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

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 optics2. 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 xzWx

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.

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• 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

Cen

troid

Cen

troid

vs.

dis

plac

emen

t for

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eren

t spo

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• 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

Curvature sensing

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

Curvature sensing

WIWIzI

.2

IIWzWz

IIII

.2

12

12

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

)(21

21yyxx WWz

IIII

• Solution inside the boundary,

• Solution at the boundary,

)()()()(

21

21

xx

xx

zWRxHzWRxHzWRxHzWRxH

IIII

+ -

I1

I2

I1- I2

Curvature sensing

As the propagation distance, z, increases,

• Sensitivity increases.• Spatial resolution decreases.• Diffraction effects increase.• The relationship between the signal, (I1- I2)/(I1+ I2) and the curvature, Wxx + Wyy, 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

Has

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).

daass jiji

xji

xji ,,1,1,

2

d

daass jiji

yji

yji ,1,1,,

2

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.

Rsa

Has

TTTT HwwppHHR 1)(

,...]1,1,1,1,1[,...]1,1,1,1,1,1[

wp

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).

TT HCHHR 11)(

C

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, zi, (modal reconstruction).

TT HZHZHZZR )()]()[( 1

...],,[ ,321 zzzZ

Keck I I AO + NI RSPEC (J une/ 00)

HST: J,H,K composite

1.6 arc sec

Bright nucleus

Faint nucleus

Gas in betweenx

2.00 m 2.08 m 2.17 m

H2 emission line f rom shocked or ionized gas (v=580 km/ s)

Keck: K-band Active galaxyNGC6240:Max et al.

R = 12.5 NGS,35” separation

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 mirror Centroid measurements

Modeling the system dynamics

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

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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 10-20 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-9876783.

• W. M. Keck Observatory has supported this work.

Acknowledgements

90 sec integration90 sec integration

Thank you!