Metrology and Sensing

Transcription

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

2 2 Preliminary Schedule No Date Subject Detailed Content Introduction Introduction, optical measurements, shape measurements, errors, definition of the meter, sampling theorem Wave optics (ACP) Basics, polarization, wave aberrations, PSF, OTF Sensors Introduction, basic properties, CCDs, filtering, noise Fringe projection Moire principle, illumination coding, fringe projection, deflectometry Interferometry I (ACP) 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 Hartmann-Shack WFS Hartmann method Miscellaneous methods

4 4 Hartmann Shack Wavefront Sensor Lenslet array divides the wavefront into subapertures Every lenslet generates a simgle spot in the focal plane The averaged local tilt produces a transverse offset of the spot center Integration of the derivative matrix delivers the wave front W(x,y) array detector wavefront D array h f D meas u x spot offset D sub refractive index n

5 5 Spot Pattern of a HS - WFS Aberrations produce a distorted spot pattern Calibration of the setup for intrinsic residual errors Problem: correspondence of the spots to the subapertures

6 6 Hartmann Shack Wavefront Sensor Typical setup for component testing detector test surface lenslet array fiber illumination Lenslet array collimator beamsplitter telescope for adjustment of the diameter

7 7 Array Signal Lenslet array ideal signal array of phase spot pattern cross section of the spot pattern Real signal: 1. discretization 2. quantization 3. noise original discretized and quantized with noise

8 8 HS - WFS : Size of Sub-Apertur Dynamic range: ratio of spot diameter to size of sub-aperture Averaging of wavefront slope inside sub-aperture x p lens focal length f point spread function I(x) D spot /2 D sub incoming wave front W(x p ) D spot /2

9 9 Setup of a Hartmann-Shack - WFS 1. Simple setup detector test surface lenslet array fiber illumination 2. With telecope and relay lens collimator beamsplitter telescope for adjustment of the diameter pupil plane straylight stop telescope array focal plane relay lens sensor D pup D arr f arr m rel

10 10 General Setup of a HS - WFS Generalized setup: adaptation of diameter Wavefront is scaled pupil plane W W, sens pup D sens D pup sensor plane D pup D sens W pv f 1 f 1 f 2 f 2 W' pv Relay lens Adaptation of sensor size array focal plane relay lens sensor x x' f arr

11 11 Real Measurement of a HS-WFS Problem in practice: definition of the boundary

12 12 Real Measurement of a HS-WFS Problem in practice: exact determination of the spot centroid: - noise - discretization - quantization - broadening by partial coherence - broadening by local curvature - error by centroid affecting coma - error by partly illuminated pixels original discretized and quantized with noise

13 13 Parametrization of a HS-WFS Layout parametrization: Fresnel number Fill factor Spot size N F D D D D D spot sub spot 2 2 Dmeas D f N 4 f meas array 1 2N F sub sub F sub sensitivity 2 f Dsub 2 f N D 2N D meas sub f [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 f mrel max D h 2 f sub 2

14 14 Parametrization of a HS-WFS Wavefront detectability f [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 f

15 15 Problems with a Hartmann-Shack - Sensor Die Wellenfläche wird über die Subapertur gemittelt Fresnelzahl der Arraylinse, Spotgröße / Subaperturdurchmesser Auflösung, mit Pixelgröße p und Subaperturzahl N N W F min 2 sub 4 f 4p N N mess f Zuordnung der Spots zu den Subaperturen, Dynamikbereich Spotablage subpixelgenau messen x f n W x Probleme mit teilausgeleuchteten Subaperturen Keine Probleme mit spektraler Breite, Kohärenz und Polarisation

16 16 Errors in the HS - Wavefrontsensor Tilted sensor plane Rotated sensor in the azimuth Scattering of focal lengths of the lenslets Average of slope inside the subaperture area Errors in the wavefront reconstruction algorithms Coma of lenses Wrong focal length due to dispersion for different wavelength Sensor plane not exactly matched with focal plane Partly illuminated lenslets Electronical noise Zernike errors due to bad known normalization radius / edge of pupil Geometrical distortions of the array Truncation of spot by the corresponding subaperture / cross talk Discrete finite number of pixels Quantization of signal on the detector

17 17 Dynamic Range due to Local Curvature Theoretical largest curvature: R = f Real size of point spread function: Larger curvature: cross talk generates errors R min f N F a) b) R R D sub D sub D spot f f

18 18 HS-WFS : Discretization Errors Signal errors due to finite pixel size discretization of the point spread function N: number of pixels per sub-aperture x c N = 4 N = 6 N = 8 N = 12 N = D spot

19 19 Averaging Error by Subapertures Averaging of the wavefront over the finite size of the subaperture Number of subapertures (linear) : ns index of the Zernikes: n Error decreases with growing ns Error larger for 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

20 0.06 ns = 100 Averaging Error by Subapertures 0.04 Larger errors of Zernike polynomials (radial order 0.15n) for small number of sub-apertures (ns) 0.1 Larger gradients of high order Zernikes suffers strongly from 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

21 21 Averaging of Subapertures: Example Determination of wavefront of a microscopic lens Number ns of subapertures (linear): 16, 32, 64, 100 Calculated: 1. gradient of wavefront 2. reconstructed wavefront 3. errors Zernikes Errors due to averaging and shifted center of the subaperture dw / dr c j in 16 5*r(AP) 16 W in real Nr j *r(AP)

22 22 Fresnel Number and Crosstalk Relative size of the spot in a HS WFS: determined by Fresnel number Small NF: large PSF, crosstalk of neighbouring apertures Larger error of centroid calculation for subapertures at the edge D D spot sub 2 f D 2 sub 1 2N F I(x) x s /D sub N f = 1 N f = 2 N f = 5 N f = all subapertures only 1 neighbouring subaperture N F x

23 23 HS-WFS : Partly Illuminated Sub-Apertures Partly illuminated sub-aperture: change of centroid and error of signal partial illuminated lens focal plane detector centroid Wrong signal for constant phase plateaus o x o x o subapertures f z partly illuminated illumination

24 24 HS-WFS : Partly Illuminated Sub-Apertures Example Change of point spread function due to partly illumination

25 25 Comparison HS-Sensor - Interferometer Feature HS- WFS PSIinterf. 1 Complexity of the setup + 2 Cost of 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 for reconstructing the wavefront + 8 Test systems with central obscuration + 9 Speed of measurement + 10 Absolute accuracy + 11 Measurement possible independently of wavelength, polarization and coherence +

26 26 Hartmann Method Similar to Hastmann Shack Method with simple hole mask and two measuring planes Measurement of spot center position as geometrical transverse aberrations Problems: broadening by diffraction s' y s' 1 ( s' 2s' 1 ) y 1 y 1 y 2 Hartmann screen with hole mask test optic reference plane first measuring plane focal plane second measuring plane y u' y 1 s' 1 y 2 s' y s' 2

27 27 Hartmann Method Schematic drawing of transverse aberrations Hartmann screen first measuring plane focal plane second measuring plane y' D ideal s 1 ' real with aberrations s y ' s 2 ' Distance of planes limited: overlap of spots Coherent coupling of sub-aperture fields, interference induces errors of centroid

28 28 Hartmann Method: Pinhole Array Geometry Possible geometry of the pinholes: - number of pinholes, - size of holes - distance / geometry Parameters determine the accuracy 120 points 3000 positions for a 3.6 azimuthal scan a) polar b) cartesian c) hexagonal d) Albrecht grid

29 29 Hartmann Method Properties z-positions critical for large spots diameters No dependence on spectral range and polarization Coherence is critical, interference for overlapping pinhole images Apodization not critical Averaging gives stable data evaluation

30 30 Hartmann Method Real pinhole pattern with signal Problems with cross talk and threshold

31 31 Hartmann Method Separated spots in case of diffraction d d obj d d d d ( gesamt) 2 ' s ds d1 f D 2.44 d s object plane Hartmann diaphragm ideal image plane 2. camera plane d s d 2 D obj h finite site source field of view spot intensity f d 1 broadening due to diffraction geometrical projected size of pinhole D Airy /2 d' s D' Feld /2 total pinhole defocus and diffraction convolution with field of view

32 32 Hartmann Method in Case of Apodization Apodized beam: centroid rays pass through the perfect 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 of the transverse aberrations delivers the wave aberration W ( x, y) 1 R x x' dx 0

33 33 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: diffraction spreading of light pencils D sub = 0.2 mm D sub = 0.1 mm N sub z image plane Hartmann screen z = - f z = - f/2 z = - f/4 z = 0 z = f/4 z = f/2 z = f

34 34 Hartmann Sensor Problem: diffraction spreading of light pencils

35 35 Pyramidal Wavefront Sensor Pyramidal components splits the wavefront Signal evaluation analog to 4-quadrant detector beam profile 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 focussed relay lens for pupil imaging 3 4 pyramid prism astigmatism pupil x-tilt

36 36 Knife Edge Method Moving a knife edge perpendicular through the beam cross section knife edge x movement Relationship between power transmission and intensity: Abel transform for circular symmetry beam x y P( x) 2 x I( r) r dr r 2 2 d Example: geometrical spot with spherical aberration before caustic zone rays below near paraxial focus z

37 37 Indirect Wavefront Sensing Foucault knife edge method Ref: R. Kowarschik / J. Wyant

38 38 Zonal Aberration Eyepiece with strong zonal pupil aberration instrument pupil eyepiece lens and pupil of the eye retina Illumination for decentered pupil : dark zones due to vignetting caustic of the pupil image enlarged Kidney beam effect

39 39 Indirect Wavefront Sensing Foucault knife edge method Ref: R. Kowarschik / J. Wyant

40 40 Slit-Scan-Method Method very similar to moving knife edge Integration of slit length must be inverted: - inverse Radon transform - corresponds to tomographic methods x moving slit d beam x y

41 41 General Filter Techniques Generalized concept: filtering the wave Realizations: 1. Foucualt knife edge 2. slit 3. Toepler schlieren method 4. Ronchi test 5. wire test 6. Lyot test (/4 wire)

42 42 General Filter Techniques Knife edge filter for defocussing Changing intensity distribution as a function of the filter position

43 43 Ronchi Method Measurement of surfaces by fringe deformation Grating creates reference: fringe of 1st order after Ronchi grating Evaluation of the lateral aberrations of the wavefront 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

44 44 Ronchi Method Setup Problem: superposition of perturbing higher orders y p x p y surface under test L x y x R g Ronchigrating

45 45 Ronchi Method Ronchi pattern of low order aberrations Complex evaluation of patterns