Physical Optics. Lecture 1: Wave optics Herbert Gross.

Transcription

1 Phsical Optics Lctu : Wav optics 8-4- Hbt Goss

2 Phsical Optics: Contnt No Dat Subjct Rf Dtaild Contnt.4. Wav optics G Compl filds wav quation -vctos intfnc light popagation intfomt 8.4. Diffaction G Slit gating diffaction intgal diffaction in optical sstms point spad function abations Foui optics G Plan wav pansion solution imag fomation tansf function phas imaging 4.5. Qualit citia and Raligh and Machal citia Sthl atio cohnc ffcts two-point G solution solution citia contast aial solution CTF Photon optics K ng momntum tim-ng unctaint photon statistics fluoscnc Jablonsi diagam liftim quantum ild FRT Cohnc K Tmpoal and spatial cohnc Young stup popagation of cohnc spcl OCT-pincipl Polaiation G Intoduction Jons fomalism Fsnl fomulas bifingnc componnts Las K Atomic tansitions pincipl sonatos mods las tps Q-switch pulss pow Nonlina optics K Basics of nonlina optics optical suscptibilit nd and 3d od ffcts CARS micoscop photon imaging 3.6. PSF ngining G Apodiation supsolution tndd dpth of focus paticl tapping confocal PSF.6. Scatting L Intoduction sufac scatting in sstms volum scatting modls calculation schms tissu modls Mi Scatting 7.6. Gaussian bams G Basic dsciption popagation though optical sstms abations Gnalid bams G Lagu-Gaussian bams phas singulaitis Bssl bams Ai bams applications in supsolution micoscop 4.7. Miscllanous G Coatings diffactiv optics fibs K = Kmp G = Goss L = Lu

3 3 Contnts Compl filds -vctos and plan wavs Wav quation Light Popagation Intfnc Intfomt

4 4 Mawll quations Quantitis: lctical fild t tim H magntic fild dilcticit j cunt m pmabilit chag dnsit c spd of light Mawll quations: cunt induction souc f magntic chag continuit of chags ot H j ot div H div div j t t t H Basic quations fo lctomagntism and optics

5 Altnativ fomulation of th Mawll quations in a mdium: M magnitiation B magntic induction D lctic displacmnt Th lctomagntic filds fom a tansvs wav : wav vcto indicats th diction of popagation Wavlngth (scala) Mawll quations H B i D B j i D H J J M H B P D o o n n c o c c 5

6 Spd of light in mdium in vacuum constants ng dnsit of a fild Local flow of ng: Ponting vcto Intnsit 6 ng Dnsit and Ponting vcto n c n c H u c H S c I o m c s Am Vs Vm As

7 7 Filds in Dilctic Mdia Spd of light in mdium Wavlngth in mdium c c n n Wav vcto in mdium n n Rfaction Rf: Salh / Tich

8 8 lctomagntic Filds Dsciption of lctomagntic filds: - vctoial natu of fild stngth - dcomposition of th fild into componnts A cost A cos( t ) Popagation plan wav: - fild vcto otats - pojction componnts a oscillating sinusoidal

9 9 Basic Foms of Polaisation. Lina componnts in phas. cicula phas diffnc of 9 btwn componnts 3. lliptical abita but constant phas diffnc

10 Basic Wav Optics Scala wav A( ) a( ) i phas function ( ) ( ) A Phas sufac: - fid phas fo on tim const - phas sufac ppndicula to unit vto Rf: W. Ostn

11 Hamonic Wav Wav hamonic in spac and tim R{F(t)} R{F(t)} T t U F t Rpsntation with compl ponntial function Im{F(t)} t' R{F(t)} R{U()} t t F it t R cos t i R cos U Im{F(t)} t' Coupling of tim and spac popagation/dvlopmnt ( t) with c i( t )

12 Plan Wav Condition of a plan wav const. = const. U ~ cos n U ~ cos n sin cos

13 3 Plan and Sphical Wavs Plan wav wav vcto ( t) i A ( t ) Sphical wav ( t) A i ( t ) Rf.: B. Döband

14 4 Sphical Wav Fild of a sphical wav ( t) i( t ) adius in sphical coodinats

15 5 Plan Wav pansion Basis:. supposition of solutions. spaabilit of coodinats Plan wav i t t Spctal pansion A(): plan wav spctum Dispsion lation: wald sph Tansvs pansion A ( ) 3 n co Main ida: A ( ) d d - Fild dcomposition in plan wavs - Switch into Foui spac of spatial fquncis - Popagation of plan wav as simpl phas facto - Bac tansfom into spatial domain - Supposition of plan wav with modifid phas T i d i A( ) d d i

16 6 Plan Wav pansion Popagation of plan wavs: pu phas facto. act sph. Fsnl quadatic appoimation vanscnt wavs componnts dampd in impotant onl fo na fild stups Popagation algoithm --sctions a coupld Paaial appoimation --sction dcoupld i A A A i i A i i v v Av v A i fo ( ' ') Fˆ Fˆ ( ' ') Fˆ i Fˆ i / v ( ) i v v v Fˆ Fˆ i ( ) ( ) vanscnt

17 7 Popagation b Plan / Sphical Wavs pansion fild in simpl-to-popagat wavs. Sphical wavs. Plan wavs Hugns pincipl i ' ( ') ( ) d ' ' spctal psntation Foui appoach ( ') Fˆ i Fˆ ( ) ' i i () ()

18 8 Popagation of Plan Wavs Phas of a plan wav i i ncos i i n h ; Th spctal componnt is simpl multiplid b a phas facto in duing popagation i ; h th function h is th phas function Bac-tansfoming this into th spatial domain: Popagation cosponds to a convolution with th impuls spons function ; H H i h ; d ; d Fsnl appoimation fo popagation: U P i... i i ; U ; d d i

19 9 Angula Spctum Plan wav i ( ) A( ) Wav numb n n/ Spatial fqunc: -scaling of n ( ) A i Foui tansfom to gt th plan wav spctum i d d d

20 wald Sph Assuming an objct as gating with piod L obj L Scatting of a wav at th objct with - consvation of ng in out - consvation of momntum Th outgoing -vcto must b on a sph: wald's sph fo possibl scattd wav vctos in obj out gating wald sph obj out out in obj in

21 Rfaction in -Spac Plan wav factd at a plan intfac i t o -vcto should b constant in lngth n n n n Total intnal flction: possibl if n > n. cas : factd a. cas : bginning total intnal flction 3. cas : total intnal flction n n 3...

22 vanscnt Wavs Usual cas in optics: - factiv ind n al - no damping o loss duing light popagation - spatial fqunc al n Gnal cas: n and compl - dampd o vanscnt wav - absoption of th fild along th popagation path - absoption constant Damping along : - compl factiv ind - absoption constant - Lambt-B law n~ n i I( ) I n 4 n i A I / I n / =.368 /

23 3 Mtals Compl factiv ind Altnativ fomulation: attnuation Rlation with conductivit n~ n i n i n~ n i 4 n ~ i i Tpical data of som mtals matial n in [m] gold silv aluminum tungstn platinum silicon

24 4 3D Tansf Function - Missing Con Ralistic cas: finit numical aptu illumination i Blu con: possibl incoming wav diction du to illumination con scattd wav 3D cohnt tansf function: limitd gn aa that fulfills all conditions s obs tansf function objct Missing con: ctain ang of spatial aial spatial fquncis can not b sn in th imag missing con tansf function objct

25 5 McCutchn Fomula and Aial Rsolution Imaging of a plan wav at a volum objct : minimum valu solution D: maimum intval Unctaint lation: D = P( ) tansvs pupil Radius of th wald sph gnalid 3D pupil: d aa Tansvs solution du to Abb Dv R sin n / NA / n NA n R cap light con D D aial pupil P( ) Aial solution: - hight of th cap of th con - McCutchn fomula Dv n R Rcos n / n NA n NA sin n/ wald sph

26 6 3D Tansf Function Imaging as 3D scatting phnomn Onl spcial spatial fquncis a allowd du to ng consvation and momntum psvation Gn cicl: suppotd spatial fquncis of th tansmittd wav vcto obj = s - i o-ma n/ i s obj i bacwad fowad wald sph

27 Mawll quation fo th fild vctoial Th spatial inhomognitis coupls th fild componnts Homognous without chags non-conductiv spaation of vcto componnts scala Tim indpndnc: Wav quation of Hlmholt In coodinat psntation Wav numb in mdium factiv ind n Fast -oscillation spaatd Slowl vaing nvlop appoimation ~ ~ t t i t ) ( ) ( D o o n n c i ) ( ) ( Wav quation ln o o t ) ( c n i n n n o ( ) 7

28 8 Wav quation Paaial appoimation paaial wav quation Conditions fo scala appoimation:. Dcoupling of fild componnts wavlngth small in compaison to f diamt. No lag angls du to gomt Computation of fild in lag distancs Scala Hlmholt quation a D n( ) o i

29 9 Cohnt Numical Fild Popagation. Spctal psntation o othogonal pansions: Plan wav pansion Foui mthod pansion into gaussian bams in paaial sstms pansion into sphical wavs scatting gomtis pansion into ignmods fo bounda poblms.g. in fibs intgatd optics wavguids. Intgal psntations (fild on sufacs) Kichhoff diffaction intgal Raligh-Sommfld I+II Spcial appoimations: Fsnl- Collins- Faunhof intgal Db intgal Richads-Wolf vctoial psntation Bounda dg wav appoimation Mthod of stationa phas Saddlpoint mthod 3. Dict solution of th wav quation (volum solution) Finit diffnc mthod Finit lmnt mthod Radial basis functions Potntial mthods Mthod of lins

30 3 Solution Mthods of th Mawll quations Mawllquations act/ numical diffaction intgals st appoimation nd appoimation spctal mthods dict solutions of th PD finit lmnt mthod Kichhoffintgal asmptotic appoimation mod pansion finit lmnts bounda lmnt mthod Raligh- Sommfld st ind Fsnl appoimation plan wav spctum finit diffncs hbid mthod BM + FM Raligh- Sommfld nd ind Faunhof appoimation vcto potntials Db appoimation bounda dg wav

31 3 Wav Optical Cohnt Bam Popagation Mthod Calculation Poptis / Applications Kichhoff diffaction intgal Foui mthod of i i ( ) ( ' ) ' F AP ' ˆ i v ˆ plan wavs I I ( ') F F( ) Split stp bam popagation df Wav quation: divativs appoimatd n ( ) i no Ratacing Ra lin law of faction Cohnt mod pansion Incohnt mod pansion j j j s sin i ' j n n' sin i Fild pansion into mods n ) * ( ) c n n( cn ( ) n( ) d Intnsit pansion into cohnt mods n I( ) c n ( ) n Small Fsnl numbs Numical computation slow Lag Fsnl numbs Fast algoithm Na fild Compl bounda gomtis Nonlina ffcts Sstm componnts with a abations Matials with ind pofil Smooth intnsit pofils Fibs and wavguids Patial cohnt soucs

32 Tpical chang of th intnsit pofil Nomalid coodinats Diffaction intgal 3 Fsnl Diffaction gomtical focus f a stop fa on gomtical phas intnsit a f a v f a u ; ; / ) ( ) ( d v J f ia v u u i u a f i

33 33 Split Stp Popagatos Calculating th fild aft v small D in man singl stps Algoithms: finit diffncs of plan wavs Th fild is nown in th complt volum This tim-consuming tchniqu is ncssa fo non-homognous mdia s stating plan inhomognous mdium final plan man stps Split-stp-popagato:. Finit Diffncs. Plan wav pansion

34 Classical bam popagation: split-stp appoach Wav quation two opatos: intfacs (matial) and diffaction Foui tansfom Plan wav dcomposition ) ( ˆ F i D D F D D ˆ ) '( S D D D ˆ ' H D n n i i o ˆ ˆ ) ( i D ˆ n n i S o ) ( ˆ Split Stp Popagatos 34

35 35 Finit Diffnc Popagation Paaial Hlmholt wav quation Appoimation of divativs Implicit schm lina sstm of quations fo on stps D iono j j D a D D a c n n j m o i o n o D a D D D b n n i n D D j m o o o j n j n c b j n j n o a popagation n j a j n ( ) n j n o D j j Lag Fsnlnumb of on vol Fast valuation of tidiagonal sstm Phas nal flat no lag tilt Stabilit:. plicit schms usuall unstabl. Good stabl implicit schms ists n+ n j- j j+ latal

36 36 Bam Popagation ampls Focussd gaussian bam with sphical abation - asmmt inta- vs. ta focal - sign of sphical abation has influnc - boadning of bam waist diamt - diffaction fings c 9 = Gaussian bam popagation in a paabolic gadint ind mdium Rfocussing ffcts c 9 =.5 c 9 = -.5

37 37 Intnsit of Supposd Filds CCD is not abl to dtct phas du to tim avaging Masuing of intnsit with simpl dtcto Masud intnsit is tim avag Intfomt and hologaph: coding of phas infomation into masuabl intnsit vaiation I P t A o Contast / visibilit: nomalid diffnc of two diffnt intnsitis (tpicall maimum / minimum valus) Valu btwn... I I ma min C I ma I min Gnal cas of two-wav intfnc cos I I I C Rf: W. Ostn

38 38 Intfnc of Wavs Th main popt is th phas diffnc btwn two wavs Intfnc of two wavs j j I I I I I cos spcial cas of qual intnsits I I cos Maima of intnsit at vn phas diffncs Minima of intnsit at odd phas diffncs Intfnc of plan wavs Intfnc of sphical wavs:. outgoing wavs otational hpboloids j j N (N ). on outgoing on incoming wav otational llipsoids Rf: W. Ostn

39 39 Two Bam Intfnc Two bam intfnc of two wavs: - popagation in th sam diction - sam polaiation - phas diffnc small than aial lngth of cohnc Cohnt supposition of wavs I I I I I cos D Diffnc of phas / path diffnc Numb of fings location of sam phas Conast Ds D D Ds N K I I ma I min ma I min I I I I

40 4 Intfnc of Two Wavs Supposition of two plan wavs:. Intnsit. Phas diffnc Spacing of fings Intfnc of two sphical wavs Mo complicatd gomt ) ( cos ² ² ) ( A A A A I D D D ) ( ) ( ) ( ) ( Rf.: B. Döband sin n s

41 4 Two Bam Intfnc Intfnc of two plan wavs und diffnt dictions Fing distanc s s n

42 4 Intfomt Basic ida: - spaation of a wav into two bams (tst and fnc am) -v bam supasss diffnt paths - supposition and intfnc of both bams - analsis of th pattn Diffnt stups fo: - th bam splitting - th supposition - th fncing Diffnt path lngths Diffnc quivalnt of on fing D nt nt N tw t w n Masumnt of plats: Haiding fings of qual inclination Nwton fings of qual thicnss Rf: W. Ostn

43 43 Classification of Intfomts Division of amplitud: - Michlson intfomt - Mach-Zhnd intfomt - Sagnac intfomt - Nomasi intfomt - Talbot intfomt - Point diffaction intfomt Division of wavfont: - Young intfomt - Raligh intfomt Division of souc: - Llods mio - Fsnl bipism Rf: R. Kowaschi

44 44 Localiation of Fings Intfnc volum fo a plat incidnt light font sid flctd bac sid flctd volum of intfnc fings Intfnc volum fo a wdg font sid flctd incidnt light bac sid flctd volum of intfnc fings Rf: R. Kowaschi

45 45 Tst b Nwton Fings Rfnc sufac and tst sufac with nal th sam adii Intfnc in th ai gap Rfnc flat o cuvd possibl Cosponds to Fiau stup with contact Boad application in simpl optical shop tst Radii of fings to dtcto bamsplitt m mr illumination tst sufac path diffnc fnc sufac h: flat Rf: W. Ostn

46 46 Tsting with Twman-Gn Intfomt Shot common path snsibl stup fnc mio Two diffnt opation mods fo flction o tansmission collimatd las bam Alwas facto of btwn dtctd wav and componnt und tst bam splitt objctiv lns stop. mod: lns tstd in tansmission auilia mio fo autocollimation. mod: sufac tstd in flction auilia lns to gnat convgnt bam dtcto

47 47 Tsting with Fiau Intfomt Long common path quit insnsitiv stup Autocollimating Fiau sufac quit na to tst sufac shot cavit lngth Imaging of tst sufac on dtcto Stalight stop to bloc unwantd light Cuvd tst sufac: auilia objctiv lns (aplanatic doubl path) Highst accuac collimato auilia lns conv sufac und tst bam splitt light souc stop dtcto Fiau sufac

48 48 Intfogams of Pima Abations Sphical abation Astigmatism Coma Dfocussing in

49 49 valuation of Fings Intnsit of fings I( t) I( ) V ( ) cos W( ) ( t) R ( ) I ( t) S R I(t) V() W() (t) R s () I R (t) intnsit of fings contast of pattn phas function to b found fnc phas multiplicativ spcl nois additiv nois Tacing of fings: - tim consuming mthod intpolation inding of fings missing lins Foui mthod: -wavlt mthod - FFT Mthod - gadint mthod - fit of modal functions

50 5 Ral Masud Intfogam Poblms in al wold masumnt: dg ffcts Dfinition of bounda Ptubation b cohnt sta light Local sufac o a not wll dscibd b Zni pansion Convolution with motion blu Rf: B. Döband

51 5 Intfogam - Dfinition of Bounda Citical dfinition of th intfogam bounda and th Zni nomaliation adius in alit

52 5 Intfomt Gnal dsciption of th masumnt quantit: suppostion of spatiall modulatd signal and nois I( ) I ( ) T ( ) cos ( ) I ( ) I o : basic intnsit souc T: tansmission of th sstm including spcl : phas to b found I N : nois snso lctonics digitiation Signal pocssing SNR impovmnt: - filting - bacgound subtaction N oiginal signal filtd signal bacgound pocssd signal Rf: W. Ostn

53 53 Intfomt pfct intfogam ducd contast du to bacgound intnsit with spcl with nois Rf: W. Ostn