Library of Congress Cataloging-in-Publication Data

Tyson, Robert K., 1948-

Field guide to adaptive optics / Robert K. Tyson, Benjamin W. Frazier. -- 2nd ed

p. cm. -- (Spie field guides series; v. FG24)

Includes bibliographical references and index.

ISBN 978-0-8194-9017-9

1. Optics, Adaptive. 2. Optical detectors. 3. Optical measurements. I. Frazier, Benjamin W. (Benjamin West) II. Title.

TA1522.T93 2012

621.36′9--dc23

2011050389

Published by

SPIE

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First printing

## Introduction to the Series

Welcome to the * SPIE Field Guides*—a series of publications written directly for the practicing engineer or scientist. Many textbooks and professional reference books cover optical principles and techniques in depth. The aim of the

*is to distill this information, providing readers with a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. A significant effort will be made to provide a consistent notation and style between volumes in the series.*

**SPIE Field Guides**Each * SPIE Field Guide* addresses a major field of optical science and technology. The concept of these

*is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page, and provides full coverage of that topic on that page. Highlights, insights and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each*

**Field Guides***provide additional information such as related material outside the main scope of the volume, key mathematical relationships and alternative methods. While complete in their coverage, the concise presentation may not be appropriate for those new to the field.*

**Field Guide**The * SPIE Field Guides* are intended to be living documents. The modular page-based presentation format allows them to be easily updated and expanded. We are interested in your suggestions for new

*topics as well as what material should be added to an individual volume to make these Field Guides more useful to you. Please contact us at*

**Field Guide****fieldguides@SPIE.org**.

John E. Greivenkamp, **Series Editor**

College of Optical Sciences

The University of Arizona

## The Field Guide Series

Keep information at your fingertips with all of the titles in the Field Guide Series:

* Adaptive Optics*,

*, Robert K. Tyson & Benjamin W. Frazier*

**Second Edition*** Atmospheric Optics*, Larry C. Andrews

* Binoculars and Scopes*, Paul R. Yoder, Jr. & Daniel Vukobratovich

* Diffractive Optics*, Yakov G. Soskind

* Geometrical Optics*, John E. Greivenkamp

* Illumination*, Angelo Arecchi, Tahar Messadi, & R. John Koshel

* Image Processing*, Khan M. Iftekharuddin & Abdul A. Awwal

* Infrared Systems, Detectors, and FPAs*,

*, Arnold Daniels*

**Second Edition*** Interferometric Optical Testing*, Eric P. Goodwin & James C. Wyant

* Laser Pulse Generation*, Rüdiger Paschotta

* Lasers*, Rüdiger Paschotta

* Microscopy*, Tomasz Tkaczyk

* Optical Fabrication*, Ray Williamson

* Optical Fiber Technology*, Rüdiger Paschotta

* Optical Lithography*, Chris A. Mack

* Optical Thin Films*, Ronald R. Willey

* Polarization*, Edward Collett

**Probability, Random Processes, and Random Data*** Analysis*, Larry C. Andrews & Ronald L. Phillips

* Radiometry*, Barbara G. Grant

* Special Functions for Engineers*, Larry C. Andrews

* Spectroscopy*, David W. Ball

* Visual and Ophthalmic Optics*, Jim Schwiegerling

* Wave Optics*, Dan Smith

## Field Guide to Adaptive Optics, Second Edition

There have been a number of books and thousands of papers published with descriptions and mathematical expressions regarding adaptive optics. The material in this * Field Guide* is a summary of the methods for determining the requirements of an adaptive optics system, the performance of the system, and requirements for the components of the system. This second edition has a greatly expanded presentation of adaptive optics control system design and operation. Discussions of control models are accompanied by various recommendations for implementing the algorithms in hardware.

This book is not just another book on adaptive optics. There are already many fine volumes. This volume is intended for students, researchers, and practicing engineers who want a “go to” book when the calculation was “needed yesterday” (by a customer who won’t be paying for it until the next fiscal year).

Many of the expressions are in the form of integrals. When that is the case, we show the results graphically for a variety of practical values. Some of the material in this volume duplicates similar expressions found in other volumes of the * Field Guide* series. We have attempted to remain consistent with symbols of the other volumes. In some cases, however, we chose different symbols because they are well known within the adaptive optics literature.

Descriptions of the operation of subsystems and components and specific engineering aspects remain in the citations of the Bibliography.

This * Field Guide* is dedicated to the late Horace Babcock, whose pioneering ideas created the field of adaptive optics.

Robert K. Tyson

University of North Carolina at Charlotte

Ben W. Frazier

AOA Xinetics,

Northrop Grumman Aerospace Systems

## Table of Contents

**Glossary x**

**Introduction 1**

Conventional Adaptive Optics System 1

Image Spread with Atmospheric Turbulence 2

The Principle of Phase Conjugation 3

Point Spread Function for an Astronomical Telescope 4

**Modeling the Effect of Atmospheric Turbulence 5**

Fried’s Coherence Length 5

Isoplanatic Angle 6

Kolmogorov Model 7

Atmospheric Turbulence Models 8

Coherence Length for Various Wavelengths and Turbulence Models 9

Greenwood Frequency 10

Wind Models 11

Scintillation 12

Zernike Polynomials 13

Legendre Polynomials 15

Angle of Arrival (Tilt) Fluctuations (Image Motion) 17

Modulation Transfer Function 18

**Beam Propagation 19**

System Performance Estimation 19

Modal and Zonal Fitting Error 20

**Wavefront Sensors 21**

Partial Correction 21

Shack–Hartmann Wavefront Sensor and Error 22

Shack–Hartmann Lenslet Array Selection 24

Curvature Wavefront Sensor and Error 25

Pyramid Wavefront Sensor and Error 26

**Deformable Mirrors 27**

Photodiodes 27

Photodiode Noise 28

Lateral-Effect Position-Sensing Detectors 29

Quad Cells 30

Noise Equivalent Angle 32

The Strehl Ratio: Laser Beam Propagation to the Far Field with Wavefront Error 33

Strehl Ratio 34

Laser “Brightness” 35

Laser Beam Quality 36

Astronomical “Brightness” 37

Spot Size for a Gaussian Beam 38

Spot Size for a Uniform Circular Aperture 39

Temporal Error 40

Focal Anisoplanatism (the “Cone Effect”) 41

Laser Guide Stars 42

Subsystem Requirements: The Wavefront Sensor 44

Angular Isoplanatic Error 45

Subsystem Requirements: Tilt Mirror 46

Subsystem Requirements: How Many Actuators?

Zonal or Modal Control 47

Subsystem Requirements: Deformable Mirror 48

Deformable Mirror Actuator Configurations 49

Ferroelectric Actuators 50

Electrostatic Actuators 52

Voice Coil Actuators 53

Deformable Mirror Influence Function Models 54

Bimorph and MEMS Mirrors 55

Segmented Deformable Mirrors 56

**Control and Reconstruction 57**

Actuator and Wavefront Sensor Layouts 57

Correctability and Flattening of a Deformable Mirror 58

Adaptive Optics System Feedback Configuration 59

Deformable Mirror Dynamic Model 60

Controller Dynamic Model 61

Wavefront Sensor Dynamic Model 62

Latency 63

One-Dimensional Sampling 64

Two-Dimensional Sampling 66

Temporal Sampling Rate Selection 67

System Stability 68

General Control-System Parameters 69

Sensitivity Functions 70

Bandwidth Estimation from Controller Gains 72

Poke Matrix 73

Example System Geometry 74

Singular-Value Decomposition of the Poke Matrix 76

Actuator and Subaperture Observability 77

Identification of Actuator Locations 79

Poke Matrix Smoothing 80

Actuator Slaving: Active Actuator Identification 81

Actuator Slaving: Slave Logic 82

Adding Slaving into the Reconstructor 83

Tilt Removal 84

Piston and Waffle Removal 85

Reconstructor Generation: Least Squares 86

Reconstructor Generation: Regularization 87

T-Filter 89

Modal Suppression 90

Interactuator Shear Suppression 91

Nullspace Suppression 92

Weighting Matrices 93

Modal Feedback 94

Reconstructor Generation: Procedure 96

Reconstructor Comparison 97

Slope Discrepancy 98

Offloads and Woofer-Tweeter Systems 99

Open-Loop Wavefront Estimation 100

Kalman Filtering 101

Multivariable System Performance 102

Disturbance Injection 104

Wavefront Sensor Calibration 105

Centroiding and Thresholding 106

Misregistration 107

Subaperture Spillover 108

**Equation Summary 109**

**Bibliography 117**

**Index 121**

#### Glossary

a

Width of segment gap

A

Structure constant at the surface

ADC

Analog-to-digital converter

**A**_{Infl}

Influence function amplitude

**a**_{0}

Piezoelectric constant

b

Size of mirror segment

B

Laser brightness

**B**_{Astro}

Astronomical brightness

**B**

Poke matrix

**B**_{c}

Calibrated poke matrix

**B**_{r}

Reduced poke matrix

**B**_{s}

Smoothed poke matrix

c

Speed of light (=3×10^{8} m/s)

**c**_{a}

Interactuator coupling

**c**_{nm}

Fourier–Legendre coefficient

${C}_{n}^{2}$

Atmospheric turbulence structure constant

CCD

Charge-coupled device

d

Size of subaperture (in object space)

d

Separation of the membrane and the addressing electrode

**d**_{0}

Characteristic distance of a laser guide star

D

Aperture diameter

DAC

Digital-to-analog converter

**d**

Vector of wavefront disturbances

**e**

Vector of calculated wavefront errors

E

Pulse energy of laser

**e**_{n}

Read-noise in electrons per pixel

f

Focal length

**f**_{BW}

Closed-loop bandwidth

**f**_{c}

Crossover frequency

**f**_{G}

Greenwood frequency

F

Focal length of the system

**F**_{Rayleigh}

Return flux for Rayleigh guide star

**F**_{Sodium}

Return flux for sodium guide star

**g**_{1}

Loop gain

**g**_{2}

Leak gain

G

Gain

**G**

Generalized system plant model

h

Planck’s constant (=6.626×10^{−34} J $\cdot $ s)

h

Altitude

**H**_{T}

Height of the tropopause

H–V

Hufnagel–Valley

* I*(

*)*

**r**Intensity distribution

**I**_{Aper}

Intensity at the circular aperture

**I**_{d}

Dark current

**I**_{Gap}

Diffracted energy from gaps

**I**_{J}

Johnson noise current

**I**_{n}

Noise current

**I**_{p}

Photocurrent

**I**_{s}

Shot noise current

**I**_{0}

On-axis intensity

**I**

Identity matrix

**J**_{1}

Bessel function

k

Wavenumber

k

Sample time index

**k**_{B}

Boltzmann constant (= 1.38 × 10^{−23} J/K)

K

Aperture shape parameter for beam propagation

**K**_{g}

Increase in error at the null

**K**

Controller model

**l**_{0}

Inner scale of turbulence

L

Propagation distance

**L**_{T}

Thickness of the tropopause

**L**_{0}

Outer scale of turbulence

m

Azimuthal index for Zernike polynomials

m

Number of wavefront measurements in influence matrix

**m**_{v}

Visual magnitude of a star

M

Magnification

**M**^{2}

Beam quality

**M**

Modal feedback matrix

**M**_{r}

Reduced modal feedback matrix

MTF

Modulation transfer function

n

Radial index for Zernike polynomials

**n**

Vector of measurement noise

**n**_{a}

Number of actuators

**n**_{B}

Number of detected background photo-electrons per subaperture

**n**_{m}

Number of measurements

**n**_{p}

Number of detected photoelectrons per subaperture

**n**_{s}

Number of slaved actuators

**n**_{v}

Number of active actuators

**n**_{R}

Rayleigh scattering density

**N**_{Act}

Number of actuators

**N**_{D}

Number of pixels in a subaperture

NEA

Noise equivalent angle

NEP

Noise equivalent power

**N**_{p}

Photon count

**N**_{Zern}

Number of Zernike modes

**O**

Offload matrix

OPD

Optical path difference

p

Curvature sensor image plane offset

P

Optical power

**P**_{n}

Legendre polynomial of order **n**

**P**

Error propagation covariance matrix

**P**_{p}

Piston projection matrix

**P**_{t}

Tilt projection matrix

**P**_{w}

Waffle projection matrix

PSD

Position sensing detector

PSF

Point spread function

q

Electron charge (= 1.6 × 10^{−19} C)

**Q**_{d}

Atmospheric disturbance covariance

**Q**_{n}

Measurement noise covariance

r

Radial coordinate

**r**_{a}

Actuator pad radius

**r**_{c}

Interactuator spacing

**r**_{m}

Mirror radius

**r**_{s}

Radius of supporting ring

**r**_{0}

Coherence length of the atmosphere

RMS

Root mean square

**r**

Vector coordinate in the wavefront

R

Radius of circle

R

Photodiode responsivity

**R**

Reconstructor

**R**_{r}

Reduced reconstructor

RSS

Root sum square

$\Re $

Zernike radial polynomial

s

Shear distance

s

Summing index in the Zernike radial polynomial

s

Laplace transform variable

**s**_{Act}

Distance between actuators

S

Strehl ratio

**S**

Sensitivity function

**S**_{d}

Slope discrepancy matrix

**S**_{l}

Slave logic matrix

**S**_{w/jit}

Strehl ratio including effects of jitter

SLC

Strategic laser communications

SNR

Signal-to-noise ratio

SVD

Singular value decomposition

t

Thickness of the bimorph

**t**_{d}

Latency

T

Matrix transpose operator

T

Transmission of the optics

**T**

Complementary sensitivity function

**T**_{A}

Transmission of the atmosphere

**T**_{m}

Tension of the membrane

**T**_{s}

Sampling time

**u**

Vector of actuator commands

**v**_{G}

Wind velocity at low altitude

**v**_{T}

Wind velocity at the tropopause

**v**_{w}

Wind velocity as a function of altitude

* v*(

*)*

**z**_{Bufton}

Wind velocity of the Bufton model

V

Applied voltage

w

Gaussian beam radius

**w**_{0}

Gaussian beam waist

* W*(

*,*

**x***)*

**y**Wavefront

W

Wind velocity aloft

**W**

Weighting matrix

WCE

Wavefront control experiment

WFS

Wavefront sensor

* x*,

**y**Cartesian coordinates

**y**

Corrected output wavefront

z

Coordinate along propagation path

z

Altitude (if propagation path is vertical)

z

Z-transform variable

α_{jit}

Root-mean-square average jitter

$(\mathrm{\Delta}\phi {)}^{2}$

Wavefront error variance (distance squared)

δ

Hartmann spot shift

Δλ

Spectral bandwidth

ε_{0}

Permittivity

${\epsilon}_{0}$

Phase

$\phi $

Deformable mirror influence function (surface deflection)

$\phi $

Wind direction relative to the telescope aperture

Φ(* r*)

Phase

η

Detector efficiency

κ

Fitting constant

λ

Wavelength

λ_{LGS}

Laser wavelength

ν

Spatial frequency

ν_{G}

Wind velocity at low altitude

θ

Angular size of reference source

θ_{d}

Half-angle beam divergence

θ_{0}

Isoplanatic angle

ρ

Discrete-time equivalent latency

ρ_{Col}

Sodium column abundance

σ_{fitting}

Wavefront fitting error

σ_{Na}

Resonant backscatter cross section

σ_{R}

Rayleigh scattering cross section

σ_{Temp}

Temporal wavefront error

σ_{Tilt}

Wavefront tilt

σ_{WFS}

Wavefront sensor measurement error

σ^{2}

Wavefront error variance (radians squared)

${\sigma}_{I}^{2}$

Intensity variance

${\sigma}_{n}^{2}$

Measurement error variance

${\sigma}_{\chi}^{2}$

Log-amplitude variance

${\sigma}_{\phi}$

Root-mean-square wavefront error over a subaperture

τ

Actuator rise time

Ω_{N}

Nyquist frequency

ζ

Zenith angle