHZDYHIURQWLVJLYHQE\ 

/ V xo  yo  zo  xs  ys  zs  V xo  yo  zo   zc 

 

ZKHUH2  xoyozo 6  xsyszs DQG&  zc ZKLOHVLVWKHpoint characteristic>%RUQDQG:ROI@LQWURGXFHGE\+DPLOWRQDQGLVVLPSO\WKHRSWLFDO SDWKEHWZHHQWKHWZRH[WUHPHVRIWKHUD\IURPZKLFKLWLVGHWHUPLQHG(T  

*HRPHWULFDO2SWLFV



 yo

ys xo

yi C

O

xs

xi Oi

z

Oo

object plane

plane of the entrance pupil

I

plane of the exit pupil

image plane

z



)LJ&KDUDFWHULVWLFSODQHVIRUGHWHUPLQLQJDEHUUDWLRQV

DOORZVXVWRKLJKOLJKWWKHGHSHQGHQFHRI /IURPWKHFRRUGLQDWHVRIWKHSRLQWV2 DQG62QWKHRWKHUKDQGWKHFRRUGLQDWHVRI6DUHQRWLQGHSHQGHQWEHFDXVHWKH\ KDYHWRUHVSHFWWKHFRQGLWLRQ

xs  xi   ys  yi   zs  zi 



R  

 

ZKHUH ,  xi yi zi  LV WKH SDUD[LDO LPDJH SRLQW RI 2 DQG R LV WKH UDGLXV RI WKH VSKHUHS ,IZHIL[WKHSRVLWLRQRIWKHREMHFWSODQHWKDWRIWKHLPDJHSODQHLVDOVRGHWHU PLQHG /HW XVWDNH VHSDUDWH RULJLQVRQ WKHVH SODQHVUHVSHFWLYHO\ 2o DQG 2iIRU WKHREMHFWVSDFHDQGWKHLPDJHVSDFH )LJ 6RZHKDYHzo zi DQGzc ±dZKHUHdLVWKHGLVWDQFHEHWZHHQLPDJHDQGSXSLOSODQH,QFRQFOXVLRQ/FDQ EHGHVFULEHGRQO\LQWHUPVRIWKHWUDQVYHUVDOFRRUGLQDWHVRI2DQG6 

/

/ xo  yo  xs  y s 

 

2QWKHRWKHUKDQGxoyoDQGxsysUHSUHVHQWWKHFRRUGLQDWHVRIWZRYHFWRUV ro DQGrsUHVSHFWLYHO\)RUWKHURWDWLRQDOV\PPHWU\DVVXPHGIRUWKHV\VWHPZHKDYH WKDW/GRHVQRWYDU\LIWKHVHYHFWRUVVLPXOWDQHRXVO\URWDWHDURXQGWKHRSWLFDOD[LV 6R/FDQRQO\GHSHQGRQ 

ro 

xo   yo   rs 

xs   y s   ro x rs

ro rs FRV T

xo xs  yo y s   

ZKHUH TLVWKHD]LPXWKDODQJOHEHWZHHQWKHWZRYHFWRUV7KHUHIRUH /PXVWEHD IXQFWLRQRIHYHQGHJUHHLQWKHFRRUGLQDWHV,IZHGHYHORSLWLQVHULHVZHKDYHRQO\ 

1RUPDOO\LQWH[WVWKDWGHDOZLWK WKHDEHUUDWLRQWKHRU\LQGHSWKWKHVSHFLDO6HLGHOYDULDEOHV DUHXVHGZKLFKLQSDUWLFXODUUHSODFHWKHRUGLQDU\FRRUGLQDWHVWDNHQRQWKHSXSLOVDQGWKHREMHFW DQGLPDJHSODQHVVXLWDEO\UHVFDOLQJWKHP)RUWKHVDNHRIVLPSOLFLW\WKLVUHVFDOLQJLVQRWXVHG KHUH DQG WKHUHIRUH WKH SRO\QRPLDO FRHIILFLHQWV RI WKH DEHUUDWLRQ IXQFWLRQ GR QRW FRLQFLGH ZLWK WKRVHUHSRUWHGLQWKHVHWH[WVEXWPXVWDOVREHUHVFDOHGIRUDFRPSDULVRQ

$EHUUDWLRQV



)LJ $EHUUDWLRQ RI WKH ZDYHIURQW DQG DEHUUDWLRQ RI WKH UD\V 7KH ),' UD\ GRHV QRW QHF HVVDULO\ OLH RQ WKH PHULGLDQ SODQHRIWKHSRLQW,

F



S

F

I¢ I

R

S

Oi

zc

z

C pupil S

image plane

F

d



WHUPVRIHYHQGHJUHH 

/

/   /   /   /   ! 

 

,QSDUWLFXODUWKHGHYHORSPHQWRI/LQSRODUFRRUGLQDWHVXSWRWKHIRXUWKGHJUHHLV / # /   /  ro   /  ro  



/  rs   / ro rs FRV T  /  rs   / ro rs  FRV T  /  ro  rs  FRV T 



 

/  ro  rs   /  rors FRV T

ZKHUHWKHLQGH[HVRIWKHFRHIILFLHQWV /ijkLQGLFDWHWKHSRZHURIHDFKRIWKHYDULD EOHVrorsFRV T7KHpiston WHUPVLQ /  /  / DUHQRWJHRPHWULFDEHUUD WLRQVWKH\DUHQHFHVVDULO\QXOOLIZHDGKHUHWRWKHK\SRWKHVLVWKDWWKHVXUIDFHV F DQG S LQWHUVHFW WKH RSWLFDO D[LV RQ WKH FHQWHU & RI WKH H[LW SXSLO +RZHYHU WKH\ DSSHDUIRUJUHDWHUJHQHUDOLW\LIWKHYDULRXVIURQWVFHDFKJHQHUDWHGE\RQHRIWKH SRLQWVRQWKHREMHFWSODQHDUHLQVWHDGGHWHUPLQHGE\WDNLQJWKHVDPHRSWLFDOSDWK IRUDOORIWKHSRLQWVHTXDOWRWKDWZKLFKRQWKHRSWLFDOD[LVMRLQVWKHRULJLQ2o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point characteristicLV

*HRPHWULFDO2SWLFV  GHILQHGDVWKHRSWLFDOSDWKXQLTXHO\GHWHUPLQHGE\DUD\WUDFHGEHWZHHQWZRDV VLJQHG SRLQWV 3 DQG 4 VXSSRVLQJ WKHQ WKDW WKHUH LV RQO\ RQH RI WKHVH UD\V ZH KDYH 

V 34



L 4  L 3 

ZKHUHWKHUD\EHWZHHQ3DQG4EHORQJVWRDIDPLO\RIUD\VDVVRFLDWHGZLWKDQHL NRQDOL)RUH[DPSOHWKHVHUD\VFDQDOOSDVVWKURXJK4RUWKURXJK3,IZHFRQ VLGHU D IDPLO\ RI KRPRFHQWULF UD\V IURP 3 RU YLFH YHUVD IURP 4  DQG ZH GLIIHUHQWLDWHVE\IL[LQJWKHSRVLWLRQRI3 4 DQGFKDQJLQJWKDWRI4 3 IRUWKH HLNRQDOHTXDWLRQ  LQWKHWZRFDVHVZHREWDLQUHVSHFWLYHO\ ’ 4 V



n4 V4  ’ 3 V

 n3 V 3  

 

ZKHUH rn VUHVSHFWLYHO\LQGLFDWHWKHSRVLWLRQWKHUHIUDFWLYHLQGH[DQGWKHYHUVRU RIWKHUD\RQWKHSRLQW4RURQWKHSRLQW3ZKLOHWKHVXIIL[LQSDUHQWKHVHVLQGL FDWHVWKHSRLQWRQZKLFKZHFDOFXODWHWKHJUDGLHQW$VVXPLQJDFRQVWDQWUHIUDFWLYH LQGH[niIRUWKHLPDJHVSDFHZHFDQQRZHDVLO\FRPSXWHIURP(T  WKHLQ WHUVHFWLRQRIWKHUD\RQWKHLPDJHSODQH,QGHHGLIZHDVVRFLDWHWKHSRLQW3ZLWK DQREMHFWSRLQW2DQGWKHSRLQW4ZLWKDSRLQW6RQWKHUHIHUHQFHVSKHUH KHQFH QRWWRDSRLQW)RQWKHDEHUUDWHGZDYHIURQW GLIIHUHQWLDWLQJWKHIXQFWLRQ /RI(T  ZHKDYH





w/ wxs

wV wV wzs   wxs wzs wxs

w/ wy s

wV wV wzs   wy s wzs wy s



 

)URP(T D ZLWKVs  FRVDFRVEFRVJ ZHWKHQKDYH



wV wxs

ni FRV D

wV wzs

ni FRV J

x c  xs wV ni i  Rc wys z ni s  Rc

ni FRV E

y c  ys ni i  Rc



 

ZKHUH R‫ މ‬LV WKH GLVWDQFH EHWZHHQ 6 DQG WKH SRLQW , ‫ މ‬ZLWK FRRUGLQDWHV x‫މ‬i y‫މ‬i   )LJ 0RUHRYHUIURP(T   ZLWKzi  ZHKDYH 

wzs wxs

xi  xs wzs  zs wy s

yi  y s  zs

,QVHUWLQJ(TV  LQWR(T  ZHILQDOO\JHW

 

$EHUUDWLRQV





R c w/  ni wxs

xic  xi

yic  yi

R c w/  ni wy s

 

7KHUHIRUHWKHODWHUDODEHUUDWLRQVRQWKHSDUD[LDOLPDJHSODQHDUHH[SUHVVHGE\ DSRO\QRPLDOLQZKLFKHDFKWHUPLVGHULYHGE\ORZHULQJWKHFRUUHVSRQGLQJWHUPLQ WKHGHYHORSPHQWRIWKHDEHUUDWLRQVRIWKHZDYHIURQWRQWKHH[LWSXSLOE\RQHGH JUHH +RZHYHU (TV   FRQWDLQ DQ LPSOLFLW GHSHQGHQFH RQ WKH FRRUGLQDWHV x iy iDQGx ‫މ‬iy ‫މ‬iDVZHOODVxsysWKURXJKWKHSUHVHQFHRIR ‫މ‬ 



ª x c  yic   xs xic  xi   y s yic  yi º R c d «  i » d ¬ ¼



 

*LYHQWKDWxiyiDQGx‫މ‬iy‫މ‬iDOVRGHSHQGRQxoyoDQGxsysHYHQR‫މ‬FDQEHH[ SUHVVHGDVDIXQFWLRQRIWKHVHFRRUGLQDWHV R c xo  yo  xs  y s 

Rc



8VXDOO\WKHWUDQVYHUVHFRRUGLQDWHVRQWKHLPDJHSODQHDUHVPDOOLQUHODWLRQWR dDQGWKXVRQHFDQSURFHHGWRDSRO\QRPLDOGHYHORSPHQWRIR'ZKLFKIRUWKHF\ OLQGULFDOV\PPHWU\RIWKHRSWLFDOV\VWHPPXVWRQO\FRQWDLQWKHSURGXFWVRIHYHQ GHJUHHRIWKHFRRUGLQDWHVDOUHDG\GHVFULEHGE\(TV  ,QSDUWLFXODUWKHILUVW WZRWHUPVRIWKLVGHYHORSPHQWDUH 

Rc | d 

mA d

xo  yo # R 

 

ZKHUHmALVWKHODWHUDOPDJQLILFDWLRQRIWKHV\VWHP ,QFRQFOXVLRQWKHSRO\QRPLDOGHYHORSPHQWRI(TV  FRQWDLQVRQO\RGG GHJUHHWHUPVLQWKHWUDQVYHUVHFRRUGLQDWHVRIWKHREMHFWSRLQWDQGWKHUD\SRVLWLRQ RQWKHH[LWSXSLO/HWXVQRZSXWWKHSRLQWREMHFWRQWKHyD[LVVRWKDWxo DQG ZHFRQVLGHUWKHYDULDEOHVrsDQGTDVDIXQFWLRQRIxsDQGysZKHUHE\ 

xs

rs VLQ T

ys

rs FRV T 

7KHUHIRUHLQSRODUFRRUGLQDWHVWKHRSHUDWRUV w wxs DQG w wy s EHFRPH



 w w  FRV T  wrs rs wT

w wxs

VLQ T

w wy s

 w w FRV T  VLQ T  wrs rs wT



 

*HRPHWULFDO2SWLFV  VR WKDW IURP (TV   DQG   ZH KDYH WKDW WKH ODWHUDO DEHUUDWLRQ H[ SUHVVHGXSWRWKHWKLUGRUGHULV 



x i  xi #  '  rs VLQ T   D   '  rs  VLQ T  ' yo rs  VLQ T   '  yo  rs VLQ T

y i  yi # '  rs FRV T  ' yo 



 '  rs  FRV T  ' yo rs    FRV T  

E 

 '   '  yo  rs FRV T  '  yo 

ZKHUHWKHQHZFRHIILFLHQWV'ijkDUHDVVRFLDWHGZLWKWKHSUHYLRXVRQHVE\ 

' ijk |

d / ijk  ni

 

7KLVH[SUHVVLRQFRPHVIURPKDYLQJDSSUR[LPDWHGR‫މ‬ZLWKdLIZHFRQVLGHUWKH GHYHORSPHQW RI R' ZH ILQG WKDW LW LV FRUUHFW IRU WKH FRHIILFLHQWV RI WKH ILUVW DQG WKLUGRUGHU H[FHSW'DQG' ZKLOHLWLVRQO\DSSUR[LPDWHIRUWKHFRHIILFLHQWV RIWKHKLJKHURUGHUWHUPV,WVKRXOGDOVREHQRWHGWKDWWKHDEHUUDWLRQIXQFWLRQVKDYH EHHQH[SUHVVHGKHUHLQWHUPVRIWKHSRVLWLRQRIWKHEHDPRQWKHH[LWSXSLOZKLOH LQWKHWUDFLQJRIWKHUD\VLWLVPRUHQDWXUDOWRFRQVLGHUWKHORFDWLRQRQWKHHQWUDQFH SXSLO+RZHYHURQFHWKHREMHFWSRLQWIRUZKLFKZHZDQWWRFDOFXODWHWKHDEHUUD WLRQVLVFKRVHQWKHWZRSRVLWLRQVDUHXQLTXHO\OLQNHGWRHDFKRWKHULQILUVWDSSUR[ LPDWLRQ IURP D OLQHDU UHODWLRQVKLS ZKRVH FRHIILFLHQW LV WKH ODWHUDO PDJQLILFDWLRQ EHWZHHQWKHWZRSXSLOV,QDQ\FDVHWKHSRO\QRPLDOGHYHORSPHQWRIWKHDEHUUDWLRQ IXQFWLRQVGRHVQRWFKDQJHVKDSHZLWKRQO\WKHFRHIILFLHQWVYDU\LQJ ,WLVDOVRDXVHIXOH[SUHVVLRQIRUWKHSRLQWVRILQWHUVHFWLRQZLWKDGLIIHUHQWEXW VWLOOQHDUE\SODQHIURPWKDWLPDJH-XVWDVWKHGHYLDWLRQVRI(TV  DUHWDNHQ IURP WKH SDUD[LDO LPDJH SRLQW , RQ WKLV QHZ SODQH ZH FDQ WDNH WKHP IURP WKH SRLQWRILQWHUVHFWLRQZLWKWKHSULQFLSDOUD\FRPLQJIURP2ZKLFKE\GHILQLWLRQ FURVVHVWKHRXWSXWSXSLOLQLWVFHQWHU&DQGUHDFKHV,6LQFHZHDUHFRQVLGHULQJD KRPRJHQHRXVLPDJHVSDFHZKHUHWKHUD\VDUHVWUDLJKWWKHGHYLDWLRQYDULHVOLQHDU O\ZLWKWKHSRVLWLRQzRIWKHFRQVLGHUHGSODQHDQGWKHUHIRUHZHVLPSO\KDYH

> xic  xi @z # ¨§   

©

z· z ¸ > xic  xi @  xs  d¹ d

z z > yic  yi @z # ¨§   ¸· > yic  yi @  ys  d d © ¹



 

ZKHUHWKHGHYLDWLRQVDUHLQGLFDWHGDVDIXQFWLRQRIz )LJ ,WVKRXOGEHQRWHG WKDWLQWKHVHHTXDWLRQVWKHUHLVDOLQHDUGHSHQGHQFHRQFRRUGLQDWHVWDNHQRQWKH

$EHUUDWLRQV



ys

)LJ (IIHFW RI WKH LPDJH SODQHWUDQVODWLRQ

y¢i

yi¢ -(z/d )( ys-yi¢) (1+z/d ) yi

yi C -d

Oi

z



SODQHRIWKHSXSLOWKLVMXVWLILHVZKDWKDVEHHQVWDWHGHDUOLHUE\DVVHUWLQJWKDWWKH WHUPVLQ /DQGWKHQLQ 'FRUUHVSRQGWRDWUDQVODWLRQRIWKHLPDJHDORQJWKH RSWLFDOD[LV 7KHWKLUGRUGHUWHUPVFRUUHVSRQGWRWKHILYHPRQRFKURPDWLFSeidel aberrations spherical aberrationcomaastigmatismfield curvatureDQGdistortionZKLFKZH ZLOOGLVFXVVEULHIO\ODWHU1RUPDOO\LWLVQRWSRVVLEOHWRFDQFHODOORIWKHVHDEHUUD WLRQVVLPXOWDQHRXVO\KRZHYHULWLVXVHIXOWRFRQVLGHUWKHPRQHDWDWLPHDVVXP LQJ WKDW ZH KDYH EHHQ DEOH WR FDQFHO WKH RWKHUV E\ DGMXVWLQJ WKH RSWLFDO V\VWHP DSSURSULDWHO\

2.9.2 The cosine theorem, Abbe’s sine condition and Herschel’s condition 3DUD[LDO 2SWLFV LV EDVHG RQ WKH DVVXPSWLRQ WKDW WKH IROORZLQJ TXDQWLWLHV DUH VPDOOHQRXJKZKHUHE\LQWKHWUDFLQJRIWKHUD\VWKHDQJOHVFDQEHDSSUR[LPDWHG E\WKHLUWDQJHQWVRUWKHLUVLQHV7KH\DUH L WKHVL]HRIWKHREMHFWVZLWKUHVSHFWWR WKHORQJLWXGLQDOGLVWDQFHVWKDWLVWKHangular field of viewRIWKHLQVWUXPHQWDQG LL WKHaperture angleRIWKHUD\VDGPLWWHGE\WKHLQVWUXPHQW1HYHUWKHOHVVWKHUH DUHFDVHVLQZKLFKWKHILHOGRIYLHZLVOLPLWHGWRVPDOOREMHFWVQHDUWKHRSWLFDOD[ LV ZKLOH ZH ZDQW WR FROOHFW DV PXFK OLJKW DV SRVVLEOH IURP WKHVH REMHFWV ZLWK D ODUJHDSHUWXUH7RWKLVHQGWZRVLPSOHFULWHULDNQRZQDV Abbe’s sine condition DQGHerschel’s conditionDUHKHOSIXO7KH\DUHUHODWHGWRWKHVWLJPDWLVPRILPDJHV WKDWFDQEHREWDLQHGLQWKHVHFDVHV%RWKFRQGLWLRQVFDQHDVLO\EHGHULYHGIURPD PRUHJHQHUDOWKHRUHPDOVRYDOLGIRULQKRPRJHQHRXVPHGLD /HW XV FRQVLGHU WKH FKDUDFWHULVWLF IXQFWLRQ V 34  GHWHUPLQHG EHWZHHQ WZR SRLQWV 3 DQG 4 DQG GLIIHUHQWLDWH LW RQ ERWK H[WUHPHV )URP (TV   ZH FDQ REWDLQWKHUHODWLRQVKLS 

dV

n4 V4 ˜ dr4  n3 V 3 ˜ dr3 

 

  7KLV UHODWLRQ LV OLPLWHG DW WKH ILUVW RUGHU IRU WKH GLVSODFHPHQWV RI 3 DQG 4 DQG GRHV QRW

FRQWUDGLFWWKH(TRI&ROOLQV  ZKLFKLQVWHDGFRQVLGHUVWKHVHFRQGRUGHURILWZKLOHWKH ILUVWRUGHULVVXEWUDFWHGE\DOLJQLQJWKHzD[LVZLWKWKHPDLQEHDPDQGWKHGLVSODFHPHQWVDUHRQ WUDQVYHUVHSODQHV

*HRPHWULFDO2SWLFV



 IURPZKLFKZHFDQGHULYHWKHcosine theorem 6XSSRVH WKDW WKH SRLQW 4 LV WKH VWLJPDWLF LPDJH RI 3 :H FRPELQH WKHVH SRLQWVZLWKDQDUELWUDU\UD\ZKRVHGLUHFWLRQLQ3DQG4LVH[SUHVVHGE\WKH YHUVRUV VSDQGVTUHVSHFWLYHO\/HWXVWKHQFRQVLGHUWZRPRUHSRLQWV3'DQG 4' LQILQLWHVLPDOO\ FORVH WR 3 DQG 4 UHVSHFWLYHO\ ,Q RUGHU IRU 4‫ މ‬WR EH WKH VWLJPDWLFLPDJHRI3‫މ‬HYHQLQWKHFDVHRIDZLGHDSHUWXUHLWLVQHFHVVDU\DQG VXIILFLHQWWKDWWKHGLIIHUHQFH n4 VT ˜ drTc  n3 VS ˜ drSc 

 JJJG 33c drTc FKRVHQWRMRLQ3DQG4

ZKHUH drSc

 

JJJJG 44c  LV LQGHSHQGHQW RI WKH GLUHFWLRQ RI WKH UD\

7RSURYHWKDWWKLVFRQGLWLRQLVQHFHVVDU\VXSSRVHWKDW4‫މ‬LVWKHVWLJPDWLFLPDJH RI3‫މ‬,QWKLVFDVHQHLWKHUV 34 QRUV 3‫މ‬4‫ މ‬GHSHQGRQWKHGLUHFWLRQRIWKHUD\ FKRVHQWRMRLQWKHUHVSHFWLYHSDLURISRLQWVDQGWKHUHIRUHWKHLUGLIIHUHQFHdVIRU ZKLFKWKHH[SUHVVLRQ  PXVWEHLQYDULDQWZLWKUHVSHFWWR VSDQG VTGRHVQ¶W HLWKHU)LQDOO\WRSURYHWKDWWKHFRQGLWLRQLVDOVRVXIILFLHQWZHMRLQ3'DQG4'ZLWK DUD\3'4'DQGWUDFHIURP3'DVHFRQGUD\3'4''ZLWKWKHVDPHRSWLFDOOHQJWKDVWKH ILUVWRQH6LQFHWKHWKHRUHPDVVXPHVWKDW4LVWKHVWLJPDWLFLPDJHRI3IRUUHDVRQV JJJJG RI FRQWLQXLW\ HYHQ WKH YHFWRU 44cc drTcc  PXVW EH LQILQLWHVLPDO DQG IURP (T  ZHKDYH 

V 3c4cc  V 3 4

n4 VT < drTcc  n3 VS < drSc 

2QWKHRWKHUKDQGZHDOVRKDYH 

V 3c4c  V 3 4

n4 VT < drTc  n3 VS < drSc 

6LQFH WKH RSWLFDO SDWK RI WKH WZR UD\V KDV EHHQ FKRVHQ DV EHLQJ WKH VDPH VR WKDW V 3c4c   V 3c4cc  ZH KDYH VT 6HQDUPRQW HWDO9RO,,DOQƒ;/,;S@



+LVWRULFDOQRWHV)URP1HZWRQWR)UHVQHO

WHOHVFRSHRULHQWHGWRZDUGVLWWKHZDYHIURQWFDSWXUHGE\WKHOHQVLVPDGHFRQYHU JHQWRQ WKHIRFDO SODQHRI WKH REMHFWLYH 6LQFH WKH WHOHVFRSH PRYHV WUDQVYHUVHO\ WKHZDYHGRHVQRWFRQYHUJHRQWKHVDPHD[LVEXWLVVKLIWHGLQWKHRSSRVLWHGLUHF WLRQ RI WKH PRWLRQ 7R REVHUYH WKH VWDU LQ D[LV WKH WHOHVFRSHPXVW WKHQ EH WLOWHG VOLJKWO\IRUZDUG )UHVQHODOVRVROYHGWKHHQLJPDRIWKHH[SHULPHQWZLWKWKHSULVPVGHGXFLQJWKDW WKHDHWKHUZDVWREHpartially draggedIURPWKHPHGLXPLQPRWLRQLQSURSRUWLRQ WRWKHLQFUHDVHLQWKHGHQVLW\RIWKHDHWKHULQWKHWUDQVSDUHQWPHGLXP7KDWLVLQ PRGHUQWHUPVDVnZKHUHnLVWKHUHIUDFWLYHLQGH[DVZHZLOOVHHLQ† )UHVQHOULJKWO\FRQFOXGHGWKDWHYHQWKHH[SHULPHQWSURSRVHGE\%RVFRYLFKZRXOG QRWKDYHVKRZQDGLIIHUHQFHEHWZHHQDWHOHVFRSHILOOHGZLWKZDWHUDQGRQHZLWK RXW:HZLOOFRQWLQXHWKLVVWRU\RIWKHDHWKHUFRQFHSWODWHU

Diffraction and Interference :LWKKLVLVRODWLRQLQ0DWKLHX)UHVQHOKDGILQDOO\IRXQGWKHRSSRUWXQLW\WRGH YRWHKLPVHOIHQWLUHO\WRWKHGLIIUDFWLRQH[SHULPHQWVWKDWKHGHVFULEHGLQDPHPRLU VHQWRQ2FWREHUWR'HODPEUH6HFUHWDU\RIWKH$FDGpPLHHQWLWOHGLa diffraction de la lumière.$V5RQFKLQRWHGWKHWLWOHDOUHDG\VD\VLWDOOWKHRULJL QDOQDPHRIWKHSKHQRPHQRQLVUHVWRUHGDVRSSRVHGWRWKHWHUPXVHGE\1HZWRQ 7KHILUVWSDUWRIWKLVPHPRU\LVGHGLFDWHGWRKLJKOLJKWLQJKRZ©ODWKpRULH1HZWR QLHQQHFRQGXLWjSOXVLHXUVK\SRWKqVHVLPSUREDEOHVªFRQWHVWLQJWKHVHK\SRWKHVHV RQHE\RQHZLWKDSDVVLRQWKDWRQO\D\RXQJPDQFDQPXVWHU:LWKDOLWWOHELWRI LPDJLQDWLRQKHDOVRXVHGDQHZDUJXPHQW+HDUJXHGWKDWLIOLJKWZHUHFRPSRVHG RIPROHFXOHVWKH\ZRXOGEHRIWKHVDPHQDWXUHDVWKRVHK\SRWKHVL]HGIRUKHDWDQG ZRXOGEHUHMHFWHGE\WKHP6LQFHWKHQWKHDWPRVSKHUHFRQWDLQV©WDQWGHPLOOLDUGª RIFDORULFPROHFXOHVKRZLVLWSRVVLEOHWKDWWKHVSHHGRIDOLJKWPROHFXOHLVQRWDO WHUHG" ,QVWHDG IRU WKH ZDYH WKHRU\ KH VDZ RQO\ RQH VHULRXV REMHFWLRQ WKH RQH WKDW FRPHVIURPWKHFRPSDULVRQRIOLJKWDQGVRXQGZKLFKE\SDVVHVWKHREVWDFOHV%XW KH ZURWH WKDW QRWKLQJ SURYHV WKDW RQH FDQ FRPSDUH WKH YLEUDWLRQ RI DLU D KHDY\ IOXLGZLWKWKRVHRIWKH©FDORULTXHªDWKLQIOXLG7KHPDUFKRIOLJKWLVPXFKIDVWHU WKDQWKDWRIVRXQGVRLWVKRXOGEHGHIOHFWHGWRDIDUOHVVHUGHJUHHE\DQREVWDFOH $QGKHDGGHG©&HWWHREMHFWLRQODVHXOHjODTXHOOHLOPHSDUDLVVHGLIILFLOHGHUp SRQGUHFRPSOqWHPHQWP¶DFRQGXLWjP¶RFFXSHUGHVRPEUHVSRUWpHVª +HWKHQORRNHGDWWKHGLIIUDFWLRQIULQJHVDV*ULPDOGLKDGGRQHEXWKHZDQWHG  $)UHVQHOLa

Diffraction de la lumière, où l’on examine particulièrement le phénomène des franges colorées que présentent les ombres des corps éclairés par un point lumineux7KLVILUVW PHPRU\ZDVGHSRVLWHGDWWKH$FDGpPLHRQ2FWREHU>6HQDUPRQWHWDO9RO1ƒ ,,S@  ©7KLV REMHFWLRQ WKH RQO\ RQH WR ZKLFK LW VHHPV GLIILFXOW WR DQVZHU FRPSOHWHO\ OHG PH WR LQYHVWLJDWHWKHVKDGRZVª

)UHVQHO



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honey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¶VHGJHZDVVKDUSRUQRWDVZLWKWKH ZLUH RU WKHEDFN RI D UD]RU RU LI WKH HGJHZDV SROLVKHGRUEODFNHQHG+RZHYHU WKH\ DOO VHHPHG WR FRPH IURP WKH HGJH LWVHOI +H WKHQ UHSRUWHG D VHULHV RI PHDVXUHVLQZKLFKDVDQREVWDFOHKHXVHGDQLURQZLUHWKDWKHNQHZZDVH[DFWO\ PP LQ GLDPHWHU :LWK WKLV KH GHWHUPLQHG WKH VHSDUDWLRQ EHWZHHQ WKH IULQJHV RI WKH ILUVW RUGHU DW WKH WZR VLGHV RI WKH VKDGRZ ZLWK WKH SUHFLVLRQ RI D WHQWK RI D PPZLWKWKHZLUHSODFHGDWYDULRXVGLVWDQFHVIURPWKHVRXUFHXSWRPRIGLV WDQFHDQGWKHVFUHHQIURPWKHZLUHIURPDERXWWHQFPWRP$V*ULPDOGLKDGDO UHDG\REVHUYHGWKHIULQJHVKDYHDQRQUHFWLOLQHDUWUHQG)UHVQHOZDVDEOHWRHVWDE OLVKWKDWVXFKDWUHQGLVYHU\QHDUWRDK\SHUEROHZLWKRQHIRFXVFRLQFLGHQWZLWK WKHVRXUFHDQGWKHRWKHUZLWKWKHHGJHRIWKHZLUH7KHGLIIHUHQFHZLWKWKHWKHRU\ ZDVYHU\VPDOOEXWKHZDQWHGWREHVXUHRIWKLVIDFW+HWKHQEXLOWDVLPSOH³PL FURPHWHU´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interferenceEHWZHHQWKHZDYHVFRP LQJIURPWKHWZRHGJHV)UHVQHOGLGQRW\HWNQRZWKDW/DQGVEHUJ,S@ 7KHGHYLFHLQ)LJLVWKDWW\SLFDOO\XVHGLQRSWLFZRUNVKRSVWRFKHFNWKH VXUIDFHVXQGHUZRUNIRUFRPSDULVRQZLWKUHIHUHQFHVXUIDFHV6RGLXPRUPHUFXU\ VSHFWUDOODPSVDUHQRUPDOO\XVHGDVVRXUFHV:LWKODVHUVRXUFHVYHU\VLPLODUGH YLFHVDUHXVHGZLWKWKHDGYDQWDJHRIEHLQJDEOHWRSRVLWLRQWKHRSWLFDO HOHPHQWV HYHQDWJUHDWGLVWDQFHVEHWZHHQWKHP

7ZRZDYHLQWHUIHUHQFH



3.2.6 Localization of interference fringes ,QWKHSUHYLRXVVHFWLRQVZHZHUHLQWHUHVWHGLQWZRW\SHVRIIULQJHWKRVHWKDW FDQEHREVHUYHGLQDODUJHYROXPHRIVSDFHZKLFKDUHWKHQFDOOHGnon-localized DQGWKRVHWKDWDUHREVHUYHGRQSDUWLFXODUVXUIDFHVVXFKDVWKHIULQJHVRI+DLGLQJHU DQG)L]HDXZKLFKDUHFDOOHGlocalized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†ZHZLOODQDO\]HWKHSUREOHPRIVSDWLDODQGWHPSRUDOFRKHUHQFHLQGH WDLOKHUHZHFRQWLQXHWRFRQVLGHURQO\WKHFDVHRIVSDWLDOO\LQFRKHUHQWVRXUFHV,Q JHQHUDOLWLVREVHUYHGWKDWDVZHKDYHDOUHDG\VHHQLQ†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¶VELSULVPRUGRXEOHPLUURUDOLQHDUVRXUFHSDUDOOHOWRWKHHGJHPD\EH XVHGFRQWLQXLQJWRKDYHQRQORFDOL]HGIULQJHV,QIDFWIRUHDFKREVHUYDWLRQSRLQW 3WKHVRXUFHDSSHDUVSDUDOOHOWRWKHHGJHLWVHOIZLWKRXWDWUDQVYHUVHH[WHQVLRQRU WKRJRQDOWRWKHxzSODQHGHILQHGLQ)LJLQZKLFKWKHGLUHFWLRQRIWKHxD[LVLV WKDWRIWKHHGJHZKLOHWKHz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singleLQFLGHQWUD\IURPWKHVRXUFH$URXQGWKHVHSRLQWVJLVQHJOLJLEOHVR DFFRUGLQJWR(T  ZHKDYH



,QWHUIHUHQFH

' r c  'r



§   ·    ¨ ¸ x  y   © R$ R% ¹





 

ZKHUH,UHFDOOWKDWxDQGyDUHFRRUGLQDWHVFHQWHUHGRQDUHIHUHQFHSRLQW6RQWKH VRXUFHWDNHQLQDGLUHFWLRQSHUSHQGLFXODUWRWKHUD\WKDWIURP6UHDFKHVWKHRE VHUYDWLRQSRLQW3FRQVLGHUHGR$DQGR%DUHWKHFXUYDWXUHUDGLLLQ6RIDUHYHUVH ZDYH RQWKHWZRSDWKV$DQG% VWDUWLQJIURP37KHQLQJHQHUDOHYHQIRUWKHVH SRLQWV/WKHIULQJHVGLVDSSHDUZKHQWKHWUDQVYHUVHGLPHQVLRQVRIWKHVRXUFHDUH VXFK WKDW 'r‫ މ‬± 'r H[FHHGV D GLVWDQFH RQ WKH RUGHU RI O ,I LQ SDUWLFXODU WKH ZDYHIURQWVW$DQGW% )LJ DUHVSKHULFDOR$DQGR%DUHLQGHSHQGHQWRIx DQGyDQGWKHUHGXFWLRQLQYLVLELOLW\LVLQGHSHQGHQWRIWKHGLUHFWLRQLQZKLFKWKH VRXUFHH[WHQGVWUDQVYHUVHO\)RUDFLUFXODUVRXUFHLIZHFRQVLGHUDOYDULDWLRQ IURPWKHFHQWHUWRWKHSHULSKHU\WKHYLVLELOLW\UHPDLQVJRRGDVORQJDVWKHVRXUFH GLDPHWHUUHPDLQVEHORZ 

lD



O R A RB   R A  RB

 

,QHYDOXDWLQJWKHYLVLELOLW\RIIULQJHVKRZHYHURQHPXVWFRQVLGHUZKDWWKHDF WXDOH[WHQVLRQRIWKHVRXUFHXVHGE\WKHRSWLFDOV\VWHPLV7KHIULQJHVFDQWKHUH IRUHEHYLVLEOHHYHQLIWKHVRXUFHLVYHU\ZLGHSURYLGHGWKDWWKHDQJXODUILHOGZLWK ZKLFKWKH\DUHREVHUYHGLVVXIILFLHQWO\OLPLWHG ,QWKHFDVHRIDODPLQDZLWKSODQHSDUDOOHOIDFHVWKHORFDOL]DWLRQUHJLRQLVDWLQ ILQLW\IRUZKLFKR$DQGR%DUHERWKLQILQLWH7KXVIRU(T  WKHSDWKGLI IHUHQFHLVFRQVWDQWIRUDOOYDOXHVRIxDQGyLHLWGRHVQRWGHSHQGRQWKHSRVLWLRQ RQ WKH VRXUFH +DLGLQJHU¶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

$/

$&

FRV -  M d VLQ -c  D FRV -  M  # VLQ M FRV -c  D VLQ M

7ZRZDYHLQWHUIHUHQFH



b

locus of L

N L

r J A

O

1 n

J

J

J

1 n

A

O

1

1

r

L locus of L N

b



)LJ6WXG\RIWKHORFDOL]DWLRQRIWKHIULQJHVIRUDSODQHZDYH

1RZIRUMLQVWHDGRIDSSO\LQJ(T  ZHXVH(T  DSSUR[LPDW LQJVLQ M ZLWK VLQM 7KHUHIRUH 

$/ #

d VLQ -c  D FRV -  M FRV -  M   VLQ D n FRV -c  D FRV -c  D

,IIRUMZHQRZXVH(T  GHYHORSLQJWKHVHFRQGIUDFWLRQDWWKHVWRU GHULQDZLWKDIHZVWHSVZHKDYH 

$/ #

d VLQ -c FRV - d FRV    WDQ  -  WDQ  -c  D n FRV -c n FRV -c





 

ZKLFKIRUVPDOODQJOHV-ZHFDQDSSUR[LPDWHZLWK 

d VLQ - d ª  ·º § $/ #   «  VLQ  - ¨    ¸ »  n¬ n D n © ¹¼

 

2QWKHRWKHUKDQGU dDLVWKHGLVWDQFHRI$IURPSRLQW2ZKHUHWKHWZRIDF HV LQWHUVHFW )LJ   VR IRU DQ LQFLGHQW SODQH ZDYH $/  LV SURSRUWLRQDO WR U DQGWKHORFXVRISRLQWV/LVDSODQHSDVVLQJWKURXJKWKHSRLQW2ZKRVHLQFOLQDWLRQ EZLWKWKHZDYHIURQWUHIOHFWHGE\WKHILUVWIDFHLVREWDLQDEOHE\  WDQ E 

D FRV $1  $/ § FRV - · WDQ -    WDQ  -   WDQ  -c  # ¨    -c ¸ FRV n n FRV -c 21 © ¹   



ZKHUH1LVWKHSURMHFWLRQRI2RQWKHUHIOHFWHGUD\LQ$





,QWHUIHUHQFH

S )LJ 6WXG\ RI WKH LPDJHV RI WKH ORFDOL]DWLRQSRLQW/$LVWKH LPDJH RI / UHIOHFWHG E\ WKH LBt ILUVW IDFH RI WKH ODPLQD DQG /%W LV WKH WDQJHQWLDO LPDJH UHIOHFWHGE\WKHVHFRQGIDFH )RU VPDOO DQJOHV - WKH GLI IHUHQFH

LA

 / $ $  / %W $  LV SRVLWLYH DQG EHFRPHV QHJDWLYH IRU VXIILFLHQWO\ ODUJH DQJOHV DV LQ WKH FDVH RIWKHILJXUH

A

C

1 n

J

j

d

L

B

a 

7KHORFDOL]DWLRQVXUIDFHYDULHVZLWKWKHDQJOHRILQFLGHQFH-DQGLQSDUWLFXODU IRU- LQWKHH[DPSOHFRQVLGHUHGKHUHRIDODPLQDZLWKLQGH[nVXUURXQGHGE\ DLU LW LV ORFDWHG RQ DQ LQWHUPHGLDWH SODQH EHWZHHQ WKH IDFHV RI WKH ODPLQD LWVHOI 1RWHKRZHYHUIURP(T  WKDWWKHORFDOL]DWLRQSODQHPRYHVDZD\LQSUR SRUWLRQWR|-D 7KH RSWLFDO SDWK GLIIHUHQFH IRUWKH SRLQWV / FDQ QRZ HDVLO\ EH REWDLQHG IURP (TV  DQG  WDNLQJh   $/ FRV - VRWKHWHUPVLQdDHOLGHHDFK RWKHU1HJOHFWLQJWKRVHRIWKHVXSHULRURUGHUZHVLPSO\KDYH 

'L /

nd FRV -c 

ZKHUHdLVWKHWKLFNQHVVLQ$WDNHQSHUSHQGLFXODUWRWKHQGIDFH ,Q WKH SUHVHQW PRPHQW LW ODFNV GHWHUPLQDWLRQ DV WR WKH GHJUHH WR ZKLFK WKH VRXUFHFDQEHH[WHQGHGZLWKRXWDSSUHFLDEO\UHGXFLQJWKHYLVLELOLW\RIWKHIULQJHV 7R WKLV HQG LQ RUGHU WR DSSO\ (T   LW LV QHFHVVDU\ WR GHWHUPLQH RQ WKH VRXUFHZKLFKDUHWKHUDGLLRIFXUYDWXUHR$DQGZKLFKDUHWKHUDGLLRIFXUYDWXUHR% RQWKHWZRSDWKVRIDQLQYHUVHZDYHHPLWWHGE\DVRXUFHSRLQWFRUUHVSRQGLQJWRD JLYHQSRLQW/ GHVSLWHLWEHLQJYLUWXDO 7KHZDYHUHIOHFWHGIURPWKHILUVWIDFHLV VSKHULFDO ZLWK D FXUYDWXUH FHQWHU ORFDWHG LQ WKH LPDJH /$ RI / UHIOHFWHG E\ WKH VDPHIURQWIDFH7KHUHIUDFWHGZDYHLVQRWH[DFWO\VSKHULFDOIRUWKHDVWLJPDWLVP LQWURGXFHGE\WKHUHIUDFWLRQV,IZHFRQVLGHUDQDUURZOLJKWSHQFLOFHQWHUHGRQWKH GLUHFWLRQ-RIWKHVRXUFHWUDFLQJRQO\WKHUD\VO\LQJRQWKHSULQFLSDOSODQHRIWKH ODPLQDWKDWRUWKRJRQDOWRERWKIDFHVZHFDQVWLOOORFDWHDQLPDJH/%RI/IURP ZKLFKWKHZDYHGLYHUJHVDQGWKHFRUUHVSRQGLQJUDGLXVRIFXUYDWXUHRQWKLVSODQH /HWXVWKHUHIRUHFRQVLGHUWKHORFDOL]DWLRQSRLQW/DVVRFLDWHGZLWKDUD\6$ )LJ  7KHWDQJHQWLDOLPDJH/%WRI/FDQEHGHWHUPLQHGE\FRQVLGHULQJDQLQYHUVH UD\GLUHFWHGWRZDUGV/DQGLQFOLQHGDWDVPDOODQJOHGZLWKUHVSHFWWRWKHUHIUDFWHG UD\RQWKHSULQFLSDOSODQHRIWKHZHGJHRUVLPSO\E\XVLQJ(T D RIWKH

7ZRZDYHLQWHUIHUHQFH



ZDYHIURQW WUDQVIRUPDWLRQ IURP D dioptre :LWK D UDWKHU FRPSOLFDWHG FDOFXODWLRQ ZLWKLQWKHOLPLWRIDoUHFDOOLQJWKDW $/ v  D DQGWKDWdYDULHVZLWKWKHSRVL WLRQHOLPLQDWLQJWKHUHVLGXDOWHUPVLQDWKHWDQJHQWLDOVHSDUDWLRQ / $ / %W  R$ R%WLV 

'RW

º d ª FRV  VLQ  -   WDQ  -c FRV -»  n FRV -c «¬ FRV -c ¼

D 

,QVWHDGIRU(T E WKHVDJLWWDOVHSDUDWLRQLV 

'RV

d FRV -  n FRV -c

E 

:KHQWKHRXWHUDQGLQQHUPHGLDKDYHWKHVDPHUHIUDFWLYHLQGH[LHIRUn  WKHVHH[SUHVVLRQVUHGXFHWRd FRV- 2QWKHRWKHUKDQGLQ)L]HDX¶VFRQILJXUDWLRQ IRUVPDOO-'R#dn 0RUHRYHU ZH KDYH VHHQ WKDW WKH IULQJHV UHPDLQ YLVLEOH DV ORQJ DV WKH VRXUFH KDV D GLDPHWHU OHVV WKDQ  OR$ R%  'R  )RU D GLVWDQW HQRXJK VRXUFH IRU ZKLFKR$|R%!!'RWKDWZKLFKPDWWHUVIRUWKHYLVLELOLW\RIIULQJHVLVWKHDQJX ODURSHQLQJRIWKHVRXUFHVHHQIURP/$)RUVPDOO-IULQJHVDUHWKHUHIRUHYLVLEOH LIIRUWKLVRSHQLQJLWLV 

/D 

 R$

R$ R% On  #  'R d

 

)RUd PPO QPDQGn WKH)L]HDXIULQJHVDUHYLVLEOHRQO\LI WKHVRXUFHRUWKHSDUWRILWDFWXDOO\XVHGKDVDQDQJXODUDSHUWXUH/DOHVVWKDQ| q ,Q RWKHU ZRUGV ZKLOH WKH )L]HDX IULQJHV DUH FRPPRQO\ REVHUYHG RQ D IHZ PLFURQ WKLQ VKHHWV IRU PDFURVFRSLF WKLFNQHVVHV HYHQ ZLWK WKH PRQRFKURPDWLF OLJKWRIDVSHFWUDOODPSWKH\DUHREVHUYDEOHRQO\ZLWKDKLJKGHJUHHRIFROOLPDWLRQ RIWKHOLJKWEHDPV

3.2.7 Two-wave interferometers ,QWHUIHUHQFHLVVXLWDEOHIRUKLJKSUHFLVLRQPHDVXUHPHQWVDQGWKHUHDUHPDQ\LQ WHUIHURPHWHUV WKDW KDYH EHHQ GHVLJQHG IRU YDULRXV SXUSRVHV LQ IXQGDPHQWDO DQG DSSOLHGSK\VLFVDVLQVSHFWURVFRS\LQWKHPHDVXUHPHQWRIGLVWDQFHVDQGVSHHGV LQ PHDVXUHV RI ZDYHOHQJWKV UHIUDFWLYH LQGH[ TXDOLW\ RI RSWLFDO FRPSRQHQWV 

)URPDQXPHULFDOYHULILFDWLRQZLWKDQRQDSSUR[LPDWHWULJRQRPHWULFFDOFXODWLRQRIUD\WUDFLQJ WKLVH[SUHVVLRQLVPXFKPRUHDFFXUDWHWKDQWKDWJLYHQE\%RUQDQG:ROI  ZKLFKGRHVQRW PDWFKWKHOLPLWIRUDo



,QWHUIHUHQFH

0DODFDUD HWF:HZLOOQRZORRNDWVRPHRIWKHtwo-waveLQWHUIHURPHWHUV DQGWKHXVHWKDWFDQEHPDGHRIWKHP

0LFKHOVRQLQWHUIHURPHWHU

,Q0LFKHOVRQGHYHORSHGZKDWEHFDPHDORQJZLWKLWVYDULDQWVWKHPRVW IDPRXVDQGYHUVDWLOHRIDPSOLWXGHGLYLVLRQLQWHUIHURPHWHUV+HXVHGLWIRUWKHH[ SHULPHQWWKDWSURYHGWKHFRQVWDQF\RIWKHVSHHGRIOLJKW>0LFKHOVRQDQG0RUOH\ 0LFKHOVRQ@DQGLQIRUWKHILUVWLQWHUIHURPHWULFPHDVXUHPHQWRI WKHVWDQGDUGVDPSOHPHWHUZLWKUHVSHFWWRDFDGPLXPOLQH>0LFKHOVRQDQG%HQRLW @ ,WHVVHQWLDOO\FRQVLVWVRIDSODQHVHPLUHIOHFWLYHPLUURU' beam splitter ZKLFK VHUYHVERWKWRGHFRPSRVHDQGWRUHFRPELQHWKHUDGLDWLRQRIDVRXUFH6DQGWZR RWKHUSODQHPLUURUV0DQG0ZKLFKUHIOHFWEDFNUHVSHFWLYHO\WKHUHIOHFWHGDQG WUDQVPLWWHGZDYHVFRPLQJIURP' )LJ *HQHUDOO\EXWQRWQHFHVVDULO\'LV SODFHGDWƒZLWKUHVSHFWWRWKHLQFLGHQWUDGLDWLRQDQGWKHUHIRUHPLUURUV0DQG 0IRUPZLWK'WZRDUPVDWƒ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‫މ‬RIWKHPLUURU0UHIOHFWHGE\'LVUHSUHVHQWHGDV ZRXOGEHVHHQE\DQREVHUYHULQ27KHLQWHUIHUHQFHSDWWHUQVWKDWFDQEHREVHUYHG ZLWK WKLV WRRO DUH WKHUHIRUH WKH VDPH SURGXFHG E\ WZR IODW DQG IDFLQJ UHIOHFWLYH VXUIDFHVVHSDUDWHGE\DLUZLWKWKHGLIIHUHQFHWKDWKHUHZHGRQRWKDYHPXOWLSOH UHIOHFWLRQVEHWZHHQWKHP7KHUHIRUHIRUWKHLUVWXG\WKDWZKLFKZHGHWHUPLQHGLQ WKH SUHYLRXV SDUDJUDSKV LV YDOLG ZLWK WKH VLPSOLILFDWLRQ RI WDNLQJ WKH UHIUDFWLYH LQGH[RIWKHODPLQDDVEHLQJHTXDOWRDQGWKHVDPHVLJQIRUERWKUHIOHFWLRQV,Q VKRUWZLWKDQH[WHQGHGVRXUFHLI0‫މ‬LVSDUDOOHOWR0+DLGLQJHU¶VFLUFXODUIULQJ HVORFDOL]HGDWLQILQLW\DUHREVHUYHG,IRQWKHRWKHUKDQGWKHVHWZRVXUIDFHVDUH LQFOLQHGWRHDFKRWKHUDQGFORVHWRJHWKHU)L]HDX¶VIULQJHVDUHREVHUYHGDORQJOLQHV RIHTXDOVHSDUDWLRQDQGORFDOL]HGQHDU0DQG0‫މ‬,IILQDOO\WKHVXUIDFHVDUHLQ

7ZRZDYHLQWHUIHUHQFH

)LJ 0LFKHOVRQ¶V LQWHUIHURPHWHU



M1

M¢2

L

S

D M2

O



FOLQHGDQGGLVWDQWIRUWKHYDULRXVSRLQWVLQWKHILHOGRIYLHZWKHUHLVDGLYHUVLW\RI LQFLGHQFHDQJOHVRQWKHPLUURUVWKDWLVQRORQJHUQHJOLJLEOH7KHQWKHIULQJHVEH FRPHFXUYHGZLWKWKHFRQYH[SRUWLRQIDFLQJWKHHGJHRIWKHDLUZHGJHEHWZHHQ0 DQG0‫މ‬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³DFKURPDWLF´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coherence time $ VSHFLDO YDULDQW RI 0LFKHOVRQ¶V LQWHUIHURPHWHU LV ZKDW LV FDOOHG D lambdameter LQ ZKLFK RQH RU ERWK RI WKH PLUURUV DUH UHSODFHG E\ D FRUQHU FXEH DQG PRXQWHGRQ DPRYDEOH FDUULDJH VR DV WR FRQWLQXRXVO\ YDU\ WKH UHODWLYH OHQJWKRI WKH DUPV 7\SLFDOO\ D IUHTXHQF\VWDELOL]HGQP +H1H ODVHU EHDP WKH IUH TXHQF\YDOXHRIZKLFKLVZHOONQRZQLVVHQWWRWKHLQWHUIHURPHWHUDQGWKHIULQJHV DUHGHWHFWHGZLWKDSKRWRGLRGHGXULQJWKHPRWLRQRIWKHFDUULDJH$VHFRQGODVHU EHDPLVVHQWLQSDUDOOHOWRWKHILUVWDQGLWVIULQJHVDUHUHYHDOHGE\DQRWKHUSKRWRGL



,QWHUIHUHQFH

RGH )URP WKH UDWLR RI WKH IULQJH FRXQWV UHYHDOHG EHWZHHQ WZR DOPRVW  H[WUHPH SRVLWLRQVRIWKHFDUULDJHZHFDQREWDLQWKHIUHTXHQF\RIWKHODVHUWREHPHDVXUHG ZLWKDQDFFXUDF\WKDWFDQJRGRZQWRDIHZWHQVRI0+]

7Z\PDQ*UHHQLQWHUIHURPHWHU

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‫މމ‬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

 )LJ 7Z\PDQ  *UHHQ LQWHUIHURPHWHU

M

D S C

M¢R L C

R

P

M¢¢R



7ZRZDYHLQWHUIHUHQFH



)L]HDXLQWHUIHURPHWHU

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¶V LQWHU IHURPHWHU

D

P

d LASER A A R

C reference surface





,QWHUIHUHQFH

UHVSHFWWRWKHIRUZDUGZDYHSURSDJDWHGE\WKHOHQV7KHUHIRUHLQWHUIHURJUDPVDUH DFRGLQJRIWKHRSWLFDOSDWKYDULDWLRQ23' 2SWLFDO3DWK'LIIHUHQFH EHWZHHQWKH YDULRXVSRLQWVLQWKHWUDQVYHUVHEHDPSURILOHDFKDQJHLQGHSWK'zRQWKHWHVWVXU IDFHSURGXFHVRQH23' nD'zRSHUDWLQJLQDLUDUHIUDFWLRQLQGH[nD LVXVX DOO\DVVXPHG7KLVFRUUHVSRQGVWRDSKDVHGHOD\ 

G

S 23'  OR

 

ZKHUHORLVWKHZDYHOHQJWKLQYDFXXPRIWKHODVHUHPSOR\HG%\QHJOHFWLQJPXOWL SOH UHIOHFWLRQV WKH ZDYH LQWHQVLW\ UHDFKLQJ WKH GHWHFWRU DW WKH WUDQVYHUVH FR RUGLQDWHSRLQW xy LVJLYHQE\ 

I x y

I  ª¬  M FRV G x y º¼ 

 

ZKHUH I LV WKH VXP RI WKH LQWHQVLWLHV WKDW LQGHSHQGHQWO\ UHWXUQ IURP WKH GLYLGHU DQGWKHRSWLFXQGHUWHVWDQGMLVWKHPRGXODWLRQFRHIILFLHQWRIWKHIULQJHVWKHVH FRHIILFLHQWVDUHLGHDOO\FRQVWDQWZLWKLQWKHPHDVXULQJUDQJHRIWKHLQWHUIHURPHWHU 7KHIULQJHVWKDWDUHIRUPHGDUHWKHUHIRUHOLNHFRQWRXUOLQHVZLWKDSDFHHTXDOWRS LQWKHSKDVHZKLFKFRUUHVSRQGVWRD 23'  ORDQGDKHLJKWYDULDWLRQ 'z  OR nD  7KHILUVWHVWLPDWHRIWKHTXDOLW\RIDVXUIDFHFDQEHVLPSO\REWDLQHGE\GUDZLQJ DVWUDLJKWOLQHWKDWMRLQVWKHH[WUHPHSRLQWVRIDFHQWUDOIULQJHZLWKLQWKHPHDVXULQJ UDQJHDQGE\GHWHUPLQLQJWKHPD[LPXPIULQJHGHYLDWLRQIURPWKLVOLQHLQUHODWLRQ WRWKHSDFHRIWKHIULQJHV)RUDPRUHDFFXUDWHDQDO\VLVWKHUHFRQVWUXFWLRQRIWKH SKDVH PDS G xy  LV XVHG LQVWHDG RU HTXLYDOHQWO\ WKDW RI 23' xy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phase-shifting interferometry7KLVHVVHQWLDOO\FRQVLVWVLQWUDQVODWLQJWKHUHIHUHQFH

7ZRZDYHLQWHUIHUHQFH



SODWH ORQJLWXGLQDOO\ LQ RUGHU WR YDU\ WKH SKDVH LQ D ZHOOFRQWUROOHG PDQQHU 7KH IULQJHLPDJHLVGHWHFWHGE\D&&'VHQVRUDQGE\QHJOHFWLQJWKHHIIHFWRIPXOWLSOH UHIOHFWLRQVWKHLQWHQVLW\WKDWUHDFKHVLWVSL[HOVLVJLYHQE\





I xi  y j  I





^

`

I  xi  y j   M FRV ª¬ G x y  Iº¼ 

 

ZKHUH xi,yj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



Sn xi  y j

³

S  S n   n





I xi  y j  I d I

­S ½ I  xi  y j ®  M > sn VLQ G  sn  FRV [email protected]¾  ¯ ¿





ZKHUHsn ±±±±7KHIROORZLQJUHODWLRQVDUHIRXQG 

S  S

S  S 

 I  M VLQ G

S  S

S   S

 I  M FRV G



DQGVRZHKDYH 

 S  S  S  S S  S  S  S

 I  M VLQ G  I  M FRV G

WDQ G 

 

$OVRNQRZLQJWKHVLJQVRIVLQGDQGFRVGVHSDUDWHO\IURPWKLVHTXDWLRQZHFDQ GHULYH WKH YDOXH RI G‫  މ‬GS PRGXOH S FRUUHVSRQGLQJ WR HDFK SL[HO ZLWKRXW DPELJXLW\RIVLJQ%\DSSURSULDWHO\XQZUDSSLQJWKHSKDVHLQWKH]RQHVGHOLPLWHG E\WKHH[WUHPHYDOXHVDQGSRIG‫މ‬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



,QWHUIHUHQFH

-DPLQDQG0DFK=HKQGHULQWHUIHURPHWHUV

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

'L

nd FRV -c$  FRV -c% 

 

6LQFH WKH LQGH[ n DQG WKH WKLFNQHVV d RI WKH ODPLQDH DUH FRQVWDQW WKHVH DUH IULQJHVRIHTXDOLQFOLQDWLRQEXWQRZWKHSKDVHVKLIWGHSHQGVRQWKHGLIIHUHQFHEH WZHHQ WKH FRVLQHV RI WKH LQWHUQDO LQFLGHQFH DQJOHV LQWR WKH SODWH 6R WKDW LI WKH SODWHVDUHSDUDOOHOWKHSKDVHVKLIWLV]HUR 6LQFH DOO RI WKH RSWLFDO VXUIDFHV DUH SODQH WKH DSSDUHQW VRXUFHV RI WKH UD\V N DQG O HPHUJLQJIURP WKH VHFRQGSODWHRI WKH LQWHUIHURPHWHU DUH VHSDUDWHG LQ SUR SRUWLRQWRWKHURWDWLRQDQJOHDQGDUHUHFLSURFDOO\DOLJQHGLQWKHGLUHFWLRQSHUSHQ S

)LJ -DPLQ¶V LQWHU IHURPHWHU

B A

T1

a

C1

i

d

b c

h

T2

e C2 j

f

k l

g m

L 

7ZRZDYHLQWHUIHUHQFH



M1

D1

D1

S

M1

S

(b)

(a) M2

D2

M2

D2



)LJ0DFK=HKQGHULQWHUIHURPHWHULQWZRGLIIHUHQWDUUDQJHPHQWVRIWKHODPLQDH

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ƒWKH\EHFRPHRI KLJKRUGHU>%RUQDQG:ROIS@ $W\SLFDODSSOLFDWLRQRI-DPLQ¶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¶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¶VFRQILJXUDWLRQLVUHSURGXFHGE\SURYLGLQJWKDWWKHGLYLGHU'DQGWKHPLUURU6



,QWHUIHUHQFH

RI HDFK SDLU DUH ZHOO SDUDOOHO DQG HTXLGLVWDQW ,Q WKLV FDVH E\ WLOWLQJ WKH VHFRQG SDLURIDVPDOODQJOHZLWKUHVSHFWWRWKHILUVWDQGXVLQJDQH[WHQGHGVRXUFHIULQJHV RIHTXDOLQFOLQDWLRQFDQEHREVHUYHG,QVWHDGLQWKHFDVHRI)LJ E WKHPLU URU0DQGWKHGLYLGHU'DUHURWDWHGIURPHDFKRWKHUE\DVPDOODQJOH8VLQJD EHDPRIFROOLPDWHGOLJKWWKHIULQJHVRIHTXDOWKLFNQHVVDUHREVHUYHGORFDWHGLQDQ LQWHUPHGLDWHSRVLWLRQEHWZHHQ0DQG' &RPPRQSDWKLQWHUIHURPHWHUV

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sRIWKHLQFRPLQJUD\DQGWKDWRIWKHWZR HPHUJLQJUD\VFDQEHFDOFXODWHGLQJHQHUDOE\WKHPDWULFHVFRQVLGHUHGLQ† ,Q SDUWLFXODU (TV   DQG   H[SUHVV WKH PDWULFHV M DQG M FRU UHVSRQGLQJ WR WKH WZR SDWKV LQ ZKLFK WKH EHDP VSOLWWHU DQG WKH PLUURUV DUH WUDY HUVHGZLWKWZRRSSRVLWHVHTXHQFHVRIUHIOHFWLRQV7KXVWKHGLIIHUHQFHLQGLUHFWLRQ EHWZHHQWKHWZRUD\VHPHUJLQJIURPWKHEHDPVSOLWWHULQWKHVDPHVLGHRIWKHLQSXW UD\FDQEHH[SUHVVHGDV 

's

M  M s 

 

0RUHRYHUWKHFRPSOHWHRYHUODSRIWKHWZRSDWKVLVRQO\SRVVLEOHLIWKHPDWULFHV M DQG M KDYH DW OHDVW RQH HLJHQYDOXH HTXDO WR  2Q WKH RWKHU KDQG WKHVH SDWKVDUHVXSHULPSRVHGLIDQGRQO\LIWKH\DUHERWKFORVHGRQWKHLQSXWSRLQWDWWKH GLYLGHU ([SHULPHQWDOO\ WKLV FDQ EH DFKLHYHG E\ VHQGLQJ D FROOLPDWHG EHDP WKURXJK WKH GLYLGHU DQG DOLJQLQJ WKH PLUURUV VR DV WREULQJ WKHEHDP WUDQVPLWWHG

7ZRZDYHLQWHUIHUHQFH



AII

II

AI II

b B II

C A

I

a

B

c

CI AI A II I



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omitWKHSUHVHQFHRIUHIUDFWLRQV LQWKHGLYLGHUVXEVWUDWH ,QWKHFDVHRI)LJZHKDYHHLJHQYDOXHVA B C IRUERWKPD WULFHVVRM  MDQG's IRUDQ\GLUHFWLRQRI s,IWKHPLUURUVDUHDOLJQHGWR IRUPWKHFORVHGSDWK$%&$IRUDJLYHQLQFRPLQJUD\SDUDOOHOWRWKHHLJHQYHFWRUa FRUUHVSRQGLQJWRWKHHLJHQYDOXHADOOUD\VSDUDOOHOWRaIRUPDFORVHGSDWK,QWKLV FDVHWKHLQWHUIHUHQFHILHOGUHPDLQVXQLIRUPIRUDQ\LQFRPLQJVSKHULFDOZDYH,Q GHHG IRU DQ LQFRPLQJ UD\ ZKRVH GLUHFWLRQ GRHV QRW FRLQFLGH ZLWK a WKH WZR



,QWHUIHUHQFH

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aFRUUHVSRQGLQJWRAUHDSSHDUVLQWKHRSSRVLWHGLUHFWLRQIRUERWKSDWKV ,QVWHDG DQ LQFOLQHG UD\ ZLWK UHVSHFW WR a SURGXFHV WZR HPHUJLQJ UD\V WLOWHG EH WZHHQWKHPEHFDXVHMDQGMLQGXFHRSSRVLWHURWDWLRQVDURXQGa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a DUHUHIOHFWHG EDFN ZDUGVDQGWKHWZRSDWKVLQVLGHWKHLQWHUIHURPHWHUDUHSDUDOOHOWRHDFKRWKHU

6KHDULQJLQWHUIHURPHWHUV

7KHUHDUHYDULRXVLQWHUIHURPHWULFWHFKQLTXHVWKDWGRQRWXVHDUHIHUHQFHZDYH VXEVWDQWLDOO\IUHHRIGHIHFWVVXFKDVWKDWXVHGLQWKH)L]HDXDQG7Z\PDQ*UHHQ LQWHUIHURPHWHUV,QSDUWLFXODUWKH0DFK=HKQGHULQWHUIHURPHWHUDOORZVIRUWKHXVH RI D VLQJOHZDYH LQZKLFK WKH WZREHDPV SURGXFHGE\ WKH GLYLVLRQ LQ DPSOLWXGH FDQ EH PDGH WR LQWHUIHUH RYHUODSSLQJ WKHP RQO\ SDUWLDOO\ E\ PHDQV RI D ODWHUDO WUDQVODWLRQ shear  NHHSLQJ WKHP SDUDOOHO WR HDFK RWKHU RU QRW %DWHV LQWURGXFHG WKLVWHFKQLTXHXVLQJD0DFK=HKQGHULQWHUIHURPHWHUWRGHWHUPLQHWKHRSWLFDOTXDOL W\RIDFRQYHUJLQJZDYHIURQW>%DWHV@+HZDVDUUDQJLQJWKHLQWHUIHURPHWHUVR WKDW IRU HDFK RI WKH WZR SDWKV WKH ZDYH FRQYHUJHG DW WKH VDPH SRLQW ) RQ WKH VHPLUHIOHFWLYHVXUIDFHRIWKHODVWGLYLGHU'LQ)LJ7KHILUVWGLYLGHU'DQG

7ZRZDYHLQWHUIHUHQFH



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tWKHZHGJHWKLFNQHVVDWWKHLQFLGHQFHSRLQWDQGn LWVUHIUDFWLYHLQGH[7KHQZHWDNHDVRXUUHIHUHQFHV\VWHPDVHWRIWKUHHD[HVZLWK RULJLQDWWKHSRLQWRILQFLGHQFH$7KHzD[LVLVFKRVHQLQWKHGLUHFWLRQRIWKHUH IOHFWHGEHDPIURPWKHILUVWIDFHZLWKWKH xD[LVO\LQJRQWKHSODQHRILQFLGHQFH 7KHFRRUGLQDWHVRIWKHH[LWSRLQW&RIWKHUHIUDFWHGUD\DUHWKHQ



x&

t FRV - WDQ -c t

y&

t WDQ D

z&

t VLQ - WDQ -c t

VLQ n

s

 VLQ  -



# t D VLQ  n   VLQ  -

 



ZKHUH s LV WKH shear distance DQG LQ WKH VHFRQG HTXDWLRQ WKH DSSUR[LPDWLRQ LV YDOLGIRUVPDOOD1HJOHFWLQJWHUPVRIKLJKHURUGHULQDLWIROORZVWKDWEHWZHHQ WKHWZRUHIOHFWHGEHDPVWKHRSWLFDOSDWKGLIIHUHQFHRQWKHxySODQHZLWKz LV 

D # t n   VLQ  - 

 



,QWHUIHUHQFH

s

y a side view d

d

K j

x z

s

C top view

J

A

(b)

(a)



)LJ 6FKHPDWLFUHSUHVHQWDWLRQRID0XUW\LQWHUIHURPHWHUDQGLQWHUIHUHQFHSDWWHUQRE WDLQHGZLWKDVSKHULFDOZDYHIURQW7KHUD\VDQGDQJOHVWKDWGRQRWOLHRQWKHSODQHRIWKH GUDZLQJDUHVKRZQLQJUD\7KHLQFOLQDWLRQEHWZHHQWKHIDFHVRIWKHODPLQDLVH[DJJHUDWHG VRDVWRKLJKOLJKWWKHSDWKRIWKHUHIOHFWHGUD\IURPWKHVHFRQGVXUIDFH

ZKLOH WKH YHUVRU RI WKH UD\ HPHUJLQJ IURP & LV DSSUR[LPDWHO\ JLYHQ E\ G  ZKHUH G # D n   VLQ  - 



 

7KHWZRUHIOHFWHGUD\VDUHWKHUHIRUHVKLIWHGEHWZHHQWKHP LQWKHGLUHFWLRQRI WKH xD[LV  E\ D GLVWDQFH s WKDW GHSHQGV RQ WKH WKLFNQHVV RI WKH ZHGJH DQG ZLWK PXWXDOLQFOLQDWLRQGWKDWGHSHQGVRQD%RWKYDULDEOHVGHSHQGRQ-DQGLQSDUWLF XODUsKDVDPD[LPXPWKDWIRUDJODVVZLWKLQGH[FRUUHVSRQGVWRDQDQJOH RILQFLGHQFHRIDERXWƒ /HW XV FRQVLGHU D VHFRQG V\VWHP RI &DUWHVLDQ D[HV ZLWK WKHz'D[LV FRLQFLGHQW ZLWKWKHSULQFLSDOUD\RIWKHUHIUDFWHGEHDPZHWDNHDVWKHRULJLQSRLQW sa RQ WKHSODQHz RIWKHSUHYLRXVV\VWHPZKHUH 

a





y&  Dz& # tD   VLQ  - 

 

7KHUHODWLRQVKLSEHWZHHQWKHWZRD[LVV\VWHPVLVGHILQHGE\WKHWUDQVIRUPDWLRQ xc



§ yc· ¨ zc ¸ © ¹

x  s § FRV G VLQ G · § y  a · §  G · § y  a ·  ¨  VLQ G FRV G ¸ ¨ z ¸ # ¨ G  ¸ ¨ z ¸  © ¹© ¹ © ¹© ¹

 

,QWKHUHVSHFWLYHUHIHUHQFHV\VWHPVWKHILHOGVRIWKHWZRZDYHVUHIOHFWHGE\WKH ODPLQDDUHJLYHQE\

7ZRZDYHLQWHUIHUHQFH





E

Eo x y  z e ikz i) x  y  z 

E

Eo x c y c z c  D e ik zc D i) xc yc zc D 



 

ZKHUH)UHSUHVHQWVDSKDVHYDULDWLRQLQVSDFHZLWKUHVSHFWWRDSODQHZDYH7KH VHFRQGEHDPDOVRSUHVHQWVLQJHQHUDODFHUWDLQDPRXQWRIDVWLJPDWLVPLQWURGXFHG E\WKHUHIUDFWLRQVEXWDVORQJDVDLVYHU\VPDOOFRPSDUHGWRWKHUDGLXVRIFXUYD WXUHRIWKHZDYHIURQWRQWKHODPLQDWKLVHIIHFWLVVPDOODQGZHKDYHFKRVHQWRQH JOHFWLWKHUH+RZHYHUWKHFRQVHTXHQFHVRIWKLVDEHUUDWLRQFDQEHPLQLPL]HGE\ DGMXVWLQJWKHLQFOLQDWLRQRIWKHODPLQDIRUWKHYDOXH-WKDWPD[LPL]HVWKHGLVWDQFH s>+HJHGXVHWDO@ 1RUPDOO\WKHWZRUHIOHFWHGZDYHVDUHDOPRVWRIWKHVDPHLQWHQVLW\DQGLIWKH YDULDWLRQVLQILHOGDPSOLWXGHDUHVPDOOZLWKLQWUDQVYHUVHGLVWDQFHVFRPSDUDEOHWRs WKHLQWHUIHUHQFHILJXUHYLVLEOHRQDVFUHHQLQzKDVDPRGXODWLRQRILQWHQVLW\JLYHQ E\ 

E  E



|  Eo



  FRV 4 

 

ZKHUHWKHDUJXPHQWRIWKHPRGXODWLRQLVJLYHQE\ 

4 | k Gy  kD  ) x  y  z  ) x  s y  a  Gz  z  Gy  D 

 

ZKHUH(TV  KDYHEHHQXVHGQHJOHFWLQJWKHWHUPVRIRUGHUKLJKHUWKDQWKH ILUVWLQD /HWXVFRQVLGHULQSDUWLFXODUWKHFDVHLQZKLFKWKHSKDVH)UHSUHVHQWVDSDUD EROLFZDYHIURQWZLWKV\PPHWU\RIURWDWLRQDURXQGWKHzD[LVWKDWLV 

) x y  z

k x   y   R z





 

ZKHUHRLVWKHORFDOFXUYDWXUHUDGLXVRIWKHZDYHIURQW7KHQPDNLQJDVHULHVRI UHDVRQDEOHDSSUR[LPDWLRQVWKDWQHJOHFWWKHGLVSODFHPHQWVDVVRFLDWHGZLWKWKHDQ JOHDEHWZHHQWKHZHGJHIDFHVDQGWKHGLIIHUHQFHRIRSWLFDOSDWKDIRUZKLFK 

a  Gz  s  w z  Gy  R z  D  R z 

 

ZKHUHwLVWKHKDOIZLGWKRIWKHEHDPDWWKHILUVWRUGHULQGZHKDYH 4 | k Gy  kD 



k s    xs   ya   y Gz R z



ª ks z º | x  k G «  » y  )R  R z ¬ R z ¼



 



,QWHUIHUHQFH

ZKHUH)RFRQWDLQVWHUPVWKDWGRQRWGHSHQGRQxRUy(TXDWLQJWKLVH[SUHVVLRQWR PXOWLSOHVRISZHJHWWKHVSDFLQJEHWZHHQWKHIULQJHVDQGWKHLULQFOLQDWLRQMZLWK UHVSHFWWRWKHxD[LV 

d

O  WDQ M G ª¬  z R z º¼



sd  VLQ M OR z



sK  OR z

 

ZKHUHdLVWKHVHSDUDWLRQRIWKHIULQJHVLQWKHGLUHFWLRQyDQGKLVWKDWQRUPDOWR WKHVDPHIULQJHV)URPWKHPHDVXUHPHQWRIRQHRIWKHVHVHSDUDWLRQVDQGRIWKHDQ JOHMDQGE\WKHNQRZOHGJHRIWKHWUDQVODWLRQVZHFDQWKHQREWDLQWKHUDGLXVRI FXUYDWXUHRIWKHZDYHIURQWDWWKHSRVLWLRQzZKHUHWKHLQWHUIHUHQFHLVREVHUYHG 

R z



sK  O VLQ M

 

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sZKLFKLQWXUQGHSHQGVRQWKHDQJOHRILQFLGHQFH7KHUHIRUHZKHQWKH UDGLXVRIFXUYDWXUHRIWKHZDYHIURQWLVUHODWLYHO\VPDOOWKHIULQJHVEHFRPHFXUYHG DQGORVHWKHLUSDUDOOHOLVP+RZHYHUDVPHQWLRQHGDERYHWKLVGLVWRUWLRQLVUHGXFHG XVLQJWKHDQJOHRILQFLGHQFHWKDWSURGXFHVWKHPD[LPXPVKHDU 6WHOODULQWHUIHURPHWU\

$QLQWHUHVWLQJDSSOLFDWLRQRIWKHGHSHQGHQFHRIYLVLELOLW\RIWKHIULQJHVZLWKWKH VL]H RI WKH VRXUFHV LV UHSUHVHQWHG E\ WKH PHDVXUHPHQW RI WKH GLDPHWHU RU RI WKH VHSDUDWLRQRIFHOHVWLDOERGLHV,WZDVLQLWLDOO\VXJJHVWHGE\)L]HDXDQGWKHQUHDO L]HG E\ 0LFKHOVRQ ZLWK KLV VWHOODU LQWHUIHURPHWHU PRXQWHG RQ WKH UHIOHFWLQJ WHO HVFRSH ZLWK  P DSHUWXUH RI WKH 0RXQW :LOVRQ 2EVHUYDWRU\ >%RUQ DQG :ROI  @7KHVKDSHRIWKLVGHYLFHZDVVLPLODUWRWKDWRI0LFKHOVRQDQG3HDVH@ 6WHOODULQWHUIHURPHWHUVEDVHGRQWKLVSULQFLSOHKDYHDOVREHHQFRQWUXFWHGLQUH FHQWWLPHVZLWKWZRRUPRUHWHOHVFRSHVVHSDUDWHGE\GLVWDQFHVRIRYHUP7KH PRVWSRZHUIXODUHWKHVery Large TelescopeFRQVLVWLQJRIWHOHVFRSHVZLWKP DSHUWXUHDQGDPD[LPXPVHSDUDWLRQRIPDWWKH2EVHUYDWRU\RI&HUUR3DUDQDO LQ &KLOH DQG WKH Large Binocular Telescope RQ 0RXQW *UDKDP LQ $UL]RQD FRQ VLVWLQJRIWZRPPLUURUVVHSDUDWHGE\P>&RQWL@$QDOWHUQDWLYHPHWKRG LV WR FRUUHODWH WKH intensity IOXFWXDWLRQV RI WKH VLJQDOV REWDLQHG ZLWK KLJKVSHHG SKRWRGHWHFWRUVORFDWHGLQGLVWDQWWHOHVFRSHVDLPHGDWDVWDUWREHPHDVXUHG>+DQ EXU\%URZQDQG7ZLVV@ ,QWKHPLFURZDYHUHJLRQLWLVLQVWHDGSRVVLEOHWRGHWHFWWKHUDGLDWLRQILHOGLWVHOI HOHFWURQLFDOO\,QWKLVFRQWH[W0LFKHOVRQ¶VVWHOODULQWHUIHURPHWHUKDVLQVSLUHGWKH GHYHORSPHQWRIWKHaperture synthesisWHFKQLTXHZKLFKDOORZVIRUDKLJKUHVROX WLRQE\PHDQVRIUDGLRWHOHVFRSHVDUUDQJHGWRIRUPDOLQHDURUPDWUL[DUUD\RULQ WZRFURVVGLUHFWLRQV+LJKUHVROXWLRQVDUHWKHQREWDLQHGE\FRPELQLQJa posteriori VLJQDOVRIYDULRXVUDGLRWHOHVFRSHVRQDQLQWHUFRQWLQHQWDOEDVLV Very Large Baseline Interferometry  HTXLSSHG ZLWK DWRPLF FORFNV LQ SDUWLFXODU K\GURJHQ PDVHU ZKLFKSURYLGHDORFDORVFLOODWRUVLJQDO>/LSVRQ HWDO@7KHSquare Kilometer ArrayLVFXUUHQWO\EHLQJUHDOL]HGFRQVWLWXWHGE\DFHQWUDOFOXVWHURIDQWHQQDVH[ WHQGHGRQDGLDPHWHURI.PDQGIURPWKRXVDQGVRIRULHQWDEOHDQWHQQDVDUUDQJHG LQ D PDWUL[ RI  .P LQ GLDPHWHU DQG LQ VSLUDOV XS WR  .P RI GLVWDQFH RU PRUH

0XOWLSOHZDYHLQWHUIHUHQFH 3.3.1 Fabry-Perot interferometers ,QWZR)UHQFKSK\VLFLVWVIURPWKH8QLYHUVLW\RI0DUVHLOOH$OIUHG3HURW DQG&KDUOHV)DEU\GHVFULEHGDQHZLQWHUIHUHQFHGHYLFHWKDWVRRQEHFDPHUHYROX WLRQDU\LQWKHILHOGRIVSHFWURVFRS\DQGLVQRZDIXQGDPHQWDOFRPSRQHQWRIODVHUV ,WPDGHXVHRIWKHLQWHUIHUHQFHSURGXFHGE\WKHPXOWLSOHUHIOHFWLRQVWKDWH[LVWEH WZHHQWZRSHUIHFWO\DOLJQHGVHPLVLOYHUHGSODQHPLUURUV )LJ ,QWKLVZD\ RQHFDQVHH+DLGLQJHU¶VIULQJHV † EXW ZLWKDOPRVWPRQRFKURPDWLFOLJKW  WKHVHULQJVDUHQRZPXFKWKLQQHU)DEU\DQG3HURWVRRQUHDOL]HGWKDWWKLVVXEWOHW\ ZDVZHOOVXLWHGWRSUHFLVHWKLFNQHVVPHDVXUHPHQWVLQZDYHOHQJWKPHDVXUHPHQWV DQG IRU UHVROYLQJ QHLJKERULQJ VSHFWUDO OLQHV ZLWK JUHDW DFFXUDF\ 7KH\ WKHUHIRUH GHVLJQHG DQG EXLOW VRSKLVWLFDWHG LQWHUIHURPHWHUV ZLWK ZKLFK WKH\ GHYRWHG WKHP



)LJ )DEU\3HURW LQ WHUIHURPHWHU

,QWHUIHUHQFH

II

IT

l



VHOYHVWRPDQ\VSHFWURVFRSLFPHDVXUHV,QWKHLUKRQRUWKHGHYLFHVRIWKLVW\SHDUH FDOOHG Fabry-Perot interferometers )3  %HIRUH WKHP *HRUJH $LU\ KDG DQD O\WLFDOO\ VROYHG WKH SUREOHP RI PXOWLSOH UHIOHFWLRQV EHWZHHQ WZR SDUDOOHO JODVV ODPLQDH REWDLQLQJ DQ H[SUHVVLRQ WKDW KH SXEOLVKHG LQ  LQ KLV Mathematical Tracts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| y DQGDWOHDVWLQWKHRU\ ZLWKHTXDOPLUURUV WKH\KDYHDWUDQVPLVVLRQHTXDOWRIRUDSODQHZDYHRQ WKHUHVRQDQFHSHDN7KH\FDQDFFHSWODUJHGLDPHWHUOLJKWEHDPVWKRXJKVPDOOLQ

0XOWLSOHZDYHLQWHUIHUHQFH



GLYHUJHQFHWKLVLVDOLPLWLQWKHXVHRIFODVVLFDOVRXUFHVEXWLVQRWVRZLWKODVHUV +RZHYHUDWFRPSDUDEOHUHVROXWLRQSRZHUWKH\DUHPXFKEULJKWHUWKDQGLIIUDFWLRQ JUDWLQJPRQRFKURPDWRUV 7KH)DEU\3HURWZLWKSODQHPLUURUVZDVWKHSURJHQLWRURIWKHopen optical resonator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¶K\SHUILQH VWUXFWXUHHWF$ZLGHGLVFXVVLRQRIWKH)DEU\3HURWLQWHUIHURPHWHUVDQGWKHLUDS SOLFDWLRQLVUHSRUWHGE\9DXJKDQ  ZLWKPDQ\KLVWRULFDOUHIHUHQFHV,QWKLV VHFWLRQ,RQO\FRQVLGHUWKH)DEU\3HURWWKHRU\ZLWKSODQHPLUURUVDQGWKHLUSURS HUWLHV $OWKRXJK VRPH RI WKHLU OLPLWDWLRQV DSSHDU WR KDYH EHHQ RYHUFRPH ZLWK FXUYHG PLUURU FDYLWLHV PDQ\ RWKHU DVSHFWV UHPDLQ FRPPRQ DQG WKHLU VWXG\ LV SURSDHGHXWLF

3.3.2 Transmission function (Airy function) of a Fabry-Perot with plane and parallel mirrors 8VXDOO\WKHUHDUHWZRZD\VWRFDOFXODWHWKHWUDQVPLVVLRQRID)3WKHILUVWFRQ VLGHUV WKH VXEVHTXHQW UHIOHFWLRQV EHWZHHQ WKH WZR PLUURUV DQG FRXOG EH JHQHU DOL]HGIRUFDVHVRIDQRQFRQVWDQWZDYHRYHUWLPHZKLOHWKHVHFRQGH[SORLWVWKH ERXQGDU\FRQGLWLRQVJLYHQE\WKHPLUURUVWKHPVHOYHVEXWRQO\DSSOLHVWR³FRQWLQ XRXV´ZDYHVWKDWLVVWDWLRQDU\RYHUWLPH7RLQWURGXFHDELWRIJHQHUDOLW\LQVWHDG RIFRQVLGHULQJWKHPRVWWULYLDOFDVHOHWXVFRQVLGHUWKHFDVHWKDWWKHPLUURUVFDQEH GLIIHUHQWDQGZLWKDQDEVRUSWLRQQRWQHFHVVDULO\]HURDQGODVWO\WKDWWKHLQWHUSRVHG PHGLXPFDQLQWURGXFHDGGLWLRQDODEVRUSWLRQ7KLVHQWDLOVOLWWOHHIIRUWPRUHDQGWKH VLPSOLILFDWLRQVFDQDOVREHGRQHLQWKHHQG)RUVLPSOLFLW\ZHWDNHLQVWHDGHTXDO WRXQLW\WKHUHIUDFWLYHLQGH[RQERWKVLGHVRIWKHUHIOHFWLQJVXUIDFHV

,QWHUIHUHQFHZLWKPXOWLSOHSODQHZDYHV

)RUWKHH[SODQDWLRQRIWKLVPHWKRGZKLFKIROORZVWKHDQDO\VLVE\$LU\LQ)LJ



,QWHUIHUHQFH

1

t1 r1

2

r2 t2

tm

tm e idc

 )LJ'LDJUDPRIWKHPXOWLSOHUHIOHFWLRQVEHWZHHQWKHPLUURUV

WKHVXEVHTXHQWUHIOHFWHGZDYHVDUHVFKHPDWL]HGZLWKUD\VVRDVWRGLVWLQJXLVK WKHP7KHPHQWLRQHGFRHIILFLHQWVDUHWKRVHWKDWDSSO\WRWKHILHOG tt rr tP GF

PLUURUWUDQVPLVVLRQFRHIILFLHQWV UHIOHFWLRQFRHIILFLHQWV WUDQVPLVVLRQFRHIILFLHQWRIWKHPHGLXPLQDVLQJOHSDVV SKDVHGLIIHUHQFHLQWKHSDWKEHWZHHQWZRUHIOHFWLRQV

(DFKRIWKHWUDQVPLWWHGEHDPVKDVLQFRPPRQWKHIDFWRUtttPH[S iGF DQGWKH WRWDOILHOGRIWKHWUDQVPLWWHGZDYHLVXOWLPDWHO\WKHVXPRIWKHILHOGVRIWKHVLQJOH RXWSXWZDYHV

E7 



E, tttP ei GF    rrtP ei GF  



f j E, tttP ei GF  ¦ rrtP ei GF j 



t t t ei GF  E,   P    rrtP eiGF









 

6HOIFRQVLVWHQF\RIWKHILHOGZLWKERXQGDU\FRQGLWLRQV

6FKHPDWLFDOO\WKHSUREOHPFDQEHGHVFULEHGE\)LJZKHUHEDQGEUH VSHFWLYHO\ UHSUHVHQW WKH DPSOLWXGHV IRU ZDYHV WUDYHOLQJ IRUZDUG DQG EDFNZDUG ZKLOH WKH RWKHU WHUPV DUH WDNHQ DV LQ WKH SUHYLRXV FDVH 2Q PLUURU  WKH WUDQV PLWWHGIRUZDUGILHOGZLOOVXPZLWKWKHUHIOHFWHGILHOGWKDWUHWXUQVIURPPLUURU 

E 

tE,  rE  

 

,QVWHDGRQPLUURUIRUWKHWUDQVPLWWHGZDYHDQGWKHUHIOHFWHGZDYHZHKDYH

0XOWLSOHZDYHLQWHUIHUHQFH



E+

EI

ET

E_ 1

2



)LJ%RXQGDU\FRQGLWLRQVRQWKHPLUURUVDQGVHOIFRQVLVWHQF\RIWKHILHOG

E7  t E  



E 

r E  



 

)RUWKHSURSDJDWLRQEHWZHHQWKHPLUURUVZHDOVRKDYH E 



E 

tP ei Gc  E    t P ei G c  E   

 

)URPDOORIWKHVHHTXDWLRQVLWIROORZVWKDW E  ttP ei GF  E,  rrtP ei GF E  



DQGZHILQDOO\DJDLQREWDLQ 

E7

t t t ei GF  E,   P  i G    rrtP e F

‫ މ‬

:LWKDIHZPRUHVWHSVZHILQGWKHUHIOHFWHGZDYHILHOG 

E5

§ t ct r t  eiGF · E, ¨ rc     P  i G ¸   rrtP e F ¹ ©

§ t ct r r t  eiGF E, rc ¨       P  i G © rcr   rrtP e F

· ¸  ¹

 

ZKLFKFDQDOVREHREWDLQHGWKURXJKWKHVXPPLQJPHWKRGRIWKHSUHYLRXVVHFWLRQ +HUH r‫ މ‬DQG t‫ މ‬DUH WKH UHIOHFWLRQ DQG WUDQVPLVVLRQ FRHIILFLHQWV UHVSHFWLYHO\ RI WKHILUVWPLUURUVHHQIURPWKH RXWHUVLGHRIWKHFDYLW\VLQFHLQJHQHUDOWKH\DUH QRWHTXDOWRrDQGt:HZLOOGHDOZLWKWKHUHODWLRQVEHWZHHQWKHVHFRHIILFLHQWV ODWHUZKHQZHWUHDW PXOWLSOHOD\HU PLUURUVLQSDUWLFXODUWKHUDWLRt‫މ‬tr‫މ‬rWXUQV RXW WR EH UHDO DQG QHJDWLYH IRU D PLUURU ZLWKRXW DEVRUSWLRQ OLNH WKRVH REWDLQHG ZLWKDVXFFHVVLRQRIWUDQVSDUHQWGLHOHFWULFOD\HUV



,QWHUIHUHQFH

7UDQVPLWWHGZDYHLQWHQVLW\

&RQVLGHUWKHFDVHLQZKLFKWKHH[WHUQDOPHGLXPKDVWKHVDPHUHIUDFWLYHLQGH[ RQERWKVLGHVRIWKHLQWHUIHURPHWHUDQGLVDOVRHTXDOWRWKHLQWHUQDORQH7KHQWR JHW WKH WUDQVPLWWHG ZDYH LQWHQVLW\ LW LV HQRXJK WR GR WKH VTXDUH PRGXOXV RI (T   

tttP I,   rrtP ei GF

I7





6R IDU QR K\SRWKHVLV KDV EHHQ PDGH RQ WKH r DQG t FRHIILFLHQWV RI PLUURUV ZKLFKLQJHQHUDODUHFRPSOH[KRZHYHULQWKHFDOFXODWLRQRIWKHWUDQVPLWWHGLQ WHQVLW\ZHFDQLQWURGXFHWKHUHDOSDUDPHWHUVRDQGTPE\MRLQLQJLQGWKHSKDVH VKLIWVGUDQGGPUHVSHFWLYHO\GXHWRWKHUHIOHFWLRQVDQGWKHWUDQVPLVVLRQRIWKHLQ WHUQDOPHGLXPZKHUHE\ 

G

GF  G U  G P 

 

7KHQOHW 





t 

t  T

T

 

EHWKHWUDQVPLVVLYLW\RIWKHPLUURUV 

rr 

R

 

WKHJHRPHWULFPHDQRIWKHUHIOHFWLYLW\RIWKHPLUURUVDQG 



tP 

TP

 

WKHWUDQVPLVVLYLW\RIWKHPHGLXPLQWHUSRVHGEHWZHHQWKHPLUURUVIRUDVLQJOHSDVV ,IWKHUHIUDFWLYHLQGH[HVRIWKHYDULRXVPHGLDDUHGLIIHUHQW(TV  VKRXOGEH PRGLILHGDGDSWLQJWKHPZLWKWKHVDPHFRQVLGHUDWLRQVXVHGWRGHULYHWKHHTXDWLRQV RI†+RZHYHULIWKHILUVWDQGODVWPHGLDDUHHTXDOZHFDQZULWH I7

I, I,



TTTP   RTP

 ei G

I,

TTTP    RTP FRV G  R TP

TTTP

  RTP

IURPZKLFKZHILQDOO\JHW



  RTP VLQ  G 





0XOWLSOHZDYHLQWHUIHUHQFH

)LJ $LU\ IXQFWLRQ IRU D FRQWUDVWSDUDPHWHUF    7KHFRUUHVSRQG LQJILQHVVH FLVDERXW 



1.0 0.8 0.6

T 0.4 0.2 0.0 m2p



I7

I,

d

( m +1) 2 p 

 TTTP     F VLQ G    RT  P

 

ZKHUHWKHFRQWUDVWSDUDPHWHUFLVGHILQHGE\WKHIRUPXOD 

F

 RTP

  RTP 



 

,I WKH LQWHUQDO PHGLXP LV FRPSOHWHO\ WUDQVSDUHQW DQG WKH PLUURUV DUH HTXDO KHQFHT T TDQG 

R  T  A  

 

ZKHUHALVWKHDEVRUSWLRQRIPLUURUVZHVLPSO\KDYH 

I7

I,

 T    F VLQ G    R 



I,

 A · § ¨  ¸       F VLQ G  ©   R ¹

ZKHUHQRZ 

F

R

  R 



 

)RUJRRGGLHOHFWULFILOPPLUURUVWKHWHUPALVODUJHO\UHSODFHGE\ORVVHVAVGXH WRGLIIXVLRQFDXVHGE\GHIHFWVLQWKHGLHOHFWULFOD\HUV:KHQHYHQWKLVWHUPLVQHJ OLJLEOH(T  LVUHGXFHGWR



,QWHUIHUHQFH



I7

I,

    F VLQ  G 

 

7KHIUDFWLRQDOVRSUHVHQWLQ(TV  DQG  LVDSHULRGLFIXQFWLRQLQ G DQG LV FDOOHG WKH Airy form function )LJ   7KH WUDQVPLVVLRQ PD[LPD DUH REWDLQHGZKHQWKHRUGHURILQWHUIHUHQFHmGHILQHGDV 

G  S

m

 

DVVXPHV LQWHJHU YDOXHV DV     DQG WKH PLQLPD ZKHQ DVVXPLQJ KDOILQWHJHU YDOXHVDV7KHcontrastZKLFKLVWKHUDWLREHWZHHQWKHPD[LPXPDQG WKHPLQLPXPLVF,WLVLPSRUWDQWWRQRWHWKDWLIWKHPLUURUVDUHHTXDODQGWKH ORVVHVE\DEVRUSWLRQRUGLIIXVLRQDUHQXOOWKHWUDQVPLVVLRQLVXQLWDU\RQWKHSHDNV 2WKHUZLVHWKHUHLVDQDWWHQXDWLRQRIWKHWUDQVPLWWHGOLJKWEXWIRUWKHVDPHFWKH WUDQVPLVVLRQ FXUYH REWDLQHG E\ YDU\LQJ G LV RQO\ VFDOHG WR WKH QHZ YDOXH RI WKH WUDQVPLVVLRQRQWKHSHDN

5HIOHFWHGZDYHLQWHQVLW\

6LPLODUO\ZHFDQFDOFXODWHWKHLQWHQVLW\RIWKHUHIOHFWHGZDYHVR



I5

t t c RTP eiG I , rc     rrc   RTP ei G 



§ t tc ·   ¨     ¸ RTP ei G  © rrc ¹ I , rc    RTP ei G



  

ZKHUHLWVKRXOGEHQRWHGWKDWGLQFRUSRUDWHVWKHSKDVHRIrrtPDVLQWKHFDVHRI WUDQVPLVVLRQ 7KH UHIOHFWHG LQWHQVLW\ LV HVVHQWLDOO\ FRPSOHPHQWDU\ WR WKDW RI WKH WUDQVPLWWHGLQWHQVLW\DQGSUHVHQWVDEDFNJURXQGYDOXHHTXDOWRI_r‫_މ‬IXUURZHGE\ DVXFFHVVLRQRISHULRGLFGLSV%RWKWKHQXPHUDWRUDQGGHQRPLQDWRURIWKHIUDFWLRQ KDYHPLQLPDZLWKSHULRGSLQG:KLOHWKHWUDQVPLVVLRQSHDNVDUHV\PPHWULFDO LQGWKHGLSVDUHJHQHUDOO\DV\PPHWULFDOLQIDFWEHFDXVHRIWKHWHUPtt‫ މ‬rr‫ މ‬ aib = s H[S iE ZKHUHsLVLWVPRGXOXVDQGELWVSKDVHLIELVQRW]HURWKHUHLV DVKLIWRIWKHPLQLPDRIWKHQXPHUDWRUDQGDFRQVHTXHQWDV\PPHWU\RIWKHGLSVLQ WKHUHIOHFWHGLQWHQVLW\+RZHYHUWKHUHLVDSK\VLFDOOLPLWWRWKHH[WHQWRIWKLVSKDVH VKLIWWKDWLVQRWPDGHH[SOLFLWLQ(T   ,QSDUWLFXODU(T  FDQEHUHZULWWHQDV   7KLV V\PPHWU\ LV SUHVHQW IRU WKH LGHDO FDVH RI D SODQH ZDYH DQG XQOLPLWHG SODQH PLUURUV ,Q SUDFWLFH D ODVHU EHDP LV VSDWLDOO\ GHOLPLWHG DQG D JRRG DOLJQPHQW LV DOVR UHTXLUHG IRU D WLOWHG EHDPWKHSHDNVDUHDV\PPHWULFDO

0XOWLSOHZDYHLQWHUIHUHQFH

 

I5

     ª    a RTP º¼     a RTP VLQ G   b R TP  bRTP VLQ G rc ¬    RTP    RTP VLQ  G 

,QDGGLWLRQLWKDVDVKLIW'GRIWKHUHIOHFWLRQPLQLPDZLWKUHVSHFWWRWKHWUDQVPLV VLRQPD[LPD*LYHQWKDWWKHWUDQVPLVVLRQRIWKHPLUURUVRID)DEU\3HURWLVQRU PDOO\VPDOOZHFDQDVVXPHVPDOOGa,DQGbVRWKDWWKHVLQHVFDQEHUHSODFHGE\ WKHLU DUJXPHQWV 7KHUHIRUH GLIIHUHQWLDWLQJ ZLWK UHVSHFW WR G WKHQ QHJOHFWLQJ WKH WHUPV RI KLJKHU RUGHU WKDQ WKH ILUVW LQ G a DQG b DQG ILQDOO\ VROYLQJ IRU G ZH ILQGWKHWUDQVODWLRQEHWZHHQWKHUHIOHFWLRQPLQLPDDQGWKHWUDQVPLVVLRQVPD[LPD 

'G # E

  RTP    RTP

)LQHVVH

$VZHFDQLPPHGLDWHO\XQGHUVWDQGIURP(TV  DQG  WKHWUDQV PLVVLRQ SHDNV DUH EHFRPLQJ LQFUHDVLQJO\ QDUURZ ZLWK LQFUHDVLQJ SDUDPHWHU F KHQFH ZLWK WKH UHIOHFWLYLW\ RI WKH PLUURUV ,W LV LQWHUHVWLQJ QRZ WR FDOFXODWH WKH ZLGWK[RIWKHSHDNVDWKDOIKHLJKW ):+0full width half maximum DVDIXQF WLRQRIF/HWXVWKHUHIRUHFRQVLGHURQHRIWKHSHDNVIRUZKLFKG mSZLWKLQWH JHUm$URXQGRXUSHDNWKHWUDQVPLVVLRQLVRIWKHPD[LPXPIRUYDOXHVRIG HTXDOWR 

mS r

Gr

[  

VR 

   F VLQ  [ 

  

 

,I F LV VXIILFLHQWO\ ODUJH WKLV HTXDWLRQ LV YHULILHG IRU YDOXHV [ VXFK WKDW WKH VLQH FDQEHFRQIXVHGZLWKLWVDUJXPHQWDQGWKHUHIRUHVROYLQJIRU[ZHJHW 

[

  F

 

$Q\ZD\WKHSDUDPHWHUWKDWPRVWFKDUDFWHUL]HVWKHTXDOLW\RID)DEU\3HURWLV WKH UDWLR ZKLFK LV FDOOHG WKH finesse RI WKH VHSDUDWLRQ EHWZHHQ WZR VXFFHVVLYH SHDNVDQGWKH):+0ZLGWKRIDSHDN6LQFHWKHGLVWDQFHEHWZHHQWZRSHDNVLVS LQWKHSKDVHGWKHILQHVVHLVJLYHQE\



,QWHUIHUHQFH

F



S F  

 

,QWHUPVRIWKHILQHVVHWKH$LU\IXQFWLRQFDQWKHUHIRUHEHUHIRUPXODWHGDV 

I7 I,

TPD[     F  VLQ  G  S



 

ZKHUHTPD[UHSUHVHQWVWKHSHDNWUDQVPLVVLRQ ,ILQ(T  ZHUHSODFHFZLWK(T  ZHJHWZKDWLVFDOOHGWKHfinesse for reflection FR



S R   R

 

7KLVGLVWLQFWLRQLVQHFHVVDU\EHFDXVHYDULRXVIDFWRUVVXFKDVWKHLPSHUIHFWLRQVRI PLUURUIODWQHVVGLIIUDFWLRQWKHZD\LQZKLFKWKHLQWHUIHURPHWHULVXVHGDQGPRUH FRQWULEXWHWRWKHDFWXDOILQHVVHDVZHVKDOOVHHODWHU

2SWLFDOSDWKDQGSKDVHGLIIHUHQFHEHWZHHQWZRVXFFHVVLYHUHIOHFWLRQV

$VZHDOUHDG\NQRZIURP†WKHGLIIHUHQFHLQRSWLFDOSDWKVEHWZHHQWZR VXFFHVVLYHUD\VLQWKHVHULHVRIUHIOHFWLRQVEHWZHHQWZRSODQDUDQGSDUDOOHOPLUURUV LVJLYHQE\

 L nl FRV -c 



 

ZKHUH-‫މ‬LQGLFDWHVKHUHWKHLQFLGHQFHDQJOHZLWKLQWKHLQWHUIHURPHWHUnLVWKHUH IUDFWLRQLQGH[SUHVHQWWKHUHDQGlLVWKHVHSDUDWLRQEHWZHHQWKHPLUURUV7KHSKDVH VKLIWGXHWRWKLVGLIIHUHQFHLQSDWKOHQJWKLVWKHQ 

GF

S

L OR

S

nl FRV -c  OR

 

1RWHWKDW GF  lk c FRV -c LVMXVWWKHSKDVHGLIIHUHQFHWKDWRQHKDVWUDYHOLQJIURP RQHPLUURUWRWKHRWKHUDORQJWKHGLUHFWLRQQRUPDOWRWKHPLUURUVWKDWLVWKHSUR MHFWLRQLQWKLVGLUHFWLRQRIWKHZDYHYHFWRUk‫މ‬PXOWLSOLHGE\WKHGLVWDQFHl ,QJHQHUDOWKHWZRUHIOHFWLRQVJLYHULVHWRDIXUWKHUSKDVHVKLIWGUZKLFKDOVR GHSHQGVRQWKHDQJOHRILQFLGHQFHDQGWKHZDYHOHQJWK7KHRYHUDOOSKDVHLVWKHUH IRUH 

G

GF  G U  G P 

 

0XOWLSOHZDYHLQWHUIHUHQFH



KRZHYHULWLVQRUPDOO\GPGUGF EXWnotDOVRGPGUS DQGLQSUDFWLFDODS SOLFDWLRQVGPGUFDQEHFRQVLGHUHGFRQVWDQWRUHYHQQHJOLJLEOH,QWKHODWWHUFDVH UHFDOOLQJ(T  ZHFDQDOVRZULWH 

L

nl FRV -c 

mOR

 

3.3.3 Applications of Fabry-Perot interferometers 7KHV\VWHPRIIULQJHVLQWKHWUDQVPLVVLRQ

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

m S

S

nl FRV -cm  G U  OR

FDOOLQJmRWKHLQWHUIHUHQFHRUGHULQWKHFHQWHUVRWKDW 

mR S

S

nl  GU  OR

DQGVXEWUDFWLQJWKHVHHTXDWLRQVEHWZHHQWKHPLWLVIRXQGWKDWWKHSKDVHGLIIHUHQFH ZLWKWKHFHQWHULV 

 mR  m S

S

nl   FRV -cm  OR

 

IURPZKLFKIRUVPDOO-‫މ‬ 

OR mR  m # -cm  nl

 

,IZHQRZQXPEHUWKHULQJVDVZHGLGIRUWKH+DLGLQJHUIULQJHVWKHUDGLXVRIWKH ULQJVLQWKHIRFDOSODQHRIWKHOHQVLV



,QWHUIHUHQFH

f

(a)

piezoelectrics

photocell diaphragm f

(b)



)LJ $UUDQJHPHQWXVHGZLWKD)DEU\3HURWLQWKHWZRFRQILJXUDWLRQV D HWDORQDQG E VFDQQLQJLQWHUIHURPHWHU



rp # f

n nH

OR p    e  nl

 

ZKHUHeLVWKHfractional order in the centerfLVWKHIRFDOGLVWDQFHRIWKHOHQVLQ WKHH[WHUQDOPHGLXPDQGnHLVWKHUHIUDFWLYHLQGH[RXWVLGHRIWKHLQWHUIHURPHWHU 8QGHULGHDOFRQGLWLRQVDVZHOODVIRUWKHSXUSRVHRIEHFRPLQJGHQVHUWKHULQJV DUHSURJUHVVLYHO\WKLQQHUPRYLQJDZD\IURPWKHFHQWHU)RUDnullIUDFWLRQDORUGHU WKHFHQWUDOIULQJHLVDVSRWRIOLJKWZLWKWKHPD[LPXPLQWKHFHQWHU$V-‫މ‬LQFUHDV HVWKHSKDVHGLIIHUHQFHZLWKWKHFHQWHUYDULHVDSSUR[LPDWHO\DV 

G-

S

nl  -c  OR

 

ZKHUHE\ E\ VXEVWLWXWLQJ WKLV H[SUHVVLRQ LQSODFH RI[ LQ (T   DQG WKHQ VROYLQJIRU-‫މ‬ZHILQGWKDWWKHZLGWKDWKDOIPD[LPXP ):+0 RIWKHVSRWLV 

a

f

n nH

 OR  F nl

ZKHUHFLVWKHILQHVVHGHILQHGE\(T  

 

0XOWLSOHZDYHLQWHUIHUHQFH



7KHQXPEHURIYLVLEOHULQJVLVOLPLWHGE\WKHDQJXODUILHOGRIWKHLQFLGHQWUDGLD WLRQ )RU D SRLQW VRXUFH WKH LQWHUIHUHQFH IULQJHV DUH QRW ORFDOL]HG KRZHYHU LQ JHQHUDOWKH\WDNHRQDPRUHFRPSOLFDWHGDVSHFWWKDQ+DLGLQJHU¶VIULQJHV,QGHHG IRUQRQ]HURDQJOHVRILQFLGHQFHIURPDVLQJOHLQFLGHQWUD\ ZHREWDLQDVHULHVRI UD\VWKDWDUHKRZHYHUVKLIWHGEHWZHHQWKHP7KHUHIRUHIRUH[DPSOHLIRQHXVHVD GLYHUJHQWODVHUEHDPWRREVHUYHVKDUSIULQJHVLWLVQHFHVVDU\WRXVHDOHQVORFDWHG GRZQVWUHDP RI WKH LQWHUIHURPHWHU WKDW UHFRPELQHV WKH YDULRXV UD\V LQ LWV IRFDO SODQH0RYLQJDZD\IURPWKLVSODQHWKHULQJVDSSHDUWRPXOWLSO\DQGWKHLULQWHU SUHWDWLRQLVFRPSOLFDWHG 2QH ZRXOG WKLQN WKDW HYHQ WKH OHQV LQ IURQW RI WKH LQWHUIHURPHWHU LV HVVHQWLDO KRZHYHU WKLV LV QRW FRPSOHWHO\ WUXH ,W LV XVHG WR FROOLPDWH WKH UDGLDWLRQ RI WKH VRXUFHWKHUHE\WKHUDGLDWLRQHPLWWHGE\HDFKVRXUFHSRLQWLVFRQYHUWHGLQWRDSODQH ZDYHRQZKLFKWKHDERYHWKHRU\LVEDVHG,QDGGLWLRQWKHOHQVOLPLWVWKHDQJXODU ILHOGRIUDGLDWLRQLPSLQJLQJRQWKHLQWHUIHURPHWHUWRWKHEHQHILWRIWKHEULOOLDQFH RIWKHIULQJHVWKDWUHPDLQSUHVHQW,QFRQVWUXFWLQJWKHEULJKWIULQJHVRIWKHLQWHUIHU HQFHILJXUHDSODQH)DEU\3HURWKDVLQIDFWWKHDELOLW\WRVHOHFWUD\VZLWKSDUWLFX ODUGLUHFWLRQV,IWKHVRXUFHLVQRWLQIRFXVRUWKHOHQVLVDEVHQWWKHYDULRXVUD\V HPLWWHGE\DVLQJOHVRXUFHSRLQWDUHGLUHFWHGWRGLIIHUHQWSRLQWVRQWKHIRFDOSODQH RIWKHVHFRQGOHQV%XWDJDLQIRUWKHPXOWLSOHLQWHUIHUHQFHRIWKHUHIOHFWLRQVHULHV RQWKHPLUURUVRIHDFKLQFLGHQWUD\WKHEULJKWQHVVDWHDFKSRLQWRQWKLVSODQHGH SHQGVRQWKHDQJOHRILQFLGHQFHUHVXOWLQJLQDVHULHVRIOXPLQRXVULQJV'LIIHUHQW VRXUFH SRLQWV JLYH ULVH WR WKH VDPH ILJXUH DQG LI WKH VRXUFH LV VSDWLDOO\ LQFR KHUHQW WKHRYHUDOOLQWHQVLW\LVWKHLQFRKHUHQWVXPRIWKHFRQWULEXWLRQVRIHDFKRI WKHVRXUFHSRLQWV /DVWO\WZRZRUGVRQWKHIULQJHV\VWHPLQUHIOHFWLRQ7KHVHDUHFRPSOHPHQWDU\ WRWKRVHLQWUDQVPLVVLRQZLWKWKLQGDUNULQJVLQDEULJKWILHOGDJDLQREVHUYHGLQ WKHIRFDOSODQHRIDOHQV+RZHYHUWKHWUHQGRIWKHLQWHQVLW\RIWKHVHULQJVLVFULWL FDOO\GHSHQGHQWRQWKHSUHVHQFHRIDSKDVHVKLIWLQWKHUHIOHFWLRQRQWKHPLUURUVLQ WKH SUHVHQFH RI ZKLFK WKH\ EHFRPH DV\PPHWULFDO 7KHLU YLVLELOLW\ DOVR EHFRPHV QHJOLJLEOH ZKHQ WKH ORVVHV RQ WKH PLUURUV DUH FRPSDUDEOH WR WKHLU WUDQVPLVVLRQ >9DXJKDQ†@

)UHHVSHFWUDOUDQJHDQGRYHUODSRIRUGHUV

$VZHKDYHMXVWVHHQLQWKHSUHYLRXVVHFWLRQGLVDIXQFWLRQRIVHYHUDOIDFWRUV lnO RUWKHIUHTXHQF\Q -‫މ‬DQGE\YDU\LQJRQHZHFDQGRDVFDQRIWKH$LU\ IXQFWLRQ)RUH[DPSOHLIZHPHDVXUHWKHLQWHQVLW\RIDPRQRFKURPDWLFODVHUEHDP WUDQVPLWWHGE\WKH)3ZLWKDSKRWRGLRGHDQGSXWWKLVVLJQDORQWKHYD[LVRIDQRV FLOORVFRSHZKLOHRQWKHXD[LVZHVHQGDVLJQDOFRUUHVSRQGLQJWRWKHSDUDPHWHU WKDW LV YDULHG RQ WKH VFUHHQ ZH REVHUYH WUDFHV VLPLODU WR WKH RQH LQ )LJ  :KHQWKHLQWHUIHUHQFHRUGHUGHILQHGE\(T  DVVXPHVWZRVXFFHVVLYHLQ WHJHUYDOXHVWKHGLVWDQFHEHWZHHQWZRFRUUHVSRQGLQJSHDNVWKDWRQHKDVLQWHUPV RIWKHSDUDPHWHUWKDWYDULHV DOWKRXJKRQHDWDWLPH LV



,QWHUIHUHQFH

IRUWKHSKDVHGLWLVQDWXUDOO\

'G 65



 S 

 

IRUWKHRSWLFDOSDWKL n l FRV-‫މ‬ZLWKORQFRQVWDQWLWLV 

' L 65

c  Q

OR 

Lm   Lm

 

IRUWKHZDYHOHQJWKZLWKLFRQVWDQWLWLV 

'O R 65

O  O m  O m  # R  L

 

IRUWKHIUHTXHQF\ Q c O R ZLWKLFRQVWDQWLWLV

'Q 65



Qm   Qm 

c  L

 

,QSDUWLFXODUIURP(TV    DQG  LWLVIRXQGWKDWWKHFDYLW\ UHVRQDQFHIUHTXHQF\FRPEDWWKHYDULDWLRQRImZLWKDVVLJQHGYDOXHVRIWKHRWKHU SDUDPHWHUVLVH[SOLFLWO\ 

Qm

mS  G U  G P c  Snl FRV -

 

,WLVWREHQRWHGWKDWLQ(T   'Q 65LVLQGHSHQGHQWRIQ+RZHYHUWKLVLV LQJHQHUDORQO\DQDSSUR[LPDWLRQDVVXPLQJWKDWWKHVXPRISKDVHVKLIWVGXHWRWKH PLUURUV DQG WKH LQWHUQDO PHGLXP GU  GP GRHV QRW GHSHQG RQ Ȟ 7KLV FDQ EH DFKLHYHGE\UHFLSURFDOO\FRPSHQVDWLQJWKHGLVSHUVLRQRIWKHPLUURUVDQGWKHPHGL XP 7KH 65 LQGH[ LQ (TV   PHDQV WKDW WKHVH GLIIHUHQFHV FRUUHVSRQG WR WKHfree spectral range7RXQGHUVWDQGLWVPHDQLQJOHWZHORRNDWDQH[DPSOH 6XSSRVHZHZDQWWRPHDVXUHWKHK\SHUILQHVWUXFWXUHRIDFHUWDLQDWRPLFVSHF WUDOOLQHHPLWWHGE\DVRXUFHZKRVHOLJKWEHDPLVZHOOFROOLPDWHGRQWKHLQWHUIHU RPHWHU7KHQZHVFDQWKHRSWLFDOSDWKLRIWKHLQWHUIHURPHWHUZKLOHPHDVXULQJWKH OLJKWWUDQVPLWWHGZLWKDGHWHFWRUDVLQWKHFRQILJXUDWLRQRI)LJ E ZLWK-‫ މ‬ ,Ig Q LVDIXQFWLRQWKDWUHSUHVHQWVWKHVSHFWUDOGLVWULEXWLRQRIWKHLQFLGHQWUDGLD WLRQWKHVLJQDOWKDWLVREVHUYHGLVWKHFRQYROXWLRQRIgZLWKWKH$LU\IXQFWLRQ f



S L



³ g Q   F VLQ SQ  L c d Q  

 

0XOWLSOHZDYHLQWHUIHUHQFH



6LQFHWKH$LU\IXQFWLRQLVSHULRGLFWKLVLQWHJUDOFDQEHUHSKUDVHGDVDVXPPDWLRQ RILQWHUIHUHQFHRUGHUmWDNHQDVDQLQWHJHU 

S L

f

c L

c ·  § g ¨ 'Q  m d 'Q  ¸  L © ¹   F VLQ  § S'Q  L · L ¨ ¸ c ¹ ©

¦ ³

m  c

 

>ZKHUHWREHVWULFWZLWKWKLVH[SUHVVLRQZHDVVXPHg Q  IRUQc[email protected] *HQHUDOO\RQO\VPDOOVFDQVRIWKHOHQJWKLDUHXVHGDURXQGDPHDQYDOXHLR DQGLQWKLVFDVHWKHZLGWKRIWKH$LU\IXQFWLRQSHDNVRQWKHD[LVRIWKHSDUDPHWHU 'QFKDQJHVOLWWOHZLWKLQWKHUDQJHRILYDOXHVWKDWDUHFRQVLGHUHG,IWKHILQHVVHLV VXIILFLHQWO\KLJKWKH$LU\IXQFWLRQDVVXPHVQHJOLJLEOHYDOXHVDURXQGWKHLQWHJUD WLRQH[WUHPHVRI(T  ZKLFKFDQWKHQEHUHSODFHGZLWKr c LR 6RWKH LQWHJUDOFDQEHZULWWHQDV 

Sc g Lc Q  F  LR R

c  LR

³

g 'Q  Qc

c  LR

 d 'Q    L · §   F VLQ  ¨ S'Q c R ¸ © ¹

ZKHUHWKHLQLWLDOFRHIILFLHQWLVLQWURGXFHGIRUDVXLWDEOHUHQRUPDOL]DWLRQZLWKLQWKH OLPLWRIDODUJHILQHVVHF 7KH QHZ IXQFWLRQ g Lc R Q  UHSUHVHQWV WKH FRQYROXWLRQ RI g Q  ZLWK D VLQJOH SHDNRIWKH$LU\IXQFWLRQZKLOHWKHLQGH[LRUHPLQGVXVWKDWWKLVIXQFWLRQLVFDO FXODWHGIRUDSUHGHWHUPLQHGYDOXHDURXQGZKLFKWRPRGXODWHWKHRSWLFDOSDWKEH WZHHQWKHPLUURUV(T  FDQQRZEHUHZULWWHQDV 

S L

f Sc § c · g Lc ¨ m ¦ ¸   F  LR m  R ©  L ¹

 

,I WKH ILQHVVH RI WKH LQWHUIHURPHWHU LV QRZ VXIILFLHQWO\ KLJK VR WKDW WKH $LU\ IXQFWLRQ DWWKHYDULDWLRQRIQ H[KLELWVYHU\WKLQSHDNVZLWKUHVSHFWWRWKHWUHQGRI g Q ZHFDQDSSUR[LPDWH g‫މ‬LR Q ZLWKg Q LWVHOI2QWKHRWKHUKDQGIRUVPDOOYDU LDWLRQV'LDURXQGLRLWLV 

mc mc mc #  'L  L  LR  LR

Qm  'Q 

 

DQG KHQFH WKH REVHUYHG VLJQDO LV JLYHQ E\ WKH VXSHUSRVLWLRQ RI VSHFWUDO RUGHUV HDFKRIZKLFKFDQEHDWWULEXWHGWRDVFDOHLQQRULHQWHGLQWKHRSSRVLWHGLUHFWLRQ IURPWKHRQHRQL )LJ 6RWKHRUGHUmLVDVVRFLDWHGZLWKDVFDOHLQZKLFK WKH IUHTXHQF\ Qm  mc LR  FRUUHVSRQGV WR WKH SRVLWLRQ LR LQ WKH OHQJWK VFDOH



,QWHUIHUHQFH

m

m+1

Lo nm+2 nm+1

nm+1

L np

order m

nm

np

order m +1



)LJ6XSHUSRVLWLRQRIVSHFWUDORUGHUV

ZLWKDVFDOHUHQRUPDOL]DWLRQHTXDOWR 

'Q 'L



m c  LR  LR

 

7KHYDULRXVVFDOHVLQQDUHVKLIWHGIURPHDFKRWKHUE\DQDPRXQW 

Q m   Qm

c  LR

'Q 65 

 

,IWKHUHIRUHWKHVSHFWUXPRIUDGLDWLRQWREHVWXGLHGLVFRQWDLQHGZLWKLQDIUHH VSHFWUDO UDQJH WKH SDWWHUQ RI WKH WUDQVPLWWHG VLJQDO ZLOO VKRZ VHYHUDO UHSHDWHG FRSLHVRILWLQVXFFHVVLRQ,IRQWKHRWKHUKDQGWKHVSHFWUXPLVZLGHUWKHYDULRXV FRSLHVRYHUODSDQGLWPD\QRWEHHDV\WRUHFRQVWUXFWWKHRULJLQDOVSHFWUXP7KHVH DPELJXLWLHVFDQVWLOOEHUHVROYHGE\UHFRUGLQJWKHVSHFWUDIRUWZRRUPRUHGLIIHUHQW YDOXHVRILR

5HVROYLQJSRZHURID)DEU\3HURW

:H VSRNH HDUOLHU RI WKH DELOLW\ RI D )DEU\3HURW LQWHUIHURPHWHU WR UHVROYH WKH VSHFWUDOOLQHV&RQVLGHUWKHFDVHRIDVSHFWUXPFRQVLVWLQJRIWZRQHLJKERULQJOLQHV ZLWKIUHTXHQF\ Q DQGQ WKH VHSDUDWLRQRIZKLFK LV VPDOO FRPSDUHG WR WKH IUHH VSHFWUDOUDQJHDQGZLWKWKHVDPHLQWHQVLW\$VZHKDYHVHHQEHIRUHWKHLQWHQVLW\ WUDQVPLWWHGE\YDU\LQJWKHRSWLFDOSDWKOHQJWKLLVJLYHQE\WKHFRQYROXWLRQZLWK WKH $LU\ IXQFWLRQ ZKLFK LQ WKH FDVH RI WZR QHDUO\ PRQRFKURPDWLF FRPSRQHQWV QDUURZHUWKDQWKH$LU\SHDNVIURP(T  FDQEHH[SUHVVHGDVWKHVXP

0XOWLSOHZDYHLQWHUIHUHQFH



S

)LJ 7D\ORU¶V FULWHULRQ IRU UHVROYLQJ WZR QHDUE\ VSHFWUDOOLQHV

1 Io

0.83

Io /2

n



S L

I

  F VLQ 

   I     F VLQ SQ  L c SQ  L c



 

,I ZH QRZ OLPLW RXUVHOYHV WR FRQVLGHULQJ D JLYHQ LQWHUIHUHQFH RUGHU m WKH WZR SHDNVOLHDWWKHYDOXHV L mc  Q DQG L mc  Q  7KHUHIRUH 

S L

I

    I § L  L ·  § mS L  L ·   F VLQ  ¨ mS   F VLQ ¸ ¨ ¸ L ¹ L ¹ © ©

1RZLWLVVDLGWKDWWKHWZRIUHTXHQFLHVQDQGQDUHUHVROYHGLQWKHVFDQRILLID GHSUHVVLRQEHWZHHQWKHFRUUHVSRQGLQJSHDNVLQWKHWUDQVPLWWHGVLJQDOLVYLVLEOH$ XVHIXOFRQYHQWLRQIRUTXDQWLI\LQJWKLVFRQFHSWLVWKDWNQRZQDVTaylor’s criterion )LJ ,WVWDWHVWKDWWZRVSHFWUDOOLQHV RIHTXDOLQWHQVLW\ EHJLQWREHUHVROYHG ZKHQWKHLUVHSDUDWLRQLVVXFKWKDWWKHLULQGLYLGXDOFXUYHVLQWHUVHFWDWKDOIKHLJKW,Q WKLV FDVH ZLWK L KDOIZD\ EHWZHHQ WKH WZR OLQHV ZKHUHE\ L  L L  L L  L  ZLWKLL|LRZHFDQZULWH 

S

I

 § L   F VLQ  mS  ¨ ©

 L · ¸  LR ¹

I  

IURPZKLFKIRUELJFZHILQGWKDW 

L  L LR

  F mS

  mF

 

ZKHUH FLVWKHILQHVVHGHILQHGE\(T  )RUWKLVFRQGLWLRQRIVHSDUDWLRQLQ FRUUHVSRQGHQFHWRRQHRIWKHWZRSHDNVWKHVLJQDOLVLQVWHDG





,QWHUIHUHQFH

S

I  I

  # I  I    § · L  L   F VLQ  ¨ m S   ¸ LR ¹ ©

 I  

+RZHYHULQWKHVXPRIWKHWZRFXUYHVWKHPD[LPDDUHDOLWWOHFORVHUWRHDFKRWKHU DQGWKHVLJQDODWWKHPD[LPXPLVDERXWI :LWK 7D\ORU¶V FULWHULRQ WKH UDWLR EHWZHHQ WKH FHQWHUWRSHDN VLJQDOV LV DERXW 7KLVYDOXHLVDOPRVWWKHVDPHDVRayleigh’s criterionZKLFKFRQVLGHUVWZR OLQHVUHVROYHGZKHQWKLVUDWLRLV,WGHULYHVIURPWKHFKDUDFWHULVWLFVRIDSULVP RUJUDWLQJVSHFWURVFRSHZKRVHLQVWUXPHQWDOIXQFWLRQLQDGGLWLRQWRDFHQWUDOSHDN KDVDVXFFHVVLRQRIVHFRQGDU\PD[LPDDQGPLQLPDVRWKDWWKHWZROLQHVDUHFRQ VLGHUHGUHVROYHGZKHQWKHPDLQSHDNRIRQHIDOOVRQWKHILUVWPLQLPXPRIWKHRWK HU )LQDOO\ LQ WHUPV RI IUHTXHQF\ WKH PLQLPXP VHSDUDWLRQ IRU FRQVLGHULQJ WZR OLQHVUHVROYHGLV 

GQ

Q  Q

mc mc mc L  L  #  L  L  LR LR

'Q 65 F



 

7KHresolving powerLVGHILQHGDV 

RP

Q GQ

O  GO

 

7KHQILQDOO\IRUD)DEU\3HURWLWUHVXOWVWKDW 

RP

mF  

 

DQGWKHUHIRUHLWLVGHWHUPLQHGE\WKHRUGHURILQWHUIHUHQFHDQGWKHILQHVVH,I5D\ OHLJK¶VFULWHULRQLVXVHGWKHUHVROYLQJSRZHUEHFRPHVmF 7KHUHVROYLQJSRZHUFDQLQSULQFLSOHWKHUHIRUHEHLQFUHDVHGDWZLOOE\JURZLQJ WKHRUGHURILQWHUIHUHQFHDQGWKLVFDQEHDFKLHYHGE\LQFUHDVLQJWKHVHSDUDWLRQEH WZHHQ WKH PLUURUV +RZHYHUDW WKH VDPH WLPH WKH IUHH VSHFWUDO UDQJHGHFUHDVHV DQG WKLV PD\ UHQGHU SUREOHPDWLF WKH LQWHUSUHWDWLRQ RI WKH WUDQVPLWWHG VLJQDO ,Q SUDFWLFHRQHKDV WR ORRNIRUD VXLWDEOH FRPSURPLVH IRU HDFK DSSOLFDWLRQ DQG LQ JHQHUDOLWLVJRRGWKDWWKHIUHHVSHFWUDOUDQJHUHPDLQVFRQVLGHUDEO\ODUJHUWKDQWKH VSHFWUDOZLGWKRIWKHOLQHVWREHVWXGLHG

5HFRUGLQJRIVSHFWUD

,QWKHWZRFRQILJXUDWLRQVRI)LJWKHILUVWPHWKRGRIUHFRUGLQJVSHFWUDXV LQJDQetalonFRQVLVWVLQSKRWRJUDSKLQJWKHIULQJHV\VWHPWKLVFDQEHGRQHZLWKD

0XOWLSOHZDYHLQWHUIHUHQFH



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l EHWZHHQ PLUURUV DQG WKH UHIUDFWLYH LQGH[ n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l7KHDGYDQWDJHRIWKHSLH]RHOHFWULFPHWKRGLVKRZHY HUWRIDFLOLWDWHWKHUHFRUGLQJRIWKHHQWLUHVSHFWUXPLQDYHU\VKRUWWLPH,WVPDLQ GLVDGYDQWDJHLVGXHWRWKHK\VWHUHVLVRISLH]RHOHFWULFPDWHULDOVRWKDWWKHUHLVDO ZD\VDVOLJKWGHYLDWLRQIURPWKHSURSRUWLRQDOLW\EHWZHHQGLVSODFHPHQWDQGDSSOLHG YROWDJH )RU KLJKSUHFLVLRQ PHDVXUHPHQWV LW LV SUHIHUDEOH WR XVH WKH SUHVVXUH VFDQ LQ ZKLFKWKHGLVWDQFHEHWZHHQWKHPLUURUVLVNHSWFRQVWDQWEXWWKHLQWHUIHURPHWHULV HQFORVHGLQDFRQWDLQHULQZKLFKWKHSUHVVXUHRIDJDVYDULHVZKLFKWKHQYDULHVLWV UHIUDFWLYHLQGH[n7KLVW\SHRIVFDQLVPXFKVORZHUWKDQWKHSLH]RHOHFWULFRQHEXW WKHFKDQJHRInZLWKWKHJDVSUHVVXUHFDQEHNQRZQDQGWDEXODWHGZLWKJUHDWSUHFL VLRQE\PHDQVRID0LFKHOVRQRU-DPLQLQWHUIHURPHWHU)URP(T  ZHKDYH 

mO R

nl 

7KHQIRUDQDVVLJQHGGLVWDQFHlDQGDQRUGHUmGLIIHUHQWLDWLQJZHJHW





,QWHUIHUHQFH

m GO R

l Gn 

VRE\GLYLGLQJPHPEHUWRPHPEHUZHKDYH 

GO R OR

Gn # Gn  n

2QWKHRWKHUKDQGLQWHUPVRIORDIUHHVSHFWUDOUDQJHLVJLYHQE\(T   DQGKHQFHWKLVVDPHLQWHUYDOLVREVHUYHGZLWKDYDULDWLRQRInHTXDOWR 

'n

OR OR #  nl l

 

,QWKHFDVHRIDLULWLVn±|˜IRUDQDWPRVSKHUHDQGLIIRUH[DPSOHOR  QPDQGl FPDZKROHVSHFWUDORUGHUFDQEHREVHUYHGE\YDU\LQJWKHSUHV VXUHE\DERXWPEDU

/LPLWDWLRQVWRWKHILQHVVHRID)DEU\3HURW

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fiDQGLWVUHODWLYHILQHVVH FiJLYHQE\WKHUDWLREHWZHHQWKHIUHHVSHFWUDOUDQJH DQGWKHLU):+0FKDUDFWHULVWLFZLGWK7KHRYHUDOOLQVWUXPHQWDOIXQFWLRQf Q LVQR ORQJHUWKHVLPSOHIXQFWLRQRI$LU\EXWLVUDWKHUREWDLQHGIURPWKHFRQYROXWLRQEH WZHHQWKHVHIXQFWLRQV7KHZLGWKRIf Q LVWKHUHIRUHJUHDWHUWKDQWKDWRIWKHVLQJOH fiIXQFWLRQVEXWOHVVWKDQWKHVXPRIWKHLUZLGWKV7KHYDOXHRIWKHRYHUDOOILQHVVH

0XOWLSOHZDYHLQWHUIHUHQFH



FGHSHQGVRQWKHVKDSHRIWKHfiLIIRUH[DPSOHZHDVVLPLODWHWKHPWR*DXVVLDQV ZHZRXOGKDYH  F



¦i  Fi 

ZKLOHLIZHDVVLPLODWHWKHPWR/RUHQW]LDQVZHZRXOGKDYHDVLPLODUH[SUHVVLRQ EXWZLWKRXWWKHVTXDUHV 1RUPDOO\WKHIDFWRUVLQSRLQWV D DQG E DUHSUHGRPLQDQWLQGHWHUPLQLQJWKH ILQDOYDOXHRIWKHILQHVVH:LWKRJLYHQE\(T  DQGTPE\(T  IURP (TV  DQG  WKHILQHVVHFRUUHVSRQGLQJWR D LV S RTP    RTP

FD



 

ZKLFKIRUDPHGLXPZLWKRXWORVVHVLVUHGXFHGWRWKHILQHVVHE\UHIOHFWLRQRI(T  7KLVILQHVVHFDQEHLQFUHDVHGE\HQKDQFLQJWKHUHIOHFWLRQRIWKHPLUURUV KRZHYHUZKHQ5EHFRPHVFRPSDUDEOHWRWKHORVVHVWKH)DEU\3HURWWUDQVPLV VLRQEHJLQVWRIDOOUDSLGO\IRUH[DPSOHLQWKHFDVHRI(TV  WKHWUDQVPLV VLRQ RQ WKH SHDN LV UHGXFHG WR  ZKHQ  ±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qSY RU WKH DYHUDJH 506 YDOXH RQ WKH VXUIDFH Oq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

FE

qSY

 RU FE

 

q506  

6HHIRUH[DPSOHWKHWKUHHSRLQWVXSSRUWV\VWHPDGRSWHGE\%XUOHLJK

 



,QWHUIHUHQFH

ZKHUHqLQGLFDWHVWKHRYHUDOOTXDOLW\RIWKHpairRIPLUURUV7KHSUREOHPWKDWXVX DOO\DULVHVLVZKDWUHDVRQDEOHOLPLWVKRXOGEHJLYHQWRWKHILQHVVHIRUUHIOHFWLRQWR EH UHTXLUHG IRU D JLYHQ SDLU RI ODPLQDH" :LWK FD ! FE WKH RYHUDOO LQVWUXPHQWDO IXQFWLRQLVKHDYLO\LQIOXHQFHGE\VXUIDFHGHIHFWVZLWKYLUWXDOO\QRJDLQLQUHVROX WLRQ EXW UDWKHU E\ UHGXFLQJ WKH WUDQVPLWWHG LQWHQVLW\ ,Q SUDFWLFH WKH UHDVRQDEOH OLPLWYDOXHLVEHWZHHQFEDQGFE 3RLQWV F DQG G DUHVLPLODUWRWKHSUHYLRXVRQH,IDLVWKHDQJOHEHWZHHQWKH WZRPLUURUVDQGDWKHLUGLDPHWHU IRUFLUFXODUPLUURUV ZHKDYH FF



 O   DD

 

(YHQWKHLQKRPRJHQHLW\RIWKHPHGLXPEHWZHHQWKHPLUURUVFDQEHH[SUHVVHGDVD IUDFWLRQRIDZDYHOHQJWKDIWHUZKLFKZHSURFHHGDVIRUSRLQW E  ,QWKHFDVHRIDQH[WHQGHGVRXUFHWKHRSHQLQJLQSRLQW H GHOLPLWVWKHDQJXODU ILHOGXVHGIRUUHFRUGLQJ)RUDZHOOFHQWHUHGDQGIRFXVHGDSHUWXUHZLWKUDGLXVb WKHDFFHSWHGPD[LPXPLQWHUQDODQJOHLVDSSUR[LPDWHO\HTXDOWR n b -cP # H  n f



 

ZKHUHnDQGnHDUHWKHLQWHUQDODQGH[WHUQDOUHIUDFWLYHLQGH[HVDWWKH)DEU\3HURW DQGfLVWKHIRFDOOHQJWKLQWKHH[WHUQDOPHGLXPRIWKHOHQVWKDWIRUPVWKHIULQJHV )URP(TV  DQG  ZHDOVRQRWLFHWKDWWKHGLVWULEXWLRQRIWKHSKDVH GLIIHUHQFHVGHOLPLWHGE\WKHDSHUWXUHKDVWKHVKDSHRIDUHFWDQJOHVLQFHIRUHDFK LQFUHPHQWRIWKHRSHQLQJareaWKHUHLVDSURSRUWLRQDOLQFUHDVHLQG-)URPWKHVH HTXDWLRQV WKHRSHQLQJILQHVVH FDQ WKHUHIRUH EH DVVXPHG WR EH WKH LQYHUVH RI WKH VSHFWUDOIUDFWLRQDOORZHGE\WKHRSHQLQJ



FH

S G-P

OR nl



§ nf · ¨ ¸  © nHb ¹

 

ZKHUHlLVVWLOOWKHSK\VLFDOVHSDUDWLRQEHWZHHQWKHPLUURUVLQSDUWLFXODULIbLV FKRVHQDVHTXDOWRWKHGLDPHWHUaRIWKHFHQWUDOVSRWJLYHQE\(T  LWLVa FRZKHUH FRLVWKHILQHVVHDVVRFLDWHGZLWKWKHRWKHUSURFHVVHV7KHRSHQLQJIL QHVVHKRZHYHUGHJUDGHVUDSLGO\LIWKHDSHUWXUHLVQRWZHOOFHQWHUHGRUIRFXVHGVR LW QHHGV WR EH SRVLWLRQHG ZLWK JUHDW FDUH 0RUHRYHU WKH UHFRUGHG VLJQDO KDV LWV PD[LPXPZKHQWKHIUDFWLRQDORUGHUDWWKHFHQWHURIWKHIULQJHV\VWHPLVQRW]HUR EXWUDWKHUKDOIRI G-P  S  )RUWKHDQDO\VLVRIDZHOOFROOLPDWHGODVHUEHDPWKHGLVFRXUVHEHFRPHVGLIIHU HQW EHFDXVH WKH DQJXODU ILHOG LV GHOLPLWHG E\ WKH GLYHUJHQFH RI WKH EHDP LWVHOI ZKLFKFDQEHYHU\VPDOO /DVWO\ZHFRQVLGHUWKHHIIHFWUHVXOWLQJIURPGLIIUDFWLRQ6LQFHPLUURUVKDYHD

'LHOHFWULFILOPZLWKPXOWLSOHOD\HUV



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

'LHOHFWULFILOPZLWKPXOWLSOHOD\HUV 3.4.1 Reflection and transmission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plane and parallelLQWHUIDFHV EHWZHHQ GLHOHFWULF PHGLD DQG WKH ILHOG FRQVLGHUHG LV FRQVWLWXWHG E\ PRQRFKUR PDWLFSODQHZDYHVVLPLODUWRZKDWZHGLGLQ&KDSWHUIRUWKHGHWHUPLQDWLRQRI )UHVQHO¶VIRUPXODV 0RUHJHQHUDOO\DOD\HUHGPHGLXPFDQDOVRKDYHDFRQWLQXRXVWUHQGRIWKHUH IUDFWLYHLQGH[DQGRIWKHHOHFWULFSHUPLWWLYLW\DQGPDJQHWLFSHUPHDELOLW\>%RUQDQG



,QWHUIHUHQFH

:ROI@+LVWRULFDOO\)$EHOqVFRQGXFWHGWKHILUVWVWXGLHVRIWKHSURSDJDWLRQ RIHOHFWURPDJQHWLFZDYHVLQWKHVHPHGLD>$EHOqV@DQGDOOVXEVHTXHQWDQDO\ VLVFDQEHWUDFHGEDFNWRWKHP )RU DQ DVVLJQHG LQFLGHQW SODQH ZDYH RQ D GLVFRQWLQXRXV OD\HUHG PHGLXP WKH DQJOHVRILQFLGHQFH-jZLWKLQHDFKRIWKHOD\HUVFDQEHUHDGLO\GHWHUPLQHGE\WKH UHODWLRQ 

n j VLQ - j

n VLQ - 

 

ZKHUHnjLVWKHUHIUDFWLYHLQGH[RIWKHjWKOD\HUDQGn-DUHWKHLQLWLDOUHIUDFWLYH LQGH[ DQG LQFLGHQFH DQJOH )URP WKHVH DQJOHV ZH FDQ WKHQ GHWHUPLQH WKH UH IOHFWLRQDQGWUDQVPLVVLRQFRHIILFLHQWVRIWKHLQGLYLGXDOLQWHUIDFHVYLDWKH)UHVQHO¶V IRUPXODV  DQG   :LWKUHIHUHQFHWR)LJIRUHDFKRIWKH7(DQG70ZDYHVOHWXVFDOO E j   E cj DQG E j  E cj  WKHDPSOLWXGHRIWKHILHOGVDFURVVWKHjWKLQWHUIDFH MXVWEH IRUHZLWKRXW WKHSULPH DQGMXVW DIWHU ZLWK WKH SULPH  ZKHUH ZLWK LQGH[HV ³´ DQG³±´ZHGHQRWHUHVSHFWLYHO\WKHZDYHWKDWSURSDJDWHVIRUZDUGDQGWKHZDYH WKDWSURSDJDWHVEDFNZDUG,QDGGLWLRQZHDVVXPHWKDWWKHVHZDYHVDUHWKHUHVXOW RIDVLQJOHZDYHLQFLGHQWRQWKHPXOWLOD\HU7KHUHIRUHWKHWUDQVPLWWHG UHIOHFWHG  ZDYHJHQHUDWHGE\ E j  ZLOOVXPFRKHUHQWO\in the same directionRISURSDJDWLRQ ZLWKWKHUHIOHFWHG WUDQVPLWWHG ZDYHJHQHUDWHGE\ E j  E cj



E j 

r j  E cj  t j  E j   t j  E cj   r j E j  



 

ZKHUH t j   t j   r j   r j   DUH WKH WUDQVPLVVLRQ t  DQG UHIOHFWLRQ r  FRHIILFLHQWV IRU WKH IRUZDUG DQGEDFNZDUGZDYHVRQ WKH jWK LQWHUIDFH JLYHQ E\ WKH)UHVQHO¶V (TV  WKH\are differentIRU7(DQG70ZDYHVEXWWKHADQG)LQGH[HV DUHRPLWWHGWRDYRLGHQFXPEHULQJWKHQRWDWLRQ 5HVROYLQJIRU E j  DQG E j ZHKDYH E j 

 E j 

 t j   t j 

E cj 

r j  t j 

t j  t j  

E cj  r j  r j  r j  E cj   t j 





 

E cj 

2QWKHRWKHUKDQGIURPWKH)UHVQHO¶VHTXDWLRQVZHFDQLPPHGLDWHO\VHHWKDWIRU ERWK7(DQG70ZDYHVZHKDYH  DQGDOVRWKDW

r j 

 r j  

 

'LHOHFWULFILOPZLWKPXOWLSOHOD\HUV



)LJ )LHOGV RQ WKH jWK LQ WHUIDFH EHWZHHQ WKH j WK PHGLXP DQG WKHjWKPHGLXP

E j-

-1

+

E¢ j

E¢j -

+ E j-1

n j -1

nj j



t j   t j   r j  r j 

 

 

7KHUHIRUHWKHUHODWLRQVKLSEHWZHHQWKHILHOGVDWWKHjWKLQWHUIDFHFDQEHSXWLQWR WKHYHFWRULDOIRUP § E j  · ¨ ¸ ¨ E j  ¸ © ¹



§ E j · ¸  Tj ¨ ¨ E j ¸ © ¹

 

§ ¨¨ © rj

 

ZKHUHTLVWKH[PDWUL[ Tj



 tj

rj · ¸   ¸¹

ZKHUHIRUEUHYLW\LWLVIXUWKHUXQGHUVWRRGWKDWtjDQGrjUHIHUKHUHWRWKHIRUZDUG ZDYH   /HWXVQRZFRQVLGHUWKHSKDVHGLIIHUHQFHWKDWRFFXUVLQJRLQJWKURXJKWKHjWK OD\HUEHWZHHQWKHLQWHUIDFHVjDQGj $VZHOHDUQHGZLWK(T  WKLVSKDVH VKLIWLVJLYHQE\ 

Ij

Gj

 Sn j h j



Oo

FRV - j 

 

ZKHUHnjhi-jDUHUHVSHFWLYHO\WKHUHIUDFWLYHLQGH[WKHWKLFNQHVVDQGWKHDQJOH RILQFLGHQFHIRUWKHjWKOD\HU7KHILHOGVDWWKHEHJLQQLQJDQGHQGRIDOD\HUDUH WKHQOLQNHGWRHDFKRWKHUE\WKHUHODWLRQVKLS



§ E cj · ¨ ¸ ¨ E cj ¸ © ¹

§ E j · ¸  Hj ¨ ¨ E j ¸ © ¹

 



,QWHUIHUHQFH

ZKHUHHjLVWKHPDWUL[

Hj



§ e i I j ¨ ¨  ©

 · ¸  ei I j ¸¹

 

7RHDFKOD\HUWKHUHIRUHZHFDQDVVRFLDWHWKHtransfer matrix Lj



Tj H j

§ i I j ¨ e t j ¨ r e i I j © j

r j ei I j · ¸  ei I j ¸¹

 

,QFRQFOXVLRQLQWKHSUHVHQFHRImOD\HUVWDNLQJIRUWKHILQDOPHGLXP Lm ZLWKIm DQGE\LQWURGXFLQJWKHLQGH[f mIRUEUHYLW\WKHRYHUDOOWUDQV IHUPDWUL[LVJLYHQE\ §a b· L ¨ ¸ ©c d¹



LL ˜˜˜ L f 

 

DQGWKHLQLWLDODQGILQDOILHOGVDUHUHODWHGE\ § E · ¨ ¸ © E ¹



§ E f · ¸  L¨ ¨ E f ¸ © ¹

 

2Q WKH RWKHU KDQG ZLWK DQ LQSXW ZDYH LQ WKH VROH IRUZDUG GLUHFWLRQ ZH KDYH E f  DQGLQWKHRSSRVLWHFDVHRIDEDFNZDUGZDYHLWLV E  6RWKHDPSOL WXGHFRHIILFLHQWVIRUWKHRYHUDOOWUDQVPLVVLRQDQGUHIOHFWLRQRIWKHYDULRXVLQWHUIDF HVDUHJLYHQUHVSHFWLYHO\E\

 t

E f E

 r a

E E

c  t a

E E f

E f

ad  bc  r a

E f

b     a

ZKHUHaDQGcDUHWKHHOHPHQWVRIWKHILUVWFROXPQRILLWVKRXOGEHQRWHGWKDWWKH FRHIILFLHQWVDUHQRWHTXDOIRUWKHIRUZDUGDQGEDFNZDUGZDYHV)RUWKHLQWHQVLW\ ZHKDYHILQDOO\WKDWWKHWUDQVPLVVLYLW\DQGUHIOHFWLYLW\DUHUHVSHFWLYHO\JLYHQE\ T

n f P FRV - f    R n P f FRV - a

T

 n P f FRV - ad  bc  R n f P FRV - f a





c  a b a





  

'LHOHFWULFILOPZLWKPXOWLSOHOD\HUV



ZKHUHnP-DQGnfPf-fDUHUHVSHFWLYHO\WKHUHIUDFWLYHLQGH[WKHPDJQHWLF SHUPHDELOLW\DQGWKHDQJOHRILQFLGHQFHLQWKHLQLWLDODQGWKHILQDOPHGLXP $QDOWHUQDWLYHZD\WRGHDOZLWKWKLVSUREOHPLV WRGLUHFWO\UHVRUWWRERXQGDU\ FRQGLWLRQVIRUWKHILHOGVRQWKHLQWHUIDFHVHVSHFLDOO\WKRVHUHODWLQJWRWKHWDQJHQ WLDOFRPSRQHQWVRIILHOGV(DQG+$VZHKDYHDOUHDG\GRQHZLWK(T  KHUH LQGLFDWLQJWKHVHFRPSRQHQWVZLWKDEDUIRUHDFKRIWKH7(DQG70ZDYHVDFURVV WKHjWKLQWHUIDFHLWLVVLPSO\ 

Ej

E cj  H j

H c j 

 

ZKHUHRQERWKVLGHVRIWKHLQWHUIDFHWKHVHQHZYDULDEOHVDUHQRZWKHVXPRIWKH WDQJHQWLDOFRPSRQHQWVIRUWKHZDYHVLQWKHWZRGLUHFWLRQVRIWUDYHO 

Ej

E j   E j 

E cj  E cj  H

H j   H j 

j

H cj  H cj   

ZKHUH ZH KDYH WR QRWH WKDW JHQHUDOO\ E j  z E cj  HWF 2Q WKH RWKHU KDQG IRU (TV  LQWKHOD\HUjWKHDPSOLWXGHVRIWKHPDJQHWLFILHOGRIWKHZDYHDQG DUHUHODWHGWRWKHDPSOLWXGHVRIWKHHOHFWULFILHOGRIWKHVDPHZDYHKHQFH 

H

j

H cj  H cj





K j E cj  E cj 

 

ZKHUHKjLVWKHeffective admittanceRIWKHjWKOD\HUGHILQHGE\



Kj

­ nj FRV - j ° ° Pjc ®  ° n j ° P j c FRV - j ¯

IRUWKH7(ZDYH 

 

IRUWKH70ZDYH

7KHGLIIHUHQFH EHWZHHQ WKH7( DQG 70ZDYHVGHULYHVVLPSO\ IURP WKH IDFW WKDW WKHHOHFWULFDQGPDJQHWLFILHOGVDUHSHUSHQGLFXODUWRHDFKRWKHUDQGWKDWWKHYHF WRUVLQWKHRUGHU E H kIRUPDULJKWKDQGHGVHWIRUERWK7(DQG70ZDYHV ZLWK WKHFRQYHQWLRQVFKRVHQLQWKLVWH[W ZKLOHWKHPLQXVVLJQLQIURQWRI E cj LQ(T  GHULYHVIURPWKHFKDQJHRIWUDYHOGLUHFWLRQ 7KHYDULDEOHV E j DQG H j DUHWKHUHIRUHDOLQHDUFRPELQDWLRQRI E cj DQG E cj  DQGYLFHYHUVD DQGLQYHFWRUIRUPZHFDQZULWH



§Ej · ¨ ¸ ¨H j ¸ © ¹

§ E c · j ¸ Nj ¨ ¨ E c ¸ © j¹

§ E c · ¨ j¸ ¨ E c ¸ © j¹

§Ej · N j  ¨ ¸  ¨H j ¸ © ¹

 



,QWHUIHUHQFH

ZKHUH §  ¨ ©Kj

Nj



 ·  N j  K j ¹¸

 §  K j · ¨ ¸   ©¨    K j ¹¸

 

,Q WXUQ WKH WDQJHQWLDO FRPSRQHQWV E j  E j  DUH UHODWHG WR WKH DPSOLWXGHV E j  E j E\ § E · ¨ j¸ ¨ E ¸ © j¹



§ E j · ¸  cj ¨ ¨ E j ¸ © ¹

 

ZKHUH 

cj

­°  ®FRV j °¯

IRUWKH7(ZDYH IRUWKH70ZDYH



 

$OVR IRU WKH SDLU RI YDULDEOHV E  H  ZH FDQ ZULWH D UHFXUVLYH UHODWLRQVKLS EH WZHHQRQHLQWHUIDFHDQGWKHQH[WDWWKHH[WUHPHVRIDVLQJOHOD\HU7KDQNVWR(TV  DQG  ZHKDYH



§Ej· ¨ ¸ ¨ ¸ ©H j ¹

§  ¨  ©Kj

 · § e i I j ¨ K j ¹¸ ¨  ©

 · §   K j · § E j  · ¸¨ ¸  ¸¨ ei I j ¹¸ ©¨    K j ¹¸ ©¨ H j  ¹¸

 

DQGWKHUHIRUH §Ej· ¨ ¸ ¨H j ¸ © ¹



§ E j  · Mj ¨ ¸  ¨ H j  ¸ © ¹

 

ZKHUHMLVWKHPDWUL[



Mj

§ ¨ FRV I j ¨ ¨ iK j VLQ I j ©

i

 · VLQ I j ¸ Kj ¸  FRV I j ¸¹

D 

ZKLFKLVFDOOHGWKHcharacteristic matrixRIWKHjWKOD\HURUHYHQWKHinterference matrix ,IWKHOD\HUjLVDQLGHDOGLHOHFWULFZLWKRXWDEVRUSWLRQWKHMjPDWUL[KDVDXQL WDU\GHWHUPLQDQWDQGDOVRKDVUHDOGLDJRQDOHOHPHQWVZKLOHWKHQRQGLDJRQDOHOH PHQWV DUH LPDJLQDU\ 7KLV DOVR DSSOLHV ZKHQ WKH LQGH[ RI WKH LQLWLDO PHGLXP LV JUHDWHUWKDQWKDWRIWKHOD\HUjDQGWKHLQLWLDODQJOHRILQFLGHQFHLVVXIILFLHQWO\ODUJH VRWKDWIRU(T  VLQ-jH[FHHGVWKHXQLW7KHZDYHWKDWSHUPHDWHVWKHOD\HUj

'LHOHFWULFILOPZLWKPXOWLSOHOD\HUV



LV WKHQ DQ HYDQHVFHQW ZDYH ZLWK SXUH LPDJLQDU\ FRV-j ,Q WKLV FDVH IRU (TV   DQG   Ij  i I‫މމ‬j DQG Kj  iK‫މމ‬j DUH LPDJLQDU\ DQG (T D  EH FRPHV § ¨ FRVK Iccj ¨ ¨ © iKccj VLQK Iccj

Mj



i

 · VLQK Iccj ¸ Kccj ¸  FRVK Iccj ¸¹

E 

6LPLODUWR(T  WKHLQWHUIHUHQFHPDWUL[IRUDVHWRImOD\HUVLVJLYHQE\ WKHSURGXFWRIWKHPDWULFHVRIWKHLQGLYLGXDOOD\HUV §A B· M ¨ ¸ ©C D¹



M M ˜˜˜ M m 

 

,IWKHUHIRUHWKHPXOWLOD\HULVPDGHIURPLGHDOGLHOHFWULFVZLWKRXWORVVHVHYHQWKH RYHUDOOPDWUL[MLVXQLWDU\DQGLWVHOHPHQWVADQGDDUHUHDOZKLOHBDQGCDUH LPDJLQDU\ 7KH PDWUL[ RI LQWHUIHUHQFH HYHQ LI LW LV RI OHVV LPPHGLDWH XQGHUVWDQGLQJ KDV WKH DGYDQWDJH RI EHLQJ H[SUHVVHG GLUHFWO\ DV D IXQFWLRQ RI WKH UHIUDFWLYH LQGLFHV DQGWKHWKLFNQHVVRIWKHYDULRXVOD\HUVDQGDOORZVIRUDIDVWHUFRPSXWDWLRQRIWKH RYHUDOOEHKDYLRURIWKHVHWRIOD\HUV)RUWKHDPSOLWXGHVEDQGE±RIWKHILHOGVDW WKHHQGVRIWKHVWDFNRIOD\HUVZHXOWLPDWHO\KDYH § E · ¨ ¸ © E ¹



§ Ef · ¸  c N M N f c f ¨ ¨ Ef ¸ © ¹

 

,WLVWKHUHIRUHDSSDUHQWWKDW

L c N M N f c f 



 

ZKHUHWKHHOHPHQWVa,bcdRILH[SUHVVHGLQWHUPVRIWKHHOHPHQWVABCD RIMDUHJLYHQE\ a

 K

cf

c

 K

cf



c c

AK  BKK f  C  DK f  AK  BKK f  C  DK f  

b

 K

cf

d

 K

cf

c c

AK  BKK f  C  DK f  

AK  BKK f  C  DK f   

DQG WKH UHIOHFWLRQ DQG WUDQVPLVVLRQ FRHIILFLHQWV FDQ VWLOO EH FDOFXODWHG ZLWK (TV  DQG  





,QWHUIHUHQFH

3.4.2 Propagation in periodic structures 2I SDUWLFXODU LPSRUWDQFH DUH WKH SHULRGLF VWUXFWXUHV ZKRVH DQDO\WLFDO VWXG\ GDWHVEDFNWRWKHKLVWRULFDOZRUNVRI)$EHOqVPHQWLRQHGDWWKHEHJLQQLQJ>$EHOqV @ DQG VXEVHTXHQWO\ WR WKRVHRI $ 3HUYDN @ 7KH*''LVDSSUR[LPDWHO\JLYHQE\WKHUDWLR'W'ZZKHUH'ZLVWKHVSHFWUDO UHIOHFWLYLW\EDQGRIWKHPLUURUDQG'WLVWKHGHOD\GLIIHUHQFHIRUSXOVHVFHQWHUHGRQ WKHH[WUHPHVRIWKLVEDQG

%LEOLRJUDSKLFDOUHIHUHQFHV $EHOqV)Recherche sur la propagation des ondes electromagnétiques sinusoidales dans les milieux stratifiés. Applications aux couches minces$QQGH3K\VLTXH  DQG   %DWHV:-A wavefront shearing interferometer3URF3K\V6RF   %RUQ0DQG:ROI(Principles of Optics3HUJDPRQ3UHVV3DULV   +DQEXU\ %URZQ 5 DQG 7ZLVV 54 Correlation between photons in two beams of light 1DWXUH   &RQWL$Gli occhiali sull’Universo/H6FLHQ]H PD\  'HODQR(DQG3HJLV5- Methods of synthesis for dielectric multilayer FiltersLQ3URJUHVV LQ2SWLFV9RO9,,S  



,QWHUIHUHQFH

'LWFKEXUQ5:Light9RO,DQG,,$FDGHPLF3UHVV,QF/RQGRQ   'UHZ5/A Simplified Shearing Interferometer3URF3K\V6RF   )RUN5/0DUWLQH]2(DQG*RUGRQ-3Negative dispersion using pairs of prisms2SW /HWW   )RZOHV *5 Introduction to Modern Optics +ROW 5LQHKDUW DQG :LQVWRQ 1HZ %RUQDQG:ROI@&DVHVRIWKLV W\SHPD\EHLQWHUHVWLQJLQWKHVSHFWUDOUHJLRQIURPWKHUDGLRZDYHVWRWKHIDULQIUD UHGEXWDUHREYLRXVO\RIOLWWOHSUDFWLFDOXWLOLW\LQWKHYLVLEOHUHJLRQ +RZHYHU5D\OHLJKDQG6RPPHUIHOGPRGLILHG.LUFKKRII¶VWKHRU\ DSSUR[LPDWH EXWUHDVRQDEOH E\PHDQVRIWKHIXQFWLRQVWKHRU\RI*UHHQHOLPLQDWLQJWKHQHHGWR PDNHERWKRIWKRVHWZRDVVXPSWLRQVPHQWLRQHGDERYH,QWKLVZD\WKH\UHVROYHG WKHLQFRQVLVWHQF\SUHVHQWLQWKH.LUFKKRIIPRGHOE\LQWURGXFLQJZKDWLVFDOOHGWKH Rayleigh-Sommerfeld theory of diffraction,WVKRXOGEHQRWHGKRZHYHUWKDWWKHVH WKHRULHV FRQWDLQ DQ LPSRUWDQW VLPSOLILFDWLRQ WKH OLJKW LV WUHDWHG DV D VFDODU SKH QRPHQRQQHJOHFWLQJWKHIDFWWKDWWKHYDULRXVFRPSRQHQWVRIWKHHOHFWULFDQGPDJ 

6FDODUWKHRU\RIGLIIUDFWLRQ



QHWLFILHOGVDUHFRXSOHGWRRQHDQRWKHULQ0D[ZHOO¶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

6FDODUWKHRU\RIGLIIUDFWLRQ 4.1.1 General premises 7KHW\SLFDOGLIIUDFWLRQWRSRORJ\FRQVLGHUVWZRVSDWLDOUHJLRQV,DQG,,VHSDUDW HGE\DVXUIDFHS SSDVVKRZQLQ)LJ5HJLRQ,FRQWDLQVWKHVRXUFHVRI UDGLDWLRQ ZKLOH UHJLRQ ,, LV WKH VSDFH ZKHUH WKH GLIIUDFWLRQ LV WR EH GHWHUPLQHG 7KH VXUIDFH S LV JHQHUDOO\ WDNHQ DW ³LQILQLW\´ ZKLOH WKH S VXUIDFH GHOLPLWV WKH VRXUFHVDQGWKHVFUHHQZKLFKFRQVLVWVRIRSDTXHDUHDVDQGRSHQLQJV7KLVVFUHHQ LQWHUDFWVZLWKWKHILHOGJHQHUDWHGLQUHJLRQ,DEVRUELQJDELWRIHQHUJ\UHIOHFWLQJD ELWRILWDQGOHWWLQJVRPHRIWKHZDYHDOWHUHGE\WKHLQWHUDFWLRQJRHVLQWRUHJLRQ ,,,QWKHIROORZLQJZHZLOOH[DPLQHWKLVVLPSOHVWRIVLWXDWLRQ7KHUHPD\DOVREH FRPSOH[ VLWXDWLRQV LQ ZKLFK WKHUH DUH UHIOHFWLRQV DQG UHIUDFWLRQV IRU ZKLFK WKH ILHOGFDQEHLGHQWLILHGE\FRPELQLQJYDULRXVPHWKRGVVXFKDVUD\WUDFLQJLQWHU IHUHQFHHWFDQGWKRVHRIGLIIUDFWLRQ 7KHGLIIUDFWLRQWKHRU\UHOLHVLQSDUWLFXODURQDJHQHUDOUHODWLRQVKLSNQRZQDV *UHHQ¶VWKHRUHPRU*UHHQ¶VVHFRQGLGHQWLW\ /HW U 3  DQG G 3  EH WZR FRPSOH[ IXQFWLRQV RI SRVLWLRQ 3 DQG S EH D FORVHGVXUIDFHWKDWFRQWDLQVDYROXPHV,IUGDQGWKHLUILUVWDQGVHFRQG SDUWLDOGHULYDWLYHVDUHXQLTXHDQGFRQWLQXRXVLQVDQGRQSWKHQ 

§

wU

wG ·

³³³ G’U U ’G d V w ³³ ¨© G w n  U w n ¸¹ d S  V

S

 

'LIIUDFWLRQ





Opaque objects

Sources

I

I

I

S¢1 II (a)

Sources and objects

Sources and objects

III S1¢¢ S2

S1

S1

II (b)

II

S2

(c)

S2



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

ZKHUH w wn Q(KUOLFK HWDO@VKRZVLPLODURVFLOODWLRQVRIWKHILHOGRQWKHRSHQLQJZLWKDYDOXHDWWKH FHQWHUWKDWDFFRUGVEHWWHUZLWKU.7KHDJUHHPHQWZLWK.LUFKKRII¶VLQWHJUDOLVH[ SHFWHGWREHHYHQEHWWHUZLWK³EODFN´GLDSKUDJPVDVZHZLOOGLVFXVVODWHU 7KHIDFWWKDWU,,H[KLELWVVWURQJRVFLOODWLRQVLVQRWLQFRQWUDGLFWLRQZLWKWKHDV VXPSWLRQRIDXQLIRUPYDOXHIRUwUwnDQGLQGHHGwU,,wnUHWXUQVWKHXQLIRUP VWDUWLQJYDOXHRQ A7KHEHKDYLRURI8,,FDQEHXQGHUVWRRGWKURXJKWKHGHYHORS PHQWLQSODQHZDYHVRIWKHILHOGWUDQVPLWWHGIURPWKHGLDSKUDJPLWLVIRXQGWKDW GXHWRWKHGLDSKUDJPWKHVSHFWUXPRIZDYHYHFWRUV kRSHQVXSPDNLQJWKHSUHV HQFHRIHYDQHVFHQWZDYHVFORVHWRWKHRSHQLQJLWVHOIUHOHYDQWDVZHOO7KHVHDUHLQ SDUWLFXODUWKHZDYHVZLWKkQ|WKDWJLYHWKHJUHDWHVWFRQWULEXWLRQWRWKHRVFLOOD WLRQVRIU,,RQA /LNHZLVHWKHIDFWWKDWU.GRHVQRWUHWXUQDXQLIRUPDPSOLWXGHRQ ALVQRWD SURRIRILWVPDWKHPDWLFDOLQIHULRULW\DVLWFDQEHVSOLWLQWRWZRFRQWULEXWLRQV,DQG ,, HDFK RI ZKLFK UHWXUQV WKH XQLIRUP YDOXH RI LWV LQWHJUDQG 7KH FRQVLVWHQF\ RI .LUFKKRII¶V WKHRU\ ZDV ILUVW GHPRQVWUDWHG E\ 0DUFKDQG DQG :ROI ZKR VKRZHG WKDW WKH U. ILHOG RQ WKH DSHUWXUH FDQ EH ZULWWHQ DV WKH VXP RI D GLUHFW OLJKWLQJ ZDYH geometrical-optics wave DQGDboundary-diffracted wave %': IURPWKH RSHQLQJ HGJH ZKLFK FDQ EH H[SUHVVHG DV D OLQH LQWHJUDO DORQJ WKH RSHQLQJ HGJH >0DUFKDQGDQG:ROI@,QSDUWLFXODUIRUDQRSHQLQJ ALQDSODQHVFUHHQQRU PDOWRWKHzD[LVLWLV



U , x y 

H x  y U x y   

U . x y 

H x y U x y   U %':  

U ,, x y 

H x y U x y   U %': 

ZKHUHH xy LVLI xy ADQGRWKHUZLVHDQGZKHUH 

U %': x y 

ª wU x c y c z º   eikR H x y  dx cdy c  « » ³³ wz  S A ¬ ¼z  R

7KHUHIRUHLQWKHVDPHFRQGLWLRQU,LVFRPSRVHGRQO\RIWKHGLUHFWZDYHDQGWKXV LWDSSHDUVSK\VLFDOO\OHVVSODXVLEOH>6WDPQHV@

'LIIUDFWLRQ  ,QPDQ\FDVHVRISUDFWLFDOLQWHUHVWLQZKLFKWKHDSHUWXUHVL]HVDUH!!O DVRI WHQ KDSSHQV IRU WKH RSWLFDO UHJLRQ RI WKH VSHFWUXP  WKH GLIIUDFWHG SRZHU LV FRQ ILQHGWRDQDUURZUDQJHRIDQJOHVDURXQGWKHRULJLQDOGLUHFWLRQRIWKHZDYH RQA  7KLV IDFW LV PRVWO\ D FRQVHTXHQFH RI WKH WHUP H[S ikr  UDWKHU WKDQ WKH FRVLQH WHUPV,QGHHGDVZHZLOOVHHODWHULIU r LVDSSUR[LPDWHO\XQLIRUPLQDPSOLWXGH DQGSKDVHRQWKHDSHUWXUHZKHQWKHGLUHFWLRQRIREVHUYDWLRQLVYHU\LQFOLQHGZLWK UHVSHFWWRWKHD[LVRIWKHRSHQLQJH[S ik r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rWDNHQRQA DQGEURXJKWRXWRIWKHLQWHJUDO,IWKHQLQSDUWLFXODUZHGR QRWPRYHWRRIDUDZD\IURPWKHD[LVRIWKHRSHQLQJDOOREOLTXLW\IDFWRUVFDQEH DSSUR[LPDWHG WR  DQG WKH GLVWLQFWLRQ EHWZHHQ WKH YDULRXV PRGHOV RI .LUFKKRII DQG5D\OHLJK6RPPHUIHOGGLVDSSHDUV 

4.1.5 The Huygens-Young-Fresnel principle 7KLVSULQFLSOHZDVLQLWLDOO\LQWURGXFHGE\PHQRIJUHDWJHQLXVIURPVRPHVLP SOH SK\VLFDO FRQVLGHUDWLRQV UDWKHU WKDQ WKURXJK WKH VRPHZKDW DEVWUDFW IRUPDOLW\ RI.LUFKKRII¶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boundary value WRWKHILHOGUDQGLWVGHULYDWLYHwUwnRQWKHVXUIDFHRILQWHJUDWLRQWKHVHVKRXOG LQVWHDGEHFRQVLGHUHGDVDdiscontinuityRIUDQGwUwnDWWKHWZRVLGHVRIWKHVXU IDFHWKXVWUDQVIRUPLQJWKHDVVXPSWLRQVLQWRDSUREOHPRIsaltus7KHUHVWULFWLRQLV WKDWWKHGLIIUDFWLQJERG\LVblack,QKLVORQJGHIHQVH.RWWOHULQWURGXFHVWKHLGHD WKDWDEODFNERG\FDQEHDVVLPLODWHGWRDQopen doorEHWZHHQWZR5LHPDQQVSDF HVRQHSK\VLFDODQGRQHQRQSK\VLFDOLQZKLFKWKHZDYHHTXDWLRQVDUHVWLOOYDOLG 7KHUDGLDWLRQWKDWVWULNHVLWVLPSO\GLVDSSHDUVIURPWKHSK\VLFDOVSDFHWRHQWHUWKH RWKHUVSDFHZLWKRXWWKHQHHGIRUDEVRUSWLRQ,QWXUQWKHGRRUFRUUHVSRQGLQJWRWKH LOOXPLQDWHGSDUWRIWKHEODFNERG\LVDQRSHQVXUIDFHWKDWEHORQJVWRERWKVSDFHV DQG DSSHDUV WR KDYH D QXOO WKLFNQHVV LQ HDFK RI WKHP 7KH QHZ LQWHUSUHWDWLRQ RI .LUFKKRII¶VDVVXPSWLRQVLVWKHQWRFRQVLGHUDGLVFRQWLQXLW\EHWZHHQWKHWZRVLGHV RI WKH GRRU LQWKH SK\VLFDO VSDFH 6R LIUV LV D UHSUHVHQWDWLYH FRPSRQHQW RI WKH ILHOGRIWKHJHRPHWULFDOZDYHWKDWLOOXPLQDWHVWKHEODFNERG\WKHFRQWRXUFRQGL WLRQVDUHUHSODFHGE\WKHGLVFRQWLQXLW\ 

UD  U E

UV 

wU D wU E  wn wn

wU V  wn

 

ZKHUH UD DQG UE UHSUHVHQW WKH OLPLW YDOXH RI WKH ILHOG UHVSHFWLYHO\ DW WKH WZR

6FDODUWKHRU\RIGLIIUDFWLRQ



VLGHVDDQGERIWKHGRRUDQG QLVDYHUVRUQRUPDOWRLWVVXUIDFH$QDORJRXVUHD VRQLQJLVDOVRGRQHIRUWKHVXUIDFHLQWKHJHRPHWULFVKDGRZRIWKHEODFNERG\IRU ZKLFKKHDVVXPHVWKDW 

UD  U E



wU D wU E  wn wn

 

 

,QRWKHUZRUGVWKHUHLVQRGLVFRQWLQXLW\EHWZHHQWKHWZRVLGHV H[WHUQDODQGLQWHU QDOWRWKHERG\ RIWKHVXUIDFHLQVKDGRZQHLWKHULVWKHUHDQ LPSRVLWLRQWKDWWKH ILHOGWKHUHSUHVHQWVQRULWVGHULYDWLYHVDUHQXOO 7U\LQJWRWUDQVODWH.RWWOHU¶VLGHDZHFDQWKXVLPDJLQHZLWKUHIHUHQFHWR)LJ  D  WKDW WKH VXUIDFH S ‫ މ‬HQFORVHV D VLQJOH SRLQW VRXUFH ZLWK FRRUGLQDWHV rV ZKLOH WKH VXUIDFH S " FORVHO\ HQFORVHV WKH EODFN ERG\ DQG LV GLYLGHG LQWR WZR SDUWV RQH LOOXPLQDWHG E\ WKH VRXUFH L DQG WKH RWKHU LQ WKH VKDGH O (DFK RI WKHVHLVDQRSHQGRRULQDQRQSK\VLFDOVSDFHZKHUHWKHLQFLGHQWUDGLDWLRQLVORVW IRUHYHU 6LQFH *UHHQ¶V WKHRUHP GRHV QRW DSSO\ WR VSDFHV FRQQHFWHG E\ GRRUV branch cut LWLVQHFHVVDU\WRLVRODWHHDFKRIWKHWZRGRRUVZLWKDFORVHGVXUIDFH VXUURXQGLQJ LW DQG ZKLFK FDQ EH EURXJKW LQGHILQLWHO\ FORVH WR WKH QXOO WKLFNQHVV VXUIDFHWKDWFRQVWLWXWHVWKHGRRU:LWKLQWKHOLPLWRIDQLQILQLWHVLPDOGLVWDQFHWKH DSSOLFDWLRQRI*UHHQ¶VWKHRUHPWRWKHFDVHRI)LJ D OHDGVWRWKHH[SUHVVLRQ



wU · wG º ª§ wU U rR U V rR  w ³³ «¬¨© w nD  w nE ¸¹ G  UD  U E w n »¼ d L L wU · wG º ª§ wU w ³³ «¬¨© w nD  w nE ¸¹ G  UD  U E w n »¼ d O O



 

ZKHUHUV r UHSUHVHQWVWKHVSKHULFDOZDYHFHQWHUHGLQ rVJHQHUDWHGE\WKHVRXUFH DQG G LV VWLOO JLYHQ E\ (T   ,Q WKH WZR LQWHJUDOV WKH FRQWULEXWLRQV WR WKH VLGHVD H[WHUQDOWRWKHERG\ DQGE LQVLGHRIWKHERG\ RIHDFKGRRUDUHVXPPHG WDNLQJ WKHYHUVRU n  nD  nE RULHQWHG LQ WKH GLUHFWLRQ DoE:LWK.RWWOHU¶V DV VXPSWLRQV  DQG  WKLVH[SUHVVLRQLVUHGXFHGWR 

wG º ª wU U rR U V rR  w ³³ «¬G w nV  UV w n »¼ d L  L

 

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

'LIIUDFWLRQ  GRRU ARQDQRWKHUVSDFHLQZKLFKWKHVRXUFHVDUHORFDWHG6XUURXQGLQJWKHGRRU ZLWKDFORVHGVXUIDFHZLWKLQWKHOLPLWFRLQFLGLQJZLWKWKHVLGHVDDQGERI AZH JHW 



U rR

wG º ª wU V  UV d A  wn w n »¼ A

w ³³ «¬G

 

ZKHUH WKH YHUVRU n LV KHUH RULHQWHG EDFN WRZDUGV WKH VRXUFH VSDFH ,Q WKLV ZD\ WKHUHLVQRK\SRWKHVLVWRFUHDWHIRUWKHILHOGLQWKHVKDGRZRIWKHVFUHHQEHFDXVH LQWKLVVSDFHWKHVFUHHQLVVLPSO\QRWWKHUHDQGWKHVXUIDFH S RI)LJ WREH H[WHQGHGDWLQILQLW\FDQQRZEHDVSKHUHWKDWDOVRLQFOXGHVWKHGRRU7KHPDWKH PDWLFDO LQFRQVLVWHQFLHV RI .LUFKKRII¶V DVVXPSWLRQV DUH WKHUHIRUH HOLPLQDWHG LQ WKHLUQHZLQWHUSUHWDWLRQ .RWWOHUDUJXHVWKDWZLWKKLVLQWHUSUHWDWLRQWKHUHLVQRQHHGWRDVVXPHWKDWWKH EODFNERG\LVWKLQ\HWKHGRHVQRWPHQWLRQWKHSRVVLELOLW\RIIXUWKHUGLIIUDFWLRQRI WKH GLIIUDFWHG ZDYHV 7KH SUREOHP RI PXOWLSOH GLIIUDFWLRQ DW WKH HGJH RI VPDOO RSHQLQJVRQEODFNVFUHHQVZDVDGGUHVVHGE\0DUFKDQGDQG:ROIWKURXJKDGLIIHU HQWDSSURDFKEDVHGRQWKH%':PHWKRG>0DUFKDQGDQG:ROI@

4.1.8 Outline on the calculation methods of the diffraction integrals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

6FDODUWKHRU\RIGLIIUDFWLRQ



GLUHFWLRQZKLOHXVLQJWKH*DXVV/HJHQGUHLQWHJUDWLRQIRUPXODIRUWKHRWKHUGLUHF WLRQ ,W LV ZRUWK PHQWLRQLQJ WKH WHFKQLTXHV WKDW DV VWDWHG DERYH H[SORLW WKH YHU\ WKLQJ WKDW PDNHV LW GLIILFXOW WR FDOFXODWH WKH LQWHJUDOV 7KHVH DUH WKH DV\PSWRWLF PHWKRGVZKRVHPDWKHPDWLFDOEDVHVDUHGHVFULEHGLQWKHFODVVLFDOWH[WE\(UGpO\L 7KHUHWKHDXWKRUGHVFULEHVDPRQJRWKHUWKLQJVWKHDV\PSWRWLFH[SDQVLRQRIIXQF WLRQV GHILQHG E\ LQWHJUDOV FRQWDLQLQJ D ³ELJ´ SDUDPHWHU >(UGpO\L @ 7KHVH PHWKRGVDUHDSSOLFDEOHWRFDVHVLQZKLFKWKHDSHUWXUHVDUHODUJHFRPSDUHGWRWKH ZDYHOHQJWK ZKHUHE\ ZLWK WKH ZDYH YHFWRU k o f WKH GLIIUDFWLRQ LQWHJUDO DS SURDFKHVDQDV\PSWRWLFH[SDQVLRQWKDWW\SLFDOO\FRQVLVWVRIDVHULHVRISRZHUVLQ k7RREWDLQWKHTXDOLWDWLYHEHKDYLRURIWKHGLIIUDFWLRQLQWHJUDOLWLVXVXDOO\VXI ILFLHQWWRFRQVLGHURQO\WKHILUVWWHUPRIWKHH[SDQVLRQZKLOHIRUDTXDQWLWDWLYHDV VHVVPHQWLWPD\EHQHFHVVDU\DOVRWRFRQVLGHUVRPHRIWKHVXEVHTXHQWWHUPVZLWK RXW D SDUWLFXODU LQFUHDVH LQ WKH FRPSXWDWLRQ WLPH 7KH PRVW ZHOO NQRZQ DQG LPSRUWDQWDPRQJWKHDV\PSWRWLFPHWKRGVLVWKDWZKLFKLVVDLGWREHof the stationary phaseZKLFKLQWKHPLGVWRIVRPXFKPDWKHPDWLFDOFRPSOH[LW\DWOHDVWKDV WKHPHULWRIEHLQJLQWXLWLYHO\XQGHUVWDQGDEOHZLWKSK\VLFDOFRQVLGHUDWLRQV )RUWKUHHGLPHQVLRQDOZDYHVDQGDQDVVLJQHGREVHUYDWLRQSRLQW KHUHLPSOLFLW  DW\SLFDOLQWHJUDORIGLIIUDFWLRQKDVWKHIRUP 

U

³³A g x y eik f x y dx dy 

 

ZKHUHWKHUHDOIXQFWLRQk fUHSUHVHQWVWKHSKDVHRIWKHILHOGRQWKHDSHUWXUHADQG WKHIXQFWLRQgUHSUHVHQWVLWVDPSOLWXGH DVWKH\DUHVHHQDWWKHREVHUYDWLRQSRLQW  7KHVH IXQFWLRQV DUH DVVXPHG FRQWLQXRXV DQG FRQWLQXRXVO\ GHULYDEOH )RU VXI ILFLHQWO\ODUJHYDOXHVRIkWKHH[SRQHQWLDOWHUPRVFLOODWHVVRUDSLGO\ZLWKUHVSHFW WRgWKDWWKHLQWHJUDWLRQRQDVPDOOILQLWHVXUIDFHLVQHJOLJLEOHH[FHSWLQWKHQHLJK ERUKRRGRISRLQWVZKHUHWKHSKDVHk f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first species IRU ZKLFKWKHILUVWGHULYDWLYHVRIfZLWKUHVSHFWWRWKHYDULDEOHVxDQGyDUHQXOO)XU WKHUFDVHVDUHWKRVHLQZKLFKDSRLQWRIILUVWVSHFLHVLVRQWKHHGJHRIWKHRSHQLQJ &ULWLFDOSRLQWVRIWKHsecond speciesDUHWKRVHDORQJWKHHGJHRIWKHDSHUWXUHZKHUH WKH GHULYDWLYH RI f LV FDQFHOHG RQO\ LQ WKH GLUHFWLRQ WDQJHQW WR WKH HGJH DQG WKH FULWLFDOSRLQWVRIWKHthird speciesFRLQFLGHZLWKWKHFRUQHUVRIWKHFRQWRXUZKHUH WKHUHLVDGLVFRQWLQXLW\LQWKHGLUHFWLRQWDQJHQWWRWKHHGJH7KLVLVZK\VRPHSDUWV RIWKHDSHUWXUHHGJHDSSHDULOOXPLQDWHGLQYLVXDOREVHUYDWLRQDQGRWKHUVGRQRW ,QSDUWLFXODUIRUDFULWLFDOSRLQWRIWKHILUVWNLQGERWKIXQFWLRQVfDQGgDUHGH YHORSHGDURXQGLWLQD7D\ORUVHULHVDQGWKHH[SRQHQWLDOWHUPLVDOVRH[SDQGHGLQD

'LIIUDFWLRQ  VXLWDEOH IRUP 7KH RYHUDOO VHULHV REWDLQHG ZLWK WKHVH GHYHORSPHQWV LV WKHQ LQWH JUDWHGWHUPE\WHUPE\WDNLQJWKHOLPLWVRILQWHJUDWLRQDWLQILQLW\DQGWKLVLVSRV VLEOHLIWKHFULWLFDOSRLQWLVVXIILFLHQWO\LVRODWHGIURPRWKHUVDQGIDUIURPWKHHGJH )LQDOO\FROOHFWLQJWKHUHVXOWVRIWKHYDULRXVLQWHJUDOZLWKWKHVDPHSRZHULQk RQHREWDLQVWKHDV\PSWRWLFVHULHVVRXJKW6LPLODUGHYHORSPHQWSURFHGXUHVDORQJ ZLWKFRRUGLQDWHWUDQVIRUPDWLRQVDUHDOVRXVHGLQWKHRWKHUFDVHV7KHDV\PSWRWLF GHYHORSPHQWRIWKHILUVWVSHFLHVSRLQWVFDQEHFRQVLGHUHGWKHFRQWULEXWLRQRIWKH JHRPHWULFZDYHSDVVLQJWKURXJKWKHDSHUWXUHZKLOHWKHRWKHULVWKDWRIWKHZDYH GLIIUDFWHG IURP WKH HGJH ,W WXUQV RXW WKDW WKH ILUVW WHUP RI DV\PSWRWLF GHYHORS PHQWVIRUWKHILUVWVHFRQGDQGWKLUGVSHFLHVLVSURSRUWLRQDOWRkkDQGk :H DOVR QHHGVHSDUDWHO\ WR FRQVLGHU WKH VLWXDWLRQV LQ ZKLFK WZRRU PRUH FULWLFDO SRLQWV RI YDULRXV VSHFLHV DUH SODFHG FORVH WRJHWKHU 7KH FROOLVLRQ EHWZHHQ WKH SRLQWVRIILUVWVSHFLHVFRUUHVSRQGVWRWKHFDXVWLFRIWKHJHRPHWULFDOZDYHZKLOHLI SRLQWVRIVHFRQGDQGWKLUGVSHFLHVDUHLQYROYHGWKHUHDUHFDXVWLFVZLWKWKHGLIIUDFW HGZDYH7KHVHFDVHVDUHWUHDWHGXVLQJZKDWDUHFDOOHGuniform asymptotic approximations>6WDPQHV@ ,QFDVHVLQZKLFKWKHGLIIUDFWLRQLQWHJUDOLVUHGXFHGWRDOLQHLQWHJUDORIWKHW\SH 



U

³C g x eik f x dx 

 

DV DQ DOWHUQDWLYH WR WKH VWDWLRQDU\ SKDVH PHWKRG WKH PHWKRG RI steepest descent FDQEHXVHGLQZKLFKWKHIXQFWLRQRIDUHDOYDULDEOHWREHLQWHJUDWHGLVH[WHQGHGWR WKH FRPSOH[ SODQH DQG LVUHQGHUHGKRORPRUSKLF E\ H[FOXGLQJ WKHQHLJKERUKRRGV RIVLQJXODUSRLQWVDQGLQVHUWLQJDEUDQFKFXW7DNLQJDGYDQWDJHRIWKHSURSHUWLHVRI KRORPRUSKLFIXQFWLRQVLQWHJUDWLRQFDQEHFDUULHGRQWKHFRPSOH[SODQHRQDSDWK IRU ZKLFK WKH UHDO SDUW RI f LV FRQVWDQW ZKLOH LWV LPDJLQDU\ SDUW KDV WKH KLJKHVW VORSH ,QWHJUDWLRQ LV WKHQ FRPSOHWHG ZLWK WKH FRQWULEXWLRQ RI WKH VLQJXODU SRLQWV DQGWKHEUDQFKFXWV:LWKWKLVPHWKRGRQHFDQJHWDEHWWHUDV\PSWRWLFDSSUR[LPD WLRQ>6ROLPHQRHWDO@ ,Q WKH QH[W FKDSWHU ZH ZLOO LQVWHDG GLVFXVV WKH PHWKRGV EDVHG RQ WKH SODQH ZDYHVVSHFWUXPRIWKHILHOGRQDQDSHUWXUHZLWKWKHLULPSOHPHQWDWLRQLQSURJUDPV WKDWWDNHDGYDQWDJHRIWKHFast Fourier Transform

4.1.9 Diffraction with quasi-monochromatic radiation 6R IDU ZH KDYH FRQVLGHUHG WKH FDVH RI DQ LGHDO PRQRFKURPDWLF LOOXPLQDWLRQ :KDW KDSSHQV LQVWHDG ZKHQ WKH UDGLDWLRQ HPSOR\HG KDV D ILQLWH EDQGZLGWK 'Q DURXQGDFDUULHUIUHTXHQF\QR",QSDUWLFXODUOHWXVFRQVLGHUWKHFDVHLQZKLFKWKH LQFLGHQWZDYHKDVDQDPSOLWXGHJLYHQE\ 

u r t

f

³ U r Q ei SQt d Q 

 

6FDODUWKHRU\RIGLIIUDFWLRQ



ZKHUHULVWKH)RXULHU¶VWUDQVIRUPLQWKHVSDFHRIWKHIUHTXHQFLHVRIWKHUHDOILHOG XRIZKLFKuLVWKHDQDO\WLFIXQFWLRQ7KLVWUDQVIRUPDWLRQEHWZHHQtDQGQZLOOEH EHWWHUGHILQHGLQWKHQH[WFKDSWHU)ROORZLQJ*RRGPDQ  ZHWKHQDSSO\WKH LQWHJUDWLRQRQWKHIUHTXHQFLHVWRWKHILUVWGLIIUDFWLRQLQWHJUDORI5D\OHLJK6RPPHU IHOG)RURtZR!!OZHKDYH 

u , rR  t

i ³

f



Q Z R i  SvQR i  SQt e U r  Q d S d Q  v ³³ R 

 

S

ZKHUHv  cnLVWKHVSHHGRIOLJKWLQWKHFRQVLGHUHGPHGLXPO  vQR  ro  r  DQGZRLVWKHGLVWDQFHRIWKHREVHUYDWLRQSRLQWIURPWKHSODQHRIWKHDSHUWXUH2Q WKHRWKHUKDQGGLIIHUHQWLDWLQJ(T  RYHUWLPHZHKDYH w u r t wt



f

³ i SQ U r Q ei SQt d Q 

 

7KHUHIRUH H[FKDQJLQJ WKH LQWHJUDWLRQ RUGHU LQWR (T   DQG DSSO\LQJ (T  ZHJHW u , rR  t



f ZR  d S ³ i SQ e i  SQ t  R v U r  Q d Q ³³    Sv R

S

ZR w  u r t  R v d S  ³³ Sv R  wt



 

S

ZKHUHKHUHWKHIXQFWLRQuLQWKHLQWHJUDOLVWDNHQDWWKHdelayed time t  ro  r v  VRDVWRWDNHLQWRDFFRXQWWKHSURSDJDWLRQWLPHEHWZHHQrDQGro ,QSDUWLFXODUFRQGLWLRQVZHFDQILQGDVLPLODUH[SUHVVLRQWRWKHGLIIUDFWLRQLQWH JUDOVRIWKHPRQRFKURPDWLFFDVH/HWXVUHVWDUWIURP(T  LIUKDVDOLP LWHGH[WHQVLRQLQDQLQWHUYDO'QDURXQGQRLQIUHTXHQF\VSDFHDQGLIWKHDSHUWXUH KDVDOLPLWHGH[WHQVLRQIRUZKLFKRYDULHVEHWZHHQDPLQLPXPRDDQGDPD[LPXP REDURXQGDQDYHUDJHYDOXH R rR WKHDUJXPHQWRIWKHH[SRQHQWLDOFDQEHUHZULW WHQDV 

ª º i  S « Q c  Q R c  R  Qt » ¬v ¼

ª  º § R·  i  S « QcRc  Q ¨ t  ¸  QR  QR »   v ¼ © v¹ v ¬v

ZKHUHR‫ މ‬R R DQGQ ‫ މ‬Q± Q %\LQYHUWLQJWKHLQWHJUDWLRQRUGHUDQGDSSUR[ LPDWLQJWKHWHUPRIDPSOLWXGHQvZLWK Q v(T  EHFRPHV 

 %RUQ DQG :ROI   LQVWHDG VWDUWV IURP .LUFKKRII¶V GLIIUDFWLRQ LQWHJUDO ,Q WKLV ZD\ WKH

LQWHJUDWLRQ VXUIDFH FDQ KDYH DQ\ VKDSH 8QIRUWXQDWHO\ WKH H[SUHVVLRQV WKDW DUH REWDLQHG DUH PXFKPRUHFRPSOLFDWHG

'LIIUDFWLRQ



 u , rR  t



n



i Q i  S QR Z R i c e ³³ R e v

S f QR v U

³ r Q e

S

i

§ R· S  QcRci  SQ ¨ t  ¸ v © v ¹ d S d Q

 



,IILQDOO\WKHVSHFWUDOEDQGRIXLVVXIILFLHQWO\OLPLWHGVRWKDW 

'Q 

v  RE  RD

 

WKHWHUPLQQ‫މ‬R ‫މ‬FDQEHQHJOHFWHGDQGZLWK(T  WKHGLIIUDFWLRQLQWHJUDO EHFRPHV 

u, rR  t



S S i Q  i QR Z R i QR § R· e v ³³  e v u ¨ r  t  ¸ d S  v v¹ R ©

 

S

7KH WHUP t ± R v LV W\SLFDO RI WKH UHWDUGHG SRWHQWLDOV DQG R v UHSUHVHQWV DQ DYHUDJHWLPHGHOD\ZKLOHWKHDUJXPHQWRIWKHH[SRQHQWLDORXWRIWKHLQWHJUDOLVWKH DYHUDJHSKDVHVKLIWRIWKHZDYHWKDWDUULYHVLQ rR7KLVSKDVHVKLIWLVFRPSHQVDWHG ZLWKWKDWGXHWRWKHWLPHGHOD\,QGHHGIDFWRUL]LQJuDURXQGWKHFDUULHUIUHTXHQF\ Q DV SOHDVHQRWHWKHFKDQJHRIIRQW  u r  t u r  t e i  SQt 



 

ZHFDQUHZULWH(T  DV 

u rR  t



S i Q Z R i QR § R· v e u ¨ r  t  ¸ d S  v ³³ R  v¹ ©

 

S

,QWKHPRQRFKURPDWLFFDVH u r  t LVDFRQVWDQWDPSOLWXGHRYHUWLPHDQGWKXV ZHDJDLQILQGWKHGLIIUDFWLRQLQWHJUDO ‫ މ‬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†ZKLFKLVGHGLFDWHGWRFR KHUHQFH

)UHVQHODQG)UDXQKRIHUGLIIUDFWLRQ



)UHVQHODQG)UDXQKRIHUGLIIUDFWLRQ 4.2.1 Evolution of the diffraction pattern with the distance from the aperture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raunhofer’sdiffractionRUfar field 7KHUHJLPHDWLQWHUPHGLDWHGLVWDQFHVLVFDOOHG Fresnel’s diffractionZKLOHWKDW QHDUWKHDSHUWXUHZKLFKFDQVWLOOUHDVRQDEO\EHWUHDWHGZLWK*HRPHWULFDO2SWLFVLV FDOOHGWKHnear field7KHERUGHUEHWZHHQWKHVHW\SHVRIGLIIUDFWLRQLVEOXUUHGDQ\ ZD\DQGZHZLOOFRPHEDFNWRLWVKRUWO\ ,Q WKH )UDXQKRIHU¶V UHJLRQ WKH GLIIUDFWLRQ SDWWHUQ FRUUHVSRQGV WR WKH DQJXODU VSHFWUXP RI WKH ILHOG GLDSKUDJPHG E\ WKH DSHUWXUH,W VKRXOG EH QRWHGKRZHYHU WKDW LQ WKH FDVH RI D WULDQJXODU RSHQLQJ VXFK DV WKDW LQ )LJ  WKLV OLPLW LV UHDFKHG VORZO\ ZKLOH WKH VLPLODULW\ ZRXOG EH UHDFKHG IDVWHU IRU D UHFWDQJXODU RSHQLQJ RU JHQHUDOO\ IRU DSHUWXUHV GHOLPLWHG E\ SDLUV RI SDUDOOHO VLGHV EHWZHHQ WKHP ,QWKHSUHFHGLQJVHFWLRQVWKHUHVXOWVRIWKHVFDODUWKHRU\KDYHEHHQSUHVHQWHGLQ WKHLUPRVWJHQHUDOIRUPKRZHYHULQFDOFXODWLQJWKHGLIIUDFWLRQILJXUHVVRPHIXU WKHU DSSUR[LPDWLRQV DUH QRUPDOO\ LQWURGXFHG 7KH\ DUH RI D GLIIHUHQW GHJUHH GH SHQGLQJRQWKHGLVWDQFHIURPWKHRSHQLQJVDQGWKXVDUHVWLOOFDOOHGE\WKHQDPHV RI)UHVQHODQG)UDXQKRIHU7KHDGYDQWDJHWKDWLVREWDLQHGRQ(TV     DQG   ZKLFKDUHKRZHYHUUHODWLYHO\VLPSOH LVWKHSRVVLELOLW\RISHUIRUP LQJFDOFXODWLRQVXVLQJWKH)DVW)RXULHU7UDQVIRUP

4.2.2 Initial approximations 7KHQOHWAEHDQDSHUWXUHSUDFWLFHGRQDSODQHRSDTXHVFUHHQDQGSRDSODQHRI REVHUYDWLRQSDUDOOHOWR ADQGSODFHGDWDGLVWDQFH ZRIURPLW)ROORZLQJWKHFRQ VLGHUDWLRQVPDGHDWWKHHQGRI†ZHOLPLWRXUDWWHQWLRQWRZKHQ

'LIIUDFWLRQ





)LJ6LPXODWLRQRIWKHGLIIUDFWLRQSDWWHUQ SURGXFHGE\DWULDQJXODUDSHUWXUHRQDPRQ RFKURPDWLF SODQH ZDYH 7KH KHLJKW RI WKH WULDQJOH LV  PP DQG WKH ZDYHOHQJWK LV  PP 7KH ILJXUHV DUH WDNHQDW GLVWDQFHV IURP WKH RSHQLQJ LQGLFDWHG DW WKH WRS DQG WKHLU ZLGWKLVPPuPP  IRUWKHILUVWHLJKWDQG PP u PP IRU WKH RWKHU WZR 7KH ILJXUH KHUH WR WKH ULJKW UHSUHVHQWV LQVWHDG WKH DQ JXODU VSHFWUXP RI WKH ILHOG RQ WKH RSHQLQJ DQGLVDERXWPUDGZLGH,QWKHILJXUHV WKH JUD\VFDOH UHSUHVHQWV WKH PRGXOH RI WKH ILHOG DPSOLWXGH UDWKHU WKDQ WKH LQWHQVLW\ VR DVWRKLJKOLJKWWKHZHDNHUGHWDLOVWKDWZRXOG RWKHUZLVHQRWEHDSSUHFLDEOH0RUHRYHUWKH PD[LPDDUHQRUPDOL]HGWRWKHVDPHYDOXH 

















 





)UHVQHODQG)UDXQKRIHUGLIIUDFWLRQ



i

WKHVRXUFHVDQGWKHREVHUYDWLRQSRLQWVDUHDSSUR[LPDWHO\DOLJQHGZLWKDQD[ LVSDVVLQJWKURXJKWKHDSHUWXUH A:LWKRXWWRRPXFKORVVRIJHQHUDOLW\ZH FDQDVVXPHWKDWWKLVD[LVLVQRUPDOWR ADQG SRDQGFRQVLGHULWDVWKH zD[LV RID&DUWHVLDQUHIHUHQFHV\VWHPxyzFHQWHUHGLQA ii WKHFRRUGLQDWHVxyRIWKHREVHUYDWLRQSRLQWVDUHVPDOOFRPSDUHGWRZR iii WKHVL]HRIALVVPDOOFRPSDUHGWRZREXWDOVR!!O iv WKHUDGLLRIFXUYDWXUHRIWKHLQFLGHQWZDYHIURQWRQ ADUHODUJHFRPSDUHGWR WKHVL]HRIALHWKHVRXUFHVDUHGLVWDQWIURPA 7KHUHIRUHZLWKLQWKHOLPLWVRIWKHVHDVVXPSWLRQVZHFDQDSSUR[LPDWHWKHREOLTXL W\IDFWRUVWRDQGWKHGLIIUDFWLRQLQWHJUDO  EHFRPHV U R xR  y R





i eikR U x y dx dy  O ³³ R

 

A

ZKHUH 

rR  r

R



ª x  x   y  y   z  z  º R R ¬« R ¼»



 

)RUEUHYLW\,KDYHRPLWWHGWKH zFRRUGLQDWHVLQ URDQG U(T  UHSUHVHQWV WKHWUDQVIHURSHUDWLRQRIWKHILHOGIURPWKHSODQHRIWKHDSHUWXUHWRWKHREVHUYDWLRQ SODQH(YHQ(TV  DQG  FDQEHDSSUR[LPDWHGWR(T  LWLVVXI ILFLHQWWRQRWHWKDWWKDQNVWRWKHFRQGLWLRQV i DQG iv RQWKHDSHUWXUHZHKDYH wUwn wUw z  ikU$VDQDGGLWLRQDODSSUR[LPDWLRQWKHWHUP RLQWKHGH QRPLQDWRUFDQEHUHSODFHGZLWK ZR

zR  z 

 

DQGEURXJKWRXWRIWKHLQWHJUDO

4.2.3 Fresnel’s approximation 7KHVLPSOHUHSODFHPHQW Ro ZRFDQQRWEHGRQHLQWKHH[SRQHQWVLQFHWKHUH VXOWLQJHUURUZRXOGEHPXOWLSOLHGE\ kREWDLQLQJDYHU\ODUJHQXPEHU kR !!  7RVLPSOLI\WKHLQWHJUDQGIXUWKHUKRZHYHUZHFDQGHYHORS RZLWKDELQRPLDOVH ULHVRIWKHURRWLQ(T   

 b 

    b  b  " YDOLGIRU b      

+HQFHLI xR  x  yR  y  RR 

'LIIUDFWLRQ



 R

ZR  

xR  x   y R  y  Z R

 ZR 

xR  x



 yR  y



ZR



 º ª « xR  x  yR  y ¼» ¬   " Z R



:KHQZRLVODUJHHQRXJKIRUZKLFKWKHWHUPVDIWHUWKDWLQZRFDQEHQHJOHFWHGLW LVVDLGWREHDFDVHRI)UHVQHO¶VGLIIUDFWLRQ8QGHUWKHVHFRQGLWLRQVZHDVVXPHWKH UHODWLRQ 

ZR 

R

 ª xR  x   yR  y  º»¼   Z R «¬

 

ZKLFKLVFDOOHG Fresnel’s approximation2QWKHRWKHUKDQGLIWKHHUURUPDGHLQ QHJOHFWLQJWKHWHUPLQZRZHUHDOUHDG\H[FHVVLYHLWZRXOGEHEHWWHUWRGHDOZLWK WKHSUREOHPLQDQRWKHUZD\$SSO\LQJ(T  WR(T  RQHREWDLQV



U R xR  y R



ieikZ R OZ R

i

k ª xR  x   yR  y  º¼»

« ³³ e ZR ¬

U x y dxdy 

 

A

$VDFRQGLWLRQRIVXIILFLHQWDFFXUDF\IRUWKLVIRUPXODZHPD\UHTXLUHWKDWWKHHU URUPDGHLQ RVWRSSLQJDWWKHWHUPLQZRLVO,IIRUH[DPSOH ZR FP DQGO QPWKHDQJOHVXEWHQGHGE\RVKRXOGEHƒ ƒLIWKHWHUPV LQZRZHUHDOVRLQFOXGHG )RUWXQDWHO\WKLVFRQGLWLRQLVQRWUHDOO\QHFHVVDU\,Q RUGHU WKDW WKH )UHVQHO¶V DSSUR[LPDWLRQ UHPDLQ YDOLG LW LV RQO\ UHTXLUHG WKDW WKH WHUPVRIKLJKHURUGHULQWKHGHYHORSPHQWRIRGRQRWDSSUHFLDEO\FKDQJHWKHYDOXH RIWKHGLIIUDFWLRQLQWHJUDO)RUGLVWDQFHV ZRVXIILFLHQWO\VPDOOWRYLRODWHWKHFRQGL WLRQWKDWWKHHUURULQRLVOIRUH[DPSOHZKHQWKHWHUPLQZRLVOWKDWLV 

U Z R 

O  ZLWK U 

xR  x   yR  y  

WKHWHUPLQZRLV 

U Z R

Z R U U  Z R

Z R O  U

DQGLVWKHUHIRUHJHQHUDOO\YHU\ODUJHFRPSDUHGWRO LQWKHDERYHPHQWLRQHGH[ DPSOHLWLVO 7KXVZKHQWKHSRLQW x y H[SORUHVWKHLQWHJUDWLRQGRPDLQ WKH IXQFWLRQ R ‫  މ‬U   ZR   >SKDVH RI U x y @ k JHQHUDOO\ FKDQJHV E\ PDQ\ ZDYHOHQJWKVDQGWKHQWKHH[SRQHQWLDOXQGHUWKHLQWHJUDWLRQVLJQRVFLOODWHVPDQ\

)UHVQHODQG)UDXQKRIHUGLIIUDFWLRQ



WLPHV E\ FKDQJLQJ WKH VLJQ RI WKH UHDO DQG WKH LPDJLQDU\ SDUW RI WKH LQWHJUDQG &RQVHTXHQWO\WKHFRQWULEXWLRQVIURPWKHYDULRXVHOHPHQWVRIVXUIDFHRIWKHDSHU WXUHJHQHUDOO\FDQFHOHDFKRWKHURXW destructive interference  +RZHYHU WKH VLWXDWLRQ LV GLIIHUHQW IRU D VXUIDFH HOHPHQW WKDW LV FHQWHUHG RQ D SRLQW FDOOHGDcritical pointRUSROH ZKHUHIRUDJLYHQSRLQWRIREVHUYDWLRQR ‫މ‬LV VWDWLRQDU\+HUHWKHLQWHJUDQGYDULHVPXFKPRUHVORZO\DQGZHFDQH[SHFWWKDWLW JLYH D VLJQLILFDQW FRQWULEXWLRQ KHUH 7KHUHIRUH WKH LQWHJUDO YDOXH LV VXEVWDQWLDOO\ GHWHUPLQHG E\ WKH EHKDYLRU RI R ‫ މ‬LQ WKH QHLJKERUKRRG RI WKH SRLQWV ZKHUH R ‫ މ‬LV VWDWLRQDU\ 7KLV LV WKH SULQFLSOH RI WKH stationary phase method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¶VDSSUR[LPDWLRQZHFDQFRQWLQXHLQWKHVWXG\ RI(T  ,WFDQEHLQWHUSUHWHGDVWKHFRQYROXWLRQRIU x y ZLWKWKHIXQFWLRQ k

h x y



eikZ R i  Z ª¬ x e R i OZ R

  y º

¼



 

(T   FDQ DOVR EH WUDQVFULEHG LQ DQ HTXLYDOHQW IRUP E\ GHYHORSLQJ (T   

R

ZR 

xR  yR  xxR  yyR x   y     ZR ZR ZR

 

VRWKDWWKHGLIIUDFWLRQLQWHJUDOEHFRPHV k

U R xR  yR



ieikZ R i  Z xR  e R OZ R u³³ e

i

 y  R



u

k S x  y xxR  yyR i Z OZ R  e R







 

U x y dx dy

A

7KHQ DSDUW IURP WKH SKDVHDQG DPSOLWXGH WHUPV RXWVLGH RI WKH LQWHJUDO VLJQ WKH ILHOG U xRyo FDQEHGHULYHGIURPWKH)RXULHUWUDQVIRUPRI 

ª k H[S «i x  y  Z R ¬



º

» U x y  ¼

'LIIUDFWLRQ



 HYDOXDWHGDWWKHspatial frequencies 

xR OZ R

fx

fy

yR  OZ R

 

$QDOWHUQDWLYHWRWKHDSSUR[LPDWLRQRIRJLYHQE\(T  LVWRFDOFXODWHWKH GLIIUDFWHGILHOGQRWRQDSODQHVXUIDFHORFDWHGDWDGLVWDQFH ZRIURPWKHDSHUWXUH EXWRQDKHPLVSKHULFDOVXUIDFHRIUDGLXV SZLWKLWVFHQWHURQWKHFRRUGLQDWHV x  y z  LQWKHFHQWHURIWKHRSHQLQJZKHUHE\ xR  yR  zR  R

S

x  xR   y  yR   S   xR  yR

x   y    xxR  yyR  S 



DQG(T  LVUHSODFHGZLWK 

R#S

x   y  xxR  yyR   S S

 

ZKLOH(T  LVUHSODFHGZLWK 

U R xR  y R



ieikS OS

³³ e

i

S k xxR  yyR i x   y  OS e S



U x y dx dy 

 

A

7KLVH[SUHVVLRQLVPRUHSUHFLVHDQGDOORZVIRUFDOFXODWLQJWKHGLIIUDFWHGILHOG HYHQZLWKODUJHDQJXODURSHQLQJVZLWKRXWLQWURGXFLQJFRPSOLFDWLRQVLQWKHFDOFX ODWLRQRIWKHGLIIUDFWLRQLQWHJUDOZKLFKFDQVWLOOEHH[SUHVVHGLQWHUPVRID)RXULHU WUDQVIRUPHYDOXDWHGDWWKHIUHTXHQFLHV 

fx

xR  OS

fy

yR  OS

 

+RZHYHUWKHGLIIUDFWLRQSURILOHLVREWDLQHGRQDVSKHULFDOVXUIDFH

4.2.4 Self-consistency of Fresnel’s diffraction :H KDYH VHHQ EHIRUH WKDW )UHVQHO¶V DSSUR[LPDWLRQ LV YDOLG IRU GLVWDQFHV ZR PXFKODUJHUWKDQWKHVL]HRIERWKWKHDSHUWXUHDQGWKHWUDQVYHUVHFRRUGLQDWHVRIWKH REVHUYDWLRQ SRLQW +RZHYHU LW DOVR KDSSHQV WKDW WKLV DSSUR[LPDWLRQ LV DSSOLHG ZLWKQRFRQFHUQLQRWKHUFDVHV$VSDUWLDOMXVWLILFDWLRQIRUWKLVXVHZHFDQLQYRNH

)UHVQHODQG)UDXQKRIHUGLIIUDFWLRQ



DVHOIFRQVLVWHQF\SURSHUW\RIWKHLQWHJUDO  6XSSRVHWKDWZHGLYLGHWKHGLV WDQFHLQWRWZRSDUWVIRUZKLFK ZR  Z ZDQGGXSOLFDWHWKHLQWHJUDWLRQLQWZR RSHUDWLRQVWREHGRQHLQVXFFHVVLRQ eikZ R   O ZZ 

U R xR  y R



u³³ e

i

f f

³ ³ dxcdy ce

i

k ª xR  xc   yR  yc  ¼»º  Z  ¬«

f f

k ª xc x   yc y  º»¼  Z «¬ U



 

x y dx dy

A

ZKHUHDQLQILQLWHLQWHJUDWLRQLVDGGHGLQDQLQWHUPHGLDWHSODQHx ‫މ‬y ‫މ‬%\FROOHFWLQJ WKHDUJXPHQWVRIWKHH[SRQHQWLDOVXQGHUWKHVLJQRILQWHJUDWLRQZLWKVRPHVWHSV ZHJHW i



k  ª     Z xR  x c  Z yR  y c  Z  x c  x  Z  y c  y º» « ¬ ¼  ZZ  i

k  ª xR  x   yR  y  º»¼  Z R «¬

k ZR i  ZZ 

  

  ª§ Z · §Z · º Z Z «¨  xR   x  x c ¸  ¨  yR   y  y c ¸ »  ZR ZR «© Z R ¹ © ZR ¹ »¼ ¬

:LWKDQDSSURSULDWHWUDQVODWLRQRIWKHYDULDEOHVx ‫މ‬DQGy ‫(މ‬T  EHFRPHV

U R xR  y R



eikZ R   O ZZ 

³³ e

i

k ª xR  x   yR  y  º¼»  Z R ¬« U

i

 ZR S  xcc  ycc OZZ  

f f

u

³ ³

x y dx dy

A

dx ccdy cce







 



f f

7KHVHFRQGGRXEOHLQWHJUDOFDQEHIDFWRUL]HGLQWZRLQGHSHQGHQWWHUPVLQ xүүDQG yүү(DFKRIWKHPFDQLQWXUQEHH[SUHVVHGE\)UHVQHOLQWHJUDOVDQGLVHTXDOWR f

³

f

dx cce

i

ZR S  xcc OZZ  



OZZ  ZR

f

ª

§S

·

§S

·º

³ dt «¬FRV ¨©  t  ¸¹  i VLQ ¨©  t  ¸¹»¼ 

  i OZZ   ZR 

%\ UHSODFLQJ WKLV YDOXH LQ (T   ZH DJDLQ ILQG (T   RI WKH )UHVQHO GLIIUDFWLRQ ,Q FRQFOXVLRQ WKH VXEGLYLVLRQ RI ZR LQWR Z DQG Z LV DUELWUDU\ DQG HDFKRIWKHGLIIUDFWLRQLQWHJUDOVUHVXOWLQJIURPWKLVGHFRPSRVLWLRQPD\QRWUHVSHFW

'LIIUDFWLRQ  WKHYDOLGLW\FRQGLWLRQVRI)UHVQHO¶VDSSUR[LPDWLRQ+RZHYHULIZRLVODUJHHQRXJK WKHRYHUDOOUHVXOWLVVWLOOYDOLG 2QWKHRWKHUKDQGDVZLOOEHVHHQODWHUWKH+X\JHQV)UHVQHOGLIIUDFWLRQLQWH JUDOLVPDWKHPDWLFDOO\HTXLYDOHQWWRWKHSDUD[LDOHTXDWLRQXVHGWRGHULYHWKHSURS DJDWLRQRIODVHUEHDPV7KHUHIRUHDV6LHJPDQ  ZULWHVLWLVWKHSDUD[LDOQD WXUHRIWKHZDYHWDNHQLQWRDFFRXQWWKDWPDNHV)UHVQHO¶VDSSUR[LPDWLRQYDOLG©WKH SDUD[LDORU)UHVQHODSSUR[LPDWLRQLVDSK\VLFDOSURSHUW\RIWKHRSWLFDOEHDPQRWD PDWKHPDWLFDOSURSHUW\RIWKH+X\JHQV)UHVQHOIRUPXODWLRQª 

4.2.5 Fraunhofer’s approximation :KHQZRLVYHU\ODUJHZLWKUHVSHFWWRWKHDSHUWXUHVL]HIRUZKLFK

x   y  PD[



ZR



O  

 

(T  FDQEHIXUWKHUVLPSOLILHGE\REWDLQLQJ 

R

ZR 

xR  yR xxR  yyR   ZR Z R

 

ZKLFKLVFDOOHGFraunhofer’s approximation(T  LVUHGXFHGWR k



U R xR  yR



ieikZ R i  Z xR e R OZ R

 y  R



³³ e

i

S xxR  yyR OZ R U

x y dx dy   

A

ZKLFKDOORZVIRUREWDLQLQJWKHILHOGULQWKH)UDXQKRIHU¶VUHJLRQ $SDUW IURP WKH WHUPV RXW RI WKH LQWHJUDO (T   LV VLPSO\ WKH )RXULHU WUDQVIRUPRIWKHILHOGRQWKHRSHQLQJDQGLVHYDOXDWHG DVEHIRUH DWWKHIUHTXHQ FLHV fx  xR O ZR  fy  yR O ZR  7KH TXDOLWDWLYH EHKDYLRU RI WKH GLIIUDFWLRQ GH VFULEHGLQ†LVWKXVXQGHUVWRRG)RUODUJHGLVWDQFHVWKHGLIIUDFWLRQVKDSHKDV DQ DVSHFW IL[HG E\ WKH )RXULHU WUDQVIRUP EXW LWV VL]H JURZV ZLWK ZR ,QGHHG IRU XQFKDQJHGVSDWLDOIUHTXHQFLHVfxfyWKHFRRUGLQDWHVxRyRYDU\DVZRO fxZRO fyUH VSHFWLYHO\ ,QVWHDG WKH YDULDELOLW\ RI WKH GLIIUDFWLRQ SDWWHUQ LQ WKH LQWHUPHGLDWH )UHVQHO¶V UHJLRQFRUUHVSRQGVWRWKHGHFUHDVHZLWKLQFUHDVLQJZRRIWKHFRQWULEXWLRQRI i



e

x  y ZR



)UHVQHODQG)UDXQKRIHUGLIIUDFWLRQ

)LJ 0HWKRG IRU REVHUYLQJ )UDXQKRIHU¶V GLIIUDF WLRQILJXUHV



A

S

f



$W RSWLFDO ZDYHOHQJWKV WKH FRQGLWLRQV UHTXLUHG IRU WKH YDOLGLW\ RI (T   PD\EHYHU\VHYHUH,IDLVWKHZLGWKRIWKHDSHUWXUH HJWKHGLDPHWHULIALVFLU FXODU (T  FDQEHUHZULWWHQDV 

Z R !!

D  O

 

DQGPDQ\DXWKRUVFRQVLGHUDFRQYHQWLRQDOWKUHVKROGYDOXH Z)  DO)RUH[DP SOHZLWK D PPDQGO QP Z) P+RZHYHU)UDXQKRIHU¶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x yLQWKHLQWHJUDQG,QWKHFDVHRIWKH FRQYHUJLQJZDYHWKLVLVTXLWHREYLRXVZKLOHWKHHIIHFWRIDOHQVZLOOEHVWXGLHGLQ GHWDLO LQ D VHFWLRQ RI &KDSWHU  +HUH , DQWLFLSDWH RQO\ WKDW WKH QRUPDO H[SHUL PHQWDODUUDQJHPHQWIRUREVHUYLQJ)UDXQKRIHU¶VGLIIUDFWLRQLVRIWKHW\SHVKRZQLQ )LJ7KHILUVWOHQVSURGXFHVDZDYH DWOHDVWDSSUR[LPDWHO\ SODQHRQWKHDSHU WXUHA7KHVHFRQGOHQVFRQYHUWVWKHGLUHFWLRQRIHDFKRIWKHSODQHZDYHVRIZKLFK ZHFDQWKLQNRIWKHGLIIUDFWHGZDYHEHLQJFRPSRVHGDWDSRLQWLQWKHIRFDOSODQH 7KHWZROHQVHVPXVWEHODUJHHQRXJKWRQRWLQWXUQEHDGLIIUDFWLQJGLDSKUDJP

4.2.6 Debye’s approximation ,Q WKH FDVH RI D FRQYHUJHQW ZDYH IRFXVHG ZLWK ODUJH QXPHULFDO DSHUWXUH RQH JHQHUDOO\XVHVZKDWLVFDOOHGDebye’s approximationDVDQDOWHUQDWLYHWR)UHVQHO¶V DSSUR[LPDWLRQZKLFKLQVWHDGLVYDOLGIRUDSHUWXUHVVPDOOHUWKDQWKHREVHUYDWLRQ GLVWDQFHDQGWKHUHIRUHZLWKVPDOOQXPHULFDODSHUWXUHV,Q'HE\H¶VDSSUR[LPDWLRQ WKH ILHOG LQ WKH IRFDO UHJLRQ LV WUHDWHG DV D VXSHUSRVLWLRQ RI SODQH ZDYHV ZKRVH

'LIIUDFWLRQ





)LJ 'LIIUDFWLRQ ZLWK D FRQYHUJLQJZDYH

S n

Po R

Jo

p F

f

 ZDYH YHFWRUV IDOO ZLWKLQ WKH JHRPHWULF FRQH KDYLQJ DV D YHUWH[ WKH IRFDO SRLQW DQGDVDEDVLVWKHRSHQLQJ,QRWKHUZRUGVWKLVDSSUR[LPDWLRQFRQVLVWVRIDWUXQ FDWLRQLQWKHDQJXODUVSHFWUXPRIWKHGLIIUDFWHGILHOG /HW XV FRQVLGHU WKHUHIRUH WKH FDVH LQ ZKLFK DQ DSSUR[LPDWHO\ VSKHULFDO FRQ YHUJLQJZDYHLOOXPLQDWHVDQRSHQLQJDQGKDVLWVFHQWHUDWDSRLQW)$VWKHLQWHJUD WLRQVXUIDFHIRUWKHFDOFXODWLRQRIWKHGLIIUDFWLRQZHWDNHDVSKHULFDOVXUIDFHRIUD GLXV fZLWKDFHQWHULQ)GHOLPLWHGE\WKHHGJHRIWKHDSHUWXUHLWVHOI )LJ 2Q WKLVVXUIDFHWKHLQFLGHQWZDYHILHOGFDQEHH[SUHVVHGDV 

U

P r

e ikf  f

 

ZKHUH WKH FRPSOH[IXQFWLRQ P WDNHV LQWR DFFRXQWERWK WKH ILHOG DPSOLWXGHYDULD WLRQVDQGDVPDOOGHYLDWLRQRIWKHZDYHIURQWIURPWKHVSKHUHDVWKHSRVLWLRQ rRQ WKHLQWHJUDWLRQVXUIDFHFKDQJHV:HDOVRDVVXPHWKDWWKHDSHUWXUHVL]HDQGWKHGLV WDQFHfDUHPXFKODUJHUWKDQWKHZDYHOHQJWKDQGZHDUHOLPLWHGWRWKHREVHUYDWLRQ SRLQWV3RZKRVHGLVWDQFHIURP)LVVPDOOFRPSDUHGWR f7KHYHFWRUV Q R f pDUH WKXVGHILQHGLQ)LJ %\DSSO\LQJWKHVHFRQGLWLRQVWR.LUFKKRII¶VGLIIUDFWLRQLQWHJUDODVVXPLQJWKH SRLQW)DVWKHRULJLQIRUWKHSRVLWLRQVDQGLJQRULQJWKHWHUPLQwPwnLQWKHGHULYD WLYHRIWKHILHOGZHJHW 

U. p

ik S

³³ P r S

eik R  f ª  FRV Q R º d S  ¼ ¬ Rf

 

,IWKHGLVWDQFH pEHWZHHQ)DQG3RLVDOVRVPDOOFRPSDUHGWR fWKHFRVLQHFDQEH DSSUR[LPDWHGE\ZKLOHWKHWHUPR fFDQEHDSSUR[LPDWHGDV 

R f

f   p    p x f  f # p x V 

ZKHUH V r  LV WKH YHUVRU RI f )LQDOO\ WDNLQJ dS Rf  # dSf d: WKH LQWHJUDO

)UHVQHODQG)UDXQKRIHUGLIIUDFWLRQ



  EHFRPHV DQ LQWHJUDO RQ WKH VROLG DQJOH : VXEWHQGHG E\ WKH DSHUWXUH ZKHUHE\ 

U. p

i P r ei k˜ p d :  O ³³ :

 

ZKHUH r DQG k  kV DUH QRZ IXQFWLRQV RI WKH DQJXODU FRRUGLQDWHV 7KLV LV FDOOHG 'HE\H¶V GLIIUDFWLRQ LQWHJUDO DQG WKH DSSUR[LPDWLRQV XVHG IRU LWV GHULYDWLRQ DUH FDOOHGDebye approximations  7KHH[SUHVVLRQ  FDQEHFRQVLGHUHGDVWKHVXSHUSRVLWLRQRISODQHZDYHV ZKRVH DPSOLWXGH LV SURSRUWLRQDO WR WKH ILHOG DW LQGLYLGXDO SRLQWV RI WKH VSKHULFDO VXUIDFHRILQWHJUDWLRQ,QRWKHUZRUGVHDFKSODQHZDYHFRUUHVSRQGVWRDVLQJOHJH RPHWULFUD\SDVVLQJWKURXJKWKHDSHUWXUHZLWKWKHSKDVHDQGLQWHQVLW\DVVRFLDWHG ZLWK LW2Q WKH RWKHUKDQG LW QHJOHFWV WKH FRQWULEXWLRQ RISODQH ZDYHVGLIIUDFWHG RXWVLGHRIWKHJHRPHWULFDOFRQHZKLFKFDQRQO\EHQHJOHFWHGZKHQWKH Fresnel’s number LV !!  7KLV QXPEHU LV GHILQHG DV WKH QXPEHU RI )UHVQHO ]RQHV N) a  OZo FRUUHVSRQGLQJWRDFLUFXODUDSHUWXUHRIGLDPHWHU aLOOXPLQDWHGE\D SODQHZDYHDQGREVHUYHGDWWKHORQJLWXGLQDOGLVWDQFHZR VHH† +HUH ZRFRLQ FLGHVZLWKf

4.2.7 Diffraction through an optical system described by an ABCD matrix ,QKLVPDMRUDUWLFOHIURP6WXDUW&ROOLQVLQWURGXFHGDGLIIUDFWLRQLQWHJUDO WKDWLVWKHJHQHUDOL]DWLRQRI)UHVQHO¶VLQWHJUDOWKURXJKDQRSWLFDOV\VWHPE\OLQN LQJWKHGLIIUDFWLRQWKHRU\ZLWK*HRPHWULF2SWLFV>&ROOLQV@,QGHHGLQ)UHV QHO¶V LQWHJUDO WKH DPSOLWXGH RI WKH ILHOG DW HDFK SRLQW RQ WKH RXWSXW SODQH LV WKH VXPRIWKHHOHPHQWDU\FRQWULEXWLRQVIURPWKHSRLQWVRIWKHLQSXWSODQHZHLJKHG E\ D SKDVH DQG DQ DPSOLWXGH WKDW GHSHQG RQ WKH JHRPHWULF RSWLFDO SDWK EHWZHHQ WKHWZRSRLQWV7KHVDPHUHVXOWZDVSXEOLVKHGLQGHSHQGHQWO\VKRUWO\DIWHUZDUGVE\ -$$UQDXGLQDKDUGWRUHDGZRUN>$UQDXG@,Q&ROOLQV¶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

'LIIUDFWLRQ  WKHSURSDJDWLRQRIODVHUEHDPV 7KHNHUQHORIWKHLQWHJUDOWREHDSSOLHGWRWKHDPSOLWXGHRIWKHILHOGLVWKHUH IRUHH[SUHVVHGDV 

K x  y x  y



a x  y x  y eikR L x y x  y 

 

ZKHUHDJDLQ kR SORLVWKHZDYHYHFWRUPRGXOHLQWKHYDFXXPZKLOH LLVWKH RSWLFDO SDWK DQG a LV D FRPSOH[ DPSOLWXGH %RWK GHSHQG RQ WKH SRVLWLRQV RI WKH SRLQWVFRQVLGHUHGRQWKHLQLWLDODQGILQDOSODQHV$IWHUGHWHUPLQLQJWKHYDOXHRI L WKURXJKWKHV\VWHP VHH† &ROOLQVDSSOLHVWKHVDPHDSSUR[LPDWLRQVXVHGWR GHULYHWKH)UHVQHOGLIIUDFWLRQLQWHJUDOHVVHQWLDOO\DVVXPLQJWKDW aLVFRQVWDQWZLWK UHVSHFWWRWKHSRVLWLRQVDOWKRXJKGHSHQGLQJRQWKHSDUDPHWHUVRIWKHRSWLFDOV\V WHP 7KLV LV HTXLYDOHQW WR UHSODFLQJ WKH DPSOLWXGH WHUP R ZLWK ZR DQG WKHQ KDYLQJLWPRYHGRXWRIWKHLQWHJUDO  7KHUHIRUHWKHDPSOLWXGHRIWKHHOHFWULF ILHOGDWWKHREVHUYDWLRQSRLQWLV 

E x  y

f ik L x  y  x  y e R     E f

a ³³

x y dx dy



 

)LQDOO\IRUWKHGHWHUPLQDWLRQRI a&ROOLQVDSSOLHVWKHSULQFLSOHRIHQHUJ\FRQVHU YDWLRQ,IWKHLQLWLDODQGILQDOPHGLDDUHRUGLQDU\GLHOHFWULFVIRU(T  ZH FDQDVVXPHWKDW 

f

³³f n E x  y



dx dy

f

³³f n E x y



dx dy 

 

ZKHUH n DQG n DUH UHVSHFWLYHO\ WKH UHIUDFWLYH LQGH[HV LQ WKH LQLWLDO DQG ILQDO VSDFHV(T  GLIIHUVIURP&ROOLQV¶RULJLQDORQHEHFDXVHKHLPSOLFLWO\FRQ VLGHUVDXQLWDU\LQGH[IRUERWKRIWKHVHVSDFHV VHHIRRWQRWHLQ†  5HSODFLQJ(T  LQ  DQGDSSO\LQJ(TV DQG IRUWKH RSWLFDO SDWK &ROOLQV ILQGV WKH PRGXOH RI a H[FHSW IRU D FRPSOH[ PXOWLSOLFDWLYH FRQVWDQWRIXQLWPRGXOH)RUFRPSDULVRQZLWKWKH)UHVQHOLQWHJUDOIRUWKHFDVHLQ ZKLFKWKHRSWLFDOV\VWHPLVWKHVLPSOHSURSDJDWLRQLQDKRPRJHQHRXVVSDFHLWLV IRXQGWKDWWKLVFRQVWDQWLVi)LQDOO\ZHKDYH 

a

i

n nko  n S B

ZKHUH B GHQRWHVWKH determinantRIBDQGWKHSUHVHQFHRIWKHUHIUDFWLYHLQGLFHV  7KHDVVXPSWLRQRIVXFKSUHVHUYDWLRQPHDQVWKDWWKHYDULRXVWUDQVPLWWLQJRSWLFDOVXUIDFHVKDYH

JRRGDQWLUHIOHFWLRQWUHDWPHQWDQGWKDWWKHPLUURUVDUHKLJKO\HIILFLHQWRWKHUZLVHWKHGLVFUHSDQ FLHVPD\EHVLJQLILFDQW

([DPSOHVRI)UDXQKRIHU¶VGLIIUDFWLRQ



JHQHUDOL]HV &ROOLQV¶ UHVXOW IRU WKH FDVH RI LQLWLDO DQG ILQDO LQGH[HV RWKHU WKDQ  &ROOLQV¶VGLIIUDFWLRQLQWHJUDOLVWKHUHIRUH E  x  y 

n nkR ik L e R R n S B

i



f ik L x  y  x  y e R      E f

u³³



 

x y dx dy 

ZKHUHLLVGHFRPSRVHGLQWKHVXPRIWZRWHUPVL LRLRIZKLFK

L

 7 7 § n B A U  U ¨ 7  © n B





n B · § U · ¸ ¨ ¸  U n DB ¹ © U ¹

§ x · ¨ y ¸  U © ¹

§ x · ¨ y ¸    © ¹

ZKLOH LRLVWKHRSWLFDOSDWKDORQJWKHSULQFLSDOUD\ VHH†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x%x&x'x DQG$y%y&y'y(T  WDNHVWKHIROORZLQJIRUP E  x  y 



i

f nkR n eikR LR ³³ E x  y dx dy f n S Bx B y

§x x y y · §A Dy  · Ay  · º k ª §D i R « n ¨ x x  y ¸  n¨      ¸ n¨ x x  y ¸ » ¨ ¸ ¨ ¸ ¨ ¸» B By B y ¹ © Bx By  ¹ © Bx ¹¼  ue «¬ © x



 

([DPSOHVRI)UDXQKRIHU¶VGLIIUDFWLRQ ,QWKHIROORZLQJZHZLOOFRQVLGHUIRUVLPSOLFLW\RQO\WKHFDVHRIDSODQHZDYH LQFLGHQWRQDQDSHUWXUH AWKDWLVDZDYHZLWKFRQVWDQWPRGXOHDQGSKDVHRQ A DQGZLWKLQWHQVLW\HTXDOWRIA+HUH,LQWURGXFHVRPHDEEUHYLDWLRQVFRPPRQWRDOO RIWKHH[DPSOHV(T  RIWKH)UDXQKRIHUGLIIUDFWLRQFDQEHUHZULWWHQDV

'LIIUDFWLRQ





y

)LJ 'HILQLWLRQRIYDULDEOHVIRUD UHFWDQJXODUDSHUWXUH

b 2 x

b 2 a 2



U R xR  yR C ³³ e



i  S f x x  f y y

a 2

dx dy 

 

A

ZKHUHWKHVSDWLDOIUHTXHQFLHV f x DQG f y DUHJLYHQE\(T  DQG



C

U A

k   i ikZ i  Z xR  yR  e Re R OZ R

 

ZKHUHUALVWKHDPSOLWXGHRIWKHILHOGRQWKHDSHUWXUH

4.3.1 Rectangular aperture )RUDUHFWDQJXODUDSHUWXUHZLWKWKHVL]HDQGSRVLWLRQRIWKHD[HVGHILQHGDVLQ )LJ(T  EHFRPHV )LJ ,QWHQVLW\ SURILOH DORQJ DQ D[LV LQ WKH )UDXQKRIHU¶V GLIIUDFWLRQ IURP D UHFWDQJXODU DS HUWXUH

æ sin b ö è b ø

1.0

2

0.4 0.3 0.2 0.008 -4 p

0.017 -3p

0.1

0.047 - 2p

-p

- 3.47 p - 2.46p -1.43p

0

p 1.43p

b 2p

3p

4p

2.46p 3.47p



([DPSOHVRI)UDXQKRIHU¶VGLIIUDFWLRQ



)LJ )UDXQKRIHU GLIIUDFWLRQ SDWWHUQ SUR GXFHG E\ D UHFWDQJXODU DSHUWXUH 7KLV LPDJH LV REWDLQHG IURP D QX PHULFDO VLPXODWLRQ LQ WKH VDPH FRQGLWLRQV DV WKRVH XVHG LQ )LJ   ,WUHSUHVHQWVWKHDQJXODUVSHF WUXPJHQHUDWHGE\DQDSHUWXUHZLWK VLGHV RI  PP u  PP  LOOXPL QDWHG E\ D SODQH ZDYH ZLWK  O  PP 7KH ILJXUH H[WHQGV IRU DERXW PUDGIRUHDFKVLGH



 a  b

U R xR  y R C

³ ³

e



i  S f x x  f y y

dxdy





abC VLQF S f x a VLQF S f y b   

a  b 

7KHOXPLQRXVLQWHQVLW\RIWKHGLIIUDFWLRQSDWWHUQLV 

I xR  yR

IA

a  b VLQF S f x a VLQF S f y b  O  Z R





 

ZKHUHWKHGHSHQGHQFHRQHDFKRIWKHxDQGyFRRUGLQDWHVLVH[SUHVVHGE\WKHIXQF WLRQVLQF E UHSUHVHQWHGLQ)LJ7KHW\SLFDORYHUDOODSSHDUDQFHRIWKHGLIIUDF )LJ *UDSKLFDO GHWHUPL QDWLRQRIWKHPD[L PXPSRVLWLRQ

10 8 6 4 2

tan b 0 2 4 6 8 10

p

p/2

0

p/2

b

p

3p/2

2p

5p/2



'LIIUDFWLRQ  WLRQSDWWHUQLVVKRZQLQ)LJLQSDUWLFXODULWLVVHHQWKDWLWLVIXUURZHGE\D UHJXODU JULG RI QXOO LQWHQVLW\ OLQHV GHILQHG E\ S fx a  rS r S "  DQG E\ S fy b rS r S " 0RUHRYHULWVH[WHQVLRQDQGWKHSDFHRIWKHJULGLQ xDQG y YDULHV inversely proportionalWR aDQG bUHVSHFWLYHO\,QWHUPVRIREVHUYDWLRQGL UHFWLRQ WKH DQJXODU VHSDUDWLRQ EHWZHHQ WKH FHQWHU DQG WKH ILUVW PLQLPXP IRU VPDOODQJOHV LVJLYHQE\ 



O ER a

DR

O  b

 

7KHSRVLWLRQEPRIWKHLQWHQVLW\PD[LPDFDQEHREWDLQHGE\GHULYLQJWKHIXQF WLRQVLQF E IURPZKLFKZHKDYH 

WDQ E P

E P 

 

ZKLFKFDQEHUHVROYHGJUDSKLFDOO\DVLQ)LJ7KHFHQWUDOSHDNFRQWDLQVDERXW RIWKHSRZHUWKDWSDVVHVWKURXJKWKHDSHUWXUHDQGWKHHLJKWVLGHSHDNVWKDW VXUURXQGLWFRQWDLQDERXWRIWKLVSRZHUDOWRJHWKHU

4.3.2 Circular aperture 7RFDOFXODWHWKHGLIIUDFWLRQSDWWHUQIURPDFLUFXODUDSHUWXUHRIUDGLXV aZHFDQ H[SORLWWKHV\PPHWU\RIWKHSUREOHPE\FKRRVLQJDGLUHFWLRQRIREVHUYDWLRQVXFK WKDW xR DQGyR  ZRWDQ DZKHUHDLVWKHDQJOHRIREVHUYDWLRQ)RUDOOSRLQWV xR yR  ZLWK WKH VDPH YLHZLQJ DQJOH WKH UHVXOW ZLOO EH WKH VDPH 7KHUHIRUH ZH FDQ WDNH fx   VR WKDW WKDQNV WR WKH FLUFXODU V\PPHWU\ ZH FDQ DVVXPH IRU fy D VRPHZKDWPRUHSUHFLVHYDOXHWKDQZHJHWZLWK)UHVQHO¶VDSSUR[LPDWLRQWKDWLV 

fy

 VLQ D  O

 

ZKLFK FDQ HDVLO\ EH GHGXFHG IURP )LJ  DQG (TV   ,Q SDUWLFXODU VLQ D yR S UHSODFHV WDQ D yR Z R KRZHYHUDLVQRUPDOO\VPDOODQGWKHGLI y

)LJ *HRPHWU\ IRU WKH GLIIUDFWLRQ IURPDFLUFXODUDSHUWXUH dy

y

x

a

z

ysina 2Ö a2 y2



([DPSOHVRI)UDXQKRIHU¶VGLIIUDFWLRQ



IHUHQFHLVRIWHQQHJOLJLEOH7KHILHOGWKDWLVREVHUYHGDWWKHDQJOHDLVWKHUHIRUH a



U D C

³e

i U ay

 a   y  dy 

a

ZLWK 

U

S

VLQ D a  O

 

1RZH[SORLWLQJRQHRIWKHNQRZQLQWHJUDOUHSUHVHQWDWLRQVRIWKH%HVVHOIXQFWLRQ -ZKHUHE\ 

- U

U S



³ e  i Uu

  u  du 

 



DQGWDNLQJy auZHILQDOO\ILQGWKDW 

U D C Sa 

 - U  U

 

ZKHUHU IXQFWLRQRID LVJLYHQE\(T   7KHOXPLQRXVLQWHQVLW\RIWKHGLIIUDFWLRQSDWWHUQLVQRZJLYHQE\ )LJ ,QWHQVLW\SURILOHDORQJD UDGLDOD[LVRIWKH)UDXQ KRIHU¶V GLIIUDFWLRQ SDW WHUQIURPDFLUFXODUDS HUWXUH :LWK WKH SHDN QRUPDOL]HGWRXQLW\WKH KHLJKW RI WKH ILUVW VHF RQGDU\PD[LPXPLVRQ O\  DQG RQO\ IRUWKHVHFRQG

é2J1 (r)ù ë r û

1.0

2

0.5 0.4 0.3

- 10

-5.14 -5

-3.83

-7.02

-8.42

0.2 0.1 0

5

10

r = 2 pa sin a l 

'LIIUDFWLRQ





)LJ $LU\¶VULQJVZLWKDFLUFXODUDSHU WXUH(YHQWKLVLPDJHLVREWDLQHG E\ D QXPHULFDO VLPXODWLRQ XQGHU WKHVDPHFRQGLWLRQVXVHGIRU)LJ  ,W UHSUHVHQWV WKH DQJXODU VSHFWUXP SURGXFHG E\ D FLUFXODU DSHUWXUH RI UDGLXV  PP LOOXPL QDWHG E\ D SODQH ZDYH ZLWK O  PP 7KH ILJXUH H[WHQGV IRU DERXWPUDGIRUHDFKVLGH

 





§ Sa  · §  - U · IA ¨ ¸  ¸ ¨ © ORo ¹ © U ¹

I D

 

ZKLFK LV FDOOHG$LU\¶VIRUPXOD7KH LQWHQVLW\ GLVWULEXWLRQ LV FKDUDFWHUL]HGE\ WKH IXQFWLRQ> - U [email protected]ZKLFKLVVKRZQLQ)LJQRWLFHLWVUHVHPEODQFHWRIXQF WLRQ VLQF E  7KHGLIIUDFWLRQVKDSHLVQRZFLUFXODUO\V\PPHWULFDO )LJ DQGFRQVLVWVRI DFHQWUDOOLJKWGLVNZKLFKLVFDOOHGWKHAiry’s diskVXUURXQGHGE\FRQFHQWULFULQJV RIUDSLGO\GHFUHDVLQJLQWHQVLW\7KHSRVLWLRQRIWKHILUVWEODFNULQJLVJLYHQE\WKH ILUVW]HURRIWKH%HVVHOIXQFWLRQ-WKDWLVIRUU#7KHDQJXODUUDGLXVRIWKLV ILUVWEODFNULQJLVWKXVJLYHQE\ 

D # VLQ D #

 O O #  S a a



O  D

 

ZKHUH D aLVWKHGLDPHWHURIWKHDSHUWXUH7KHQH[WWZREODFNULQJVRFFXUDWD #ODDQGODDQGWKHVHSDUDWLRQ ' VLQ D EHWZHHQWZRQHLJKERULQJ ULQJVWHQGVDV\PSWRWLFDOO\WROD,IWKHGLIIUDFWLRQSDWWHUQLVREVHUYHGRQWKHIR FDOSODQHRIDOHQVDVLQ)LJWKHUDGLXVRIWKHILUVWEODFNULQJLVJLYHQE\ 

r$

Df



Of  a

 

7KH LQWHJUDWLRQ RI WKH QRUPDOL]HG IXQFWLRQ > - U [email protected]S RQ D FLUFXODU DUHD ZLWKUDGLXVURKDVDVDUHVXOWDIRUPXODFUHGLWHGWR5D\OHLJK

([DPSOHVRI)UDXQKRIHU¶VGLIIUDFWLRQ



)LJ 5HFWLOLQHDUVOLW

x xo a

Zo

z

 

L UR   -  UR  - UR 

 

ZKLFKUHSUHVHQWVWKHIUDFWLRQRIHQHUJ\WKDWIORZVZLWKLQWKHUDGLXVURDQGZKHUH - LV WKH %HVVHO IXQFWLRQ RI WKH ILUVW NLQG RI RUGHU  7KXV WKH HQHUJ\ IORZLQJ WKURXJKWKH$LU\¶VGLVNLVDERXWRIWKHWRWDOIORZZKLOHWKDWRIWKHILUVWULQJ LVDERXWDQGWKDWRIWKHVHFRQGULQJLVDERXWRIWKHWRWDOIORZ

4.3.3 Single slit ,QWKHFDVHRIDUHFWDQJXODURSHQLQJKDYLQJVLGHVaDQGbWKDWDUHYHU\GLIIHUHQW IURPHDFKRWKHUVXFKDV b!! a )LJ LW PD\EH SRVVLEOHWKDWIRUDJLYHQ GLVWDQFH ZRRIREVHUYDWLRQ)UDXQKRIHU¶VFRQGLWLRQLVRQO\UHVSHFWHGLQRQHGLUHF WLRQVD\ xSDUDOOHOWRWKHVKRUWHUVLGH a,QVWHDGLQWKHGLUHFWLRQ ySDUDOOHOWRWKH ORQJHUVLGHbWKHIULQJHVPD\EHSRRUO\GHYHORSHGDQGWKHOLJKWWKDWDUULYHVRQWKH REVHUYDWLRQSODQHFDQVWLOODSSHDUDVWKHJHRPHWULFSURMHFWLRQRIWKHRSHQLQJ8Q GHUWKHVHFRQGLWLRQVWKHSUREOHPFDQEHWUHDWHGLQDRQHGLPHQVLRQDOZD\:LWKD IHZFDOFXODWLRQVZKLFK,GRQRWVKRZKHUHIRUEUHYLW\IRUDSODQHZDYHSHUSHQ GLFXODUO\ LQFLGHQW WR WKH DSHUWXUH A WKH ILHOG REVHUYHG DW WKH DQJOH D IURP WKH QRUPDOWR$LVZLWKJRRGDSSUR[LPDWLRQJLYHQE\ )LJ 'RXEOHVOLW

a a

d a

dsin a 

'LIIUDFWLRQ



 

U D

S kZ U a i i R § Sa · i A e  FRV D VLQF ¨ VLQ D ¸  OZ R © O ¹

 

7KHFRUUHVSRQGLQJLQWHQVLW\LV

I D



IA

a § Sa · VLQF ¨ VLQ D ¸  OZ R © O ¹

 

$Q H[DPSOH RI GLIIUDFWLRQ SDWWHUQ SURGXFHG E\ WKLV W\SH RI DSHUWXUH LV JLYHQ LQ )LJ D 

4.3.4 Double slit 7KLVKLVWRULFDOO\LPSRUWDQWFDVHZDVSURSRVHGE\2VWHUEHUJ DQG 6PLWK6KHSSDUGDQG+U\QHY\FK@



I z

ª § krA  z z ¨ FRV I A «      ¨ « z  rA  z   rA  © z  z  rA ¬

·º ¸ »  ¸ ¹ ¼»

ZKHUHzLVWKHGLVWDQFHIURPWKHRSHQLQJDQGrALVLWVUDGLXV)URPWKLVH[SUHVVLRQ LWIROORZVWKDWIRU z oWKHDPSOLWXGHRIWKHLQWHQVLW\RVFLOODWLRQRQWKHD[LVGH FD\VWR]HURZKLOHIRU z!! rALWFDQEHDSSUR[LPDWHGE\(T  FDOFXODWHG ZLWKLQWKHOLPLWRIDVRXUFHDWLQILQLW\

4.4.3 Poisson’s paradox $Q DVWRQLVKLQJ GLIIUDFWLRQ HIIHFW FDQ EH REVHUYHG ZLWK FLUFXODU REVWDFOHV DQG WKHVWRU\RILWVGLVFRYHU\LVTXLWHLQWHUHVWLQJ:KHQLQ)UHVQHOSUHVHQWHGKLV WKHRU\RIGLIIUDFWLRQWRDFRPSHWLWLRQRUJDQL]HGE\WKH)UHQFK$FDGHP\WKHMXU\ LQFOXGHGDPRQJRWKHUHPLQHQWVFLHQWLVWVRIWKHWLPH6'3RLVVRQZKRVXSSRUWHG WKH HPLVVLRQ FRUSXVFXODU  WKHRU\ )URP )UHVQHO¶V WKHRU\ 3RLVVRQ GHGXFHG D UH PDUNDEOHDQGVHHPLQJO\DEVXUGFRQFOXVLRQKHVKRZHGWKDWLQWKHPLGGOHRIWKH VKDGRZSURMHFWHGE\DQRSDTXHGLVNRQHZRXOGKDYHWRREVHUYHDUDWKHUEULOOLDQW VSRW RI OLJKW $W WKLV REMHFWLRQ )UHVQHO DQG $UDJR ZKR ZDV DQRWKHU PHPEHU RI WKHMXU\UHDGLO\UHVSRQGHGZLWKDQH[SHULPHQWDOWHVWWKHVSRWRIOLJKWZDVDFWXDOO\ SUHVHQW )UHVQHO¶V ZRUN ZRQ WKH SUL]H DQG WKH WLWOH RI Mémoire Couronné 7KLV

'LIIUDFWLRQ





)LJ 9DULDEOHV XVHG IRU WKH 3RLVVRQSDUDGR[

J r

R

r

a

Zo

P

SUHYLVLRQRI)UHVQHO¶VWKHRU\PDGHDVWURQJLPSUHVVLRQLQWKHVFLHQWLILFZRUOGDW WKHWLPHDQGZDVRQHRIWKHGHFLGLQJIDFWRUVLQIDYRURIWKHZDYHWKHRU\RIOLJKW 7KH3RLVVRQ$UDJRVSRWDVLWLVQRZFDOOHGKDGEHHQREVHUYHGPDQ\\HDUVEHIRUH  E\*LDFRPR)LOLSSR0DUDOGL  EXWKLVZRUNKDGEHHQIRUJRWWHQ :H FDQ UHIRUPXODWH 3RLVVRQ¶V UHDVRQLQJ DV IROORZV 2QFH D VRXUFH 6 DQG DQ REVHUYDWLRQSRLQW3DUHIL[HGRQWKHD[LVRIDFLUFXODUREVWDFOHZHFDQWUDFHDVEH IRUHDVHULHVRI)UHVQHO]RQHVZKRVHHGJHVPHHWWKHUHODWLRQVKLS O 643 64 A 3  m  m   "  



 

ZKHUH4LVDSRLQWRQWKHRXWHUHGJHRIWKH]RQH mFRQVLGHUHGDQG4ALVDSRLQW RQ WKH HGJH RI WKH REVWDFOH :H FDQ WKHQ UHDSSO\ WKH UHDVRQLQJ WKDW OHG WR (T  WKHILHOGLQ3LVKDOIRIWKHFRQWULEXWLRQRIWKH)UHVQHO¶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

R

Z R  r   rdr

)LJ ,QWHQVLW\ LQ WKH FHQWHU RI WKH 3RLVVRQ$UDJR VSRW YHUVXV WKH GLVWDQFH IURP WKH REVWDFOH IRU D SODQH ZDYH LQFLGHQW ZLWK LQWHQ VLW\ I$ 7KH REVWDFOH LV FLUFXODUZLWKUDGLXVa

RdR  FRV -

Z R  a  

I IA

1

0

ZR  U R

0

a

Z o

([DPSOHVRI)UHVQHO¶VGLIIUDFWLRQ



)LJ 'LIIUDFWLRQ ILJXUH SURGXFHG E\ D FLUFXODU REVWDFOH 1XPHULFDO VLPXODWLRQ IRU D  PP UDGLXV REVWDFOHRQWKHZDLVWSODQHRID *DXVVLDQ LQFLGHQW EHDP ZLWK w PPDQGO PP7KHILJ XUH LV  PP ZLGH DQG LV SUR GXFHG RQ D VFUHHQ SODFHG DW  FPIURPWKHREVWDFOH*UD\VFDOH UHSUHVHQWDWLRQ LV OLQHDU ZLWK LQ WHQVLW\





VRWKDW 

U 3

i  UA O

f  S ikR e

³ ³

U 

ZR S R dR d I i Z RUA R R O

f ikR e

³

U

R

dR 

ZKHUH UALVWKHILHOGDPSOLWXGHRQWKHSODQHRIWKHREVWDFOH7KHUHPDLQLQJLQWH JUDOFDQEHH[SUHVVHGWKURXJKWKHVLQHDQGFRVLQHLQWHJUDOIXQFWLRQV 6L x  &L x  >$EUDPRZLW]DQG6WHJXQ@ 

U P i

S Z RUA O

Z RUA ikU Sº ª  «¬&L kU  i 6L kU  i  »¼ # i U e

ZKHUHWKHODVWHTXDOLW\LVYDOLGIRU kU!!7KHUHIRUHWKHLQWHQVLW\REVHUYHGLQ3 LV )LJ  

I 3

IA

Z R  Z R  a 

 

ZKHUHIALVWKHLQWHQVLW\RIWKHZDYHRQWKHSODQHRIWKHREVWDFOH)RUZRofWKLV HTXDWLRQJLYHVI 3  oIRWKLVLVGXHWRKDYLQJWDNHQDSODQHZDYHDVWKHLQFLGHQW ILHOG ,QRUGHUIRUWKH3RLVVRQ$UDJRVSRWWREHYLVLEOHWKHGLVNLUUHJXODULWLHVVKRXOG EHOHVVWKDQWKHZLGWKRILWVILUVW)UHVQHO]RQH,QDGGLWLRQDSSURDFKLQJWKHREVWD FOH WKH VSRW EHFRPHV VPDOOHU DQG VPDOOHU GRZQ WR WKH VL]H RI WKH ZDYHOHQJWK ZKHQ WKH GLVWDQFH GURSV EHORZ a $ VLPXODWLRQ RI WKH GLIIUDFWLRQ SURGXFHG E\ D VPDOOVSKHUHLVVKRZQLQ)LJ

'LIIUDFWLRQ





4.4.4 Zone plate ,I DQ RSHQLQJ LV FRQVWUXFWHG VR DV WR REVWUXFW ZLWK UHVSHFW WR D SRLQW 3 WKH )UHVQHO]RQHVLQDQDOWHUQDWLQJPDQQHUVXFKDVWKRVHWKDWDUHHYHQWKHQDOORIWKH UHPDLQLQJ RQHV JLYH D FRQWULEXWLRQ LQ SKDVH EHWZHHQ WKHP DQG (T   EH FRPHV 

U3

U  U   U   " 

U 3 DQGWKHUHIRUHDOVRWKHLQWHQVLW\LQ3EHFRPHV PXFKODUJHUWKDQLQWKHDE VHQFHRIWKLVNLQGRIDSHUWXUH7KLVLVFDOOHGD zone plateDQGLWDFWVLQDPDQQHU VLPLODUWRDOHQVLQIDFW(T  FDQEHUHZULWWHQDV

mO  rm 

   Z R ZV



 

ZKLFKKDVWKHVDPHVKDSHDVWKH*DXVV¶V(T D ZKHUHWKHYDULDEOHVsoDQG siKDYHEHHQUHSODFHGZLWK ZVDQG ZRUHVSHFWLYHO\7KXVD]RQHSODWHKDVDIRFDO OHQJWKFKDUDFWHUL]HGE\ 

f

rm   mO

 

IRUZKLFKWKHUDGLDWLRQHPLWWHGE\DVRXUFHSODFHGRQWKHD[LVRIWKHODPLQDDWD GLVWDQFH ZVLVUHIRFXVHGDWDGLVWDQFH ZRFDOFXODEOHIURP(T  7KLV³OHQV´ KRZHYHUKDVDVWURQJFKURPDWLFDEHUUDWLRQIURPWKHGHSHQGHQFHRI fRQ O,QDG GLWLRQXQOLNHQRUPDOOHQVHVWKHODPLQDKDVDPXOWLWXGHRIIRFDOGLVWDQFHV r f r f  r f VRWKDWLQDGGLWLRQWRWKHPDLQIRFXVWKHOLJKWLVDOVRIRFXVHGWRD OHVVHUH[WHQWRQRWKHUSRLQWVIRUZKLFKWKHWUDQVSDUHQWULQJVRIWKHODPLQDFRUUH VSRQGWRDQRGGQXPEHURIDGMDFHQW)UHVQHO¶V]RQHV$ORQJZLWKWKHFRQYHUJLQJ GLIIUDFWHG ZDYHV WKHUH DUH DOVR GLYHUJLQJ ZDYHV WKDW FRUUHVSRQG WR WKH QHJDWLYH IRFDOOHQJWKV

4.4.5 Rectangular aperture /HWXVVXSSRVHWKDWDUHFWDQJXODUDSHUWXUHZLWKWKHVLGHVRIDPSOLWXGH aDQG b DVVLJQHG DV LQ )LJ  LV LOOXPLQDWHG ZLWK QRUPDO LQFLGHQFH E\ D SODQH ZDYH )URP(T  RQHJHW 

U R xR  y R C

a 

b

³

³

 a  b 

i

e

k ª xR  x   yR  y  º»¼  Z R «¬ dx dy 

 

([DPSOHVRI)UHVQHO¶VGLIIUDFWLRQ



S (l )

)LJ &RUQX¶VVSLUDOWKLVFXUYH LVSORWWHGWDNLQJDVFRRU GLQDWHV WKH )UHVQHO LQWH JUDOV E\ YDU\LQJ WKH SD UDPHWHUlZKLFKLQWXUQ PHDVXUHV WKH SDWK WDNHQ RQ WKH VDPH FXUYH IURP WKHRULJLQLLVWKHOHQJWK RIDYHFWRUWKDWMRLQVWZR SRLQWVRIWKHFXUYH

1.6

1.2 ¥

0.5 L

2.0

0

-0.4 -0.8

l

0.4

-0.5

0.8

0.5

C(l )

-2.0 -0.5

-¥ -1.2

-1.6



ZKHUHCLVQRZJLYHQE\ 

C

i

eikZ R UA  OZ R

ZKHUHLQDGGLWLRQ UALVWKHDPSOLWXGHRIWKHILHOGRQWKHDSHUWXUH(T   FDQEHVLPSOLILHGE\WUDQVODWLQJDQGVFDOLQJWKHYDULDEOHVxDQGy 

[

k x  xR  K SZR

k y  yR  S ZR

 

ZLWKDFRUUHVSRQGLQJFKDQJHLQWKHLQWHJUDWLRQOLPLWV [





k S ZR

§a · ¨  xR ¸  K © ¹

k S ZR

[



§a · ¨  xR ¸  K © ¹

k S ZR

§b · ¨  yR ¸  © ¹

k S ZR

§b · ¨  yR ¸  © ¹

IURPZKLFKRQHJHWV [

U xR  yR C



S

K

S

 i K S Z R  i [  d[ e e ³  dK k [³ K 





 

'LIIUDFWLRQ





 )LJ   )UHVQHO GLIIUDFWLRQ SDWWHUQV IRU UHFWDQJXODU DSHUWXUHV1XPHULFDO VLPXODWLRQ IRU DQ LQFLGHQW SODQH ZDYH ZLWK O   PP 7KH QXPEHU LQGLFDWHV WKH DSHUWXUH KHLJKW LQ PL FURQV WKH ZLGWK RI HDFK DSHUWXUH LV  WLPHV WKH KHLJKW 7KH ILJXUHV DUH SURGXFHG RQ D VFUHHQSODFHGDWFPIURPWKHDSHUWXUHV7KHJUD\VFDOHUHSUHVHQWDWLRQLVOLQHDUZLWKLQWHQ VLW\

IURPZKLFK 

U xR  yR

i

eikZ R UA  u

^ ª¬& [  & [ º¼  i ª¬6 [  6 [ º¼ `    ^ ª¬& K  & K º¼  i ª¬6 K  6 K º¼ `

ZKHUHWKHIXQFWLRQV&DQG6DUH)UHVQHO¶VLQWHJUDO VHHDSSHQGL[%DWWKHHQGRI WKLVERRN 7KHFRUUHVSRQGLQJGLVWULEXWLRQRILQWHQVLW\LV I xR  y R



^ ¬ª& [  & [ ¼º  ¬ª6 [  6 [ ¼º `  u^ ª¬& K  & K º¼  ª¬6 K  6 K º¼ `

IA 























 



DQGDVXVXDO IALVWKHLQWHQVLW\RQWKHRSHQLQJ7KHEHKDYLRURI)UHVQHOLQWHJUDOV FDQ EH LOOXVWUDWHG ZLWK D JHRPHWULF UHSUHVHQWDWLRQ LQWURGXFHG E\ $ &RUQX C l  DQGS l DUHWDNHQDVWKH&DUWHVLDQFRRUGLQDWHVRIDSRLQWRIDSDUDPHWULFFXUYHDV WKHSDUDPHWHUlFKDQJHVIURP±fWRfDGRXEOHVSLUDOLVIRUPHGFDOOHG&RUQX¶V VSLUDO )LJ   6LQFH C    S     WKH FXUYH JRHV WKURXJK WKH RULJLQ DQG VLQFH 

& l

& l  6 l

6 l  

LWLVDQWLV\PPHWULFZLWKUHVSHFWWRERWKD[HV3DUDPHWHUlKDVWKHLQWHUHVWLQJSURS HUW\RIEHLQJHTXDOWRWKHOHQJWKRIWKHFXUYHPHDVXUHGIURPWKHRULJLQ,QGHHGLI

([DPSOHVRI)UHVQHO¶VGLIIUDFWLRQ



1.5

)LJ 'LIIUDFWLRQIULQJHVSUR GXFHG E\ D UHFWLOLQHDU HGJH

geometric shadow edge ®

1.0

I IA 0.5

0.0 4

8

0

h

4

8



dsLVDFXUYHDUFHOHPHQWZHKDYH 

ds  d&   d6 

ª §S ·   § S  ·º «FRV ¨  l ¸  VLQ ¨  l ¸ » dl © ¹ © ¹¼ ¬

dl  

7KHLQWHQVLW\FDQWKXVEHHYDOXDWHGDVWKHSURGXFW 

I xR  yR

IA  L [  [ L K  K  

 

ZKHUHWKHIXQFWLRQ L [ [  LVWKHOHQJWKRIWKHYHFWRU LMRLQLQJWZRSRLQWVWKDW DUH IRXQG UHVSHFWLYHO\ JRLQJ DORQJ WKH &RUQX¶V VSLUDO IRU WKH WZR GLVWDQFHV [ DQG[  IURPWKHRULJLQ /HWXVFRQVLGHUIRUH[DPSOHWKHFDVHRIVFDQQLQJWKHFRRUGLQDWHxRNHHSLQJyR )RUxR  LKDVLWVWZRH[WUHPHVDWWZRGLDPHWULFDOO\RSSRVLWHSRLQWVRQWKH VSLUDO$V xRJURZV [DQG [  FKDQJHOLQHDUO\ZLWK xREXWZKLOHRQHRIWKHWZR SRLQWV $ ZUDSV DZD\ IURP WKH RULJLQ WKH RWKHU % XQUROOV DSSURDFKLQJ WKH RULJLQ$VDUHVXOWWKHREVHUYHGLQWHQVLW\LVRVFLOODWLQJZLWKLQFUHDVLQJDPSOLWXGH )LQDOO\ZKHQ xRUHDFKHVWKHJHRPHWULFDOVKDGRZHGJHLHIRU xR  aSRLQW% FRPHVWRWKHRULJLQ)URPWKDWSRLQWRQJURZLQJxRDJDLQERWKSRLQWVZUDSWKHP VHOYHVLQWKHVDPHGLUHFWLRQDZD\IURPWKHRULJLQ:KHQZRLVVPDOOWKHSRLQW$LV DOUHDG\ YHU\ LQWHUQDO WR WKH VSLUDO IRU ZKLFK WKH LQWHQVLW\ GHFD\V PRQRWRQLFDOO\ :KHQLQVWHDG ZRLVODUJHLHFORVHWRRULPPHUVHGLQWKH)UDXQKRIHUUHJLRQWKH LQWHQVLW\GHFD\VRVFLOODWLQJ ,QSDUWLFXODUIRUYHU\VPDOOYDOXHVRI ZRFRPSDUHGWRWKHVL]HRIWKHDSHUWXUH DQGIRUSRLQWVRIREVHUYDWLRQLQWKHJHRPHWULFSURMHFWLRQDUHDRIWKHDSHUWXUHWKH WZRSRLQWV$DQG%DUHERWKYHU\FORVHWRRQHDQGWKHRWKHURIWKHWZRRSSRVLQJ FHQWHUV ±ò±ò  DQG òò UHVSHFWLYHO\DURXQGZKLFKWKHVSLUDOZUDSVZKHUHE\ L|  VRWKDWI|IA,QVWHDGLIWKHREVHUYDWLRQSRLQWLVLQWKHJHRPHWULFVKDGRZ

'LIIUDFWLRQ





 )LJ)LJXUHRIGLIIUDFWLRQIURPDKDOISODQHVFUHHQ6LPXODWLRQIRUDQLQFLGHQWSODQH ZDYHZLWKO PPDQGDQREVHUYDWLRQVFUHHQFPDZD\7KHILJXUHKDVGLPHQVLRQV PP[PP/LQHDUUHSUHVHQWDWLRQRILQWHQVLW\LQWKHJUD\VFDOH

RIWKHVFUHHQSRLQWV$DQG%DUHFORVHWRHDFKRWKHUDQGI |7KHUHIRUHWKHUHLV DJRRGDJUHHPHQWZLWKZKDWLVH[SHFWHGLQVSLWHRIWKHODUJHDSSUR[LPDWLRQVPDGH DVLQWKLVFDVH RQWKHREOLTXLW\IDFWRUDQGRQWKHGLVWDQFHV6RPHH[DPSOHVRI )UHVQHO¶VGLIIUDFWLRQRQUHFWDQJXODUDSHUWXUHVDUHJLYHQLQ)LJ

4.4.6 Slit and straight edge $)UHVQHOGLIIUDFWLRQIURPDORQJVOLWFDQEHWUHDWHGDVLQWKHFDVHRIDUHFWDQJX ODURSHQLQJWDNLQJ[ ±fDQG[ fLQ(T  ZHJHW 

U xR  yR

eikZ R UA ª¬& K  & K º¼  i ª¬6 K  6 K º¼     i

^

`

7KHUHFWLOLQHDUHGJHLVLQWXUQDOLPLWFDVHRIWKHVOLWWDNLQJ [ ± fVRDV VXPLQJa ZHKDYH 

U yR

eikZ R UA  i

­° ª  §  ·º ª  §  · º ½° yR ¸ »  i «  6 ¨ yR ¸ » ¾    ® «  & ¨¨ ¸ ¨ OR ¸ R © ORR ¹ »¼ «¬  © ¹ »¼ °¿ °¯ «¬ 

$GUDZLQJRIKRZWKHLQWHQVLW\YDULHVDVDIXQFWLRQRI K

 OZ R yR LVJLYHQ

LQ)LJDQGDQLPDJHRIWKLVW\SHRIGLIIUDFWLRQLVVKRZQLQ)LJ

%LEOLRJUDSKLFDOUHIHUHQFHV $UQDXG-$Nonorthogonal optical waveguides and resonators7KH%HOO6\VWHPV7HFKQL FDO-RXUQDO 1RYHPEHU  $EUDPRZLW]0DQG6WHJXQ,$Handbook of Mathematical Functions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± DQG+XPSKU\'DY\  VXFFHHGHGLQUHSURGXFLQJLPDJHV RYHUOD\HUVRIVLOYHUQLWUDWHEHWZHHQDQG+RZHYHUWKHVHLPDJHVZHUH QRWIL[HGDQGDIWHUDVKRUWWLPHWKH\ZHUHORVWEHFDXVHWKHOLJKWGLIIXVHGIURPWKH HQYLURQPHQWSURGXFHGDXQLIRUPEODFNHQLQJ6LQFHVFLHQWLVWVGRQRWXVXDOO\KDYH JUHDW SUDFWLFDO VHQVH 'DY\ DEDQGRQHG WKHVH LQWHQWV ,QVWHDG DQ ROG SDVVLRQDWH OLWKRJUDSKHU -RVHSK 1LFpSKRUH 1LHSFH   XVHG -XGHD¶V ELWXPHQ WR SURGXFHVWDEOHLPDJHVLQUHOLHIDQGDSSOLHGWKLVPHWKRGWR]LQFRJUDSK\LQ KHVXFFHHGHGLQREWDLQLQJLPDJHVE\PHDQVRIWKHGDUNURRPFDSWXULQJSRVHVRYHU DSHULRGRIKRXUVLQIXOOVXQ+HXVHGDSODWHRIVLOYHUSODWHGFRSSHUFRDWHGZLWK D ELWXPHQ OD\HU ([SRVHG WR OLJKW WKH ELWXPHQ SRO\PHUL]HV DQG WKH QRQ LOOXPLQDWHG SDUWV DUH UHPRYHG LQ D EDWK RI SHWUROHXP DQG ODYHQGHU HVVHQFH 7KH OLJKWLVJLYHQE\WKHUHPDLQLQJELWXPHQZKLOHWKHGDUNSRUWLRQVDUHUHIOHFWHGE\ WKH VLOYHU SODWH $IWHU OHDUQLQJ WKLV WHFKQLTXH LQ  /RXLV -DFTXHV 0DQGp 'DJXHUUH ± ZKRZDVDSDLQWHURIWKHDWULFDOVFHQHU\HQWHUHGLQWRSDUW QHUVKLSZLWK1LHSFHLQ'DJXHUUHXVHGVLOYHUFRDWHGFRSSHUSODWHVH[SRVHGWR LRGLQH YDSRU LQ GDUNQHVV WKDW IRUPHG D WKLQ OD\HU RI SXUSOH LRGLGH VLOYHU RQ WKH VODE([SRVHGWROLJKWLQDGDUNURRPWKLVOD\HUFRXOGIRUPLPDJHVZLWKH[SRVXUHV RIKRXUVEXW'DJXHUUHZDVDEOHWRUHGXFHWKHH[SRVXUHWLPHVGUDVWLFDOO\GRZQWR  RU  PLQXWHV ZLWK WKH GLVFRYHU\ RI WKH latent image ZKLFK ZDV GHYHORSHG E\ WUHDWLQJWKHSODWHZLWKPHUFXU\YDSRU:LWKWZRVXEVHTXHQWZDVKLQJVLQDNLWFKHQ VDOWVROXWLRQWKHXQH[SRVHGVLOYHULRGLGHZDVUHPRYHGIL[LQJWKHLPDJH,Q DIWHU D SURSRVDO E\ $UDJR WKH )UHQFK JRYHUQPHQW DFTXLUHG WKH PDQXIDFWXULQJ ULJKWVWRWKHdaguerreotypeDQG$UDJRKLPVHOIPDGHWKHILUVWSXEOLFH[SODQDWLRQ

 +LVWRULFDOQRWHV)URP)RXULHUWR)L]HDX  RILWLQIURQWRIWKHJDWKHUHGPHPEHUVRIWKH$FDGHPLHVRI6FLHQFHVDQG)LQH$UWV 7KHGDJXHUUHRW\SHZDVDQLPDJHZLWKJUHDWGHILQLWLRQDQGZDVJUHDWO\VXFFHVVIXO EXWLWKDGWKHGHIHFWRIEHLQJDXQLTXHXQUHSURGXFLEOHSLHFH,Q-RKQ+HU VFKHO GLVFRYHUHG WKDW VLOYHU KDOLGHV GLVVROYH LQ D VRGLXP WKLRVXOSKLGH EDWK 7KLV VXEVWDQFHZDVWKHQXVHGLQSODFHRIVDOWWRIL[WKHLPDJHV ,PSRUWDQWSURJUHVVKDGDOVREHHQPDGHLQGHSHQGHQWO\E\WKH(QJOLVKSK\VLFLVW :LOOLDP+HQU\)R[7DOERW  ,QKHZDVDEOHWRIL[LPDJHVRQD VKHHWRISDSHUEUXVKHGZLWKDVLOYHUQLWUDWHVROXWLRQ+RZHYHUWKHTXDOLW\RIWKHVH LPDJHV ZDV QRW JRRG DQG KH GLVFRQWLQXHG WKH ZRUN XQWLO KH KHDUG WKH QHZV RI $UDJR¶VUHSRUW+HWKHQUHVXPHGKLVLQYHVWLJDWLRQVDQGLQKHILOHGDSDWHQW GLIIXVLQJKLVSURFHVVZKLFKKHFDOOHGphotographyWKURXJKRXW)UDQFHZLWKDOHW WHUDGGUHVVHGWRWKHAcadémie des sciences,WFRQVLVWHGRIWZRVWHSV,QWKHILUVW 7DOERWZRXOGH[SRVHDVKHHWRIWUDQVOXFHQWSDSHULPSUHJQDWHGZLWKVLOYHULRGLGH LQ WKH LPDJH SODQH RI WKHREMHFWLYH RI D GDUNURRP 2Q WKLV VKHHW D ODWHQW LPDJH ZRXOGIRUPZKLFKKHWKHQIL[HGLQDEDWKRIVRGLXPWKLRVXOIDWHREWDLQLQJDQHJD WLYHLPDJHDcalotype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±  7KXV WKH SKRWRJUDSK EHFDPH YHU\ VKDUS GHILQLWLYHO\ VXSSODQWLQJ WKH GDJXHUUHRW\SH ,Q D VKRUW WLPH SKRWRJ UDSK\ KDG EHFRPH D FRPPRQ SUDFWLFH DQG VLQFH WKHQ LW KDV EHHQ LQFUHDVLQJO\ SHUIHFWHG,QSDUDOOHOWRWKHSURJUHVVRIFKHPLVWU\WKHUHZDVDVWURQJSXVKWRLP SURYHWKHTXDOLW\RIWKHOHQVHVDQGEHJLQWKHGHVLJQRIVRSKLVWLFDWHGREMHFWLYHVWR SURGXFHVWURQJQXPHULFDODSHUWXUHVVRDVWRUHGXFHH[SRVXUHWLPHV

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reflex cameraZKRSKRWRJUDSKHGDVWDSOHPDGHZLWKDFRORU IXO 6FRWWLVK IDEULF ZLWK VHSDUDWH H[SRVXUHV WKURXJK FRORUHG ILOWHUV RQH UHG RQH EOXH DQG RQH JUHHQ 7KH UHVXOW ZDV LPSHUIHFW DOWKRXJK DSSDUHQWO\ HIIHFWLYH DV WKHILOPZDVLQVHQVLWLYHWRUHGEXWVHQVLWLYHWRXOWUDYLROHWZKLFKWKHUHGIDEULFUH IOHFWHGDQGWRZKLFKWKHUHGILOWHUZDVWUDQVSDUHQW :H PXVW DOVR UHPHPEHU +HUPDQQ /XGZLJ )HUGLQDQG YRQ +HOPKROW]   ZKRZDVDQLPSRUWDQW*HUPDQSK\VLFLDQDQGSK\VLFLVWWRZKRPZHDOVR RZHWKHIRUPXODWLRQRIWKHSULQFLSOHRIFRQVHUYDWLRQRIHQHUJ\DQGWKHFRQFHSWRI IUHHHQHUJ\LQWKHUPRG\QDPLFV$VDSK\VLFLDQKHDWWHQGHGWRWKHSK\VLRORJ\RI WKHH\HDQGLQLQYHQWHGWKHophthalmoscopeZLWKZKLFKKHVWXGLHGWKHUHWL QD DQG LWV QHUYRXV V\VWHP KH DOVR LQYHQWHG WKH ophthalmometer ZLWK ZKLFK KH VWXGLHGWKHFU\VWDOOLQHFXUYDWXUHDQGDFFRPPRGDWLRQSUREOHPV,QKHSXE OLVKHG D ERRNRQ WKHVH WRSLFVHandbuch der Physiologischen OptikZKLFK ZDV RYHUYDULRXVHGLWLRQVWKHUHIHUHQFHWH[WIRUSK\VLRORJLFDORSWLFVIRUPDQ\GHFDGHV

6XJJHVWHGUHDGLQJV 7KHKLVWRULFDOGDWDLQWKHVHQRWHVDUHGHULYHGIURPWKHIROORZLQJWH[WV 9DULRXVDXWKRUV(QFLFORSHGLDGHOODVFLHQ]DHGHOODWHFQLFD0RQGDGRULHG0LODQR   LQWKHfotografiaHQWU\ %RWWD]]LQL8I “politecnici” francesi, e La “moderna analisi”; E. Bellone, L’esposizione del sistema del mondo,Q6WRULDGHOOD6FLHQ]D9ROGLUHFWHGE\35RVVL*UXSSR HGO¶(VSUHVVR,VWLWXWR*HRJUDILFR'H$JRVWLQLVSD   %RUQ0DQG:ROI(Principles of Optics3HUJDPRQ3UHVV3DULV   %UHZVWHU'Memoir of the life of M. le Chevalier Fraunhofer7KH(GLQEXUJK-RXUQDORI 6FLHQFH9RO9,,  $FFHVVLEOHDWKWWSERRNVJRRJOHFRP  'H 3D] $ L’occhio della modernità. Pittura e fotografia dalle origini alle avanguardie storiche&RRSHUDWLYD/LEUDULD8QLYHUVLWDULD(G%RORJQD   )L]HDX+Sur les hypothèses relatives à l’éther lumineux, et sur une expérience qui paraît démontrer que le mouvement des corps change la vitesse avec laquelle la lumière se propage dans leur intérieur&RPSWHUHQGXV  $QQDOHVGH&KLPLH HWGH3K\VLTXH   )UDXQKRIHU-Bestimmung des Brechungs- und Farbenzer-streuungs-Vermögens verschiedener Glasarten, in bezug auf die Vervollkommnung achromatischer Fernröhre 'HQNVFKULIWHQGHU.|QLJO$NDGHPLHGHU:LVVHQVFKDIWHQ]X0QFKHQIUGLH-DKUH XQG  Neue modification des Lichtes durch gegenseitige Einwirkung und Beugung der Strahlen, und Gesetze derselben 'HQNVFKULIWHQ GHU .|QLJO $NDGHPLH GHU :LVVHQVFKDIWHQ ]X 0QFKHQ IU  XQG   

6XJJHVWHGUHDGLQJV



   )UHUFNV-Fizeau’s research program on ether drag: a long quest for a publishable experiment3K\VLFVLQ3HUVSHFWLYH   *UDVVPDQQ+*Theory of Compound Colors3KLORVRSKLFDO0DJD]LQH   7UDQVODWHGIURP$QQDOHQGHU3K\VLNXQG&KHPLH   0D[ZHOO-&OHUNExperiments on colours, as perceived by the eye7UDQV5R\6RF(GLP EXUJKSDUW,,  On colour vision1DWXUH   3HUX]]L*Maxwell, dai campi elettromagnetici ai costituenti ultimi della materiaVHULHV ³,JUDQGLGHOODVFLHQ]D´DQQR,Q1RYHPEHU/H6FLHQ]H6S$HG   3URYHUELR(The production of achromatic objectives in the first half of the nineteenth century: the contribution of Giovanni Battista Amici 0HP 6RF $VWURQRPLFD ,WDOLDQD   1$6$$VWURSK\VLFV'DWD6\VWHP 6HJUq( From falling bodies to radio waves:+)UHHPDQDQG&RPSDQ\HG1HZ g h @ x >h g @ x ³ g [ h x  [ d [ ³ h [ g x  [ d [    f

f

,QVWHDGWKHFRUUHODWLRQRIgDQGhIXQFWLRQVis notFRPPXWDWLYHDQGPD\EHLQGL FDWHGE\WKHV\PEROEXWLVXVXDOO\H[SOLFLWO\LQGLFDWHGE\DQLQWHJUDO f

 > gh @ x

³

f

f

g [ h x  [ d [

³ g  x  [ h [ d [ > gr [email protected] x    

f

ZLWK gr x  g x  ,Q ERWK FDVHV WKH QRWDWLRQ >g«[email protected] x  LV SUHIHUDEOH WR g x «h x ZKHUH«VWDQGVIRU RU HVSHFLDOO\ZKHQWKHYDULDEOHxLVUHSODFHG E\DUHVFDOLQJVXFKDVaxDQGHYHQZRUVHLIWKHUHVFDOLQJLVGLIIHUHQWIRUgDQGh ,QGHHGLQ(TV  xUHSUHVHQWVWKHWUDQVODWLRQWREHDSSOLHGEHWZHHQWKH WZRIXQFWLRQVDQGWKHLQWHJUDWLRQYDULDEOHLVFRPPHQVXUDEOHZLWKx,QWKHFDVHRI D VFDOH FKDQJH LW LV DPELJXRXV DV WRZKHWKHU RQH KDV WR LQWHJUDWH LQ [ RU LQ a[ 7KHEHVWFKRLFHZRXOGEHWRXVHDX[LOLDU\IXQFWLRQVZKHUHWKHVFDOLQJLVHPEHGGHG LQWKHIXQFWLRQLWVHOI+HUH,ZLOOXVHIRUEUHYLW\WKHQRWDWLRQaāxWRLQGLFDWHWKDW WKHFRHIILFLHQWaLVDQLQWHJUDOSDUWRIWKHIXQFWLRQWKDWDFWVRQx,QDQ\FDVHZKHQ >g«[email protected] ax LVZULWWHQLWVKRXOGEHXQGHUVWRRGWKDWWKHYDULDEOHRILQWHJUDWLRQKDV WKHVDPHGLPHQVLRQVDVaxDQGWKHUHIRUHLILQ(TV  xLVUHSODFHGE\ ax,[LVDOVRUHSODFHGE\a[,QWKHVWXG\RIWKHH[SUHVVLRQVLWLVXVHIXOWRPDNHD GLPHQVLRQDOFRQWUROUHPHPEHULQJWKDWLQWKHFRQYROXWLRQDQGFRUUHODWLRQWKHUH VXOWKDVWKHGLPHQVLRQVRIWKHSURGXFWRIWKRVHRIWKHWZRIXQFWLRQVDQGDVPHQ

0DWKHPDWLFDOSUHOLPLQDULHV



WLRQHGRIWKDWRIWKHLUDUJXPHQW $VLPLODUGLVFRXUVHVKRXOGDOVREHPDGHIRUWKH)RXULHU¶VWUDQVIRUP,IWKHYDU LDEOHxLVUHSODFHGE\axLWLVWREHXQGHUVWRRGWKDWfxLVUHSODFHGE\fxaDQGWKDW WKH GLPHQVLRQV RI WKH WUDQVIRUP DUH WKRVH RI WKH RULJLQDO IXQFWLRQ PXOWLSOLHG E\ WKRVHRIaxJLYHQWKDWdxLVDOVRUHSODFHGE\adx,I\RXGRQRWZDQWWRXVHDX[LO LDU\ IXQFWLRQV WKDW LQFRUSRUDWH WKH VFDOLQJ \RX VKRXOG DOZD\V H[SOLFLWO\ LQGLFDWH WKHFRRUGLQDWHVRIWKHVSDFHGRPDLQDQGWKDWRIWKHIUHTXHQFLHVEHWZHHQZKLFKWKH WUDQVIRUPDWLRQLVSHUIRUPHG

5.1.3 Hankel’s transform $V ZH VKDOO VHH LQ WKH QH[W VHFWLRQ WKH +DQNHO¶V WUDQVIRUP LQWHUYHQHV LQ WKH FDOFXODWLRQ RI D WZRGLPHQVLRQDO )RXULHU¶V WUDQVIRUP ZKHQ XVLQJ SRODU FRRUGL QDWHV,QLWVJHQHUDOIRUPWKH+DQNHO¶VWUDQVIRUPRIRUGHUnLV 

f

Hn  gÖ n w

³

rg r - n wr dr 

 

ZKHUH-nLVWKH%HVVHO¶VIXQFWLRQRIRUGHUn7KHLQYHUVHWUDQVIRUPLV 

Hn 

g r

f

³

wgÖ n w - n wr dw 

 

DQGWKLVFDQEHGHPRQVWUDWHGWKDQNVWRWKHSURSHUWLHVRIWKH)RXULHU¶VWUDQVIRUP HnDQG Hn DUHFRPSOHWHO\V\PPHWULFDOVRLI gÖ n w LVWKHWUDQVIRUPRI g r  g w LVWKHWUDQVIRUPRI gÖ n r 

5.1.4 Fourier’s transform in polar coordinates :KHQWKHRSWLFDOV\VWHPDQGWKHILHOGKDYHDFLUFXODUV\PPHWU\LWPD\EHXVH IXOWRXVHDSRODUFRRUGLQDWHV\VWHPIRUFDOFXODWLRQRIWKH)RXULHU¶VWUDQVIRUP,Q SDUWLFXODUZHFDQUHIRUPXODWH(T D ZLWKWKHIROORZLQJUHSODFHPHQWV 

x

r FRV I 

fx

f r FRV M 

y

r VLQ I 

fy

f r VLQ M 



 

LQDGGLWLRQZHGHQRWHgDQGGLQWKHQHZFRRUGLQDWHVDV 

g x y

g S r I 



G fx f y



GS f r  M  

 



)RXULHU2SWLFV

7KXV(T D EHFRPHV



GS f r  M

f S

³ ³

g S r  I e  Si f r r FRV IM r dr d I



 

7KHIXQFWLRQ g S r  I FDQEHGHYHORSHGLQD)RXULHU¶VVHULHVDVIROORZV g S r I



f

¦n gn r einI  f

 

ZKHUHLQWXUQ  S g S r  I e inId I  S ³

gn r



 

5HSODFLQJ(T  LQ(T  ZHKDYH 

GS f r  M

f

f

¦n einM ³ f

rg n r dr ³

 S  Si f r FRV IM in IM r e d I 



 

7KHDUJXPHQWRIWKHVHFRQGLQWHJUDOLVSHULRGLFZLWKSHULRGSIRUZKLFKERWK WKH YDULDEOH RI LQWHJUDWLRQ DQG WKH H[WUHPHV RI LQWHJUDWLRQ FDQ EH LQGHSHQGHQWO\ VKLIWHG,QSDUWLFXODU(T  FDQEHVLPSOLILHGE\PHDQVRIWKH%HVVHO¶VIXQF WLRQV-nGHVFULEHGLQ†%RIDSSHQGL[%7KXVWKHWUDQVIRUPDWLRQLQSRODUFRRU GLQDWHVLVILQDOO\REWDLQHG 

FS  GS f r  M

f

f

S¦n i  n einM ³ rg n r - n Sf r r dr  

f

 

ZKHUH WKH IXQFWLRQV gn DUH JLYHQ E\ (T   ,W FDQ EH LPPHGLDWHO\ QRWLFHG WKDWHYHQGS frI FDQEHH[SUHVVHGLQDVHULHVVLPLODUWRWKDWRIgS rM  

GS f r  M

f

¦n Gn f r einM  f

 

ZKHUHGn fr LVi ±n SWLPHVWKH+DQNHO¶VWUDQVIRUPRIRUGHUnRIgn r  f



Gn f r i  n S ³ rgn r - n Sf r r dr  

 

0DWKHPDWLFDOSUHOLPLQDULHV







F ^g x y ` 

g x y 

G fx f y

ª § x  y  ·º  H[S « S ¨    ¸ »  ab b ¹¼ ¬ ©a

H[S ª S a  f x   b f y  º  ¬ ¼

 §x· § y· UHFW ¨ ¸ UHFW ¨ ¸  ab ©a¹ ©b¹

VLQF Saf x VLQF Sbf y 

 §x· § y· WUL ¨ ¸ WUL ¨ ¸  ab © a ¹ © b ¹

VLQF Saf x VLQF Sbf y 

 §x y· G¨  ¸  ab © a b ¹



 §x· § y· FRPE ¨ ¸ FRPE ¨ ¸  ab ©a¹ ©b¹

FRPE af x FRPE bf y 

gS r 

GS f r

 §r· FLUF ¨ ¸  a ©a¹

- Saf r  af r













^

§ i Sf  · i H[S ¨  r ¸  E E ¹ ©

g x 

G fx

§· P¨ ¸ ©x¹

i S VJQ f x 

VWHS x 

§   G fx P ¨  © Si f x

x

 







fx



H[S iSEx  

 

`

B gS fr 

H[S iSEr  





F ^g x ` 

· ¸ ¹



§ i Sf  · i H[S ¨  x ¸  E E ¹ ©

7DEOH7UDQVIRUPVIRUVRPH VHSDUDEOH IXQFWLRQVLQ&DUWHVLDQFRRUGLQDWHVRULQFLUFXODU FRRUGLQDWHV>VHHDOVR %DUUHWWDQG0\HUV  ,QSDUWLFXODUSDJHRQWKHGLVFXVVLRQ RIWKHTXDGUDWLFSKDVHIDFWRULQWKHODVW[email protected]7KHFRQVWDQWVabDQGEDUHUHDODQGSRVLWLYH



)RXULHU2SWLFV

7KHLQYHUVH)RXULHU¶VWUDQVIRUPLQSRODUFRRUGLQDWHVFDQEHVLPLODUO\REWDLQHG IURP(T E  

FS   g S r I

f

f

f



S¦n i n einI ³

f r Gn f r - n Sf r r d U 

 

7KHWZRHTXDWLRQVIRU FSDQG FS±DUHWKHUHIRUHVLPLODUH[FHSWIRUDVXEVWLWX WLRQRIi nIRUi ±n $SDUWLFXODUO\LQWHUHVWLQJFDVHLVZKHQgS rM KDVFLUFXODUV\PPHWU\IRUZKLFK gS rM  gS r VRWKDW(TV  DQG  DUHUHGXFHGWRH[SUHVVLQJ DSDUW IURP IDFWRUV S IURP WKH VWDQGDUG GHILQLWLRQ  WKH ]HUR RUGHU +DQNHO¶V WUDQVIRUP EHWZHHQgS r DQGGS fr 



f

B  GS f r

S ³ rg S r - Sf r r dr

B  g S r

S ³



f





 

f r GS f r - Sf r r d U

7KH\ DUH JHQHUDOO\ LQGLFDWHG ZLWK WKH QDPH RI )RXULHU%HVVHO WUDQVIRUPV 6RPH IXQFWLRQVZLWKWKHLUWUDQVIRUPVDUHJLYHQLQ7DEOH

5.1.4 Uncertainty relation ,QWKH)UDXQKRIHU¶VGLIIUDFWLRQZHKDYHVHHQWKDWWKHUHLVDQLQYHUVHUHODWLRQ VKLSEHWZHHQWKHUHODWLYHZLGWKRIWKHRSHQLQJVdODQGWKHDQJXODUGLYHUJHQFHRI WKHGLIIUDFWHGZDYH,QRWKHUZRUGVWKHSURGXFWEHWZHHQWKHVHWZRTXDQWLWLHVGRHV QRWGHSHQGRQWKHVL]HRIWKHVFDOH7KLVIDFWLVDGLUHFWFRQVHTXHQFHRIWKHVLPL ODULW\WKHRUHP  7KHVDPHEHKDYLRUZLOODOVREHIRXQGZLWK*DXVVLDQODVHU EHDPVWKHPRUHWKLQWKH\DUHWKHPRUHWKH\GLYHUJH3K\VLFDOO\WKLVHIIHFWDULVHV IURP WKH DSSOLFDWLRQ WR WKH UDGLDWLRQ RI Heisenberg’s uncertainty principle IRU ZKLFKWKHSURGXFWRIWKHXQFHUWDLQWLHVLQWKHPHDVXUHPHQWRIWZRFRQMXJDWHYDULD EOHVVXFKDVSRVLWLRQDQGPRPHQWXPFDQQRWIDOOEHORZDPLQLPXPYDOXH2QWKH RWKHUKDQGLQ4XDQWXP0HFKDQLFVDSK\VLFDOV\VWHPFDQEHGHVFULEHGE\DZDYH IXQFWLRQDQGWKHUHIRUHWKHXQFHUWDLQW\SULQFLSOHKDVDQLQWHUSUHWDWLRQEDVHGRQWKH PDWKHPDWLFDO)RXULHU¶VWUDQVIRUP>%DUUHWWDQG0\HUV S@ /HWXVOLPLWRXUVHOYHVKHUHWRWKHRQHGLPHQVLRQDOFDVH7RILQGWKHJHQHUDOUH ODWLRQEHWZHHQWKH³ZLGWK´RIDIXQFWLRQg x LQ LLQWKHVSDFHRISRVLWLRQVDQG WKDW RI LWV WUDQVIRUP G f  LQ WKH IUHTXHQF\ VSDFH ZH PXVW ILUVW GHILQH ZKDW LV PHDQWE\width)RUWKLVSXUSRVHXVLQJWKHFRQFHSWVRIVWDWLVWLFDODQDO\VLVZHFDQ WDNH~g x ~DQG~G x ~DVZHLJKWIXQFWLRQVDQGGHILQHDFHQWURLGDV

0DWKHPDWLFDOSUHOLPLQDULHV



f



x

f



³f x g x dx  f  ³f g x dx



³f f G f df f  ³f G f df

f



 

DQGDvarianceDV f



Vx





³f x  x g x f  ³f g x dx



f

dx Vf







³f f  f G f f  ³f G f df



df

  

7KHUHVSHFWLYHZLGWKVDUHWKHUHIRUHWKHVTXDUHURRWRIWKHVHYDULDQFHV ,QWKHVHWZRH[SUHVVLRQVZLWKDFKDQJHRIYDULDEOHZHFDQWDNHERWK]HUR x  DQG f  ZLWKRXW ORVV RI JHQHUDOLW\ 7KHUHIRUH XVLQJ WKH GHULYDWLYH WKHRUHP D DQGWKH3DUVHYDO¶VWKHRUHPZHKDYH f

 Vf





f



³f f  G f df f  ³f G f df

  S

³f F ^g c x ` df f  ³f g x dx

f

 S



³f g c x dx  f  ³f g x dx

ZKHUH g ‫ މ‬x  LV WKH ILUVW GHULYDWLYH RI g x  ,Q DGGLWLRQ IRU &DXFK\6FKZDU] LQH TXDOLW\  ZHKDYH f



VxV f t

 S

³ f g x x g c x dx f

³f





 

g x dx

7KDQNVWRWKHUHODWLRQVKLS d ª xg x º¼ dx ¬



xg c x  g x 

 

ZH FDQ UHSODFH xgү x  LQ (T   ,QWHJUDWLQJ E\ SDUWV DQG UHFDOOLQJ WKDW g x oIRUxorfWKHLQWHJUDOLQWKHQXPHUDWRURIWKLVLQHTXDOLW\EHFRPHV 

f

³ f g x xg c x dx



f f

g x xg c x dx  ³

f f



g x dx 

 

IURPZKLFK  )RUWKHVKLIWWKHRUHPDSKDVHWHUPDSSHDUVLQWKHQHZIXQFWLRQgDQGLWVWUDQVIRUPGKRZHYHU

LWGLVDSSHDUVLQWKHVTXDUHPRGXOHLQWKHLQWHJUDOVRI(TV  





)RXULHU2SWLFV

ƒe ³

f f

g x xg c x dx



f f



g x dx 

 

6LQFHWKHQWKHPRGXOHRIDFRPSOH[QXPEHULVJUHDWHUWKDQWKHDEVROXWHYDOXHRI LWVUHDOSDUWZHILQDOO\ILQGWKHGHVLUHGLQHTXDOLW\ 

V x Vf t

  S

 

ZKLFKH[SUHVVHVWKHPLQLPXPYDOXHWKDWFDQKDYHWKHSURGXFWRIWKHZLGWKVLQWKH VSDFHRISRVLWLRQVDQGIUHTXHQFLHV7KHVLJQtEHFRPHVDQHTXDOLW\ZKHQgү x LV SURSRUWLRQDOWRxg x WKDWLVZKHQg x LVD*DXVVLDQIXQFWLRQ

6DPSOLQJWKHRUHPV 5.2.1 Fourier’s series ,QWKHWKHRU\RIGLIIUDFWLRQWKDWZHKDYHH[DPLQHGVRIDUWKHILHOGFDQEHDS SUR[LPDWHGE\FRQWLQXRXVIXQFWLRQVRIWKHFRRUGLQDWHVDQGLQSDUWLFXODUZHKDYH VHHQ WKDW GLIIUDFWLRQ LQWHJUDOV DUH HVVHQWLDOO\ )RXULHU¶V LQWHJUDOV 7KHLU DQDO\WLFDO VROXWLRQ LV DEVHQW LQ PRVW FDVHV RQ WKH RWKHU KDQG DQ\ QXPHULF SURFHGXUH UH TXLUHVXVWREHFRQWHQWWRSHUIRUPWKHFDOFXODWLRQRQDQGZLWKDILQLWHQXPEHURI FRRUGLQDWH YDOXHV 7KXV WKH )RXULHU¶V WUDQVIRUP JRHV IURP EHLQJ FRQWLQXRXV WR EHLQJdiscreteVLQFHVDPSOLQJQHHGVWREHSHUIRUPHGERWKLQWKHYDULDEOHVRIWKH IXQFWLRQWREHWUDQVIRUPHGDQGLQWKRVHRILWVWUDQVIRUP 7\SLFDOO\WKHVHVDPSOLQJVDUHGRQHDWDFRQVWDQWSDFHDQGLQWKLVFDVHZHZLOO VHHWKDWWKHGLVFUHWHWUDQVIRUPDWLRQZHQHHGLVFORVHO\UHODWHGWRWKHFourier’s seriesWKDWDUHGHILQHGIRUSHULRGLFIXQFWLRQV)RUWKLVSXUSRVHZHVHHKRZWKHVHVH ULHVFDQEHWUDFHGEDFNWRDSDUWLFXODUFDVHRI)RXULHU¶VLQWHJUDO)RUEUHYLW\ZH FRQILQHRXUVHOYHVKHUHWRWKHRQHGLPHQVLRQDOFDVH>2UDQ%ULJKDP@ $SHULRGLFIXQFWLRQpH[SUHVVHGDVD)RXULHU¶VVHULHVLVJLYHQE\WKHH[SUHVVLRQ 

p x

f

¦

cn ei  Sn fo x 

 

n f

ZKHUHfoLVWKHIXQGDPHQWDOIUHTXHQF\ZKLFKFRUUHVSRQGVWRWKHSHULRGXo foLQ WKH FRRUGLQDWH x RI WKH IXQFWLRQ p ,Q WKLV H[SUHVVLRQ LW ZDV SUHIHUDEOH WR XVH D FRPSOH[H[SRQHQWLDOQRWDWLRQLQVWHDGRIWKHPRVWIDPLOLDUVXPPDWLRQRQVLQHVDQG FRVLQHVWKHFRHIILFLHQWVcnDUHJHQHUDOO\FRPSOH[DQGDUHJLYHQE\WKHLQWHJUDOV

6DPSOLQJWKHRUHPV





cn

 Xo  p x e i  Sn x X o dx  X o ³ X o 

 

:HQRZGHYHORSWKHIXQFWLRQpLQWHUPVRILWV)RXULHU¶VWUDQVIRUP:HFDQLP DJLQH WKDW p LV WKH FRQYROXWLRQ RI D IXQFWLRQ q x  ZKLFK LV GHILQHG DV GLIIHUHQW IURP]HURRQO\LQWKHLQWHUYDOXoxXoZKHUHqLVHTXDOWRpZLWKWKHSH ULRGLFIXQFWLRQ /Xo FRPE x/Xo  p x



§  ·  q x FRPE ¨ ˜ x ¸  Xo © Xo ¹

 

ZKHUHWKHDVWHULVNLQGLFDWHVWKHFRQYROXWLRQLQWHJUDORQWKHYDULDEOHxWKDWPXVWEH H[HFXWHGEHWZHHQWKHWZRWHUPV7KHWUDQVIRUPRIqLV 

Q f

f

Xo 

³f q x ei S x f dx ³ X o  p x ei S x f dx 

 

,QSDUWLFXODUIRUf n fo n /XoIURP(T  ZHKDYH § n · Q¨ ¸ © Xo ¹



X o cn 

 

2QWKHRWKHUKDQGIRUWKHWKHRUHPVRIVLPLODULW\DQGFRQYROXWLRQWKHWUDQVIRUPRI pLVJLYHQE\ P f Q f FRPE X o f Q f



f

¦ G Xo f  n

f

n f f

n f

n f

¦ Q n fo G X o f  n ¦



 

cn G f  n fo 

IURPZKLFKZHVHHWKDWWKHVSHFWUXPRIpLVFRQVWLWXWHGE\DVXFFHVVLRQRISXOVHV ZLWKDFRQVWDQWSDFHfoZKRVH³DUHD´ LQWHJUDWLQJLQf LVJLYHQE\WKHFRHIILFLHQWV cn RI WKH GHYHORSPHQW LQ )RXULHU¶V VHULHV RI p 7KLV IDFW LPSOLHV VRPHWKLQJ WKDW RQHVKRXOGDOZD\VNHHSLQPLQGVRDVWRDYRLGVXUSULVHV HYHU\WLPHWKDWZHVDPSOHDVSHFWUXPLQWKHIUHTXHQF\GRPDLQZLWKDFRQ VWDQW SDFH fo WKH VDPSOHG YDOXHV FRUUHVSRQG WR D SHULRGLF IXQFWLRQ LQ WKH GRPDLQRIWKHSRVLWLRQVZLWKSDFHXo  fo )RUWKHSURSHUWLHVRIV\PPHWU\RI)RXULHU¶VWUDQVIRUPWKHRSSRVLWHDOVRDSSOLHV WKHVSHFWUXPRIDIXQFWLRQVDPSOHGZLWKDFRQVWDQWSDFHLQWKHGRPDLQRIWKH SRVLWLRQVLVSHULRGLF



)RXULHU2SWLFV

5.2.2 Sampling theorem in Cartesian coordinates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exactly LI WKH LQWHUYDO EHWZHHQ VDPSOHV GRHV QRW H[FHHG D FHUWDLQ OLPLW >:KLWWDNHU @7KLVWKHRUHPZDVXVHGE\6KDQQRQLQKLVVWXGLHVRQWKHLQIRUPDWLRQWKHRU\ >6KDQQRQ @ )RU KLV GHPRQVWUDWLRQ ZH ZLOO IROORZ *RRGPDQ   ZKR FRQVLGHUVDWZRGLPHQVLRQDOH[WHQVLRQWRWKHFDVHGLVFXVVHGE\6KDQQRQ,Q)LJ WKHVWHSVRIWKLVSURFHGXUHDUHUHSUHVHQWHG 6XSSRVHWKHUHIRUHWKDWDIXQFWLRQgLVVDPSOHGRQDUHFWDQJXODUJULGRISRLQWV DQGZHGHILQHDVDPSOLQJIXQFWLRQ § x gV x y FRPE ¨ © 'x



§ y · ¸ FRPE ¨¨ ¹ © 'y

· ¸ g x y  ¸ ¹

 

ZKLFKFRQVLVWVRIDODWWLFHRIIXQFWLRQVG x/'x ±ny/'y±m îg n'xm'y UHJXODUO\ VSDFHG ZLWK SLWFK 'x DQG 'y LQ WKH GLUHFWLRQV RI WKH D[HV x DQG y UHVSHFWLYHO\ +HUHZHKDYHFKRVHQWRPDWFKWKH³YROXPH´ DQGLQWKHRQHGLPHQVLRQDOFDVH WKH³DUHDV´ RIWKHSXOVHVQDPHO\WKHLQWHJUDWLRQRIgVRQHDFKGZLWKWKHYDOXH RIgRQWKHFRUUHVSRQGLQJODWWLFHSRLQWPXOWLSOLHGE\'x'y)RUWKHWUDQVIRUPDWLRQ SURSHUWLHVRIWKHFRPEIXQFWLRQ VHH7DEOH WKHVSHFWUXPGVRIgVLVWKHFRQ YROXWLRQRQWKHIUHTXHQFLHVfxDQGfy 



GV f x  f y











' x ' y ªFRPE ' x ˜ f x FRPE ' y ˜ f y º G f x  f y  ¬ ¼

 

7KHUHIRUHWKHVSHFWUXPRIWKHVDPSOHGIXQFWLRQLVHTXDOWRWKHUHSHWLWLRQRIWKH VSHFWUXPRIgDURXQGHDFKSRLQW n 'xm 'y LQWKHIUHTXHQF\VSDFH 



GV f x  f y

f

f

¦ ¦

n f m f

§ n m · G ¨ fx   fy  ¸  ¨ 'x ' y ¸¹ ©

 

,IWKHIXQFWLRQgKDVDOLPLWHGEDQGLWVVSHFWUXPGLVQXOORXWVLGHRIDILQLWHUH JLRQ SLQWKHIUHTXHQF\VSDFH,IWKHUHIRUHWKHVDPSOLQJgLVVXIILFLHQWO\GHQVH WKHYDULRXVUHSOLFDVRIGWKDWFRQVWLWXWHGVDUHVHSDUDWHGIURPHDFKRWKHUZLWKRXW RYHUODSSLQJ

6DPSOLQJWKHRUHPV



g

G

x

f

multiplication

convolution comb x D

...

comb fD ...

D

x

...

... 1 D

1 D

gs

f

Gs

sampling

x

1 f D multiplication

1 D

convolution 1 sinc p x D D

rect fD

x

g

1 2D

1 2D

f

G truncation

reconstruction

x

f 

)LJ4XDOLWDWLYHJUDSKLFDOUHSUHVHQWDWLRQRIWKHVDPSOLQJWKHRUHP7KHILJXUHVRQWKH OHIWGHVFULEHIXQFWLRQVLQWKHVSDFHRISRVLWLRQVZKLOHWKRVHWRWKHULJKWLQGLFDWHWKHLUWUDQV IRUPVLQIUHTXHQF\VSDFH



)RXULHU2SWLFV

,QRUGHUIRUWKLVWRKDSSHQLIfjDQGfjDUHWKHH[WUHPHIUHTXHQFLHVWKDWFRQ WULEXWHWRWKHVSHFWUXP ZLWKj xy LWQHHGVWREHfjfj'jVRWKHVDPSOLQJ UDWH'jPXVWexceedWKHNyquist’s frequency  ! F1 j 'j



fj   fj  

D 

7KHUHIRUHWKHRULJLQDOVSHFWUXPFDQEHREWDLQHGH[DFWO\E\PXOWLSO\LQJGVIRU DILOWHUIXQFWLRQWKDWLVXQLWDU\RQIRUH[DPSOHDUHFWDQJOH RWKDWLQFOXGHVWKHUH JLRQ SDQGLV]HURRXWVLGHVRDVWRH[FOXGHWKHRWKHUUHSOLFDVRIG)XUWKHUPRUH ZHDVVXPHIRUVLPSOLFLW\WKDWWKHVSHFWUXPRIgLVFHQWHUHGRQWKHRULJLQLQWKH IUHTXHQF\VSDFH,QWKLVFDVHfj fj fjEDQGWKH1\TXLVW¶VIUHTXHQF\LV  ! F1 j 'j



Wj 

E 

ZKHUHWjLVWKHbandwidthRIg 2WKHUZLVHLWLVQHFHVVDU\WRVXEWUDFWWKHFDUULHUDQGDSSO\WKHWUDQVODWLRQWKHR UHP E  WR REWDLQ D PRUH HIILFLHQW VDPSOLQJ WKH UDSLGO\ RVFLOODWLQJ IXQF WLRQVUHTXLUHDJUHDWGHDORISUHFLVLRQLQWKHFDOFXODWLRQDVZHOODVDKLJKUHVROX WLRQRIWKHDFWXDODSSDUDWXV ,QPDQ\FDVHVRISK\VLFDOLQWHUHVWWKHIXQFWLRQgLVUHDODQGWKHUHIRUHLWVVSHF WUXPKDVDQHYHQUHDOSDUWDQGDQRGGLPDJLQDU\SDUWDURXQGWKHRULJLQ(YHQLIDW WKHSRVLWLYHIUHTXHQFLHVWKHVSHFWUXPRIWKHVLJQDOLVFHQWHUHGRQDFDUULHUfSZLWK H[WUHPHYDOXHVHTXDOWRfS“WLQWKHRYHUDOOEDQGLWLVDOVRQHFHVVDU\WRFRQVLGHU LWVFRQMXJDWHGFRS\DWWKHQHJDWLYHIUHTXHQFLHVIRUZKLFKWKH1\TXLVWIUHTXHQF\LV  fSW 7KHVXEWUDFWLRQRIWKHFDUULHUZLWKSK\VLFDOPHDQVWKHQDOORZVXVWRUH GXFHWKHVDPSOLQJDWWKH1\TXLVWIUHTXHQF\JLYHQE\(T E  /HWFxDQGFyEHWKHPLQLPXPYDOXHVRIWKHKDOIZLGWKRIWKLVUHFWDQJOH7KH VDPSOLQJJULGRIgPXVWWKHQKDYHVSDFLQJ 

'x d

 Fx

   'y d Wx Fy

  W y

 

DQGWKHILOWHUIXQFWLRQFDQEHFKRVHQDV 

H

§ f UHFW ¨ x © Fx

§ fy · ¸ UHFW ¨¨ ¹ © Fy

· ¸  ¸ ¹

 

ZKHUHE\ 











GV f x  f y H f x  f y { G f x  f y 

6DPSOLQJWKHRUHPV



ZKRVHWUDQVIRUPDWLRQLV ª º §  · §  · ˜ x ¸ FRPE ¨ ˜ y ¸ g x y » ¬ª Fx Fy VLQF SFx ˜ x VLQF SFy ˜ y ¼º «FRPE ¨ ¨ 'y ¸  «¬  »¼ © 'x ¹ © ¹ g x y 





%\GHYHORSLQJWKHFRQYROXWLRQLQWHJUDOZHREWDLQ g x y



f

u

Fx Fy ' x ' y f

¦ ¦ g n' x  m' y VLQF ª¬ SFx x  n' x º¼ VLQF ª¬ SFy y  m' y º¼ 

  

n f m f

ZKLFK LQ WKH FDVH RI WDNLQJ WKH LQWHUYDOV 'x DQG 'y DV EHLQJ HTXDO WR FxDQG FyUHVSHFWLYHO\EHFRPHV  g x y

f

n f m



f

§ n m· ª § n g¨  ¸ VLQF « SFx ¨ x  ¨ Fx Fy ¸ Fx  © ¬ f © ¹

¦ ¦

ª § ·º m ¸ » VLQF « SFy ¨¨ y  Fy  ¹¼ © ¬«



·º ¸»   ¸ ¹ ¼»

 

7KLV LV WKH UHVXOW RI WKH Whittaker-Shannon sampling theorem 1RWLFH WKDW IRU HDFKSDLU xy WKLVRSHUDWLRQLQYROYHVallgVDPSOHV,WLVWKHUHIRUHQRWDVLPSOH LQWHUSRODWLRQ EHWZHHQ QHLJKERULQJ SRLQWV ZKLFK ZRXOG QRW EH DEOH WR HOLPLQDWH WKHEHDWEHWZHHQWKHFRPSRQHQWVRIWKHYDULRXVUHSOLFDVRIWKHgVSHFWUXP,I\RX ZDQWHG WR SURFHHG IRU LQWHUSRODWLRQ LW ZRXOG EH QHFHVVDU\ WR VDPSOH DW D IUH TXHQF\DWOHDVWWZLFHWKDWRI1\TXLVW

5.2.3 Sampling theorem in polar coordinates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x KDVDQLQILQLWHQXPEHURI]HURVFRUUHVSRQGLQJWRWKHYDOXHVxnIRU ZKLFK- xn  7KHVHxnDUHFDOOHGWKHzeros of -5HGHILQLQJWKHIXQFWLRQDU JXPHQWxZLWKx DrDQGDVVLJQLQJWRDWKHYDOXHV



)RXULHU2SWLFV



xn  ZLWK n  !  a

Dn

 

ZKHUHaLVDSUHGHWHUPLQHGYDOXHIURP-RQHFDQEXLOGDQLQILQLWHVHWRIIXQF WLRQVRIrWKDWLVWKHIXQFWLRQV- Dnār ,WWXUQVRXWWKDWWKLVVHWRIIXQFWLRQVFRQ VWLWXWHVDEDVLVRIRUWKRJRQDOIXQFWLRQVIRUWKHJHQHULFIXQFWLRQVf r GHILQHGZLWK LQDUDQJH a WKDWVDWLVI\VRPHUHJXODULW\FRQGLWLRQV DEVROXWHLQWHJUDELOLW\DQG OLPLWHGYDULDWLRQV>3DSRXOLV@ ,QSDUWLFXODULWLV a



³ r-  Dn r -  Dm r dr 

­ a ° - Dn a IRU n m  ® ° n m IRU  z ¯

 

:LWKWKLVEDVLVWKHIXQFWLRQf r KDVDGHYHORSPHQW 

f r

f

¦ Cn -  Dn r 

ZLWK   r  a 

 

n 

ZKHUH WKH FRHIILFLHQWV Cn FDQ EH FDOFXODWHG E\ PXOWLSO\LQJ ERWK VLGHV RI (T  IRUr - Dnr DQGLQWHJUDWLQJEHWZHHQDQGa$SSO\LQJ(T  DQG (T  IRUWKH]HURRUGHU+DQNHO¶VWUDQVIRUPFDVHZHKDYH 

Cn

 fÖR Dn a  - Dn a

 fÖR Dn  a  -n 

 

ZKHUH-nDUHWKHYDOXHVRI-FRUUHVSRQGLQJWRWKH]HURVxnRI-TXRWHGDERYH DQG fÖR w LVWKH+DQNHO¶VWUDQVIRUPRIWKHIXQFWLRQ 

fR r

­ f r IRU   r d a  ® IRU r ! a ¯

 

7KDQNVWRWKHV\PPHWU\SURSHUWLHVRIWKH+DQNHO¶VWUDQVIRUPDVLPLODUGHYHO RSPHQWFDQDOVREHFRQGXFWHGIRUWKHWUDQVIRUP fÖ w RIf r /HWXVFRQVLGHULQ SDUWLFXODU WKH FDVH RI D IXQFWLRQ f r  WKDW KDV D ]HURRUGHU +DQNHO¶V WUDQVIRUP fÖ w ZLWKDlimited bandwidth 

fÖ w  IRU

w ! b 

 

ZKHUHbQRZPHDQVWKHOLPLWRIDQLQWHUYDO b LQWKHVSDFHRIIUHTXHQFLHVw :LWKWKHVDPHSURFHGXUHDVEHIRUHZHILQGWKDWWKHFRHIILFLHQWV CÖ n UHZULWWHQIRU WKHGHYHORSPHQWRI fÖ w DUH

6DPSOLQJWKHRUHPV



 f rn  b -n 

CÖ n



 

ZKHUH rn  xn b DUH WKH SRLQWV ZKHUH f r  LV VDPSOHG 7KHUHIRUH ZH REWDLQ WKDW fÖ w FDQEHGHULYHGIURPWKLVVDPSOLQJE\WKHIRUPXOD  f f rn -  rn w Pb w  ¦ b n  -n 

fÖ w



 

ZKHUHPb w LVDQLPSXOVHIXQFWLRQWKDWLVPb w  IRUwdbDQGPb w    IRU w ! b :LWK WKLV OLPLWHG VSHFWUDO EDQG K\SRWKHVLV WKH ZKROH IXQFWLRQ f r  FDQ EH GHULYHG IURP LWV VDPSOLQJ GRLQJ WKH WUDQVIRUP RI ERWK VLGHV RI (T  ZHKDYH 

f r

f  f f rn w -  rn w Pb w -  rw dw  ¦ b n  -n  ³ 

:LWKDIHZFDOFXODWLRQV>3DSRXOLV@DQGVWLOOH[SORLWLQJWKHIDFWWKDW- r nb   ZHILQGWKDW 

f r

f r f r n n -  br   ¦ b n  -n rn   r 

 

1RWHWKDWIRUr rnLWLV- r b   :H FDQ ILQDOO\ UHIRUPXODWH (T   IRU WKH FDOFXODWLRQ RI WKH )RXULHU %HVVHO WUDQVIRUP IURP WKH FRPSDULVRQ RI (TV   DQG D  ZH ILQG WKDWIRUDJLYHQIXQFWLRQg r  GS f r



SgÖ  Sf r 

ZKHUH GS f r DQG gÖ  w DUHWKH)RXULHU%HVVHODQG+DQNHO¶VWUDQVIRUPVRI]HUR RUGHU UHVSHFWLYHO\ RI WKH IXQFWLRQ g r  ZLWK w  Sfr 7KHUHIRUH WKH FRQGLWLRQ  EHFRPHV 

GS f r  IRU

f r ! c ZLWK c

b S 

)LQDOO\IRUfrdcZHKDYH 

GS f r

 f g rn -  Srn f r  ¦ Sc  n  -n 

 



)RXULHU2SWLFV

ZKHUH , XOWLPDWHO\ UHPHPEHU WKDW rn DUH WKH SRLQWV ZKHUH WKH IXQFWLRQg LV VDP SOHGFKRVHQVRWKDW- Sr n c  

5.2.4 The Discrete Fourier’s Transform /HWXVILQDOO\FRQVLGHUDSDLURIFRQWLQXRXVIXQFWLRQVgDQGGWKDWDUHHDFKRWK HUV¶ )RXULHU¶V WUDQVIRUP %\ H[SORLWLQJ WKH VDPSOLQJ WKHRUHP XQGHU WKH DSSUR SULDWHFRQGLWLRQVLQSODFHRIgDQGGZHFDQREWDLQDFRXSOHRIGLVFUHWHIXQFWLRQV ZKLFKDUHDVDWLVIDFWRU\DSSUR[LPDWLRQRIWKHRULJLQDORQHV:HZLOOQRZVHHKRZ WRFRQYHUWWKH)RXULHU¶VWUDQVIRUPLQWRDQXPHULFSURFHGXUHDSSOLFDEOHWRDGLJLWDO FRPSXWHU 7KH RSHUDWLRQ WKDW LV REWDLQHG LV FDOOHG WKH discrete Fourier’s transform )RU VLPSOLFLW\ ZH ZLOO OLPLW RXUVHOYHV KHUH WR WKH FDVH RI WKH RQHGLPHQ VLRQDOWUDQVIRUPDWLRQLQ&DUWHVLDQFRRUGLQDWHV>2UDQ%ULJKDP@7KHVWHSVRI WKLVSURFHGXUHDUHVKRZQLQ)LJ 7KHILUVWVWHSLVWRVDPSOHWKHIXQFWLRQg x ZLWKFRQVWDQWSDFH'x)RUWKLVSXU SRVHZHXVHDQRQQRUPDOL]HGIXQFWLRQFRPE x'x  

gs x

§ x · g x FRPE ¨ ¸ © 'x ¹

f

¦n

f

§ x ·  n ¸  g n' x G ¨ ' © x ¹

 

:LWKWKLV DUELWUDU\ FKRLFHRIWKHVDPSOLQJIXQFWLRQWKHDUHDEHORZHDFKSXOVH RIgsLVHTXDOWR'xWLPHVWKHFRUUHVSRQGLQJYDOXHRIg,WVWUDQVIRUPLVWKHSHULRGLF UHSHWLWLRQRIWKHVSHFWUXPGRIg 

Gs f

' x G f FRPE ' x ˜ f

f

¦n

f

§ n · G¨ f  ¸  ' x¹ ©

 

,I 'x LV QRW WDNHQ WR EH VXIILFLHQWO\ VPDOO WKH FRSLHV RI G RYHUODS DQG WKLV JLYHV ULVH WR WKH SKHQRPHQRQ RI aliasing VRPHWKLQJ WKDW DSSHDUV WR EH FRQWULE XWLQJWRWKHIUHTXHQF\fFDQLQVWHDGEHXQGHUDIDOVHQDPHDFRS\RIWKDWDWDIUH TXHQF\f±n 'x ZLWKnDVDSRVLWLYHRUQHJDWLYHLQWHJHU 7KHUHPHG\LVWRVDPSOHDFFRUGLQJWRWKHUXOHVRXWOLQHGLQ†DERYH+RZ HYHULIgGRHVQRWKDYHDOLPLWHGEDQGZLGWKEXWLWVVSHFWUXPGRHVDWOHDVWGHFOLQH DZD\IURPLWVFHQWHUZHVKRXOGWU\WRWKLFNHQWKHVDPSOLQJXQWLODOLDVLQJLVEHDUD EOH RU ILOWHU g EHIRUH VDPSOLQJ LW VRPHWKLQJ WKDW PXVW DOZD\V EH GRQH LQ UHDO HTXLSPHQWWRUHGXFHWKHHIIHFWVRIQRLVH,WLVFOHDUWKDWLQWKHVHFDVHVZHZLOORQ O\KDYHDQDSSUR[LPDWHUHSUHVHQWDWLRQRIRXUIXQFWLRQRUVLJQDO 7KHVHFRQGVWHSLVWRPDNHDWUXQFDWLRQLQWKHVDPSOLQJLQGHHGZHFDQKDQGOH RQO\DILQLWHQXPEHUNRIVDPSOHV7RGRWKLVZHPXOWLSO\WKHVDPSOHIXQFWLRQE\ DUHFWDQJOHIXQFWLRQ

6DPSOLQJWKHRUHPV



g

G

f

x comb x D

D comb fD

N

...

... D

...

...

x

gs

f

1/D

1/D Gs

...

aliasing

... x

1/D

rect x 1 + 1 ND 2 2N

D 2

N 12 D

ND sinc p ND f

f

1/D exp[-ipD(N-1) f ]

x

f

gt

Gt leakage x

N 1 comb x ND ND

comb( f ND) ...

...

ND

ND

1/D

1/D

f

N

...

...

x

f

1/(ND)

gs

Gt ... N

x

... 1/D

)LJ4XDOLWDWLYHJUDSKLFGHULYDWLRQRIWKHGLVFUHWH)RXULHU¶VWUDQVIRUP

N

1/D

f





)RXULHU2SWLFV

r x



§ x   ·   UHFW ¨ ¸  © X o  N ¹

 

ZKHUHXo N'xLVWKHZLGWKRIWKHWUXQFDWLRQIXQFWLRQDQGrLVQRQ]HURRQO\IRUx EHWZHHQ±'x DQGXo±'x  ,QWKLVZD\WKHNVDPSOHGYDOXHVOLHLQWKHFHQWHURIWKHNLQWHUYDOVLQZKLFKLW LVSRVVLEOHWRGLYLGHWKHZLGWKXo7KLVWUXQFDWLRQKDVWKHUHVXOW



gW x

§ x · g x FRPE ¨ ¸ r x © 'x ¹

N  g 

¦n

§ x ·  n ¸  © 'x ¹

n' x G ¨

 

,QWKLVH[SUHVVLRQWKHLQGH[nUXQVYDOXHVIURPWRNDQGWKLVDV\PPHWU\ZLWK UHVSHFW WR  LV SUHIHUUHG E\ PDWKHPDWLFLDQV ,W LV DOVR SUHVHQW LQ WKH ³FDQRQLFDO´ DOJRULWKPVRIWKH))7DQGZHZLOOKDYHWRFRQVLGHULWLQWKHIROORZLQJ 7KHWUXQFDWLRQLQWURGXFHVWKHVHFRQGFKDQJHLQWKHRULJLQDOVSHFWUXPRIg,Q IDFWLQWKHIUHTXHQF\GRPDLQWKHPXOWLSOLFDWLRQE\WKHIXQFWLRQrFRUUHVSRQGVWRD FRQYROXWLRQIRULWVWUDQVIRUP 

R fx

X o VLQF SX o f x e iS  N X o f x 

 

ZKLFKGHIRUPVWKHVSHFWUXPDQGLQWURGXFHVDQRVFLOODWLRQ ripple 7RUHGXFHWKLV HIIHFWLWLVJRRGWRH[WHQGWKHVDPSOLQJLQWHUYDOXoDVPXFKDVSRVVLEOHIRUZKLFK RDSSUR[LPDWHVDQLPSXOVH:HZLOOUHWXUQWRWKLVSRLQWDWWKHHQGRIWKHVHFWLRQ 7KH WKLUG VWHS LV WR VDPSOH WKH VSHFWUXP RI WKH H[SUHVVLRQ   )RU WKLV SXUSRVHZHFKRRVHDVDPSOLQJVWHSWKHUHOHYDQFHRIZKLFKZHZLOOVRRQVHH 

'f

 Xo

  N'x

 

DQGDVDPSOLQJIXQFWLRQHTXDOWRFRPE Xo f 7KLVFKRLFH DOVRDUELWUDU\ DOORZV XVWRPDLQWDLQWKHVDPHVFDOHLQWKHGRPDLQRIVSDFHZKHUHWKLVVDPSOLQJHTXDOV WKHFRQYROXWLRQ  g V x

ª N  §  ·º ª  ˜ x  n ¸» « « ¦ n  g n' x G ¨ © 'x ¹¼ ¬ ' x ¬

f

¦l

f

§  ·º ˜ x  lN ¸ »    G¨ © 'x ¹¼

ZKHUH g V LQGLFDWHVDQDSSUR[LPDWLRQRIWKHVDPSOHIXQFWLRQgVDQGZKHUHWKHGHO WDSURSHUW\  ZDVXVHG:LWKDIHZVWHSVZHILQGWKDW

6DPSOLQJWKHRUHPV

 f

g V x



¦l

f

N  g 

¦n

§ x ·  n  lN ¸  © 'x ¹

n' x G ¨

 

:HFDQQRWHWKDWWKLVH[SUHVVLRQLVSHULRGLFZLWKSHULRGXo N'xDQGLQSDU WLFXODUWKDWLWFRQVLVWVRIDUHJXODUVXFFHVVLRQRISXOVHVZLWKSLWFK'xLWZRXOGQRW KDYHKDGWKLVSURSHUW\ZLWKDGLIIHUHQWFKRLFHRI'f%HVLGHVWKHRUGHUl RI g V  FRLQFLGHVZLWKgVMXVWDVZHZDQWHG /HWXVQRZUHVXPHWKHUHVXOWVRIWKH)RXULHU¶VVHULHVGLVFXVVLRQVLQFH g V LVSH ULRGLFZHFDQDSSO\(T  WRLWVRWKDWLWVWUDQVIRUPLVJLYHQE\ G V f



f

Xo

¦

G m G X o f  m 

 

m f

ZKHUHIRU(T  WKHLQGLYLGXDOSXOVHVRI G V KDYHDUHD G m



 Xo

X o ' x 

³

g V x e i  Smx

X o dx



 

' x 

5HSODFLQJ(T  LQ  DQGFRQVLGHULQJWKDWWKHLQWHJUDWLRQLVRQO\RYHU DSHULRGZHJHW  N

G m



N  g 

¦n

n' x ei Snm N



 

(YHQWKLVH[SUHVVLRQLVSHULRGLFLQ mZLWKSHULRG NMXVWUHSODFH mZLWK mүNWR FKHFNLWVLQFHH[S iSn  IRUWKHnLQWHJHU 7KHUHIRUH ERWK g V  DQG LWV )RXULHU¶V WUDQVIRUP GV DUH SHULRGLF IXQFWLRQV RI SXOVHWUDLQV6LPLODUO\ZHGHQRWHE\ g n



' x g n' x  

 

WKHSXOVHDUHDRI g V DVFDQEHGHGXFHGIURP(T  DVZHOO:HFDQWKHQ HVWDEOLVKWKHUHODWLRQVKLSEHWZHHQWKHVHDUHDVDV 

G m

 Xo

N  g e i  Snm N  n

¦n

ZLWKm N±

 

ZLWKn N±

 

7KHLQYHUVHUHODWLRQVKLSLV 

g n

N  ' x ¦ m  G m ei  Snm

N



)RXULHU2SWLFV

DVFDQEHVHHQZLWKVRPHVLPSOHVWHSV 7KHVH UHODWLRQVKLSV FRXOG EH FRQVLGHUHG WR EH WKH GLVFUHWH WUDQVIRUPDWLRQ DQG LWVLQYHUVHWKDWZHDUHORRNLQJIRU+RZHYHULWVWLOOODFNVDUHODWLRQVKLSEHWZHHQ WKHVHFRHIILFLHQWVDQGWKH)RXULHU¶VWUDQVIRUPRIWKHRULJLQDOIXQFWLRQ g7KLVUHOD WLRQVKLSLVPDGHTXLWHFRPSOLFDWHGE\WKHWUXQFDWLRQ 7RXQGHUVWDQGLWVHIIHFWVFRQVLGHUWKHFDVHLQZKLFKgLVDVLPSOHVLQXVRLGZLWK IUHTXHQF\ f,WVVSHFWUXPLVWKHQPDGHXSRIWZR'LUDF¶VGHOWDVDWIUHTXHQFLHVrf 7KHFRQYROXWLRQZLWK RSURGXFHVWZRSKHQRPHQDWKHWZRSXOVHVZLGHQDQGDVH ULHV RI VDWHOOLWH VSLNHV LQYDGHV WKH HQWLUH VSHFWUXP 7KLV HIIHFW ZKLFK LV FORVHO\ UHODWHGWRWKHGLIIUDFWLRQSURGXFHGE\DQRSHQLQJ LVFDOOHGleakage 2QHZD\WRFRXQWHULWLVWRFKRRVHDWUXQFDWLRQLQWHUYDOWKDWLVDZKROHQXPEHU RISHULRGVRIWKHVLQXVRLG,QIDFWLQWKLVFDVHWKHVDPSOLQJRIWKHVSHFWUXPPDGH ZLWKWKHUXOHVGLVFXVVHGDERYHSURGXFHVRQO\WZRQRQQXOOYDOXHVDWWKHIUHTXHQ FLHVrf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g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

6SDWLDOO\ERXQGHGIXQFWLRQVZLWKOLPLWHGEDQGZLGWK

/HWXVFRQVLGHUWKHFDVHLQZKLFKWKHIXQFWLRQgLV]HURRXWVLGHRIWKHWUXQFDWLRQ LQWHUYDOWKLVRSHUDWLRQKDVQRHIIHFWKHUHDQGVLQFHZLWKLQWKHLQWHJUDWLRQLQWHUYDO RI(T  LWLV g s g s  DVPHQWLRQHGDERYHDFFRUGLQJWR(T  LWUH VXOWV WKDW GV >m  N'x @  X oG m  ,I g DOVR WKHQ KDV D OLPLWHG VSHFWUXP DQG WKH VDPSOLQJLVVXIILFLHQWO\GHQVHWRPDNHWKHDOLDVLQJQHJOLJLEOHIURP(T   ZHKDYH § m · § m ·  G ¨ ¸ # X oG m  WKDWLV G m # ' f G ¨ ¸  IRU m © N'x ¹ © N 'x ¹

"  N     

6DPSOLQJWKHRUHPV



,QWKHFDVHZHDUHGHDOLQJZLWKKHUHIRU(T  WKHdiscrete Fourier’s transformLVWKHQH[SUHVVHGE\ 

§ m · G¨ ¸ © N'x ¹

N  g 

' x ¦n

n' x e i Snm N  

 

DQGLWVinverseFDQEHREWDLQHGIURP(T   

 N'x

g n' x

N  § G  ¨

m · i  Snm ¸e © N'x ¹

¦m

N



 

7KHV\PPHWU\EHWZHHQWKHVHWZRUHODWLRQVKLSVFDQEHVHHQVLPSO\E\UHPHPEHU LQJWKDW N'x  'f7KHUHIRUHWKHGLYLVLRQIRU NWKDWDSSHDUVEHIRUHWKHVXPRI (T  LVQRWVXUSULVLQJ

3HULRGLFIXQFWLRQVZLWKOLPLWHGEDQGZLGWK

/HWXVQRZFRQVLGHUWKHFDVHLQZKLFKWKHIXQFWLRQ gLVSHULRGLFZLWKDSHULRG Xo,WV)RXULHU¶VWUDQVIRUPFRQVLVWVRISXOVHVDWIUHTXHQFLHV fm  m  XoZKRVHDU HDIRU(TV  DQG  LV



Gm

 Xo

X o ' x 

g x e i  Smx

³

X o dx



 

' x 

6DPSOLQJWKHVSHFWUXPDWIUHTXHQFLHVfmDQGE\H[SORLWLQJWKHSHULRGLFLW\RIg ZHFDQEULQJ(T  WR  ZHVWLOOKDYH g V   gVEXWWKLVWLPHRQWKH ZKROH xD[LV 7KHUHIRUH WKH HIIHFW RI WUXQFDWLRQ LV DOVR FDQFHOHG LQWKLV FDVH ,Q SDUWLFXODU LW LV QRZ G V   GV 2Q WKH RWKHU KDQG (T   VWDWHV WKDW LI WKH VSHFWUXPRIgKDVDOLPLWHGEDQGZLGWKDQGWKHVDPSOLQJLQWKHVSDFHLVVXIILFLHQWO\ GHQVHWRUHQGHUDOLDVLQJQHJOLJLEOHWKH]HURRUGHURI GVFRLQFLGHVZLWK G7KHUH IRUHIRU(T  ZHDOVRKDYH 

Gm

G m

 N

N  g 

¦n

n' x ei Snm N 

 

7KLVUHVXOWFDQDOVREHREWDLQHGE\WUDQVIRUPLQJWKHLQWHJUDORI(T  LQD VXPPDWLRQZLWKWKHUHSODFHPHQWVxon'xdxo'xDQGXooN'x+RZHYHULQ WKLVZD\WKHSURRIZRXOGKDYHPDGHWKHERXQGVLPSRVHGE\WKHDOLDVLQJOHVVYLVL EOH7KHFRUUHVSRQGLQJLQYHUVHWUDQVIRUPLV





)RXULHU2SWLFV

g n' x

N  G ei  Snm N  m

¦m



 

5.2.5 The Fast Fourier Transform 0DWKHPDWLFDOO\WKHFDQRQLFDOGLVFUHWH)RXULHU¶VWUDQVIRUPLVH[SUHVVHGLQWKH GLPHQVLRQOHVVIRUP 

N  g 

G m

¦n

n e i Snm N



 

DQGLWVLQYHUVHE\ 

g n

 N

N  G 

¦m

m ei  Snm N 

 

WR ZKLFK IRU H[DPSOH (TV   FDQ EH WUDFHG EDFN UHVSHFWLYHO\ E\ FKRRVLQJDVFDOHRI xVXFKWKDW'x DQGFRQVHTXHQWO\IRU(T  DVFDOH IRUfVXFKWKDW'f NVLPSOLI\LQJWKHQRWDWLRQIRUWKHDUJXPHQWRIG7KHFDOFX ODWLRQRIWKHVHH[SUHVVLRQVLQYROYHVNRXWSXWYDOXHVHDFKRIZKLFKUHTXLUHVDVXP RI NLQSXWYDOXHVHDFKPXOWLSOLHGE\DFRPSOH[QXPEHU7KHUHIRUHWKHGLUHFWDS SOLFDWLRQ RI WKH WUDQVIRUPDWLRQ IRUPXOD UHTXLUHV N N±  VXPV DQG N FRPSOH[ PXOWLSOLFDWLRQV ,QWKHWZRGLPHQVLRQDOFDVHDVIRUWKHFDOFXODWLRQRIWKHGLIIUDFWLRQWKHVLWXD WLRQLVDJJUDYDWHGHYHQPRUHIRUZKLFKDJULGRINîNYDOXHVZRXOGUHTXLUHN N±  VXPVDQG NFRPSOH[PXOWLSOLFDWLRQV)RU N ZHZRXOGKDYHDQH[RU ELWDQWQXPEHURIELOOLRQVXPVDQGPXOWLSOLFDWLRQV )RUWXQDWHO\-:&RROH\DQG-:7XNH\DVZHOODVRWKHULQGHSHQGHQWDXWKRUV LQWURGXFHGDPDWKHPDWLFDODOJRULWKPFDOOHGWKH Fast Fourier Transform ))7 WR FDOFXODWHWKHGLVFUHWH)RXULHU¶VWUDQVIRUP>&RROH\DQG7XNH\@,WHOLPLQDWHV WKH UHGXQGDQW RSHUDWLRQV RI WKH GLUHFW PHWKRG DQG UHGXFHV WKH QXPEHU RI QHFHV VDU\ RSHUDWLRQV WR DSSUR[LPDWHO\ N ORJ N :H KDYH QR QHHG KHUH WR HQWHU LQWR WKHFRPSOH[GHWDLOVRIWKHGHULYDWLRQRIWKLVDOJRULWKPRIZKLFKLQDGGLWLRQWKHUH DUHYDULRXV³FDQRQLFDO´YHUVLRQV>2UDQ%ULJKDP@+RZHYHUWKHYDOXHVWKDW ZHFDQDWWULEXWHWRNDUHQRWDUELWUDU\LQRUGHUWRREWDLQPD[LPXPHIILFLHQF\LWLV QHFHVVDU\WKDW NEHKLJKO\IDFWRUL]DEOH7KHPRVWFRPPRQDOJRULWKPLVWREDVH VRWKDW NPXVWEHDSRZHURIWKHUHDUHDOVREDVHDQGYHUVLRQVWKDWDUH PRUHHIILFLHQWDQGLQZKLFKNPXVWEHDSRZHURIWKHVHEDVHV:KDWPDWWHUVLVWKDW WKH\VROYHWKHFDOFXODWLRQRI(TV  HIILFLHQWO\7KHEDVHDOJRULWKPFDQ EHIRXQGLQDOOPDWKHPDWLFDOVRIWZDUHSDFNDJHVVXFKDVWKH Numerical Recipes ZKDW LV VWULNLQJ LV WKDW WUDQVFULEHG IRU H[DPSOH LQ )RUWUDQ LW RQO\ UHTXLUHV RQH   ,Q DQ\ FDVH LW LV DOZD\V JRRG WR FKHFN WKH H[DFW FRUUHVSRQGHQFH ZLWK WKH FDQRQLFDO (TV

 

$SSOLFDWLRQVRIWKH)RXULHUWUDQVIRUPWRWKHGLIIUDFWLRQ



SDJHRILQVWUXFWLRQVIDUOHVVWKDQZKDWLWWDNHVWRH[SODLQLW )UHTXHQWO\ERWKWKHVLJQDODQGWKHFRUUHVSRQGLQJVSHFWUXPDUHFHQWHUHGRQ]H URLQWKHLUUHVSHFWLYHVFDOHV,QVXFKFDVHVWRDSSO\WKH))7LWLVDOVRQHFHVVDU\WR SHUIRUPD WUDQVODWLRQRIWKHVDPSOHVEHIRUHDQGDIWHUWKHWUDQVIRUPDWLRQ,QGHHG WKHGLVFUHWHWUDQVIRUPDWLRQDVVXPHVWKDWERWKLQWKHGRPDLQRIVSDFHDQGWKDWRI WKHIUHTXHQFLHVWKHIXQFWLRQV g n DQG G m DUHSHULRGLFZLWKSHULRG N&RQVLGHU WKHPDWKHPDWLFDOO\PRUHHOHJDQWFDVHRIWKH&ODQJXDJHWKH))7DOJRULWKPSODFHV WKH]HURYDOXHRISRVLWLRQDQGIUHTXHQF\LQFRLQFLGHQFHZLWKWKHLQGLFHVnDQGm  UHVSHFWLYHO\,QDGGLWLRQWKHYDOXHV m 1±FRUUHVSRQGWRWKHSRVLWLYH IUHTXHQFLHV ZKLOH IRU WKH SHULRGLFLW\ RI WKH VSHFWUXP WKH QHJDWLYH RQHV DUH DV VLJQHGWRWKHUHPDLQLQJYDOXHVZLWK m  N N7KHVDPHKDSSHQVIRUWKH SRVLWLRQV7KHUHIRUHEHIRUHH[HFXWLQJWKH))7WKHYDOXHVg n DWWKHQHJDWLYHSR VLWLRQV n ±1±PXVWEHWUDQVODWHGLQWR g nN :LVKLQJWKHQWRUHFRP SRVH WKH IUHTXHQF\ VFDOH LW LV QHFHVVDU\ WR WUDQVODWH WKH YDOXHV G m  LQ G m±N  IRU m 11±,QWKHFDVHRISURJUDPPLQJLQ)RUWUDQPXVWEHDGGHGWR DOOLQGH[HVnDQGm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f

$SSOLFDWLRQVRIWKH)RXULHU¶VWUDQVIRUPWRWKHGLIIUDFWLRQ $VZHKDYHVHHQLQWKHFKDSWHUGHGLFDWHGWRGLIIUDFWLRQLQWKH)UHVQHODSSUR[L PDWLRQ WKH SURSDJDWLRQ RI DQ HOHFWURPDJQHWLF ZDYH LQ D KRPRJHQHRXV PHGLXP FDQEHGHWHUPLQHGE\PHDQVRID)RXULHU¶VWUDQVIRUPRIWKHSURGXFWRIWKHILHOG ZKLFKLVGHILQHGRQWKHSODQHFURVVHGE\WKHUDGLDWLRQE\DSKDVHWHUPWKDWYDULHV UDGLDOO\DFFRUGLQJWR(T  7KLVIDFWLVH[SORLWHGWRFDOFXODWHWKHSURSDJD WLRQQXPHULFDOO\WKHLQSXWILHOGLVVDPSOHGRQDUHJXODUJULGRISRLQWVDQGWKHRXW SXWILHOGLVUHWXUQHGRQDVLPLODUJULGZLWKWKHVDPHQXPEHURISRLQWV $QRWKHU DSSURDFK FRQVLVWV LQ FDOFXODWLQJ WKH DQJXODU VSHFWUXP RI WKH ILHOG RQ WKH DSHUWXUH DQG IURP WKLV UHFRQVWUXFWLQJ WKH GLIIUDFWHG ILHOG LQ WKLV FDVH LW LV QHFHVVDU\WRFDOFXODWHWZR)RXULHU¶VWUDQVIRUPVWRJHWWKHILHOGDWDJLYHQGLVWDQFH z IURP WKH RSHQLQJ $V ZH ZLOO VHH WKLV PHWKRG LV HIIHFWLYH LQ WKH FDVH RI QHDU ILHOGGLIIUDFWLRQZKHUHWKH)UHVQHODSSUR[LPDWLRQIDOOVEXWEHFRPHVXQIHDVLEOHLQ )UDXQKRIHU¶VGLIIUDFWLRQ 9DULRXVFRQGLWLRQVDUHUHTXLUHGIRUWKHVXFFHVVRIWKHVHDSSOLFDWLRQV7KHILUVW RQHFDQEHHDVLO\RYHUORRNHGRUPD\VHHPRYHUO\REYLRXVEXWLWLVQRWLWLVQHFHV VDU\WKDWWKHLQWHJUDOVZKLFKDUHWREHFRPSXWHGUHDOO\KDYHWKHVKDSHRID)RXUL



)RXULHU2SWLFV

HU¶V WUDQVIRUP )RU H[DPSOH .LUFKKRII¶V GLIIUDFWLRQ LQWHJUDO GRHV QRW KDYH WKLV SURSHUW\,QWKHELQRPLDOGHYHORSPHQWRIWKHURRWZHKDYHDW\SLFDOWHUPRIWKH WUDQVIRUPZKLFKLVWKHPL[HGELOLQHDUSURGXFWRIWKHFRRUGLQDWHVRIWKHSRLQWVRQ WKH DSHUWXUH DQG WKRVH RQ WKH REVHUYDWLRQ SODQH EXW WKHUH DUH DOVR RWKHU PL[HG WHUPV RI KLJKHU SRZHU ,QVWHDG LQ D WUXH )RXULHU¶V WUDQVIRUP WKH IXQFWLRQ WR EH WUDQVIRUPHGFDQQRWFRQWDLQDQ\GHSHQGHQFHRQWKHYDULDEOHVFRQMXJDWHGWRWKRVH RQZKLFKLWLVLQWHJUDWHGXQOHVVWKH\PD\EHIDFWRUL]HGDQGUHPRYHGIURPWKHLQ WHJUDO7KHUHIRUHWKHGLIIUDFWLRQLQWHJUDOFDQEHWUDFHGEDFNWRD)RXULHU¶VWUDQV IRUP RQO\ LQ WKH )UHVQHO¶V DQG )UDXQKRIHU¶V DSSUR[LPDWLRQV LQ ZKLFK WKH DGGL WLRQDO PL[HG WHUPV DUH QHJOHFWHG 7KLV IDFW VKRXOG EH NHSW LQ PLQG ZKHQ IRU H[DPSOHRQHZDQWVWRGHWHUPLQHWKHUHVROYLQJSRZHURIDQRSWLFDOLQVWUXPHQW VHH † ZLWKDODUJHQXPHULFDODSHUWXUH $QRWKHU WKLQJ WR UHPHPEHU LV WKDW WKH )RXULHU¶V WUDQVIRUP PHWKRG DSSOLHV WR HQWLWLHVWKDWVXPXSVFDODUO\DVLVWKHFDVHZLWKDFRXVWLFORQJLWXGLQDOZDYHVLQD JDVLQZKLFKWKHZDYHDPSOLWXGHLVJLYHQE\WKHGLIIHUHQFHLQSUHVVXUHFRPSDUHG WRWKDWLQWKHDEVHQFHRIVRXQG+RZHYHUZHZLOODSSO\WKLVWUDQVIRUPIRUDsingle &DUWHVLDQ FRPSRQHQW RI WKH DPSOLWXGH RI WKH ILHOG RI DQ HOHFWURPDJQHWLF ZDYH ZLWKLQWKHOLPLWRIWKHSDUD[LDODSSUR[LPDWLRQ ,Q WKH FDVHRI)UHVQHO GLIIUDFWLRQ WZR RWKHU FRQGLWLRQV DUH DOVR UHTXLUHG 7KH ILUVWLVWKDWthe field on the starting plane is spatially delimitedLQIDFWQXPHULFDO O\ZHFDQVDPSOHRQO\DILQLWHSRUWLRQRIVSDFH7KHVHFRQGFRQGLWLRQLVWKDW the field on the arrival plane is also spatially delimitedVRDVWRVDWLVI\(T   LQGHHGWKHVSDFLQJEHWZHHQWKHVDPSOHVPXVWQHFHVVDULO\EHILQLWH7KLVZLOODO ORZIRUDSSO\LQJWKH:KLWWDNHU6KDQQRQVDPSOLQJWKHRUHP 6RPHWLPHVLQVWHDGRIFRQVLGHULQJDILHOGWKDWGHFD\VWR]HURRXWVLGHRIDILQLWH UHJLRQ RI VSDFH D ILHOG LV FRQVLGHUHG WKDW H[WHQGV SHULRGLFDOO\ WKURXJKRXW WKH VSDFH DORQJ ERWK WUDQVYHUVDO FRRUGLQDWHV ,Q WKLV FDVH LWV VSHFWUXP FRQVLVWV RI D WZRGLPHQVLRQDO JULG RI 'LUDF¶V GHOWD )RU UHDVRQV RI WUDQVYHUVH V\PPHWU\ DOW KRXJK WKH ILHOG SURILOH FKDQJHV WKH VDPH SHULRGLFLW\ PXVW EH PDLQWDLQHG DW DQ\ SURSDJDWLRQGLVWDQFH,QWKLVFDVH)UHVQHO¶VGLIIUDFWLRQFDQQRWEHDSSOLHGZKHUH WKHSHULRGLFLW\RIWKHIXQFWLRQWREHWUDQVIRUPHGLVHUDVHGE\WKHSKDVHWHUPWKDW YDULHV ZLWK WKH VTXDUH RI WKH GLVWDQFH IURP WKH D[LV:LWK VRPH FRQWULYDQFH ZH FDQLQVWHDGDSSO\WKHDQJXODUVSHFWUXPPHWKRGWRWKLVFDVH,WVKRXOGDOVREHQRWHG WKDW FRQWUDU\ WR )UDXQKRIHU¶V GLIIUDFWLRQ WKH GLIIUDFWLRQ RI D SHULRGLF ILHOG GRHV QRWWHQGWRLWVVSHFWUXPDVWKHGLVWDQFHLQFUHDVHV,WLVDOVRWUXHWKDWDWUXO\SHULRG LFILHOGDQGWKHUHIRUHDQLQILQLWHO\H[WHQGHGRQHLVQRWSK\VLFDOO\IHDVLEOH

5.3.1 Application of the FFT to the Fresnel diffraction /HWXVQRZFRQVLGHUWKHFDVHWKDWRQWKHGHSDUWXUHSODQHWKHILHOGu WUDQVPLWWHG IURPDSRVVLEOHDSHUWXUH LV]HURRUDWOHDVWVXIILFLHQWO\QHJOLJLEOHRXWVLGHDUHFWDQ JOHZLWKVLGHV axDQG ayFHQWHUHGRQWKHRULJLQDVQRWWRJLYHDSSUHFLDEOHRYHUDOO

$SSOLFDWLRQVRIWKH)RXULHUWUDQVIRUPWRWKHGLIIUDFWLRQ



FRQWULEXWLRQV WR WKH GLIIUDFWLRQ LQWHJUDO IURP WKHUH 7R SURFHHG ZLWK WKH DQDO\VLV ZHLQWURGXFHWKHVSDWLDOIUHTXHQFLHV 

xo  OZ o

fx

fy

yo  OZ o

 

7KHGLIIUDFWLRQLQWHJUDO  EHFRPHV



uR f x OZ R  f y OZ R

 u³

ay

ax 

³

 ay

a  x   





i ikZ iSOZ R f x   f y  e Re u OZ R

 ª iS  x   y  º  i  S f x x  f y y « e OZ R » u x y e dxdy « » ¬ ¼

 

/HWXVJREDFNWRWKHDQDO\VLVRIWKHSUHYLRXVVHFWLRQ2EVHUYLQJWKHVWUXFWXUH RI(T  ZHVHHWKDWWKHIXQFWLRQ 



V fx f y



iOZ R e ikZ R e



iSOZ R f x   f y 

uR f x OZ R  f y OZ R 

 

FRUUHVSRQGVWRWKH)RXULHU¶VWUDQVIRUPRIWKHIXQFWLRQ v x y



iS

e

 x  y OZ R





u x y 

 

,QDGGLWLRQWRWKHVFDOHWUDQVIRUPDWLRQRIYDULDEOHV uRGLIIHUVIURP VRQO\IRUWKH WHUPOZRDQGDSKDVHIDFWRU7KHUHIRUHLI uR xRyR LVVXIILFLHQWO\OLPLWHGZLWKLQD UHFWDQJOHZLWKVLGHV bx by centered around the originWKHLQWHJUDWLRQFDQEHUH GXFHGWRDVXPPDWLRQRQDJULGRI Nu MSRLQWVZLWKFRRUGLQDWHV xn ym ZKHUH WKHFRUUHVSRQGLQJVSDFLQJ'xDQG'yLQWKHxDQGyFRRUGLQDWHVPHHWWKHFRQGLWLRQV 

'x d

OZ R OZ  ' y d R  bx by

 

DULVLQJIURP(TV  DQG  ,QSDUWLFXODUZHWDNHWKDW N N d n d    



xn

n' x 



n  ] ZLWKNHYHQ

D 



ym

m' x  

M M dmd   m  ] ZLWKMHYHQ  

E 



)RXULHU2SWLFV

2QWKHRWKHUKDQGWKHVDPSOLQJJULGPXVWFRYHUWKHDUHDaxuayLQZKLFKu xy LV FRQILQHG6RLWLVQHFHVVDU\WKDW a y a y by a a b Nt x t x x Mt t  ' x OZ R ' y OZ R



 

7KHVH FRQGLWLRQV LPSRVH D PLQLPXP OLPLW RQ WKH QXPEHU RI JULG VDPSOHV ZKLFKZHZLOOGLVFXVVDWWKHHQGRIWKLVVHFWLRQ,QWKHIROORZLQJZHGHQRWHWKH VLGHOHQJWKVRIWKHVDPSOHGDUHDZLWK 

XD

N ' x t a x  YD

M ' y t a y 

 

,QWXUQvLQ(T  UHSUHVHQWVWKHVSHFWUXPRIVDQGGLIIHUVIURPuRQO\IRU DSKDVHIDFWRULWLVEDQGOLPLWHGJLYHQWKDWZHKDYHDVVXPHGuWREH]HURRXWVLGH RIDGHOLPLWHGUHJLRQLQWKHWUDQVYHUVHFRRUGLQDWHV7KHUHIRUH uRFDQDOVREHVDP SOHGZLWKRXWORVVRILQIRUPDWLRQ 2QWKHRWKHUKDQGWKH))7DOJRULWKPSURGXFHVDOLQHDUWUDQVIRUPDWLRQEHWZHHQ VHWV ZLWK WKH VDPH QXPEHU RI HOHPHQWV KHQFH WKH FDOFXODWLRQ RI uo REWDLQHG WKURXJKWKLVDOJRULWKPZLOOVWLOOEHH[HFXWHGRQDJULGRI NSRLQWV fxnүE\MSRLQWV fymүGHILQHGE\ 

f x n



f y  mc

F nc x  N

mc

Fy M



 

N N d n c d   n c  ] IRUNHYHQ  

D 

M M d mc    mc  ] IRUMHYHQ  

E 

ZKHUH 

Fx

  Fy 'x

  'y

 

DUHWKHPD[LPXPIUHTXHQFLHVWKDWGHULYHIURPWKHVDPSOLQJRIWKHLQWHUYDOVXDDQG YDUHVSHFWLYHO\7KHUHIRUHWKHYDOXHVDVVXPHGE\fxnүDQGfymүDUH 

f x  nc

nc   N 'x

f y  mc

mc   M 'y

 

)RUWKHFKRLFHPDGHRQ'xDQG'yZLWKWKHFRQGLWLRQV  ZHKDYH 

XE

OZ R Fx

OZ R t bx  YE 'x

OZ R Fy

OZ R t by  'y

 

$SSOLFDWLRQVRIWKH)RXULHUWUDQVIRUPWRWKHGLIIUDFWLRQ



ZLWKZKLFKLWRFFXUVWKDWWKHVDPSOLQJDWWKHDUULYDOSODQHFRYHUVWKHUHJLRQZKHUH WKHILHOGLVVLJQLILFDQWO\GLIIHUHQWIURP]HUR :LWK(TV  (T  EHFRPHV

§ nc mc · uR ¨ OZo  OZ o ¸ ¨ N'x N ' y ¸¹ ©

 u

N  

¦

M  

¦

N M n  m   

ª nc  mc  º  iSOZo « »     ¬« N ' x M ' y ¼» ieikZ R e

ª iS n ' xm' y « e OZ o u n ' x  m' y « ¬





'x' y OZ o

º i  S§ nnc  mmc · ¨ ¸ »e © N M ¹ » ¼

  

,QWKHIDUILHOGUHJLPH )UDXQKRIHU¶VGLIIUDFWLRQ  bxaOZRFxDQG byaOZRFy 7KHUHIRUH WKH QHFHVVDU\ QXPEHU NuM RI JULG VDPSOHV JLYHQ E\ WKH LQHTXDOLWLHV  LVLQGHSHQGHQWRI ZR,QVWHDGLQWKHQHDUILHOGUHJLPHLWLVXVXDOO\ ba a DQGLQVXFKFDVHNuMLQFUHDVHVZLWKGHFUHDVLQJZR %HWZHHQWKHVL]HRIWKHLQLWLDODQGILQDOJULGDPDJQLILFDWLRQLQWHUYHQHVJLYHQ E\ 

mA x

N

OZ R  mA y X D

M

OZ R  YD 

 

,ILQSDUWLFXODUZHUHTXLUHDXQLWDU\PDJQLILFDWLRQZKLFKFDQIRUH[DPSOHRFFXU LQWKHVWXG\RIUHVRQDQWFDYLWLHVLWLVQHFHVVDU\WKDWWKHLQLWLDOSODQHEHVDPSOHGRQ DUHFWDQJOHRIVLGHVXF PD[ XDXE DQGYF PD[ YDYE ZKHUHE\ 

N

X F  M OZ R

YF   OZ R

 

7RFRQFOXGH(T  FDQXOWLPDWHO\EHUHVROYHGZLWKWKHIROORZLQJVWHSV i PXOWLSO\LQJHDFKYDOXHRIILHOGuIRUWKHFRUUHVSRQGLQJFXUYDWXUHWHUP ii WUDQV ODWLQJWKHVDPSOHVRIWKHPDWUL[WKXVREWDLQHG iii DSSO\LQJWKHWZRGLPHQVLRQDO ))7iv WUDQVODWLQJWKHPDWUL[EDFNv PXOWLSO\LQJHDFKYDOXHIRUWKHFRUUHVSRQG LQJWHUPRIFXUYDWXUHDQGDPSOLWXGH ,WVKRXOGEHQRWHGWKDWWKHIUDFWLRQVWKDWDSSHDULQ(TV  UHSUHVHQWDJHQ HUDOL]DWLRQ RI WKH FRQFHSW RI )UHVQHO¶V QXPEHU PXOWLSOLHG E\   GHILQHG E\ (T   7KLV YDOXH FRQVLVWV RI WKH QXPEHU RI )UHVQHO ]RQHV IRU WKH REVHUYDWLRQ GLVWDQFHZRLQZKLFKWKHDSHUWXUHKDVWREHGLYLGHGVRDVWRREWDLQXQLIRUPFRQWUL EXWLRQVLQWKHGLIIUDFWLRQLQWHJUDOZLWKDSODQHLQFLGHQWZDYH+HUH NDQG MLQGL FDWHWKHQXPEHURIVDPSOHVUHTXLUHGDOVRWDNLQJLQWRDFFRXQWWKHILHOGVWUXFWXUHRQ WKHLQSXWZLQGRZDQGWKHUHIRUHLWVVSHFWUDOEDQG



)RXULHU2SWLFV

5.3.2 Angular spectrum of the field on an opening :HZLOOQRZVHHDGLIIHUHQWDSSURDFKWRGLIIUDFWLRQWKHRU\ZKLFK,ZRXOGFDOO ³)RXULHU¶V GLIIUDFWLRQ´ :KLOH WKH )UHVQHO.LUFKKRII GLIIUDFWLRQ XVHV D GHYHORS PHQW RI WKH ILHOG LQ VSKHULFDO wavelets WKDW DUH LQGLYLGXDOO\ JHQHUDWHG E\ HDFK SRLQWRQWKHVXUIDFHRIDQRSHQLQJKHUHZHLQVWHDGFRQVLGHUWKHGHYHORSPHQWLQ SODQHZDYHVWKDWDUH³RYHUDOOJHQHUDWHG´E\WKHILHOGRQWKHZKROHRSHQLQJ%RWK PHWKRGVFDQEHFRQVLGHUHGDVOLPLWLQJFDVHVRIDILHOGGHFRPSRVLWLRQLQSURSDJD WLRQPRGHV /HWXVWDNHWKHFDVHRIDPRQRFKURPDWLFZDYHWKDWFURVVHVDQxySODQHLPPHGL DWHO\ SODFHG EHKLQG DQ RSHQLQJ LQ WKH SRVLWLYH GLUHFWLRQ RI WKH zD[LV $V LQ WKH FDVHRI.LUFKKRIIOHWXVVXSSRVHWKDWZHNQRZWKHFRPSOH[ILHOG u xy RQWKLV SODQHDQGZDQWWRGHWHUPLQHWKHILHOGu xyz LQWKHKDOIVSDFHz!)XUWKHUVXS SRVHWKDWWKHVSDFHLQZKLFKWKHZDYHVSURSDJDWHLVKRPRJHQHRXVDQGLVRWURSLF 7KURXJK WKH xySODQH WKHIXQFWLRQ UKDV D WZRGLPHQVLRQDO )RXULHU¶V WUDQVIRUP JLYHQE\ 

f f



A fx f y z



³ ³ u x y eSi f x x  f y y dx dy 

 

f f

DQGUHFLSURFDOO\ZHKDYH f f



u x y 

 Si f x  f y ³ ³ A f x  f y  e x y df x df y 

 

f f

2QWKHRWKHUKDQGDPRQRFKURPDWLFSODQHZDYHRIXQLWDPSOLWXGHWKDWSURSDJDWHV ZLWKGLUHFWLRQFRVLQHV DEJ LVVLPSO\H[SUHVVHGE\ 

eik Dx Ey Jz 

o x y  z

ZKHUHk SOZLWKOHTXDOWRWKHZDYHOHQJWKLQWKHFRQVLGHUHGPHGLXPDQG 

J

  D  E 

7KHUHIRUHWKHILHOG u xy FDQEHFRQVLGHUHGDVWKHVXSHUSRVLWLRQLQ z RIWKH ILHOGVRIDSODQHZDYHVSHFWUXPZKRVHGLUHFWLRQFRVLQHVDUH 

D Of x  E

Of y  J

DQG(T  FDQEHUHZULWWHQDV





  Of x  Of y





 

$SSOLFDWLRQVRIWKH)RXULHUWUDQVIRUPWRWKHGLIIUDFWLRQ



f f



u x y 

D E §D E · A ¨   ¸ eik Dx Ey d d  O O O O © ¹ f f

³ ³

 

ZKHUH A DOEO LVFDOOHGWKH angular spectrumRI uEHFDXVHLWUHSUHVHQWVWKH VSHFWUDODPSOLWXGHRIWKHSODQHZDYHVWKDWFRPSRVHu7KHFDOFXODWLRQRIAFDQEH SHUIRUPHGLQJHQHUDORQHYHU\SODQHSDUDOOHOWRWKHxySODQHDWDGLVWDQFH zIURP LW 

§D E · A¨   z ¸ ©O O ¹

f f

³ ³ u x y z eik Dx Ey dx dy 

 

f f

5.3.3 Reconstruction of the diffracted field ,QSULQFLSOHDFFRUGLQJWRRXUNQRZOHGJHRIWKHDQJXODUVSHFWUXPZHFDQFDO FXODWHuIRUHYHU\GLVWDQFHz f f



u x y  z

D E §D E · A ¨   z ¸ eik Dx Ey d d  O O O O © ¹ f f

³ ³

 

2QWKHRWKHUKDQGIRUDJLYHQSODQDUFRPSRQHQWRIWKHVSHFWUXPWKHSKDVHGLIIHU HQFHEHWZHHQWZRSRVLWLRQVDWDGLVWDQFH zEHWZHHQWKHP ZKLOHNHHSLQJWKHYDO XHVRIxDQGyXQFKDQJHG LVVLPSO\ SO JzZKHUHE\ 

§D E · A¨   z ¸ ©O O ¹

§D E · A ¨   ¸ eikz ©O O ¹

D E



 

DVZHFDQDOVRFKHFNE\DSSO\LQJWKH+HOPKROW]¶VHTXDWLRQ’u ku WR(T  >*RRGPDQ@ 7KHSODQHZDYHFRPSRQHQWVDUHRUGLQDU\IRUDDQGEYDOXHVVXFKWKDW 

D  E   

 

+RZHYHULIWKHILHOGWKDWDIIHFWVWKHRSHQLQJLVUDSLGO\PRGXODWHGLQVSDFHVXFK DVFDQKDSSHQZLWKGLIIUDFWLRQJUDWLQJVWKHVSHFWUXPRI uFDQEHH[WHQGHGWRYDO XHVRIDDQGEIRUZKLFK 

D  E !  

 

,QWKLVFDVHWKHFRUUHVSRQGLQJSODQHZDYHFRPSRQHQWLVLQKRPRJHQHRXVWKDWLV DQHYDQHVFHQWZDYH+RZHYHULQWKLVW\SHRIVLWXDWLRQDVZHOODVLQWKHFDVHLQ



)RXULHU2SWLFV

ZKLFKDE DYHFWRUWUHDWPHQWRIGLIIUDFWLRQLVQHHGHG )URPWKHNQRZOHGJHRIWKHDQJXODUVSHFWUXPAIRUz WKHILHOGuLQWKHKDOI VSDFHz!FDQEHFDOFXODWHGDV f f

§ D E · ik A ¨   ¸ e ©O O ¹ f f

u x y  z

³ ³



f f

³ ³ A fx  fy  e

DxEy  D E z d D d E 



O





ikz  O fx  O fy

O

  



e



i S f x x  f y y

d fx d fy 

f f

,IzLVODUJHHQRXJKWKHFRQWULEXWLRQRIHYDQHVFHQWZDYHVFDQEHQHJOHFWHGDQG LQWHJUDWLRQFDQEHGHOLPLWHG ZLWKLQDFLUFXODUDUHDGHILQHGE\  7KHILHOG IRU FRRUGLQDWHV z !! O FDQ WKHUHIRUH EH FDOFXODWHG E\ WZR )RXULHU¶V WUDQVIRUPV :LWKWKHILUVWZHREWDLQWKHDQJXODUVSHFWUXPRIu xy ZKLFKLVWKHQPXOWLSOLHG E\WKHtransfer function 



Tz fx  fy

­ ikz ° ®e °¯







 O fx  O fy





IRU fx   fy    O  

 

RWKHUZLVH

DQGILQDOO\ZHDSSO\DQLQYHUVHWUDQVIRUPWRREWDLQu xy]  ,Q SDUWLFXODU ZH FRQVLGHU WKH FDVH RI D VSKHULFDO ZDYH H[SUHVVHG E\ eikrr ZKHUHrLVWKHGLVWDQFHIURPWKHRULJLQRIFRRUGLQDWHV8VLQJLWVV\PPHWU\DQGDS SO\LQJ(T  WRWKLVIXQFWLRQRQWKHSODQHz LWV)RXULHU%HVVHOWUDQVIRUP LV>$EUDPRZLW]DQG6WHJXQ(T@ f

i

S



r

S ³ e O -  SUr dr



U   O 





ZKHUHULVWKHUDGLDOFRRUGLQDWHRIWKHVSDWLDOIUHTXHQF\GHILQHGE\(TV   7KHUHVXOWLVUHDOLIU!ODQGLPDJLQDU\IRUU!O5HWXUQLQJLQ&DUWHVLDQFRRU GLQDWHVWKHSODQHZDYHVSHFWUXPRIRXUVSKHULFDOZDYHLVJLYHQE\ 



A fx f y 

iO







  Of x  O f y



iO  J

DQGLWVGHYHORSPHQWLQSODQHZDYHVLVJLYHQE\>:H\O6WDPQHV@



e

i

S r O

r

§

f f

³ ³

f f

i

J ·

O i  S¨© f x x  f y y  O z ¸¹ e d fx d fy  J

$SSOLFDWLRQVRIWKH)RXULHUWUDQVIRUPWRWKHGLIIUDFWLRQ



7KH PHWKRG RI WKH SODQH ZDYH VSHFWUXP GRHV QRW VROYH WKH SUREOHPV DULVLQJ IURPWKHHGJHHIIHFWVWKDWZHKDYHDOUHDG\GLVFXVVHGIRU.LUFKKRII¶VGLIIUDFWLRQ VLQFHHYHQLQWKLVFDVHILHOGNQRZOHGJHLVUHTXLUHGRQDSODQHEH\RQGWKHRSHQLQJ LW FDQ EH REWDLQHG RQO\ WKURXJK WKH DSSOLFDWLRQ RI rigorous YHFWRU PHWKRGV  +RZHYHULWDOORZVIRULJQRULQJYDULRXVDSSUR[LPDWLRQVLQWURGXFHGIRUWKHFDOFXOD WLRQRIWKH)UHVQHO¶VDQG)UDXQKRIHU¶VGLIIUDFWLRQ0RUHRYHULQWKLVZD\ZHFDQ EULQJEDFNIURPWKHZLQGRZWKDWZKLFKZHKDGH[FOXGHGLQGHSDUWLQJWKDWLVWKH DELOLW\WRUHEXLOGWKHILHOGYHFWRULDOO\DOEHLWDSSUR[LPDWHO\ 6XSSRVHLQIDFWWKDWZHNQRZERWKWKHDPSOLWXGHVExDQGEyRIWKHILHOGRQWKH SODQHz )URPWKHWUDQVIRUPDWLRQ  ZHFDQWKHQGHULYHWZRVSHFWUDODP SOLWXGHV AExDQG AEyIRUHDFKSODQHZDYHRIWKHILHOGVSHFWUXP6LQFHIRUHDFKRI WKHPRQHNQRZVWKHZDYHYHFWRU kWKHFRPSRQHQW AEzFDQEHGHULYHGIURPWKH 0D[ZHOO¶VHTXDWLRQIRUWKHILHOGGLYHUJHQFHLHIURPk ‡ AE IRUZKLFK 



AEz fx  fy 













AEx fx  fy  O fx  AEy fx  fy  O fy 



  O fx  O fy





 

,QDGGLWLRQWKHVSHFWUDODPSOLWXGHVIRUWKHPDJQHWLFILHOG B H FDQEHREWDLQHG IURP WKH )DUDGD\¶V HTXDWLRQ kuAE  ZAB ZAH/P  7DNLQJ WKH LQYHUVH WUDQV IRUPJLYHQE\(T  RIWKHWKUHHFRPSRQHQWVRI AEDQGRI ABLWLVWKXV SRVVLEOHWRREWDLQWKHDPSOLWXGHVRIWKHHOHFWULFDQGPDJQHWLFILHOGVRIWKHZDYHLQ WKHLUWKUHH&DUWHVLDQFRPSRQHQWV)LQDOO\ZHFDQGHULYHWKH3R\QWLQJYHFWRU SR  ERuHRWKDWLVWKHDFWXDOGHSRVLWDU\RIWKHGLIIUDFWHGZDYHHQHUJ\IORZ

5.3.4 Application of the FFT to a spatially delimited field with limited bandwidth 6XPPLQJXSOHWXVDVVXPHWKDWRXUILHOGLVZHOOVSDWLDOO\OLPLWHGDQGZLWKOLP LWHGEDQGZLGWKDQGWU\WRVROYHWKHGLIIUDFWLRQFDOFXODWLRQZLWKWKH))77KHILUVW WKLQJWRGRLVWRFDOFXODWHWKHGLVFUHWHDQJXODUVSHFWUXPWKDWLV § j l A¨  ¨ N'x M ' y ©

· ¸ ¸ ¹

'x' y ¦

N  

n N 

¦

M  

m M 





u$ n ' x  m ' y e

§ jn lm ·  Si ¨  ¸ ©N M¹

  

ZKHUHj ±NN±l ±MM±DQGu$LVWKHILHOGRQWKHRSHQLQJ 7KH DSSOLFDELOLW\ FRQGLWLRQV RI WKLV IRUPXOD DUH WKRVH DOUHDG\ GLVFXVVHG IRU WKH VDPSOLQJWKHRUHP7KXVIRUH[DPSOHLQWKHFDVHRIDSODQHZDYHGLIIUDFWHGE\DQ DSHUWXUH RI GLDPHWHU d LW VKRXOG EH WDNHQ WKDW 'x 'y ! p d ZKHUH p LV WKH QXPEHURIULQJVWKDWZHZDQWWRUHEXLOG7RWKLVHQGLWVKRXOGEHQRWHGWKDWZLVK



)RXULHU2SWLFV

LQJ WR XVH WKH DQJXODU VSHFWUXP WR UHFRQVWUXFW WKH ILHOG DW DQ\ GLVWDQFH IURP WKH RSHQLQJLWLVQHFHVVDU\WKDWWKHHQHUJ\DVVRFLDWHGZLWKWKHVHWRIDOORIWKHH[FOXG HGULQJVEHQHJOLJLEOHWRUHQGHUWKHDOLDVLQJKDUPOHVV,QDGGLWLRQLIVDPSOLQJRI WKHRSHQLQJHGJHLQWKH&DUWHVLDQJULGLVLQVXIILFLHQWO\GHQVHWKHUHVXOWLVVKRZLO\ DOWHUHGE\³IORUDO´IULQJHVWKDWFDQEHYHU\EHDXWLIXOEXWQRWWUXH,IWKHQWKHILHOG LQWKHRSHQLQJLVVSDWLDOO\PRGXODWHGZHKDYHWRYHULI\WKDWZHHIIHFWLYHO\LQFOXGH LWVVSHFWUXP ,QSDUWLFXODULQRSWLFDOLQVWUXPHQWVZHXVXDOO\KDYHWRFRQVLGHUWKHILHOGGLI IUDFWHG E\ DQ DSHUWXUH RI UDGLXV rA ZKRVH ZDYHIURQW LV DSSUR[LPDWHO\ VSKHULFDO ZLWK FXUYDWXUH C 7KH IDVWHVW RVFLOODWLRQV LQ WKH ILHOG DUH QHDU WKH RSHQLQJ HGJH ZLWKDVSDWLDOSHULRGRIaO CrA  _C_rA )RUDQLQFLGHQWSODQHZDYH0DQ VXULSXUVXJJHVWVDVDPSOLQJ'x'ydrADQGLQWKHFDVHRIDFXUYDWXUHGLIIHUHQW IURP]HUR>0DQVXULSXU@



­° r   C  rA  ' x  ' y  PLQ ® A  O   C rA ¯°



½°  ¾ ¿°

 

(T   FDQ EH FDOFXODWHG E\ WKH ))7 WDNLQJ LQWR GXH FRQVLGHUDWLRQ WKH QHFHVVDU\WUDQVODWLRQVDQGYDULDWLRQVRIVFDOH7KLVRSHUDWLRQFDQEHGXSOLFDWHGIRU WKH xDQG yFRPSRQHQWVRIWKHILHOGDQGILQDOO\WKH zFRPSRQHQWRIWKHVSHFWUXP FDQDOVREHGHULYHGIURP(T   /DVWO\WKHILHOGDWDGLVWDQFHzIURPWKHRSHQLQJLVJLYHQE\



u n ' x  m' y  z N  M   

¦ ¦ j 

N M l   



 eikz u NM ' x ' y

§ j l A¨  ¨ N 'x M ' y ©

· ¸ ¸ ¹

ª º  § jO · § lO · » «    ikz «  ¨ ¸ » ¸ ¨¨ § jn lm · © N ' x ¹ © M ' y ¹¸ » i  S¨  ¸ « ¼ e © N M ¹ e ¬

8QIRUWXQDWHO\WKHWHUPZLWKWKHVTXDUHEUDFNHWJURZVLQGHILQLWHO\ZLWKLQFUHDVLQJ z DQG WKHUHIRUH LWV H[SRQHQWLDO EHFRPHV UDSLGO\ RVFLOODWLQJ 7R DYRLG DOLDVLQJ SUREOHPVLWLVWKHQQHFHVVDU\WRLQFUHDVHLQSURSRUWLRQWR zWKHQXPEHU NîMRI VSHFWUXPVDPSOHV2QWKHRWKHUKDQGWKHGLIIUDFWHGZDYHH[SDQGVDQGLWLVWKHUH IRUHQHFHVVDU\WRH[WHQGWKHVDPSOHGDUHDN'xîM'yDURXQGWKHRSHQLQJRUUHFRQ VWUXFWDGHQVHUVSHFWUXPE\PHDQVRI(T   ,QWKHSUHYLRXVH[DPSOHRIDFLUFXODURSHQLQJLWLVQHFHVVDU\WRWDNHWKDW>0DQ VXULSXU@ 

­ ½  zVPD[ ° ° N ' x  M ' y ! PD[ ®rA   ¾  ° °   V PD[ ¯ ¿





 

$SSOLFDWLRQVRIWKH)RXULHUWUDQVIRUPWRWKHGLIIUDFWLRQ



ZKHUH 

VPD[

C rA    C  rA 







O  rA

 

LVWKHDQJXODUFRPSRQHQWWKDWGHOLPLWVWKHVSHFWUXP

5.3.5 Extending the range of application of the angular spectrum method 2QH PHWKRG WR H[WHQG WKH DSSOLFDELOLW\ RI ))7 ZDV LQWURGXFHG E\ 0 0DQ VXULSXU7KLVFRQFHUQVWZRFDVHVWKHILUVWRQHH[WHQGVWKHYDOLGLW\RIWKHFDOFXOD WLRQLQWKH)UDXQKRIHUUHJLRQDOORZLQJXVWROLPLWWKHVDPSOHGDUHDRQWKHRSHQLQJ SODQH7KHVHFRQGFRQFHUQVDYHU\LPSRUWDQWFDVHLQRSWLFVWKHRQHLQZKLFKWKH ILHOGRQWKHRSHQLQJLVFRQVWLWXWHGE\DQDSSUR[LPDWHO\VSKHULFDOZDYH7KLVLVLQ GHHGWKHW\SLFDOIRUPRIWKHILHOGRQWKHRXWSXWSXSLORIDJRRGRSWLFDOLQVWUXPHQW )LUVW FDVH H[WHQVLRQ RI WKH PHWKRG LQ WKH )UDXQKRIHU UHJLRQ :H IROORZ LQ SDUW0DQVXULSXUUHVWDUWLQJIURP(T  DQGUHZULWLQJLWDV f f

u x y  z

eikz

iSOz N fx ³ ³ Ac fx  fy  z e

 f  y

e i S f x x  f y y d fx d fy   

f f

ZKHUH 



Ac fx  fy  z





A fx  fy 



½   º° Nª °­   ikz ®  O fx  O fy  « O fx  O fy » ¾  ¬ ¼ ° ° ¿    e ¯





LVD PRGLILHGDQJXODUVSHFWUXPLQZKLFKWKHFDUULHUH[S ikz KDVEHHQVXEWUDFWHG DQGDSDUDEROLFFXUYDWXUHWHUPLVDGGHG7KHSKDVH kzGLQWKHH[SRQHQWLDOZLWK 

G

  D  E   

N  D  E  





 

ZKHUHDDQGEDUHVWLOOJLYHQE\(TV  UHSUHVHQWVWKHGLIIHUHQFHEHWZHHQ WKHVSKHULFDODQGSDUDEROLFZDYHIURQWWKDWDSSURDFKHVLW7KHFRHIILFLHQWNLVDIUHH 

 &RPSDUHG WR 0DQVXULSXU¶V H[SUHVVLRQ >0DQVXULSXU @ , DGGHG D IDFWRU  WR WKH VHFRQG

WHUP , UHFDOO WKDW O rA  LV WKH DQJXODU UDGLXV RI WKH ILUVW EODFN ULQJ LQ )UDXQKRIHU¶V GLIIUDFWLRQ RI D FLUFXODU DSHUWXUH ZLWK UDGLXV rA DQG LW PXVW EH DEXQGDQW ZLWK UHVSHFW WR WKLV YDOXH



)RXULHU2SWLFV

SDUDPHWHUWRPLQLPL]HWKHPD[LPXPVORSHRIG ODWHULQGLFDWHGZLWK G PD[ ZLWK WKHFKDQJHRIDDQGENLVRQWKHRUGHURIXQLW\DQGLWVRSWLPXPYDOXHZLOOEHGLV FXVVHGDWWKHHQGRIWKLVVHFWLRQ7KLVRSWLPL]DWLRQFDQDOVREHHIIHFWHGWDNLQJLQWR DFFRXQWWKHFXUYDWXUHFRQWULEXWLRQWKDWLVSUHVHQWLQWKHVSHFWUXPA (T  KDVWKHIRUPRIWKHLQYHUVHWUDQVIRUPRIWKHSURGXFWRIWZRIXQF WLRQV DQG WKHUHIRUH FDQEH FRQYHUWHG LQWR WKH FRQYROXWLRQ RI WKHLU LQYHUVH WUDQV IRUPV fxfyoxy UHVSHFWLYHO\ a c x y z





F



^Ac f x  f y  z ` DQG

F



^

iS

i x  y e Oz N Oz N



iSOzN f x   f y  e



`



  



WKDWLV iS

f f



u x y  z

eikz

³ ³

a c x c y c z

f f

ª x  x c   y  y c  º i »¼ dx c dy c    e OzN «¬ Oz N

%\GHYHORSLQJWKHH[SRQHQWLDOZHKDYH u x y  z



e

ikz 

iS   x y Oz N



iS

f f



³

a c x c y c z

f f

i S

i xc  yc e OzN xxc yyc dxc dy c e Oz N Oz N



:HQRZUHSODFHWKHYDULDEOHVx ‫މ‬y ‫މ‬ZLWKX ‫ މ‬x ‫ މ‬OzN Y ‫ މ‬y ‫ މ‬OzN 7KHUHIRUH



u x y  z

i O z N e

ikz 

iS   f f x y Oz N





³ ³ ac OzNX c OzNX c z 

 

f f

ue



iSOzN X c Y c

ei S xX c yY c dX c dY c

ZKHUHWKHLQWHJUDOQRZKDVWKHIRUPRID)RXULHU¶VWUDQVIRUPIURPWKHVSDFHRIWKH ³SRVLWLRQV´X ‫މ‬Y ‫މ‬WRWKDWRIWKH³IUHTXHQFLHV´xy u x y  z



 iO z N e

ikz 

iS   x y Oz N





F

X cY co x  y

^

a c Oz NX c Oz NX c z e

,QFRQFOXVLRQWRXVHWKLVPHWKRGRQHPXVW



iSOzN X c Y c



`



  

$SSOLFDWLRQVRIWKH)RXULHUWUDQVIRUPWRWKHGLIIUDFWLRQ



i VHWWKHLQWHUYDOVDQGWKHVDPSOLQJH[WHQVLRQRIWKHILHOGDWWKHLQLWLDOSODQH,Q SDUWLFXODUWKHVDPSOLQJVKRXOGH[WHQGRYHUDVXUIDFHZLWKVLGHV XD YDZKHUHE\ WKHVDPSOLQJLQWHUYDOVLQWKHVSDFHRIWKHIUHTXHQFLHV fxDQG fyDUHHTXDOWRXD YD7RFKRRVHWKHDSSURSULDWHYDOXHVRIXDDQGYDLWLVQHFHVVDU\WRHVWDEOLVKWKH PD[LPXPIUHTXHQFLHVOVPD[RIWKHLQLWLDOILHOG,QWKHFDVHRIDQDSHUWXUHZLWKD VSKHULFDOZDYHIURQWVPD[LVJLYHQE\(T  7KHQZHFDOFXODWHWKHYDOXHV RI N DQG G PD[  ZLWK WKH SURFHGXUH GHVFULEHG DW WKH HQG RI WKH VHFWLRQ 7KH H[ WHQVLRQWREHVDPSOHGLVILQDOO\JLYHQE\

^

`

X D  YD ! PD[ rA   zG PD[ 



 

7KHVDPSOLQJLQWHUYDOVGHSHQGRQWKHEHKDYLRURIWKHLQLWLDOILHOGDQGIRUH[DP SOH IRU D VSKHULFDO ZDYHIURQW WKH\ PXVW UHVSHFW WKH FRQGLWLRQ   7R WKLV PXVWEHDGGHGWKHFRQGLWLRQ 

'x ' y 

Oz N   rA  zG PD[



 



VRWKDWWKHVDPSOLQJLQWKHIUHTXHQF\VSDFHLVVXIILFLHQWO\H[WHQGHGWRSHUIRUPWKH ODVWWUDQVIRUP1RWHWKDWIRUODUJHzWKLVFRQGLWLRQEHFRPHVLQGHSHQGHQWRIz

ii FDOFXODWHZLWKWKHILUVW))7WKHDQJXODUVSHFWUXPADVLQWKHSUHYLRXVFDVH iii  FDOFXODWH WKH IXQFWLRQ A‫ މ‬IURP (T   DQG WKHQFDOFXODWH a ‫މ‬ZLWK DQ LQ YHUVH))7 iv ODVWO\FDOFXODWHWKHILHOGuIURP(T  ZLWKDQRWKHU))7IURPWKHVFDOHG FRRUGLQDWHV X ‫ މ‬Y ‫މ‬WRWKHFRRUGLQDWHV x y7KHH[WHQVLRQRIVDPSOLQJLQ X ‫ މ‬Y ‫މ‬LV JLYHQE\ XD OzN  YD OzN UHVSHFWLYHO\7KHQWKHILQDOVDPSOLQJLQ x yH[WHQGV RQDUHFWDQJOHRIVLGHVNOzNXDDQGMOzNYDUHVSHFWLYHO\7KLVDOORZVRQHWRFDO FXODWH WKHILHOGRQ ODUJHUGLVWDQFHV WKDQ WKH RUGLQDU\ PHWKRGDV ORQJ DV WKH LQH TXDOLW\  LVUHVSHFWHG 6HFRQGFDVHVSKHULFDOZDYHFDUULHURIWKHILHOGRQWKHRSHQLQJ6XSSRVHWKDW WKHILHOGRQWKHLQLWLDOSODQH WDNHQZLWK z  KDVDVSKHULFDOZDYHFDUULHUZLWK FHQWHURQDSRLQW&ZLWKFRRUGLQDWHV xFyFzF DQGWKHUHIRUHZHIDFWRULWDV



u x y  z

 uo x y e

ikzF 

x  xF   y  yF  zF 



 

ZKHUHWKHDPSOLWXGHV uDQG uoUHSUHVHQWDILHOGFRPSRQHQWRQWKHSODQH xy ,Q RWKHUZRUGVWKH\LQGLFDWHWKHprojectionRIWKHZDYHILHOGRQWKLVSODQH7KHDP  7KLV³SUHFDXWLRQ´GRHVQRWDSSO\LQWKHFDVHRIDORQJLWXGLQDODFRXVWLFZDYH



)RXULHU2SWLFV

)LJ *HRPHWULFVWXG\RID ZDYH ZLWK D VSKHUL FDOFDUULHU

xC

C RS

O

zs

zC

z

xs x



SOLWXGH uoPD\DOVRLQFOXGHDEHUUDWLRQWHUPVUHODWLYHWRDVSKHULFDOZDYHDQGWKH WUDQVPLVVLRQHIIHFWVRIDQDSHUWXUHDW z 7KLVPHWKRGLVXVHIXOIRUFDOFXODWLQJ WKHILHOGQHDUWKHFHQWHURIFXUYDWXUHRIWKHVSKHULFDOZDYHFDUULHU ,QWKHFDVHLQZKLFKWKHILHOGLVQRWDVVLJQHGRQWKHSODQHRIWKHDSHUWXUHEXW UDWKHURQWKHVSKHUHFHQWHUHGDW&DQGSDVVLQJWKURXJKWKHRULJLQZLWKDUDGLXV 

 RS

xF   yF   zF  

 

VRWKDWLWLVNQRZQWKDW us xs  y s u x s  y s  z



zs 

 

ZKHUH 

zs

zc  zc   x s   xs xc  ys    ys yc 

 

WKHDPSOLWXGH uoFDQEHDSSUR[LPDWHO\REWDLQHGIURPWKHDPSOLWXGHRQWKHVSKHUH WKURXJKWKH³JHRPHWULF´SURMHFWLRQ  uo x y T x y

RV ik R  Rs § RV R · e uV ¨ x  xF  xF  V y  yF  yF ¸    R R R © ¹

ZKHUHLWLVWREHQRWHGWKDWWKHGLVWDQFHRLVVWLOODIXQFWLRQRIxDQGy 

R

x  xF   y  yF   zc 



 

ZKLOHWKHFRHIILFLHQW RV RVFDOHVWKHDPSOLWXGHLQIXQFWLRQRIWKHGLVWDQFH RIURP WKHFHQWHU&DQGWKHIXQFWLRQ TUHSUHVHQWVWKHWUDQVPLVVLRQRIDQRSHQLQJRQWKH

$SSOLFDWLRQVRIWKH)RXULHUWUDQVIRUPWRWKHGLIIUDFWLRQ



SODQHz  6LQFHWKHVSKHULFDOFDUULHUFDQJLYHUDSLGRVFLOODWLRQVWRWKHDPSOLWXGHIRUODUJH QXPHULFDODSHUWXUHVZHSHUIRUPDIXUWKHUIDFWRUL]DWLRQVLPLODUWRWKDWRIWKHSUH YLRXVFDVHE\LQWURGXFLQJDQHZDPSOLWXGH

uS x y uo x y

ª x  xF   y  yF   N x  xF   y  yF  »º ikzF «  « »  zF  zF «¬ »¼  e ikzF e

 

ZKHUH DJDLQ N LV D SDUDPHWHU WKDW FDQ EH RSWLPL]HG IRU D PLQLPXP QXPEHU RI VDPSOHVWREHWDNHQRQWKHLQLWLDOSODQHDQGLVRQWKHRUGHURIWKHXQLW7KHH[SUHV VLRQ  LVUHSODFHGZLWK 

G

  D  E    

N  D  E  





 

ZKHUHQRZD  xxF zFE  yyF zF(T  WKHQEHFRPHV

u x y  z



 uS x  y e

iS

N ª x  xF   y  yF  ¼»º OzF ¬«



 

7KHVSHFWUXPARIuLVWKHQWKHFRQYROXWLRQRIWKHWUDQVIRUPV





U p fx f y



^

`

F u p x y 

 

DQG



§ iS N ª x  xF   y  yF  º · « ¼» ¸ F ¨ e OzF ¬ ¨ ¸ © ¹

i

O zc i S e N

O zc f x  f y N



ei S xc f x  yc f y 

ZKHUHE\



A fx f y  z

 i

O zc N





³³ U p [ K e

º  Oz ª  i  S ª xc f x [  yc f y K º iS c « f x [  f y K » ¬ ¼ N ¬ ¼ d [d K









%\GHYHORSLQJWKHDUJXPHQWRIWKHH[SRQHQWLDOWKHLQWHJUDOFDQEHWUDFHGEDFNWR DGLUHFW)RXULHU¶VWUDQVIRUPZKHUHE\



)RXULHU2SWLFV



A fx f y  z





Oz

i

O z c  i  S xc f x  y c f y i S c f x   f y  § O z c Oz · e e N H ¨ fx  fy c ¸ N N N ¹ ©



 

ZKHUH H X Y

F

f x  f y o X Y





h fx f y



^ h f x  f y `





U p fx f y e





i  S xc f x  y c f y  i S

O zc f x  f y N







 

)LQDOO\IURP(T  ZHFDQJHWWKHILHOGDWWKHGLVWDQFH zE\XVLQJWKHLQ YHUVH)RXULHU¶VWUDQVIRUPRI S  O z c § O zc O z · i z  zc  O fx  O fy  fy c ¸e O H ¨ fx N N N ¹ ©    



A fx f y  z



i

ue



i

 ½  S ­°  ª zc ® O fx   O fy º»   O fx   O fy °¾ i  S x f  y f O ¯°  N «¬ ¼ c x c y ¿° e









IRUZKLFK 

u x y  z

F







ª A f x  f y  z º  ¬ ¼

 

%\VXPPLQJXSWKLVPHWKRGDOVRUHTXLUHVWKHDSSOLFDWLRQRIWKUHHWUDQVIRUPV ZLWKWKHIROORZLQJVWHSV

i NQRZLQJWKHDPSOLWXGH uRRIWKHILHOGRQWKHSODQHRIWKHRSHQLQJWRZKLFKWKH VSKHULFDOFDUULHURIWKHHIIHFWLYHDPSOLWXGH uKDVEHHQH[WUDFWHGZLWK(T   WRFDOFXODWHupRQDJULGRISRLQWVZKRVHVSDFLQJVDWLVILHVWKHLQHTXDOLW\ 

­r O ½ '  PLQ ® A   ¾  ¯ GPD[ ¿

 

7KHYDOXHVRINDQG G PD[ ZLOOEHGLVFXVVHGVKRUWO\7KHH[WHQVLRQRIWKHIXQFWLRQ hLQWKHSODQHfxfyIROORZVWKDWRIUpDQGLVJLYHQE\ 

VPD[ O

G PD[    O rA

 

ZKHUH,VWLOOWRRNDPRUHFRQVHUYDWLYHFRQGLWLRQWKDQ0DQVXULSXU 2QWKHRWKHU KDQGIRUWKHH[SRQHQWLDOWHUPhKDVRVFLOODWLRQVZKRVHPD[LPXPIUHTXHQF\LV

$SSOLFDWLRQVRIWKH)RXULHUWUDQVIRUPWRWKHGLIIUDFWLRQ



v

z rc  c V PD[  FRQ rc N



xc   yc  

 

DQGWKHUHIRUHWKHH[WHQVLRQRIWKHVDPSOLQJLQ x yPXVWEHJUHDWHUWKDQWZLFHWKDW RIWKLVYDOXH 

­° z § O · ½° X D  YD ! PD[ ®rA  rc  c ¨ G PD[  ¸ ¾  N© rA ¹ ¿° °¯

 

ii FDOFXODWLQJWKHWUDQVIRUPRIup iii  FDOFXODWLQJ WKH IXQFWLRQ h DQG LWV WUDQVIRUP IURP fx fy WR X Y IURP (TV   $OWKRXJK WKLV WUDQVIRUP LV GLUHFW WKH H[WHQVLRQ RI WKH H VDPSOLQJ LQ WKHVHQHZYDULDEOHVLVDJDLQ XD YD7KHUHIRUHWKHVDPSOLQJRIVSHFWUXP ARIWKH ILHOGLQLWVYDULDEOHVfxfyH[WHQGVRQXDN OzF YDN OzF UHVSHFWLYHO\ iv  )LQDOO\ FDOFXODWLQJ WKH VSHFWUXP A DW GLVWDQFH z ZLWK(T   DQG LWV LQ YHUVHWUDQVIRUP7KHQWKHILQDOVDPSOLQJRIWKHILHOGuLQxyH[WHQGVRQDUHFWDQ JOHRIVLGHVNOzF NXD DQGMOzF NYD UHVSHFWLYHO\ &DOFXODWLRQSURFHGXUHRI NDQG G PD[ 7KLVLVQRWWKHVDPHDVPDNLQJDILWEH WZHHQDVSKHUHDQGDSDUDERORLGEXWUDWKHUPLQLPL]HVWKHDEVROXWHPD[LPXPYDO XHRIWKHVORSHRIG)RUGHWDLOVVHHWKHZRUNE\0DQVXULSXU  ,QVKRUWLWLV QHFHVVDU\WRHVWDEOLVKWKHFRUUHVSRQGLQJLQWHJUDWLRQLQWHUYDOLQWKHYDULDEOHVDDQG ERI(TV  DQG  IRUGGHSHQGLQJRQWKHFDVHFRQVLGHUHG,QSUDF WLFHIRUWKHSRODUV\PPHWU\RIGLWLVVXIILFLHQWWRFRQVLGHUDVLQJOHUDGLDOYDULDEOH UZLWK”U”U”U”ZKHUHUDQGUDUHWKHOLPLWVRILQWHJUDWLRQ'LIIHUHQ WLDWLQJGZLWKUHVSHFWWRURQHJHWVDQHZIXQFWLRQ G :LWKDIXUWKHUGHULYDWLRQLQ UZHKDYHGHWHUPLQHGWKHSRVLWLRQUPRIWKHPD[LPXP G P RI G ZKLFKPD\RU PD\ QRW IDOO LQWR WKH LQWHJUDWLRQ LQWHUYDO ,Q DGGLWLRQ WKH YDOXHV G  G U  G  G U DUHDOVRGHWHUPLQHG(DFKRIWKHVHTXDQWLWLHVGHSHQGVRQWKHSDUDPH WHUN2QHVKRXOGWKHUHIRUHORRNIRUWKHYDOXHRINVRWKDWWKHPD[LPXPEHWZHHQ G   G  DQGHYHQ G P ZKHQUPLVEHWZHHQUDQGULVWKHPLQLPXPSRVVLEOH

5.3.6 Talbot’s effect :KHQDZDYHLVWUDQVPLWWHGE\DVHWRIHTXDODSHUWXUHVGLVWULEXWHGWRIRUPDSH ULRGLFILJXUHRUPRUHJHQHUDOO\ZKHQWKHZDYHLVLPSULQWHGZLWKDSHULRGLFPRG XODWLRQ LQ DPSOLWXGH DQGRU SKDVH HLWKHU IRU WUDQVPLVVLRQ RU IRU UHIOHFWLRQ WKLV VDPHPRGXODWLRQLVSHULRGLFDOO\UHSURGXFHGDORQJWKHGLUHFWLRQRISURSDJDWLRQRI WKHZDYH7KLVHIIHFWZDVGHVFULEHGLQE\+)7DOERWDQGEHDUVKLVQDPH >7DOERW@ &RQVLGHUWKHFDVHLQZKLFKRQWKHSODQH z WKHILHOGLVSHULRGLFDORQJWZR



)RXULHU2SWLFV

&DUWHVLDQGLUHFWLRQV xDQG y RUWKRJRQDOWRWKH zD[LV ZLWKSHULRGV XoDQG YoUH VSHFWLYHO\,ILWVVSHFWUDOEDQGLVOLPLWHGLWZLOOFRQVLVWRIDILQLWHQXPEHURISXOVHV DWIUHTXHQFLHV fn  n XoDQG fm m YoZLWK n mLQWHJHU7KHQ(T  FDQEH UHGXFHGWRWKHVXPRIDILQLWHQXPEHURISODQHZDYHV  U x y  z

N

M

¦ ¦

  An m eikz  O n X o  O m Yo ei  S xn X o  ym Yo   

n N m M

ZKHUHAnmDUHWKHDPSOLWXGHVDVVRFLDWHGZLWKWKHVHZDYHVDQGZKHUHNDQGMDUH LQWHJHUV ODUJH HQRXJK WR LQFOXGH DOO VSHFWUXP IUHTXHQFLHV )URP WKLV H[SUHVVLRQ ZH FDQ REVHUYH WKDW IRU HDFK SRVLWLRQ z WKH ILHOG UHWDLQV LWV SHULRGLFLW\ LQ WKH WUDQVYHUVHGLUHFWLRQVEXWWKHDPSOLWXGHVRIHDFKZDYHDUH PXOWLSOLHGE\DSKDVH WHUPWKDWGHSHQGVRQ zDQGWKHUHIRUHWKHILHOGSURILOHYDULHVZLWK zDVLQDNLQGRI NDOHLGRVFRSH ,IKRZHYHU N Xo M YoOWKDWLVLIWKHVSHFWUXPLVVXIILFLHQWO\OLPLWHGVR WKDWWKHSDUD[LDODSSUR[LPDWLRQLVYDOLG(T  FDQEHVLPSOLILHGLQ

 U x y  z eikz

N

 i Sn 

M

¦ ¦

Anm e

zO zO iSm  X o Yo  ei  S xn X o  ym Yo

  

n N m M

&RQVLGHUQRZWKHFDVHLQZKLFKWKHSHULRGV XoDQG YoDUHLQDUDWLRQDOUHOD WLRQVKLS 

QX o

P

PYo 

ZKHUHQPDUHFRSULPHLQWHJHUVVRWKDWPLVWKHOHDVWFRPPRQPXOWLSOHRIWKHWZR SHULRGV7KHQDWWKHGLVWDQFH]qVXFKWKDW 

zq

q

P ZLWKqLQWHJHU O

 

ZHKDYHWKDWWKHILUVWH[SRQHQWLDOLQ(T  EHFRPHV 

Tnm

e



iSq n  Q  m P 



 

7KHUHIRUHIRU q LHHYHQZHKDYHWKDW Tnm XQLIRUPO\IRUDOORI WKHSODQHZDYHVRIWKHVSHFWUXPLQRWKHUZRUGVDWWKHFRUUHVSRQGLQJGLVWDQFHV]q WKHILHOG ULVLGHQWLFDOWRWKHLQLWLDORQH7KLVPHDQVWKDWWKHILHOGKDVDSHULRGLF EHKDYLRUDORQJWKHzD[LVZLWKSHULRGPO 2WKHU q YDOXHV DOVR KDYH LQWHUHVWLQJ VLWXDWLRQV )RU VLPSOLFLW\ OHW XV FRQVLGHU WKHRQHGLPHQVLRQDOFDVHZLWK Q DQG P ZKHUHPRGXODWLRQLVDORQJWKH y

$QDO\VLVRIRSWLFDOV\VWHPVE\PHDQVRIWKHWKHRU\RIOLQHDUV\VWHPV



D[LV6RLIqLVDQRGGLQWHJHUHJZKHQz YROZHKDYHWKDWTm   m)XU WKHU VXSSRVH WKDW RXU SHULRGLF IXQFWLRQ f y  LV FRPSRVHG RI WKH UHSHWLWLRQ RI DQ LVRODWHGILJXUH fR y RQDQXOOEDFNJURXQG,WVVSHFWUXP FFDQEHFRQVLGHUHGWREH WKHSURGXFWRIWKHFRPEIXQFWLRQIRUWKHVSHFWUXP FRRIWKHVLQJOHILJXUH/DVWO\ ZHDVVXPHWKDWFRLVZLGHDQGUHJXODUZLWKUHVSHFWWRWKHGHOWDVDPSOLQJ$WzGLV WDQFHVIRUZKLFK qLVDQRGGLQWHJHUWKHVSHFWUXPFDQWKHQEHGHFRPSRVHGLQWRD VXPRIWZRFRPEVRQHIRUmHYHQDQGRQHIRUmRGGHDFKPXOWLSOLHGE\WKHVSHF WUXP FR7KHWZRFRPEVWKHUHIRUHKDYHSHULRGYRWKDWFRUUHVSRQGLQJWRRGG m LVVKLIWHGE\YRDQGLWVVLJQLVLQYHUWHGE\WKHPXOWLSOLFDWLRQZLWK Tm,QWKH y VSDFHHDFKRIWKHWZRFRPEVLVWUDQVIRUPHGLQWRDVHULHVRIGHOWDVZLWKSHULRGYR DQGWKHPXOWLSOLFDWLRQE\WKHFRVSHFWUXPEHFRPHVDFRQYROXWLRQRIWKHGHOWDZLWK fR JLYLQJ ULVH WR WKH UHSHWLWLRQ RI ILJXUHV ZLWK WKH VDPH ZLGWK DQG VKDSH DV WKH RULJLQDORQHEXWZLWKDKDOYHGSHULRG+RZHYHULQRXUFDVHWKHGHOWDLQ yFRUUH VSRQGLQJWRWKHRGG mFRPEKDVDPLQXVVLJQGXHWR TDQGDSKDVHSiyYRGXH WRWKHWUDQVODWLRQZKLFKEHFRPHVDIXUWKHUDOWHUQDWLQJVLJQEHWZHHQDGHOWDDQGWKH QH[W )LQDOO\ WKH ILHOG DW GLVWDQFH z FRQVLGHUHG KHUH LV WKH VXP RI WKHVH WZR VH TXHQFHVRIILJXUHV7KHUHIRUHIRUWKHK\SRWKHVLVPDGHRQFRWKHUHLVDVXEVWDQWLDO FDQFHOODWLRQLQWKHVXPRIWKHILJXUHVWKDWDUHORFDWHGLQWKHRULJLQDOSRVLWLRQDQG ZKDWDSSHDUVLVDILJXUHVLPLODUWRWKHLQLWLDORQHEXWVKLIWHGE\DKDOISHULRG ,QWKHFDVHWKHQLQZKLFKqLVKDOILQWHJHUIRUH[DPSOHZKHQz YR O WKH VSHFWUXPFFDQVWLOOEHGHFRPSRVHGLQWRVHWVRIGHOWDZLWKSHULRGYRDQG 

Tm

 e iSm  ZKHUHE\Tj Tj iZLWKjLQWHJHU

 

$JDLQLQWKLVFDVHWKHILHOGRQWKHREVHUYDWLRQSODQHFDQEHFRQVLGHUHGDVWKHVXP RIWZRSHULRGLFUHSHWLWLRQVRIfR1RZWKHVXPGRHVQRWLQYROYHDGHOHWLRQEXWWKH YDULRXVILJXUHVDOWHUQDWHZLWKDPSOLWXGHò i fRDQGò i fRZLWKDKDOYHGSH ULRGFRPSDUHGWRWKHILJXUHRQWKHLQLWLDOSODQH

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coherentLOOXPLQDWLRQKRZHYHUZHVKDOOVHHWKDW HYHQ LQ WKH FDVH RI incoherent LOOXPLQDWLRQ WKH SULQFLSOH RI OLQHDU VXSHUSRVLWLRQ



)RXULHU2SWLFV

FDQDSSO\IRUWKHLQWHQVLW\UDWKHUWKDQWKHDPSOLWXGH 7KHUHIRUHWKHLPDJHIRUPLQJSURSHUWLHVRIDQRSWLFDOV\VWHPFDQEHDQDO\]HG DFFRUGLQJWRWKHWKHRU\RIOLQHDUV\VWHPVIRUZKLFKWKHIROORZLQJSURSHUW\LVUH TXLUHG/HWXVVXSSRVHWKDWWKHIXQFWLRQ g$ x$y$ UHSUHVHQWVWKHLQSXWVLJQDODQG g% x%y% WKHRXWSXWRQHE\DV\VWHPGHILQHGE\DQRSHUDWRUTVRWKDW g % x%  y % T ^g $ x$  y$ ` 



 

7KHV\VWHPLVVDLGWREHOLQHDULIIRUDQ\IXQFWLRQfDQGgDQGDQ\FRQVWDQWVaDQG bZHKDYH 

T ^ af x$  y$  bg x$  y$ ` aT ^ f x$  y$ `  bT ^ g x$  y$ `   

7KLV SURSHUW\ DOORZV IRU WUHDWLQJ WKH V\VWHP LQ WHUPV RI D UHVSRQVH WR FHUWDLQ ³HOHPHQWDU\´IXQFWLRQVWKDWIRUPDEDVLVLQZKLFKWKHLQSXWVLJQDOVFDQEHGHFRP SRVHG ,Q SDUWLFXODU ZH DUH DOUHDG\ DFFXVWRPHG WR GHFRPSRVLQJ DQ LQSXW LPDJH LQWRLWVREMHFWSRLQWVDQGWKLVUHVXOWVLQUHSUHVHQWLQJWKHJHQHULFIXQFWLRQ g$ZLWK WKHFRQYROXWLRQLQWHJUDO>*RRGPDQ@ f f



g $ x$  y $

³ ³

g $ [ K G x$  [ y$  K d [d K 

 

f f

ZHFDQWKHUHIRUHFRQVLGHU g$ [K DVDZHLJKWDVVLJQHGWRWKHHOHPHQWDOIXQFWLRQ G x$[ y$K ,IZHQRZDSSO\WKHWUDQVIRUPDWLRQ TWR g$ x$y$ WKDQNVWRWKH OLQHDULW\RI TH[SUHVVHGE\(T  WKLVRSHUDWRUFDQEHEURXJKWZLWKLQWKHLQ WHJUDOWRDFWRQWKHLQGLYLGXDOGDQGZHJHW f f



g % x%  y %

³ ³

g $ [ K T ^G x$  [ y$  K ` d [d K 

 

f f

)LQDOO\ ZH LQWURGXFH WKH IXQFWLRQ h x% y%[ K  UHSUHVHQWLQJ WKH V\VWHP UH VSRQVHDWSRLQW x% y% LQWKHLPDJHVSDFHWRDQLPSXOVHVLJQDOFHQWHUHGRQWKH REMHFWSRLQW [K 7KLVIXQFWLRQLVFDOOHGWKHLPSXOVHUHVSRQVHRIWKHV\VWHPLWLV 

h x%  y%  [ K T ^G x$  [ y$  K ` 

 

IRUZKLFKE\H[FKDQJLQJWKHYDULDEOHV[Klx$ y$(T  EHFRPHV f f



g % x%  y %

³ ³

f f

g $ x$  y$ h x%  y %  x$  y$ dx$ dy$ 

 

$QDO\VLVRIRSWLFDOV\VWHPVE\PHDQVRIWKHWKHRU\RIOLQHDUV\VWHPV



,QWKHSDUWLFXODUFDVHLQZKLFKWKHLPSXOVLYHUHVSRQVH hLV spatially invariant ZKHUHE\ 

h x %  y %  x$  y $

h x%  mA x$  y%  mA y$ 

 

ZKHUHmALVWKHPDJQLILFDWLRQFRQVWDQWEHWZHHQWKHFRRUGLQDWHVRQWKHREMHFWVDQG LPDJH SODQHV (T   VLPSO\ EHFRPHV WKH FRQYROXWLRQ RI WKH LQSXW VLJQDO ZKRVHVFDOHLVHQODUJHGE\DIDFWRU mAZLWKWKHLPSXOVHUHVSRQVHIXQFWLRQRIWKH V\VWHP f f



g % x%  y %

³ ³

g $ x $  y $ h x%  x $  y%  y $ dx $ dy $ 

 

f f

ZKHUH,LQWURGXFHGWKHWUDQVIRUPDWLRQV x $



mA x$ 

y $

mA y$ 

 

DQGWKHIXQFWLRQ 

g $ x $  y $

§ x  y · g $ ¨ $  $ ¸   mA © mA mA ¹

 

7KHVHWUDQVIRUPDWLRQVDUHXVHIXOLQWKHVWXG\RILPDJHIRUPDWLRQDQGZLOOEHXVHG LQWKHIROORZLQJ $QRSWLFDOV\VWHPZLWKDVSDWLDOO\LQYDULDQWLPSXOVHUHVSRQVHLVFDOOHG isoplanatic+RZHYHUGXHWRDEHUUDWLRQVWKHLPSXOVHUHVSRQVHJHQHUDOO\YDULHVZLWKWKH SRVLWLRQ RI WKH REMHFW SRLQW FRQVLGHUHG VR RSWLFDO V\VWHPV DUH UDUHO\ LVRSODQDWLF DORQJWKHLUHQWLUHILHOGRIYLHZWKLVFDQEHUHPHGLHGE\GLYLGLQJWKHILHOGLQWRDS SUR[LPDWHO\LVRSODQDWLFUHJLRQVRQZKLFKWRHVWLPDWHWKHSHUIRUPDQFHRIWKHV\V WHP (T  DOORZVIRUDQDO\]LQJDQRSWLFDOLVRSODQDWLFV\VWHPLQVSHFWUDOWHUPV LQGHHGLQWKHVSDFHRIVSDWLDOIUHTXHQFLHVLWVLPSO\EHFRPHVWKHSURGXFW 



G% f x  f y









c f x  f y  7) f x  f y G$

 

c UHSUHVHQWVWKHWUDQVIRUPRI g $ x $  y $ DQGWKHIXQFWLRQ7)FDOOHG ZKHUH G$ WKH transfer functionLVWKHWUDQVIRUPRI h7KLVWUDQVIHUIXQFWLRQJHQHUDOO\FRP SOH[ UHSUHVHQWV WKH HIIHFW RI ILOWHULQJ RI WKH VSDWLDO IUHTXHQFLHV WKDW WKH V\VWHP PDNHV RQ LQSXW VLJQDOV VLPLODUO\ WR ZKDW KDSSHQV IRU H[DPSOH LQ DQ HOHFWURQLF DPSOLILHURQWKHWHPSRUDOIUHTXHQFLHVRIWKHDPSOLILHGVLJQDO ,QWKHQH[WVHFWLRQVVHYHUDOIXQFWLRQVZLOOEHLQWURGXFHGWRFKDUDFWHUL]HWKHLP DJH TXDOLW\ SURGXFHG E\ RSWLFDO V\VWHPV LQ WZR OLPLW FDVHV RI LOOXPLQDWLRQ FR



)RXULHU2SWLFV

KHUHQW DQG LQFRKHUHQW )RU FRKHUHQW LOOXPLQDWLRQ ZH ZLOO VWXG\ WKH amplitude point spread function $36)  FORVHO\ OLQNHG WR WKH LPSXOVH UHVSRQVH DQG LWV WUDQVIRUPDWLRQWKH coherent transfer function &7) ,QWKHFDVHRILQFRKHUHQWLO OXPLQDWLRQZHKDYHWKHIXQFWLRQVFDOOHGWKH point spread function 36) DQGLWV WUDQVIRUPWKH optical transfer function 27) ZKRVHPRGXOHLVFDOOHGWKH modulation transfer function 07)  DQG LWVSKDVH WKH phase transfer function 37)  :KHQ WKH V\VWHP LV QRW LVRSODQDWLF WKH 36) LV UHSODFHG E\ WKH point response function 35)  7KH HYDOXDWLRQ RI WKH SHUIRUPDQFH RI RSWLFDO LQVWUXPHQWV LQ WHUPV RI VSDWLDO IUHTXHQF\DQDO\VLVDQGOLQHDUV\VWHPVWKHRU\LVUHODWLYHO\UHFHQW7KHILUVWVWXGLHV LQ WKLV GLUHFWLRQ ZHUH PDGH E\ (UQVW $EEH   DQG E\ /RUG 5D\OHLJK  ZKRUHFRJQL]HGWKDWWKHILQLWHRSHQLQJRIDQLQVWUXPHQWFRQVWLWXWHGD FXWRIWKHKLJKHUVSDWLDOIUHTXHQFLHVJHQHUDWHGE\DQREMHFWILJXUH6XEVHTXHQWO\ GXULQJWKHVWKHXVHRIVLQXVRLGDOWHVWSDWWHUQVIRUWKHDQDO\VLVRIRSWLFDOV\V WHPVEHFDPHFRPPRQDQG30'XIILHX[ZDVWKHILUVWWRSXEOLVKDERRNLQ RQ WKH XVH RI )RXULHU¶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ayleigh’s criterion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ptical Signal RUInformation ProcessingZKLFKH[SORLWV WKHSDUDOOHOQDWXUH LQWKHVSDWLDOFRRUGLQDWHV RIWKHVLJQDOWUDQVPLVVLRQ,PSRUWDQW FRQWULEXWLRQV WR WKLV GLVFLSOLQH ZHUHJLYHQE\ 1RUEHUW:LHQHU>:LHQHU@ 3 (OLDV >(OLDV @ 3HWHU )HOOJHWW >)HOOJHWW @ *LXOLDQR 7RUDOGR GL )UDQFLD >7RUDOGRGL)UDQFLD@DQG(/2¶1HLOO>2¶1HLOO@ 7KHUHVXOWRIWKHVHVWXGLHVZDVWKHGHYHORSPHQWRIVSDWLDORSWLFDOILOWHULQJWHFK QLTXHVLPDJHUHFRJQLWLRQLPDJHUHVWRUDWLRQKRORJUDSK\DSHUWXUHV\QWKHVLVRSWL FDOSURFHVVLQJPHWKRGVRIQRQRSWLFDOVLJQDOVHWFRQZKLFKH[WHQVLYHOLWHUDWXUH H[LVWV+HUHLQDGGLWLRQWR*RRGPDQ  ,UHIHUUHDGHUVWRWKHERRNE\$QWKR Q\9DQGHU/XJW Optical Signal Processing >9DQGHU/XJW@DQGWKDWE\)UDQ FLV76:LOOLDPVDQG%HFNOXQG@

$QDO\VLVRIRSWLFDOV\VWHPVE\PHDQVRIWKHWKHRU\RIOLQHDUV\VWHPV



5.4.1 Degrees of freedom of optical signals 7KHUHDUHVHYHUDOZD\VWRFKDUDFWHUL]HDVLJQDO7KHILUVWLVWRJLYHWKHLUIRUPLQ VSDFH RU LQ WLPH WKRVH WKDW DUH SK\VLFDOO\ UHDOL]DEOH KDYH D OLPLWHG H[WHQVLRQ LQ VSDFHDQGWLPHWKRXJKWKH\DUHRIWHQLGHDOL]HGZLWKXQOLPLWHGH[WHQVLRQVVXFKDV SODQHZDYHV $QLPSRUWDQWSDUDPHWHULVWKHUHIRUHWKHWLPHGXUDWLRQ TRRUWKHH[WHQVLRQ XR YR«RIWKHVLJQDOLQWKHVSDFH$QRWKHUZD\LVWRGHVFULEHLWVVSHFWUXPZKLFK LQWXUQFDQEHFKDUDFWHUL]HGE\DOLPLWHGEDQGZLGWK WtLQWKHWHPSRUDOIUHTXHQ FLHVDQG Wx Wy«LQWKHVSDWLDORQHVLQWURGXFHGLQ†7KLVOLPLWDWLRQPD\ DOVREHGXHWRWKHILQLWHUHVROXWLRQRIWKHGHWHFWRU ,Q WKH SUHFHGLQJ VHFWLRQV ZH KDYH DSSUHFLDWHG WKHVH FRQFHSWV LQ FRQQHFWLRQ ZLWK WKH VDPSOLQJ WKHRUHP IRU ZKLFK VLJQDOV ZLWK OLPLWHG VSHFWUDO EDQG FDQ EH UHFRQVWUXFWHG H[DFWO\ E\ VDPSOLQJ WKH VLJQDO ZLWK D IUHTXHQF\ KLJKHU WKDQ WKH 1\TXLVW¶VIUHTXHQF\GHILQHGE\(TV  ,IWKHQWKHVLJQDOLVDOVRGHOLPLWHG LQWKHWLPH RUVSDFH GRPDLQLWFDQEHGHWHUPLQHGE\DILQLWHQXPEHU NRIVDP SOHVLQRWKHUZRUGVLWFDQEHVDPSOHGZLWKRXWLQIRUPDWLRQORVV7REHULJRURXV GHOLPLWHGVLJQDOVLQERWKGRPDLQVRIWLPH RUVSDFH DQGIUHTXHQFLHVDUHQRWSRV VLEOHEXWLQSUDFWLFHWKLVDVVXPSWLRQFDQEHREWDLQHGZLWKUHDVRQDEOHDFFXUDF\ 7KHQIURPWKHVDPSOLQJWKHRUHPZHNQRZWKHPLQLPXPQXPEHURIVDPSOHVUH TXLUHGWRUHSUHVHQWWKHVLJQDODFFXUDWHO\,WLVJLYHQIRUHDFKFRRUGLQDWHE\ 

Nt

ToWt  N x

 X oWx  N y

YoW y  " 

 

)RUDQLPDJHRQWKHxySODQHWKHWRWDOQXPEHURIVDPSOHVLVWKHQ 

N

 X oWxYoW y 

 

7KH SURGXFWV EHWZHHQ WKH VSDFH H[WHQVLRQ DQG WKH FRUUHVSRQGLQJ EDQGZLGWK XoWx YoWy ZKLFK DUH FDOOHG WKH length bandwidth product, height bandwidth product DUH D VWDQGDUG PHDVXUH RI WKH FRPSOH[LW\ RI D VLJQDO DQG WKH\ SOD\ DQ LPSRUWDQWUROHLQRSWLPL]LQJRSWLFDOV\VWHPV ,QDGGLWLRQHDFKVDPSOHFDQEHFKDUDFWHUL]HGE\DYDULHW\RIRWKHUSDUDPHWHUV )RU H[DPSOH WR GHILQH DQ LPDJH LW LV QHFHVVDU\ WR JLYH LWV LQWHQVLW\ ZKLFK LQ WXUQ FDQ EH TXDQWL]HG LQ D ILQLWH QXPEHU a RI GLVFUHWH OHYHOV WKDW FRQVWLWXWH WKH grayscale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



)RXULHU2SWLFV

FRQWUROV\VWHPWKDWSHUIRUPVWKLVRSHUDWLRQ ,QDGGLWLRQWRWKHLQWHQVLW\ZHFDQWKHQDVVLJQcOHYHOVWRFRORUsOHYHOVWRVDWX UDWLRQ pOHYHOVWRSRODUL]DWLRQKHQFH g  acspYDOXHVDUHUHTXLUHGWRFKDUDFWHUL]H HDFKVDPSOH )LQDOO\FRQVLGHULQJDOORIWKHVHVDPSOHVZHKDYHWKDWWKHQXPEHURIdegrees of freedomRIDVLJQDOLVJLYHQE\gN

5.4.2 Behavior of a lens in the paraxial approximation ,QWKHVWXG\RIWKHEHKDYLRURIDQRSWLFDOV\VWHPZLWKLQWKHWKHRU\RIOLQHDUV\V WHPVZHWDNHDVDPRGHOWKHFDVHRIDWKLQOHQVLPPHUVHGLQDLU7KHUHVXOWVZH ZLOOREWDLQKHUHDUHLQSDUD[LDODSSUR[LPDWLRQDQGPRQRFKURPDWLFUDGLDWLRQ7KH VSDWLDOO\ LQFRKHUHQW TXDVLPRQRFKURPDWLF LOOXPLQDWLRQ FDVH ZLOO EH GLVFXVVHG LQ †DQGWKHUHDIWHU ,Q WKH VWLJPDWLF FDVH LQ ZKLFK D SODQH ZDYH LV WUDQVIRUPHG LQWR D VSKHULFDO ZDYHZHFRXOGUHSUHVHQW WKHHIIHFWRI DOHQV RQWKHZDYHDPSOLWXGH DWWKHRXW SXWSXSLOSODQHZLWKWKHIXQFWLRQ



i

e



S nD f  f  r  f  Oo

p r 

ZKHUH OoLVWKHYDFXXPZDYHOHQJWK nDLVWKHUHIUDFWLRQLQGH[RIDLU fLVWKHGLV WDQFHRIWKHFRQYHUJHQFHSRLQWIURPWKHSODQHRIWKHSXSLOSRVLWLYHLIWKHZDYHLV FRQYHUJHQWDQGQHJDWLYHLILWLVGLYHUJHQWrLVWKHGLVWDQFHIURPWKHRSWLFDOD[LVRQ WKH SODQH RI WKH SXSLO ILQDOO\ p r  UHSUHVHQWV WKH SXSLO WKDW LV WKH WUDQVPLVVLRQ IXQFWLRQDVVRFLDWHGZLWKWKHOHQVDSHUWXUH $ UD\ RI OLJKW SDVVLQJ WKURXJK D WKLQ OHQV HQWHULQJ ZLWK FRRUGLQDWHV x  y  HPHUJHV URXJKO\ ZLWK WKH VDPH FRRUGLQDWHV WKHUHIRUH ZH FDQ FROODSVH WKH WZR SULQFLSDOSODQHVRQDVLQJOHVXUIDFH6 KRZHYHULWLVVXEMHFWHGWRDSKDVHGHOD\ SURSRUWLRQDOWRWKHWUDYHUVHGWKLFNQHVVZKLFKLQSDUD[LDODSSUR[LPDWLRQLVJLYHQ E\ 

I x y

Sn S n' R  D x   y   Oo Oo f





 

ZKHUHnLVWKHUHIUDFWLRQLQGH[RIWKHOHQV'RLVWKHWKLFNQHVVRQLWVD[LVZKLFKLV DWSRLQW   ZKLOHfLVWKHIRFDOGLVWDQFH6LQFHWKHWHUPLQ'RLVFRQVWDQWLHLW GRHVQRWGHSHQGRQWKHWUDQVYHUVHFRRUGLQDWHVLQZKDWIROORZVLWZLOOEHQHJOHFW HGIRUVLPSOLFLW\ /HWXVQRZH[DPLQHWKHFRQILJXUDWLRQVKRZQLQ)LJLQGHWDLO:KHQFDOFX ODWLQJWKHZD\LQZKLFKWKHILHOGFKDQJHVIURPWKHSODQH S$WRWKHSODQHS%WKUHH

$QDO\VLVRIRSWLFDOV\VWHPVE\PHDQVRIWKHWKHRU\RIOLQHDUV\VWHPV

SA



SB

S

P ( x,y )

ZA

ZB

f

f

sB

sA



)LJ'LVWDQFHVIRUWKHFDOFXODWLRQRIWKHGLIIUDFWLRQEHWZHHQWKHSODQHVS$DQGS%

RSHUDWLRQVPXVWEHSHUIRUPHG7KHILUVWLVWKHSURSDJDWLRQIURP S$WRSWKHSODQH RI WKH OHQV 7KH VHFRQG LV WKH PXOWLSOLFDWLRQ E\ WKH GHSKDVLQJWHUP SURGXFHGE\ WKHOHQVDQGWKHDPSOLWXGHWHUPRILWVSXSLO p x y e



i

S   x y Of







 

ZKHUHO  OonDDQGpLVHTXDOWRLQWKHDUHDRIWKHOHQVDQGRXWVLGH7KHWKLUG RSHUDWLRQ LV WKH SURSDJDWLRQ IURP S WR S% 2YHUDOO XVLQJ WKH )UHVQHO GLIIUDFWLRQ LQWHJUDO  ZHKDYH i

 u% x%  y %

S s$  s % O

e  O s$ s %

³³ ³³ p x y u$ x$  y$ e



i I x I y

dx$ dy$ dx dy 

S S





 

ZKHUHIxDQGIyDUHIXQFWLRQVRIWKHYDULRXVFRRUGLQDWHV)RUIxZHKDYH



Ix

 x  x%  x  º» S ª x  x$ «    O« s$ s% f » ¬ ¼



%\UHDUUDQJLQJWKHWHUPVZHKDYH 

Ix

§x S ª x$  x% x  x ·º     x ¨ $  % ¸ »  « O ¬ s$ s% S © s$ s % ¹ ¼

ZKHUHIRUEUHYLW\ZHLQWURGXFHGWKHSDUDPHWHUSZKHUHE\

 





)RXULHU2SWLFV

 S

f   Z$ Z%  s$ s% f

     s$ s% f

S s$ s%

f  f  Z$Z%

 



$QH[SUHVVLRQVLPLODUWR(T  DSSOLHVWRIy

5.4.3 Image formation ,QWKHVSHFLDOFDVHLQZKLFKS WKHSODQHVS$DQGS%DUHWZRSODQHVFRQMX JDWHGE\WKHOHQVVRWKDWs$ sos% si(T  EHFRPHV



u% x%  y %

e

i

S so  si O C%

x%  y%

u³³ $36) x%  y %  x $  y $ C $ x $  y $ uo x $  y $ dx $ dy $ 



So





 

ZKHUHWKHWUDQVIRUPDWLRQV  DQG  ZHUHUHVSHFWLYHO\XVHGIRUWKHFR RUGLQDWHVRIWKHREMHFWSRLQWDQGIRUWKHDPSOLWXGHRIWKHILHOGRQWKHREMHFWSODQH ,QDGGLWLRQIRUEUHYLW\,LQWURGXFHGWKHWHUPVRIILHOGFXUYDWXUH C $



i

S

 e soOmA

x$  y $

S

e siO i

 C%

x%  y %





 

$36)LVWKH amplitude point spread functionRIWKHOHQVREWDLQHGE\LQWHJUDWLQJ RQO\WKHWHUPVLQxDQGy  $36) x%  y %  x $  y $

  O so si

³³ p x y e



i  S x f x  y f y

dx dy

S









P fx f y  O  so si

 

ZKHUHQRZ 

fx

 § x $ x ·  % ¸ ¨ O © mA so si ¹

fy

 § y $ y ·  % ¸ ¨ O © mA so si ¹

 

  8QOLNH RWKHU WH[WV , SUHIHUUHG H[SOLFLWO\ WR GLVWLQJXLVK KHUH WKH LPSXOVLYH UHVSRQVH IURP WKH

$36),QIDFWWKHWHUPVRIWKHFXUYDWXUHRIILHOGLQDGGLWLRQWRWKHSKDVRURISURSDJDWLRQDUH RIWHQLPSOLFLWO\LQFRUSRUDWHGLQWKLVUHVSRQVHRULQWKHILHOGDPSOLWXGHV+HUHLQVWHDG,FRQVLGHU WKHVHWHUPVDVVHSDUDWH

$QDO\VLVRIRSWLFDOV\VWHPVE\PHDQVRIWKHWKHRU\RIOLQHDUV\VWHPV



DQGPLVWKH)RXULHU¶VWUDQVIRUP F RIp,QRWKHUZRUGVIRUHDFKREMHFWSRLQWr$  x $ y $  WKH point spread function ZKLFK LV WKH LPDJH RI r$ LV JLYHQ E\ WKH )UDXQKRIHU¶VGLIIUDFWLRQSDWWHUQRIWKHSXSLOFHQWHUHG ZLWKfx  fy  DWSRLQW r%  x % y %  m Ar$ :LWKWKHFKDQJHLQYDULDEOHV  UHPHPEHULQJWKDWWKHODWHUDOPDJQLILFDWLRQ LVJLYHQE\mA sisoDQGXVLQJ(TV  (T  EHFRPHV

u% x%  y %





S i O so  si e

O  so si

f f

u

³ ³

f f

C% x%  y %



§ x  x y  y $ ·  P¨ % $  % ¸ C$ x $  y $ u$ x $  y $ dx $ dy $  Osi ¹ © Osi





 

7KHWHUPV C $ DQG C%FRQVWLWXWHDQRQWULYLDOFRPSOLFDWLRQ C%GLVDSSHDUVLQ WKHLQWHQVLW\FDOFXODWLRQ,QVWHDG C $ LVQRWWULYLDOO\HOLPLQDEOH+RZHYHULIWKH SXSLO LV ODUJH WKH WUDQVIRUP P RI p KDV D QDUURZ FHQWUDO SHDN DQG E\ YDU\LQJ x $  y $  ZLWKLQ WKDW SHDN WKH WHUP RI ILHOG FXUYDWXUH YDULHV OLWWOH DQG FDQ EH DS SUR[LPDWHGDV iS  C $ # e O so mA



x%  y%



 

DQG EH EURXJKW RXW RI WKH LQWHJUDO ,Q (T   ZH FDQ UHFRJQL]H WKH FRQ YROXWLRQSURGXFWEHWZHHQWKHILHOG u PXOWLSOLHGE\ C $ DQGWKHGLIIUDFWLRQSDW WHUQ UHVXOWLQJ IURP WKH OHQV SXSLO :LWK WKH DSSUR[LPDWLRQ   (T   FDQEHUHZULWWHQDV S



u% x  y



i L x  y ª § x º  y · Oo e  «P ¨ ¸ u$ x y »  O O s s O  so si i¹ ¬ © i ¼

 

ZKHUHFRQYROXWLRQLVSHUIRUPHGRQWKHYDULDEOHVxyDQG   7KH FRQGLWLRQ IRU ZKLFK WKLV DSSUR[LPDWLRQ LV YDOLG LV REWDLQHG IURP WKH FRPSDULVRQ RQ WKH

LPDJH SODQH EHWZHHQ WKH ZLGWK RI WKH 36) SHDN ZKLFK IRU D FLUFXODU SXSLO ZLWK UDGLXV a LV FRQWDLQHG LQ D UDGLXV HTXDO WR O_si_ a  DQG WKH GLVWDQFH 'r ZLWKLQ ZKLFK WKH SKDVH S x$  y$   O so mA KDV D YDULDWLRQ HTXDO WR S 7KHUHIRUH LW LV QHFHVVDU\ WKDW a !!  O~Vo~  IRUDQREMHFWSRLQWLQD[LVDQGa!!rAIRUSRLQWVRXWVLGHRIWKHD[LVDWD GLVWDQFHr$



)RXULHU2SWLFV

nD so  si 

L x y



nD  si

§  ·   ¨  ¸ x  y  > 3o 3i @  m A¹ ©





 

ZKHUH,KDYHDOVRKHUHUHLQWHJUDWHG>[email protected] 'R n  nD ZKLFKLVWKHRSWLFDOSDWK EHWZHHQWKHSULQFLSDOREMHFWDQGLPDJHSODQHV

5.4.4 As a lens does the Fourier’s transform /HWXVQRZJREDFNWRH[DPLQH(TV  ,ISLVGLIIHUHQWIURP]HUR LHZKHQWKHSODQHV S$DQG S%DUHQRWFRQMXJDWHGZHFDQUHZULWH(T  LQ WKHIRUP 

Ix

§x §x x $  x %  ª x ·º x ·   « x  S ¨ $  % ¸»  S ¨ $  % ¸ s$ s% S ¬ s s s % ¹¼ © $ © $ s% ¹ x$



ª º S §  ·  S §    ·   S x $ x %   x  S § x$  x % · « ¨  ¸  x% ¨ ¸ ¨ ¸» s$ © S s$ ¹ s% © S s% ¹ s$ s % S ¬ © s$ s % ¹ ¼







§x Z % x $   Z $ x %   f x $ x %  ª x ·º  « x  S ¨ $  % ¸»   S¬ f  Z$Z% © s$ s % ¹ ¼ 

 

   

7KHLPSXOVHUHVSRQVHRIWKHOHQVIRUWKHDPSOLWXGHLVQRZ h x$  y $  x %  y % 



 O  s$ s %

i

f f

³ ³

p x y e

­ § x$ x % · º ª §y S °ª y ·º  ¸»  « y  S ¨ $  % ¸» ®« xS ¨ OS ° ¬ s s s s © $ % ¹¼ ¬ © $ % ¹¼ ¯



°  ¾ °¿ dx dy

 

f f

/HWXVFRQVLGHUIRUVLPSOLFLW\WKHFDVHLQZKLFKWKHSXSLOLVLQILQLWHO\H[WHQG HGWKDWLVp HYHU\ZKHUH,QWKLVOLPLWFDVHZLWKDVXLWDEOHFKDQJHRIYDULDEOHV (T  EHFRPHV 

h

 O  s$ s %

f f

³ ³

f f

e

i

S   x y OS



dx dy

i S O s$ s %

i f    O f   Z$ Z%

ZKHUHWKHLPSXOVHUHVSRQVHhLVQRZUHGXFHGWRDFRQVWDQW7KHUHIRUH(T   EHFRPHV

$QDO\VLVRIRSWLFDOV\VWHPVE\PHDQVRIWKHWKHRU\RIOLQHDUV\VWHPV

S

u% x%  y %





³ ³

SZ

i f i s$  s % i $ x %   y %  eO D e O OD

f f

u



u$ x $  y $ e

i

S Z% x$   y $  O D



ei  OS Df x$ x%  y$ y% dx

   $ dy$ 

f f



f   Z $ Z % 

ZLWK D

ZKLFKLVYHU\VLPLODUWR(T  IRUWKH)UHVQHOGLIIUDFWLRQ,IZHFRQVLGHUWKH SDUWLFXODUFDVH Z% LHZKHQ S%FRLQFLGHVZLWKWKHVHFRQGIRFDOSODQHRIWKH OHQV WKHQ D LV UHGXFHG WR f  DQG WKH WHUP LQ x$  y$ LV FDQFHOHG 7KHUHIRUH DSDUWIURPDSKDVHDQGDPSOLWXGHWHUPWKHILHOGLQS%LVWKH)RXULHU¶VWUDQVIRUPRI WKHILHOGLQS$HYDOXDWHGDWWKHVSDWLDOIUHTXHQFLHV 

x%  Of

fx

fy

y%  Of

 

)LQDOO\LIDGGLWLRQDOO\Z$ LH S$FRLQFLGHVZLWKWKHILUVWIRFDOSODQHHYHQ WKHUHPDLQLQJILHOGFXUYDWXUHWHUPGLVDSSHDUVDQGZHKDYH u% x%  y %





i i e Of i e Of

i

S f f f O

³ ³

u$ x $  y $ e

 i S

 x$ x %  y $ y % Of dx

f f

$ dy$

  

S f §x O U$ ¨ %

y ·  % ¸ O O f f ¹ ©

ZKHUH 



U$ f x  f y



F ^u$ x$  y$ ` 

 

7KHUHIRUH WKHDPSOLWXGH RI WKH ILHOGRQ WKH LPDJH IRFDOSODQHFRUUHVSRQGV WR WKHWUDQVIRUPRIWKHILHOGDPSOLWXGHRQWKHREMHFWIRFDOSODQH

5.4.5 Behavior in three dimensions )LQDOO\ZHFRQVLGHUWKHFDVHLQZKLFKWKHSODQHZKHUHWKHLPDJHLVUHYHDOHGLV RXWRIIRFXV7KXVOHWXVZULWH(T  LQWKHIRUP 

 S

      so  z$ si  z% f

 



)RXULHU2SWLFV

ZKHUHsoDQGsiUHSUHVHQWWKHSRVLWLRQVRIWZRPXWXDOO\FRQMXJDWHSODQHVZKLOHz$ DQGz%DUHWKHGHYLDWLRQVIURPWKHVHSRVLWLRQV,Iz$soDQGz%siZHKDYH WKDW   |  mA z$  z%  S si







 

ZKHUHmALVWKHODWHUDOPDJQLILFDWLRQSURGXFHGE\WKHOHQVEHWZHHQWKHDERYHFRQ MXJDWHGSODQHV7KLVDSSUR[LPDWLRQDOORZVIRULQWURGXFLQJDVSDWLDOLQYDULDQFHIRU WKH D[LDO GLUHFWLRQ DV ZHOO ,QGHHG E\ DSSO\LQJ (T   WR (T   DS SUR[LPDWLQJWKHWHUPVsDQGsLQWKHGHQRPLQDWRUUHVSHFWLYHO\ZLWKsoDQGsi DQGDSSO\LQJWKHVXEVWLWXWLRQV  ZHREWDLQ u% x%  y % # 



i  e  O so si

S so  si  z$  z% O C%

x%  y %

u ³³ u$ x $  y $ C $ x $  y $ $36) x%  x $  y%  y $  z%  z$ dx$ dy $ 



6$





 

ZKHUH C $ DQGC%DUHVWLOOJLYHQE\(TV  PRUHRYHU z$



mA  z$ 

 

DQG 

$36) x c y c z c

³³ p x y e

i

º S ª   xxc  yy c z c x   y  » Osi «¬ si ¼ dx dy 





 

6

UHSUHVHQWVWKHWKUHHGLPHQVLRQDOpoint spread functionIRUWKHDPSOLWXGH$QH[ WHQVLYHGLVFXVVLRQRIWKHVHWKUHHGLPHQVLRQDOSURSHUWLHVLVJLYHQE\0LQ*X>*X @

5.4.6 Image formation with a generic optical system ,QWKHFDVHRIDQRSWLFDOV\VWHPFRPSRVHGRIYDULRXVHOHPHQWVZHFDQVWLOOUH VRUWZLWKLQFHUWDLQOLPLWVWRWKHUHVXOWVIURP*HRPHWULFDO2SWLFV,IWKHV\VWHPLV ZHOOGHVLJQHGZHFDQGHILQHDQLQSXWSXSLODQGDQRXWSXWSXSLO%RWKDUHLPDJHV RIWKHDSHUWXUHRIWKHV\VWHPZKLFKLVVXIILFLHQWO\VPDOOWRUHQGHUWKHGLIIUDFWLRQ SURGXFHGE\WKHRWKHUGLDSKUDJPVSUHVHQWQHJOLJLEOH$WWKHVDPHWLPHLWLVVXIIL

$QDO\VLVRIRSWLFDOV\VWHPVE\PHDQVRIWKHWKHRU\RIOLQHDUV\VWHPV



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sinecondition6XS SRVHWKDWWKHREMHFWKDVDSHULRGLFVWUXFWXUHZLWKSHULRGqoDORQJWKHyGLUHFWLRQ IRUH[DPSOHLWLVFRQVWLWXWHGE\DVHULHVRIOLQHVSDUDOOHOWRWKHxD[LVDQGXQLIRUP O\VSDFHG:KHQLWLVLOOXPLQDWHGE\DFRKHUHQWSODQHZDYHLWSURGXFHVDGLIIUDFW HGILHOGFRQVWLWXWHGE\DVHULHVRISODQHZDYHVFDOOHGGLIIUDFWHGRUGHUVZKRVHGL UHFWLRQRQWKHzySODQHLVJLYHQE\ 

VLQ - j

j O qo 

ZKHUHj rrr«LVWKHLQGH[RIWKHGLIIUDFWHGRUGHU,QSDUD[LDODSSUR[ LPDWLRQHDFKRIWKHVHSODQHZDYHVLVFRQYHUWHGE\WKHRSWLFDOV\VWHP LQWKHQRQ WHOHVFRSLFFDVH LQDVSKHULFDOZDYHZLWKLWVRULJLQRQDOLQHSDUDOOHOWRWKHyD[LV RQWKHLPDJHIRFDOSODQH+HUHKRZHYHUIRUHDFKGLIIUDFWHGRUGHU,FRQVLGHURQO\ WKHORFDOEHKDYLRURIWKHZDYHDURXQGRQHRILWVUD\VSDVVLQJWKURXJKWKHREMHFW SRLQW2RQWKHRSWLFDOD[LVDQGUHDFKLQJWKHD[LDOLPDJHSRLQW, )LJ 7KLVUD\ IRUPVDQJOHV-jDQG-‫މ‬jZLWKWKHRSWLFDOD[LVUHVSHFWLYHO\LQWKHREMHFWDQGLPDJH VSDFHV,IZHFRQVLGHUDSDLURIRUGHUVZLWKjHTXDOLQDEVROXWHYDOXHDQGRSSRVLWH LQVLJQWKHLULQWHUIHUHQFHILJXUHRQWKHLPDJHSODQHKDVDSLWFKHTXDOWR 

qi j

O  VLQ -cj

7KHSLWFKRIWKHIULQJHVGHFUHDVHVZLWKWKHLQFUHDVHRIjDQGWKHUHIRUHZHFDQVD\ WKDW WKHILQHVWGHWDLOVREVHUYHGLQDQLPDJHDUHJLYHQE\WKHKLJKHVWGLIIUDFWLRQ RUGHUVWUDQVPLWWHGE\WKHV\VWHP 2QWKHRWKHUKDQGIRUWKHFRUUHFWUHSURGXFWLRQRIWKHREMHFWqiPXVWEHLQGHSHQG HQWRIj7KHUHIRUHLWLVQHFHVVDU\WKDW



)RXULHU2SWLFV

)LJ *UDSKLF UHSUHVHQWDWLRQ RI $E EH¶V VLQH FRQGLWLRQ 7KH SULQ FLSDO REMHFW DQG LPDJH SODQHV DUH UHSODFHG ZLWK VSKHULFDO VXUIDFHV ZLWK FHQWHU LQ 2 DQG FHQWHULQ,UHVSHFWLYHO\

J’j

Jj O

I

so

si =m^so 

qi qo



VLQ - j VLQ -cj

mA 

ZKHUHmALVWKHWUDQVYHUVHPDJQLILFDWLRQDQGZKHUHWKHVHFRQGHTXDOLW\H[SUHVVHV WKH VLQH FRQGLWLRQ ZKLFK LV UHTXLUHG WR DYRLG DEHUUDWLRQV SDUWLFXODUO\ ZKHQ WKH V\VWHPKDVDODUJHDQJXODURSHQLQJDQGWKHSDUD[LDODSSUR[LPDWLRQLVLQYDOLG 7KHVHFRQGZD\WRXQGHUVWDQGWKHV\VWHP¶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xo yo RQWKDWSODQHDQGWKHRYHUDOOILHOGDPSOLWXGHFDQEHGHFRP SRVHGLQWRDQLQWHJUDORIGIXQFWLRQVWKDQNVWRWKHLUVLIWLQJSURSHUW\ 

u xo  yo

³³ u xoc  yoc G xo  xoc  yo  yoc dxoc dyoc 

6o

:LWKOLPLWHGDEHUUDWLRQVIRUHDFKZDYHOHWWKHZDYHWKDWHPHUJHVIURPWKHV\V WHPFDQEHOLNHQHGWRDSRUWLRQRIDVSKHULFDOZDYHZKRVHFDUULHUFRQYHUJHVLQWKH SDUD[LDOLPDJHSRLQWRIFRRUGLQDWHV xo  mAxo y o  mAyo,QSDUD[LDODSSUR[LPD WLRQ ZH FDQ H[SUHVV WKH DPSOLWXGH RI WKH ZDYH JHQHUDWHG E\ HDFK SRLQW RQ WKH RXWSXWSXSLOSODQHDV

$QDO\VLVRIRSWLFDOV\VWHPVE\PHDQVRIWKHWKHRU\RIOLQHDUV\VWHPV i

u p x y xo  y o



e

S Lo xo  y o Oo uo S

i i Od ª¬ x i u e Od i



xo  yo p x y xo  yo





   xx  y    yy º o o ¼



 



ZKHUHWKHVHFRQGSDUWRIWKHH[SUHVVLRQUHSUHVHQWVWKHVSKHULFDOZDYHFDUULHUDQG ZKHUHi Odi UHSUHVHQWVLWVDPSOLWXGHLQSDUD[LDODSSUR[LPDWLRQO OoniLVWKH ZDYHOHQJWKniLVWKHUHIUDFWLYHLQGH[LQWKHLPDJHVSDFHDQGdiLVWKHGLVWDQFHEH WZHHQWKHRXWSXWSXSLODQGWKHLPDJHSODQH/DVWO\H[S iSLoOo uo p UHSUHVHQWV WKHDPSOLWXGHRIWKHUHFRQVWUXFWHGILHOGRQWKHSODQHRIWKHRXWSXWSXSLOWRZKLFK WKHSDUD[LDOFDUULHUZDVVXEWUDFWHG,QSDUWLFXODUSLoOoFRUUHVSRQGVWRWKHSKDVH VKLIWGXHWRWKHRSWLFDOSDWKEHWZHHQWKHREMHFWSRLQWDQGWKHFHQWHURIWKHH[LWSX SLODQGWKHUHIRUHDOUHDG\LQFOXGHVWKHWHUPCRIILHOGFXUYDWXUHWKDWZHHQFRXQ WHUHGLQGHDOLQJZLWKWKHFDVHRIDOHQV+HUHpLVDVVXPHGQXOORXWVLGHRIWKHSX SLOEXWDGPLWVDPSOLWXGHPRGXODWLRQLQVLGHDQGLQFOXGHVDEHUUDWLRQVLQFOXGLQJWKH JDSEHWZHHQWKHSDUD[LDOFXUYDWXUHDQGWKHLGHDOVSKHUHZLWKWKHVHDGGLWLRQVWKH IXQFWLRQpLVFDOOHGWKHgeneralized pupil $JDLQDSSO\LQJWKHGLIIUDFWLRQLQWHJUDO  IRUWKHGLVWDQFHdiDQGLQWHJUDW LQJRQWKHREMHFWSODQHZHJHWWKHDPSOLWXGHRIWKHILHOGDWWKHLPDJHSODQH

ui xi  yi

i

e

S iS di  xi   yi  O Od i

 u³³ e



i

S Lo xi  yi Oo uo

  

xo  yo $36) xi  yi  xo  y o dxo dy o 

S o

ZKHUH S o HVVHQWLDOO\UHSUHVHQWVWKHREMHFWVXUIDFHVFDOHGIRUWKHWUDQVYHUVHPDJ QLILFDWLRQDQGWKH$36)LVJLYHQE\ $36) xi  yi  xo  y o



 O  di 

i

S

ª x xi  xo  y yi  y o º¼

¬ ³³ p x y xo  yo e Odi

dxdy 



 

A

ZKHUHALVWKHVXUIDFHRIWKHSXSLODQGWKH$36)KDVWKHLQYHUVHVL]HRIDQDUHD ,IWKHV\VWHPLVLVRSODQDWLFWKDWLVLIpLVQRWGHSHQGHQWRQWKHFRRUGLQDWHVRI WKHREMHFWSRLQWZHFDQGHILQHDQHZSRLQWVSUHDGIXQFWLRQ 

7KHFRHIILFLHQWi Odi LVLQWURGXFHGLQDFFRUGDQFHZLWKWKHQRWDWLRQRI%RUQDQG:ROI  

†XQOLNH*RRGPDQ  &KDS



)RXULHU2SWLFV

F o

uo xi  yi 

mF

REMHFWDPSOLWXGH

 $36) xiyi 







U o f x  f y 



REMHFWDPSOLWXGHVSHFWUXP

u

 F o

point spread function IRUWKHDPSOLWXGH



mF  F o

  ui xi  yi 

mF

LPDJHDPSOLWXGH







&7) f x  f y 



coherent transfer function

  Ui f x  f y 







LPDJHDPSOLWXGHVSHFWUXP

 7DE)RUPDWLRQRILPDJHVZLWKFRKHUHQWUDGLDWLRQ+HUHWKHIXQFWLRQV uo DQG ui DUH H[SUHVVHGLQWKHVDPHVSDWLDOFRRUGLQDWHV xiyiZKLFKDUHWKRVHRIWKHLPDJHVSDFH

 $36) xic  yic

  O di 

³³ p x y e

i

S xxic  yyic Od i dxdy

A



 O  di 

§ xc y c · P ¨ i  i ¸  © Od i Od i ¹



 

ZKHUH xic xi  xo DQG yic yi  y o DUHWKHGLIIHUHQFHEHWZHHQWKHFRRUGLQDWHVRI WKHREVHUYDWLRQSRLQWIURPWKHLGHDOLPDJHSRLQW,QRWKHUZRUGVWKH$36)LVSUR SRUWLRQDOWRWKHWUDQVIRUPPRIWKHSXSLOLQFOXGLQJDQ\DPSOLWXGHDQGSKDVHPRG XODWLRQV,ILQSDUWLFXODUWKHDEHUUDWLRQVDUHQHJOLJLEOHDQGWKHSXSLOLVFLUFXODUVR WKDWZHFDQZULWHp r  FLUF ra ZKHUHaLVLWVUDGLXVZHKDYH $36) r c



§ a ar c · - ¨ S ¸  c Od i r © Od i ¹

 

)XUWKHUPRUHDSSO\LQJWKHVDPHDSSUR[LPDWLRQWRLoDVZDVPDGHWR C A LQ(T  ZHFDQUHZULWH(T  DV



ui xi  yi

i

e

S L xi  yi Oo

³³ $36) xi  xo  yi  yo uo xo  yo dxodyo   

S o

ZKHUH 

L xi  yi

Lo xi  yi 

Sni S ni di  xi   yi   Oo Oo di





 

(T   UHSUHVHQWV D FRQYROXWLRQ SURGXFW VR E\ SHUIRUPLQJ WKH )RXULHU

$QDO\VLVRIRSWLFDOV\VWHPVE\PHDQVRIWKHWKHRU\RIOLQHDUV\VWHPV



WUDQVIRUPZHJHW  Ui f x  f y













&7) f x  f y U o f x  f y 

 

  ZKHUH U i LVWKHWUDQVIRUPRI ui H[S ikL ui7KHIXQFWLRQ&7)LVFDOOHGWKHcoherent transfer functionDQGLWLV 



&7) f x  f y



F ^$36) xic  yic `

 O di 

°­ § x c y c F ®P ¨ i  i °¯ © Odi Odi

· °½ ¸ ¾  ¹ ¿°

 

$SSO\LQJ WKH WKHRUHPV RI VLPLODULW\ DQG V\PPHWU\ ZH ILQDOO\ JHW WKH LPSRUWDQW HTXDWLRQ 



&7) f x  f y







p Odi f x  Odi f y 

 

:LWK(TV  DQG  ZHKDYHWUDFHGWKHLPDJHIRUPLQJSURFHVVWR WKDWRIDVSDWLDOIUHTXHQF\ILOWHULQZKLFKWKHILOWHUIXQFWLRQLVJLYHQ³VLPSO\´E\ WKHƒrotated ERWKD[HVLQYHUWHG JHQHUDOL]HGSXSLORIWKHRSWLFDOV\VWHP:H FDQUHFRJQL]HLQWKLVIDFWDMXVWLILFDWLRQRI'HE\H¶VGLIIUDFWLRQLQWHJUDOZKLFKZH KDYHVWXGLHGLQ†)RUDQLGHDOV\VWHPdiffraction-limitedWKHSXSLOIXQFWLRQ LVQXOORXWVLGHRIWKHV\VWHPH[LWSXSLODQGLVFRQVWDQWLQLWVLQWHULRU,QWKHFDVHRI DFLUFXODUSXSLOZLWKUDGLXVrWKHILOWHULQJLVUHGXFHGWRDWUXQFDWLRQRIWKHVSDWLDO IUHTXHQFLHVDWWKHFXWRIIIUHTXHQF\ 

fr

r  Od i

 

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

5.4.7 Images with incoherent illumination 6R IDU ZH KDYH FRQVLGHUHG WKH FDVH RI PRQRFKURPDWLF LOOXPLQDWLRQ 0RUH FRPPRQO\ LPDJHV DUH IRUPHG E\ LQFRKHUHQW QDWXUDO UDGLDWLRQ DQG WKH SK\VLFDO



)RXULHU2SWLFV

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†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specklesWRWKHSRLQWZKHUHREMHFWVLQWKHVFHQHDUHGLIILFXOWWRSHUFHLYHLQGHWDLO ,Q WKH FDVH RI quasi-monochromatic LOOXPLQDWLRQ ZH FDQ H[SUHVV WKH ILHOG DV WKHSURGXFWRIDFDUULHUIRUDQDPSOLWXGHVORZO\YDU\LQJRYHUWLPH7KHWHPSRUDO FRKHUHQFHWKHQKDVDQH[WHQVLRQOLPLWHGWRDQLQWHUYDO'tRIDERXW S'Q ZKHUH 'QLVWKHLOOXPLQDWLRQEDQGZLGWK>%RUQDQG:ROI†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¶PRYHPHQWFDQQRORQJHUEH IROORZHGE\WKHGHWHFWRUVXFKDVDQH\HRUDFDPHUDZLWKDJLYHQRSHQLQJWLPHRI LWVVKXWWHU ,Q † ZH VDZ D FRQGLWLRQ UHTXLUHG IRU TXDVLPRQRFKURPDWLF UDGLDWLRQ WR DSSO\ GLIIUDFWLRQ LQWHJUDOV ,W SODFHV D FRQVWUDLQW EHWZHHQ WKH EDQGZLGWK DQG WKH

$QDO\VLVRIRSWLFDOV\VWHPVE\PHDQVRIWKHWKHRU\RIOLQHDUV\VWHPV



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polychromatic,QWKLVFDVHZHFDQGLYLGHLWVVSHFWUXPLQWRDILQLWH QXPEHURITXDVLPRQRFKURPDWLFEDQGVHDFKRIZKLFKLVWKHQWUHDWHGVHSDUDWHO\ )LQDOO\ZHPDWKHPDWLFDOO\DQDO\]HWKHIRUPDWLRQRILPDJHVZLWKspatiallyincoherent illuminationE\DJHQHULFRSWLFDOV\VWHP&RQVLGHUWKHFDVHRIDQREMHFW WKDWGLIIXVHVTXDVLPRQRFKURPDWLFUDGLDWLRQVRWKDWWKHDPSOLWXGHRIWKHILHOGRQ LWVVXUIDFHLVH[SUHVVHGDVDIXQFWLRQRIWLPHDVZHOODVRIFRRUGLQDWHV6XSSRVHLQ DGGLWLRQ WKDW WKH GLIIHUHQFHV RI WKH SDWK EHWZHHQ WKH VXUIDFH REMHFW DQG LPDJH SODQHDUHVPDOOHQRXJKVRDVWRVWLOOEHDEOHWRDSSO\WKHGLIIUDFWLRQLQWHJUDODVZH GLVFXVVHG LQ † WKLV FRQGLWLRQ LV JHQHUDOO\ YHULILHG IRU SDLUV RI FRQMXJDWH SRLQWVE\WKHV\VWHP%HVLGHVZHQHJOHFWWKHWLPHGHOD\DQGSKDVHVKLIWGXHWRWKH SURSDJDWLRQVXFKDVWKRVHSUHVHQWLQ(T  LQGHHGWKHVHWHUPVKDYHQRHI IHFWRQWKHLQWHQVLW\REWDLQHGZLWKLQFRKHUHQWLOOXPLQDWLRQ/DVWO\ZHDVVXPHWKDW WKHV\VWHPLV DWOHDVWORFDOO\ LVRSODQDWLF:LWKDVPDOOFKDQJH(T  EH FRPHV 

 ui xi  yi  t

³³ $36) xi  xo  yi  yo u xo  yo  t dxo dyo 

 

S o

 ZKHUH ui LVZLWKRXWWKHSKDVHWHUPPHQWLRQHGDERYHDQGWKH$36)LVVWLOOFRP SXWHGE\(T  EXWZLWKOUHSODFHGE\WKHPHDQZDYHOHQJWKRIWKHUDGLDWLRQ LQLWVVSHFWUDOEDQG7KHLQWHQVLW\RIWKHLPDJHLVWKHQJLYHQE\ ii xi  yi

a ³³ dxo dy o S o

u³³ $36) xi  xo  yi  y o $36) xi  xoc  yi  y oc 



 

S o

u u xo  y o  t u xoc  y oc  t dxoc dy oc 

ZKHUHWKHWULDQJXODUEUDFNHWVLQGLFDWHWKHWLPHDYHUDJH aLVWKHFRHIILFLHQWWREH DSSOLHGIRUWKHWUDQVLWLRQIURPWKHILHOGDPSOLWXGHuWRWKHLQWHQVLW\iDQG 

ii xi  yi

  a ui xi  yi  t ui xi  yi  t

a ui xi  yi  t ui xi  yi  t 

:HDOVRDVVXPHWKDWWKHILHOGRQWKHVXUIDFHREMHFWLVVSDWLDOO\LQFRKHUHQWDQGLQ



)RXULHU2SWLFV

SDUWLFXODUWKDW 

a u xo  y o  t u xoc  y oc  t

io xo  y o AF G xo  xoc  y o  y oc 

 

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† RQHILQGVWKDWZLWKDSULPDU\FLUFXODUVRXUFHWKHVSDWLDOFRKHUHQFHDUHD RIDVHFRQGDU\VRXUFHKDVDGLDPHWHURIOP-+HUHOPLVWKHDYHUDJHZDYH OHQJWKRIWKHUDGLDWLRQDQG-LVWKHDQJXODUKDOIDSHUWXUHZLWKZKLFKWKHSULPDU\ VRXUFHLVVHHQIURPWKHVHFRQGDU\RQH>YDQ&LWWHUW=HUQLNH%RUQDQG :ROI†@)RUH[DPSOHDVFHQHLOOXPLQDWHGE\WKH6XQZLWKOPa PPKDVDFRKHUHQFHGLDPHWHURQWKHRUGHURIPPDQGWKHSDUWLDOFRKHUHQFHDU HDH[WHQGVXSWRDGLDPHWHUDERXWWLPHVJUHDWHU 5HSODFLQJ(T  LQ(T  ZHJHW 

ii xi  yi

AF ³³ 36) xi  xo  yi  y o io xo  y o dxo dy o  

 

S o

ZKHUH 

36) xi  xo  yi  y o



$36) xi  xo  yi  y o 

 

LVWKHpoint spread functionIRUWKHLQWHQVLW\ZLWKLQFRKHUHQWLOOXPLQDWLRQDQGKDV WKHGLPHQVLRQVRIWKHLQYHUVHVTXDUHRIDQDUHD

5.4.8 Resolution 7KH SUREOHP RI HYDOXDWLQJ WKH SHUIRUPDQFH RI DQ RSWLFDO V\VWHP DQG LPDJH TXDOLW\LVRQHRIWKHPRVWIHOWLQRSWLFVEXWDPRQJWKHPDQ\YDULHWLHVRIPHDV XUHPHQWV SURSRVHG QRQH RI WKHVH LV TXLWH VDWLVIDFWRU\ $Q LPSRUWDQW FRQFHSW LV WKDW RI fidelity RI UHSURGXFWLRQ IRUZKLFKYDULRXV PHWKRGVRI PHDVXUHPHQW KDYH EHHQSURSRVHGVXFKDVWKHURRWPHDQVTXDUHGHYLDWLRQSRLQWE\SRLQWEHWZHHQWKH UHDOREMHFWDQGLPDJH,QLWVHOIWKLVVKRXOGEHDQREYLRXVFULWHULRQ+RZHYHULWLV DIIHFWHGE\HUURUVWKDWFDQEHFRQVLGHUHGLPSRUWDQWRUQRWVXFKDVDPDJQLI\LQJ HUURUDVPDOOURWDWLRQRUGLVWRUWLRQDYDULDWLRQLQLQWHQVLW\RUFRQWUDVWRUDFRORU

$QDO\VLVRIRSWLFDOV\VWHPVE\PHDQVRIWKHWKHRU\RIOLQHDUV\VWHPV



FKDQJH7KHUHIRUHWKHFRQFHSWRIILGHOLW\DOVRHQGVXSGHSHQGLQJRQWKHSXUSRVH IRUZKLFKWKHV\VWHPLVXVHG %HVLGHVRQHRIWHQUHVRUWVWRWKHFRQFHSWRIresolutionWKDWLVWKHDELOLW\WRGLV FULPLQDWHE\PHDQVRIWKHLPDJHSURGXFHGE\WKHV\VWHPWKHUHDOREMHFWVWKDWGH WHUPLQHLWDQGWKHLUVKDSH,QDGGLWLRQLWLVUHTXLUHGWKDWWKHHYDOXDWLRQEH³REMHF WLYH´ GHWHUPLQHG E\ WKH FKDUDFWHULVWLFV RI WKH V\VWHP LWVHOI DQG WKHQ JLYHQ a priori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ayleigh’s criterionZKLFKVWDWHVWKDW>5D\OHLJK@ WZRSRLQWVRXUFHVDUHMXVWUHVROYHGRQWKHLPDJHSODQHLIWKHFHQWUDOPD[L PXPLQWHQVLW\RIWKHGLIIUDFWLRQSDWWHUQSURGXFHGE\DVRXUFHFRLQFLGHVZLWK WKHILUVW]HURRIWKHGLIIUDFWLRQILJXUHSURGXFHGE\WKHRWKHUVRXUFH ,QWKHFDVHRIDQLGHDOV\VWHPLQZKLFKWKH36)LVJLYHQE\$LU\¶VILJXUHWKH UHVROXWLRQLVH[SUHVVHGE\WKHUDGLXVRIWKHILUVWGDUNULQJDURXQGWKHFHQWUDOSHDN :LWKPRUHFRPSOLFDWHGGLIIUDFWLRQILJXUHVIRUH[DPSOHLQWKHSUHVHQFHRIDEHUUD WLRQVWKLVFULWHULRQLVJHQHUDOL]HGUHTXLULQJWKDWEHWZHHQWKHWZRPD[LPDWKHUHLV DGHSUHVVLRQVXFKWKDWWKHVXPRIWKHLQWHQVLWLHVSURGXFHGE\WZRHTXDOVRXUFHVLV RIWKDWRQWKHPD[LPD 7KHSparrow criterionLVOHVVUHVWULFWLYHDQGFRQVLGHUVWKDW>6SDUURZ@ WZRSRLQWVRXUFHVRIHTXDOLQWHQVLW\DUHMXVWUHVROYHGLIWKHVHFRQGDU\GHULYD WLYH RI WKH LQWHQVLW\ KDOIZD\ DORQJ WKH FRQQHFWLRQ EHWZHHQ WKH SRVLWLRQ RI WKHLUFHQWHUVLVFDQFHOHG ,QRWKHUZRUGVWKHVRXUFHVDUHVWLOOUHVROYHGZKHQWKHLQWHQVLW\GHFUHDVHEHWZHHQ WKHWZRPD[LPDLVFDQFHOHG %RWKRIWKHVHFULWHULDPD\EHXVHIXOIRUDFRPSDULVRQEHWZHHQRSWLFDOV\VWHPV EXWGRQRWJLYHDQDEVROXWHPHDVXUHRIWKHUHVROXWLRQOLPLW)RUH[DPSOHXVLQJD UHDVRQDEOHPRGHOVXFKDVWKHVXPRIWZR$LU\IXQFWLRQV\RXFDQPDNHDfitRIWKH REVHUYHG LPDJH DQG JHW WKH GLVWDQFH EHWZHHQ WKH WZR VRXUFHV 7KH OLPLW WR WKLV GLVWDQFHPHDVXUHPHQWLVJLYHQE\WKHQRLVHSUHVHQWLQWKHLPDJHVRWKHPHDVXUH PHQWHUURUFDQEHPXFKORZHURUPXFKKLJKHUWKDQWKHHVWLPDWHGUHVROXWLRQZLWK WKHVHFULWHULD 7KHUHIRUH LI WKH LQVWUXPHQW SXSLO LV FLUFXODU ZLWK GLDPHWHU ' IRU H[DPSOH D WHOHVFRSHDQJXODUUHVROXWLRQLVODZLWKWKH5D\OHLJKFULWHULRQDQGOD ZLWKWKDWRI6SDUURZ 2QWKHRWKHUKDQG5D\OHLJKDQG6SDUURZ¶VFULWHULDFDQZRUNZHOOIRUHYDOXDW



)RXULHU2SWLFV

LQJWKHSHUIRUPDQFHRIDWHOHVFRSHLQVROYLQJWKHVWUXFWXUHRIDELQDU\VWDUEXWDUH HVVHQWLDOO\LQDGHTXDWHWRHYDOXDWHWKHTXDOLW\RIUHSURGXFWLRQRIH[WHQGHGREMHFWV )RU WKLV VLWXDWLRQ ZH UHO\ RQ WKH VSHFWUDO DQDO\VLV VXPPDUL]HG E\ WKH Optical Transfer Function

5.4.9 The Optical Transfer Function (YHQ(T  KDVWKHIRUPRIDFRQYROXWLRQDQGLIZHFDOFXODWHLWV)RXUL HU¶VWUDQVIRUPZHJHWWKHSURGXFW





Ii f x  f y









AF 2 f x  f y Io f x  f y 

 

ZKHUHWKHIXQFWLRQ2LV 



2 fx f y



^

`

F $36) xic  yic $36) xic  yic 

 

7KLVUHSUHVHQWVWKHVSDWLDOIUHTXHQF\ILOWHULQJRSHUDWHGE\WKHRSWLFDOV\VWHPLQWKH UHSURGXFWLRQ RI WKH REMHFW IRU WKH FDVH RI LQFRKHUHQW LOOXPLQDWLRQ $SSO\LQJ WKH DXWRFRUUHODWLRQWKHRUHP  ZLWKWKHVXEVWLWXWLRQ[ x Odi K y Odi  DQG(TV  ZLWKWKHUHSODFHPHQWfx xp Odi fy yp Odi ZKHUHxpDQG ypDUHWKHWUDQVYHUVHFRRUGLQDWHVRIWKHWUDQVODWLRQWDNHQRQWKHSXSLO¶VSODQHZH KDYH 

§ xp yp · 2¨  ¸ © Od i Od i ¹

 O  di 

f f

³ ³ p x y p x  x p  y  y p dx dy 

 

f f

7KHIXQFWLRQ2FRUUHVSRQGVWRWKHautocorrelationRIWKHJHQHUDOL]HGSXSLO )LJ )LJ *HRPHWULF LQWHUSUHWD WLRQRIWKH27)E\WKH DXWRFRUUHODWLRQ RI WKH SXSLO  JLYHQ E\ WKH LQWHJUDO RQWKHRYHUODS DUHD  RI WKH SURGXFW RI WKHSXSLOIXQFWLRQZLWK LWV FRS\ VKLIWHG DQG FRQMXJDWHG

overlapping area h P

h

P* x

x

y

x 

$QDO\VLVRIRSWLFDOV\VWHPVE\PHDQVRIWKHWKHRU\RIOLQHDUV\VWHPV



F o



i o xi  y i 

mF

REMHFWLQWHQVLW\











I o f x  f y 



VSHFWUXPRIWKHREMHFWLQWHQVLW\

u



36) xiyi 

F o

point spread function IRUWKHLQWHQVLW\

mF





 F o

i i xi  yi 

mF

LPDJHLQWHQVLW\





mF  F o

u $36) xiyi 



36) xiyi 

mF

optical transfer function QRWQRUPDOL]HG





I i fx f y 



VSHFWUXPRIWKHLPDJHLQWHQVLW\







p Od i f x  Odi f y 



mF  F o







F o

$36) xiyi 



2 fx f y 









p Odi f x  Odi f y 











2 fx f y 

7DE)RUPDWLRQRILPDJHVZLWKLQFRKHUHQWUDGLDWLRQ(YHQKHUHWKHIXQFWLRQV i o DQG iiDUHH[SUHVVHGLQWKHVDPHVSDWLDOFRRUGLQDWHVxiyiZKLFKDUHWKRVHRIWKHLPDJHVSDFH 1RWHWKDWWKHFRQYROXWLRQIRUWKHFDOFXODWLRQRI2LVEHWZHHQWKH180°-rotated ERWKD[HV inverted SXSLODQGWKHconjugatedSXSLO VHHWKHV\PPHWU\DQGFRQMXJDWLRQWKHRUHPVDQG (T  DQGLVWKXVHTXDOWRWKHDXWRFRUUHODWLRQRIWKHSXSLOLWVHOI+HUHFRQYROXWLRQLV XQGHUVWRRGE\LQWHJUDWLQJRQWKHVFDOHRI fx DQGfy

  ,W LV QRUPDOL]HG WR LWV YDOXH DW IUHTXHQFLHV fx   fy   REWDLQLQJZKDW LV FDOOHGWKHOptical Transfer Function 27) ZKLFKLVJLYHQE\ f



§ xp yp · 27) ¨  ¸ © Od i Od i ¹

f

³ f ³ f p x y p x  x p  y  y p dx dy f

f

³ f ³ f





 

p x y dx dy

,WKDVWKHIROORZLQJSURSHUWLHV 27)  



27) f x  f y   27) f x  f y d 27)   

27)  f x   f y

 



)RXULHU2SWLFV 1.0 0.8 0.6 0.4 0.2 0.0 -30 30 0 0 30



-30

)LJ$[RQRPHWULFUHSUHVHQWDWLRQRIWKH27)RIDFLUFXODUSXSLO7KLVILJXUHLVREWDLQHG E\WKHDXWRFRUUHODWLRQRIDFLUFOHZLWKDGLDPHWHUHTXDOWRVWHSVRIWKHJULG

7KHILUVWRQHWULYLDOO\IROORZVWKHQRUPDOL]DWLRQGRQH7KHVHFRQGLVDGLUHFWFRQ VHTXHQFHRIWKHIDFWWKDWIRU(T  WKH27)LVSURSRUWLRQDOWRWKHWUDQVIRU PDWLRQ RI D UHDO IXQFWLRQ DQG KHQFH LV +HUPLWLDQ 0RVW LPSRUWDQW LV WKH WKLUG SURSHUW\ WKDW FDQ EH GHPRQVWUDWHG E\ DSSO\LQJ 6FKZDU]¶V LQHTXDOLW\ WR (T   ,QWKHFDVHLQZKLFKWKHSXSLOIXQFWLRQLVV\PPHWULFDOIRUinversionIRUZKLFK p [ K



p [ K 

WKH FRUUHVSRQGLQJ 27) LV UHDO 7KLV V\PPHWU\ H[LVWV IRU FLUFXODU RU UHFWDQJXODU RSHQLQJVEXWQRWIRUWULDQJXODURSHQLQJV $VXPPDU\RIWKHUHODWLRQVEHWZHHQWKHIXQFWLRQVWKDWDUHXVHGLQWKHDQDO\VLV RILQFRKHUHQWLPDJLQJLVJLYHQLQ7DEOH )UHTXHQWO\WKH27)LVH[SUHVVHGLQWHUPVRIPRGXOHDQGSKDVH 



27) f x  f y







07) f x  f y e



i 37) f x  f y



 

ZKHUH07)DQG37)DUHUHDOIXQFWLRQVWKHVHDUHFDOOHGWKHModulation Transfer FunctionDQGWKHPhase Transfer FunctionUHVSHFWLYHO\ /HW XV QRZ FRQVLGHU WKH FDVH LQ ZKLFK WKHUH DUH QR DEHUUDWLRQV DQG WKH SXSLO IXQFWLRQLVFRQVWDQWDQGHTXDOWRLQVLGHRIWKHRXWSXWSXSLODQGRXWVLGH7KH 27)VLPSOLILHVLQDVLPSOHUHDOJHRPHWULFIXQFWLRQRIWKHSXSLO¶VVKDSHDQGWZR

$QDO\VLVRIRSWLFDOV\VWHPVE\PHDQVRIWKHWKHRU\RIOLQHDUV\VWHPV



1.0 0.8 0.6 0.4 0.2 0.0 -30 30 0 0 30

-30



)LJ$[RQRPHWULFUHSUHVHQWDWLRQRIWKH27)RIDVTXDUHSXSLOZLWKVLGHVSDUDOOHOWRWKH xDQGyD[HV7KLVILJXUHLVREWDLQHGIURPWKHDXWRFRUUHODWLRQRIDVTXDUHZLWKVLGHVHTXDOWR VWHSVRIWKHJULG

H[DPSOHVDUHVKRZQLQ)LJVDQGIRUFLUFXODUDQGVTXDUHSXSLOFDVHV7KH VKDSHRIWKHVH27)VLVVLPLODUWRDFRQHRUDS\UDPLGUHVSHFWLYHO\ZKRVHYHUWH[ FRUUHVSRQGVWRQXOOVSDWLDOIUHTXHQFLHV:KDWLVPRVWLPSUHVVLYHLVWKDWWKHUHLVD OLQHDU DWOHDVWLQLWLDOO\ GHFD\RIWKHV\VWHP¶VIUHTXHQF\UHVSRQVHWRWKHJURZLQJ LQPRGXOXV RIWKHVSDWLDOIUHTXHQF\7KHH[WHQWWRZKLFKWKH27)LVQRQ]HURLV KHUHOLPLWHGWRDFLUFOHRUDVTXDUH ,QWKHFRPPRQFDVHRIDFLUFXODUSXSLORIUDGLXVrWKHUDGLXVRIWKHEDVHRIWKH FRQHRIWKH27)LHLWVFXWRIIIUHTXHQF\LVHTXDOWR 

fr

r  Od i

 

ZKLFKLVWZLFHWKDWRIWKH&7)IRUWKHVDPHSXSLO2QHPLJKWWKHUHIRUHWKLQNWKDW WKHLQFRKHUHQWLOOXPLQDWLRQJLYHVDEHWWHUUHVROXWLRQWKDQWKHFRKHUHQWRQH$SDUW IURPWKHIDFWWKDWWKH&7)LVFRQVWDQWLQLWVEDQGZKLOHWKH27)GHFUHDVHVVRWKDW WKHUHLVDVRUWRIFRPSHQVDWLRQWKHWZRWUDQVIHUIXQFWLRQVDUHQRWFRPSDUDEOH,Q GHHGWKH&7)UHIHUVWRWKHamplitudeRIWKHILHOGZKLOHWKH27)UHIHUVWRWKHintensityZKLFKDUHWKHTXDQWLWLHVWRZKLFKWKHOLQHDUVXSHUSRVLWLRQSULQFLSOHFDQEH DSSOLHGLQDPXWXDOO\H[FOXVLYHZD\LQWKHWZRFDVHVRILOOXPLQDWLRQ 7KLV EHKDYLRU SRLQWHG DURXQG WKH RULJLQ RI WKH IUHTXHQFLHV LV DOVR WKHUH IRU SXSLOVRIPRUHFRPSOLFDWHGIRUPPDGHZLWKULQJPDVNVRUFRPSRVHGRIVHSDUDWH DSHUWXUHV,QVSLWHRIDOORIWKDWLWLVGXHWRWKHZDYHQDWXUHRIWKHUDGLDWLRQHYHQ LQ WKH FRQWH[W RI LQFRKHUHQW LOOXPLQDWLRQ ZKHUH WKH ZDYHOHQJWK GRHV QRW DSSHDU



)RXULHU2SWLFV

OTF

1.0

0

0.5 0.5 1 1.5 2 3

4

0.0

0.0

0.5

1.0

ldi f/r

1.5

2.0



)LJ5DGLDOSURILOHVRIWKH27)ZLWKDFLUFXODUDSHUWXUHRIUDGLXVrIRUWKHFDVHRIdefocusingDVDIXQFWLRQRIWKHQRUPDOL]HGVSDWLDOIUHTXHQF\7KHQXPEHUVDVVRFLDWHGZLWKHDFK FXUYH UHSUHVHQWWKH GHYLDWLRQ IURP WKH LGHDO VSKHUH DW WKH HGJHRI WKH SXSLOH[SUHVVHG LQ XQLWVRIZDYHOHQJWKO7KHFXUYHPDUNHGZLWKFRUUHVSRQGVWRWKHLGHDOFDVH

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

&RKHUHQFH :HKDYHUHSHDWHGO\HQFRXQWHUHGWKHFRQFHSWRIcoherenceLQWKHLOOXPLQDWLRQ FRQGLWLRQVIURPDQH[WHQGHGVRXUFHWRGHWHUPLQHWKHYLVLELOLW\RIWKHIULQJHVRILQ WHUIHUHQFH DQG GLIIUDFWLRQ DQG LQ WKH DQDO\VLV RI WKH TXDOLW\ RI RSWLFDO V\VWHPV +RZHYHUWKHRQO\WZRFDVHVKDQGOHGVRIDUZHUHWKHH[WUHPHFRQGLWLRQVRIcoherentDQGincoherentLOOXPLQDWLRQLQZKLFKZHKDYHDSSOLHGUHVSHFWLYHO\WKHlinear VXSHUSRVLWLRQSULQFLSOHVRIWKHamplitudesDQGWKDWRIintensity7KHVHFDVHVGRQRW



)RXULHU2SWLFV

UHSUHVHQWWKHIXOOVWRU\DVVXFKFRQGLWLRQVDUHRQO\PDWKHPDWLFDOLGHDOL]DWLRQVWKDW EHFRPHLQDGHTXDWHLQYDULRXVVLWXDWLRQVVXFKDVLQPLFURVFRS\HVSHFLDOO\WKDWRI SKDVH FRQWUDVW DQGGDUNILHOG LOOXPLQDWLRQLQ ³FRKHUHQW´RSWLFDO V\VWHPV VXFK DV VSDWLDOILOWHULQJRSWLFDOSURFHVVLQJDQGKRORJUDSK\LQVSHFWURVFRS\DQGLQWHUIHU RPHWU\DQGDOVRLQSKRWRJUDSK\+HUHZHZLOOGHILQHFRKHUHQFHLQDPRUHJHQHUDO VHQVHDVcorrelationEHWZHHQWKHDPSOLWXGHVRIWKHILHOGDQGZHZLOODQDO\]HWKH LQWHUPHGLDWHVLWXDWLRQVRIpartial coherencetemporal coherenceDQGWKHpropagation of coherence 7KHILUVWH[SHULPHQWVRQFRKHUHQFHZHUHFDUULHGRXWE\(9HUGHWZKRLQ PHDVXUHG WKH VSDWLDO OHQJWK RI FRKHUHQFH RI DQ H[WHQGHG VRXUFH ,Q  KH VKRZHG WKDW IULQJHV RI LQWHUIHUHQFH FRXOG EH REVHUYHG ZLWK WZRSLQKROHV LOOXPL QDWHGGLUHFWO\E\WKH6XQZKHQWKHLUGLVWDQFHZDVDOUHDG\ȝ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photon bunching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aser HandbookPart A$UHFFKL6FKXOW]'X %RLVHG  

&RKHUHQFH



5.5.1 Random processes $ JHQHULF random process LV GHVFULEHG E\ D FROOHFWLRQRI IXQFWLRQVRIRQHRU PRUHYDULDEOHVLQGH[HGE\DYDOXH]UHSUHVHQWLQJWKHYDULRXVLQGHSHQGHQWUHDOL]D WLRQV RI WKH VDPH SURFHVV IRU H[DPSOHg rt;] ZKHUH r LV WKH SRVLWLRQLQ VSDFH DQGtLVWKHWHPSRUDOFRRUGLQDWH>%DUUHWWDQG0\HUV@ ([SHULPHQWDOO\LWLVQRUPDOWRFRQVLGHUWKHYDOXHRIgLQDGLVFUHWHQXPEHUN RISRLQWVIRUH[DPSOHWKHYDOXHVtttNIRUDIXQFWLRQRIDVLQJOHYDULDEOHt 7KHQg t g t g tN DUHDVHWRIUDQGRPYDULDEOHVFKDUDFWHUL]HGE\DKLHUDU FK\RIMRLQWSUREDELOLW\GLVWULEXWLRQV probability density function3') >$UHFFKL @ 

3U g  3U g  g   !  3U g  g !  g N 

 

ZKHUHIRUH[DPSOH3U gg LVWKHMRLQWSUREDELOLW\RIREWDLQLQJWKHYDOXHVgLQ tDQGgLQtLQWKHVDPHUHDOL]DWLRQ7KHVWDWLVWLFDOSURSHUWLHVRIDJHQHULFUDQ GRPSURFHVVDUHDOVRGHVFULEHGE\VWDWLVWLFDODXWRFRUUHODWLRQIXQFWLRQVRILQFUHDV LQJRUGHU 

gg ! g N { ³ g g ! g N 3U g g  !  g N dgdg  ! dg N  f

 

ZKHUHWKHWULDQJXODUEUDFNHWVLQGLFDWHWKHexpectation valueREWDLQHGIURPWKHensemble average RQ WKH YDULRXV UHDOL]DWLRQV DQG WKH LQWHJUDO H[WHQGV EHWZHHQ f DQGfRQDOOYDULDEOHVgm7KHILUVWRIWKHVHH[SHFWDWLRQVLVVLPSO\WKHVWDWLVWLFDO DYHUDJHRIWKHYDULDEOHDWWLPHt g t { ³ g t 3U g t dg 



f

 

7KHVHFRQGH[SHFWDWLRQYDOXHLVWKHVWDWLVWLFDODXWRFRUUHODWLRQRIWKHIXQFWLRQgEH WZHHQWZRYDOXHVtDQGt 

R t  t

gg  { ³ gg  3U g g  dgdg   f

 

:KHQgLVDFRPSOH[IXQFWLRQLQWKHSURGXFWRIWKHYDULRXVYDOXHVgmRQHDOVR WDNHV FRPELQDWLRQV LQ ZKLFK VRPH WHUPV DUH WKH FRUUHVSRQGLQJ FRPSOH[ FRQMX JDWH)RUH[DPSOHIRUDQHOHFWURPDJQHWLFZDYHSDUWLFXODULPSRUWDQFHLVKHOGE\ WKHVWDWLVWLFDODXWRFRUUHODWLRQIXQFWLRQ 

R t  t { E E 

ZKLFKIRUt tFRUUHVSRQGVWRWKHLQWHQVLW\

 



)RXULHU2SWLFV



I{

nc HR E E  

 

$SURFHVVLVVDLGWREH WHPSRUDULO\ stationary in the strict senseZKHQDOODX WRFRUUHODWLRQIXQFWLRQVDQGWKHUHIRUHDOORILWVVWDWLVWLFDOSURSHUWLHVDUHLQGHSHQG HQWRIWKHRULJLQRIWLPH8VXDOO\ZHOLPLWRXUVHOYHVWRstationarySURFHVVHV in the broad senseIRUZKLFKLWLVRQO\UHTXLUHGWKDWWKHH[SHFWDWLRQ¢g t ²LVQRWGHSHQG HQWRQtDQGWKDW¢g t g t ²GHSHQGVRQO\RQWKHGLIIHUHQFHW tt $VWDWLRQDU\SURFHVVLVWKHQVDLGWREHergodicLIWKHDYHUDJHRQWKHVHWRIUHDOL ]DWLRQV FDQ EH UHSODFHG ZLWK DQ DYHUDJH RQ WKH WLPH RI D VLQJOH UHDOL]DWLRQ 7KH GHJUHHRIHUJRGLFLW\IROORZVWKDWRIVWDWLRQDULW\JLYHQE\WKHRUGHURIDXWRFRUUHOD WLRQXSWRZKLFKWKHUHLVLQGHSHQGHQFHIURPWKHRULJLQ,QSDUWLFXODUWKHILUVWWZR H[SHFWDWLRQVRIDVWDWLRQDU\DWOHDVWLQWKH EURDGVHQVHDQGHUJRGLFSURFHVVDUH JLYHQE\ 



g

R W

g ]o

R W ]o

 OLP T T of

 OLP T T of

T 

³

g t ]o dt 



D 

T 

T 

³

g t ]o g t  W ]o dt  

E 

T 

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

5.5.2 The power spectrum of random processes 7KH VLJQDOV REWDLQHG SK\VLFDOO\ DUH DOZD\V VXEMHFW E\ YDU\LQJ GHJUHHV WR D QRLVHWKDWDOWHUVWKHLUXVHIXOLQIRUPDWLRQFRQWHQWDQGW\SLFDOO\FKDQJHVIURPPHDV XUHWRPHDVXUH7KHUHIRUHLWLVQHFHVVDU\WRNQRZWKHVWDWLVWLFDOFKDUDFWHULVWLFVRI

&RKHUHQFH



WKLV QRLVH VR WKDW LW FDQ EH VXEWUDFWHG RU DWWHQXDWHG $ XVHIXO LQGLFDWLRQ FRPHV IURPLWVVSHFWUXPEXWLWKDVWREHWUHDWHGZLWKSDUWLFXODUPHWKRGVWRZKLFKZHZLOO QRZVHHDQRGIRUWKHRQHGLPHQVLRQDOFDVH 7KH)RXULHU¶VWUDQVIRUPRIDVLJQDOLVRIWHQLQGLFDWHGE\WKHQDPHRIamplitude spectrumDQGLWVVTXDUHPRGXOHLVFDOOHGWKHpower spectrum,QGHHGWKH3DUVH YDO¶VWKHRUHPDVVXUHVXVWKDWWKHWRWDOHQHUJ\RIWKHVLJQDOREWDLQHGE\LQWHJUDWLQJ WKHVTXDUHPRGXOHRIWKHILHOGRQWKHFRRUGLQDWHVLVHTXDOWRWKHLQWHJUDORQWKHIUH TXHQFLHVRIWKHVTXDUHPRGXOHRIWKH)RXULHU¶VWUDQVIRUPRIWKHILHOG7KHDXWRFRU UHODWLRQWKHRUHPWKHQH[SUHVVHVWKHLPSRUWDQWFRQFHSWWKDWWKHSRZHUVSHFWUXPLV WKH )RXULHU¶V WUDQVIRUP RI WKH DXWRFRUUHODWLRQ RI WKH VLJQDO LWVHOI )RU D RQH GLPHQVLRQDOVLJQDOH[WHQGHGRYHUWLPHZHKDYH 

F



f

f

`

g [ g [  t d [



G Q 

 

$OORIWKLVLVSRVVLEOHLIWKHVLJQDOLVGHWHUPLQLVWLFIRUZKLFKWKHVHLQWHJUDOVDUH ILQLWH +RZHYHU LQ WKH FDVH RI QRQGHWHUPLQLVWLF VLJQDOV VXFK DV QRLVH ZLWKRXW VSDWLDORUWHPSRUDOOLPLWVZHFDQUHVRUWWRWKHFRQFHSWRIPHGLD7KXVIRUDstationaryUDQGRPVLJQDO1RUEHUW:LHQHUGHILQHGDQDXWRFRUUHODWLRQIXQFWLRQJLYHQ E\>:LHQHU%DUUHWWDQG0\HUV9DQGHU/XJW@ 

sg t

 T ³ g t c g t c  t dt c  T of T T OLP

 

6LQFHsgLVJHQHUDOO\DIXQFWLRQLQLZHFDQFDOFXODWHLWV)RXULHU¶VWUDQVIRUP 

S: Q

f

³ f sg t eSi Qt dt 

 

ZKLFKLVFDOOHGWKHspectral power density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¶VGHILQLWLRQRQHFDQXVHWKHSUHYLRXVRQHRI$6FKXVWHUWKDWSURSRVHG VSHFLI\LQJWKHVSHFWUXPRIg t E\WKHperiodogram GHILQHGE\>6FKXVWHU@ 

SS

  GT Q  T T of OLP

 



)RXULHU2SWLFV

ZKHUH 

T 

GT Q

³T  g t eSiQt dt 

 

LVWKH)RXULHU¶VWUDQVIRUPRIWKHIXQFWLRQg t WUXQFDWHGLQDQLQWHUYDORIZLGWKT )RUHDFKIXQFWLRQgIRUZKLFKS:LVILQLWHS:DQGSSFRLQFLGH :LWK LQFUHDVLQJ T LW LQFUHDVHV WKH IUHTXHQF\ UHVROXWLRQ FDOOHG WKH resolution bandwidth ZLWK ZKLFK WKH VSHFWUXP LV REWDLQHG EXW WKH SHULRGRJUDP GRHV QRW FRQYHUJHXQLIRUPO\IRUDVLPSOHVSHFWUDOIXQFWLRQHLWKHUEXWUDWKHULWLQFUHDVHVWKH VSHHG RI LWV RVFLOODWLRQV )RU WKLV UHDVRQ HOHFWURQLF VSHFWUXP DQDO\]HUV KDYH DQ DGGLWLRQDOILOWHUZLWKDZLGWKLQGLFDWHGE\WKHWHUPvideo bandwidth$IXUWKHUZD\ RI LPSURYLQJ WKH FRQYHUJHQFH LV WR PHGLDWH RQ PDQ\ GLVWLQFW UHDOL]DWLRQV RI WKH IXQFWLRQg t  $VWDWLRQDU\UDQGRPSURFHVVLQWKHEURDGVHQVHg t WKDWJRHVWKURXJKDVSDWLDO O\LQYDULDQWOLQHDUV\VWHPSURGXFHVDQHZUDQGRPSURFHVVg‫ މ‬t JLYHQE\ 

gc t

f

³f g t c h t  t c dt c 

 

ZKHUHWKHLQWHJUDWLRQFDQEHH[WHQGHGEHWZHHQfDQGfJLYHQWKDWWKHLPSXOVH UHVSRQVHhRIWKHV\VWHPLV]HURRXWVLGHRIDILQLWHLQWHUYDO,WFDQEHGHPRQVWUDWHG >9DQGHU/XJW@WKDWWKHVSHFWUDOSRZHUGHQVLW\DWWKHV\VWHPRXWSXWLVVLPSO\ JLYHQE\ 



SRXW Q

H Q SLQ Q 

 

ZKHUHSLQLVWKHLQSXWGHQVLW\DQGHLVWKHWUDQVIRUPRIh

5.5.3 Theory of partial coherence &RQVLGHUDSRO\FKURPDWLF UHDO VFDODUILHOG X rt ZKLFKFDQEHUHSUHVHQWHG E\LWV FRPSOH[ DQDO\WLFIXQFWLRQ 

u r t

f

³ U r Q eSi Qt d Q 

 

ZKHUHU rQ LVWKHtemporal)RXULHU¶VWUDQVIRUPRIWKHUHDOILHOGX r t ,IWKHILHOG LVTXDVLPRQRFKURPDWLFLWVDQDO\WLFDOIXQFWLRQFDQEHIDFWRUHGDV  7KHYHFWRUWKHRU\RISDUWLDOFRKHUHQFHIRUWKHHOHFWURPDJQHWLFILHOG LQFOXGLQJSRODUL]DWLRQ LV

UHSRUWHGE\7HUYR6HWlOlDQG)ULEHUJ  

&RKHUHQFH





u r  t u r  t e  SiQ t 

 

ZKHUH Q LVWKHFDUULHUIUHTXHQF\DQGuLVDFRPSOH[DPSOLWXGH WDNHQIURPX WKDW YDULHVrandomlyLQVSDFHDQGWLPH SOHDVHQRWHWKHFKDQJHRIIRQW 7KLVGHFRPSR VLWLRQFDQEHGRQHZLWKRXWORVVRIJHQHUDOLW\HYHQLIWKHILHOGVSHFWUXPLVEURDG EDQG,QWKLVFDVHDVXLWDEOHYDOXHFDQEHJLYHQWR Q IRUH[DPSOHWKHFHQWHUIUH TXHQF\RIWKHVSHFWUXP*LYHQWKDWWKHSKDVHRIuYDULHVUDQGRPO\LWVPHDQYDOXH LV]HUR 

u r t

 

 

7KHWULDQJXODUEUDFNHWVGHQRWHDQHQVHPEOHDYHUDJHEXWZHDVVXPHKHUHWKDWWKH HPLVVLRQ LV HUJRGLF VR ZH FDQ DOVR LQWHUSUHW WKHP DV DQ DYHUDJH RYHU WLPH ,I KRZHYHU WKH ZDYH LV PRQRFKURPDWLF WKH WHPSRUDO DYHUDJH LV QRW ]HUR DQG WKH FDVHLQZKLFKWKHZDYHLVWKHVXPRIWKHPRQRFKURPDWLFFRPSRQHQWVZLWKDSKDVH UHODWLRQVKLSHVWDEOLVKHGEHWZHHQWKHPVKRXOGEHWUHDWHGZLWKFDXWLRQ &RQVLGHUWZRSRLQWV3DQG3LQVSDFHWLPHZLWKFRRUGLQDWHV rt DQG rt  ,I WKH SURFHVV LV WHPSRUDULO\ VWDWLRQDU\ LWV FRPSOH[ VSDFHWLPH DXWRFRUUHODWLRQ IXQFWLRQRuGHSHQGVRQO\RQWKHGLIIHUHQFHW ttDQGGHILQHVWKHmutual coherence functionDV 

* r r W { Ru r t  W r t

u r t  W u r t 

 

$SSO\LQJ WKH IDFWRUL]DWLRQ   D VHFRQG YHUVLRQ RI WKH PXWXDO FRKHUHQFH LV DOVRGHILQHGZKLFK*RRGPDQ  FDOOVWKHmutual intensity 

J r r W { Ru r t  W r t

u r t  W u r t 

 

%HWZHHQWKHVHWZRGHILQLWLRQVWKHUHLVWKHUHODWLRQVKLS 

* r r W

J r r W e i  SW 

 

)RUDTXDVLPRQRFKURPDWLFZDYHZLWKUHVSHFWWR*WKHIXQFWLRQJKDVWKHDG YDQWDJHRIYDU\LQJPRUHVORZO\ZLWKWMXVWDVuUHSUHVHQWVDVORZO\YDULDEOHDP SOLWXGHPRGXODWLQJWKHFDUULHU )XUWKHUKHUHDQGEHORZZHDVVXPHWKDWWKHSURFHVVLVHUJRGLFVRWKHHQVHPEOH DYHUDJHLVHTXDOWRWKHWLPHDYHUDJH,QDGGLWLRQWRWKHVHIXQFWLRQVDQRUPDOL]HG YHUVLRQFDOOHGWKHcomplex degree of mutual coherenceLVDOVRGHILQHG>%RUQDQG  )RUW WKHWZRIXQFWLRQVFRLQFLGH%RUQDQG:ROI  GHILQHJRQO\IRUWKLVYDOXHRIW

,QFRQWUDVW%DUUHWWDQG0\HUV  GHILQHDVWKHPXWXDOFRKHUHQFHIXQFWLRQZKLFKWKH\FDOO *ZKDW*RRGPDQ  FDOOVJVXFKDVKHUH



)RXULHU2SWLFV

)LJ 'LDJUDP RI WKH HYDOXDWLRQ RI WKH PXWXDO FRKHUHQFH IXQFWLRQ

(r2,t-|r -r2|/c) P2

S

O (r,t)

P1 (r1,t-|r -r1|/c)



:ROI@ J r r W

J r r W {



u r t



u r t





 

ZKLFKDVZHVKDOOVHHQRZFRUUHVSRQGVWRWKHYLVLELOLW\RIWKHIULQJHVRILQWHUIHU HQFH EHWZHHQ WZR ZDYHV FRPLQJ IURP WZR GLIIHUHQW RULJLQV 7KH GHQRPLQDWRU VKRZVH[FHSWIRUDFRQVWDQWIDFWRUWKHJHRPHWULFPHDQRIWKHLQWHQVLW\SURGXFHG E\WKHWZRVRXUFHVVHSDUDWHO\ /HWXVUHVWDUWIURP9DQGHU/XJW@ 7KHIXQFWLRQJLVIRUr rDQGW DQGRUGLQDULO\LWLVVLJQLILFDQWO\GLIIHU HQWIURP]HURRQO\LQDILQLWHQHLJKERUKRRGRIWIURPDQGLQDILQLWHQHLJKERU KRRGRIrrIURPZLWKDJUDGXDOGHFD\DZD\IURPWKHRULJLQ:HFDQFKDUDF WHUL]H LW LQ WZR FRPSOHPHQWDU\ ZD\V )LJ   DV DQ DLG WR XQGHUVWDQGLQJ WKH SULQFLSOHVRIFRKHUHQFH7KHFRPSOH[GHJUHHRIFRKHUHQFHLVDIXQFWLRQRIVHYHQ SDUDPHWHUVRIZKLFKVL[VSDWLDODUHWKHFRRUGLQDWHVRI rDQG rRIWKHWZRKROHV DQGRQHWHPSRUDOW2QWKHRWKHUKDQGWFRUUHVSRQGVWRWKHWUDYHOWLPHGLIIHUHQFH WRZDUGWKHREVHUYDWLRQSRLQWIURPWKRVHWZRSRVLWLRQVDQGWKHUHIRUHGHSHQGVRQ WKHJHRPHWU\RIWKHV\VWHP,Q:DOWKHU@





J r  r





VLQ k r  r

k r  r



 

ZKHUH k  SO LV WKH PRGXOXV RI WKH ZDYH YHFWRU FRUUHVSRQGLQJ WR WKH FDUULHU >%DUUHWWDQG0\HUV@8QIRUWXQDWHO\IRUWKLVIXQFWLRQWKHYDULDQFHGHILQHGDV LQ(T  GLYHUJHV2QWKHRWKHUKDQGWKHIXQFWLRQJRIDQH[WHQGHG/DPEHU



)RXULHU2SWLFV

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specklesZKLFKDUHFKDUDFWHUL]HGE\UDSLGIOXF WXDWLRQVLQWKHLQWHQVLW\DQGSKDVHRIWKHZDYHILHOGDVWKHSRVLWLRQFKDQJHV,IOD VHUGLIIXVHUDQGVFUHHQDUHVWXFNWRHDFKRWKHUWKHVHVSHFNOHVDUHLPPRELOH%H \RQG WKH WZR KROHV ZH QRZ FRQVLGHU DQRWKHU VFUHHQ WR REVHUYH WKH LQWHUIHUHQFH IULQJHVRIWKHUDGLDWLRQWKDWSDVVHVWKURXJKWKHP%\PRYLQJWKHKROHVWKHIULQJHV ZLOOFKDQJHWKHLUSRVLWLRQUDQGRPO\DQGWKHLUYLVLELOLW\ZLOOIOXFWXDWHLQDUHPDUND EOHZD\EXWZLWKRXWGHFD\2XUPHDVXUHZLOOWKHQGHWHUPLQHDKLJKGHJUHHRIspatial coherenceHYHQWKRXJKWKHVSDWLDO correlation lengthRIWKHILHOGRQWKHILUVW VFUHHQPD\EHSDUWLFXODUO\VPDOO>6KDPLU@ ,IWKHGLIIXVLQJREMHFWLVSODFHGLQUDSLGPRWLRQWKHVSHFNOHVPRYHDQGWKHUH IRUHWKHYDOXHRIJREWDLQHGZLWKWKHWHPSRUDODYHUDJHRI(T  FDQJLYHWKH FRUUHFWYDOXHRIWKHVSDWLDOFRUUHODWLRQRIWKHILHOG2QWKHRWKHUKDQGWKHILHOGRQ WKHILUVWVFUHHQLVUDSLGO\PRGXODWHGLQDPSOLWXGHDQGSKDVHE\WKHPRYHPHQWRI WKHREMHFWDQGLVQRORQJHUPRQRFKURPDWLF

:DYHHTXDWLRQVIRUWKHPXWXDOFRKHUHQFH

)ROORZLQJ%RUQDQG:ROI  OHWXVQRZFRQVLGHUDVWDWLVWLFDOO\VWDWLRQDU\ ZDYHLQWKHYDFXXPDQGOHWu rt u rt EHWKHZDYHDPSOLWXGHVLQWZRGLIIHU HQWVSDFHWLPHSRVLWLRQV:LWKWKHDVVXPSWLRQRIHUJRGLFLW\WKHPXWXDOFRKHUHQFH IXQFWLRQLVJLYHQE\ * r r t  t



u r t  t u r t  t

 T of T /LP

t

T

³ u r t  t u r t  t dt



T

$SSO\LQJWR*WKH/DSODFLDQRSHUDWRU 

’

w w w    wx wy wz

ZKHUH x y zDUHWKHFRRUGLQDWHVRI rDQGH[FKDQJLQJWKHRUGHURIWKHRSHUD WRUVZHKDYH

&RKHUHQFH





’ * r r t  t

  T of T

T

/LP

³

ª¬’ u r t  t º¼ u r t  t dt 

T

$VZHNQRZIURP†WKHZDYHHTXDWLRQ  LVDOVRDSSOLFDEOHWRWKHDQD O\WLFIXQFWLRQ uRIWKHILHOG7KHUHIRUHZHFDQUHSODFHLQWKHLQWHJUDO ’ ZLWK v wwt ZKHUHvLVWKHYHORFLW\LQWKHPHGLXP%\DJDLQLQYHUWLQJWKHRUGHU RIRSHUDWLRQVZHFDQZULWH ’ * r r t  t



 w * r r t t  v  wt

D 

5HSHDWLQJWKHSURFHGXUHIRUSRLQWZHDOVRKDYH ’ * r r t  t



 w * r r t  t  v  wt

E 

)LQDOO\KDYLQJDVVXPHGWKHVWDWLRQDULW\IRUZKLFK*GHSHQGVRQO\RQWKHGLIIHU HQFHEHWZHHQWKHWLPHVW ttZHKDYHw*wt w*wt w*wW,QFRQ FOXVLRQWKHIXQFWLRQRIPXWXDOFRKHUHQFHREH\VWZRZDYHHTXDWLRQVDVWKHVSDFH WLPHSRVLWLRQRIWKHWZRSRLQWVFRQVLGHUHGLVYDULHG  9DQ&LWWHUW=HUQLNHWKHRUHP

$V D FRQVHTXHQFH RI (TV   ZH FDQ WKHQ DOVR DSSO\ WKH PDWKHPDWLFDO PHWKRGV RI WKH WKHRU\ RI GLIIUDFWLRQ WR WKH FDOFXODWLRQ RI WKH PXWXDO FRKHUHQFH ZKHQWKLVLVNQRZQRQDVXUIDFH&RQVLGHUWKHSDUWLFXODUFDVHRIDSULPDU\LQFR KHUHQWH[WHQGHGVRXUFHRQDJLYHQSODQHVXUIDFHSWDNHQDVSHUSHQGLFXODUWRWKHz D[LV )LJ $SSO\LQJWKHILUVW5D\OHLJK6RPPHUIHOGGLIIUDFWLRQLQWHJUDOIRUD QRQPRQRFKURPDWLFZDYH  DQGWDNLQJLQWRDFFRXQWWKDWZHKDYHWRVDWLVI\ WKHWZRZDYHHTXDWLRQV  ZHWKHQKDYH 



* r rc  t  t



 z w * rc  rc  t  R v  t drc  Sv ³³ R wt S





DQGLQVXFFHVVLRQ  7KHFKRLFHRIWKLVLQWHJUDOKHUHLVRQO\IRUPDWKHPDWLFDOFRQYHQLHQFH%RUQDQG:ROI  

UHSRUWDVLPLODUSURFHGXUHEXWXVLQJWKH.LUFKKRIIGLIIUDFWLRQLQWHJUDODSSOLFDEOHWRVXUIDFHVRI DQ\IRUP8QIRUWXQDWHO\WKHILQDOH[SUHVVLRQLVPXFKPRUHFRPSOLFDWHGLQFOXGLQJPDQ\WHUPV ZKLFKDUHLQVWHDGUHGXFHGKHUHWRRQO\WKHVHFRQGGHULYDWLYHRYHUWLPH



)RXULHU2SWLFV

y R1

x

S

P1 z2

S1 O

z1

S2

R2

P2

z



)LJ3URSDJDWLRQRIWKHPXWXDOFRKHUHQFH

 z w * r rc  t  t  R  v drc  ³³ S  v R  wt

* r r t  t





S



ZKHUHZLWKUHVSHFWWRWKHRULJLQ2 rDQG rLQGLFDWH twoREVHUYDWLRQSRLQWV3 DQG3LQIURQWRIWKHVXUIDFHSzDQGzDUHUHVSHFWLYHO\WKHGLVWDQFHRI3DQG 3IURPWKLVVXUIDFH LQWKH zGLUHFWLRQ ZKLOH r‫މ‬DQG r‫މ‬LQGLFDWHVWZRSRLQWV6 DQG6RQSODVWO\R _rr‫_މ‬DQGR _rr‫)_މ‬LQDOO\ * r r t  t



  S v 

z

z

w

³³ R drc ³³ R wtwt * rc  r c  t  R S



v  t  R v drc

S

  

:HFDQFRQVLGHUWKHVRXUFHUDGLDWLRQTXDVLPRQRFKURPDWLFZKHQWKHFRQGLWLRQ  LVUHVSHFWHG 

'Q 

v  RD  RE

 

ZKHUH RDDQG REDUHWKHPLQLPXPDQGPD[LPXPGLVWDQFHV_r r‫ _މ‬RU_r r‫ _މ‬ IURP WKH SRLQWV RQ WKH VXUIDFH S EHORQJLQJ WR WKH VWDWLRQDU\ SKDVH DUHDV PHQ WLRQHGLQ† DQGWKXVVLJQLILFDQWO\FRQWULEXWHWRWKHILHOGDWWKHREVHUYDWLRQ SRLQW3 RU3 ,QWKLVFDVHZHFDQDSSO\WKHDSSUR[LPDWLRQVDGRSWHGIRUWKHLQ WHJUDO  ZKHUHWKHIDFWRUL]DWLRQ  LVXVHGREWDLQLQJ J r r t  t Q     Z Z  i  S R  R v e J rc  rc  t  R v  t  R v drcdrc  ³³ ³³    v R R

 Q 

S S





ZKHUH R DQG R UHSUHVHQWWKHDYHUDJHYDOXHRIRDQG RDWWKHYDULDWLRQRIWKH

&RKHUHQFH



SRLQWV6DQG6RQWKHLQWHJUDWLRQVXUIDFHUHVSHFWLYHO\ )LQDOO\ LI WKH VRXUFH LV VSDWLDOO\ LQFRKHUHQW ZLWK LWV GHJUHH RI FRKHUHQFH DS SUR[LPDWHGE\(T  ZHFDQUHGXFH(T  WRDVLQJOHVXUIDFHLQWHJUDO ,QSDUD[LDODSSUR[LPDWLRQWDNLQJWKDWW DQGDVVXPLQJWKDW R  R v  WF  ZHKDYH * r r



J r r

Q  Ac v  Zo 

³³ e

i S

Q R  R v I

r  dr 

 

S

ZKHUHR _rr_R _rr_rLVQRZWKHSRVLWLRQRIDSRLQW6RQWKHVXUIDFHS RQZKLFKWKHSRLQWV6DQG6FRQYHUJHDQGZoLVWKHJHRPHWULFPHDQEHWZHHQz DQG z 7KLV LV WKH Van Cittert-Zernike theorem ZKLFK FDQ EH LQWHUSUHWHG DV WKH VROXWLRQWRDGLIIUDFWLRQSUREOHP/HWXVWDNHWKHSRLQW3DVIL[HGDQGWKHSRLQW3 DV YDULDEOH LPDJLQH WKDW WKH VRXUFH LV DQ RSHQLQJ LQ D VFUHHQ LOOXPLQDWHG E\ D VSKHULFDOZDYHWKDWFRQYHUJHVLQ3,QWKHRSHQLQJWKHUHLVWKHQDWUDQVSDUHQF\ WKDWPRGXODWHVWKHZDYH amplitudeLQSURSRUWLRQWR I r 7KHPXWXDOFRKHUHQFH IXQFWLRQDVWKHSRVLWLRQRI3FKDQJHVLVWKHUHIRUHSURSRUWLRQDOWRWKHDPSOLWXGH RIWKHZDYHGLIIUDFWHGE\WKDWDSHUWXUH$VDUHVXOWZHKDYHWKDWLIZHFDQDOVR FRQVLGHU DSULPDU\ VRXUFH DV FRPSOHWHO\ LQFRKHUHQW D VHFRQGDU\ VRXUFH PD\ LQ VWHDGEHSDUWLDOO\FRKHUHQWHVSHFLDOO\ZKHQWKHSULPDU\VRXUFHDSSHDUVDWWKHVHF RQGDU\RQHXQGHUDVPDOOVROLGDQJOH2QWKHRWKHUKDQGLIWKHVHFRQGDU\VRXUFHLV PXFKZLGHUWKDQLWVVSDWLDOFRKHUHQFHZLGWKWKHREMHFWVDUHVWLOOLQFRKHUHQWO\LO OXPLQDWHGE\WKLV $VDIXUWKHUVLPSOLILFDWLRQZHFRQVLGHUWKHFDVH z  z  ZR,QSDUD[LDODS SUR[LPDWLRQ DV IRU )UHVQHO¶V GLIIUDFWLRQ LQWHJUDOV IRU WKH DUJXPHQW LQ WKH H[SR QHQWLDORI(T  ZHKDYH>%RUQDQG:ROI@ 

R  R #

x  y  x  y  x  x x  y  y y    Zo Zo

 

ZKHUH x y x yDQG x yDUHUHVSHFWLYHO\WKHWUDQVYHUVHFRRUGLQDWHVRI r r DQGr(T  EHFRPHV * r r



Q  Ac i\  i  S f x x  f y y e ³³ e I r  dr  v  Zo 

 

S

ZKHUH  \

S

Q x  y  x  y   v  Zo

fx

Q x  x  v Zo

fy

Q y  y    v Zo

,QWKHSDUWLFXODUFDVHRIDXQLIRUPFLUFXODUVRXUFHZLWKUDGLXV aDQGLWVFHQWHU



)RXULHU2SWLFV

LQ2WKHcomplex degree of coherenceRQWKHREVHUYDWLRQSODQH ZKLFKLVQRUPDO L]HGWRLQ3 3 LV VHH7DEOH  

ZKHUH f r



J

 - Saf r i\ e  Saf r

f x   f y  7KHUHIRUHZLWK Q v  O DQG a Zo

J

- SDr O SDr O

ei\ 

 

D ZHKDYH

 

ZKHUH rLVWKHGLVWDQFHEHWZHHQ3DQG37DNLQJDVWKHUDGLXVRIWKHFRKHUHQFH DUHDWKDWFRUUHVSRQGLQJWRWKHILUVW]HURRIWKH%HVVHOIXQFWLRQ-ZHKDYH 

r # 

O  D

 

,Q H[SHULPHQWV LQ ZKLFK LW LV QHFHVVDU\ WR KDYH D KLJK GHJUHH RI FRKHUHQFH EH WZHHQWKHSRLQWV3DQG3%RUQDQG:ROI  WDNHDVWKHOLPLWGLVWDQFHWKH YDOXHWRZKLFKLWFRUUHVSRQGVWRJ WKDWLVE\VROYLQJ(T  IRUr 

rc # 

O  D

 

:LWK WKLV YDOXH ZH KDYH MXVWLILHG WKDW ZKLFK ZDV DQWLFLSDWHG LQ † IRU VXQ OLJKW

5.5.4 Hanbury Brown and Twiss’ interferometer $QDOWHUQDWLYHWR0LFKHOVRQ¶VVWHOODULQWHUIHURPHWHUZKLFKUHTXLUHVKLJKSKDVH VWDELOLW\RQWKHWZRRSWLFDOSDWKVFDQEHIRXQGLQRIWKHLQWHUIHURPHWHUGHYHORSHG E\ 5REHUW +DQEXU\ %URZQ DQG 5LFKDUG 4 7ZLVV >+DQEXU\ %URZQ DQG 7ZLVV @,WRQO\UHTXLUHVWKHPHDVXUHRIWKHLQWHQVLW\E\WZRGHWHFWRUVWKDWPD\EH YHU\GLVWDQW,QWKHLUH[SHULPHQWWKH\REVHUYHGWKH6LULXVVWDUZLWKWZRVPDOOWHOH VFRSHV WKDW ZHUH DERXW  PHWHUV DSDUW %\ FRUUHODWLQJ WKH LQWHQVLW\ IOXFWXDWLRQV UHFRUGHGLQWKHWZRVLJQDOVLWLVSRVVLEOHWRGHWHUPLQHWKHVL]HRIWKHVWDUVRXUFH 7KHFRUUHODWLRQIXQFWLRQXVHGLV 

E r t E r t E r t  W E r t  W 

&RKHUHQFH



ZKHUHrDQGrDUHWKHSRVLWLRQVRIWKHWZRGHWHFWRUV7KHFRUUHVSRQGLQJFRPSOH[ GHJUHHRIIRXUWKRUGHUFRKHUHQFHLV I r t I r t  W

J  r r W



I r t I r t



)RUDVWDWLRQDU\WKHUPDOVRXUFHZHKDYHWKDWWKHGLVWULEXWLRQRIWKHLQWHQVLWLHVP I  LV*DXVVLDQDURXQGWKHPHDQYDOXH IR,QWKLVFDVHLWKDVEHHQVKRZQWKDW>:ROI @ J  r r W  



  J r r W  

:HDOVRKDYHWKDWWKHFRUUHODWLRQRIWKHLQWHQVLW\IOXFWXDWLRQVGHILQHGE\ 

GI

I I 

LVJLYHQE\



GI r t GI r t  W

ª¬ I r t  I r t º¼ ª¬ I r t  W  I r t º¼  I r t I r t  W  I r t I r t 

WKDWLV 

GI r t GI r t  W

I r t I r t

J  r r W  

  J r r W  

7KHUHIRUHE\PHDVXULQJWKHFRUUHODWLRQRIWKHLQWHQVLW\IOXFWXDWLRQVLWLVSRVVLEOH WR GHWHUPLQH WKH PRGXOXV RI WKH FRPSOH[ GHJUHH RI FRKHUHQFH RI WKH UDGLDWLRQ PHDVXUHGRQWKH(DUWKHPLWWHGIURPWKHVWDUDQGWKHQSURFHHGWRGHWHUPLQHWKH GLDPHWHURIWKHVWDUDVLQWKHFDVHRIWKHVWHOODULQWHUIHURPHWHU

5.5.5 A classic experiment ,QWKHH[SHULPHQWZLWKWKH)UHVQHO¶VELSULVP VHH† ZHIRXQGWKDWWKHYLV LELOLW\RIWKHIULQJHVLVUHGXFHGGRZQWR]HURDQGLVWKHQSDUWLDOO\UHFRYHUHGZLWKD SHULRGLFWUHQGDVWKHVRXUFHVL]HLQFUHDVHV7KHVDPH³UHYLYDO´RIYLVLELOLW\LVDOVR REVHUYHGLQ5H\QROGV HWDO@

5.5.6 Propagation of coherence in an optical system 7KHUHVXOWVRIWKHSUHFHGLQJVHFWLRQVFDQEHJHQHUDOL]HGIRUDQRSWLFDOV\VWHP WKDWSURGXFHVLPDJHV7KHFDVHRIQRQPRQRFKURPDWLFZDYHLQDV\VWHPWKDWLV DW OHDVWORFDOO\ LVRSODQDWLFPD\EHWUHDWHGE\WKHDQDO\WLFIXQFWLRQRIWKHILHOGRQWKH LPDJHSODQHFDOFXODWHGE\LQWHJUDWLQJ(T  RQWKHIUHTXHQFLHVQ

&RKHUHQFH



pi

po

~ Si º S o

So

I2

O1

I1

O2

 )LJ6WXG\RIWKHSURSDJDWLRQRIWKHPXWXDOFRKHUHQFHSoDQGSiDUHUHVSHFWLYHO\WKH REMHFWDQGLPDJHSODQHVZKLOHpo DQG pi DUHWKHFRUUHVSRQGLQJSXSLOV

f

u x y  t

³



ª  º i  SQ «t  L x  y Q » ¬ c ¼ dQe





 

u³³ $36) x  xo  y  y o  Q U o xo  y o  Q dxo dy o  S o

ZKHUH U o LVWKHVSHFWUXPRIWKHILHOGRQWKHREMHFWSODQHUHVFDOHGWRWKHFRRUGL QDWHV xo DQG y o DFFRUGLQJWR(TV  DQGcLVWKHVSHHGRIOLJKWLQWKHYDF XXP+HUHLSOD\VWKHUROHRI R LQ(T   7KH GHSHQGHQFH RQ IUHTXHQF\ Q DSSHDUV LQ YDULRXV SRVLWLRQV KRZHYHU LI WKH VSHFWUDOEDQGZLGWKLVVPDOOHQRXJKWRRYHUORRNHYHQWKHFKURPDWLFDEHUUDWLRQVRI WKHV\VWHPZHFDQLJQRUHWKLVGHSHQGHQFHLQ LDQGLQWKH$36):HFDQWKHQUH YHUVHWKHLQWHJUDWLRQRUGHUDQGLQWHJUDWHZLWKUHVSHFWWRQXVLQJ(T  RE WDLQLQJ  u x y  t



§

·

$36) x  xo  y  y o  Q uo ¨ xo  y o  t  L x y Q ¸ dxo dy o  ³³ c © ¹ 



So





 

:LWKWKHIDFWRUL]DWLRQ  ZHWKHQKDYH u x y  t

e

Q i  S L x  y Q c

§





·

$36) x  xo  y  y o  Q uo ¨ xo  y o  t  L x y Q ¸ dxo dy o  ³³ c © ¹ 

So



 

/HWXVQRZFRQVLGHUWZRSRLQWV,DQG,RQWKHVDPHLPDJHSODQHDQGWKHLURE MHFWSRLQWV2DQG2 )LJ 7KHIXQFWLRQJRIPXWXDOFRKHUHQFHHYDOXDWHGRQ WKHLPDJHSODQHRQSRLQWV,DQG,LVWKHQJLYHQE\



)RXULHU2SWLFV

J x  y x  y  W

u x  y  t  W u* x  y  t

t

Q Q i  S L x yQ i  S L x  y Q c e c

 § · u³³ $36) x  xo y  y o Q uo ¨ xo  y o  t  L x  y Q ¸ dxody o c © ¹ S



o

 § · u³³ $36) x  xo   y  y o   Q uo  ¨ xo   y o   t  L x  y  Q ¸ dxo  dy o  c © ¹ S o





 t

 

7DNLQJWKHDYHUDJHRYHUWLPHLQVLGHWKHLQWHJUDOVZHILQDOO\JHW Q i  S 'L e c

J x  y x  y  W

u³³ ³³ $36) x  xo  y  yo  Q $36) x  xo  y  y o  Q   



S o S o

u J xo  yo  xo  yo  W  'L c dro dro 

ZKHUH 

'L

L x  y Q  L x  y  Q  

 

8VXDOO\LQDJRRGRSWLFDOV\VWHPLWLVUHTXLUHGWKDWDPRQJDOORIWKHJHRPHWULF SDWKV WKDW FRQQHFW DQ REMHFW SRLQW WR LWV LPDJH WKH SDWK FKDQJHV DUH VPDOO RU DW OHDVWQRELJJHUWKDQWKHW\SLFDOZDYHOHQJWKIRUZKLFKWKHV\VWHPLVEXLOW,QRWKHU ZRUGVWKH$36)LVDYHU\QDUURZIXQFWLRQDQGWKHFRQWULEXWLRQVWRWKHLQWHJUDORQ S o HVVHQWLDOO\GHULYHRQO\IURPDVPDOOQHLJKERUKRRGRI,DQG,0RUHRYHUWKH RSWLFDOSDWKGLIIHUHQFH'LEHWZHHQWKHWZRREMHFWSRLQWVDQGWKHUHVSHFWLYHLP DJHVLVJHQHUDOO\VPDOOFRPSDUHGWRWKHSDWKOHQJWK7KHUHIRUHLIWKHWZRREMHFW SRLQWV DUH FORVH HQRXJK WR HDFK RWKHU ZLWK TXDVLPRQRFKURPDWLF UDGLDWLRQ ZH FDQKDYH'LFWF,QWKDWFDVHZHFDQLJQRUHWKHWHUPLQ'LLQWKHDUJX PHQWRIJLQWKHLQWHJUDO ,IWKHQWKHLOOXPLQDWLRQRQWKHREMHFWSODQHLVWRWDOO\LQFRKHUHQWIRUZKLFKZH FDQDSSO\WKHDSSUR[LPDWLRQ  IRUW ZHKDYH



J x  y x  y 

Ac e

Q i  S 'L c

u³³ $36) x  xo  y  y o  Q $36) x  xo  y  y o  Q I xo  y o dro  S o

  

&RKHUHQFH



DQGLIWKHWZRREVHUYDWLRQSRLQWVFRLQFLGHZHDJDLQILQG(T   ,IWKHLQWHQVLW\RQWKHREMHFWSODQHLVFRQVWDQWZHFDQLQWHUSUHWWKHLQWHJUDORI (T   DV DQ DXWRFRUUHODWLRQ WKLV WLPH RI WKH $36) LWVHOI 7KHUHIRUH LI WKH V\VWHPLVLVRSODQDWLF JGHSHQGVRQO\RQWKHGLIIHUHQFHV x  x x y  y y DQG LI WKH $36)V DUH OLPLWHG WR D VPDOO H[WHQVLRQ LQ WKH LQWHJUDWLRQ VXUIDFH FKDQJLQJWKHLQWHJUDWLRQYDULDEOHLQWRr ‫މ‬o rorZHFDQUHZULWHJDV



J x  y

Ac e

Q i  S 'L c Io

u³³ $36)  xoc   yoc  Q $36) x  xoc  y  yoc  Q droc 



 

f

)RU(TV  DQGWKHDXWRFRUUHODWLRQWKHRUHP  LWV)RXULHU¶VWUDQV IRUPLVWKHQ 



F ^J x  y ` v p Odi f x  Odi f y







 

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aDQGLWLVLOOX PLQDWHGXQLIRUPO\ZHFDQDVVXPHWKDW 



p Odi f x  Odi f y





§ Od f · v FLUF ¨ i r ¸  © a ¹

 

%\ DQWLWUDQVIRUPLQJ VHH 7DEOH   WKH PXWXDO LQWHQVLW\ RQ WKH LPDJH SODQH LV JLYHQE\



§ a · J r v S ¨ ¸ © Od i ¹



§ ar ·  - ¨ S  ¸ Od i ¹ ©  ar S  Od i

 

ZKHUH rLVWKHGLVWDQFHEHWZHHQWKHREVHUYDWLRQSRLQWV,DQG,DQGZKHUH-LV WKH%HVVHOIXQFWLRQRIWKHVDPHQDPH



)RXULHU2SWLFV

5.5.7 Partial coherence and microscopy $OWKRXJKJHQHUDOO\ZLWKLQFRKHUHQWLOOXPLQDWLRQWKHFRUUHODWLRQOHQJWK LFRQ WKH VXUIDFH RI WKH LOOXPLQDWHG REMHFWV LV YHU\ VPDOO WKH VLWXDWLRQ FKDQJHV ZKHQ WKHVHREMHFWVDUHREVHUYHGZLWKDPLFURVFRSHHVSHFLDOO\ZKHQLWLVFDSDEOHRIUH VROYLQJGLVWDQFHVFORVHUWRRUORZHUWKDQ LF,QWKLVVLWXDWLRQWKHFRKHUHQFHRIWKH LOOXPLQDWLRQPDNHVWKHRSWLFDOV\VWHPQRQOLQHDUZLWKUHVSHFWWRLQWHQVLW\FRQVH TXHQWO\ LQ WKH HQODUJHG LPDJHZHKDYH WKH SUHVHQFHRI³IULQJHV´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condenserRUIURPWKHIURQWWKURXJKWKHVDPHREMHFWLYH epi-illumination 7KHUHDUHHVVHQWLDOO\WZRZD\VWRFRQFHQWUDWHWKHVRXUFHUDGLDWLRQRQWKHRE MHFWSODQH )LJ 7KHILUVWFDOOHGcritical illuminationLVWRXVHDVOLJKWO\H[ WHQGHGVRXUFHDQGWRSURGXFHDVKUXQNLPDJHRILWRQWKHREMHFWSODQH+RZHYHU WKHQRQXQLIRUPLW\RILQWHQVLW\RIWKHVRXUFHLVUHSURGXFHGRQVXFKDSODQH7KH VHFRQGPHWKRGFDOOHG Köhler’s illuminationLVWRSURGXFHDQHQODUJHGLPDJHRI WKHVRXUFHRQWKHILUVWIRFDOSODQHRIWKHFRQGHQVHUZKLFKIRFXVHVWKHUDGLDWLRQDW LWVVHFRQGIRFDOSODQHZKHUHWKHREMHFWSODQHRIWKHPLFURVFRSHLVORFDWHG>.|KOHU @,QWKLVZD\WKHQRQXQLIRUPLW\RIWKHVRXUFHGRHVQRWJLYHULVHWRDVSDWLDO QRQXQLIRUPLW\ RI LOOXPLQDWLRQ EXW WKH\ DIIHFW LWV DQJXODU VSHFWUXP ,Q PRGHUQ PLFURVFRSHVWKHODPSLVFRQWDLQHGLQDERG\VHSDUDWHIURPWKHPLFURVFRSHDQG WKHUDGLDWLRQLVFRQGXFWHGWRWKHFRQGHQVHUWKURXJKDEXQGOHRIRSWLFDOILEHUVWKDW DUHPL[HGWRSURGXFHDVPXFKXQLIRUPLW\DVSRVVLEOH 7KHVXUSULVLQJUHVXOWRIWKHSUHYLRXVVHFWLRQMXVWLILHVDQRWKHUURDGIRUWKH=HU QLNH¶VFRQFOXVLRQUHVXPHGE\%RUQDQG:ROI  ZKHQKHVWDWHVWKDWWKHPX WXDOLQWHQVLW\RQWKHLPDJHSODQH is independent of the aberrations of the system DQGWKDWthe condenser aberrations have no influence on the resolution power of a microscope7KLVLV FRQWUDU\WRWKHRFFDVLRQDODVVHUWLRQWKDWWKHFRQGHQVHUPXVWEH ZHOOFRUUHFWHGWRLQFRKHUHQWO\LOOXPLQDWHWKHREMHFWSODQH7KLVDSSOLHVWRFDVHVRI ERWKFULWLFDODQG.|KOHU¶VLOOXPLQDWLRQ>%RUQDQG:ROI@,QWKHH[DPSOHRI WKHIRFXVLQJHUURUDORQHZLWKFULWLFDOLOOXPLQDWLRQWKHFRQGHQVHUSURGXFHVDORQ JLWXGLQDOO\VKLIWHGLPDJHRIWKHVRXUFHEXWIRUWKHYDQ&LWWHUW=HUQLNHWKHRUHPLW VXIILFHVWKDWWKLVVRXUFHLVODUJHFRPSDUHGWRLWVFRKHUHQFHDUHDDQGWKDWLWORRNV DQJXODUO\ODUJHIURPWKHSODQHRIWKHPLFURVFRSHVRDVWRJHQHUDWHLQFRKHUHQWLO

&RKHUHQFH



condenser aperture diaphragm

condenser aperture diaphragm

(a)

(b)

collector lamp

field diaphragm

lamp

field diaphragm



)LJ D FULWLFDOLOOXPLQDWLRQ E .|KOHU¶VLOOXPLQDWLRQ

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¶VILJXUHJHQHUDWHG E\ WKH GLIIUDFWLRQ IURP WKH FLUFXODU HGJH RI WKH SXSLO KDV D UDGLXV JLYHQ E\ (T   

r$i



Oo si  ni a

 

ZKHUH aLVWKHUDGLXVRIWKHSXSLODQG siLVWKHGLVWDQFHRIWKHLQWHUPHGLDWHLPDJH SODQHIURPWKHSULQFLSDOSODQHRIWKHREMHFWLYHORLVWKHDYHUDJHZDYHOHQJWKLQWKH YDFXXP DQG ni LV WKH UHIUDFWLYH LQGH[ RI WKH LQWHUPHGLDWH VSDFH $OWKRXJK WKH SURSDJDWLRQLVQRWSDUD[LDOLQWKHREMHFWVSDFHRIWKHREMHFWLYHZHFDQLQYRNHWKH VLQHFRQGLWLRQIRUZKLFKEHWZHHQWKHKHLJKWVhoDQGhiRIDQREMHFWDQGLWVLPDJH WKHUHIUDFWLYHLQGH[HVnoDQGniDQGWKHDQJOHVDoDQGDiRIDPDUJLQDOUD\SDVVLQJ WKURXJKWKHEDVHRIWKHREMHFWWKHIROORZLQJUHODWLRQKROGV 

ho no VLQ Do

hi ni VLQ Di # hi ni

a  si



)RXULHU2SWLFV

5HSODFLQJhiZLWKWKHYDOXHRIr$iIURP(T  ZHREWDLQWKHDQDORJRXV$LU\ UDGLXVLQREMHFWVSDFH 

r$i # 

OR  no VLQ Do

 

7KLV GLVWDQFH UHSUHVHQWV WKH UHVROXWLRQ RI WKH REMHFWLYH DFFRUGLQJ WR 5D\OHLJK¶V FULWHULRQZLWKLQFRKHUHQWLOOXPLQDWLRQ7KHTXDQWLW\ no VLQDoLVWKH numerical aperture1$RERIWKHREMHFWLYHDQGLVJHQHUDOO\PDUNHGRQLWVEDUUHOXQGHUWKHLQGL FDWLRQRIPDJQLILFDWLRQ*RRGREMHFWLYHVLQWKHDLUW\SLFDOO\KDYHYDOXHVRI1$RE DURXQGDQGFDQUHDFK7KHLPPHUVLRQREMHFWLYHVFDQKDYH1$XSWR $QDQDORJRXVUHDVRQLQJFDQEHFRQGXFWHGWRGHWHUPLQHWKHFRKHUHQFHDUHDSUR GXFHGE\WKHVRXUFHRQWKHSODQHRIWKHPLFURVFRSH7KHUHIRUHWKHQHHGIRULQFR KHUHQW LOOXPLQDWLRQ UHTXLUHV WKDW WKH FRQGHQVHU KDYH D QXPHULFDO DSHUWXUH 1$FR JUHDWHUWKDQWKDWRIWKHREMHFWLYH 

m

1$FR !   1$ RE

,QSDUWLFXODU+++RSNLQVDQG30%DUDKDPVKRZHGWKDWWKHEHVWUHVROXWLRQ DFFRUGLQJ WR 5D\OHLJK  LV REWDLQHG ZLWK D UDWLR m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dark field illuminationVRWKDW RQO\GLIIUDFWHGUDGLDWLRQIURPREMHFWVLVFROOHFWHGE\WKHREMHFWLYH2QHDOWHUQDWLYH LVWRSODFHDFLUFXODUVWRSRQWKHFHQWHURIWKHVHFRQGIRFDOSODQHRIWKHREMHFWLYHLQ RUGHUWREORFNWKHGLUHFWUDGLDWLRQ central dark field +RZHYHUERWKV\VWHPVDOWHU WKHFRQWULEXWLRQRIVSDWLDOIUHTXHQFLHVDQGWKHUHVXOWLQJLPDJHLVQRWIDLWKIXO,Q DQ\FDVHWKHGLDSKUDJPSURGXFHVDQDOWHUDWLRQRIWKH36)DWWULEXWDEOHWRWKHFRQ GHQVHUZLWKQRQWULYLDOFRQVHTXHQFHVRQWKHLOOXPLQDWLRQFRKHUHQFH 7KHGHFLVLYHVROXWLRQIRUPDNLQJWKHWUDQVSDUHQWELRORJLFDOREMHFWVYLVLEOHZDV SURSRVHG E\ )ULW] =HUQLNH WKURXJK WKH LQWURGXFWLRQ RI D UHWDUGLQJ ODPLQD RQ WKH

6SDWLDOILOWHULQJ



LPDJH IRFDO SODQH RI WKH REMHFWLYH 7KLV WHFKQLTXH LV FDOOHG phase contrast microscopy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

6SDWLDOILOWHULQJ 7KHSHFXOLDUSURSHUW\RIDQRSWLFDOV\VWHPWRSURGXFHWKH)RXULHU¶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reduces WKH LQIRUPDWLRQ FRQWDLQHG LQ WKH VLJQDO 6RPH WLPHVWKLVLVMXVWWKHVLPSOLILFDWLRQZHZDQWWRJHWD³YHOYHW\´\RXWKIXOIDFH RUD EHDXWLIXO 7(0 PRGH  LV JHQHUDOO\ PRUH ³DWWUDFWLYH´ WKDQ D ³ZULQNOHG´ RQH PDUNHGE\GLIIUDFWLRQULQJV ZKLFKRQWKHRWKHUKDQGFRQWDLQVPRUHLQIRUPDWLRQ DERXWD³OLYHGOLIH´ RIGXVWDQGVFUDWFKHVRQWKHRSWLFV +RZHYHULQPDQ\FDVHV RQHZDQWVWRH[WUDFWSDUWLFXODUGHWDLOVIURPDVFHQHWKDWLVFRPSOH[RUDIIHFWHGE\ QRLVHRUDEHUUDWLRQV7KHILOWHULQJPXVWWKHQEHZHOOFDOLEUDWHGWRDYRLGIDOVHGH WDLOVRUIDOVHDWWULEXWLRQVWKDWLVDQ\LQIRUPDWLRQRILWVRZQWKDWLWDGGV ,QJHQHUDOWKHWHUPspatial filteringPHDQVWKHPXOWLSOLFDWLRQRIWKHVSHFWUXPRI DVLJQDOZLWKDJLYHQWUDQVPLVVLRQIXQFWLRQ,WFDQEHUHDOL]HGLQWKHPRVWYDULHG IRUPV 7KH VLPSOHVW FDVH LV WKDW RI WKH GLDSKUDJPV RU PDVNV ZLWK DSSURSULDWH VKDSHZKLFKDUHFDOOHG binary filtersVLQFH WKHLUWUDQVPLVVLRQDVVXPHVRQO\WZR YDOXHVDQG7KHILOWHULQJFDQDOVREHSHUIRUPHGZLWKPXOWLSOHGLVFUHWHOHYHOV



)RXULHU2SWLFV

y

(a)

y

t(x,y) x

f

f y

(b)

f

x

2f y

(c)

f

2f y u 2

us

x

x So R Z1

t¢(x,y) x

y

F(x,y) x

2f ut

t¢(x,y) x

f y

t(x,y)

y

F(x,y) x

Fo

Fi

Si

Zs Z2



)LJ6FKHPHVRIRSWLFDOVSDWLDOILOWHULQJ D ³f ´ILOWHULQJV\VWHPWKHILUVWOHQVSUR GXFHVDFROOLPDWHGEHDPRQWKHWUDQVSDUHQF\tDQGWKHRWKHUWZROHQVHVSHUIRUPWZRWUDQV IRUPVLQVXFFHVVLRQRQWKHLQWHUPHGLDWHIRFDOSODQHDVSDFHILOWHUFLVSODFHGDQGILQDOO\ RQWKHLPDJHIRFDOSODQHRIWKHODVWOHQVRQHJHWVWKHILOWHUHGVLJQDOt' E ILOWHULQJV\VWHP ZLWK FRQYHUJHQW LOOXPLQDWLRQ F  VWXG\ RI WKH )RXULHU¶V WUDQVIRUP SURSHUWLHV ZLWK D FRQ YHUJHQWZDYH

RU LQ FRQWLQXRXV IRUP IRU H[DPSOH ZLWK WUDQVSDUHQFLHV DOVR REWDLQHG SKRWR JUDSKLFDOO\ZKLFKPRGXODWHWKHLQWHQVLW\WUDQVPLWWHGDQGLQWKLVFDVHRQHVSHDNV RI intensity filters$QRWKHUZD\LVWRYDU\WKHUHODWLYHSKDVHRIWKHVSHFWUXPFRP SRQHQWVIRUH[DPSOHE\GHSRVLWLQJWKLQILOPVZLWKJLYHQVKDSHVDQGWKLFNQHVVRQ WUDQVSDUHQW RSWLFDO ODPLQDH ZKLFK DUH WKHUHIRUH phase filters )LQDOO\ WKH PRVW FRPSOH[FDVHLVWKDWRIintensityDQGphase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¶VWUDQVIRUPFDQEHREWDLQHGLQPRUHYDULHGIRUPVZLWKUH VSHFWWRWKDWZKLFKKDVEHHQGLVFXVVHGLQ†,QGHHGLQPDQ\DSSOLFDWLRQVWKH VLJQDOWREHSURFHVVHGLVUHFRUGHGRQDWKLQODPLQDDWUDQVSDUHQF\VXFKDVDSKR WRJUDSKLF ILOP D VSDWLDO LQWHQVLW\ DQGRU SKDVH PRGXODWRU XQLIRUPO\ LOOXPLQDWHG

6SDWLDOILOWHULQJ



E\DSODQHRUVSKHULFDOPRQRFKURPDWLFZDYHZLWKUDGLXVRIFXUYDWXUHRDQGZDYH OHQJWK O 7KH FRQFHSWXDOO\ VLPSOHVW FDVH LV WKDW VKRZQ LQ )LJ  D  ZKLFK LV FDOOHGD³f ´ILOWHULQJV\VWHPWKLVPHWKRGKDVWKHGLVDGYDQWDJHRISURGXFLQJYL JQHWWLQJ PDLQO\ LQWKHILUVWWUDQVIRUPDWLRQVXQOHVVRQHXVHVDODUJHHQRXJKOHQV WR FROOHFW GLIIUDFWHG UDGLDWLRQ HYHQ IURP WKH PDUJLQDO DUHDV RI WKH WUDQVSDUHQF\ 2QHV\VWHPWKDWXVHVRQHOHQVOHVVDQGWKDWGRHVQRWVXIIHUIURPYLJQHWWLQJLVWKDW RI)LJ E >*RRGPDQ@,QWKLVFDVHWKHLOOXPLQDWLRQRQWKHWUDQVSDUHQF\ LVFRQYHUJHQW /HW XV FRQVLGHU IRU H[DPSOH WKH FDVH LQ ZKLFK WKH VLJQDO LV LPSULQWHG RQ D WUDQVSDUHQF\SODFHGRQWKHSODQH1ZLWKDWUDQVPLVVLRQIXQFWLRQIRUWKHILHOGHTXDO WRt xy DQGWKDWWKHWUDQVSDUHQF\LVOLWE\DZDYHZLWKUDGLXVRIFXUYDWXUHR IRU D FRQYHUJLQJZDYH R LVQHJDWLYH  DV LQ )LJ  F  7KHWUDQVPLWWHG DPSOLWXGH LV JLYHQE\ uW x  y t x y uo e



i

S x  y OR





 

$SSO\LQJWKLVVLJQDOWRWKH)UHVQHOGLIIUDFWLRQLQWHJUDO  IRUSURSDJDWLRQLQD KRPRJHQHRXVPHGLXPXSWRWKHGLVWDQFH d RIURPWKHWUDQVSDUHQF\RQSODQH SZKHUHWKHLOOXPLQDWLQJZDYHFRQYHUJHVWKHDPSOLWXGHLV



uV Odf x  Odf y



i i  e Od



S d iSOd f   f  x y O e



uo

f

f

³f ³f t x y



i  S x f x  y f y e



   dx dy

7KHUHIRUHWKHDPSOLWXGHGLIIUDFWLRQILJXUHuV xVyV WKDWLVIRUPHGRQSODQH SFRU UHVSRQGVDSDUWIURPDILHOGFXUYDWXUHWHUPWRWKHWUDQVIRUPRI t xy YDOXHGDW WKHVSDWLDOIUHTXHQFLHVfx xVOdfy yVOd 1RZZHDGGWKHHIIHFWVRIDOHQVZLWKIRFDOfDVVKRZQLQ)LJ F ,ILQ(T  ZKLFKGHWHUPLQHVWKHILHOGDIWHUWKHOHQVRQDJHQHULFSODQHVFUHHQ 2ZH UHSODFHWKHILHOGuRQWKHREMHFWSODQHZLWKuWZHJHW SZ

u x  y





i f ik s  s i  x  y e OH e OH

f f

uuo

³ ³ t x y

S§ Z  · i ¨   ¸ x  y e O© H R ¹





e

i 

S f xx  y y OH dx dy 



 

f f

ZLWK H

f   ZZ  

ZKHUHZLVWKHGLVWDQFHRIWKHSODQHRIWUDQVSDUHQF\IURPWKHREMHFWIRFDOSODQHRI WKHOHQVDQGZLVWKHGLVWDQFHRIWKHVFUHHQ2IURPWKHLPDJHIRFDOSODQH :LWKVRPHSDVVDJHVLWLVIRXQGWKDWWKHWHUPZ HRLVFDQFHOHGZKHQ



)RXULHU2SWLFV

f   Z  R Z 



 

LH ZKHQ WKH SODQH 2 RQ ZKLFK WKH ILHOG LV FDOFXODWHG FRLQFLGHV ZLWK WKH LPDJH SODQHRIWKHZDYHVRXUFHSRLQWWKDWLOOXPLQDWHVWKHWUDQVSDUHQF\7KHUHIRUHDVLGH IURPDWHUPRIFXUYDWXUHRIILHOGuLVKHUHJLYHQE\WKH)RXULHU¶VWUDQVIRUPRIuW HYDOXDWHGDWWKHIUHTXHQFLHV 

fx

f x  O f  ZZ 







fy

f y  O f  ZZ 



 





DQGLQWKHVHFRQGLWLRQV(T  EHFRPHV



§ OH · OH u ¨ fx fy ¸ f © f ¹

i f ik s  s iS e  e OH

ZH f x  f y f

f f

uuo

³ ³ t x y e







i  S x f x  y f y

 

dx dy

f f

,I Z WKHQWKHWHUPRIFXUYDWXUHRXWRIWKHLQWHJUDOLVDOVRUHGXFHGWRXQLW\ +RZHYHU ZLWK UHVSHFW WR WKH FDVH UHSUHVHQWHG E\ (TV   ZH FDQ QRZ FKDQJH WKH WUDQVIRUPDWLRQ VFDOH E\ YDU\LQJ Z DQG NHHSLQJ WKH VRXUFH SRVLWLRQ IL[HGZLWKUHVSHFWWRWKHOHQVVRWKDWZ±R ZVLVFRQVWDQW :HFDQXQGHUVWDQGWKDWWKLVSURSHUW\DOVRDSSOLHVWRPRUHFRPSOH[RSWLFDOV\V WHPVE\LQVHUWLQJWKHWUDQVSDUHQF\RQWKHSDWKRIUDGLDWLRQEHWZHHQDVRXUFHSRLQW DQGLWVLPDJHSODQH2QWKLVSODQHDVLGHIURPDWHUPRIILHOGFXUYDWXUHZHKDYH WKH)RXULHU¶VWUDQVIRUPRIWKHVLJQDOUHFRUGHGRQWKHWUDQVSDUHQF\,WLVWKHUHIRUH QRWQHFHVVDU\WROLPLWRXUVHOYHVWRRSWLFDOV\VWHPVLQZKLFKWKHplane of the transformOLHVRQDIRFDOSODQH,QJHQHUDOLWLVORFDWHGRQWKHSODQHZKHUHWKHVSKHULFDO FDUULHUZDYHILQGVLWVFHQWHURIFRQYHUJHQFHZKLFKLWVKRXOGEHUHPHPEHUHGFRU UHVSRQGVWRWKHRULJLQRIWKHVSHFWUXPZLWK fx  fy RQZKLFKWKHVSDWLDOILOWHU LVWREHFHQWHUHG ,QWKHGLDJUDPIURP)LJ E ZHKDYHWKHUHIRUHWKDWWKHVSHFWUXPRIWKH WUDQVSDUHQF\LVIRUPHGRQWKHLPDJHSODQHRIWKHVRXUFHSURGXFHGE\WKHILUVWOHQV DQGWKHVSDWLDOILOWHULVSODFHGWKHUHLQ%HKLQGWKLVSODQHZHSODFHWKHVHFRQGOHQV ZKLFKSURGXFHVWKHILOWHUHGLPDJHRIWKHWUDQVSDUHQF\RQWKHILQDOSODQH

5.6.1 Binary filters 7KH ILUVW VWXGLHV RQ VSHFWUXP PDQLSXODWLRQ RI DQ LPDJH ZHUH PDGH E\ (UQVW $EEHLQDQGWKHQE\$%3RUWHULQZLWKWKHDLPRIYHULI\LQJ$EEH¶V WKHRU\RILPDJHIRUPDWLRQZLWKDPLFURVFRSH>$EEH3RUWHU@7KHLUHV

6SDWLDOILOWHULQJ













)LJ1XPHULFDOVLPXODWLRQRIVSDWLDOILOWHULQJSURGXFHGE\DQRSWLFDOV\VWHPZLWKWZR OHQVHVLQFRQILJXUDWLRQf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

VHQFHLVWRLOOXPLQDWHDWKLQSHULRGLFJULGRIZLUHVZLWKDFRKHUHQWDQGFROOLPDWHG EHDPDQGUHSURGXFHWKHLPDJHZLWKDOHQV 2Q WKH IRFDO SODQH RI WKH OHQV RU LQ WKH FHQWHU RI FRQYHUJHQFH RI WKH FDUULHU ZDYH WKHVSHFWUXPRIWKHJULGLVIRUPHGFRQVLVWLQJRIDVHULHVRILVRODWHGUHJ XODUO\VSDFHGFRPSRQHQWVZLWKDGLIIXVLRQGHJUHHDQGDVWUXFWXUHWKDWGHSHQGVRQ WKHOHQVSXSLODQGWKHH[WHQVLRQRIWKHJULGRUWKHLOOXPLQDWLRQ%\SODFLQJYDULRXV W\SHVRIdiaphragmRQWKLVSODQHRQHFDQJHWGLIIHUHQWVSDWLDOILOWHULQJHIIHFWV



)RXULHU2SWLFV

7KH HIIHFWLYHQHVV RI WKLV V\VWHP LV LOOXVWUDWHG E\ WKH H[DPSOH VKRZQ LQ )LJ LQZKLFKDJULGIRUPHGE\KRUL]RQWDODQGYHUWLFDOOLQHVLVVXSHULPSRVHGRQD *DXVVLDQEHDP+HUHWKH³f ´FRQILJXUDWLRQZLWKWZROHQVHVZDVXVHG%\LQWHU FHSWLQJLQWKHSODQHRIWKHWUDQVIRUPWKHVLJQDOVSHFWUXPZLWKDUHFWDQJXODUGLD SKUDJP WKDW RQO\ SDVVHV WKH FRPSRQHQWV DOLJQHG ZLWK WKH KRUL]RQWDO D[LV LQ WKH FHQWHURIWKHVSHFWUXPZHKDYHWKHYHUWLFDOOLQHVUHPRYHGIURPWKHEHDPSURILOH DWWKHLPDJHSODQHRIWKHV\VWHP %LQDU\ILOWHUVDUHQRWWKHEHVWLQSULQFLSOHEXWWKH\DUHRIWHQHIIHFWLYHDQGEH FDXVHRIWKHLUVLPSOLFLW\WKH\ILQGPDQ\DSSOLFDWLRQV)LOWHULQJZLWKGLDSKUDJPV FDQEHVXFFHVVIXOO\XVHGIRUH[DPSOHWRHOLPLQDWHWKHUDVWHURIWHOHYLVLRQLPDJHV DQGWKHKDOIWRQHVFUHHQLQJWKDWLVXVHGLQSKRWRSULQWLQJWRUHSUHVHQWFRQWLQXRXV WRQHLPDJH,QIDFWWKHVFUHHQLQJKDVDSHULRGLFVWUXFWXUHDQGWKHVSHFWUXPRIWKH SULQWHGSKRWRFDQEHFRQVLGHUHGDVWKHFRQYROXWLRQRIDUHJXODU'LUDF¶VGHOWDSDW WHUQZLWKWKHRULJLQDOVSHFWUXPRIWKHLPDJH7KHUHIRUHDVLQWKHVDPSOLQJWKHR UHPWKLVVSHFWUXPFDQEHUHWULHYHGE\PHDQVRIDGLDSKUDJPDOORZLQJRQO\RQHRI WKHGLIIXVHGOLJKWVSRWVWRSDVV W\SLFDOO\WKHRQHRIWKH]HURRUGHUZKLFKLVWKH EULJKWHVW  %LQDU\ ILOWHUV FDQ DOVR EH XVHG WR H[WUDFW SHULRGLF VLJQDOV LQ WKH SUHVHQFH RI QRLVHRUWRHOLPLQDWHXQZDQWHGSHULRGLFFRPSRQHQWVZKRVHIUHTXHQF\LVFRQWDLQHG ZLWKLQ WKH VSHFWUDO EDQG RI WKH LQSXW VLJQDO notches filters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

5.6.2 Schlieren filter :LWK WKH ZRUG schlieren ZKLFK PHDQV stria WKH XQHYHQQHVV RI WKH UHIUDFWLYH 

 )ROORZLQJ .LUFKKRII¶V WKHRU\ D OLQH RI GLVFRQWLQXLW\ LQ WKH LQWHQVLW\ RI WKH VLJQDO LV

UHSURGXFHGZLWKDGDUNOLQHEHWZHHQWZRVHULHVRIEULJKWIULQJHVSDUDOOHOWRWKHOLQHLWVHOI,QGHHG D ILOWHU IXQFWLRQ JLYHQ E\   UHFW fxaf\a  KDV DV LWV WUDQVIRUP G xy   aVLQF aSxaSy  &RQVLGHU D ILJXUH UHSUHVHQWLQJ D VWHS LQ WKH DPSOLWXGH 7KH FRQYROXWLRQ RI WKLV VWHS ZLWK WKH WUDQVIRUPRIWKHILOWHUSURGXFHVWKHGLIIHUHQFHEHWZHHQWKHRULJLQDOVWHSDQGLWVFRS\³EOXQW´E\ WKHFRQYROXWLRQZLWKWKHVLQF)DUDZD\IURPWKHGLVFRQWLQXLW\WKHUHVXOWLQJDPSOLWXGHLV]HUR ZKLOHQHDULWWKHDPSOLWXGHSUHVHQWVRVFLOODWLRQDURXQG]HUR7KHQWKHLQWHQVLW\KDVWZREULJKW IULQJHV RQ WKH VLGHV RI D GDUN RQH DW WKH VWHS DQG RWKHU ZHDNHU SDUDOOHO IULQJHV GXH WR WKH RVFLOODWLQJVKDSHRIWKHVLQF7KLVVLWXDWLRQPD\FKDQJHLIZHFRQVLGHUWKHHOHFWURPDJQHWLFHGJH FRQWULEXWLRQZKLFKLVQHJOHFWHGLQWKHRU\

6SDWLDOILOWHULQJ



LQGH[LQWUDQVSDUHQWPDWHULDOVLVLQGLFDWHG7KH\DUHQRWYLVLEOHZLWKRUGLQDU\RSWL FDO PHWKRGV EXW DUH D VHULRXV GHIHFW LI WKH\ DUH SUHVHQW LQ JODVV IRU RSWLFV 7KH VFKOLHUHQDUHDOVRFRPPRQO\SUHVHQWLQDLUSURGXFHGE\ZLQGIODPHKHDWRUPRY LQJ REMHFWV 7KHVH VWULDH ZHUH GHVFULEHG IRU WKH ILUVW WLPH E\ 5REHUW +RRNH LQ DIWHUKHREVHUYHGWKHHIIHFWRQWKHOLJKWE\KRWDLUULVLQJIURPWKHIODPHRID FDQGOH 7KH schlieren methodLVPDLQO\DWWULEXWHGWR$XJXVW7RHSOHUZKRGHVLJQHGDQ RSWLFDOV\VWHPIRUGHWHFWLQJWKHVWULDHSUHVHQWLQWKHJODVVWREHXVHGIRUPDNLQJ OHQVHV,WHVVHQWLDOO\FRQVLVWVLQWKHXVHRIDSRLQWVRXUFHIRULOOXPLQDWLQJLQWUDQV PLVVLRQWKHREMHFWWREHREVHUYHGDQGDVXEVHTXHQWFRQYHUJLQJOHQV$WWKHLPDJH SODQHRIWKHVRXUFHZKLFKLVWKHSODQHRIWKHWUDQVIRUPDELQDU\ILOWHULVSODFHG WKDWVLPSO\FRQVLVWVRIDEODGHLWLQWHUFHSWVWKHVLJQDOVRWKDWLWVHGJHSDVVHVWRWKH FHQWHURIWKHVSHFWUXPE\LQWHUFHSWLQJLQSDUWWKHFRQWLQXRXVFRPSRQHQW,QWKLV ZD\DERXWKDOIRIWKHVSHFWUXPLVWUDQVPLWWHGLQDJLYHQWUDQVYHUVHGLUHFWLRQ7KH VFKOLHUHQPHWKRGLQJHQHUDODOORZVIRUREVHUYLQJSKDVHREMHFWVDQGGHIHFWVLQWKH IRUPRIRSWLFVLQDVLPSOHZD\XVLQJRQO\WKHVSDWLDOFRKHUHQFHRIWKHVRXUFHDQG WKHUHIRUHZRUNVHYHQLQ³ZKLWH´OLJKW,QSDUWLFXODU schlierenSKRWRJUDSK\LQWUR GXFHGE\(UQVW0DFKLQLVQRZODUJHO\XVHGLQDHURQDXWLFDOHQJLQHHULQJWR YLVXDOL]HWKHIORZRIDLUDURXQGREMHFWV 7KHPDWKHPDWLFDOWUHDWPHQWRIWKLVV\VWHPLVQRWWULYLDO)RUVLPSOLFLW\ZHFRQ VLGHUKHUHDRQHGLPHQVLRQDOFDVHLQZKLFKWKHVLJQDOVSHFWUXPLVJLYHQE\DIXQF WLRQ U fx 7KHSUHVHQFHRIWKHEODGHOHDGVWRWKLVVSHFWUXPEHLQJPXOWLSOLHGE\D VWHSIXQFWLRQWKDWFDQEHZULWWHQDV t fx



   VJQ f x  

 

 i G x     Sx

 

ZKRVHWUDQVIRUPLVJLYHQE\ T x



2QWKHLPDJHSODQHWKHVLJQDOuiLVWKXVUHSUHVHQWHGE\WKHFRQYROXWLRQ ui x T x u x

f

³ f u xc T x  xc dxc

i f u xc  u x  dx c     S ³ f x  x c

ZKHUH u LV WKH WUDQVIRUP RI U DQG FRUUHVSRQGV WR WKH ILHOG RQ WKH LPDJH SODQH ZLWKRXWWKHEODGH7KHLQWHJUDQGGLYHUJHVWRZDUGVrf IRU xo BVRLWLVQRW WULYLDOO\WUHDWDEOH7KHLQWHJUDOFDQEHUHZULWWHQDV 



fu 

x  xc  u x  xc xc

d x c 

 



)RXULHU2SWLFV





)LJ   6LPXODWLRQ RI VFKOLHUHQ ILOWHULQJ 7KH EODGH LV DUUDQJHG YHUWLFDOO\ DQG WKH WKLQ YHUWLFDO IULQJHV DUH GXH WR DOLDVLQJ D  JUD\ VFDOH UHSUHVHQWDWLRQ EHWZHHQ  ZKLWH  DQG S EODFN RID phaseREMHFWZLWKLQDFLUFXODUDSHUWXUH E LPDJHRIWKHDSHUWXUHILOWHUHGE\WKH EODGH7KHSKDVHYDULDWLRQLVWUDQVIRUPHGLQWRDYDULDWLRQRILQWHQVLW\7KLVILJXUHLVSDUWLFX ODUO\XQIDLWKIXODQGXJO\GXHWRWKHH[FHVVLYHDPSOLWXGHRIWKHSKDVHPRGXODWLRQ 

DQGFDQEHFRQVLGHUHGWREHWKHDYHUDJHRIWKHGHULYDWLYHVRI uLQ xFDOFXODWHGDW WKHILQLWHGLIIHUHQFHRISDLUVRIYDOXHVuWDNHQV\PPHWULFDOO\IURPx7KHSUHVHQFH RIWKHLPDJLQDU\XQLWDVDIDFWRURIWKHLQWHJUDOLPSOLHVWKDW uiUHFHLYHVFRQWULEX WLRQVIURPERWKuDQGiu6XSSRVHWKDWWKHREMHFWLPSULQWVDSXUHSKDVHPRGXODWLRQ WRWKHWUDQVPLWWHGILHOGVRWKDW u x C H[S ª¬iI x º¼ 



 

ZKHUH CLVDFRPSOH[FRQVWDQWDQGILVDUHDOIXQFWLRQZLWK_I_S,WWXUQVRXW WKDWWKHLQWHQVLW\RIWKHLPDJHFDQEHDSSUR[LPDWHGDV>*RRGPDQ@ 

I x

 C ª  f M xc º dx c»  «  ³f x  xc  ¬ S ¼

 

7KHUHIRUHDSKDVHPRGXODWLRQLVFRQYHUWHGLQWRDQDPSOLWXGHPRGXODWLRQPDNLQJ LWYLVLEOHZLWKDQRUPDOGHWHFWRU$QH[DPSOHRIWKLVPHWKRGLVVKRZQLQ)LJ LQ ZKLFK D ODUJH SKDVH GHIHFW XS WR S  LV LQGXFHG RQ WKH FRQYHUJLQJ VSKHULFDO ZDYHMXVWDIWHUWKHILUVWOHQVRIWKHGLDJUDPRI)LJ E 

5.6.3 Intensity and phase filters 7KHLQWHQVLW\ILOWHUVDUHLQWXUQWUDQVSDUHQFLHVRQZKLFKWKHILOWHULQJLVUHSUH

6SDWLDOILOWHULQJ



VHQWHGE\DFRQWLQXRXVWUDQVPLVVLRQIXQFWLRQEHWZHHQDQGDQGDUHXVXDOO\FUH DWHGSKRWRJUDSKLFDOO\ )RU H[DPSOH ZLWK D VXLWDEOH DWWHQXDWLRQ RI WKH FRQWLQXRXV FRPSRQHQW ZLWK D VHPLWUDQVSDUHQW REVWDFOH \RX FDQ LQFUHDVH WKH FRQWUDVW RI WKH LPDJH 6LPLODUO\ ILOWHUVFDQDOVREHXVHGWRHTXDOL]HWKH27)RIDQRSWLFDOV\VWHPDWWKHH[SHQVHRI LWVEULJKWQHVV 7KHVHILOWHUVDUHPRVWIUHTXHQWO\XVHGLQFRPELQDWLRQZLWKSKDVHILOWHUV,ILQ SDUWLFXODUWKHSKDVHILOWHULVELQDU\ZLWKLQGXFHGSKDVHVKLIWRIRUSWKHRYHUDOO ILOWHULVVDLGWREH real-valuedDQGLWVWUDQVPLVVLRQLVH[SUHVVHGE\DUHDOIXQFWLRQ ZLWKYDOXHVEHWZHHQDQG:LWKWKLVW\SHRIILOWHURQHFDQFRUUHFWVRPHDEHU UDWLRQV IRU ZKLFK WKH 27) WDNHV QHJDWLYH YDOXHV >0DUpFKDO DQG &URFH  *RRGPDQ@$QRWKHUH[DPSOHLVRQHLQZKLFKWKHILOWHUSHUIRUPVDPDWKHPDW LFDOGLIIHUHQWLDWLRQDORQJIRUH[DPSOHWKHxGLUHFWLRQ7KHDPSOLWXGHWUDQVPLVVLRQ IXQFWLRQIRUWKLVILOWHULV



H fx f y





­ af x ® iS ¯ af x e

IRU  d af x d   IRU   d af x  

 

ZKHUH aLVDFRQVWDQWWRNHHSWKHILOWHUSDVVLYH2QWKHLPDJHSODQHWKLVILOWHULQ WHQVLILHVWKHUHJLRQVZKHUHWKHVLJQDOYDULHVDORQJWKH xGLUHFWLRQ+RZHYHUHYHQ LQWKLVFDVHRQHKDVDVWURQJDWWHQXDWLRQRIWKHUHVXOWDQWLPDJHEULJKWQHVV 2QHSDUWLFXODUO\LPSRUWDQWILOWHUEXWDOVRRQHWKDWLVQRWWULYLDOO\IHDVLEOHLVLQ GLFDWHG E\ WKH QDPH RI matched filter LQ ZKLFK ERWK LQWHQVLW\ DQG SKDVH DUH PRGXODWHG FRQWLQXRXVO\ DQG FRQVWLWXWH WKH RSWLPDO ILOWHU IRU VHSDUDWLQJ ZHOO GHILQHGVLJQDOVE\DVWDWLRQDU\DGGLWLYHQRLVH>9DQ9OHNDQG0LGGOHWRQ=D GHK DQG 5DJD]]LQL @ ,QGHHG WKH OLQHDU ILOWHU WKDW PD[LPL]HV WKH VLJQDOWR QRLVH UDWLR LQ WKH SUHVHQFH RI VWRFKDVWLF DGGLWLYH QRLVH KDV D WUDQVPLVVLRQ H[ SUHVVHGE\>9DQGHU/XJW@



H fx f y





a

 SQ f x  f y

G f x  f y

 

ZKHUHSQLVWKHVSHFWUDOSRZHUGHQVLW\RIWKHQRLVHGLVWKHWUDQVIRUPRIWKHVLJQDO EHLQJVHDUFKHGDQGaLVDQRUPDOL]DWLRQFRQVWDQW /HWXVDVVXPHWKDWWKHVLJQDOFRQVLVWVRIDVPDOOILJXUHZKHQWKLVILJXUHLVSUH VHQWFHQWHUHGDWWKHSRVLWLRQ xo, yoLQWKHVFHQHREVHUYHGE\WKHV\VWHPLWVVSHF WUXPLVIRUPHGRQWKHSODQHRIWKHWUDQVIRUPDQGWKHVSHFWUXPDPSOLWXGHWUDQV PLWWHGE\WKHILOWHULVSURSRUWLRQDOWR





 G f x  f y e



 Si xo f x  yo f y

H fx  f y





SQ f x  f y





G fx f y



e



 Si xo f x  yo f y



 



)RXULHU2SWLFV

ZKHUHWKHH[SRQHQWLDOUHSUHVHQWVDSODQHZDYHFDUULHURULHQWHGLQIXQFWLRQRIWKH SRVLWLRQRIWKHILJXUHLQWKHVFHQH6LQFHWKHLQGLFDWHGIUDFWLRQLVUHDOWKHILOWHUHG ZDYHIURQWFRLQFLGHVZLWKWKDWRIWKHSODQHZDYHRIWKHFDUULHUFRUUHVSRQGLQJWRWKH REMHFWWKHUHIRUHRQWKHLPDJHSODQHRIWKHV\VWHPRQHGRHVQRWREVHUYHWKHVDPH ILJXUHEXWUDWKHUDEULJKWVSRWLQFRUUHVSRQGHQFHZLWKWKHSRVLWLRQZKHUHWKHILJ XUHLVFHQWHUHGLQGLFDWLQJLWVSUHVHQFH

5.6.4 Phase contrast microscopes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

A xo  yo

Ao eiI xo  yo 

 

ZKHUHIUHSUHVHQWVSKDVHPRGXODWLRQ,IIKDVVPDOOYDULDWLRQV'IFRPSDUHGWRS UDGLDQVDURXQGDQDYHUDJHYDOXHIoZHFDQDSSUR[LPDWHWKHDPSOLWXGHWUDQVPLWWHG ZLWK 

A xo  yo # Ao eiIo ª¬  i'I xo  yo º¼ 

 

ZKHUHWKHWHUPVLQSRZHUVRI'IDERYHDUHQHJOHFWHG,QWKHVTXDUHEUDFNHWVWKH XQLWUHSUHVHQWVWKHLOOXPLQDWLRQFDUULHUZDYHVXSSRVHGO\XQLIRUPZKLOHWKHRWKHU WHUPLVWKHPRGXODWLRQHIIHFWRIWKHREMHFW7KHWZRWHUPVDUHLQSKDVHTXDGUDWXUH VR'IEHLQJVXSSRVHGVPDOOHYHQLQWKLVDSSUR[LPDWLRQWKHUHLVQRDSSUHFLDEOH PRGXODWLRQ RI LQWHQVLW\ 2Q WKH SODQH RI WKH )RXULHU¶V WUDQVIRUP RI WKH PLFUR VFRSHZHKDYHDOLJKWVSRWFRUUHVSRQGLQJWRWKHLPDJHRIWKHVRXUFHVXUURXQGHG E\WKHVSDWLDOVSHFWUXPRI i'IZKLFKE\GHILQLWLRQLVQXOODW]HURIUHTXHQF\)RU DSRLQWVRXUFHWKHFHQWUDOVSRWZRXOGFRUUHVSRQGWRD'LUDF¶VGHOWD\HWERWKVSHF

6SDWLDOILOWHULQJ



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

I i xi  y i # I io i  i'I xo  yo # I io ª¬  'I xo  yo º¼ 

I i xi  y i # I io i  i'I xo  yo # I io ¬ª   'I xo  yo ¼º ZLWK

xo

xi mA 

yo

IRUG

S

IRUG S   

yi mA 





 

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apodizedWKH\KDYHWZRDGGLWLRQDODE VRUELQJ ULQJV ORFDWHG LQVLGH DQG RXWVLGH ZLWK WKH SKDVH ULQJ ZLWK D UHGXFHG DE VRUSWLRQFRPSDUHGWRWKLV,QWKLVZD\DVSXULRXVGLIIUDFWLRQHIIHFWZKLFKFRQVLVWV RIDEULJKWKDORDURXQGREMHFWVRULQDUHYHUVHGFRQWUDVWLVVLJQLILFDQWO\UHGXFHG

5.6.5 VanderLugt’s filter 7KH PD[LPXP SHUIRUPDQFH RI D ILOWHU LV REWDLQHG ZKHQ LWV WUDQVPLVVLRQ PD\ YDU\LQDFRQWLQXRXVPDQQHUZLWKFRPSOH[YDOXHV,QSULQFLSOHWKLVFDQEHDFFRP SOLVKHG E\ VXSHULPSRVLQJ DQ LQWHQVLW\ ILOWHU DQG D SKDVH ILOWHU +RZHYHU D FRP SRXQGILOWHURIWKLVW\SHLVQRWVLPSOHWRUHDOL]H9DQGHU/XJWSURSRVHGLQVWHDGWR HQFRGH WKH SKDVH PRGXODWLRQ LQ WKH VDPH LQWHQVLW\ ILOWHU >9DQGHU/XJW @ LQ IDFWXVLQJDKRORJUDSKLFPHWKRGRUSXWDQRWKHUZD\RIKHWHURG\QH 7KHVFKHPHRISULQFLSOHIRUFUHDWLQJWKLVFRGLQJLVJLYHQLQ)LJ7KHUDGL



)RXULHU2SWLFV

L1

P

S

T

f

L2

f

F

)LJ5HJLVWHULQJD9DQGHU/XJW)LOWHU7LVDWUDQVSDUHQF\RQZKLFKLWLVLPSULQWHGD VDPSOH ILJXUH WR EH UHFRJQL]HG DQG ) LV D ILOP RQ ZKLFK WKH LQWHUIHUHQFH RI LWV VSHFWUXP ZLWKDSODQHZDYHGHIOHFWHGE\WKHSULVP3LVSKRWRJUDSKHG

DWLRQ RI D SRLQW VRXUFH 6 LV FROOLPDWHG E\ OHQV / $ SDUW RI WKH ZDYH SDVVHV WKURXJK WKH WUDQVSDUHQF\ 7 ZKRVH DPSOLWXGH WUDQVPLVVLRQ LV SURSRUWLRQDO WR WKH GHVLUHG LPSXOVH UHVSRQVH hREWDLQLQJ WKH VLJQDO WREH HQFRGHG 7KH OHQV /SUR GXFHV WKH )RXULHU¶V WUDQVIRUP H RI h RQ LWV VHFRQG IRFDO SODQH ZKHUH D SKRWR JUDSKLF ILOP ) LV SODFHG $ UHIHUHQFH ZDYH FRQVWLWXWHG E\ D VHFRQG SDUW RI WKDW FROOLPDWHGE\/LVGHIOHFWHGE\WKHSULVP3RQWKLVVDPHSODQH)RU(T   RQWKHILOPZHKDYHWKHUHIRUHWKHLQWHUIHUHQFHRIWZRZDYHVZLWKDPSOLWXGH 

uV

 § x y · H ¨  ¸ DQG uU e i  SKy  ZLWKK Of © Of Of ¹

VLQ  O

 

ZKHUHIRUVLPSOLFLW\IROORZLQJ*RRGPDQ  ,UHPRYHGWKHSKDVHIDFWRUVUH VXOWLQJIURPWKHSURSDJDWLRQDQGWKHWUDQVIRUPSURGXFHGE\/,QIDFWWKHSKDVH RIWKHUHIHUHQFHZDYHFDQEHYDULHGDVGHVLUHGE\PRYLQJWKHSULVPWKLVSURGXFHV DWUDQVODWLRQRIWKHIULQJHVRILQWHUIHUHQFHZKLFKKRZHYHULVLUUHOHYDQWWRWKHUH VXOW7KHFRUUHVSRQGLQJLQWHQVLW\LVFDOFXODWHGE\(T  DSSOLHGWRWKLVSDUWLF XODUFDVHREWDLQLQJ I



IV



Of 



§ x y · H ¨  ¸  IU © O f Of ¹



 

 § x y · i  SKy  § x y · i  SKy  I UV  I UV  H ¨  ¸e H ¨  ¸e Of © Of Of ¹ Of © Of Of ¹

ZKHUH IV IU IUVDUHFRQVWDQWUHDOFRHIILFLHQWVWKDWLQFRUSRUDWHWKH QRQWULYLDO JH RPHWULFFRQGLWLRQVVXFKDVWKHGLUHFWLRQZLWKUHVSHFWWRWKHILOPDQGWKHSRODUL]D WLRQRIWKHWZRLQWHUIHULQJZDYHV6LQFHFRPPRQILOPVFDSWXUHDQHJDWLYHLPDJH RQHFDQWDNHDQRWKHUSKRWRJUDSKLFVWHSRQDQRWKHUILOPWRJHWDSRVLWLYHLPDJHEXW WKLVLVQRWHVVHQWLDOWRSURGXFHWKH9DQGHU/XJWILOWHU,QDQ\FDVHWKHILOPVDUHH[ SRVHG DQG GHYHORSHG WR SURGXFH WUDQVSDUHQF\ ZLWK D UHDO FRHIILFLHQW RI WUDQV PLVVLRQ RI WKH amplitude UDWKHU WKDQ WKH LQWHQVLW\ DV LV XVXDOO\ WKH FDVH  linear

'LIIUDFWLRQJUDWLQJV



ZLWKWKHLQWHQVLW\RIWKHH[SRVXUH 

t

IV



O f





H  I U  I UV

  i  SK y  Hei  SK y  I UV H e Of Of

 

'HQRWLQJZLWKIWKHSKDVHRIHZHFDQZULWHWKLVH[SUHVVLRQDV 

t

IV



O f





H  I U   I UV

 H ª¬FRV M  SK y º¼  Of

 

IURPZKLFKZHVHHKRZWKHSKDVH HLVHQFRGHGLQWKHSRVLWLRQRIWKHIULQJHVRQD VROHO\DEVRUELQJILOWHUWKDWIXQFWLRQVDVDGLIIUDFWLRQJUDWLQJ 7KH LQWHUIHURPHWHU LQ )LJ  ZRUNV ZLWK ZDYHIURQW GLYLVLRQ DQG UHTXLUHV D ODUJHOHQV/+RZHYHUDPSOLWXGHGLYLVLRQV\VWHPVKDYHEHHQSURSRVHGWKDWXVHD EHDPVSOLWWHUDQGUHTXLUHVPDOOHUDSHUWXUHRSWLFV 7KH 9DQGHU/XJW ILOWHU LV XVHG OLNH WKH RWKHU ILOWHUV LQ RQH RI WKH GHYLFHV GH VFULEHG DERYH EXW LW LV QHFHVVDU\ WR DUUDQJH WKH V\VWHP VR WKDWRQ WKH)RXULHU¶V SODQH WKH GHVLUHG VLJQDO WUDQVIRUP FRLQFLGHV LQ VL]H RULHQWDWLRQ DQG FHQWHULQJ ZLWKWKDWUHFRUGHGRQWKHILOWHU/HW gEHWKHVLJQDOWREHDQDO\]HGDQG GLWVWUDQV IRUP7KHDPSOLWXGHWUDQVPLWWHGE\WKHILOWHULVWKHQ Gc

IV



O f





G H  IU

   G  I UV GHei  SK y  I UV GH e i  SK y    Of O f O f





 

7KHILUVWDQGVHFRQGWHUPVDUHZLWKRXWXWLOLW\DQGDWWKHLPDJHSODQHRIWKHV\V WHPWKH\JLYHULVHWRWZRVSRWVFHQWHUHGLQWKHRULJLQ,QVWHDGWKHODVWWZRWHUPV DUHWKHLQWHUHVWLQJRQHVIRUWKHGLIIUDFWLRQSURGXFHGE\WKHIULQJHVUHFRUGHGRQWKH ILOWHUWKH\SURGXFHWZRGLVWLQFWVSRWVRQWKHLPDJHSODQHGLDPHWULFDOO\RSSRVLWH RXWRIWKHD[LV7KHWHUP GHFRUUHVSRQGVWRWKH convolutionRI gDQG hFHQWHUHG RQ WKH WUDQVYHUVH FRRUGLQDWHV  KOf  ZKLOH WKH WHUP LQ GH  FRUUHVSRQGV WR WKHLUcorrelationFHQWHUHGRQ KOf RQWKHLPDJHSODQH,QRUGHUWKDWWKHVHWHUPV EHVHSDUDWHGIURPWKHRWKHUVLWLVQHFHVVDU\WKDWWKHVSDWLDOIUHTXHQF\KRIWKHUHI HUHQFHZDYHLQWKHILOWHUUHFRUGLQJEHODUJHHQRXJK$VLQWKHFDVHRIWKHPDWFKHG ILOWHUVWKHFRUUHODWLRQVSRWLQGLFDWHVWKHSUHVHQFHRIWKHREMHFWLQWKHVFHQH

'LIIUDFWLRQJUDWLQJV 7KHGLIIUDFWLRQJUDWLQJVDUHLQDEURDGVHQVHSHULRGLFRUTXDVLSHULRGLFVSDWLDO VWUXFWXUHVLQWKHLUJHRPHWULFDOVKDSHDQGLQWKHLUHOHFWURPDJQHWLFSURSHUWLHV7KH\



)RXULHU2SWLFV

DUHXVXDOO\WZRGLPHQVLRQDOLQWKHLUPDFURVFRSLFDSSHDUDQFH%XWWKH\FDQDOVREH WKUHHGLPHQVLRQDODVLQWKHFDVHRIDFRXVWRRSWLFPRGXODWRUVIRUZKLFKWKHODWWLFH LVFUHDWHGE\DWUDYHOLQJDFRXVWLFZDYHRI³YROXPH´ZLWKDSLWFKRQWKHRUGHURI PPRULQWKHFDVHRIFU\VWDOVZLWKDODWWLFHSLWFKRQWKHRUGHURIcQJVWU|PVDQG FDSDEOHRIGLIIUDFWLQJ;UD\V ,QDVWULFWVHQVHWKHGLIIUDFWLRQJUDWLQJVDUHSODQDURUFRQFDYHVXUIDFHVFRUUX JDWHGSHULRGLFDOO\7KH\DUHXVHGERWKLQUHIOHFWLRQDQGLQWUDQVPLVVLRQDQGWKHLU SHFXOLDULW\DQGLPSRUWDQFHFRQVLVWVLQVSDWLDOO\UHVROYLQJWKHVSHFWUXPRIDQLQFL GHQWZDYHLQDPDQQHUVLPLODUWRWKHFKURPDWLFGLVSHUVLRQSURGXFHGE\DSULVP EXWPXFKPRUHHIIHFWLYHO\ 7KHILUVWWRSURGXFHWKHVHGHYLFHVV\VWHPDWLFDOO\ZDV)UDXQKRIHUZKREXLOWWKH ILUVWUXOLQJPDFKLQH ruling engine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

GHVFULEHV WKHLU RSWLFDO SURSHUWLHV EDVHG RQ DFFXUDWH VLPXODWLRQV DQG WKHLU PDQXIDFWXUH ,W DOVR KDVDODUJHELEOLRJUDSK\DQGKLVWRULFDOLQWURGXFWLRQ

'LIIUDFWLRQJUDWLQJV



RP\DVUDGLDWLRQLVWKHPDLQFDUULHURILQIRUPDWLRQRQWKHQDWXUHRIWKHFRVPRV 7KHJUDWLQJVDUHDOVRXVHGWRVHOHFWWKHHPLVVLRQIUHTXHQF\RIODVHUGLRGHVDQGDUH RIWHQLQWHJUDWHGGLUHFWO\LQWRWKHVWUXFWXUHRIWKHGLRGHVWKHPVHOYHV7KH\DUHWKHQ XVHGDVEDQGSDVVILOWHUVFRPSUHVVRUVDQGH[SDQGHUVRIODVHUSXOVHVDQGLQYDULRXV RWKHURSWLFDODSSOLFDWLRQV 7KHJUDWLQJVDUHXVHGZLWKDSLWFKRIWKHLUSHULRGLFLW\WKDWLVJUHDWHUEXWRIWHQ RQO\VOLJKWO\PRUHVRWKDQWKHW\SLFDOKDOIZDYHOHQJWKRIWKHVSHFWUXPRIWKHUDGL DWLRQXVHG7KHUHIRUHWKHW\SLFDOVLPSOLILFDWLRQVRIGLIIUDFWLRQWKHRU\ZKLFKFRQ VLVWLQQHJOHFWLQJWKHHOHFWURPDJQHWLFHIIHFWVRIWKHHGJHRIWKHRSHQLQJVDQGLQWKH SDUD[LDO DSSUR[LPDWLRQ FDQQRW EH DSSOLHG WR WKHP 2Q WKH FRQWUDU\ WKH HOHFWUR PDJQHWLF SURSHUWLHV RI WKH VSHFLILF PDWHULDO RI ZKLFK WKH JUDWLQJ LV PDGH DUH RI FUXFLDO LPSRUWDQFH ,Q IDFW WR GHWHUPLQH WKHLU EHKDYLRU LW LV QHFHVVDU\ WR VWDUW IURP 0D[ZHOO¶V HTXDWLRQV DQG RQO\ LQ UHFHQW WLPHV KDYH FRPSOH[ WKHRULHV DQG VXIILFLHQWO\UHOLDEOHFDOFXODWLRQSURJUDPVEHHQGHYHORSHG+RZHYHUWKHSHULRGLFL W\RIDODWWLFHDOORZVIRUGHWHUPLQLQJLQDVLPSOHZD\FHUWDLQJHRPHWULFDVSHFWVRI WKHLUUHVSRQVHWRLQFLGHQWZDYHDQGLQSDUWLFXODUWRDVVLJQWRWKHVSHFWUXPDQDF FXUDWH ZDYHOHQJWK VFDOH 7KH JUDWLQJV KDYH WKXV IRXQG QXPHURXV DSSOLFDWLRQV LQ VSLWH RI WKH UDWKHU XQHYHQ UHVSRQVH RI WKH DPSOLWXGH GLIIUDFWHG DORQJ WKH VDPH VFDOH

5.7.1 Reflection and refraction law of the gratings $OWKRXJKWKH\DUHFDOOHGGLIIUDFWLRQJUDWLQJVWKHLUODZRIUHIOHFWLRQDQGUHIUDF WLRQLVEDVHGRQLQWHUIHUHQFH,QGHHGOHWXVFRQVLGHUDSODQHVXUIDFHSHULRGLFDOO\ FRUUXJDWHGE\LGHQWLFDOJURRYHVSDUDOOHOEHWZHHQWKHPZLWKSDFHd&RQVLGHUWKHQ DSODQHZDYHWKDWDUULYHVRQWKHJUDWLQJZLWKDQDQJOHRILQFLGHQFH-,DQGZLWKLWV SODQHRILQFLGHQFHRUWKRJRQDOWRWKHJURRYHV&RPSDUHGWRWKHFDVHRIDSODQHLQ WHUIDFH WKH JUDWLQJV KDYH ORVW WKHLU LVRWURS\ LQ WKH GLUHFWLRQ RUWKRJRQDO WR WKH JURRYHVUHPDLQLQJLGHQWLFDOWRWKHPVHOYHVRQO\ZKHQWKH\DUHVKLIWHGE\DQLQWH JHUQXPEHURIVWHSV:HFDQWKHQVXEGLYLGHWKHVXUIDFHLQWRVWULSVRIZLGWKdHDFK FRUUHVSRQGLQJWRDJURRYHDQGFRQVLGHUWKHDPSOLWXGHRIWKHUHHPLWWHGUDGLDWLRQ LQYDULRXVGLUHFWLRQVWUHDWLQJLWDVWKHGLIIUDFWLRQIURPDVOLW)LQDOO\ZHDGGWKH DPSOLWXGHFRQWULEXWLRQRIWKHYDULRXVVWULSVNQRZLQJWKDWIRUHDFKRIWKHVHWKHUH LVDSKDVHVKLIWZLWKUHVSHFWWRWKHSUHYLRXVRQH,QWKHFDVHRIreflection,LWLV )LJ   

I

VLQ -,  VLQ -5 n,

 Sd  Oo

ZKHUH-5LVWKHGLUHFWLRQRIREVHUYDWLRQ n,LVWKHUHIUDFWLYHLQGH[RIWKHPHGLXP RIWKHLQFLGHQWZDYHDQGORLVWKHZDYHOHQJWKLQWKHYDFXXP7RREWDLQFRQVWUXF WLYHLQWHUIHUHQFHLWLVWKHQQHFHVVDU\WKDWI mSZKHUH mLVDQLQWHJHUFDOOHG WKHorder of diffraction)URPWKLVFRQVWUDLQWZHJHWWKHequation of gratings



)RXULHU2SWLFV

JR

JI

x z



)LJ'LDJUDPWRGHWHUPLQHWKHSKDVHVKLIWEHWZHHQWZRGLIIUDFWHGUD\VLQUHIOHFWLRQ IURPDGMDFHQWJURRYHV



VLQ -5m

VLQ -,  m

OR  n, d

D 

ZKHUH-5mDUHWKHGLUHFWLRQVDORQJZKLFKWKHLQFLGHQWZDYHLVGLIIUDFWHG )RUWUDQVPLVVLRQZHLQVWHDGKDYH 

n7 VLQ -7m

n, VLQ -,  m

Oo  d

E 

ZKHUHQ7LVWKHUHIUDFWLYHLQGH[RIWKHPHGLXPRIWKHWUDQVPLWWHGZDYH,IWKHJUDW LQJ LV SDUWLDOO\ UHIOHFWLQJ DQG WUDQVPLWWLQJ RQH KDV WKH GLIIUDFWHG ZDYHV RI ERWK FDVHV)RU m WKHUHLVWKHRUGLQDU\UHIOHFWLRQRUWUDQVPLVVLRQZKLFKKDVQRGLV SHUVLRQ 7KH VLJQ RI m LV KHUH FKRVHQ LQ VXFK D ZD\ WKH SRVLWLYH RUGHUV DUH GLI IUDFWHGIRUZDUGZLWKUHVSHFWWRRUGLQDU\UHIOHFWLRQDQGWUDQVPLVVLRQDQGWKHQHJD WLYH RQHV EDFN 7KH DQJOHV -5m GHSHQGRQ WKH SDFH dRI WKH JUDWLQJ DQG RQ WKH ZDYHOHQJWKDVZHOODVRQ-,EXWQRWRQWKHSURILOHDQGWKHGHSWKRIWKHJURRYHRQ ZKLFKWKHUHIUDFWHGLQWHQVLW\LQVWHDGGHSHQGV /HWXVVXSSRVHWKDWZHIODWWHQWKHJURRYHVWRUHWXUQDQLVRWURSLFVXUIDFH,WWKHQ KDSSHQV WKDW LQ WKH GLIIUDFWLRQ LQWHJUDO IRU RQH RI WKH -5m YLHZLQJ GLUHFWLRQV JRLQJDFURVVDVWULSE\LWVZLGWKWKHLQWHJUDQGUHPDLQVFRQVWDQWLQPRGXOHEXWLWV SKDVHYDULHVOLQHDUO\DVLGHIURPDFRQVWDQWIURPWRmS7KHUHIRUHWKHRYHUDOO FRQWULEXWLRQRIDQ\VWULSLVQXOOLQLWVHOIH[FHSWIRUm  7KHDERYHDUJXPHQWVFDQDOVREHGHULYHGE\PDNLQJWKH)RXULHU¶VWUDQVIRUPRI WKHILHOGRIWKHUHIOHFWHGZDYHWDNHQRQDSODQHWDQJHQWWRWKHODWWLFHNQRZLQJRQ O\WKDWLWVSHULRGLFLW\LVOLQNHGWRWKHUHWLFXODUSLWFKDQGWKHSURJUHVVLYHVKLIWLVGXH WRWKHLQFOLQDWLRQRIWKHLQFLGHQWZDYH2QHIXUWKHUZD\LVWRFRQVLGHUWKHFRQGL WLRQVIRUWKHSKDVHRQWKLVVDPHSODQHVLPLODUO\WRZKDWZHVDZIRUUHIOHFWLRQDQG UHIUDFWLRQIURPSODQHLQWHUIDFHV/HW k, k5DQG k7EHWKHZDYHYHFWRUVIRUWKHLQ

'LIIUDFWLRQJUDWLQJV



JI

-6

-5

-4

-3

-2

-1

0

1

2

3

m

K

 )LJ&ODVVLFDOPRXQWLQJ*UDSKLFGHWHUPLQDWLRQRIGLIIUDFWHGRUGHUV)RUJHQHUDOLW\ ZHKHUHLQGLFDWHHYHQWKRVHLQUHIUDFWLRQZKLFKDUHDEVHQWLQWKHUHIOHFWLRQJUDWLQJVZKHQ WKH\DUHRSDTXHLQWUDQVPLVVLRQ

FLGHQWUHIOHFWHGDQGUHIUDFWHGZDYHFKRRVLQJDV&DUWHVLDQD[HV zDORQJWKHQRU PDOGLUHFWLRQWRWKHJUDWLQJ yWKDWLVSDUDOOHOWRWKHJURRYHVDQG xDVWKHSHUSHQ GLFXODURQHZLWKVXFKDGLUHFWLRQWKDWWKHFRPSRQHQWVRI k,DUHSRVLWLYHZHKDYH WKDW 

­° k xR m ® °¯ k yR

k xT  m

k xI  mK 

k yT

k yI 



 

ZKHUH K SdLVWKHPRGXOHRIWKH grating vector KWKDWOLHVRQWKHSODQHRIWKH JUDWLQJRUWKRJRQDOWRWKHJURRYHV )LJ 7KH zFRPSRQHQWVRIWKHVHYHFWRUV DUHRQWKHRWKHUKDQG 

k zj  m

sj

 S n j  O R

 k xj  m   k yj  

 

ZKHUH nLVWKHUHIUDFWLYHLQGH[RIWKHFRQVLGHUHGPHGLXPZLWK j 57DQG s5   s7 ,QRUGHUWRVSHDNRIDGLIIUDFWHGRUGHUNQRZQDVD propagating order mPXVWEHVXFKWKDW kzjLVUHDO2WKHUZLVHWKHFRUUHVSRQGLQJZDYHLVDQHYDQHV FHQW ZDYH ZKLFK FDQ EH GHWHFWHG RQO\ ZLWKLQ GLVWDQFHV IURP WKH JUDWLQJ RQ WKH



)RXULHU2SWLFV

RUGHURIO7KHVHZDYHVFDOOHG evanescent ordersWKRXJKQRWGLUHFWO\DSSUHFLD EOHKDYHDQLPSRUWDQWUROHLQFDXVLQJanomalies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ky,   ZLWK WKH LQFLGHQFH SODQH QRUPDO WR WKH JURRYHV:KHQDODWWLFHLVLQVWDOOHGLQDVSHFWURPHWHUXQGHUWKHVHFRQGLWLRQVLWLV VDLGWKDWLWVPRXQWLQJLV classicRWKHUZLVHWKHPRXQWLQJLVVDLGWREH conical,Q GHHGLWFDQHDVLO\EHVKRZQWKDWIRUky,WKHSURSDJDWLQJRUGHUVWDNHGLUHFWLRQV WKDWDUHRQWKHVXUIDFHRIDFRQHWDQJHQWWRWKHGLUHFWLRQRIWKHRUGHU-XVWFRQ VLGHU WKDW WKH YHFWRU kj IRU DQ LVRWURSLF PHGLXP OLHV RQ D VSKHUH DQG WKDW (T  ZLWKIL[HG kyjGHVFULEHVWKHHTXDWLRQRIDFLUFOHIRUWKHFRPSRQHQWV kxjDQG kzj7KHFODVVLFFDVHFRLQFLGHVZLWKWKHOLPLWRIDFRQHZLWKDQDQJXODURSHQLQJRI ƒ 'LIIHUHQWLDWLQJWKHJUDWLQJ(T D IRU-5mDQGORZHJHW 

w-5m wO R

m n, d FRV -5m

 VLQ -5m  VLQ -,  OR FRV -5m

 

7KLVPHDQVWKDWIRUDJLYHQZDYHOHQJWKWKHDQJXODUGLVSHUVLRQLVRQO\DIXQF WLRQRIWKHLQFLGHQFHDQGGLIIUDFWLRQDQJOHV 7KHDQJOHEHWZHHQWKHLQFLGHQWDQGWKHGLIIUDFWHGUD\LVFDOOHGWKH angular deviation $'  )LJ :KHQDJUDWLQJLVXVHGZLWKQXOOGHYLDWLRQIRUZKLFKZH REVHUYHWKHGLIIUDFWHGUDGLDWLRQLQWKHRSSRVLWHGLUHFWLRQWRWKHLQFLGHQWRQHWKDWLV IRU -5m  -, LW LV VDLG WR EH PRXQWHG LQ D Littrow FRQILJXUDWLRQ RU DXWR FROOLPDWLRQ,QWKLVFDVHZHKDYH 

w-5m wO R

 WDQ -5m  OR

 

ZKLFKLQFUHDVHVLQGHILQLWHO\LQFUHDVLQJ-,XSWRWKHJUD]LQJLQFLGHQFH,WVKRXOG EHQRWHGWKDWWKHGLVSHUVLRQGHSHQGVRQO\RQWKHDQJXODUFRQGLWLRQVRIHDFKRUGHU DQGWKDWmDSSHDUVRQO\DVDQLQGH[7KHUHIRUHRQFH-,KDVEHHQGHWHUPLQHGLWLV QHFHVVDU\WRFKRRVHZKHWKHUWRXVHDJUDWLQJZLWKDILQHSLWFKDQGDORZRUGHU m W\SLFDOO\RUDJUDWLQJZLWKDFRDUVHSLWFKDQGDELJRUGHULQDEVROXWHYDOXH $W\SLFDOGHIHFWRIWKHJUDWLQJVLVWKDWWKHDQJXODUGLVSHUVLRQRIWKHYDULRXVRU GHUVRYHUODSVLWFRXOGDOVRUHVXOWLQHUURUVRIDWWULEXWLRQRIWKHZDYHOHQJWK7KLV EHFRPHV SDUWLFXODUO\ UHOHYDQW ZLWK UHODWLYHO\ FRDUVH JUDWLQJV XVHG LQ KLJK RUGHU

'LIIUDFWLRQJUDWLQJV



$V LQ WKH FDVH RI LQWHUIHURPHWHUV WKH FRQFHSW RI IUHH VSHFWUDO UDQJH LV XVHG WKDW FDQEHGHGXFHGIURPWKHHTXDWLRQRIWKHJUDWLQJV,QIDFWWKHVDPHGLIIUDFWLRQDQ JOH DQG KHQFH WKH VDPH YDOXH mO IRU WZR VXFFHVVLYH RUGHUV FDQ EH REWDLQHG DW WZRZDYHOHQJWKVODQGO VWLOOLQWKHYDFXXP VXFKWKDW 

mO 

m   O 

IURPZKLFKZHJHW 

FSRO R

OR  m

 

,QWHUPVRIZDYHQXPEHUK ORWKHIUHHLQWHUYDOLV 

FSRK

   O O 

O   O K |  OO  m

 

ZKHUHWKHILQDODSSUR[LPDWLRQDSSOLHVWRODUJHRUGHUVIRUZKLFKOO|OR )LQDOO\IRU(T D WKHZDYHOHQJWKVFRUUHVSRQGLQJWRWKHSDVVDJHVRIWKH RUGHUmIURPSURSDJDWLQJWRHYDQHVFHQWDQGYLFHYHUVDZKLFKFDXVHDQRPDOLHVRQ RWKHURUGHUVYHULI\WKHFRQGLWLRQ 

VLQ -,  m

OR n, d

 IRU m

r r  " 

 

5.7.2 Diffraction efficiency :KLOHWKHFDOFXODWLRQRIWKHGLIIUDFWLRQGLUHFWLRQVRIDODWWLFHUHSUHVHQWVD³VLP SOH´JHQHUDOL]DWLRQRIUHIOHFWLRQDQGUHIUDFWLRQODZVWKHHQHUJ\WKDWIORZVLQWKH YDULRXVGLIIUDFWHGRUGHUVDQGWKXVWKHGLIIUDFWLRQHIILFLHQF\IROORZVDFRPSOLFDW HGWUHQGWKDWFDQQRWEHDWWULEXWHGWRVLPSOHPRGHOV,QWKHSDVWLWZDVQRWHYHQUH SURGXFLEOH IURP RQH JUDWLQJ WR DQRWKHU DOWKRXJK WKH\ ZHUH SURGXFHG WKH VDPH ZD\0DQ\SDUDPHWHUVQHHGWREHFRQVLGHUHGWRVROYH0D[ZHOO¶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



)RXULHU2SWLFV

detector

A.D. lo incident beam



)LJ6LPSOLILHGVFKHPHIRUPHDVXULQJWKHHIILFLHQF\RIDJUDWLQJ7KHDQJXODUGHYLD WLRQ$'LVKHOGIL[HGWRDJLYHQYDOXHZKLOHWKHJUDWLQJLVURWDWHG7KHGRWWHGFLUFOHLQGL FDWHVWKHSRVLWLRQRIWKHGHWHFWRUWRFDOLEUDWHWKHVRXUFH

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d7KHOLJKWLQJFRQGLWLRQV DUHDOVRVXFKWKDWWKHJUDWLQJFDQEHEURXJKWEDFNWRDRQHGLPHQVLRQDOVWUXFWXUH DORQJDSDWKRUWKRJRQDOWRWKHJURRYHVZLWKWKHXQLIRUPLOOXPLQDWLRQRIDSODQH ZDYHDQGFODVVLFPRXQWLQJ7KHDSHUWXUHRIWKHJUDWLQJFDQWKHQEHFRQVLGHUHGDV WKH convolution RI D ILQLWH FRPE RI M 'LUDF¶V GHOWD LQ ZKLFK ZH HPERG\ WKH SKDVHUHVXOWLQJIURPWKHSRVLWLRQRIWKHJURRYHVDQGWKHLQFLGHQFHDQJOHRIWKHLQ FLGHQWZDYHDQGDIXQFWLRQ g x UHSUHVHQWLQJWKHILHOGSUHVHQWLQD singleJURRYH LQ UHIOHFWLRQ RU LQ WUDQVPLVVLRQ 7KHUHIRUH OHW XV FDOFXODWH WKH GLIIUDFWHG SODQH ZDYHVSHFWUXPFRUUHVSRQGLQJWRWKHILHOGHPHUJLQJRQDSODQHAFORVHWRWKHJUDW LQJVXUIDFHZLWKWKHSHULRGLFLW\GHVFULEHGDERYH8VLQJWKHFRQYROXWLRQWKHRUHP ZHKDYHWKDWLWLVJLYHQE\WKHproductRIWZRIXQFWLRQVRQHLVWKHWUDQVIRUPG fx  RIg x DQGWKHRWKHULVWKHWUDQVIRUPH RIWKHGHOWDVHULHV,WFRQVWLWXWHVWKHJHQHU DOL]DWLRQRI(T  RIWKHGLIIUDFWHGILHOGIURPDGRXEOHVOLW)RUWKHFDOFXOD

'LIIUDFWLRQJUDWLQJV



WLRQRIH,LWLVHDVLHUWRIROORZDQ³RUGLQDU\´GHULYDWLRQ 

H fx

³ ei Sxf x

M 

¦

G x  qd eik,x x dx

q 

  eiM  Sf x k,x d    ei  Sf x k,x d

 

ZKHUHk,x S n,  OR VLQ-,7KHVTXDUHPRGXOHRIHLVWKH interference function 

H fx





§ Spd · VLQ  ¨ M ¸ Oo ¹ ©  § Spd ·  VLQ ¨ ¸ © Oo ¹

 

ZKHUH 

f x O R  n, VLQ -, 

p

 

([SUHVVLQJfxDQJXODUO\DFFRUGLQJWR(TV  ZHKDYH 

nG VLQ -G  OR

 

nG VLQ -G  n, VLQ -, 

 

fx p

ZKHUH-GLQGLFDWHVWKHGLIIUDFWLRQDQJOHIRUDJUDWLQJLQERWKWUDQVPLVVLRQDQGUH IOHFWLRQDQGnGLVWKHFRUUHVSRQGLQJUHIUDFWLYHLQGH[)LQDOO\WKHVSHFWUDOLQWHQVLW\ LVJLYHQE\ 

§ n VLQ-G · I¨ G ¸ © OR ¹





§ n VLQ-G · § nG VLQ-G · H¨ G ¸ G¨ ¸  O R © ¹ © OR ¹

 

ZKHUHWKHVTXDUHPRGXOHRIGLVWKHintensity function of a groove 7KHW\SLFDODVSHFWRIWKHVHIXQFWLRQVLVVKRZQLQ)LJ%RWKWKHGHQRPL QDWRUDQGWKHQXPHUDWRURI(T  VLPXOWDQHRXVO\YDQLVKIRUWKHYDOXHVRI p IRUZKLFK 

pd OR

m Ÿ nG VLQ -G  n, VLQ -,

mO R  d

 

ZKHUH mLVDQLQWHJHUDJDLQILQGLQJWKHODZRIJUDWLQJV/RFDOO\DURXQGHDFKRI WKHVH YDOXHV RI p WKH VTXDUH PRGXOH RI H EHKDYHV OLNH D VLQF ZLWK D SULQFLSDO SHDNRIKHLJKWMVXUURXQGHGE\VHFRQGDU\SHDNV7KLVLVGXHWRWKHK\SRWKHVLVRI DSUHFLVHQXPEHU MRIHTXDOO\LOOXPLQDWHGJURRYHV,WVKRXOGEHQRWHGWKDWRIDOO



)RXULHU2SWLFV

RIWKHPDLQSHDNVRI_H_RQO\VRPHFRUUHVSRQGWRSURSDJDWLQJRUGHUV %HWZHHQWZRPDLQPD[LPDWKHUHDUHM]HURVDWWKHYDOXHV 

M

pd OR

mM  q ZLWK q   "  M   

 

ZKHUH WKH QXPHUDWRU RI (T   YDQLVKHV $PLG WKHVH ]HURV WKHUH DUH M VHFRQGDU\PD[LPDZKRVHKHLJKWIqIRUMODUJHWHQGVWR>6LYRXNKLQH@ 

Iq



q    S

I SU

SHU q  M 

 

ZKHUH ISULVWKDWRIWKHFORVHVWSULQFLSDOPD[LPXP)RU q ZHKDYH I  ISUZKLFKLVQRWUHDOO\QHJOLJLEOH+RZHYHUWKHQXPEHURILOOXPLQDWHGJURRYHVLV JHQHUDOO\YHU\ODUJHXSWRKXQGUHGVRIWKRXVDQGVIRUZKLFKWKHVHFRQGDU\SHDNV JR WR FRQVWLWXWH D FRQWLQXRXV EDFNJURXQG LQZKLFK WKH PRGXODWLRQ LV ILOWHUHGRII E\YDULRXVIDFWRUVWKDWFRQWULEXWHWROLPLWLQJWKHUHVROXWLRQIRUH[DPSOHWKHILQLWH VL]HRIWKHHQWU\VOLWRUWKHVPRRWKLQWHQVLW\GHFUHDVLQJRID*DXVVLDQEHDP 6RIDUZHKDYH RQO\ FRQVLGHUHG WKH ³JHRPHWULF´ DVSHFWVRI WKHSUREOHP 7KH UHDOGLIILFXOW\DULVHVIURPWKHHYDOXDWLRQRIWKHIXQFWLRQ gRIWKHILHOGRQDVLQJOH JURRYH$YHU\URXJKH[DPSOHLVWRFRQVLGHUWKHSDWWHUQDVFRQVLVWLQJRIVOLWVZLWK DSHUWXUH s  d DQG DSSO\ WKH RUGLQDU\ DSSUR[LPDWLRQV RI GLIIUDFWLRQ WKRVH RI .LUFKKRII7KHILHOGRQWKHVOLWLVWKHUHIRUHSURSRUWLRQDOWR 

g x

­eik,x x ® ¯ 

IRU x  s    RWKHUZLVH

 

ZKHUHDJDLQk,x S n,  OR VLQ-,7KH)RXULHU¶VWUDQVIRUPRIgLV 

G fx

ª § ·º n s VLQF « Ss ¨ f x  , VLQ -, ¸ »  OR ¹¼ ¬ ©

 

$Q H[DPSOH RI WKLV FDVH LV JLYHQ LQ )LJ  DVLGH IURP D SHDN QRUPDOL]DWLRQ ZLWKs d(T  EHFRPHV § Spd · VLQ  ¨ M ¸ Oo ¹ § nd VLQ -d ·  § Sps · ©    I¨ s VLQF  ¨ ¸ ¸  Oo Oo ¹ § Spd · © ¹ ©  VLQ ¨ ¸ © Oo ¹ ZLWK p DJDLQ JLYHQ E\ (T   7KH IXQFWLRQ G PRGXODWHV WKH KHLJKW RI WKH PD[LPXPRIWKHIXQFWLRQ HDQGVRPHWLPHVGHOHWHVVRPHRIWKHP)RUH[DPSOH

'LIIUDFWLRQJUDWLQJV



100

(a)

80 60

H 2 40 20 0 0

2

6

4

8

10

1.0

(b)

0.8 0.6

G

2

0.4 0.2 0.0 0 100

2

6

4

8

10

FSR

(c)

80 60

H

2

G

2

40

resolving power

20 0 0

2

6

4

pd l o

8

10



)LJ   D  ,QWHUIHUHQFH IXQFWLRQ FDOFXODWHG IRU M   DW WKH YDULDWLRQ RI pdOR E  6TXDUH PRGXOH RI WKH VSHFWUXP IRU D VLQJOH JURRYH KHUH UHSUHVHQWHG E\ WKH IXQFWLRQ VLQF SpsOR ZLWKs d F 5HVXOWLQJLQWHQVLW\

IRUs dHYHQRUGHUVDUH]HURHGDVWKH\FRLQFLGHZLWKWKH]HURVRIG $SDUWLFXODUO\LQWHUHVWLQJFDVHLVJLYHQE\DJURRYHZLWKDULJKWWULDQJXODUSUR



)RXULHU2SWLFV

ILOHDQGDQLQFOLQHGVLGHDWWKHDQJOH-,LHSHUSHQGLFXODUWRWKHGLUHFWLRQRIWKH LQFLGHQWUDGLDWLRQDQGZLWKWKHSDFHEHLQJ 

d

Oo  n, VLQ -,

 

VRWKDWIURPRQHJURRYHWRWKHQH[WWKHUHLVDSKDVHMXPSRISUDGLDQVLQWKHSDWK RIWKHLQFLGHQWZDYH,QWKHJHRPHWULFDSSUR[LPDWLRQWKHFRUUHVSRQGLQJIXQFWLRQ gLVJLYHQE\ 

g x

­eik,x x e i  S x d ®  ¯

e iS x

d

IRU x  d   RWKHUZLVH

 

ZKRVHWUDQVIRUPLVQRZ G fx



d VLQF Sdf x  S  

 

DQGWKHVSHFWUXPRIGLIIUDFWHGUDGLDWLRQLV



§ n VLQ -G · I¨ G ¸ © OR ¹

§ Spd · VLQ  ¨ M ¸ OR ¹ § S S nG VLQ -G © d VLQF  ¨  § · Spd ©   n, VLQ -, VLQ  ¨ ¸ © OR ¹

· ¸  ¹

 

)RUVLPSOLFLW\ FRQVLGHU WKH FDVH RIUHIOHFWLRQ LQ DLU 7KHVLQF LV IRU -G  -, LHIRUDUHWURUHIOHFWHGSURSDJDWLQJRUGHUWRZKLFKFRUUHVSRQGV pdOR  m  DFFRUGLQJWR(T  DQGKDVLWVILUVW]HURVIRU



VLQ -G

­ ° VLQ -, ° ® ° VLQ , °¯

ZLWK

pd OR

pd ZLWK OR





 



$JUDWLQJRIWKLVNLQGKDGEHHQDGYRFDWHGE\5D\OHLJKZLWKRXWWKHKRSHWKDWLW FRXOGEHUHDOL]HG>5D\OHLJK@,QGHHGLWFRXOGGLIIUDFWDOPRVWDOOWKHUDGLDWLRQ LQWKHFRXQWHUSURSDJDWLQJRUGHU m HUDVLQJLQSDUWLFXODUWKHRUGHUFRUUH VSRQGLQJWRWKHRUGLQDU\UHIOHFWLRQDQGDOVRWKHRUGHU7KHGRXEWLVOHJLWLPDWH EHFDXVHWKHDERYHFDOFXODWLRQVDUHEDVHGRQDURXJKDSSUR[LPDWLRQWKDWQHJOHFWV WKHHOHFWURPDJQHWLFHIIHFWVRIWKHJUDWLQJ+RZHYHUQDWXUHKDVEHHQEHQLJQDQG DW OHDVW IRU WKH 70 SRODUL]DWLRQ WKLV EHKDYLRU LV DFKLHYHG ZLWK JRRG HIILFLHQF\ 7KHJUDWLQJVWKDWDUHDEOHWRFRQFHQWUDWHWKHLQFLGHQWUDGLDWLRQLQDVLQJOHGLIIUDFW HGRUGHUDUHFDOOHGblazedDQGLWLVQRWUHDOO\QHFHVVDU\WKDWWKH\KDYHDWULDQJXODU

'LIIUDFWLRQJUDWLQJV



TE

TM

A

Jb

E

E A

B

E

E A

B



)LJ5HSUHVHQWDWLRQRIWKHZDYHYHFWRUVRIWKHHOHFWULFILHOGIRUWKHLQFLGHQWDQGGLI IUDFWHGZDYHVDORQJWKHIDFHVADQGBRIDJUDWLQJZLWKWULDQJXODUFURVVVHFWLRQDQGYHUWH[ DQJOHRIƒ -ELVWKHEOD]LQJDQJOHRIWKH JUDWLQJ

FURVVVHFWLRQDOWKRXJKWKLVLVWKHPRVWFRPPRQO\UHDOL]HG:KHQDODWWLFHDOORZV IRURQO\WZRGLIIUDFWHGRUGHUVZLWK m DQG±LWLVSRVVLEOHWRH[WLQJXLVKWKH ]HUR RUGHU E\ DGMXVWLQJ WKH GHSWK h RI WKH JURRYHV DV LQ WKH FDVH RI VLQXVRLGDO JUDWLQJV LQ UHIOHFWLRQ ZLWK D hd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³SDUDVLWH´VXUIDFHBLVSDUDOOHOWRLWDQGWKHUHIRUHWKHERXQG DU\FRQGLWLRQVFDQQRWEHVDWLVILHGIRUWKLVSRODUL]DWLRQ2QWKH³XVHIXO´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†  DQGLQJHQHUDOZLWKLQWHUIDFHVDQGPDWHULDOVdevoid of absorptionGXHWRWKHSULQ FLSOH RI FRQVHUYDWLRQ RI HQHUJ\ DQG WKH WLPH UHYHUVDO V\PPHWU\ RI 0D[ZHOO¶V HTXDWLRQVWKHUHIOHFWLRQFRHIILFLHQWDVLGHIURPDSKDVHVKLIWLVWKHVDPHE\YLUWXH RIH[FKDQJLQJWKHLQFLGHQWDQGUHIOHFWHGZDYHV7KHVDPHDUJXPHQWVFDQDOVREH DSSOLHGWRWKHJUDWLQJZLWKWKHSUHFDXWLRQWRFRQVLGHUWKHUHODWLRQVKLSEHWZHHQWKH SRZHUVRIWKHZDYHVVLQFHWKHLQFLGHQWDQGUHIOHFWLRQEHDPVJHQHUDOO\KDYHGLI IHUHQWZLGWKV7KXV in the absence of absorptionUHIOHFWLYLW\DQGWUDQVPLVVLYLW\ LQRWKHUZRUGVWKHFRXSOLQJHIILFLHQF\ remain the same by exchanging reflected and diffracted waves



)RXULHU2SWLFV

5.7.3 Resolving power 7KHJUDWLQJVDUHXVHGIRUWKHLUDELOLW\WRGLVSHUVHWKHVSHFWUDOFRPSRQHQWVRIWKH LQFLGHQWZDYHDQJXODUO\+RZHYHUDVFDQEHVHHQIURP)LJWKHUHLVQRXQ DPELJXRXV UHODWLRQVKLS EHWZHHQ WKH DQJXODU SRVLWLRQ LQ WKH GLIIUDFWHG ZDYH DQG WKHZDYHOHQJWK7KLVLVEHFDXVHRIWKHILQLWHSHDNZLGWKRIWKHLQWHUIHUHQFHIXQF WLRQ ZLWK ZKLFK WKH UDGLDWLRQ VSHFWUXP LV FRQYROXWHG LQ WKH DQJXODU GLVSHUVLRQ SURFHVV 7KH DELOLW\ RI D JUDWLQJ WR VHSDUDWH QHLJKERULQJ VSHFWUDO OLQHV LV FDOOHG UHVROXWLRQ RU resolving power ,W LV FRQYHQWLRQDOO\ HVWLPDWHG XVLQJ WKH 5D\OHLJK FULWHULRQEDVHGRQWKHGLVWDQFHEHWZHHQWKHSRVLWLRQRIDPDMRUSHDNDQGWKDWRI WKHILUVW]HURDGMDFHQWWRWKHPD[LPXPDVVKRZQLQWKHLQVHUWRI)LJ F 7KLV GLVWDQFH'ORLVHTXDOWRWKHIUDFWLRQ)65ORMZKHUHE\IURP(T   

R

OR 'O R

mM

Md p OR

W nG VLQ -G  n, VLQ -,  OR

 

ZKHUH W  Md LV WKH ZLGWK RI WKH JUDWLQJ VXUIDFH LOOXPLQDWHG E\ WKH LQFLGHQW ZDYHDVRQWKHRWKHUKDQGZHH[SHFWVLQFHWKHZLGWKRI)UDXQKRIHU¶VGLIIUDFWLRQ SHDNVLVLQYHUVHO\SURSRUWLRQDOWRWKHZLGWKRIWKHRSHQLQJV6RLWLVQRWLPSRUWDQW KRZVPDOOWKHSLWFKRIWKHJUDWLQVLVLIQRWIRULWV)65UDWKHUZKDWFRXQWVLVLWV HQOLJKWHQHGZLGWK,QFLGHQFHDQGGLIIUDFWLRQDQJOHVDOVRRFFXULQWKHFDOFXODWLRQ DQGWKHPD[LPXPUHVROXWLRQLVZKHQWKHGLIIUDFWHGZDYHIRUWKHSURSDJDWLRQRU GHUFRQVLGHUHGSURFHHGVLQWKHRSSRVLWHGLUHFWLRQWRWKHLQFLGHQWZDYH,QDLUIRU JUD]LQJ LQFLGHQFH WKH DQJXODU IDFWRU LV DW PRVW  DQG LQ WKHVH FRQGLWLRQV ZH KDYH 

RPD[

W  OR

 

7KHUHVROXWLRQRQHJHWVFDQEHFRQVLGHUDEOHIRUH[DPSOHDWW FPDQGOR  QPZHKDYH RPD[ %\FRPSDULVRQZHFDQFRQVLGHUWKHUHVROXWLRQ REWDLQHGZLWKD6)IOLQWJODVVSULVPZLWK dndOR§PPIRUYLVLEOHUDGLD WLRQ)URP(T  ZLWKDEDVHRIFPWKHSULVPUHVROXWLRQLV 6HYHUDOIDFWRUVFRQVSLUHWRUHGXFHWKHDFKLHYDEOHUHVROXWLRQ7KHUHJXODULW\RI WKHJUDWLQJSLWFKPXVWEHPDLQWDLQHGZLWKLQORYHUWKHHQWLUHVXUIDFHZKLFK LQWXUQPXVWGHYLDWHIURPDSODQHRUVSKHUHIRUOHVVWKDQO7KLVDOVRDSSOLHVWR WKH TXDOLW\ RI WKH LQFLGHQW ZDYHIURQW 3HULRGLF YDULDWLRQV RI WKH JUDWLQJ SLWFK FDXVHG WKH DSSHDUDQFH RI ³JKRVW´ OLQHV LQ WKH VSHFWUXP ZKLFK FKDUDFWHUL]HG WKH ILUVWJUDWLQJEXLOWLQWKHSDVW 7KHUHVROXWLRQFDQEHLQFUHDVHGE\XVLQJGRXEOHJUDWLQJVSHFWURPHWHUVLQFDV FDGHEXWZKDWLVHVSHFLDOO\LPSURYHGLQWKLVZD\LVWKHH[WLQFWLRQUDWLREHWZHHQ WKHVHFRQGDU\SHDNVDQGWKHPDLQRQHV

'LIIUDFWLRQJUDWLQJV



5.7.4 Types of grating 7KHYDULHW\RIJUDWLQJVLVVRZLGHWKDWLWLVQRWHDV\WRFODVVLI\WKHP$VHQVLEOH ZD\ LV WR VXEGLYLGH WKHP DFFRUGLQJ WR WKHLU FRQVWLWXWLYH SDUDPHWHUV DQG WR WKH WHFKQLTXHXVHGWRUHDOL]HWKHPDFFHSWLQJWKHPDOPRVWLQGLYLGXDOO\IRUZKDWWKH\ DUH$PRUHDPELJXRXVZD\ZRXOGEHWRGLVWLQJXLVKWKHPE\WKHLUXVH Slit gratings RU Ronchi rulings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phase gratingsGRQRWKDYHDFRUUXJDWLRQRIWKHVXUIDFHEXWDPRGXODWLRQ RIWKHUHIUDFWLYHLQGH[RIDPRUHRUOHVVWKLQOD\HURIGLHOHFWULFPDWHULDO([DPSOHV RIWKLVW\SHDUHWKHKRORJUDPVLQZKLFKWKHUHFRUGLQJLVPDGHLQWKHYROXPHRIWKH SKRWRVHQVLWLYH PDWHULDO DQG WKH DFRXVWRRSWLFDO GHYLFHV LQ ZKLFK WKH GHQVLW\ RI WKHPDWHULDOLVPRGXODWHGE\DQDFRXVWLFWUDYHOLQJZDYH7KH\DUHDOVRXVHGLQLQ WHJUDWHGRSWLFVDQGILEHURSWLFV 7KHPRVWFRPPRQFDWHJRU\LVWKDWRI corrugated gratings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echelettes)UHQFKIRU³VPDOOVWDLUV´/DUJHU SLWFK JUDWLQJV XVHG DW GR]HQV RU KXQGUHGV RI GLIIUDFWLRQ RUGHUV DUH FDOOHG echelles7KH\ZRUNDWKLJKDQJOHVRILQFLGHQFHDQGGLIIUDFWLRQDUHOHVVVHQVLWLYH WR SRODUL]DWLRQ DQG FRYHU D ZLGHU VSHFWUDO UDQJH ZLWK JRRG HIILFLHQF\ )LQDOO\ PDFURVFRSLF JUDWLQJV XVHG ZLWK WKRXVDQGV RI RUGHUV EXLOW E\ 0LFKHOVRQ MRLQLQJ GR]HQVRIJODVVVKHHWVQRZODUJHO\GLVXVHGDUHFDOOHG echelons5HFWDQJXODUVHF WLRQJUDWLQJVDUHFDOOHGlamellarDQGDUHXVHIXODVEHDPGLYLGHUV



)RXULHU2SWLFV

7KH holographic gratings RZH WKHLU QDPH WR WKH IDFW WKDW WKH\ DUH SURGXFHG ZLWKDQLQWHUIHUHQWLDOWHFKQLTXHDQGQRWZLWKUXOLQJPDFKLQHV7KHLUPRVWFRPPRQ SURILOHLVVLQXVRLGDOZKLFKFDQRQO\EHPDGHDV\PPHWULFDODQGDWOHDVWSDUWLDOO\ EOD]HGWKURXJKFRPSOLFDWHGWHFKQLTXHV7KHLUHVVHQWLDODGYDQWDJHLVWKHLUDELOLW\WR FRUUHFWJHRPHWULFDEHUUDWLRQVRIVSKHULFDQGHYHQDVSKHULFFRQFDYHJUDWLQJV 7KH concave gratings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distributed Bragg reflector '%5  DQG KDV WZR FRUUXJDWLRQV RXW RI WKH DFWLYH ]RQH IRUPLQJ WKH PLUURUV RI WKH ODVHU FDYLW\ 7KH RWKHU RQH LV FDOOHG distributed feedback ')%  DQG KDV D VLQJOH FRUUXJDWLRQ H[WHQGHGRYHUWKHHQWLUHDFWLYH]RQH7KLVDOORZVIRUDPRQRFKURPDWLFODVHUHPLV VLRQWKDWLVPRUHHDVLO\FRQWUROODEOH

5.7.5 Production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mastersDUH WKHQXVHGWRSURGXFHYDULRXVUHSOLFDVWKURXJKFRQWDFW7KHVHODWWHUDUHREWDLQHGE\ GHSRVLWLQJ RQ WKH PDVWHU D WKLQ OD\HU RI D PDWHULDO VXLWDEOH IRU IDFLOLWDWLQJ WKH VHSDUDWLRQIURPWKHUHSOLFDDQGVXEVHTXHQWO\HYDSRUDWLQJRQLWXQGHUYDFXXPD PHWDOOLFILOPXVXDOO\VWLOODOXPLQXPRQWRZKLFKDQRWKHUVXEVWUDWHLVWKHQILUPO\ ERQGHGZLWKDORZYLVFRVLW\UHVLQWKDWLVVORZO\OHIWWRFXUH)LQDOO\WKHUHSOLFDLV FDXWLRXVO\ VHSDUDWHG IURP WKH PDVWHU (DFK UHSOLFD LQ WXUQ FRQVWLWXWHV D submaster DQG WKLV RSHUDWLRQ LV UHSHDWHG IRU VHYHUDO JHQHUDWLRQV E\ H[SRQHQWLDOO\ SURGXFLQJXSWRWHQVRIWKRXVDQGVRIJRRGUHSOLFDVIURPDVLQJOHRULJLQDO$W\SL

'LIIUDFWLRQJUDWLQJV



FDOGHIHFWLQWKHVHJUDWLQJVLVWKHSUHVHQFHRIVFDWWHULQJGXHWRQRLVHLQWKHSRVL WLRQLQJRIWKHGLDPRQGWLSDQGJKRVWLPDJHVGXHWRDSHULRGLFHUURURIWKHVFUHZ SLWFK7KHVHIOXFWXDWLRQVKDYHEHHQGUDVWLFDOO\UHGXFHGWKURXJKWKHXVHRILQWHUIH UHQWLDO FRQWURO PHWKRGV DQG DUH QR ORQJHU D VHULRXV SUREOHP EXW WKH\ KDYH QRW EHHQFRPSOHWHO\HOLPLQDWHG :LWK WKH DGYHQW RI ODVHU DQG VSHFLDO SKRWRUHVLVWV holographic gratings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diffraction-free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

5.7.6 Tools of analysis and spectral tuning 7KHJUDWLQJVILQGWKHLUQDWXUDOORFDWLRQLQYDULRXVRSWLFDOLQVWUXPHQWVERWKIRU VSHFWUDODQDO\VLVDQGDVDIUHTXHQF\VHOHFWLRQHOHPHQWLQODVHUV\VWHPV$PRQJWKH VSHFWUDO DQDO\VLV WRROV WKH PRVW LPSRUWDQW DUH spectrographs DQG monochromators7KH\DUHHVVHQWLDOO\GLVWLQJXLVKHGE\WKHZD\LQZKLFKWKHVSHFWUXPLVUHF RUGHGDQGWKH\DUHVRPHWLPHVLQFRUSRUDWHGLQWRDVLQJOHLQVWUXPHQWLQZKLFKWKH ZDYHOHQJWKLVVHOHFWHGE\WKHURWDWLRQRIDVSHFLDOPLUURU7KH spectrographsSUR GXFH DQ H[WHQGHG VSHFWUXP WKDW ZDV W\SLFDOO\ UHFRUGHG RQ D SKRWRJUDSKLF SODWH :LWKWKHDGYHQWRI&&'VHQVRUVWKH\DUHEDFNLQYRJXHLQSDUWLFXODUZKHQXVLQJ HFKHOOHVDQGUHPRYLQJWKHRYHUODSRIWKHRUGHUVE\PHDQVRIWKHGLVSHUVLRQRID SULVPRUDVHFRQGDU\JUDWLQJLQDGLUHFWLRQRUWKRJRQDOWRWKDWRIWKHPDLQJUDWLQJ



)RXULHU2SWLFV

,QWKLVZD\RQWKHVXUIDFHRIWKH&&'WKHVSHFWUXPLVGLYLGHGLQWRVHYHUDOVWULSV WKDWDOORZIRUWKHUHFRUGLQJRIDZLGHVSHFWUDOEDQGZLWKRXWKDYLQJWRURWDWHWKH JUDWLQJDQGWKXVZLWKRXWPRYLQJSDUWV 7KH monochromatorsLQVWHDGXVHDQH[LWVOLWWRVHOHFWDSDUWLFXODUZDYHOHQJWK DQGDGHWHFWRUVXFKDVDSKRWRWXEHWRPHDVXUHWKHSRZHU7KHUHJLVWUDWLRQRIWKH VSHFWUXPLVDFKLHYHGE\URWDWLQJWKHJUDWLQJIRUZKLFKWKHZDYHOHQJWKWUDQVPLWWHG E\WKHVOLWYDULHVLQDFRQWUROOHGPDQQHU6RPHLQVWUXPHQWVDUHEXLOWZLWKWZRRU WKUHH PRQRFKURPDWRUV LQ VHULHV ,Q WKLV ZD\ WKH EDFNJURXQG UHVXOWLQJ IURP VHF RQGDU\SHDNVFDQEHFRQVLGHUDEO\UHGXFHG7KH\DUHXVHIXOIRU5DPDQVSHFWURVFR S\WKDWUHTXLUHVKLJKUHVROXWLRQVDQGWKHDELOLW\WRH[WLQJXLVKWKHQHDUE\VWURQJOD VHUOLQHXVHGIRUH[FLWDWLRQ 7KH spectrophotometers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

'LIIUDFWLRQJUDWLQJV



collimation mirror

entrance slit

R1 2a grating

JI Jd

2b

exit slit

R2 focusing mirror



)LJ2SWLFDOGLDJUDPRID&]HUQ\7XUQHUPRQRFKURPDWRU

$YDULDQWRIWKLVW\SHRIVSHFWURPHWHULVWKH(EHUW)DVWLHGHVLJQ,WXVHVDVLQJOH VSKHULFDOPLUURUDQGLVWKHUHIRUHPRUHVWDEOH7KLVVROXWLRQKDVEHHQXVHGLQYDUL RXV VSDFH PLVVLRQV ZLWK LQWHUSODQHWDU\ VSDFHFUDIW >)DVWLH @ ,Q SDUWLFXODU )DVWLHQRWHGWKDWWKHUHVROXWLRQRIDVSHFWURPHWHULPSURYHVLIWKHVOLWVDUHVXLWDEO\ FXUYHG DQG LQ WKH FDVH RI WKH (EHUW GHVLJQ ZLWK WKH FHQWHU RI FXUYDWXUH RQ WKH PLUURUD[LV Concave grating spectrometers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Ö Ö 7KHVDPH D LV DOPRVW HTXDOWRWKHDQJOH 6:& FLUFXPIHUHQFHWKHDQJOH 69& Ö Ö  VR WKH UD\ UHIOHFWHG LQ : DOVR D  DQG 2:& FDQ EH VDLG IRU WKH DQJOHV 29& UHDFKHV2)LQDOO\ZHFRQVLGHUDUD\WKDWIURP9LQVWHDGRIEHLQJUHIOHFWHGLV GLIIUDFWHG DW WKH DQJOH D  E WRZDUGV ' )RU WKH VDPH ZDYHOHQJWK WKHUH LV DQ Ö Ö  D  E DQG &:' HTXDOGLIIUDFWLRQDQJOHLQ:7KHUHIRUHVLQFHDVEHIRUH &9'



)RXULHU2SWLFV

V

G

W

R

a

a

a

b b

a

D

S

O C



)LJ6WXG\LQJWKHSURSHUWLHVRIWKH5RZODQGFLUFOH

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ittrow’s mounting )LJ D  ZKHUHLQ WKH UDGLDWLRQ RXWSXW IURP WKH ODVHU LV UHWURUHIOHFWHG E\ WKH JUDWLQJ ZLWK GLIIUDFWLRQ DQJOH ]HUR IRU WKHGHVLUHG ZDYHOHQJWK 7KH JUDWLQJFRQVWLWXWHV RQH RI WKH PLUURUV RI WKH ODVHU FDYLW\ DQG WKH WXQLQJ LV SHUIRUPHG E\ URWDWLQJ LW 7R LQ FUHDVH WKH VSHFWUDO VHOHFWLRQ DQDPRUSKLF SULVPV FDQ EH XVHG WR ZLGHQ WKH EHDP

'LIIUDFWLRQJUDWLQJV



Littman mounting

Littrow mounting mirror

mirror order 0

order 0

order -1 M

M laser

grating

C

laser

B

(a)

C

grating

(b)



)LJ([WHQGHGFDYLW\ODVHUV\VWHPVZLWKJUDWLQJWXQLQJMLVWKHUHDUODVHUPLUURUCLV WKHFROOLPDWRUBLVWKHEDVHRQZKLFKWKHODWWLFHDQGPLUURUDUHMRLQHG

DQGLQFUHDVHWKHQXPEHURILOOXPLQDWHGJURRYHVRIWKHUHWLFOH7KHSDWWHUQLVDOVR XVHGDVDODVHUoutput couplerXVLQJWKH]HURRUGHUGLIIUDFWHGEHDP7KLVEXQGOHLV IXUWKHUUHIOHFWHGE\DQRUGLQDU\PLUURUWKDWLVMRLQWZLWKWKHJUDWLQJ,QWKLVZD\ ZKHQWKHJUDWLQJLVURWDWHGIRUWKHWXQLQJWKHUHLVQRGHYLDWLRQRIWKHRXWSXWEHDP IURPWKHODVHUV\VWHP $QDOWHUQDWLYHLVWKH/LWWPDQ¶VPRXQWLQJ )LJE LQZKLFKWKHODVHUUDGLD WLRQUHDFKHVWKHJUDWLQJDWJUD]LQJLQFLGHQFH -,|ƒ WKHGLIIUDFWHGRUGHULVWKHQ UHIOHFWHGEDFNWRWKHUHWLFOHE\DPLUURU7XQLQJLVDFKLHYHGE\URWDWLQJWKHPLUURU ZKLFKGRHVQRWDIIHFWWKHRXWSXWEHDPREWDLQHGDVEHIRUHE\UHIOHFWLRQLQWKH]H URRUGHU7KHVLJQDOUHWXUQLQJWRZDUGWKHODVHULVGRXEO\GLIIUDFWHGDQGWKHVHOHF WLRQLVJRRGDVWKHVXUIDFHLOOXPLQDWHGRQWKHJUDWLQJDWJUD]LQJLQFLGHQFHLVPXFK ODUJHUWKDQWKHVL]HRIWKHODVHUEHDP+RZHYHUXQGHUWKHVHFRQGLWLRQVDJUDWLQJ KDV UHODWLYHO\ ORZ HIILFLHQF\  IRU VLQJOHSDVVDJHDQG WKHUHIRUH WKH IHHG EDFNRQWKHODVHULVPXFKZHDNHU

5.7.7 Ronchi’s ruling /HWXVQRZFRQVLGHUWKHFDVHRIDVOLWJUDWLQJLQWUDQVPLVVLRQSUHYLRXVO\PHQ WLRQHG GHSRVLWHG RQ D WKLQ VXEVWUDWH DQG LOOXPLQDWHG E\ D SRLQW VRXUFH 6R )LJ   2EVHUYLQJ WKLV VRXUFH WKURXJK WKH JUDWLQJ ZH FDQ VHH LQ DGGLWLRQ WR WKH RULJLQDOVRXUFHFRUUHVSRQGLQJWRWKHGLIIUDFWLRQ]HURRUGHUDVHWRIUHSOLFDVFRUUH VSRQGLQJWRWKHRWKHURUGHUVDUUDQJHGRQDOLQHSDUDOOHOWRWKHVXUIDFHRIWKHJUDW LQJ DQG SHUSHQGLFXODU WR WKH VOLWV :H FDQ LQGHHG UHSUHVHQW WKH YDULRXV SODQH ZDYHVRIWKHSODQHZDYHVSHFWUXPRIWKHVRXUFHZLWKUD\VZKRVHGLUHFWLRQLVJLYHQ E\ WKH FRUUHVSRQGLQJ ZDYH YHFWRUV DQG ZKRVH VRXUFH LV WKH VRXUFH LWVHOI )URP (TV  IRUHDFKRUGHUmZLWKDIL[HGZDYHOHQJWKZHKDYHWKDWWKHHPHUJLQJ UD\VDOOVWLJPDWLFDOO\DSSHDUWRKDYHRULJLQDWHGIURPWKHVDPHSRLQW6m7KHGLV WDQFHdEHWZHHQWZRVXFFHVVLYHSRLQWV6mDQG6mLV



)RXULHU2SWLFV

order m diffracted x k l x ko

I

l

So z

mK l ko

order zero Sm

grating

screen



)LJ  'LDJUDP RI WKH HIIHFWV RI D JUDWLQJ LQ WUDQVPLVVLRQ ZLWK D SRLQW VRXUFH ,Q WKH RYHUODS]RQHRIWZRRUGHUVZHKDYHLQWHUIHUHQFHIULQJHVREVHUYDEOHLQWKHLQWHQVLW\IRQD VFUHHQ



d

l

K kR

l OR

K  S

ZKHUHlLVWKHGLVWDQFHEHWZHHQWKHVRXUFH6RDQGWKHJUDWLQJKLVWKHYHFWRUPRG XOHRIWKHJUDWLQJDQGORLVWKHZDYHOHQJWKFRQVLGHUHG6LQFHDOOHPHUJLQJZDYHV DUHVSKHULFDOLQWKHRYHUODSSLQJ]RQHEHWZHHQWZRRUGHUVVXFKDV]HURRUGHUDQG RUGHUWKHUHDUHIULQJHVWKDWDUH DOPRVW VWUDLJKWOLNHWKRVHGLVFXVVHGLQ† WKHVSDFLQJRIZKLFKLVSURSRUWLRQDOWRWKHUDWLRORd+RZHYHUEHLQJWKDWLQWKLV FDVH dLVSURSRUWLRQDOWRORVXFKIULQJHVDUHDFKURPDWLF7KHVHSDUDWLRQEHWZHHQ WKHIULQJHVLVDOVRLQYHUVHO\SURSRUWLRQDOWRWKHGLVWDQFHlIRUZKLFKWKHVHZLGHQXS WRJLYHDXQLIRUPILHOGZKHQl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³VRPHUVDXOW´E\PRYLQJWKHUHWLFOHIURPEHIRUHWRDIWHUWKHIRFDOSRVLWLRQ >5RQFKL@

%LEOLRJUDSKLFDOUHIHUHQFHV



%LEOLRJUDSKLFDOUHIHUHQFHV $EEH (. Beitrage zür theorie des mikroskops und der mikroskopischen wahrnehmung $UFKLY0LFURVNRSLVFKH$QDW   $EUDPRZLW]0DQG6WHJXQ,$Handbook of Mathematical Functions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fIRUpDQG\IRUU,WLVSRVVLEOHWRGHULYHWKLV

HTXDWLRQ IURP WKH FRQVLGHUDWLRQ WKDW HQWURS\ IRU D V\VWHP FKDUDFWHUL]HG E\ DQ DEVROXWH WHPSHUDWXUHTDQGDYROXPHVLVDQH[DFWGLIIHUHQWLDOVR wUwV T T wpw7 VpZKHUHULV WKHHQHUJ\RIWKHV\VWHPDQGpLVWKHSUHVVXUH>)HUPLS@WDNLQJWKDWU UV

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rrRIWKHUDGLXVrRIWKHVSKHUH>6LYRXNKLQH,,@,QRWKHUZRUGVLQWKHDG LDEDWLFFRPSUHVVLRQZHKDYHWKDWQr FRQVWDQWDQGVLQFHrvVZKHUHVLV WKH YROXPH RI WKH YHVVHO ZH DOVR KDYH WKDW QV  FRQVWDQW )URP %ROW]PDQQ¶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³UQdQ PXVW KDYH WKH IRUP UQ Q T   Qf QT >:LHQ @ 7ZR \HDUV ODWHU :LHQ SXEOLVKHG D QHZ PHPRU\ >:LHQ @ LQ ZKLFK KH SURSRVHG DQ H[SOLFLW IRUP RI WKH VSHFWUDO GLVWULEXWLRQ RI EODFN ERG\ UDGLDWLRQ LQ WKH VSDFH RI ZDYH OHQJWKVUO cOuH[S>c OT @ZKHUHccDUHWZRFRQVWDQWVZKLFKLQWKH IUHTXHQF\VSDFHLVHTXLYDOHQWWR

UQ DQH[S EQT 



 

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c OT O H[S>c OT @DQGE\5D\OHLJK RIZKLFKZHZLOOGLVFXVVODWHU ZLWKUO cTO H[S>c OT @ WR ILW WKH QHZ GDWD IRU O ODUJH ZKLOH UHVSHFWLQJ WKH 6WHIDQ %ROW]PDQQ DQG GLVSODFHPHQW ODZVJDYHYDOXHV LQ DFFRUGDQFH ZLWK WKHPHDVXUHV  f T  ³ x  f x dx  DQG VLQFH WKH LQWHJUDO LQ x LV D FRQVWDQW YDOXH

 ,QGHHG ³ f Q f Q T d Q 

6WHIDQ%ROW]PDQQ¶VODZLVUHVSHFWHG7KHFRUUHVSRQGLQJGLVWULEXWLRQLQWKHZDYHOHQJWKVSDFHRQ WKHRWKHUKDQGKDVWKHVKDSHUO OT  OM OT 

 +LVWRULFDOQRWHVIURP.LUFKKRIIWR(LQVWHLQ  >'D\DQG9DQ2UVWUDQG@ 7KH VFLHQWLVW ZKR IRXQG WKH ULJKW GLVWULEXWLRQ ZDV 0D[ .DUO (UQVW /XGZLJ 3ODQFN   EXW WKH VWRU\ RI WKLV GLVFRYHU\ LV YHU\ FRPSOH[ DQG WKH GHPRQVWUDWLRQ WKDW LV JLYHQ QRZ LV YHU\ GLIIHUHQW IURP 3ODQFN¶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¶V ZRUNDQGLQKHEHJDQDVHULHVRIILYHSXEOLFDWLRQVRQLUUHYHUVLEOHUDGLDWLRQ SURFHVVHV %ROW]PDQQ KLPVHOI KRZHYHU SRLQWHG RXW WR KLP WKDW 0D[ZHOO¶V HTXDWLRQV ZHUHLQYDULDQWIRUWHPSRUDOLQYHUVLRQDQGWKHUHIRUHRQHFRXOGQRWGHGXFHWKHLULU UHYHUVLELOLW\LQWKHLQWHUDFWLRQEHWZHHQILHOGDQGUHVRQDWRU,QRWKHUZRUGVWKHLU UHYHUVLELOLW\ KDG WR EH LPSOLFLW LQ WKH LQLWLDO FRQGLWLRQV 3ODQFN DFFHSWHG WKH FULWLFLVPDQGWKHQFRQVLGHUHGWKHFDVHRID³QDWXUDOUDGLDWLRQ´WKDWZRXOGHQVXUH LUUHYHUVLELOLW\VLPLODUO\WRWKHPROHFXODUFKDRVRI%ROW]PDQQ¶VWKHRU\,QWKLVZD\ KH FDPH WR XQGHUVWDQG DQ LPSRUWDQW UHODWLRQ EHWZHHQ WKH VSHFWUDO GHQVLW\ RI WKH ILHOGU QT DQGWKHDYHUDJHHQHUJ\uQ T RIDKDUPRQLFRVFLOODWRUZLWKIUHTXHQF\ Q%\HTXDWLQJWKHHPLVVLRQDQGDEVRUSWLRQUDWHRIWKHRVFLOODWRULPPHUVHGLQWKH QDWXUDOUDGLDWLRQILHOGKHIRXQGWKDWDWHTXLOLEULXP



UQ T   SQc uQ T 

 

ZKHUHcLVWKHVSHHGRIOLJKW $WWKLVSRLQW3ODQFNKDGRQO\WRGHWHUPLQHWKHDYHUDJHHQHUJ\RIWKHRVFLOODWRUV WRILQGWKHVSHFWUDOGHQVLW\RIWKHILHOGDVZHOO%XWLQVWHDGRIXVLQJWKHHTXLSDUWL WLRQ WKHRUHP RI VWDWLVWLFDO PHFKDQLFV KH IROORZHG D WKHUPRG\QDPLF DSSURDFK

7KHEODFNERG\UDGLDWLRQ   ORRNLQJIRUDUHODWLRQVKLSEHWZHHQHQWURS\SDQGPHDQHQHUJ\uRIWKHRVFLOODWRU ,QWKHODVWRIKLVILYHDUWLFOHVSXEOLVKHGLQWKH\HDUKHSURSRVHGWKHUHODWLRQ VKLS



S



u § u · OQ ¨ ¸  a Q © beQ ¹

 

ZKHUHeLVWKH1HSHURQXPEHUDQGu uQ T 3ODQFNGLGQRWJLYHDQH[SODQDWLRQ IRUWKHFKRLFHRIWKLVIXQFWLRQEXWLWLVSUHVXPHGWRKDYHEHHQGULYHQE\WKHIDFW WKDWKHJRW:LHQ¶VH[DFWGLVWULEXWLRQIURPLW,QDGGLWLRQKHZDVDEOHWRGHPRQ VWUDWHWKDWZLWKWKLVGHILQLWLRQWKHWRWDOHQWURS\JUHZRYHUWLPH,QSDUWLFXODUKH ZDV LPSUHVVHG E\ WKH IDFW WKDW w6wu ZDV VLPSO\ SURSRUWLRQDO WR PLQXV u 7KHUHVXOWWKDW3ODQFNIRXQGZDVWKHUHIRUH>3ODQFN@ 

UQ T   Sbc QH[S aQ7 

 

3ODQFNDWWULEXWHGWRaDQGbWKHPHDQLQJRIuniversal constantsMXVWOLNHWKHVSHHG RIOLJKW7KHFRQVWDQWbZDVDOUHDG\ZKDWKHODWHULQGLFDWHGZLWKWKHV\PEROh,DQG ZKDWZRXOGWKHQEHFDOOHGPlanck’s constant)URPWKHFRPSDULVRQZLWKWKHH[ SHULPHQWDOGDWDKHREWDLQHGWKHYDOXHVa ÂVHFÂ.b  HUJÂVHF 3ODQFN ZDV SOHDVHG ZLWK WKLV UHVXOW EXW WKH QHZ PHDVXUHV PDGH E\ /XPPHU DQG 3ULQJVKHLP DQG WKHQ PRUH DFFXUDWHO\ LQ 2FWREHU  E\ 5XEHQV DQG .XOEDXPIRUFHGKLPTXLFNO\WRUHDQDO\]HWKHZKROHTXHVWLRQ+HIRXQGWKHUH IRUHWKDWLQRUGHUWRVDWLVI\WKHFRQVWUDLQWRIDQLQFUHDVLQJHQWURS\LWZDVVXIIL FLHQW WKDW w6wu ZDV DQ\ QHJDWLYH IXQFWLRQ RI u ,I LQ SDUWLFXODU WKLV VHFRQG GHULYDWLYH ZHUH SURSRUWLRQDO WR u WKH VSHFWUDO GLVWULEXWLRQ ZRXOG EH SURSRU WLRQDOWRT6R3ODQFNIRXQGWKDWWKHIXQFWLRQ>u Ju @ZKHUHJGHSHQGVRQ WKHIUHTXHQF\ZDVEULQJLQJWRJHWKHU:LHQ¶VGLVWULEXWLRQDWVPDOOZDYHOHQJWKVDQG WKHWUHQGSURSRUWLRQDOWRTDWORQJZDYHOHQJWKVREWDLQLQJZKDWLVQRZNQRZQDV Planck’s distributionUQ T  AQ>H[S BQT @KHUHH[SUHVVHGLQWKHIUHTXHQ F\VSDFHZKHUHADQGBDUHFRQVWDQWV 6KRUWO\DIWHURQ2FWREHU3ODQFNDWWHQGHGDPHHWLQJRIWKH*HUPDQ 6RFLHW\ RI 3K\VLFV ZKHUH KH SUHVHQWHG WKLV FRQFOXVLRQ ZLWK WKH WLWOH Ueber eine Verbesserung der Wien’schen Spectral-gleichung On an improvement of the Wien spectral equation  3ODQFN KLPVHOI KDV VDLG WKDW WKH QH[W PRUQLQJ KH UH FHLYHGDYLVLWIURP5XEHQVZKRWROGKLPWKDWKHKDGFKHFNHGDQH[FHOOHQWDJUHH PHQW ZLWK KLV IRUPXOD DW HDFK SRLQW RI WKH PHDVXUHV WKDW QLJKW ,Q VXEVHTXHQW PHDVXUHPHQWVWKHPRUHDFFXUDWHWKH\ZHUHWKHEHWWHUDJUHHPHQWWKHUHZDVZLWK  )URPWKHGLVSODFHPHQWODZDQG(T  LWIROORZVWKDWu Qf Q7 )URPWKLVLWDOVRIROORZV WKDWT QF uQ 6LQFHT wSwuLQWHJUDWLQJZLWKUHVSHFWWRuLWLVIRXQGWKDWSPXVWEHD IXQFWLRQ RI uQ 'HULYLQJ (T   ZLWK UHVSHFW WR u DQG VROYLQJ IRU u ZH KDYH WKDW u  bQH[S aQ7 )LQDOO\ZLWK(T  ZHDJDLQILQGWKH:LHQGLVWULEXWLRQ  >.OHLQ@ $QDFFRXQWRIWKHGHYHORSPHQWVRIWKHEODFNERG\VRXUFHLVJLYHQE\&RWWLQJWRQ  

 +LVWRULFDOQRWHVIURP.LUFKKRIIWR(LQVWHLQ  3ODQFN¶VGLVWULEXWLRQFRQWUDU\WRWKRVHSURSRVHGE\RWKHUDXWKRUV 3ODQFN UHDOL]HG WKH VHPLHPSLULFDO VLJQLILFDQFH RI KLV GHPRQVWUDWLRQ DQG ZRXOG WKHQ VHHN D GHHSHU SK\VLFDO H[SODQDWLRQ RI KLV ODZ +H WKXV VHW DVLGH KLV UHVHUYDWLRQVDERXW%ROW]PDQQ¶VWKHRU\DQGFRQVLGHUHGWKHFDVHRIDFROOHFWLRQRIN LGHQWLFDOKDUPRQLFRVFLOODWRUVZLWKIUHTXHQF\QDPRQJZKLFKZRXOGEHVKDUHGDQ HQHUJ\ UN  Nu ZKHUH u LV WKH DYHUDJH HQHUJ\ RI DQ RVFLOODWRU 5HVXPLQJ WKH %ROW]PDQQ HTXDWLRQ WKH HQWURS\ RI WKLV FROOHFWLRQ RI RVFLOODWRUV ZRXOG WKHQ EH JLYHQE\SN k%OQWFRQVWDQWZKHUHWKH³SUREDELOLW\´WZDVSURSRUWLRQDOWR WKH QXPEHU R RI ³FRPSOH[LRQV´ ZLWK ZKLFK WKH HQHUJ\ ZDV SDUWLWLRQHG EHWZHHQ WKHRVFLOODWRUV2QWKHRWKHUKDQGLQRUGHUWRHVWDEOLVKWKLVQXPEHULWZDVQHFHV VDU\ DV %ROW]PDQQ KDG GRQH WR GLYLGH WKH HQHUJ\ LQWR DQ LQWHJHU QXPEHU P RI HTXDO HOHPHQWV H WKDW LV UN  PH 7KH FRPSOH[LRQV ZHUH WKHUHIRUH R  NP >P N @:LWKVRPHSDVVDJHVXVLQJWKH6WLUOLQJ¶VDSSUR[LPDWLRQIRU ODUJHQXPEHUVNDQGPDVLGHIURPDFRQVWDQWKHREWDLQHGWKHPHDQHQWURS\SHU RVFLOODWRU



S

k % ^   u H OQ   u H  u H OQ u H ` 

 

6LQFHIRUWKHODZRIGLVSODFHPHQWHQWURS\WKHQKDGWREHIXQFWLRQRIuQLWUH VXOWHGWKDWH hQZKHUHhZDVDQHZFRQVWDQW,QWKLVZD\KHDJDLQIRXQGKLVGLV WULEXWLRQ ODZ   ZKHUH b  h DQG a  hk )URP WKH FRPSDULVRQ ZLWK WKH H[SHULPHQWDO GDWD 3ODQFN REWDLQHG WKH YDOXHV h  Â HUJÂVHF DQG k  Â HUJ. ,W ZDV 3ODQFN KLPVHOI ZKR FDOOHG k ³%ROW]PDQQ¶V FRQVWDQW´ QRWLQJWKDWKHKDGQRWJLYHQLWDSDUWLFXODUPHDQLQJ6XEVHTXHQWO\E\WKHYDOXHRI k 3ODQFN DOVR REWDLQHG WKH $YRJDGUR QXPEHU DQG WKH FKDUJH RI WKH HOHFWURQ ZKLFKUHPDLQHGWKHEHVWGHWHUPLQDWLRQVRIWKHVHFRQVWDQWVIRUDORQJWLPH>.OHLQ @3ODQFN¶VGLVWULEXWLRQODZZDVWKHUHIRUH 

UQ T

ShQ    c H[S ¬ª hQ k BT ¼º  

 

3ODQFN¶VGHPRQVWUDWLRQZDVQRWKRZHYHUIUHHRIFULWLFLVP,QGHHGKHKDGQRW EURXJKWWKHOLPLWIRUHo1HLWKHUGLGKHIROORZ%ROW]PDQQ¶VVFKHPHVLQFHKH ZRXOG KDYH KDG WR PD[LPL]H WKH WRWDO HQWURS\ IRU WKH YDULRXV IUHTXHQFLHV ,I KH GLGKHZRXOGKDYHFRPHWRWKHVDPHFRQFOXVLRQEXWWKLVIDFWZDVRQO\XQGHUVWRRG PDQ\\HDUVODWHU 7KHFRQWUDVWEHWZHHQ3ODQFN¶VODZDQGFODVVLFDOSK\VLFVEHJDQWRHPHUJHZLWK DVKRUWQRWHSXEOLVKHGE\-RKQ:LOOLDP6WUXWW /RUG5D\OHLJK± LQ-XQH LQWKHMRXUQDO3KLORVRSKLFDO0DJD]LQH>/RUG5D\OHLJK@6WUXWWSURSRVHG DQHZGLVWULEXWLRQODZRIEODFNERG\UDGLDWLRQXVLQJDFRPSOHWHO\GLIIHUHQWUHDVRQ LQJWRWKDWRI3ODQFN+HKDGLQIDFWFRQVLGHUHGWKHUHVRQDWLQJPRGHVRIWKHHOHF WURPDJQHWLFILHOGLQDFDYLW\HVWDEOLVKLQJWKHLUQXPEHUZLWKLQDJLYHQIUHTXHQF\ UDQJH$VVXPLQJWKDWWKHUDGLDWLRQZDVLQWKHUPDOHTXLOLEULXPDWDWHPSHUDWXUHT DQGWKHQXVLQJWKHHTXLSDUWLWLRQWKHRUHPRIFODVVLFDOVWDWLVWLFDOG\QDPLFVKHKDG

7KHSKRWRQ   DWWULEXWHGWRHDFKRIWKHVHPRGHVWKHVDPHDYHUDJHHQHUJ\ZKLFKLVSURSRUWLRQDO WRT,WWXUQHGRXWWKDWVSHFWUDOGHQVLW\KDGWREHSURSRUWLRQDOWRQT2QWKHRWKHU KDQGLQRUGHUWRDYRLGDQHYLGHQWGLYHUJHQFHIRULQFUHDVLQJIUHTXHQFLHVKHVXJ JHVWHG WKDW WKH ©FRPSOHWH H[SUHVVLRQª VKRXOG EH SURSRUWLRQDO WR QTH[S aQ7  5D\OHLJK UHFRJQL]HG WKDW DW WKH KLJKHVW IUHTXHQFLHV ©DOWKRXJK IRU VRPH UHDVRQ QRW\HWH[SODLQHGªWKH0D[ZHOO%ROW]PDQQODZRIHTXLSDUWLWLRQIDLOHGKRZHYHU ©LWVHHPVSRVVLEOHWKDWLWPD\DSSO\WRWKHJUDYHUPRGHVªIRUZKLFKWKHUDGLDWLRQ GHQVLW\ZDVSURSRUWLRQDOWRTDVWKHQHZH[SHULPHQWDOPHDVXUHVZHUHEHJLQQLQJ WRKLJKOLJKW 7KLV FRQWUDVW ZDV PDGH H[SOLFLW E\ D VHULHV RI DUWLFOHV E\ 5D\OHLJK DQG E\ -DPHV +RSZRRG -HDQV   WKDW DSSHDUHG LQ 1DWXUH LQ  -HDQV LQ SDUWLFXODU ZDV WU\LQJ WR SUHVHUYH WKH JHQHUDO YDOXH RI WKH HTXLSDUWLWLRQ WKHRUHP DQGFULWLFL]HG3ODQFN¶VGHULYDWLRQRQWKHWZRDVSHFWVPHQWLRQHGDERYHKLVEUHDN ZLWK%ROW]PDQQ¶VSURFHGXUHDQGWKHIDFWWKDWKHKDGQRWEURXJKWWKHOLPLWIRUho 

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³DQQXVPLUDELOLV´(LQVWHLQEHFDPHFRQYLQFHGLQVWHDGWKDWWKH WKHUPDO V\VWHP RI WKH UDGLDWLRQ FRQWDLQHG LQ D FDYLW\ DW D FRQVWDQW WHPSHUDWXUH FRXOG KDYH GHWHFWDEOH IOXFWXDWLRQV ,Q RQH RI KLV DUWLFOHV RI  HQWLWOHG On a heuristic point of view concerning the generation and transformation of light >(LQVWHLQ[email protected]KHGHDOWZLWKWKHWKHRU\RIWKHEODFNERG\DQGZLWKWKHZRUNVRI 3ODQFNDQG:LHQ7KHQKHUHVXPHGWKH3ODQFN¶VHTXDWLRQ  WKDWOLQNVWKHVSHF WUDOGHQVLW\UWRWKHDYHUDJHHQHUJ\RIDQRVFLOODWRUEXWDFFRUGLQJWRWKHHQHUJ\ HTXLSDUWLWLRQWKHRUHPWKLVDYHUDJHHQHUJ\KDGWREHHTXDOWRk%T,QWKLVZD\KH DJDLQIRXQGWKHVRFDOOHG5D\OHLJK-HDQVIRUPXODWKDW5D\OHLJKKLPVHOIKDGIRXQG XQWHQDEOH UQ T   SQc k%T



 

(LQVWHLQ ZDV WKH ILUVW WR QRWH WKDW &ODVVLFDO 3K\VLFV FOHDUO\ OHG WR D UHVXOW LQ RSHQFRQIOLFWZLWKWKHH[SHULHQFH,QIDFW(T  LPSOLHVWKDWWKHVSHFWUDOHQHUJ\  )ROORZLQJ3ODQFNKHH[SOLFLWO\FRQVLGHUHGWKHFDVHRIWRWDOO\LQFRKHUHQWUDGLDWLRQJHQHUDWHG

E\³separateUHVRQDWRUV´

 +LVWRULFDOQRWHVIURP.LUFKKRIIWR(LQVWHLQ  GHQVLW\ VKRXOG GLYHUJH WR LQILQLW\ DV WKH IUHTXHQF\ LQFUHDVHV DFFRUGLQJ WR ZKDW 3DXO(KUHQIHVWWKHQFDOOHGultraviolet catastrophe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quanta of energyGLVWULEXWHGLQVSDFHVXFKDVDJDV WKDW©PRYHZLWKRXWGLYLGLQJDQGDUHDEVRUEHGRUJHQHUDWHGRQO\DVDZKROHª)RU WKHFDVHRIDPRQRFKURPDWLFUDGLDWLRQRIIUHTXHQF\QKHDVVXPHGWKDWWKHTXDQ WXPKDVHQHUJ\hQ,QVXSSRUWRIWKLVK\SRWKHVLVKHFRQVLGHUHGVHYHUDOH[DPSOHV LQFOXGLQJWKHREVHUYDWLRQPDGHE\3KLOLS/HQDUGLQWKDWWKHHQHUJLHVRIHOHF WURQVHPLWWHGE\WKHSKRWRHOHFWULFHIIHFWZHUHLQGHSHQGHQWRIWKHLQWHQVLW\RIWKH LQFLGHQW UDGLDWLRQ (LQVWHLQ LQWHUSUHWHG WKLV IDFW E\ IRUHVHHLQJ WKDW WKH PD[LPXP HQHUJ\ RI WKH HOHFWURQV HPLWWHG LV JLYHQ E\ EPD[  hQ  P ZKHUH WKH WKUHVKROG YDOXHPLVWKHZRUNWKDWHDFKHOHFWURQPXVWSHUIRUPWROHDYHWKHPDWHULDORQZKLFK WKHUDGLDWLRQLVLQFLGHQW7KLVODZZDVYHULILHGZLWKSUHFLVLRQRQO\\HDUVODWHU LQE\5REHUW$QGUHZV0LOOLNDQ ± SURGXFLQJDYHU\FOHDQPHWDO VXUIDFHXQGHUYDFXXPWKDWDFWHGDVDSKRWRFDWKRGH7KLVODZHDUQHG(LQVWHLQWKH 1REHO3UL]HLQ $V%HUJLD  QRWHVLQ(LQVWHLQGLGQRW\HWKDYHDGXDOLVWLFYLHZRI WKHQDWXUHRIUDGLDWLRQ+HUHFRJQL]HVWKHYDOLGLW\RIZDYHWKHRU\EXWZULWHV©RS WLFDO H[SHULPHQWV REVHUYH RQO\ WLPHDYHUDJHG YDOXHV UDWKHU WKDQ LQVWDQWDQHRXV YDOXHV +HQFH « WKH XVH RI FRQWLQXRXV VSDWLDO IXQFWLRQV WR GHVFULEH OLJKW PD\ OHDGWRFRQWUDGLFWLRQVZLWKH[SHULPHQWVHVSHFLDOO\ZKHQDSSOLHGWRWKHJHQHUDWLRQ DQG WUDQVIRUPDWLRQ RI OLJKWª +RZHYHU VRPH \HDUV ODWHU LQ  (LQVWHLQ UH VXPHG WKH IOXFWXDWLRQ LVVXH ILQGLQJ WKDW DFFRUGLQJ WR 3ODQFN¶V ODZ WKH PHDQ VTXDUH YDOXH RI WKH IOXFWXDWLRQV RI WKH HQHUJ\ RI EODFN ERG\ UDGLDWLRQ KDG WR EH JLYHQ E\ D UHODWLRQVKLS FRQVLVWLQJ RI two WHUPV ERWK SUHVHQW 7KH ILUVW SURSRU WLRQDOWRhQLVRIFRUSXVFXODUQDWXUHDQGWKHVHFRQGSURSRUWLRQDOWRcU SQ LV RI XQGXODWRU\ QDWXUH GXH WR LQWHUIHUHQFH EHWZHHQ ©OLJKW UD\Vª XQFRUUHODWHG EH WZHHQ WKHP  >(LQVWHLQ @ ,Q KLV FRQFOXVLRQ (LQVWHLQ LPDJLQHV WKDW WKH ©WKH PDQLIHVWDWLRQRIOLJKW¶VHOHFWURPDJQHWLFZDYHVLVFRQVWUDLQHGDWVLQJXODULW\SRLQWV OLNHWKHPDQLIHVWDWLRQRIHOHFWURVWDWLFILHOGVLQWKHWKHRU\RIWKHHOHFWURQªDQGLQ WKH SUHVHQFH RI PDQ\ RI WKHVH VLQJXODULWLHV GHQVHO\ DUUDQJHG ZLWK UHVSHFW WR WKH



 ,QPRGHUQODQJXDJHWKHYDULDQFHWKDWLVWKHURRWPHDQVTXDUHYDOXHRIWKHGLIIHUHQFHLQWKH QXPEHURISKRWRQVnIURPLWVPHDQYDOXHn!LVJLYHQE\>$UHFFKL@'n! n!n!

,Q RWKHU ZRUGV LQ SKRWRQ VWDWLVWLFV 3ODQFN¶V ODZ FRUUHVSRQGV WR D SUREDELOLW\ p n   n!n n! n WR GHWHFW n SKRWRQV ZLWKLQ D WLPH LQWHUYDO 't VPDOOHU WKDQ WKH LQYHUVH RI WKH VSHFWUDOEDQGXVHGWRILOWHUWKHUDGLDWLRQ7KLVSUREDELOLW\LVNQRZQDV³JHRPHWULFGLVWULEXWLRQ´ DQGn!LVWKHDYHUDJHQXPEHURISKRWRQVUHYHDOHGLQUHSHDWHGPHDVXUHPHQWVRIGXUDWLRQ't

7KHSKRWRQ   H[WHQVLRQ RI WKHLU ILHOG RQH PLJKW JHW DQ RVFLOODWLQJ ILHOG VLPLODU WR WKDW RI WKH HOHFWURPDJQHWLFWKHRU\(LQVWHLQILQDOO\GHGXFHGWKDW©WKHWZRVWUXFWXUDOSURSHUWLHV RI UDGLDWLRQ DFFRUGLQJ WR 3ODQFN¶V IRUPXOD RVFLOODWLRQ VWUXFWXUH DQG TXDQWXP VWUXFWXUH VKRXOGQRWEHFRQVLGHUHGLQFRPSDWLEOHZLWKRQHDQRWKHUª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¶V ODZZLWKDQHZDSSURDFK,QKLVSUHYLRXVZRUNVKHKDGFRQVLGHUHGLWRQO\DVDQ H[SUHVVLRQ WKDW ZDV MXVWLILHG WKH H[SHULPHQWDO PHDVXUHPHQWV EXW LW ZDV FRQWUD GLFWRU\WKDWLQLWVGHULYDWLRQLWPDGHXVHRIFODVVLFDODUJXPHQWVWKDWWKHVDPHODZ UHIXWHG7KHQKHWULHGWRJLYHDQHZGHPRQVWUDWLRQLQFOXGLQJIRUWKHILUVWWLPH WRJHWKHUZLWKWKHHQHUJ\WKHFRQFHSWWKDWHDFKTXDQWXPRIOLJKWDOVREULQJVZLWKLW DPRPHQWXPHTXDOWRhQc,QWKLVDUWLFOH(LQVWHLQFRQVLGHUHGWKHFDVHRIDFDYLW\ FRQWDLQLQJJDVDQGHOHFWURPDJQHWLFUDGLDWLRQLQWKHUPDOHTXLOLEULXPZLWKWKHZDOOV RIWKHFDYLW\5HVXPLQJWKH%RKUPRGHO(LQVWHLQVWDWHGWKDWWKHH[FKDQJHVRIHQ HUJ\EHWZHHQUDGLDWLRQDQGJDVZRXOGWDNHSODFHWKURXJKWKUHHW\SHVRISURFHVV one RI DEVRUSWLRQ RI D TXDQWXP RI UDGLDWLRQ E\ D PROHFXOH IURP D ORZHU HQHUJ\ VWDWHnWRDKLJKHURQHmDQGtwoRIHPLVVLRQRIZKLFKRQHZRXOGEHspontaneousDQGRQHstimulatedIURPWKHXSSHUWRWKHORZHUVWDWH7KHVSRQWDQHRXVSUR FHVV KDG WR KDYH D SUREDELOLW\ RI GHFD\ dWVS Amn dt  LQ WKH WLPH LQWHUYDO dt LQGHSHQGHQW RI WKH SUHVHQFH RI UDGLDWLRQ ZKLOH VWLPXODWHG DEVRUSWLRQ DQG HPLV VLRQ SUREDELOLWLHV ZHUH UHVSHFWLYHO\ JLYHQ E\ dWD UBnm dt  DQG dWVW UBmn dt  DQGWKHUHIRUHZHUHSURSRUWLRQDOWRWKHUDGLDWLRQHQHUJ\GHQVLW\U$WWKHUPDOHTXL OLEULXPDFFRUGLQJWRWKHODZRIWKH%ROW]PDQQ¶VGLVWULEXWLRQDOVRFRQVLGHULQJWKH GHJHQHUDF\ RIVWDWHVn DQGm (LQVWHLQ DJDLQ REWDLQHG 3ODQFN¶V ODZ DQGDOVR WKH UHODWLRQVKLS Amn Bmn v Q /HVVNQRZQLVWKDWLQWKLVVDPHDUWLFOHLQDGGLWLRQWR HQHUJ\(LQVWHLQDOVRDWWULEXWHGWKURXJKFRPSOH[UHDVRQLQJDQH[FKDQJHRIPR PHQWXPhQcEHWZHHQUDGLDWLRQDQGPROHFXOHVHDFKDEVRUEHGTXDQWXPWUDQVIHUV LWV LPSXOVH WR WKH PROHFXOH LQ LWV RZQ GLUHFWLRQ ZKLOH LQ VWLPXODWHG HPLVVLRQ HDFKPROHFXOHHPLWVDTXDQWXPE\JLYLQJLWLPSXOVHLQWKHVDPHGLUHFWLRQDVWKH TXDQWDWKDWVWLPXODWHLW,QVWHDGIRUWKHVSRQWDQHRXVHPLVVLRQWKHLPSXOVH\LHOGHG WRWKHTXDQWXPFDQEHGLUHFWHGLQDQ\GLUHFWLRQDQGWKHUHIRUHLWVLVRWURS\LVRE WDLQHGRQO\VWDWLVWLFDOO\,QWKLVZD\KHIRXQGWKDWWKH0D[ZHOOLDQYHORFLW\GLVWUL EXWLRQRIWKHPROHFXOHVZDVQRWDOWHUHGE\WKHWKHUPDOHTXLOLEULXPUDGLDWLRQZLWK

 +LVWRULFDOQRWHVIURP.LUFKKRIIWR(LQVWHLQ  WKHFDYLW\ 7KH\RXQJ,QGLDQSK\VLFLVW6DW\HQGUD1DWK%RVH  EHLQJLQWHUHVWHG LQWKH3ODQFNODZLQUHJDUGWRKLVWHDFKLQJRITXDQWXPSK\VLFVIRXQGWKDWDOOGHUL YDWLRQVRIWKLVODZVWLOOKDGUHFRXUVHWRFODVVLFDOWRSLFV7KH\ZHUHWKHUHODWLRQVKLS EHWZHHQWKHUDGLDWLRQGHQVLW\DQGWKHDYHUDJHHQHUJ\RIWKHRVFLOODWRUVIRXQGE\ 3ODQFNRUWKHGLVSODFHPHQWODZRI:LHQDQGWKHFRUUHVSRQGHQFHSULQFLSOHRI%RKU LQWKHFODVVLFDOOLPLWWRZKLFK(LQVWHLQZDVIRUFHGWRUHVRUWLQKLVZRUNIRUD TXDQWLWDWLYHFRPSDULVRQ7KXVWKHVHGHULYDWLRQVDSSHDUHGXQMXVWLILHGWRKLPIURP D ORJLFDO SRLQW RI YLHZ +H VXFFHHGHG LQ ILQGLQJ D IXOO\ TXDQWXP GHULYDWLRQ RI 3ODQFN¶VODZDVVXPLQJWKHTXDQWDRIOLJKWWREHSDUWLFOHVZKRVHFRRUGLQDWHVDQG PRPHQWV ZHUH UHSUHVHQWHG LQ D SKDVH VSDFH LQ VL[ GLPHQVLRQV 4XDQWL]LQJ WKLV VSDFH LQ FHOOV RI YROXPH h DQG FDOFXODWLQJ WKH QXPEHURI WKHVH FHOOV ZLWKLQ WKH UDQJHRIIUHTXHQF\ dQKHFRXOGFDOFXODWHWKHWKHUPRG\QDPLFSUREDELOLW\RIDVWDWH GHILQHG PDFURVFRSLFDOO\ 8QOLNH 3ODQFN KH IXOO\ FDUULHG RXW WKH FDOFXODWLRQ RI PD[LPL]LQJ HQWURS\ ILQDOO\ JHWWLQJ 3ODQFN¶V ODZ 2Q WKH RWKHU KDQG DV %HUJLD QRWHVLQKLVGHULYDWLRQ%RVHLQWURGXFHGWKUHHQHZHOHPHQWVRIIXQGDPHQWDOLP SRUWDQFH7KHILUVWLVZKDWZRXOGVXEVHTXHQWO\FRPHWREHFDOOHGWKHphoton spin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¶VRULJL QDOWH[W Planck’s Law and the Hypothesis of Light QuantaLQWR*HUPDQDQGZLWK DQDFFRPSDQ\LQJOHWWHUSUHVHQWLQJLWWRWKHPDJD]LQH=3K\VLNWKURXJKZKLFKKH REWDLQHGSXEOLFDWLRQRQ%RVH¶VEHKDOI>%RVH@ /DWHU(LQVWHLQH[WHQGHG%RVH¶VWUHDWPHQWWRPDWHULDOSDUWLFOHVEXWLWZDV3DXO (KUHQIHVWZKRSRLQWHGRXWWKDWWKH\KDGLQWURGXFHGDQHZVWDWLVWLFQRZFDOOHGWKH Bose-Einstein statisticLQZKLFKRQHKDVDORVVRIVWDWLVWLFDOLQGHSHQGHQFHEHWZHHQ LGHQWLFDOSDUWLFOHVQRZNQRZQDVbosonsZLWKLQWHJHUVSLQ7RWKLVTXDQWXPVWD WLVWLFZDVWKHQDGGHGWKDWRIFermi-Dirac 7KHOLJKWTXDQWDZHUHODWHUFDOOHGSKRWRQVLQE\*LOEHUW1HZWRQ/HZLVLQ DQDORJ\ZLWKSURWRQVDQGHOHFWURQVDQGWKLVQDPHKDVVLQFHUHPDLQHGLQFRPPRQ XVH+RZHYHU/HZLV¶VZRUNZDVEDVHGRQDIDXOW\SUHPLVHDVLWDVVXPHGWKHFRQ VHUYDWLRQRIWKHLUQXPEHU

 7KLVZRUNFDQEHFRQVLGHUHGDVWKHEDVLVIRULQWHUSUHWLQJWKHRSHUDWLRQRI ODVHUVDQGPDVHUV

KRZHYHULQWKHLQYHQWRURIWKH0DVHU&KDUOHV7RZQHVZDVLQVSLUHGE\WKHGHYHORSPHQW RIFRKHUHQWPLFURZDYHRVFLOODWRUVXVHGLQUDGDUIRUPLOLWDU\SXUSRVHV>%HUJLD@

7KHSKRWRQ



 (LQVWHLQ¶VLQGLFDWLRQRIRQWKHGXDOQDWXUHRIUDGLDWLRQZDVUHYLYHGRQO\ LQ  E\ /RXLV 9LFWRU 3LHUUH 5D\PRQG GH %URJOLH   ZKR UHLQ WHUSUHWHG%RKU¶VTXDQWL]DWLRQUXOHDVDVWDQGLQJZDYHFRQGLWLRQIRUHOHFWURQVDF WXDOO\H[WHQGLQJWKHZDYHSDUWLFOHGXDOLW\WRPDWHULDOSDUWLFOHVUHYROXWLRQL]LQJDOO RIWKH&ODVVLF3K\VLFVLQ4XDQWXP0HFKDQLFV %HVLGHV WKH 0LOOLNDQ H[SHULPHQW RWKHU H[SHULPHQWV DSSHDUHG WR FRQILUP WKH KHXULVWLF K\SRWKHVLV RI OLJKW TXDQWD $OUHDG\ LQ  -RVHSK -RKQ 7KRPVRQ  KDGVXJJHVWHGWR*HRIIUH\,QJUDP7D\ORU  WKDWKHVKRXOG FKHFNZKHWKHUWKHUHZHUHDQ\HQHUJ\XQLWVWKDWVHHPHGWRH[LVWLQLRQL]DWLRQH[ SHULPHQWV DQG ZLWK ;UD\V GLVFRYHUHG E\ :LOKHOP & 5|QWJHQ   LQ ,QWKLVFDVHWKH\H[SHFWHGWKDWGLIIUDFWLRQSKHQRPHQDZRXOGEHDOWHUHGZLWK DVXIILFLHQWO\ORZLQWHQVLW\7D\ORUWKHQFRQGXFWHGDQH[SHULPHQWLQZKLFKKHSKR WRJUDSKHGWKHVKDGRZRIDQHHGOHSURGXFHGE\DVOLWLOOXPLQDWHGE\DIODPHVWURQJ O\ DWWHQXDWHG ZLWK EODFNHQHG JODVVHV 7KH H[SRVXUH ODVWHG IRU WKUHH PRQWKV DQG SURGXFHGWKHVDPHIULQJHVYLVLEOHZLWKRXWDWWHQXDWLRQ7KRPVRQDQG7D\ORUZHUH XQDZDUHRIWKHZRUNRI3ODQFNDQG(LQVWHLQDQGFRQFOXGHGZLWKDQXSSHUOLPLWIRU WKHVHXQLWVRIHQHUJ\WKDWZDVPXFKODUJHUWKDQh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¶VOLIH7KRVHZKR VWDQGLQWKHUHDUJXDUGVXFKDV,P\VHOIFDQRQO\DWOHDVWKRSHWRFRQWULEXWHWRWKDW XQGHUJURZWKRIWHFKQLTXHDQGNQRZOHGJHWKDWIRVWHUVWKHVWLPXOXVLQQHZPLQGVRI QHZ H[SHULPHQWV DQG QHZ LGHDV %XW WKH RQO\ 2QH ZKR GULYHV WKH JDPH LV *RG WKURXJKSHRSOHRIJRRGZLOOZKRDOVRXQNQRZLQJO\GDUHWRUHDFK+LP



+LVWRULFDOQRWHVIURP.LUFKKRIIWR(LQVWHLQ



6XJJHVWHGUHDGLQJ 7KHLQIRUPDWLRQLQWKHVHQRWHVLVGHULYHGIURPWKHIROORZLQJWH[WV $UHFFKL)7Einstein without Relativity$QQDOLGHOO¶,VWLWXWRHPXVHRGL6WRULDGHOOD6FLHQ ]DGL)LUHQ]H\HDU,9LVVXH   %DUWROL$Sopra i movimenti prodotti dalla luce e dal calore/H0RQQLHU)LUHQ]H   %HOORQH(La teoria dinamica del caloreSL’irreversibilitàSLQ6WR ULDGHOODVFLHQ]D9ROIl corpo neroSLQ6WRULDGHOOD6FLHQ]D9ROAlbert EinsteinSLQ6WRULDGHOOD6FLHQ]D9RO6HULHVGLUHFWHGE\35RVVL *UXSSR(GLWRULDOH/¶(VSUHVVR   %HUJLD 6 Einstein: quanti e relatività, una svolta nella fisica teorica LQ ³, JUDQGL GHOOD VFLHQ]D´\HDU,Q/H6FLHQ]H6S$   %ROW]PDQQ/Ableitung des Stefan’schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie$QQDOHQ GHU3K\VLN   %RVH61Plancks Gesetz und Lichtquantenhypothese=HLWVFKULIWIU3K\VLN   'D\ $/ DQG 9DQ 2UVWUDQG &( The black body and the measurement of extreme temperatures7KH$VWURSK\VLFDO-   &RWWLQJWRQ,(Platinum and the standard of light3ODWLQXP0HWDOV5HY   (LQVWHLQ$Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt$QQDOHQGHU3K\VLN± D Zur Elektrodynamik bewegter Körper$QQDOHQGHU3K\VLN± E The development of our views on the composition and essence of radiation3K\VLNDOLVFKH=HLWVFKULIW   Zur Quantentheorie der Strahlung3K\VLNDOLVFKH=HLWVFKULIW   )HUPL(Termodinamica(G%RULQJKLHUL7RULQR   *ODXEHU5-Nobel lecture: One hundred years of light quanta5HY0RG3K\V    -DFNVRQ-'Classical Electrodynamics-RKQ:LOH\ 6RQV1HZ$OOHQHWDO@$UH FHQW UHYLHZ RQ WKLV VXEMHFW DQG LWV DSSOLFDWLRQV LV UHSRUWHG E\ 3LFFLULOOR HW DO  

6.3.3 Polarization of the modes ,QWKHSUHYLRXVGHULYDWLRQDJHQHULFFRPSRQHQWRIWKHILHOGKRZHYHURULHQWHG LVFRQVLGHUHG7KHPRGHVPD\WKXVKDYHDUELWUDU\SRODUL]DWLRQXQLIRUPDFURVVWKH

0RGHVRIKLJKHURUGHU



TEM 00

TEM10

TEM 20

TEM 00

TEM 01

TEM 02

TEM01

TEM11

TEM21

TEM10

TEM 11

TEM12

TEM02

TEM12

TEM 22

TEM20

TEM 21

TEM22

)LJ/LQHDUSRODUL]DWLRQPRGHVRI*DXVV+HUPLWHDQG*DXVV/DJXHUUHZKLFKDUHRE WDLQHGUHVSHFWLYHO\LQUHVRQDWRUVZLWKVTXDUHDQGFLUFXODUPLUURUV7KHDUURZVLQGLFDWHWKH GLUHFWLRQDQGWKHVLJQRIWKHHOHFWULFILHOGDWDGHILQLWHLQVWDQWWKDWLVDUURZVLQWKHRSSRVLWH GLUHFWLRQLQGLFDWHDSKDVHVKLIWRISLQWKHILHOGRVFLOODWLRQ7KHSRODUL]DWLRQRIWKHILHOGLV XQLIRUPRYHUWKHZKROHEHDPSURILOHEXWLQJHQHUDOLWVGLUHFWLRQGRHVQRWFRUUHVSRQGWR RQHRIWKHD[HVRIWKHPRGHDQGFDQDOVREHFLUFXODURUHOOLSWLFDO

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

+

=

+

=

+

=

+

=





3URSDJDWLRQRIODVHUEHDPLQOLQHDUPHGLD

 7KHPRGHVRIWKHVHWZRW\SHVDUHRIWHQLQGLFDWHGZLWKWKHVDPHV\PEROVDVLQ )LJVRRQHQHHGVWREHFDUHIXOQRWWRFRQIXVHWKHP&RPELQLQJPRGHVRIGLI IHUHQWRUGHUDQGZLWKGLIIHUHQWSRODUL]DWLRQDQGSKDVHZHFDQREWDLQOLJKWEHDPV ZLWKQRQXQLIRUPSRODUL]DWLRQDQGDFRQVLGHUDEOHYDULHW\RIVKDSHV )LJ 

3UDFWLFDOQRWHV 6.4.1 Intensity of a Gaussian beam TEM00 )URP(T  LWLVVHHQWKDWGHSHQGLQJRQWKHUDGLXVrWKHLQWHQVLW\RID *DXVVLDQEHDP7(0YDULHVDVH[S rw ,IPLVWKHWRWDOSRZHURIWKHEHDP LWWXUQVRXWWKDW I r



 P  r  e Sw 

w



 

ZKHUH WKH FRHIILFLHQW  Sw  FRUUHVSRQGV WR WKH QRUPDOL]DWLRQ IRU D WZR GLPHQVLRQDO*DXVVLDQ7KHSHDNLQWHQVLW\LV



IS

P  Sw 

 

6.4.2 Measure of the diameter 2w of a TEM00 mode 7RGRWKLVPHDVXUHZHFDQXVHDVFUHHQZLWKDVXFFHVVLRQRIFDOLEUDWHGKROHV VXFKDVWKDWGHVFULEHGLQ)LJ7KLVVFUHHQPXVWEHPRXQWHGRQDQ;$UQDXGDQG.R JHOQLN@7KLVKDSSHQVWKDQNVWRWKHSURSHUW\WKDWVLQMFRVM HYHQ IRUFRPSOH[DUJXPHQWV(T  FDQEHUHZULWWHQDV qxR q yR i nkR ª¬Qx x  Q y y   Qxy xy º¼ e   qx q y

< x y  z



 

ZKHUH Qx

FRV M VLQ  M   Qy qx qy

VLQ  M FRV M   qx qy

Qxy

§   VLQ M FRV M ¨  ¨ q y qx ©

 WDQ M Q y  Qx  



· ¸ ¸ ¹





 



ZKLFKLQWXUQFDQEHLPDJLQHGDVWKHFRPSRQHQWVRIDV\PPHWULFPDWUL[ 

Q

§ Qx ¨¨ © Qxy

Qxy · ¸ Q y ¸¹

 · § FRV M  VLQ M · § FRV M VLQ M · §  qx ¨  VLQ M FRV M ¸ ¨   q ¸ ¨ VLQ M FRV M ¸    y ¹© © ¹© ¹



7KHFRPSOH[UHSUHVHQWDWLRQRIWKHILHOGWKH\XVHLQWKHLUZRUNLVWKHFRQMXJDWHRIWKDWRIWKLV

WH[W  6XSSRVH LQ IDFW WKDW E xyz  LV D VROXWLRQ RI WKH SDUD[LDO HTXDWLRQ   DQG R LV D OLQHDU RSHUDWRU WUDQVIRUPLQJ WKH x y LQ x‫ މ‬y‫ މ‬WUDQVYHUVH FRRUGLQDWHV OHDYLQJ z XQFKDQJHG 7KHQ ZH KDYHWKDW



R’ A‚ ’ A E

ikR

w wz

E o R’ A‚ R R’ A R RE

ik

w wz

RE 

ZKHUHWKHV\PERO‚PHDQVWKHWUDQVSRVHGFRQMXJDWH,IWKHUHIRUHR‚ RZHKDYHWKDWRELV WKHVROXWLRQWRWKHSDUD[LDOHTXDWLRQLQWKHV\VWHPx‫މ‬y‫މ‬z,QRWKHUZRUGVLWLVVXIILFLHQWWKDWRLV DXQLWDU\WUDQVIRUPDWLRQ

3URSDJDWLRQLQDVWLJPDWLFSDUD[LDORSWLFDOV\VWHPV



ZKLFKFRUUHVSRQGVWRWKHURWDWLRQRIDGLDJRQDOPDWUL[H[SUHVVHGLQWKHV\VWHP xc ycRIWKHD[HVRIHTXLLQWHQVLW\HOOLSVHVDQGHTXLSKDVHHOOLSVHVRUK\SHUERODV ,IZHQRZFRQVLGHUWKHFDVHRIDFRPSOH[URWDWLRQMWDNLQJIRUxyDSDLURI UHDO&DUWHVLDQD[HVWKHD[HVxcycGLDJRQDOL]LQJWKHPDWUL[DUHFRPSOH[6XFKD WUDQVIRUPDWLRQFDQEHGHFRPSRVHGLQWRDUHDOURWDWLRQEDQGDQLPDJLQDU\URWDWLRQ iDZKLFKDFWLQJLQWKHVDPHSODQHWKDWLVDURXQGWKH]D[LVFRPPXWHEHWZHHQ WKHP)RUWKHUHDOURWDWLRQZHJREDFNDJDLQWRWKDWVKRZQDERYHDQGWKHUHIRUH ZHZLOOOLPLWRXUVHOYHVLQWKHIROORZLQJRQO\WRLPDJLQDU\URWDWLRQV6LQFHFRV iD  FRVKDDQGVLQ iD  i VLQKDZHKDYHWKHQ Qx



Qy Qxy

U y FRVK  D  U x VLQK  D  i V y FRVK  D  V x VLQK  D   V x  V y VLQK D FRVK D  i U y  U x VLQK D FRVK D 

U x FRVK  D  U y VLQK  D  i V x FRVK  D  V y VLQK  D 

 

7RREWDLQWKLVLQ(T  WKHIROORZLQJVXEVWLWXWLRQVZHUHPDGH M iD



 qj

U j  iV j 

x y

j



 

ZKHUH Uj DQG Vj DUH UHVSHFWLYHO\ WKH WHUPV RI FXUYDWXUH DQG H[WLQFWLRQ RI WKH EHDP7KHLUGHSHQGHQFHRQWKHFRRUGLQDWHzLVGHGXFHGIURP(T  REWDLQLQJ Uj z

 Vj z

z  z jR

z  z jR



z5j



z  z jR







 z5j 



 

  z5j 

ZKHUHzjRDUHWKHzFRRUGLQDWHVIRUWKHZDLVWVZKLOHz5jDUHWKH5D\OHLJKGLVWDQFHV 2EVHUYLQJ WKH H[SUHVVLRQ RI Qxy LQ (T   LW LV QRWHG WKDW WKH HIIHFW RI DQ LPDJLQDU\URWDWLRQLVWRPL[FXUYDWXUHDQGH[WLQFWLRQVRLWWDNHVWZRGLVWLQFWUHDO URWDWLRQVWRGLDJRQDOL]HWKHUHDOSDUWRUWKHLPDJLQDU\SDUWRIWKHPDWUL[Q



§ x3 · § FRV M 3 ¨ y ¸ ¨  VLQ M 3 © 3¹ ©

VLQ M 3 · § x · § x, · § FRV M,  FRV M 3 ¹¸ ©¨ y ¹¸ ©¨ y, ¹¸ ©¨  VLQ M,

VLQ M , · § x ·    FRV M, ¹¸ ©¨ y ¹¸

7KHURWDWLRQDQJOHVM3 z DQGM, z UHVSHFWLYHO\RIWKHD[LDOV\VWHPVRIWKHHTXL



3URSDJDWLRQRIODVHUEHDPLQOLQHDUPHGLD

SKDVHDQGHTXLLQWHQVLW\FXUYHVDUHJLYHQE\ 

WDQ M 3



Vx  V y Ux  U y

Ux  U y

WDQK D WDQ M,

Vx  V y

WDQK D 

 

:LWKUHIHUHQFHWRWKHVHWZRV\VWHPVWKHDPSOLWXGHRIWKHILHOGFDQEHUHZULWWHQDV




VHH UHIHUHQFHV TXRWHG LQ 6DQWDUVLHUR  @2QHZD\WRREWDLQDQDSSUR[LPDWH%HVVHO¶VZDYHZDVDOUHDG\SURSRVHG LQ  E\ -+ 0F/HRG ZKR KDG LGHQWLILHG LWV diffraction free SURSHUWLHV >0F/HRG@,WLVDFKLHYHGE\VHQGLQJDSODQHZDYHWRZDUGDQ axiconWKDWLV DPRQJWKHYDULRXVSRVVLELOLWLHVDJODVVFRQHZLWKDYHU\ODUJHYHUWH[DQJOH0RUH UHFHQWO\ RWKHU DXWKRUV KDYH UHSOLFDWHG WKH XVH RI DQ D[LFRQ >6FRWW DQG 0F$UGOH  $UOW DQG 'KRODNLD @ RU RI GLIIUDFWLYH HOHPHQWV >9DVDUD HW DO @ WR SURGXFHDSSUR[LPDWH%HVVHO¶VEHDPVRU%HVVHO*DXVVEHDPVRIYDULRXVRUGHU,Q SDUWLFXODU]HURRUGHU%HVVHO¶VEHDPVKDYHIRXQGLQWHUHVWLQWKHJHQHUDWLRQRIVHF RQGKDUPRQLFVLQQRQOLQHDUFU\VWDOV $JHQHUDOL]DWLRQRIWKH%HVVHO¶VZDYHFDQXOWLPDWHO\EHREWDLQHGE\DVVRFLDWLQJ HDFK SKDVH FRPSRQHQW ZLWK D SKDVH WHUP WKDW YDULHV SHULRGLFDOO\ ZLWK WKH D]L PXWKDODQJOHM,QWKLVFDVHWKHLQWHJUDORI(T  LVUHSODFHGZLWK



E x y  z  t

S  Eo eiEz iZt imM ³ e imIciDr FRV Mcd Mc S S  E ei Ez Zt  mM i m - Dr 

R

 

m

ZKHUHmLVDQLQWHJHUQXPEHUDQG-mLVWKH%HVVHO¶VIXQFWLRQRIRUGHUm7KHSRODU FRRUGLQDWHV r M  VWLOO FRUUHVSRQG WR WKH &DUWHVLDQ FRXSOH x y  ,Q WKH FDVH IRU



3URSDJDWLRQRIODVHUEHDPLQOLQHDUPHGLD

H[DPSOHRIm WKHLQWHQVLW\WUHQGKDVD]HURDWr DQGWKHFHQWUDOPD[LPXP LVUHSODFHGZLWKDULQJVXUURXQGHGDVEHIRUHE\DQLQILQLWHVXFFHVVLRQRIRWKHUGH FUHDVLQJLQWHQVLW\ULQJV7KHGDUNVSRWLQVLGHRIWKHILUVWULQJLVQDUURZHUWKDQWKH %HVVHO¶VEHDPSHDNZLWK m DQGWKHUHIRUHLWZRXOGOHQGLWVHOIHYHQEHWWHULQ DOLJQPHQWDSSOLFDWLRQVHollow beamsRIWKLVW\SHKDYHDOVREHHQSURSRVHGIRUWKH WUDSSLQJDQGJXLGDQFHRIFROGDWRPV%HVVHO¶VIXQFWLRQV ZLWKUHDODUJXPHQWV DUH UHDO DQG WKHUHIRUH WKH SKDVH SURJUHVV LV H[SUHVVHG E\ WKH H[SRQHQWLDO SUHVHQW LQ (T  ,WVKRXOGEHQRWHGDVZHOOWKDWWKHHTXLSKDVHVXUIDFHVDWDIL[HGWLPHt LHWKHZDYHIURQWVDUHQRWJHQHUDOO\SODQHEXWKHOLFDOLQDPDQQHUVLPLODUWRWKH WKUHDGRIDVFUHZ7KHSKDVHLQr LVQRWGHILQHGYDU\LQJZLWKWKHD]LPXWKDOFR RUGLQDWH MKRZHYHUIRUDOOZDYHVZLWK m zWKHILHOGLVQXOORQWKH zD[LV7KH SLWFKRIWKHVFUHZLVSURSRUWLRQDOWRmLHHDFKUHYROXWLRQDURXQGWKHzD[LVFDXV HV D SKDVH FKDQJH RI mS :H FDQ VWLOO LPDJLQH DGGLQJ WZR %HVVHO¶V ZDYHV WR JHWKHUZLWKHTXDODPSOLWXGHDQGRSSRVLWHYDOXHVRI m z ,QWKLVFDVHWKHILHOG LVJLYHQE\



E x y  z  t

ER ei Ez Zt  mM i m - m Dr  Eo ei Ez Zt mM i  m -  m Dr  ER ei Ez Zt i m FRV m M - m Dr 

  

ZKHUHZHH[SORLWHGWKHIDFWWKDW-m z    m-m z 7KHZDYHIURQWVUHWXUQWREH SODQHEXWWKHLQWHQVLW\LVPRGXODWHGDQJXODUO\ZKHUHE\HDFKULQJLVEURNHQLQWR mDUFV

%HVVHO*DXVVEHDPV 6.8.1 Zero-order Bessel-Gauss beams $SDUWLFXODUDSHUWXUHIXQFWLRQWKDWFDQEHDSSOLHGWRWKH%HVVHO¶VZDYHVLVWKDW RID*DXVVLDQIXQFWLRQ,QWKLVZD\RQHREWDLQV%HVVHO*DXVVEHDPVWKDWFDQDOVR EH FRQVLGHUHG DV WKH VXSHUSRVLWLRQ RI 7(0 *DXVVLDQ EHDPV LQVWHDG RI SODQH ZDYHVLQWKHVDPHPDQQHUDVLQWKHSUHYLRXVFDVHZLWKUHVSHFWWRWKHGLVWULEXWLRQ RIYHFWRUV kFRUUHVSRQGLQJWRWKHLUSURSDJDWLRQD[HV >*RULHWDO@7KHPDLQ GLIIHUHQFHZLWK%HVVHO¶VEHDPVLVWKDWWKHLQWHQVLW\RI%HVVHO*DXVVEHDPVUDSLGO\ GHFD\VDZD\IURPWKHLUD[LVRIV\PPHWU\WKHUHIRUHWKH\DUHHIIHFWLYHO\ORFDOL]HG LQ WKH GLUHFWLRQ WUDQVYHUVH WR WKLV D[LV 8QGHU DSSURSULDWH FRQGLWLRQV WKHLU WUDQV YHUVHLQWHQVLW\SURILOHUHPDLQVURXJKO\HTXDOWRLWVHOIEXWRQO\IRUDILQLWHGLVWDQFH FDOOHG Bessel’s distance ZKLFK SOD\V D VLPLODU UROH WR WKH 5D\OHLJK GLVWDQFH RI *DXVVLDQ EHDPV 2XWVLGH RI WKH %HVVHO¶V UDQJH WKH EHDP ZLGHQV SURJUHVVLYHO\ WDNLQJDULQJVKDSH &RQVLGHU WKHQ WKH VXSHUSRVLWLRQ RI *DXVVLDQ EHDPV RI HTXDO DPSOLWXGH DQG VL]HVZKRVHSURSDJDWLRQD[HVIRUPDFRQHGLVSRVHGDURXQGWKH zD[LVDQGDOOLQ

%HVVHO*DXVVEHDPV



WHUVHFWDWWKHSRLQW x y z  LQDGGLWLRQOHW-EHWKHKDOIDSHUWXUHDQJOH RIWKHFRQHDURXQGWKHzD[LV7KHSODQHz ZLOODOVREHWKHRQHZKHUHWKHZDLVWV RIDOOEHDPVOLH7KHUHVXOWLQJILHOGLVWKHQJLYHQE\ E x y z t

eiEz iZt

 u³

S S

EM e

  S   i z z5

iD x FRV M y VLQ M 

x  z WDQ - FRV M   y  z WDQ -VLQ M  wR iz z5 dM

  

ZKHUHEM S LVWKHDPSOLWXGHFRQWULEXWLRQRIHDFK*DXVVLDQFRPSRQHQW0RUHR YHU z5 LV WKH 5D\OHLJK GLVWDQFH WKDW ZLWKLQ WKH SDUD[LDO OLPLW LV DSSUR[LPDWHG ZLWKLWVSURMHFWLRQRQWKH zD[LVVLPLODUO\WRZKDWZDVGRQHLQ†/DVWO\ wRLV WKHEHDPZDLVWUDGLXVDQGWKHHOOLSWLFLW\RIWKHLUVHFWLRQVRQSODQHVRUWKRJRQDOWR WKHzD[LVZDVQHJOHFWHG%\UHDUUDQJLQJWKHWHUPV(T  EHFRPHV

E x y  z t

eiEz iZt

 u³

S S

EM

 r   z WDQ -     e wR iz z5  S   i z z5



 

ª  z WDQ - º «iD  » x FRV M y VLQ M w iz z5 »¼ « R e¬ dM

ZKHUH r LV WKH GLVWDQFH IURP WKH zD[LV RI WKH SRLQWV ZLWK WUDQVYHUVH FRRUGLQDWH xy %\DVVXPLQJZLWKLQWKHSDUD[LDOOLPLWWKDW EM  ERFRQVWDQWDQG - |WDQ- VLPLODUO\WRWKHFDVHRI(T  ZHJHW 





E x y  z t

ER eiEz iZt

e

r   z- wR iz z5

  i z z5

§ · iz-r - R ¨¨ Dr   ¸    wR   i z z5 ¸¹ ©

ZKHUHWKH%HVVHO¶VIXQFWLRQ-RQRZKDVDFRPSOH[DUJXPHQW,QSDUWLFXODURQWKH SODQHz WKHILHOGLVVLPSO\GHVFULEHGE\WKHSURGXFWRID*DXVVLDQIRUWKH%HV VHO¶VIXQFWLRQ 

E x y  z

 t

ER e iZt e  r

 w  R J

R

Dr 

 

7KHEHKDYLRURID%HVVHO*DXVVEHDPLVGHWHUPLQHGE\WZRFRPSHWLQJHIIHFWV 2QWKHRQHKDQGHDFKSDLURI*DXVVLDQFRPSRQHQWVWHQGVWRVHSDUDWHSURJUHVVLYH O\ZLWKLQFUHDVLQJ_z_2QWKHRWKHUKDQGWKHVSRWUDGLXVRIHDFK*DXVVLDQFRPSR QHQW LQFUHDVHV ZLWK UHVSHFW WR _z_ WKXV WHQGLQJ WR SURORQJ RU HYHQ PDLQWDLQ WKH



3URSDJDWLRQRIODVHUEHDPLQOLQHDUPHGLD

1.0

z = 0 cm 0.8 0.6 0.4 0.2 0.0 0.6

z = 25 cm

0.4

I (arb. un.) 0.2 0.0

z = 50 cm

0.2 0.0

z = 75 cm

0.01 0.00

z = 100 cm

0.01 0.00

z = 125 cm

0.01 0.00

-4

-3

-2

-1

0

x (mm)



1

2

3

4



)LJ%HVVHO*DXVVEHDPLQWHQVLW\SURILOHVDWGLIIHUHQWGLVWDQFHVzLQWKHFDVHRIO  PP- PUDGwR PP7KHVHYDOXHVFRUUHVSRQGWRDGLYHUJHQFH-*RIWKH*DXVVLDQ FRPSRQHQWVHTXDOWR S PUDGDUDWLR--* SDQGD%HVVHO¶VGLVWDQFHD FP7KH WKUHHILJXUHVDWWKHERWWRPKDYHWKHYHUWLFDOVFDOHPDJQLILHGE\WLPHVFRPSDUHGWRWKH RWKHU7KHJUD\FXUYHVUHSUHVHQWWKHWUHQGRID*DXVVLDQEHDPLQLWLDOO\RIWKHVDPHZLGWKDV WKHFHQWUDOSHDNRIWKH%HVVHO*DXVVEHDP

RYHUODSSLQJRIWKHYDULRXVEHDPV,QWKHIDUILHOG]RQHWKLVRYHUODSLVGHWHUPLQHG E\WKHUDWLREHWZHHQWKHDQJXODUDSHUWXUHRIWKHFRQHRIWKHZDYHYHFWRUVDQGWKH DQJOHRIGLYHUJHQFH-*RIHDFK*DXVVLDQEHDP

%HVVHO*DXVVEHDPV





-*

SwR § D · DwR DUFVLQ ¨ ¸ #  O  ©k¹

 

ZKHUH WKH DSSUR[LPDWLRQ LV YDOLG IRU VPDOO DQJOHV - :KHQ WKLV UDWLR LV JUHDWHU WKDQLQWKHIDUILHOGUHJLRQWKHVKDSHRIWKH%HVVHO*DXVVEHDPLVWKDWRIDULQJ VXUURXQGLQJDGDUNUHJLRQDURXQGWKHzD[LV ,QVWHDGLQWKHQHLJKERUKRRGRIz WKHYDULRXVFRPSRQHQWVRYHUODSLQWHUIHU LQJ ZLWK HDFK RWKHU DQG JLYLQJ ULVH WR D FHQWUDO SHDN WKDW UHGXFHV LQ KHLJKW  E\ PDLQWDLQLQJWKHVDPHZLGWKDVORQJDVWKHWDLOVRIWKHYDULRXV*DXVVLDQFRPSR QHQWVRYHUODSDSSUHFLDEO\ )LJ 7KLVWDNHVSODFHLQGLFDWLYHO\XSWRWKH%HV VHO¶V GLVWDQFH D GHILQHG LQ VXFK D ZD\ WKDW WKH FHQWHU RI HDFK FRPSRQHQW KDV PRYHGDZD\IURPWKHzD[LVRIDGLVWDQFHHTXDOWRwoZKHUHE\ 

D

wR  -

 

7KLVGLVWDQFHLVDFRQVHUYDWLYHHVWLPDWHRIWKHRYHUODSGLVWDQFHDVWKHUDGLXVwRI WKHEHDPFRPSRQHQWVDOVRJURZVZLWKz :KHQWKHUDWLRRI(T  LVOHVVWKDQDOO*DXVVLDQFRPSRQHQWVFRQWLQXH WRRYHUODSDQGLQWHUIHUHHYHQLQWKHIDUILHOG]RQH+RZHYHULQWKLVFDVHWKH%HV VHO*DXVVEHDPGRHVQRWDSSURDFKWKHFDVHRIDGLIIUDFWLRQIUHHEHDPLQGHHGRQ WKH SODQH z   WKH UDGLXV DW KDOIKHLJKW RI WKH IXQFWLRQ -R Dr  LV DERXW D ZKLOHIRU(T  wRD7KLVPHDQVWKDWWKHVXEVHTXHQWRVFLOODWLRQVRIWKH IXQFWLRQ-RDUHVWURQJO\GDPSHGVRWKDWWKHEHDPSURILOHLVYHU\VLPLODUWRWKDWRI D*DXVVLDQEHDPDQGDVVXFKZLGHQVSURJUHVVLYHO\ZLWKLQFUHDVLQJz ,QFRQFOXVLRQD]HURRUGHU%HVVHO*DXVVEHDPUHSUHVHQWVDSDUD[LDOZDYHJHQ HUDOL]DWLRQZKLFKKDVWKHH[WUHPHOLPLWVRID]HURRUGHU%HVVHO¶VEHDPIRUwRo fDQGD*DXVVLDQ7(0EHDPIRU-o

6.8.2 Generalized Bessel-Gauss beams 6LPLODUO\ WR WKH %HVVHO¶V ZDYHVKLJKHURUGHU %HVVHO*DXVV EHDPV FDQ EH RE WDLQHGE\DGGLQJWRWKHYDULRXV*DXVVLDQFRPSRQHQWVDSHULRGLFSKDVHPRGXODWLRQ RIWKHW\SHH[S im M DVWKHD]LPXWKDODQJOHMFKDQJHVWKHVHDUHFDOOHGBesselGauss beams of order m $Q DGGLWLRQDO LQFUHDVH LQ FRPSOH[LW\ LV LQWURGXFLQJ D UDGLDO GLVWDQFH EHWZHHQ WKH FHQWHU RI WKH *DXVVLDQ EHDPV DQG WKH zD[LV DW WKH SODQH RI WKH ZDLVWV )LJ   7KLV W\SH RI RYHUODS SURGXFHV ZKDW DUH FDOOHG generalized Bessel-Gauss beamsZKLFKFRQVWLWXWHDFODVVRIEHDPVWKDWFRQWDLQVWKRVHSUHYLRXVO\GLVFXVVHG DVVSHFLDOFDVHV>%DJLQLHWDO@$VVXPLQJDJDLQZLWKLQWKHSDUD[LDOOLPLWWKDW EM ER ei m MDQG-|WDQ-WKHUHVXOWLQJILHOGLVQRZ



3URSDJDWLRQRIODVHUEHDPLQOLQHDUPHGLD

y

j

ro

x

J z

 )LJ5HSUHVHQWDWLRQRIRQHRIWKH*DXVVLDQFRPSRQHQWVRIDJHQHUDOL]HG%HVVHO*DXVV EHDP-LVWKHKDOIDSHUWXUHDQJOHRIWKHFRQHRIWKHD[HVRIWKHLQGLYLGXDOFRPSRQHQWVDQG MLVWKHLUD]LPXWKDODQJOHODVWO\URLVWKHUDGLXVWKDWWKHVHD[HVWUDFHRQWKHZDLVWSODQH ZKLFKLVFRPPRQWRDOOFRPSRQHQWVDQGRQZKLFKWKHRULJLQRIWKHFRRUGLQDWHV LVIL[HG

E x y  z t



ei Ez iZt

S   ER eiD x FRV M y VLQ M imM ³ S   i z z5 S 

ª x  UR  z- FRV M ¼º  ¬ª y  UR  z- VLQ M ¼º ¬ wR iz z 5 ue





 

dM

IURPZKLFKUHDUUDQJLQJWKHWHUPVDQGH[SUHVVLQJWKHILHOGDFFRUGLQJWRUDGLDOFR RUGLQDWHVrDQGD]LPXWKDOFRRUGLQDWHMZHKDYHWKDW E r M z t





r   U R  z-



 e wR iz z5 ER ei Ez iZt   i z z5

imM

im-

ª º  U R  z- r» m « Dr  i  wR   i z z5 ¼ ¬



 

6SHFLILFDOO\IRUUR (T  GHVFULEHVD%HVVHO*DXVVEHDPRIRUGHUm %XWLQWKHFDVHRIWDNLQJWKDW- DQGWKHUHIRUHD WKHDUJXPHQWRIWKH IXQFWLRQ-mLVSXUHO\LPDJLQDU\DQGWKH%HVVHO¶VIXQFWLRQFDQEHUHSODFHGZLWKWKH PRGLILHG%HVVHO¶VIXQFWLRQ7KHUHVXOWLQJEHDPLVWKHQFDOOHGDmodified BesselGauss beam6LQFHWKH*DXVVLDQFRPSRQHQWVDUHQRZSDUDOOHOWRHDFKRWKHULWFDQ EH JHQHUDWHG VLPSO\ E\ VSDWLDOO\ ILOWHULQJ D *DXVVLDQ EHDP RYHU LWV ZDLVW SODQH 7KLVFDQEHDFKLHYHGE\PHDQVRIDODPLQDZKRVHWUDQVPLVVLRQLVYDULHGUDGLDOO\ LQDPSOLWXGHDQGLQWKHFDVHRImzHYHQLQSKDVHLQWKHD]LPXWKDOGLUHFWLRQ )RUm ZLWKLQFUHDVLQJSDUDPHWHUURWKHPRGLILHGEHDPWDNHVDIRUPWKDWLV WUDQVIRUPHGIURP*DXVVLDQWRIODWWHQHGDQGILQDOO\DQQXODU

%HVVHO*DXVVEHDPV



6.8.3 Propagation of a Bessel-Gauss beam through an axial optical system 5HVXPLQJWKHDQDO\VLVRI6DQWDUVLHUR  ZHFDQQRZHDVLO\GHWHUPLQHWKH SURSDJDWLRQRIDJHQHUDOL]HG%HVVHO*DXVVEHDPWKURXJKDQD[LDORSWLFDOV\VWHP ,WLVHQRXJKWRFRQVLGHUWKHHIIHFWVWKDWWKHV\VWHPKDVRQWKHLQGLYLGXDO*DXVVLDQ EHDPFRPSRQHQWVZKLFKLQWKLVFDVHKDYHDOOWKHLUSURSDJDWLRQD[LVFRUUHVSRQG LQJWRDPHULGLDQUD\)URP(T  ZHWKHUHIRUHJHW E x  y  z  t



ER

iz5 n Aq  B n

n k r U iZt inkRz z ikR LR i  R CUU  B--    wR iz z5  ue S

 u ³ dM S S



 

ª º U imM «inkR-  » x FRV M y VLQ M    w  iz z R  5  ¼» ¬«  e

ZKHUHWKHILHOGLVSURSDJDWHGEHWZHHQWKHLQLWLDOSODQH 1DQGWKHILQDOSODQH 2DW ZKLFKWKH$%&'PDWUL[RIWKHV\VWHPLVGHWHUPLQHG7KHVHSODQHVGRQRWQHFHV VDULO\FRLQFLGHZLWKWKHZDLVWSODQHVRIWKH*DXVVLDQFRPSRQHQWVVRzjZLWKj   DUH WKH GLVWDQFHV LQ WKH REMHFW VSDFH DQG LQ WKH LPDJH VSDFH IURP WKH FRUUH VSRQGLQJZDLVWSODQHVz5jDUHWKH5D\OHLJKGLVWDQFHVDQGwRjDUHWKHZDLVWUDGLXV %HVLGHVUjDUHWKHUDGLLRIWKHFLUFOHVWKDWWKHD[HVRIWKHFRPSRQHQWEHDPVWUDFH RQSODQHV 1DQG 2ZKLOH-jDUHWKHKDOIDSHUWXUHDQJOHV/DVWO\njDUHWKHUHIUDF WLYHLQGLFHVRIWKHLQLWLDODQGILQDOVSDFHVZKLOHkRLVWKHZDYHYHFWRUPRGXOHLQ WKHYDFXXPDQGLRLVWKHRSWLFDOSDWKEHWZHHQSODQHV1DQG 27KHWUDQVIRUPDWLRQ EHWZHHQWKHLQLWLDODQGILQDOYDULDEOHVLVJLYHQE\WKHUHODWLRQV  IRUUjDQG -jE\(TV  IRUWKHYDULDEOHVqjzjDQGz5jDQGE\(T  IRUwRj %\VROYLQJWKHLQWHJUDORI(T  DQGH[SUHVVLQJWKHILHOGDFFRUGLQJWRWKH UDGLDODQGD]LPXWKDOFRRUGLQDWHVrDQGMFRUUHVSRQGLQJWRWKHSDLUxyZHJHW E r M z  t n k



ER

iz5 n iZt inkRz z ikR LR i  R CUU  B--  e Aq  B n

imM 

ue

r  U  wR iz z5  i m -

m



 

ª º U r « n kR- r  i »  wR   i z z5 ¼ ¬

&RQVLGHULQSDUWLFXODUWKHSURSDJDWLRQRQO\IRUDGLVWDQFHzDORQJWKHEHDPD[ LVWKHUHIRUHWKH$%&'PDWUL[LVVLPSO\



3URSDJDWLRQRIODVHUEHDPLQOLQHDUPHGLD

§ A B · § z· ¨ C D ¸ ¨   ¸  © ¹ © ¹



:H DOVR WDNH WKH VWDUWLQJ SODQH FRLQFLGHQW ZLWK WKH ZDLVW SODQH RI WKH LQFLGHQW EHDP,QWKLVFDVH q

i SwR O  z

iz5

z z5

z5  wR

wR  -

- U

U  - z 

DQGZLWKWKHVHVXEVWLWXWLRQVZHDJDLQILQG(T   6XSSRVHLQVWHDGWKDWZHFDOFXODWHWKHWUDQVIRUPDWLRQRIWKHJHQHUDOL]HG%HV VHO*DXVV EHDP E\ WDNLQJ ERWK WKH LQLWLDO DQG ILQDO SODQHV FRLQFLGLQJ ZLWK WKH ZDLVWSODQHVLQWKHLUUHVSHFWLYHREMHFWDQGLPDJHVSDFHV/HW$ % & ' EHWKHPD WUL[RIWKHV\VWHPEHWZHHQWKHVHSODQHV § Ac Bc · ¨ C c Dc ¸ © ¹

§  z · § A B · §  z · § A  C z ¨  ¸¨C D¸¨  ¸ ¨ C © ¹© ¹© ¹ ©



Az  B  C z z  D z · ¸  C z  D ¹



 

7KHQ(T  EHFRPHV



iZt 

E r M t

ER e

r  UR imM wR im-

m

ª UR r º « n kR- r  i »    wR ¼ ¬

ZKHUH n k



ER

ER

iz5 n ikR LRc i  R C cURUR  Bc--  e  iAcz5  Bc n

 

ZLWK Loc Lo  nz  n z (T  KDVWKHVDPHVKDSHDV(T  IRUz  DQG WKXV WKH LQLWLDO EHDP LV WUDQVIRUPHG LQWR DQRWKHU JHQHUDOL]HG %HVVHO*DXVV EHDP7DNLQJWKHRULJLQRIzFRRUGLQDWHRQWKHZDLVWSODQHLQWKHREMHFWVSDFHWKH QHZEHDPLVWKHUHIRUHGHVFULEHGE\ E r M z t 

 ER

e

i E z iZt 

r   UR - z imM wR i z z5 

  iz z5

im-

ª  UR  - z r º » m « D r  i wR   iz z5 ¼ ¬

  

5HVRQDQWFDYLWLHV



ZKHUHZHWRRNWKDW D n kR-  $SDUWLFXODUO\LQWHUHVWLQJFDVHLVZKHQWKHwaist plane of the incident beam coincides with the object focal planeRIDQRSWLFDOV\VWHP&RQVHTXHQWO\WKHHPHUJ LQJEHDP KDV LWV RZQ ZDLVWSODQHRQ WKH LPDJHIRFDOSODQH DQG WKH PDWUL[ WR EH DSSOLHGKDVWKHVKDSHRI(T  ZKHUHE\A D B foDQGC fi $SSO\LQJWKHWUDQVIRUPDWLRQV  (T  EHFRPHV E r M z t

ER

iz5 fo

n n 



ueikR LR inkR-UR

e

i E z iZt 

r   fo- URz fi imM wR i z z 5 

  iz z5



 

ª n k U r  fo -  URz fi r º ui m - m «   R R  i » fi wR   iz z5 ¼ ¬

ZKHUH LR LV WKH RSWLFDO SDWK RQ WKH D[LV EHWZHHQ WKH WZR IRFDO SODQHV)URP WKH FRPSDULVRQEHWZHHQWKLVHTXDWLRQDQG(T  LWFDQEHQRWHGWKDWWKHYDULD EOHVUDQG-KDYHH[FKDQJHGWKHLUUROH7KLVFDQEHHDVLO\XQGHUVWRRGE\FRQVLGHU LQJWKHEHKDYLRURIWKHSULQFLSDOUD\VRIWKH*DXVVLDQEHDPFRPSRQHQWV,QSDUWLF XODU LIRQWKHIRFDOSODQHREMHFWWKHLQFLGHQWEHDPLVDPRGLILHG%HVVHO*DXVV EHDPWKDWLVZLWK- RQWKHLPDJHIRFDOSODQHLWEHFRPHVDQRUGLQDU\ %HVVHO*DXVVEHDPZLWKU 

5HVRQDQWFDYLWLHV 7KH*DXVVLDQEHDPVRZHWKHLUSRSXODULW\WRWKHIDFWWKDWWKH\DUHDJRRGUHSUH VHQWDWLRQ RI WKH ODVHU EHDPV JHQHUDWHG ZLWKLQ D FODVV RI UHVRQDQW FDYLWLHV 7KH\ W\SLFDOO\ IRUP WKH VROXWLRQ IRU WKH HOHFWURPDJQHWLF ILHOG WKDW LV JHQHUDWHG ZLWKLQ VXFKFDYLWLHVFRQWDLQLQJDQDFWLYHPHGLXPLQWKHSUHVHQFHRIfeedbackWKDWWKHRS WLFDOV\VWHPEULQJVEDFNRQWKHDFWLYHPHGLXP7KLVIHHGEDFNLVJHQHUDOO\DVVRFL DWHG ZLWK WKH ERXQGDU\ FRQGLWLRQV JLYHQ E\ WKH FDYLW\ PLUURUV *DXVVLDQ EHDPV W\SLFDOO\ DOVR JLYH D JRRG UHSUHVHQWDWLRQ RI WKH ILHOG ZLWKLQ VXFK SDVVLYH RSWLFDO FDYLWLHV LH ZLWKRXW DQ LQWHUQDO DFWLYH PHGLXP LQMHFWHG H[WHUQDOO\ E\ D ODVHU EHDP 7KH VWXG\ RI ODVHU FDYLWLHV LV YHU\ FRPSOH[ DQG , ZLOO RQO\ JLYH \RX VRPH YDJXHQRWLRQVKHUH)RUIXUWKHUGHWDLOVVHHWKHPRQXPHQWDOERRNVRI$QWKRQ\( 6LHJPDQLasers>6LHJPDQ@DQG$QDQ¶HY@6HHDOVR6ROLPHQRHWDO  , &KDS 9,,



3URSDJDWLRQRIODVHUEHDPLQOLQHDUPHGLD

,QWKHRSWLFDOUHJLRQRIWKHVSHFWUXPWKHUHVRQDQWFDYLWLHVZLWKWKHH[FHSWLRQ RIWKRVHRIWKHVHPLFRQGXFWRUODVHUVWKDWUHTXLUHDVHSDUDWHGLVFXVVLRQGLIIHUFRQ VLGHUDEO\IURPWKHUHVRQDQWFDYLWLHVWKDWDUHXVHGLQWKHPLFURZDYHUHJLRQ7KHVH ODWWHUDUHclosed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

'N

S  Q V 'Q  c

 

ZKHUHVLVLWVYROXPHDQG c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transverse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stableFRXOGVXVWDLQODVHUDFWLRQ ,Q$66LHJPDQZDVWKHILUVWWRUHFRJQL]HWKHEHQHILWVRIWKRVHUHIHUUHGWR DVunstableUHVRQDWRUVXVLQJDVLPSOHJHRPHWULFDQDO\VLV>6LHJPDQ@)LQDOO\ /$:HLQVWHLQFRPSOHWHGWKHWKHRU\E\FROOHFWLQJLWLQRQHRIKLVLPSRUWDQWPRQ RJUDSKV>:HLQVWHLQ@

5HVRQDQWFDYLWLHV



6.9.1 Resonant modes $V$QDQ¶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¶LQWHJUDO,QERWKFDVHVLWLVQHFHVVDU\WR FRQVLGHUWKHHIIHFWRIWKHILQLWHVL]HRIWKHPLUURUVDQGRWKHUHOHPHQWVWKDWPDNHXS WKH RSWLFDO V\VWHP 0RUHRYHU DV \HW QRWHG E\ $QDQ¶HY   LW VKRXOG EH UH PHPEHUHGWKDWWKHVHLQWHJUDOVUHIHUWRVWULFWO\PRQRFKURPDWLFZDYHVZLWKDQJXODU IUHTXHQF\ZDQGZKHQWKHWHPSRUDOSDUWH[S iZt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continuous wave LQZKLFKWKHDFWLYHPHGLXPJDLQFRPSHQVDWHVWKH ORVVHVDQGWKHIUHTXHQF\RIWKHPRGHVFDQEHFRQVLGHUHGUHDO $SSO\LQJ WKH $%&' PDWUL[ FRUUHVSRQGLQJ WR D FORVHG SDWK LQ D FDYLW\ WR WKH SURSDJDWLRQRID*DXVVLDQEHDPIURP(T  ZHFDQILQGWKHFRPSOH[FXU YDWXUHRIWKHILHOGRQWKHVWDUWLQJDUULYDOSODQHLPSRVLQJHTXDOLW\EHWZHHQ QDQG Q 7KH VLPSOHVW FDVH FRQVLVWV RI D QRQDVWLJPDWLF OLQHDU FDYLW\ IRUPHG E\ WZR VSKHULFDOPLUURUVZLWKUDGLXVRIFXUYDWXUHRDQGRVHSDUDWHGE\DGLVWDQFHL)RU ERWKRLVpositiveLIWKHPLUURULVFRQFDYH)RUWKHV\PPHWU\RIWKHUHVXOWZHFDQ LPDJLQH UHSODFLQJ HDFK FXUYHG PLUURU ZLWK D OHQV SODFHG QH[W WR D SODQH PLUURU )LJ %\FKRRVLQJWKHILUVWRIWKHVHILFWLWLRXVSODQHVDVGHSDUWXUHDQGDUULYDO WKHV\VWHPLVSDUD[LDOO\UHSUHVHQWHGE\WKH[$%&'PDWUL[



3URSDJDWLRQRIODVHUEHDPLQOLQHDUPHGLD

)LJ D   VFKHPHRIDOLQHDUUHVRQDQW FDYLW\ ZLWK WZR VSKHULFDO PLU URUV E   UHSUHVHQWDWLRQ RI WKH VDPH FDYLW\DVFRQVLVWLQJRIWZRSODQH PLUURUV DQG WZR OHQVHV 7KH IR FDOGLVWDQFHRIHDFKOHQVLVHTXDO WRWKHUDGLXVRIFXUYDWXUHRIWKH FRUUHVSRQGLQJPLUURU

(a)

L R2

R1 (b)

2

1

§A B· ¨C D¸ © ¹

·§ L· §  §  ¨  R  ¸ ¨ ¸¨  © ¹ ©   ¹ ©   R

·§   ¹