Metrology and Sensing

Transcription

1 Metology and Sensing Lectue 9: Speckle methods Hebet Goss Winte tem 017

2 Peliminay Schedule No Date Subject Detailed Content Intoduction Intoduction, optical measuements, shape measuements, eos, definition of the mete, sampling theoem Wave optics Basics, polaization, wave abeations, PSF, OTF Sensos Intoduction, basic popeties, CCDs, filteing, noise Finge pojection Moie pinciple, illumination coding, finge pojection, deflectomety Intefeomety I Intoduction, intefeence, types of intefeometes, miscellaneous Intefeomety II Examples, intefeogam intepetation, finge evaluation methods Wavefont sensos Hatmann-Shack WFS, Hatmann method, miscellaneous methods Geometical methods Tactile measuement, photogammety, tiangulation, time of flight, Scheimpflug setup Speckle methods Spatial and tempoal coheence, speckle, popeties, speckle metology Hologaphy Intoduction, hologaphic intefeomety, applications, miscellaneous Measuement of basic system popeties Bssic popeties, knife edge, slit scan, MTF measuement Phase etieval Intoduction, algoithms, pactical aspects, accuacy Metology of asphees and feefoms Asphees, null lens tests, CGH method, feefoms, metology of feefoms OCT Pinciple of OCT, tissue optics, Fouie domain OCT, miscellaneous Confocal sensos Pinciple, esolution and PSF, micoscopy, chomatical confocal method

3 3 Content Spatial coheence Tempoal coheence Speckle Speckle popeties Speckly metology

4 4 Coheence in Optics Statistical effect in wave optic: stat phase of adiating light souces ae only patially coupled Patial coheence: no igid coupling of the phase by supeposition of waves Constuctive intefeence petubed, contast educed Mathematical desciption: Aveaged coelation between the field E at diffeent locations and times: Coheence function G Reduction of coheence: 1. Sepaation of wave tains with finite spectal bandwidth Dl. Optical path diffeences fo extended souce aeas 3. Time aveaging by moved components Limiting cases: 1. Coheence: igid phase coupling, quasi monochomatic, wave tains of infinite length. Incoheence: no coelation, light souce with independent adiating point like molecules

5 5 Coheence in Phase Space coheent : evey point adiates in one diection line in phase space patial coheent :evey point has an individuell angle chaacteistic finite aea in the phase space incoheent : evey point adiates in all diections filled phase space u u u x x x

6 Coheence Function Coheence function: Coelation of statistical fields (complex) fo identical locations : intensity nomalized: degee of coheence In intefeometic setup, the amount of descibes the visibility V Distinction: 1. spatial coheence, path length diffeences and tansvese distance of points. time-elated coheence due to spectal bandwidth and finite length of wave tains t t E t E ), ( ), ( ),, ( * 1 1 z x x 1 x E(x ) E(x 1 ) Dx 1 ) ( ) ( ),, ( ),, ( ) ( I I ) ( ), ( I 6

7 7 Spatial Coheence Aea of coheence / tansvese coheence length: Non-vanishing coelation at two points with distance L c : Coelation of phase due to common aea on souce obsevation aea P 1 L c ( 1, ) Radiation out of a coheence cell of extension L c guaantees finite contast domain of coheence 1 P O The lateal coheence length changes duing popagation: stating plane eceiving plane spatial coheence gows with inceasing popagation distance 1 common aea

8 8 Spatial Cells of Coheence The numbe of speckles coesponds to the cells of coheence The numbe of cells is equivalent to the beam quality N speckle D d beam speckle M beam caustic cells speckle spots popagation 7 spot pe coss section in 1 dimension The cells of coheence ae the spatial egions in the beam coss section, which can intefee

9 9 Coheence Paamete Heuistic explanation of the coheence paamete in a system: 1. coheent: Psf of illumination lage in elation to the obsevation Lage s coheent illumination extended souce small stop of condense condense object objective lens Psf of obsevation inside psf of illumination. incoheent: Psf of illumination small in compaison to the obsevation Small s incoheent illumination extended souce lage stop of condense Psf of obsevation contains seveal illumination psfs s sin u sin u ill obs

10 10 Double Slit Expeiment of Young Fist ealization: change of slit distance D Second ealization: change of coheence paamete s of the souce Visibility / contast shinks with gowing slit spacind D sceen with slits light souce detecto z 1 V 1 D x z Dx 0 D

11 11 Double Slit Expeiment of Young Young intefeence expeiment: Ideal case: point souce with distance z 1, ideal small pinholes with distance D Intefeence on a sceen in the distance z, intensity Width of finges D lz D x I Dx ( x ) 4 I cos 0 l z x detecto souce D egion of intefeence z sceen with pinholes z

12 1 Double Slit Expeiment of Young Patial coheent illumination of a double pinhole/double slit Vaiation of the size of the souce by coheence paamete s Deceasing contast with gowing s Example: pinhole diamete D ph = D aiy / distance of pinholes D = 4D aiy s = 0 s = 0.15 s = 0.5 s = 0.30 s = 0.35 s = 0.40

13 13 Coheence Measuement with Young Expeiment Typical esult of a double-slit expeiment accoding to Young fo an Excime lase to chaacteize the coheence Decay of the contast with slit distance: diect detemination of the tansvese coheence length L c

14 ' ',0) ( 1 ),, ( ) '( ) ( d e I e z z z i z i l l l V vanishing contast 1 Van Cittet - Zenike - Theoem 1 ' ' ' ( ) a J a z a z l l 1 a z l Popagation of coheence function: in special case Van Cittet-Zenike theoem: Coheence function of an incoheent souce is the Fouie tansfom of the intensity pofile Example: cicula light souce with adius a Vanishing contast at adius 14

15 15 Tempoal Coheence Damping of light emission: wave tain of finite length Stating times of wave tains: statistical U(t) t c duation of a single tain

16 16 Tempoal Coheence Radiation of a single atom: Finite time Dt, wave tain of finite length, No peiodic function, epesentation as Fouie integal with spectal amplitude A() Example ectangula spectal distibution Finite time of duation: E( t) A( ) A( ) e i t sin Dt Dt d spectal boadening D, schematic dawing of spectal width I() D 1/ Dt

17 17 Axial Coheence Length Two plane waves with equal initial phase and diffeing wavelengths l 1, l Idential phase afte axial (longitudinal) coheence length l c c c c D l 1 l time t stating phase phase diffeence 180 in phase

18 18 Axial Coheence Length of Lightsouces Light souce l c Incandescent lamp Hg-high pessue lamp, line 546 nm Hg-low pessue lamp, line 546 nm K-isotope lamp, line at 606 nm HeNe - lase with L = 1 m - esonato HeNe - lase, longitudinal monomode stabilized.5 m 0 m 6 cm 70 cm 0 cm 5 m

19 19 Time-Related Coheence Function Time-elated coheence function: Auto coelation of the complex field E at a fixed spatial coodinate Fo puely statistical phase behaviou: = 0 Vanishing time inteval: intensity T 1 ( ) lim * ( ) ( ) * ( ) ( ) T T E t E t dt E t E t T * ( 0) E ( t) E( t) I T T Nomalized expession ( ) Usually: ( ) ( 0) deceases with gowing symmetically Width of the distibution: coheence time c * E ( t) E( t ) E( t) ( ) c

20 0 Time-Related Coheence Function Intensity of a multispectal field Integation of the powe spectal density S() I 0 S( ) d The tempoal coheence function and the powe spectal density ae Fouie-invese: Theoem of Wiene-Chintchin S( ) ( ) e i d The coesponding widths in time and spectum ae elated by an uncetainty elation 1 c D The Paceval theoem defines the coheence time as aveage of the nomalized coheence function c ( ) d The axial coheence length is the space equivalent of the coheence time l c c c

21 1 Michelson-Intefeomete Michelson intefeomete: intefeence of finite size wave tains Contast of intefeence patten allows to measue the axial coheence length/time second mio moving z signal beam wave tains with finite length elative moving z efeence beam ovelap I(z) fist mio fom souce beam splitte l c eceive

22 Intefeence Contast Supeposition of plane wave with initial phase Intensity: I m I m nm I n I m cos n m Radiation field with coheence function : Reduced contast fo patial coheence Imax Imin ( 1,, ) C I I I( ) I( ) max min Measuement of coheence in Michelson intefeomete: phase diffeence due to path length diffeence in the two ams (Fouie spectoscopy) 1 I I( ) I1( ) I( ) 1,,0 filteed signal measued signal D Dk z 4 Dl z l measued position z axial length of coheence

23 3 Young Expeiment with Boad Band Souce Deceased contast due to finite spectal bandwidth Realization with movable tiple mio efeence mio movable tiple mio contast 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0, 0,1 contast cuve x lase I(x,y) beam splitte scan x intefeogam x detecto y

24 4 Axial Coheence Contast of a 193 nm excime lase fo axial shea Red line: Fouie tansfom of spectum contast 1 0,9 0,8 measued FFT-Data 0,7 0,6 0,5 0,4 0,3 0, 0,1 0-0,8-0,6-0,4-0, 0 0, 0,4 0,6 0,8 z-shift in mm

25 5 Speckle Effect Geneation of speckles: Coheent light is efacted / eflected at a ough suface Roughness ceates phase diffeences Intefeence of all patial waves: ganulation, signatue fo a local suface patch Tansmission of andom media in a volume is also possible (atmosphee, biological) Highe ode effects: patial coheent illumination, polaization incident lase light plane of obsevation suface with oughness

26 6 Sum of Random Phasos Sum of andom phasos due to field supeposition: 1. nealy zeo esult, dominant destuctive. lage esult, dominant constuctive 3. special case of one lage contibution Ref. J. Goodman

27 7 Speckle Patten Size of objective speckles: depends on distance of obsevation z = 840 mm z = 330 mm Coloed speckles z = 160 mm z = 110 mm

28 Subjective / Objective Speckle 8 Ceating of speckle patten: 1. coheent scatteing of lase light: objective speckle incident lase light. imaging of coheent staylight: subjective speckle always be visual obsevation 1 P lens with focal length f suface with oughness p > l point of obsevation D intensity d suface with oughness z z' scheen

29 Objective Speckle Patten 9 Incident coheent light Rough suface with size D Obsevation in distance z Speckle patten with typical size of cells d z l D D Aiy incident coheent lase light intensity D ough suface d sceen z

30 30 Subjective Speckle Patten Incident coheent light Rough suface with size D lens with focal length f Obsevation in distance z Speckle size in the image: PSF, D aiy d s D intensity d Speckle patten with typical size of cells in the object d l (1 m) (1 m) NA s D aiy m: magnification Example: coase speckle fo small NA suface with oughness z F#= F#= 66 z' scheen Ref. W. Osten

31 Geneation of Speckle at a Rough Suface 31 Field eflected at ough suface with height h(x,y) Coheence of eflected waves Dependence of the coheence fom the oughness s of the suface (nomal distibution of height assumed) E ( x, y) E in ( x, y) e ik 1 cos h( x, y) ik1 cos h x, y hx1, y1 Dx, Dy e () e k (1cos ) h s 1C h Dx, Dy s = s = 10 s = 4.0 s =.0 s = / c

32 3 Speckle at Rough Sufaces The aea of coelation of the adiation A c L c shinks with the vaiance s of the suface oughness A c / 1.0 Signal to noise atio I I s N 0.5 Contast C C C s

33 Statistics of Single Speckle Poisson statistics of a single speckle: Pobability of intensity values in the patten w( I) 1 I e Contast (full developed speckle) Necessay: oughness lage than wavelength Lagest pobability: dakness I=0 I I C s I I 1 1 <I> w(i) 33 I Example

34 34 Phase Dislocations Zeo intensity points in a speckle patten Hee often votex points of the phase Found in simulation by eal and imaginay pat a) intensity b) Re and Im pats c) phase cicles: zeo I(x,y) otation aound zeos Ref. J. Goodman

35 Statistics of Supeposed Speckles 35 Incoheent supeposition of seveal speckles w(i) Pobability has intemediate maximum w( I) Zeo pobability fo dakness Deceasing contast Example 4I I 0 e I I I / I o

36 36 Speckle Statistics fo Incoheent Supeposition Reduction of speckle contast by incoheent supeposition Ovelay of lage numbe of individual fully modulated images Many images necessay to get a unifom illumination Reduction of vaiance goes with 1/ n w(i) n = n = 6 n = 1 n = 0 n = 40 n = I / I o

37 37 Speckle Reduction Coheent speckles afte diffuso plate with diffeent data stating phase spectum fa field

38 38 Speckle Contast Changing with Coheence Contast of speckle image fo changing coheence a: amplitude l co : tansvese length of coheence a/l co = 0 a/l co = 0.1 a/l co = 0.5 a/l co = 1.0 a/l co =.0 a/l co = 4.0

39 39 Reduction of Spatial Coheence Reduction of spatial coheence: - moving scatte plate - statistical mixing of phases - tempoal integation (time aveaging) - movement should be faste than detecto integation time - divesification of illumination angle (micoscopy) - divesification of wavelength (lase bandwidtz) incident coheent beam s pupil a x, a y s' moving diffuso plate lens image plane x',y'

40 40 Reduction of Spatial Coheence Reduction of spatial coheence moe effective in case of two scatte plates Only one is moving incident coheent lase beam modulated diection spatial patial coheent adiation popagation distance fixed 1st diffuso moved nd diffuso

41 41 Reduction of Speckle by Tempoal Aveaging Aveaging of speckles by time integation Moving stop in the pupil But: educed esolution coheent illumination beam educed speckle by time integation moving stop scatteing object Moving stop nea the image plane coheent illumination beam educed speckle by time integation scatteing object coheent speckle patten moving stop

42 4 Speckle Metology Usual: speckle petubs the imaging fo coheent illumination Speckle is only dependent on spatial oughness: time independent Diffeent usage of statistical speckle patten in metology: 1. Speckle photogaphy: - ecoding of two intensity images with small lateal shift of object - speckle patten invaiant but moved - compaison/evaluation by 1.1 coelation 1. calculation of diffeences 1.3 Fouie tansfom evaluation. Speckle intefeomety: - supeposition of speckle fields (both statistical o one deteministic efeence) - speckles wok as statistical stucued illumination - imaging of visualization of suface oughness - efeencing by shea: sheaogaphy 3. Speckle astonomy: - ecoding of many single speckled images (due to atmospheic changed speckles) - calculation of Fouie tansfoms - aveaging ove all images, autocoelation, statistics suppessed

43 43 Speckle Photogaphy Setup with objective speckles Displacements: subjective objective Example d d ( sub p ( obj p D ) l m d F F D ) l d Ref. W. Osten

44 44 Speckle Photogaphy Selection of a speckle cell Shift tansfom T(u,v): matched cell speckle cell Types: tanslation, otation, shea Finding the maximum coelation C(u,v) tansfom T(u,v) matched speckle cell

45 45 Speckle Photogaphy Autocoelation of diffeent shift sizes shift: 16 pix 3 pix 64 pix Ref. J. Goodman 18 pix

46 46 Speckle Intefeomety Classical setup with deteministic efeence beam Movement of diffuse object detected as phase change Setup fo in-plane displacement displacement

47 47 Speckle Intefeomety Coheence cell has finite lateal and axial dimension Depth extension of a speckle cell: - out-of plane displacement - coesponds to classic stuctued illumination fo 3D metology displacement cells speckle spots beam caustic popagation 7 spot pe coss section in 1 dimension

48 48 Speckle Sheaogaphy Double exposue afte displacement: shea intefeomety Measuement of phase gadients o slopes wedge Intensity Phase diffeence fo x shea ( x, y) ( x Dx, y) ( x, y) x S: sensitivity S( x, y) Dx object l lens Ref. W. Osten

49 49 Speckle Sheaogaphy Examples aw phase map filteed phase map unwapped phase unwapped phase 3D aw phase map filteed phase map slop map slope map 3D Ref. T. Yoshizawa

50 50 Speckle Sheaogaphy Examples fo defect detection Ref. W. Osten