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Wavefront Sensing for Adaptive Optics MARCOS VAN DAM & RICHARD CLARE W.M

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Wavefront Sensing for Adaptive Optics MARCOS VAN DAM & RICHARD CLARE W.M Wavefront sensing for adaptive optics MARCOS VAN DAM & RICHARD CLARE W.M. Keck Observatory Acknowledgments Wilson Mizner : "If you steal from one author it's plagiarism; if you steal from many it's research." Thanks to: Richard Lane, Lisa Poyneer, Gary Chanan, Jerry Nelson Outline Wavefront sensing Shack-Hartmann Pyramid Curvature Phase retrieval Gerchberg-Saxton algorithm Phase diversity Properties of a wave-front sensor Localization: the measurements should relate to a region of the aperture. Linearization: want a linear relationship between the wave-front and the intensity measurements. Broadband: the sensor should operate over a wide range of wavelengths. => Geometric Optics regime BUT: Very suboptimal (see talk by GUYON on Friday) Effect of the wave-front slope A slope in the wave-front causes an incoming photon to be displaced by x = zWx There is a linear relationship between the mean slope of the wavefront and the displacement of an image Wavelength-independent W(x) z x Shack-Hartmann The aperture is subdivided using a lenslet array. Spots are formed underneath each lenslet. The displacement of the spot is proportional to the wave-front slope. Shack-Hartmann spots 45-degree astigmatism Typical vision science WFS Lenslets CCD Many pixels per subaperture Typical Astronomy WFS Former Keck AO WFS sensor 21 pixels 2 mm 3x3 pixels/subap 200 μ lenslets CCD relay lens 3.15 reduction Centroiding The performance of the Shack-Hartmann sensor depends on how well the displacement of the spot is estimated. The displacement is usually estimated using the centroid (center-of-mass) estimator. x I(x, y) y I(x, y) s = sy = x I(x, y) I(x, y) This is the optimal estimator for the case where the spot is Gaussian distributed and the noise is Poisson. G-tilt vs Z-tilt The centroid gives the mean slope of the wavefront (G-tilt). However, we usually want the least-mean-squares slope (Z-tilt). Centroiding noise Due to read noise and dark current, all pixels are noisy. Pixels far from the center of the subaperture are multiplied by a large number: s x I(x, y) x = x = {L,3,2,1,0,1,2,3,L} The more pixels you have, the noisier the centroid estimate! Weighted centroid The noise can be reduced by windowing the centroid: Weighted centroid Can use a square window, a circular window: Or better still, a tapered window, w(x,y) s xw(x, y) I(x, y) x = s yw(x, y) I(x, y) y = Correlation (matched filtering) Find the displacement of the image that gives the maximum correlation: (sx , sx ) = arg max(w(x, y)I(x, y)) = Use FFT or quadratic interpolation to find the subpixel maximum correlation Correlation (matched filtering) Noise is independent of number of pixels Much better noise performance for many pixels Estimate is independent of uniform background errors Estimate is relatively insensitive to assumed image. Quad cells In astronomy, wavefront slope measurements are often made using a quad cell (2x2 pixels) Quad cells are faster to read and to compute the centroid and less sensitive to noise I1 + I2 I3 I4 I1 I2 + I3 I4 sx = sy = I1 + I2 + I3 + I4 I1 + I2 + I3 + I4 Quad cells These centroid is only linear with displacement over a small region (small dynamic range) Centroid is proportional to spot size Centroid vs. displacement for different spot sizes Centroid Displacement Denominator-free centroiding When the photon flux is very low, noise in the denominator increases the centroid error Centroid error can be reduced by using the average value of the denominator I I I I I1 + I2 I3 I4 1 2 + 3 4 s = sy = x E[I I I I ] E[I1 + I2 + I3 + I4 ] 1 + 2 + 3 + 4 Laser guide elongation Shack-Hartmann subapertures see a line not a spot LGS elongation at Keck Laser projected from right A possible solution for LGS elongation Radial format CCD Arrange pixels to be at same angle as spots Currently testing this design for TMT laser Pyramid wave-front sensor Aperture plane Focal plane Pyramid (glass prism) Lens to image the aperture Images of the aperture (conjugate aperture plane) Pyramid wave-front sensor Similar to the Shack-Hartmann using quad cells: it measures the average slope over a subaperture. The subdivision occurs at the image plane, not the pupil plane. Local slope determines which image receives the light Pyramid wave-front sensor non-linearity When the aberrations are large, the pyramid sensor is very non-linear. Large focus aberration 4 pupil images x- and y-slopes estimates. Modulation of pyramid sensor Without modulation: With modulation: Linear over spot width Linear over modulation width Pyramid + lens = 2x2 lenslet array + Pyramid Relay lens lenslets Duality between Shack-Hartmann and pyramid Shack-Hartmann Pyramid Object Aperture High resolution Aperture image of the object Low resolution Low resolution images of the object images of the aperture Duality between Shack-Hartmann and pyramid Shack-Hartmann Pyramid Duality between Shack-Hartmann and pyramid Aperture Focal Plane Shack-Hartmann Pyramid Pixels in Shack-Hartmann = lenslets in Pyramid Lenslets in pyramid = pixels in Shack-Hartmann Multi-sided prisms Pyramid uses 4-sided glass prism at focal plane to generate 4 aperture images Can use any N-sided prism to produce N aperture images Limit as N tends to Infinity gives the “cone” sensor Aperture Cone Relay lens Aperture image Curvature sensing Image 2 -z Aperture Wave-front at aperture z Image 1 Curvature sensing Localization comes from the short effective propagation distance, f ( f l) Aperture z = l Linear relationship between the f Defocused image I curvature in the aperture and the 1 l normalized intensity difference: Broadband light helps reduce Defocused diffraction effects. image I2 Curvature sensing Using the irradiance transport equation, I = I2W I.W z Where I is the intensity, W is the wave-front and z is the direction of propagation, we obtain a linear, first-order approximation, I I 2 I 2 1 = z W + zW. I2 + I1 I which is a Poisson equation with Neumann boundary conditions. Solution at the boundary If the intensity is constant at the aperture, I I H (x R zW ) H (x R + zW ) 1 2 = x x I1 + I 2 H (x R zWx ) + H (x R + zWx ) I1 I2 I1- I2 Solution inside the boundary Curvature I1 I 2 = z(Wxx +Wyy ) I1 + I 2 There is a linear relationship between the signal and the curvature The sensor is more sensitive for large effective propagation distances Curvature sensing As the propagation distance, z, increases, I I Sensitivity increases. 1 2 = z(W +W ) I + I xx yy Spatial resolution decreases. 1 2 Diffraction effects increase. The relationship between the signal, (I1- I2)/(I1+ I2) and the curvature, Wxx + Wyy, becomes non-linear Subaru AO system will use two different propagation distances A large distance for high sensitivity A short distance for high spatial resolution Curvature sensing Practical implementation uses a variable curvature mirror (to obtain images below and above the aperture) and a single detector. Curvature sensor subapertures Measure intensity in each subaperture with an avalanche photo-diode (APD) Detect individual photons – no read noise Wavefront sensing from defocused images There are more accurate, non-linear, algorithms to reconstruct the wavefront using defocused images with many pixels Defocused images True and reconstructed wavefronts Phase retrieval Suppose we have an image and knowledge about the pupil. Can we find the phase, , that resulted in this image? Phase retrieval Image is insensitive to: Addition of a constant to (x). Piston does not affect the image Addition of a multiple of 2 to any point on (x) Phase wrapping Replacing (x) by -(-x) if amplitude is symmetrical e.g., positive and negative defocused images look identical Called the phase ambiguity problem Gerchberg-Saxton algorithm Phase diversity Phase retrieval suffers from phase ambiguity, slow convergence, algorithm stagnation and sensitivity to noise These problems can be overcome by taking two or more images with a phase difference between them In AO, introduce defocus by moving a calibration source. Phase diversity Defocus +2 mm -2 mm -4 mm Phase diversity Poked actuators Minus poke phase Plus poke phase Difference Phase diversity Theoretical diffraction-limited image Measured image Mahalo!.
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