Fundamentals of adaptive optics and wavefront reconstruction Marcos van Dam Institute for Geophysics and Planetary Physics, Lawrence Livermore National Laboratory

Outline 1.Introduction to adaptive optics 2.Wavefront sensors Shack-Hartmann sensors Pyramid sensors Curvature sensors 3.Wavefront reconstructors Least-squares Modal reconstructors 4.Dynamic control problem

Uranus and Titan Courtesy: De Pater Courtesy: Team Keck.

Adaptive optics

Effect of the wave-front slope A slope in the wave-front causes an incoming photon to be displaced by xx z W(x)

Shack-Hartmann wave-front sensor 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 wave-front sensor The centroid (center-of-mass) is proportional to the mean slope across the subaperture. Centroid estimate diverges with increasing detector area due to diffraction and with increasing pixels due to measurement noise. Correlation or maximum-likelihood methods can be used.

Wave-front x- and y-slope measurements are usually made in each subaperture using a quad cell (2 by 2). Quad cells are faster to read and to compute the centroid. Quad cells

These centroid is only linear with displacement over a small region. Centroid is proportional to spot size. Quad cells Displacement Centroid Centroid vs. displacement for different spot sizes

Similar to the Shack-Hartmann, it measures the average slope over a subaperture. The subdivision occurs at the image plane, not the pupil plane. Less affected by diffraction. Pyramid wave-front sensor

Wave-front at aperture Aperture Image 1 z -z Image 2 Curvature sensing

Practical implementation uses a variable curvature mirror (to obtain images below and above the aperture) and a single detector.

Curvature sensing Where I is the intensity, W is the wave-front and z is the direction of propagation, we obtain a linear, first- order approximation, Using the irradiance transport equation, which is a Poisson equation with Neumann boundary conditions.

Curvature sensing Solution inside the boundary, Solution at the boundary, I1I2I1- I2I1I2I1- I2

Curvature sensing As the propagation distance, z, increases, Sensitivity increases. Spatial resolution decreases. Diffraction effects increase. The relationship between the signal, (I 1 - I 2 )/(I 1 + I 2 ) and the curvature, W xx + W yy, becomes non-linear.

Faint companions

Wave-front reconstruction There is a linear relationship between wave-front derivative and a measurement. Don’t want to know the wave-front derivative, but the wave- front or, better, the actuator commands. Need to know the relationship between actuator commands and measurement.

Actuators: Shack-Hartmann The lenslets are usually located such that the actuators of the deformable mirror are at the corners of the lenslets. Piston mode, where all the actuators are pushed up, is invisible to the wave-front as there is no overall slope. Waffle mode, where the actuators are pushed up and down in a checkerboard pattern, is also invisible.

System matrix The system matrix, H, describes how pushing each actuator, a, affects the centroid measurements, s. It is created by pushing one actuator at a time and measuring the change in centroids. Actuators Centroids

System matrix Alternatively, the system matrix can be computed theoretically using finite differences to approximate the derivatives: Another formulation is using Fourier transforms (faster than matrix multiplication). d

Actuators: Curvature Bimorph mirrors are usually used, which respond to an applied voltage with a surface curvature. The electrodes have the same radial geometry as the subapertures. Curvature sensors tend to be low order.

Reconstruction matrix We have the system matrix: We need a reconstruction matrix to convert from centroid measurements into actuator voltages: Need to invert the 2N (centroids) by N (actuators) H matrix. For well-conditioned H matrices a least-squares algorithm suffices: unsensed modes, such as overall piston, p, and waffle, w, are thrown out. Equivalently, use singular value decomposition.

Reconstruction matrix Most modes have local waffle but no global waffle. Hence, must regularize before inverting.

Reconstruction matrix 1.Penalize waffle in the inversion, e.g., using the inverse covariance matrix of Kolmogorov turbulence, and a noise-to-signal parameter, (Bayesian reconstructor). SVD Bayesian

Reconstruction matrix Comparison of reconstruction matrices SVD Bayesian

Reconstruction matrix Comparison of reconstruction matrices SVD Bayesian

Reconstruction matrix 2. Only reconstruct certain modes, z i, (modal reconstruction).

Control problem Wave-front sensing in adaptive optics is not only an estimation problem, it is a control problem. There are inherent delays in the loop due to Integration time of the camera Computation delays The AO system should minimize bandwidth errors while maintaining loop stability. The propagation of measurement noise through the loop also needs to be minimized.

Modeling the system dynamics Model the dynamic behavior of the AO system using the transfer function of each block. Deformable mirrorCentroid measurements

Modeling the system dynamics The turbulence rejection curve can be calculated from a model of the AO system.

Modeling the system dynamics We can calculate the bandwidth and noise terms from a combination of data from the telescope and modeling the system. Noise Bandwidth errors

Laser guide stars Shine a 589 nm W laser in the direction of the atmosphere. Sodium atoms at an altitude of 90 km are excited by this light and re-emit. The return can be used as a guide star.

Laser guide stars The laser is equally deflected on the way up and down, so can’t be used to measure tilt. The guide star is not at infinity, so the focus is different. Hence, need a natural guide star as well (but can be much fainter).

This work was performed under the auspices of the US Department of Energy by the University of California, Lawrence Livermore National Laboratory, under contract W-7405-Eng-48. The work has been supported by the National Science Foundation Science and Technology Center for Adaptive Optics, managed by the University of California at Santa Cruz under cooperative agreement No. AST W. M. Keck Observatory has supported this work. Acknowledgements

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