Metrology and Sensing

Transcription

1 Metrology and Sensing Lecture 1: Phase retrieval Herbert Gross Winter term 017

2 Preliminary Schedule No Date Subject Detailed Content Introduction Introduction, optical measurements, shape measurements, errors, definition of the meter, sampling theorem Wave optics Basics, polarization, wave aberrations, PSF, OTF Sensors Introduction, basic properties, CCDs, filtering, noise Fringe projection Moire principle, illumination coding, fringe projection, deflectometry Interferometry I Introduction, interference, types of interferometers, miscellaneous Interferometry II Examples, interferogram interpretation, fringe evaluation methods Wavefront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods Geometrical methods Tactile measurement, photogrammetry, triangulation, time of flight, Scheimpflug setup Speckle methods Spatial and temporal coherence, speckle, properties, speckle metrology Holography Introduction, holographic interferometry, applications, miscellaneous Measurement of basic system properties Bssic properties, knife edge, slit scan, MTF measurement Phase retrieval Introduction, algorithms, practical aspects, accuracy Metrology of aspheres and freeforms Aspheres, null lens tests, CGH method, freeforms, metrology of freeforms OCT Principle of OCT, tissue optics, Fourier domain OCT, miscellaneous Confocal sensors Principle, resolution and PSF, microscopy, chromatical confocal method

3 3 Content Introduction Algorithms Various practical aspects Accuracy

4 4 Principle of Psf Phase Retrieval Wave front determines local direction of propagation Propagation over distance z : change of transverse intensity distribution Intensity propagation contains phase information propagation z intensity caustic I(x,y,z) wave front : W(x,y) local Poynting vector direction of propagation

5 Maxwell equation for the field E, vectorial The spatial inhomogeneities couples the field components Homogeneous, without charges, non-conductive separation of vector components, scalar Time independence: Wave equation of Helmholtz In coordinate representation Wave number in medium refractive index n Fast z-oscillation separated Slowly varying envelope approximation ~ ~ t E E t i e E r t E r ) ( ), ( 0 E k E o o nk n c k ikz e z y x E z y x E ),, ( ),, ( Wave Equation 0 ln r r o o E t E E 0 ),, ( E c z y x n z E y E x E 0 E z ik E z E k n x y n n E o (, )

6 Fraunhofer Point Spread Function Rayleigh-Sommerfeld diffraction integral, Mathematical formulation of the Huygens-principle ik r r ' i e EI ( r) E( r') r r' cos dx' dy' d Fraunhofer approximation in the far field for large Fresnel number N F r p z 1 Optical systems: numerical aperture NA in image space Pupil amplitude/transmission/illumination T(x p,y p ) Wave aberration W(x p,y p ) complex pupil function A(x p,y p ) Transition from exit pupil to image plane E( x', y' ) Point spread function (PSF): Fourier transform of the complex pupil function iw ( x p, y p A( x, y ) T( x, y ) e p p AP T p i xpx' y p y' iw xp, y p RAP x, y e e p p p ) dx p dy p

7 Typical change of the intensity profile Normalized coordinates Diffraction integral 7 Fresnel Diffraction z geometrical focus f a z stop far zone geometrical phase intensity a r r f a v z f a u ; ; / 0 ) ( ), ( d e v J e f E ia v u E u i u a f i

8 8 Propagation of Intensity Transport of intensity equation couples phase and intensity I( x, y, z) k z I( x, y, z) W ( x, y) Solution with z-variation of the intensity delivers start phase at z = 0 Determine phase from intensity distribution. - Inverse propagation problem : ill posed - Boundary condition : measured z-stack I(x,y,z) Algorithm for numerical solution - IFTA / Gerchberg Saxton ( error reduction ) - Acceleration ( conjugate gradients, Fienup,...) - Modal non least square-methods - Extended Zernike method Applications: - Calculation of diffractive components for given illumination distribution - Wave front reconstruction - Phase microscopy

9 9 Principal Approach Given object I object (x) Known illumination, usually incoherent Known measured image intensities I image (x',z) for several z-values To be calculated: transfer function of the system / pupil function Relationship: convolution mask lens pupil tube lens measuring planes illumination I object (x) I pupil (x p ) = A pupil (x p )exp [ iw(x p ) ] I image (x',z)

10 10 Retrieval: Intensity z-stack Stack of transverse intensity distributions I(x,y,z=const) Criteria for z-selection: - no truncation for outer z-planes - good resolution for focal point (>6 pixel per Airy diameter) - enough redundant data for noise, discretization and quantization - enough z-planes for far field and near field: at least -R u...+r u z y x

11 11 Phase Retrieval Principle of phase retrieval for metrology of optical systems Measurement of intensity caustic z-stack Reconstruction of the phase in the exit pupil image plane x exit pupil optical system z z wave front W( x,y) y x 3D- intensity image stack I(x,y,z)

12 1 Phase Space: 90 -Rotation Transition pupil-image plane: 90 rotation in phase space Planes Fourier inverse Marginal ray: space coordinate x ---> angle ' Chief ray: angle ---> space coordinate x' Fourier plane pupil image location marginal ray x ' x' chief ray f

13 13 Phase Space Interpretation Known measurement of intensity in defocussed planes: - Several rotated planes in phase space - Information in and near the spatial domain Calculation of distribution in the Fourier plane Wave equation is valid Principle : Tomography pupil plane desired distribution x p minimal defocus plane phase space angle measuring planes image plane x' maximal defocus plane

14 14 Phase Retrieval Illumination Setups a) Pinhole and trans illumination Pinhole object - Deconvolution may be necessary - Illumination incoherent - Problems with pinhole illumination lens pinhole objective lens under test tube lens image plane Epi-illumination - Double pass with plane mirror - Only symmetrical aberrations - No field coma seen b) Epi illumination and plane mirror mirror beam splitter Trans-illumination with objective - Laser source, coherent - Calibration necessary - Speckle problem c) Calibrated illumination lens point illumination in air

15 15 Phase-Retrieval: Basic Parameter Necessary known basi cparameter for reconstruction: 1. wavelength. aperture in image space sin(u) 3. pixel size of detector 4.pinhole size in object space 5. magnification m 6. z-values of z-stack Critical data: 1. pixel size. size of pinhole (coherence, throughput) 3. deconvolution parameter of algorithm 4. background of intensity 5. selection of z-planes

16 16 Mathematical Formulation Mathematical description for Image formation, Integral equation, inverse problem Approximation with isoplanatic range: Psf shift invariant, convolution computed with Fourier methods I I I image image image ( x) I ( x, x') I ( x, x') dx' I ( x) psf psf object object ( x) I ( x) I ( x) I ( v) I ( v) I ( v) I psf object noise noise noise Discretization: pixelized image delivers a linear system Solution via optimization due noise and constraints I ( ima) jk j' k' I ( psf ) g A x n A x b min A x b j j ', kk ' I x ( obj) j', k' I min ( noi) jk

17 17 Gerchberg-Saxton-Algorithm Iterative reconstruction of the pupil phase with back-and-forth calculation between image and pupil: IFTA / Gerchberg-Saxton Substitution of known intensity startphase W(x p ) Start pupil corected intensity inpupil optional I'(x p )substituted pupil Problems with convergence: Twin-image degeneration Fourier iteration invers Fourier Modified algorithms: 1. Fienup-acceleration. Non-least-square 3. Use of pupil intensity PSF I'(x) substituted PSF corected convergence test measuring imagei(x)

18 18 Phase - Retrieval Algorithms Possible numerical algorithms: 1. Fourier algorithm. NLSQ-algorithm with Zernike coefficients ( modal ) 3. Input-Output-algorithm according to Fienup 4. Yang-Gu-algorithm 5. Ping-Pong-algorithm 6. Gerchberg-Saxton-algorithm ( error reduction ) 7. Ferwerda-algorithm 8. Gradient methods

19 19 Check of Caustik Paraxial approximation: - quadratic curve of second moment of spot size - check of wrong data possible -deviations for larger aberrations D rms [m] fit of parabola numerical result z [a.u.]

20 0 Example Phase Retrieval Evaluation of real data psf-stack measurement model phase

21 1 PSF Phase Retrieval Phase retrieval method Image z-stack Correlation of images Phase in pupil

22 Finite Size of Pinhole Image with finite size of the pinhole: convolution Characteristic diffraction structures hidden with growing size D ph Deconvolution necessary for D ph > D airy Pinholes larger than 4 D airy are not feasible D PH = 0 D PH = 0.5 D airy D PH = 0.5 D airy D PH = 0.75 D airy z = - z = z = z = 0 z = z = z = + D PH = 1.0 D airy D PH = 1.5 D airy D PH =.0 D airy

23 3 Incoherent Image of a Pinhole Logarithm of intensity Diffraction ripples disappear with growing diameter d Log I(r) d = 0.1 d = 0. d = 0.5 d = 1.0 d = 1.5 d = ,5 1 1,5 rsinu / d = 0.1 d = 0. d = 0.5 d = 1.0 d = 1.5 d =.0

24 4 Coherent Illuminated Pinhole Pinhole with coherent illumination : pupil apodization with Airy profile First zero at boundary of pupil for D 1. sin u PH D airy A(r) E( r p D J 1 1. D ) DPH 1. D airy PH airy r p r p D PH / D airy = 0.1 D PH / D airy = 0.5 D PH / D airy = 0.5 D PH / D airy = 1 D PH / D airy = D PH / D airy = r

25 5 Incoherent Image of a Circular Diske Quais point source for the a0. 4 Airy Du (relativ) 4, 3,5 Criterion: intensity threshold of Airy function at 10% / 50% 3,,5, 50 % 10 % 1, , 0,5 0, 0, 0, 0,4 0,6 0, , 1, 1,4 1,6 1,8, d

26 6 Comparison of Psf: Coherent / Incoherent Comparison of pinhole image with Airy-Psf: Rms value of deviation Coherent case: better approximation Typical limits: Incoherent : D/D airy < 0.4 Coherent : D/D airy < 0.7 I rms 0.5 Rms-limit coherent incoherent 0.01 d = d = d = d = incoherent coherent D/D airy.5

27 7 Deconvolution - Algorithms Incoherent imaging with noise I image ( x) I ( x) I ( x) I psf object noise Wiener deconvolution with fixed Tikhonov regularization I object ( v) I ( v) I * psf image I psf ( v) ( v) Wiener deconvolution with variable Tikhonov regularization Lucy-Richardson deconvolution Wavelet-Deconvolution I ( k1) obj I object ( v) I I * psf psf ( k ) x y I x, y I x, y ( v) Iimage( v) Pnoise ( v) P object I image x, y ( k x, y I x, y, obj psf ) I psf obj

28 8 Error in Phase Retrieval without Deconvolution Error of results, if no deconvolution is performed Error increases with pinhole size Deconvolution seems to be necessary for pinholes larger than 0.4 D airy c mean no deconvolution : no noise with noise with deconvolution : no noise with noise D PH / D airy

29 9 Variable Pinhole Deconvolution: Criteria Criteria for pinhole deconvolution: 1. minimal rms of stack sensitive for noise I rms j I (mod ell ) j n x n y I n z ( ist) j. Maximum correlation of the stacks robust, but less significant K I mod el I mod el I ist dx dy dz dx dy dz I ist dx dy dz 3. Minimum entropy best results S m, n I mn ln I mn

30 30 Iteration Phase Retrieval for finite sizes of Pinhole Possibilities with / without deconvolution pupil inner iteration outer iteration model parameter FFT FFT -1 FFT LSQ-fit PSF I PSF (x) substituted image of pinhole deconvolution IFTA komplex convolution I PH (x) comparison real

31 31 Measuring Setup with Magnification Transfer of coherent field I exact psf1 ( x) Apsf ( ) ( x) A x pinhole plane Lithoobjective lens intermediate focus plane Microscopic lens Sensor plane Frequency space I exact ( v) * * A ( v) A ( v) A ( v) A ( ) 1 1 v a) Focussed Therefore the phase is added A( v) T( v) e I exact ( v) T( v) T ( v) T 1 W ( v, z) W iw b) Intrafocal defocussed iw iw T ( v) e T ( v) e 1 ( v) ( v) W ( v, z) c) Extrafocal defocussed c j1 c j1 z c j c' j z cj1 c'' j

32 3 Measuring Setup with Magnification Local energy flow directed pinhole plane Litho - objective lens object plane of micro lens Microscopic lens Sensor plane locally directed energy flow

33 33 Phase Retrieval with Apodization Analysis taking apodization into account greatly improves the result input-stack analysis without apodization apodization analysis with apodization

34 34 Phase Retrieval with Apodization rms intensity If the pupil shows a significant illumination distribution: apodization must be taken into account Apodization can be fitted too Better: measured apodization used apodization not taken into account apodization taken into account 10 - input apodization not taken into account apodization taken into account iteration

35 35 Phase Retrieval with Apodization M R no A Dif R wi A Dif Retrieval without / with Apodization Correlation over z

36 36 Image Processing: Intensity Normalization Scaling / Normalization: - Intensity - Energy Problem: SNR in strong defocussed planes z = -R E z = -1R E Focus z = +1R E z = +R E intensity normalized I max = 5.1% I max = 9.8% I max = 4% energy constant

37 37 Image Processing Determination background intensity preferred: corner,every z-plane individually Truncation of outer region without signal Subtraction of underground determination of background r clipped signal undesired background signal z

38 38 Noise in Signal Noise depends on light level Defocussed z-planes more critical Modal fit with Zernikes is low pass filter probability noise strongly defocussed plane signal intensity

39 39 Centering - Line of Sight Centering of z-stack images : 1. Line of sight (centroid moves exactly on a line)). Systematic errors due to mechanical exit pupil with apodization centroid line chief ray inaccuracy (line tilt) 3. Statistical errors reference sphere intensity image plane measured data points shift correction center line axis

40 40 Truncated Beam Profile Problems for truncation 1) with truncation image difference (10%) of the beam profiles: y s 0 errors in cetroid determination Ergebnis defocus focus y s ) without truncation z [RE] defocus focus z [RE]

41 41 Object Space Defocussing a) defocussing in image space one wavefront several detector planes z object plane lens pupil b) defocussing in object space several wavefronts intensity caustic I(x,y,z) one fixed detector plane z several object planes lens pupil intensity caustic I(x,y,z obj )

42 4 Object Space Defocussing Good Linearity of Zernike coefficients Small retrace non-linearity c j ( z) c jo 1 c lin z R E C j in 0.8 image plane c 40 c 0 0. c c defocus [a.u.]

43 43 Phase Retrieval Accuracy Evaluation of real measuring data Comparison of Zernike coefficients with Hartmann test results: accuracy in the range /100 Zernike- Koeffizienten a13 a15 Retrieval A13 A15 Hartmann Koma Feld - Zone Achse + Zone + Feld

44 44 Separation of Circular Symmetric Effects Circular symmetric contributions can be separated over defocussing range: 1. apodization: increase in diameter in focal region. finite pinhole size: uniform broadening vs defocus 3. spherical aberration: asymmetric around focal plane convolution : constant offset r image plane nominal caustic apodization : focus+ / defocus - spherical spherical aberration phase: asymmetric z

45 45 Phase Retrieval Reproducability Reproducability of 3 measurements Very good agreement with uncertainties in the range of /100 for every Zernike coefficients for the first 36 terms No dependence on symmetry 0.1 c j []

46 46 Example Residuum Comparison of intensity profile cross sections x / y Blue : Input Red : Model

47 47 Estimation of Accuracy Correlation matrix of NLSQ-fit can be used to estima the accuracy Experience: largest error bars for circular symmetric aberrations

48 48 Problems and Suggestions Data not truncated Centering of stack images on common line Don t use more than 36 Zernikes Pinhole not larger than x D airy Deconvolution for nearly incoherent illumination Preferred dynamic pinhole-match and forward convolution of finite size Proposed: n > 7 z-planes in the intervall - Ru...+ Ru At least 6 detector pixels inside the Airy diameter in the focus plane Denoising of data Subtraction of background Low-pass filtering of Zernike modal functions in pupil If pupil apodization present: must be taken into account Normalization of intensity in every z-plane