Metrology and Sensing

Transcription

1 Metrology and Sensing Lecture 7: Waveront sensors Herbert Gross Winter term

2 2 Preliminary Schedule No Date Subject Detailed Content Introduction Introduction, optical measurements, shape measurements, errors, deinition o the meter, sampling theorem Wave optics Basics, polarization, wave aberrations, PSF, OTF Sensors Introduction, basic properties, CCDs, iltering, noise Fringe projection Moire principle, illumination coding, ringe projection, delectometry Intererometry I Introduction, intererence, types o intererometers, miscellaneous Intererometry II Examples, intererogram interpretation, ringe evaluation methods Waveront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods Geometrical methods Tactile measurement, photogrammetry, triangulation, time o light, Scheimplug setup Speckle methods Spatial and temporal coherence, speckle, properties, speckle metrology Holography Introduction, holographic intererometry, applications, miscellaneous Measurement o basic system properties Bssic properties, knie edge, slit scan, MTF measurement Phase retrieval Introduction, algorithms, practical aspects, accuracy Metrology o aspheres and reeorms Aspheres, null lens tests, CGH method, reeorms, metrology o reeorms OCT Principle o OCT, tissue optics, Fourier domain OCT, miscellaneous Conocal sensors Principle, resolution and PSF, microscopy, chromatical conocal method

3 3 Content Hartmann-Shack WFS Hartmann method Miscellaneous methods

4 4 Hartmann-Shack Waveront Sensor Basic principle Re: S. Merx

5 5 Hartmann Shack Waveront Sensor Lenslet array divides the waveront into subapertures Every lenslet generates a simgle spot in the ocal plane The averaged local tilt produces a transverse oset o the spot center Integration o the derivative matrix delivers the wave ront W(x,y) array detector waveront D array h D meas u x spot oset D sub reractive index n

6 6 Hartmann Shack Waveront Sensor Typical setup or component testing detector test surace lenslet array iber illumination Lenslet array collimator beamsplitter telescope or adjustment o the diameter

7 7 Spot Pattern o a HS - WFS Aberrations produce a distorted spot pattern Calibration o the setup or intrinsic residual errors Problem: correspondence o the spots to the subapertures

8 8 Array Signal Lenslet array ideal signal array o phase spot pattern cross section o the spot pattern Real signal: 1. discretization 2. quantization 3. noise original discretized and quantized with noise

9 9 HS - WFS : Size o Sub-Apertur Dynamic range: ratio o spot diameter to size o sub-aperture Averaging o waveront slope inside sub-aperture x p lens ocal length point spread unction I(x) D spot /2 D sub incoming wave ront W(x p ) D spot /2

10 10 Setup o a Hartmann-Shack - WFS 1. Simple setup detector test surace lenslet array iber illumination 2. With telecope and relay lens collimator beamsplitter telescope or adjustment o the diameter pupil plane straylight stop telescope array ocal plane relay lens sensor D pup D arr arr m rel

11 11 General Setup o a HS - WFS Generalized setup: adaptation o diameter Waveront is scaled pupil plane W W, sens pup D sens D pup sensor plane D pup D sens W pv W' pv Relay lens Adaptation o sensor size array ocal plane relay lens sensor x x' arr

12 12 Real Measurement o a HS-WFS Problem in practice: deinition o the boundary

13 13 Real Measurement o a HS-WFS Problem in practice: exact determination o the spot centroid: - noise - discretization - quantization - broadening by partial coherence - broadening by local curvature - error by centroid aecting coma - error by partly illuminated pixels original discretized and quantized with noise

14 14 Parametrization o a HS-WFS Layout parametrization: Fresnel number Fill actor Spot size N F D D D D D spot sub spot 2 2 Dmeas D N 4 meas array 1 2N F sub sub F sub sensitivity 2 Dsub 2 N D 2N D meas sub [mm] dynamic range accuracy D spot = 0.01 D spot = 0.05 D spot = 0.10 D spot = 0.20 D spot = spatial resolution N sub Accuracy: min k P mrel max D h 2 sub 2

15 15 Parametrization o a HS-WFS Waveront detectability [mm] 100 possible parameter area W min min Dynamic range D array N sub maximum spot size dynamic range limit R max min h D 2kP m sub rel minimum spatial resolution minimum spot size N sub R max min D sub D 2k P m spot rel Dsub h h k P m rel N

16 16 Problems with a Hartmann-Shack - Sensor Die Wellenläche wird über die Subapertur gemittelt Fresnelzahl der Arraylinse, Spotgröße / Subaperturdurchmesser Aulösung, mit Pixelgröße p und Subaperturzahl N N W F min 2 sub 4 4p N N mess Zuordnung der Spots zu den Subaperturen, Dynamikbereich Spotablage subpixelgenau messen x n W x Probleme mit teilausgeleuchteten Subaperturen Keine Probleme mit spektraler Breite, Kohärenz und Polarisation

17 17 Errors in the HS - Waverontsensor Tilted sensor plane Rotated sensor in the azimuth Scattering o ocal lengths o the lenslets Average o slope inside the subaperture area Errors in the waveront reconstruction algorithms Coma o lenses Wrong ocal length due to dispersion or dierent wavelength Sensor plane not exactly matched with ocal plane Partly illuminated lenslets Electronical noise Zernike errors due to bad known normalization radius / edge o pupil Geometrical distortions o the array Truncation o spot by the corresponding subaperture / cross talk Discrete inite number o pixels Quantization o signal on the detector

18 18 Dynamic Range due to Local Curvature Theoretical largest curvature: R = Real size o point spread unction: Larger curvature: cross talk generates errors R min N F a) b) R R D sub D sub D spot

19 19 HS-WFS : Discretization Errors Signal errors due to inite pixel size discretization o the point spread unction N: number o pixels per sub-aperture x c N = 4 N = 6 N = 8 N = 12 N = D spot

20 20 Averaging Error by Subapertures Averaging o the waveront over the inite size o the subaperture Number o subapertures (linear) : ns index o the Zernikes: n Error decreases with growing ns Error larger or higher order Zernikes n c in ns = 32 ns = 64 ns = 100 ns = 144 ns = c/c ns = 32 ns = 64 ns = 100 ns = 144 ns = n n Wrms Wpv

21 0.06 ns = 100 Averaging Error by Subapertures 0.04 Larger errors o Zernike polynomials (radial order 0.15n) or small number o sub-apertures (ns) 0.1 Larger gradients o high order Zernikes suers strongly rom averaging 0.02 PV value less sensitive as rms value Larger ns more accurate and stable 0 Wrms ns = 32 ns = 64 ns = 144 ns = ns = 32 ns = 64 ns = 100 ns = 144 ns = 200 n Wpv ns = 32 ns = 64 ns = 100 ns = 144 ns = 200 ns = 32 ns = 64 ns = 100 ns = 144 ns = n n n

22 22 Averaging o Subapertures: Example Determination o waveront o a microscopic lens Number ns o subapertures (linear): 16, 32, 64, 100 Calculated: 1. gradient o waveront 2. reconstructed waveront 3. errors Zernikes Errors due to averaging and shited center o the subaperture dw / dr c j in 16 5*r(AP) 16 W in real Nr j *r(AP)

23 23 Fresnel Number and Crosstalk Relative size o the spot in a HS WFS: determined by Fresnel number Small NF: large PSF, crosstalk o neighbouring apertures Larger error o centroid calculation or subapertures at the edge D D spot sub 2 D 2 sub 1 2N F I(x) x s /D sub N = 1 N = 2 N = 5 N = all subapertures only 1 neighbouring subaperture N F x

24 24 HS-WFS : Partly Illuminated Sub-Apertures Partly illuminated sub-aperture: change o centroid and error o signal partial illuminated lens ocal plane detector centroid Wrong signal or constant phase plateaus o x o x o subapertures z partly illuminated illumination

25 25 HS-WFS : Partly Illuminated Sub-Apertures Example Change o point spread unction due to partly illumination

26 26 Comparison HS-Sensor - Intererometer Feature HS- WFS PSIinter. 1 Complexity o the setup + 2 Cost o the equipment, additional high-quality components + 3 Spatial resolution + 4 Robust measurement under environmental conditions + 5 Noise due to image processing and pixelated sensor + 6 Perturbation by coherent scattering and straylight + 7 Algorithm or reconstructing the waveront + 8 Test systems with central obscuration + 9 Speed o measurement + 10 Absolute accuracy + 11 Measurement possible independently o wavelength, polarization and coherence +

27 27 Hartmann Method Similar to Hastmann Shack Method with simple hole mask and two measuring planes Measurement o spot center position as geometrical transverse aberrations Problems: broadening by diraction s' y s' 1 ( s' 2s' 1 ) y 1 y 1 y 2 Hartmann screen with hole mask test optic reerence plane irst measuring plane ocal plane second measuring plane y u' y 1 s' 1 y 2 s' y s' 2

28 28 Hartmann Method Schematic drawing o transverse aberrations Hartmann screen irst measuring plane ocal plane second measuring plane y' D ideal s 1 ' real with aberrations s y ' s 2 ' Distance o planes limited: overlap o spots Coherent coupling o sub-aperture ields, intererence induces errors o centroid

29 29 Hartmann Method: Pinhole Array Geometry Possible geometry o the pinholes: - number o pinholes, - size o holes - distance / geometry Parameters determine the accuracy 120 points 3000 positions or a 3.6 azimuthal scan a) polar b) cartesian c) hexagonal d) Albrecht grid

30 30 Hartmann Method Properties z-positions critical or large spots diameters No dependence on spectral range and polarization Coherence is critical, intererence or overlapping pinhole images Apodization not critical Averaging gives stable data evaluation

31 31 Hartmann Method Real pinhole pattern with signal Problems with cross talk and threshold

32 32 Hartmann Method Separated spots in case o diraction d d obj d d d d ( gesamt) 2 ' s ds d1 D 2.44 d s object plane Hartmann diaphragm ideal image plane 2. camera plane d s d 2 D obj h inite site source ield o view spot intensity d 1 broadening due to diraction geometrical projected size o pinhole D Airy /2 d' s D' Feld /2 total pinhole deocus and diraction convolution with ield o view

33 33 Hartmann Method in Case o Apodization Apodized beam: centroid rays pass through the perect image point A cetroid error is eliminated pinhole lens under test 1. measuring plane image plane 2. measuring plane intensity I(y) y transverse ray aberration centroid ray Hartmann diaphragm marginal ray Reconstruction o the transverse aberrations delivers the wave aberration W ( x, y) 1 R x x' dx 0

34 34 Hartmann Sensor Small power transmission transmission T 0.5 D sub = 0.5 mm D sub = 0.4 mm 0.2 D sub = 0.3 mm Problem: diraction spreading o light pencils D sub = 0.2 mm D sub = 0.1 mm N sub z image plane Hartmann screen z = - z = - /2 z = - /4 z = 0 z = /4 z = /2 z =

35 35 Hartmann Sensor Problem: diraction spreading o light pencils

36 36 Pyramidal Waveront Sensor Pyramidal components splits the waveront Signal evaluation analog to 4-quadrant detector beam proile at pyramid corresponding points pupil images in 4 quadrants 1 2 s c E E 1 1 E E 2 2 E E 3 3 E E 4 4 ocussed relay lens or pupil imaging 3 4 pyramid prism astigmatism pupil x-tilt

37 37 Knie Edge Method Moving a knie edge perpendicular through the beam cross section knie edge x movement Relationship between power transmission and intensity: Abel transorm or circular symmetry beam x y P( x) 2 x I( r) r dr r 2 2 d Example: geometrical spot with spherical aberration beore caustic zone rays below near paraxial ocus z

38 38 Indirect Waveront Sensing Foucault knie edge method Re: R. Kowarschik / J. Wyant

39 39 Zonal Aberration Eyepiece with strong zonal pupil aberration instrument pupil eyepiece lens and pupil o the eye retina Illumination or decentered pupil : dark zones due to vignetting caustic o the pupil image enlarged Kidney beam eect

40 40 Indirect Waveront Sensing Foucault knie edge method Re: R. Kowarschik / J. Wyant

41 41 Slit-Scan-Method Method very similar to moving knie edge Integration o slit length must be inverted: - inverse Radon transorm - corresponds to tomographic methods x moving slit d beam x y

42 42 General Filter Techniques Generalized concept: iltering the wave Realizations: 1. Foucualt knie edge 2. slit 3. Toepler schlieren method 4. Ronchi test 5. wire test 6. Lyot test (/4 wire)

43 43 General Filter Techniques Knie edge ilter or deocussing Changing intensity distribution as a unction o the ilter position

44 44 Ronchi Method Measurement o suraces by ringe deormation Grating creates reerence: ringe o 1st order ater Ronchi grating Evaluation o the lateral aberrations o the waveront by W x p x, R W y y R Explanation geometrical or wave-optical p lens under test Ronchigrating Relay optic +1. order 0. order -1. order

45 45 Ronchi Method Setup Problem: superposition o perturbing higher orders y p x p y surace under test L x y x R g Ronchigrating

46 46 Ronchi Method Ronchi pattern o low order aberrations Complex evaluation o patterns

47 Phase Space Analyzer Setup: y 1 slit 2s x 1 y 2 lens L x 2 y 3 toroidal lens T x 3 receiver plane y 4 a d x 4 First slit selects one cross section Lens images the slit 45 rotated toroidal lens inverts space coordinate/angle b z slit image

48 Cylindricalal lens under 45 Cylindrical lens under 45 : Focal lengths x = + T and y = - T Rotation by 45 Angle deviation u is transormed into spatial v oset Imaging o the slit: angle aberration u gives transverse aberration y x a) b) u x y v v u ocussing + T diverging - T toroidal lens T y

49 Phase Space Analyzer in Case o Sph Aberration z = 50 Image o a slit or a system with spherical aberration Image distance z is variied Cubic angle aberration more clearly seen in deocussed planes Not trivial: adjustment o proper slit width a) cross section perpendicular to slit y slit lens toroidal lens y' image plane b) cross section parallel to slit x slit lens toroidal lens x' a d b

50 Imaging o the slit Special case Slit coordinates: angle v creates deviation in x' perpendicular to the slit Special case: thin slit and toroidal lens under 45 Deocussing c 4 : slit rotates b a L d v b x b x L T L L 1 ' u b y b y L T L L 1 ' x b x L 1 ' u b y L T L ' ' AP L T L r c x b y Phase Space Analyzer

51 Phase Space Analyser Application: Measurement o microscopic lenses observation Three colored points in intermediate image plane Measurement o: - axial color - deocussing - spherical aberration objective toroidal lens azimuth 45 slit condenser pinholes collector source ilter microscopic test lens mirror