Imaging and Aberration Theory

Transcription

1 Imaging and Aberration Theor Lecture 11: Wave aberrations Herbert Gross Winter term 017

2 Preliminar time schedule Paraial imaging araial otics, fundamental laws of geometrical imaging, comound sstems Puils, Fourier otics, uil definition, basic Fourier relationshi, hase sace, analog otics and Hamiltonian coordinates mechanics, Hamiltonian coordinates Eikonal Fermat rincile, stationar hase, Eikonals, relation ras-waves, geometrical aroimation, inhomogeneous media Aberration eansions single surface, general Talor eansion, reresentations, various orders, sto shift formulas Reresentation of aberrations different tes of reresentations, fields of alication, limitations and itfalls, measurement of aberrations Sherical aberration henomenolog, sh-free surfaces, skew sherical, correction of sh, asherical surfaces, higher orders Distortion and coma henomenolog, relation to sine condition, alanatic stems, effect of sto osition, various toics, correction otions Astigmatism and curvature henomenolog, Coddington equations, Petzval law, correction otions Chromatical aberrations Disersion, aial chromatical aberration, transverse chromatical aberration, sherochromatism, secondar soectrum 10 Sine condition, alanatism and Sine condition, isolanatism, relation to coma and shift invariance, uil isolanatism aberrations, Herschel condition, relation to Fourier otics Wave aberrations definition, various eansion forms, roagation of wave aberrations Zernike olnomials secial eansion for circular smmetr, roblems, calculation, otimal balancing, influence of normalization, measurement Point sread function ideal sf, sf with aberrations, Strehl ratio Transfer function transfer function, resolution and contrast Additional toics Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertabilit

3 3 Contents 1. Definition of wave aberrations. Performance criteria 3. Primar aberrations 4. Order eansions 5. Non-circular uil shaes 6. Statistical aberrations 7. Measurement of wave aberrations

4 4 Law of Malus-Duin Law of Malus-Duin: - equivalence of ras and wavefronts - both are orthonormal - identical information wave fronts Condition: No caustic of ras ras Mathematical: Rotation of Eikonal vanish rotn s Otical sstem: Ras and sherical waves orthonormal 0 object lane 0 hase L = const L = const image lane ras s 1 z 0 z 1

5 5 Relationshi to Transverse Aberration Relation between wave and transverse aberration Aroimation for small aberrations and small aerture angles u Ideal wavefront, reference shere: W ideal Real wavefront: W real Finite difference Angle difference Transverse aberration Limiting reresentation W W W real W ideal W tan ' R W ' R W ' R reference shere wave front W( reference lane q R, ideal ra u real ra C ' z

6 Wave Aberration Eact relation between wave aberration and ra deviation General eression from geometr a describes the lateral aberration Substitution of angle b scalar roduct Eact relation is quadratic in R Aroimation for large R with 6 cosq 1 ' 1 ' 0 R a s a n s s a s s s s n W W r r r 4 1 ' R a s a R a s a n W W ' ' ' z R R R W ' ' ' z a real ra C R s s r reference shere q a A W wave front z center of reference shere real intersection oint

7 AP OE OPL r d n l (0,0, (, ( OPL OPL OPD l l R W R W ' ' W R u R s, ( ' sin ' ' ' Relationshis Concrete calculation of wave aberration: addition of discrete otical ath lengths (OPL Reference on chief ra and reference shere (otical ath difference Relation to transverse aberrations Conversion between longitudinal transverse and wave aberrations Scaling of the hase / wave aberration: 1. Phase angle in radiant. Light ath (OPL in mm 3. Light ath scaled in l ( ( ( ( ( ( ( ( ( W i k i i e A E e A E e A E OPD 7

8 8 Wave Aberration in Otical Sstems Definition of otical ath length in an otical sstem: Reference shere around the ideal object oint through the center of the uil Chief ra serves as reference Difference of OPL : otical ath difference OPD Practical calculation: discrete samling of the uil area, real wave surface reresented as matri ' ' uer coma ra otical sstem W wave aberration chief ra w' image oint z chief ra wave front object oint lower coma ra reference shere Object lane O Entrance uil EnP Eit lane EP Image lane I

9 9 Puil Samling All ras start in one oint in the object lane The entrance uil is samled equidistant In the eit uil, the transferred grid ma be distorted In the image lane a sreaded sot diagram is generated object lane oint entrance uil equidistant grid otical sstem eit uil transferred grid image lane sot diagram o ' ' o ' ' z

10 10 Wave Aberration Definition of the eak valle value W PV Reference shere corresonds to erfect imaging Rms-value is more relevant for erformance evaluation v-value of wave aberration wave aberration image lane hase front eit aerture reference shere

11 11 Wave Aberration Criteria Mean quadratic wave deviation ( W Rms, root mean square W rms W with uil area 1 W W, Wmean, AEP A EP dd Peak valle value W v : largest difference v, W W ma W, ma min General case with aodization: weighting of local hase errors with intensit, relevance for sf formation W rms 1 A ( w, W, W I w EP mean, ( EP d d d d

12 1 Wave Aberrations Sign and Reference Wave aberration: relative to reference shere Choice of offset value: vanishing mean Sign of W : 1 W (, W (, d d 0 F EP - W > 0 : stronger convergence intersection : s < 0 - W < 0 : stronger divergence intersection : s < 0 reference shere wave front W > 0 ideal ra reference lane (araial real ra U' C z R ' < 0 s' < 0

13 13 Tilt of Wavefront W Change of reference shere: tilt b angle q linear in W tilt n q Wave aberration due to transverse aberration W tilt R Re f ' Is the usual descrition of distortion wave aberration W < 0 reference shere q tilt angle ' transverse aberration ' z wave front uil lane image lane

14 14 Defocussing of Wavefront Paraial defocussing b z: Change of wavefront W Def nr R ref z' 1 nz' sin u wave aberration W > 0 ' wave front z reference shere z' defocus uil lane image lane

15 15 Primar Aberrations Reresentation of rimar aberrations Seidel terms Primar monochromatic wave aberrations Sherical aberration Coma 4 c W c W c 3 c r cos 1 1 r Surface in uil lane Secial case of chromatical aberrations Astigmatism Field curvature (sagittal W c3 c3 r cos W c4 c4 r Primar chromatic wave aberrations Aial color ~ ~ W b b r 1 1 Lateral color ~ ~ W b b r cos Distortion 3 3 W c5 c5 r cos Ref: H. Zügge

16 16 Primar Aberrations Te Wave aberration Geometrical sot Relation : wave / geometrical aberration Sherical aberration Smmetr to Periodicit W c 1 r ais constant 3 W c r cos 4 ' c 1 r ' c 1 r 3 3 sin cos oint 1 eriod ' cr (cos ' c r sin Coma Smmetr to Periodicit one lane 1 eriod W c3 r cos one straight line eriods ' 0 ' c3 r cos Astigmatism Smmetr to Periodicit Field curvature (sagittal two lanes eriod W c4 r two straight lines 1 eriod ' c4 r sin ' c cos 4 r Smmetr to Periodicit ais constant 3 W c5 r cos oint 1 eriod ' 0 ' c 5 3 Distortion Ref: H. Zügge Smmetr to Periodicit one lane 1 eriod one straight line constant

17 17 Primar Aberrations Relation : wave / geometrical aberration Te Wave aberration Geometrical sot ~ W b r 1 ~ ' b1 r sin ~ ' b1 r cos Aial color Smmetr to Periodicit ais constant ~ W b r cos oint 1 eriod ' 0 ~ ' b Lateral color Smmetr to Periodicit one lane 1 eriod one straight line constant Ref: H. Zügge

18 18 Secial Cases of Wave Aberrations Wave aberrations are usuall given as reduced aberrations: - wave front for onl 1 field oint - field deendence reresented b discrete cases ' Secial case of aberrations: 1. aial color and field curvature: reresented as defocussing term, Zernike c 4. distortion and lateral color: reresented as tilt term, Zernike c, c 3 uil lane wave front Reference shere aial chromatic z ideal image lane ' curved image shell ' wave front chief ra image height difference wave front chief ra defocus due to field curvature z z reference shere reference shere uil lane ideal image lane uil lane ideal image lane

19 19 Secial Cases of Wave Aberrations 3. afocal sstem - eit uil in infinit - lane wave as reference wave front reference lane z image in infinit 4. telecentric sstem chief ra arallel to ais ' wave front reference shere z uil lane aial chromatic ideal image lane

20 0 Eansion of the Wave Aberration Table as function of field and aerture Selection rules: checkerboard filling of the matri Image location Primar aberrations / Seidel r 6 Field Sherical Coma Astigmatism r cosq 3 r cosq 5 r cosq Distortion r 1 Tilt Distortion Distortion rimar secondar r r cos q 4 r cos q r Defocus r Aerture Astig./Curvat. 4 r r 3 cosq 3 r r 3 r 3 cos 3 q Coma rimar 3 r 3 cosq r 4 r 4 cos q r 4 Sherical rimar r 4 r 5 cosq r 5 Coma secondar Sherical r 6 secondar Secondar aberrations

21 1 Polnomial Eansion of Wave Aberrations Talor eansion of the wavefront: ' Image height inde k r Puil height inde l q Puil azimuth angle inde m Smmetr invariance: 1. Image height. Puil height 3. Scalar roduct between image and uil vector k l m W( ', r, q Wklm ' rcos q k, l, m ' r ' r ' r cosq Number of terms sum of indices in the eonent i sum i sum N i number of terms Te of aberration image location 4 5 rimar aberrations, 4th order 6 9 secondar aberrations, 6th order th order

22 Talor - Eansion The eonents of the Talor eansion on the aerture deends on the kind of reresentation of the aberrations. The eonent grows b 1 in the sequence longitudinal-transversal-wave aberrations The Seidel term 3rd order is valid onl for transverse aberrations Deendence on aerture and fiels size for the rimar aberrations: te of aberration wave aberration transverse aberration longitudinal aberration u w u w u w sherical coma astigmatism 1 0 Petzval curvature 1 0 distortion aial chromatical chromatical magnif

23 3 Talor Eansion of the Primar Aberrations Eansion of the monochromatic aberrations First real aberration: rimar aberrations, 4 th order as wave deviation W( ', r, A r A ' r A ' A ' r A s 4 c a d ' 3 Coefficients of the rimar aberrations: A S : Sherical Aberration A C : Coma A A : Astigmatism A P : Petzval curvature A D : Distortion Alternativel: eansion in olar coordinates: Zernike basis eansion, usuall onl for one field oint, orthogonalized

24 4 Raleigh Criterion The Raleigh criterion W PV l 4 gives individual maimum aberrations coefficients, deends on the form of the wave a otimal constructive interference b reduced constructive interference due to hase aberrations c reduced effect of hase error b aodization and lower energetic weighting d start of destructive interference for 90 or l/4 hase aberration begin of negative z-comonent Eamles: aberration te coefficient defocus Seidel a defocus Zernike c sherical aberration Seidel a sherical aberration Zernike c astigmatism Seidel a 0. 5 astigmatism Zernike c coma Seidel a coma Zernike c

25 5 Criteria of Raleigh and Marechal Raleigh criterion: 1. maimum of wave aberration: W v < l/4. beginning of destructive interference of artial waves 3. limit for being diffraction limited (definition 4. as a PV-criterion rather conservative: maimum value onl in 1 oint of the uil 5. different limiting values for aberration shaes and definitions (Seidel, Zernike,... Marechal criterion: 1. Raleigh crierion corresonds to W rms < l/14 in case of defocus Raleigh W rms l 19 l l 14. generalization of W rms < l/14 for all shaes of wave fronts 3. corresonds to Strehl ratio D s > 0.80 (in case of defocus 4. more useful as PV-criterion of Raleigh

26 6 PV and W rms -Values PV and W rms values for different definitions and shaes of the aberrated wavefront Due to miing of lower orders in the definition of the Zernikes, the W rms usuall is smaller in comarison to the corresonding Seidel definition

27 7 Tical Variation of Wave Aberrations Microscoic objective lens Changes of rms value of wave aberration with 1. wavelength. field osition Common ractice: 1. diffraction limited on ais for main art of the sectrum. Requirements relaed in the outer field region 3. Requirement relaed at the blue edge of the sectrum W rms [l] field zone Achrolan field edge 0.06 diffraction limit on ais l [m]

28 8 Zernike Polnomials Eansion of wave aberration surface m W ( r, c Z ( r, n n mn Zernike olnomials orders b indices: n : radial m : azimuthal, sin/cos Orthonormal function on unit circle Z m n ( r, R m n nm Direct relation to rimar aberration tes Direct measurement b interferometr Orthogonalit erturbed: 1. aodization. discretization 3. real non-circular boundar n sin ( m for ( r cos( m for 1 for m 0 m 0 m 0 m = cos sin n =

29 9 Tatian Polnomials for Ring Puils Orthogonalization of Zernike Polnomilas for ring shaed uil area Basis function deends on obsuration arameter e: no eas comarisons ossible

30 30 Polnomial for Rectangular Puil Areas Sstems with rectangular uil: Use of Legendre olnomials P n ( 1 1 P ( P n m 0 ( d n 1 if if n n m m 1. Factorized reresentation Problem: zero-crossing lines. Definition of D area-orthogonal Legendre functions W(, A P ( P ( n m nm n m General shae of the uil area: Gram-Schmidt-orthogonalization drawback: 1. Individual function for ever uil shae. no intuitive interretation 3. no comarabilit between different sstems ossible

31 31 -Dimensional Legendre Polnomials Orthogonalization of Zernike olnomials on a unit square Q k Q j d d kj Q m j l ma ma j0 l0 d m, jl j l Q m m j1 c mj Z j Gram-Schmidt-Orthogonalization rocedure Q m m 1 k1 a k Q k a m Z m c a mk m c m 1 jk mm a j c jk P mm k 1,,3,... m 1 1 m1 Tkm Tkm k1 m1 k k1 j1 c kj P mj

32 3 Legendre Polnomials D-Legendre olnomials for rectangular areas Alication: Sectrometer slit aerture First few olnomials: quite similar to Zernikes

33 33 Wave Aberration Generated at a Surface At an surface the wavefront can be comared with the ideal one before and after the refraction/reflection Due to the additivit of the hase, at an surface the contribution can be calculated Practical roblems: - collimated intermediate ra aths - change of normalization radii and grid distortion - change of wavefront/zernikes during roagation - choice of reference surface not trivial - arabasal ra calculation inaccurate real waves sherical object wave sherical image wave real ra araial ra z surface intermediate ideal object lane n n intermediate ideal image lane W. Welford, Aberrations of otical sstems, Hilger 1986 W. Welford, Ot Acta 19 ( , A new total aberration formula

34 34 Wave Aberration Additivit Comlete sstem: Additivit of hase dela at ever surface is obvious Practical roblems: - change of normalization radii - grid distortion - huge amount of information, sstematic analsis comlicated - analtical reresentation not ossible surface contributions ' W 1 W W3 total W tot arbitrar ra P ra encil surfaces 1 3 eit uil P'

35 35 Grid Distortion Real distortion of the grid in at the various mirror ositions M 1 M M 3 0 mm 110 mm 60 mm

36 Comle field with statistical hase Correlation of the hase: structural function Coherence function For gaussian statistics Auto covariance function PSD, ower sectral densit ( ( ( r i o e r E r E ( ( ( r i r i r D e e ( ( ( ( r D o o e r E r E z 1 ( c a r e r D, (, (, ( c a e C, ( ˆ, ( c v v a c e a F C v v S Statistical Wave Aberrations 36

37 37 Statistical Wave Aberrations Descrition: 1. in the satial domain: toolog of the rough surface. in the satial frequenc domain h( height, toolog C( correlation function < h 1 h > correlation Fourier transform Fourier transform A( amlitude sectrum PSD( ower sectral densit square

38 38 Satial Frequenc of Surface Perturbations Power sectral densit of the erturbation Three tical frequenc ranges, scaled b diameter D log A Four oscillation of the olishing machine limiting line sloe m = long range low frequenc figure Zernike mid frequenc 1/D 1/D 40/D micro roughness 1/l

39 39 Atmosheric Turbulence Atmosheric turbulence: statistical hase screen (k Log Cn Scale below 1 cm: Tatarski regime, viscosit PSD b Ta e k ki Scale from 1 cm to 5 m: Kolmogorov regime PSD b Ta k 11 3 Greater length scales: Karman regime PSD b Ka k 4 L o 11 6 const Karman large scale k o outer scale 11 =b Ko k 3 Kolmogorov inertial subrange k i inner scale = bta e -k viscose dissiation Tatarski k

40 40 Measurement of Wave Aberrations Wave aberrations are measurable directl Good connection between simulation/otical design and realization/metrolog Direct hase measuring techniques: 1. Interferometr. Hartmann-Shack 3. Hartmann sensor 4. Secial: Moire, Holograh, hase-sace analzer Indirect measurement b inversion of the wave equation: 1. Phase retrieval of PSF z-stack. Retrieval of edge or line images Indirect measurement b analzing the imaging conditions: from general image degradation Accurac: 1. l/1000 ossible, l/100 standard for rms-value. Rms vs. individual Zernike coefficients

41 41 Testing with Twman-Green Interferometer reference mirror Short common ath, sensible setu Two different oeration modes for reflection or transmission collimated laser beam Alwas factor of between detected wave and comonent under test beam slitter objective lens sto 1. mode: lens tested in transmission auiliar mirror for autocollimation. mode: surface tested in reflection auiliar lens to generate convergent beam detector

42 4 Interferograms of Primar Aberrations Sherical aberration 1 l Astigmatism 1 l Coma 1 l Defocussing in l

43 43 Real Measured Interferogram Problems in real world measurement: Edge effects Definition of boundar Perturbation b coherent stra light Local surface error are not well described b Zernike eansion Convolution with motion blur Ref: B. Dörband

44 44 Interferogram - Definition of Boundar Critical definition of the interferogram boundar and the Zernike normalization radius in realit

45 45 Princile of Psf Phase Retrieval Wave front determines local direction of roagation Proagation over distance z : change of transverse intensit distribution Intensit roagation contains hase information roagation z intensit caustic I(,,z wave front : W(, local Ponting vector direction of roagation

46 46 Proagation of Intensit Transort of intensit equation coules hase and intensit I(,, z k z I(,, z W (, Solution with z-variation of the intensit delivers start hase at z = 0 Determine hase from intensit distribution. - Inverse roagation roblem : ill osed - Boundar condition : measured z-stack I(,,z Algorithm for numerical solution - IFTA / Gerchberg Saton ( error reduction - Acceleration ( conjugate gradients, Fienu,... - Modal non least square-methods - Etended Zernike method Alications: - Calculation of diffractive comonents for given illumination distribution - Wave front reconstruction - Phase microsco

47 47 Phase Retrieval Princile of hase retrieval for metrolog of otical sstems Measurement of intensit caustic z-stack Reconstruction of the hase in the eit uil image lane eit uil otical sstem z z wave front W(, 3D- intensit image stack I(,,z

48 48 Gerchberg-Saton-Algorithm Iterative reconstruction of the uil hase with back-and-forth calculation between image and uil: IFTA / Gerchberg-Saton Substitution of known intensit Problems with convergence: Twin-image degeneration Modified algorithms: 1. Fienu-acceleration. Non-least-square 3. Use of uil intensit

49 49 Phase Sace Interretation Known measurement of intensit in defocussed lanes: - Several rotated lanes in hase sace - Information in and near the satial domain Calculation of distribution in the Fourier lane Wave equation is valid Princile : Tomograh uil lane desired distribution minimal defocus lane hase sace angle measuring lanes image lane ' maimal defocus lane

50 50 Eamle Phase Retrieval Evaluation of real data sf-stack measurement model hase

51 51 Phase Retrieval with Aodization M R no A Dif R wi A Dif Retrieval without / with Aodization Correlation over z

52 5 Object Sace Defocussing a defocussing in image sace one wavefront several detector lanes z object lane lens uil b defocussing in object sace several wavefronts intensit caustic I(,,z one fied detector lane z several object lanes lens uil intensit caustic I(,,z obj

53 53 Measurements Comarion hase retriev vs Hartmann test Zernikecoefficients a13 a15 Retrieval A13 A15 Hartmann Case of coma Coma field -zone ais + zone +field Ref: B. Möller