Optimizing closed-loop adaptive-optics performance with ...users.auth.gr/pitsiani/reprints/J-004-AOJOSAA94.pdf · Fig. 1. Adaptive-optics system control-loop dynamics: (a) an adaptive-optics

Vol. 11, No. 11/November 1994/J. Opt. Soc. Am. A 2871

Optimizing closed-loop adaptive-optics performancewith use of multiple control bandwidths

Brent L. Ellerbroek

Starfire Optical Range, U.S. Air Force Phillips Laboratory, Kirtland Air Force Base, New Mexico 87117

Charles Van Loan and Nikos P. Pitsianis

Department of Computer Science, Cornell University, Ithaca, New York 14853

Robert J. Plemmons

Department of Mathematics and Computer Science, Wake Forest University, Winston-Salem, North Carolina 27109

Received February 17, 1994; revised manuscript received June 24, 1994; accepted June 24, 1994

The performance of a closed-loop adaptive-optics system may in principle be improved by selection of distinctand independently optimized control bandwidths for separate components, or modes, of the wave-front-distortion profile. We describe a method for synthesizing and optimizing a multiple-bandwidth adaptive-optics control system from performance estimates previously derived for single-bandwidth control systemsoperating over a range of bandwidths. The approach is applicable to adaptive-optics systems that use eitherone or several wave-front sensing beacons and also to systems that include multiple deformable mirrors foratmospheric-turbulence compensation across an extended field of view. Numerical results are presented forthe case of an atmospheric-turbulence profile consisting of a single translating phase screen with Kolmogorovstatistics, a Shack-Hartmann wave-front sensor with from 8 to 16 subapertures across the aperture of thetelescope, and a continuous-face-sheet deformable mirror with actuators conjugate to the corners of the wave-front-sensor subapertures. The use of multiple control bandwidths significantly relaxes the wave-front-sensornoise level that is permitted for the adaptive-optics system to operate near the performance limit imposed byfitting error. Nearly all of this reduction is already achieved through the use of a control system that usesonly two distinct bandwidths, one of which is the zero bandwidth.

1. INTRODUCTION

Closed-loop adaptive-optics systems must compensatefor time-varying wave-front distortions on the basis ofnoisy sensor measurements. Time-varying distortionsare most accurately corrected by a system with a highcontrol bandwidth, but the noise in wave-front-sensormeasurements is best rejected by reduction of the controlbandwidth and temporal smoothing of the wave-front cor-rection to be applied. These trends lead to a performancetrade-off between the residual wave-front distortions thatare due to sensor noise and servo lag, and the optimumcontrol bandwidth that minimizes the sum of these twoeffects will depend on the wave-front-sensor noise leveland the temporal characteristics of the wave-front errorsto be corrected. Most adaptive-optics systems devel-oped to date have addressed this trade-off through theuse of a common control bandwidth for all componentsof the wave-front-distortion profile,' 3 but systems em-ploying several different control bandwidths for sepa-rate wave-front components have been studied 4

-7 and

recently tested in the atmosphere. 81 2 Because wave-front-sensor noise statistics and the temporal dynamicsof the wave-front distortion to be corrected vary as afunction of spatial frequency,6 13 it should in principle bepossible to improve adaptive-optics performance by usingthis more sophisticated control approach. In this paperwe describe a technique for evaluating and optimizing

the performance of adaptive-optics systems with use ofmultiple control bandwidths. We also compare the per-formance of the multiple and single-bandwidth controlapproaches for the case of atmospheric-turbulence com-pensation and a range of representative adaptive-opticssystem parameters.

One can readily achieve bandwidth optimization for aclosed-loop adaptive-optics system, using a single band-width, by using standard results for the effects of sen-sor noise and servo lag on adaptive-optics performance.Since the wave-front errors that are due to these twoeffects may be treated as statistically independent, thecombined mean-square phase error u 2

(f) to be minimizedby adjustment of the control bandwidth f takes the form

o,2 (f) = o.n2(f) + o.S2(f), (1.1)

where the terms o-,,2 (f) and or2(f) are the mean-square

phase errors that are due to sensor noise and servo lag.Explicit formulas for these two quantities depend onnumerous parameters, including wave-front-distortionstatistics, wave-front-sensor noise statistics, the geome-tries for the wave-front-sensor subapertures and thedeformable-mirror actuators, the wave-front reconstruc-tion algorithm, and the impulse response function ofthe adaptive-optics control loop.14 The value of f thatminimizes o 2

(f) may be determined numerically, andanalytical solutions may be obtained when scaling-law

0740-3232/94/112871-16$06.00 © 1994 Optical Society of America

Ellerbroek et al.

2872 J. Opt. Soc. Am. A/Vol. 11, No. 11/November 1994

approximations are assumed for the functions o- 2 (f) andoT32(f).1518

The performance of a modal adaptive-optics controlsystem5"9 may be improved further by application ofthe above optimization process separately to each dis-tinct component, or mode, of the wave-front-distortion,profile. For a system that uses an orthonormal basis ofdeformable-mirror profiles as the set of modes to be con-trolled, Eq. (1.1) is replaced by formulas of the form5 61 2

N2= E 2 (f1), (1.2)

i=12% (fi) =(°i ).(fi) + (i2)s(fi), (1.3)

where oi2 is the mean-square residual phase error inmode number i, fi is the control bandwidth for this mode,and (oi 2

).(fi) and (oi 2)8(fi) are the statistically inde-pendent contributions of sensor noise and servo lag, re-spectively, to the total mean-square phase error in modenumber i. Explicit formulas for the latter two func-tions may be computed from wave-front-distortion sta-tistics and adaptive-optics system parameters once ana priori basis of orthonormal control modes has been se-lected. The bandwidth f that minimizes Eq. (1.3) maythen be determined numerically for each i. Zernikepolynomials20 are frequently used as the control-modebasis for analytical calculations, although practical appli-cations require that this basis set be adjusted to matchthe influence-function characteristics of a particular de-formable mirror.5

A basic limitation of the above approach is that thechoice of the control-mode basis is external to the opti-mization process. Control bandwidths are determinedafter an a priori control-mode basis has been specified,and there is no way to vary both the control modes andtheir associated control bandwidths simultaneously tominimize more globally the mean-square phase error'2. There are evidently no analytical techniques yet

available for incorporating the temporal dynamics of thephase distortions to be compensated into the definitionof the wave-front control-mode basis set itself. Thisis especially true for adaptive-optics systems that useeither multiple wave-front sensing beacons or severaldeformable mirrors to compensate for atmospheric turbu-lence accross an extended field of view.

In this paper we describe a technique for improv-ing the performance of an adaptive-optics system bysimultaneously optimizing both the basis of wave-frontcontrol modes and their associated control bandwidths.The inputs to this procedure are a collection of real,symmetric n X n matrices M(1), ... , (n) that havebeen derived from the second-order statistics of single-bandwidth control systems for n different bandwidthsthat sample the bandwidth range of interest. The vari-able n denotes the number of deformable-mirror degreesof freedom. One optimizes adaptive-optics controlmodes and bandwidths by maximizing the functionalf(U) = Y.,=, max<yjn,(U T'L(i)U)ii} over all n X n uni-tary matrices U. The columns of the optimal U define theorthogonal basis of control modes in terms of deformable-mirror-actuator commands. Mode number i is controlledat bandwidth number k precisely when matrix •M (k) maxi-

mizes the quantity (UTfM(i)U)is over j = 1, .. ., n,. Theunitary U that maximizes f(U) is computed with use ofan iterative algorithm that is in some ways analogous tothe Jacobi method for diagonalizing a symmetric matrix.2

The optimal U may also be determined explicitly from theeigenvectors of M (1) in the special case where n, = 2 andM(2) = 0, because the second of the two control band-

widths is the zero bandwidth. We refer to this specialcase as reduced-range single-bandwidth control, since therange of the wave-front reconstruction matrix is in thiscase a proper subspace of the vector space of deformable-mirror-actuator commands.

It is interesting to note that these techniques areapplicable independently of the adaptive-optics systemparameters and input disturbance characteristics that de-termine the matrices N (k). These results are applicableto adaptive-optics configurations including multiple de-formable mirrors and wave-front-sensing beacons, forexample, and may also be used when matrices JN(k) in-clude the effects of time-varying telescope misalignmentsor other disturbances that are distinct from atmosphericturbulence. One limitation of the approach is that con-trol bandwidths are optimized over a finite set of discretevalues, not over the continuum of all possible controlbandwidths. For sample calculations with use of aKolmogorov turbulence spectrum, we find that 10-20distinct values are sufficient to sample the bandwidthrange of interest fully. It is in principle possible first toselect the basis of control modes and initial control band-width estimates with use of the techniques introduced inthis paper and then to refine the control bandwidth valuesfurther by means of the standard approach summarizedafter Eqs. (1.2) and (1.3) above.

The remainder of the paper is organized as follows.Analytical results are derived in Section 2, beginningwith a review of single-bandwidth adaptive-optics sys-tems in Subsection 2.A. Subsection 2.B evaluates theperformance of a multiple-bandwidth control system fora prespecified set of control modes and bandwidths.Subsection 2.C describes how the functional f(U) maybe used to optimize the performance of a multibandwidthadaptive-optics control system, and Subsection 2.D con-siders the special case of reduced-range single-bandwidthcontrol. The results on closed-loop wave-front recon-struction obtained in a previous paper are equivalent tothe material in this subsection.2 2

Sample numerical results are presented in Section 3.These results assume a single translating phase screenwith Kolmogorov statistics, a Shack-Hartmann wave-front sensor with 8-16 subapertures across the telescopeaperture diameter, and a continuous-face-sheet de-formable mirror with actuator locations conjugate tothe corners of the wave-front-sensing subapertures.Adaptive-optics performance is compared for adaptive-optics systems with use of a single optimized controlbandwidth, multiple control bandwidths, or reduced-range single-bandwidth control. The use of multiplecontrol bandwidths significantly relaxes the wave-front-sensor noise level that is permitted for an adaptive-opticssystem to operate near the performance limit imposedby fitting error. Nearly all of this reduction can beachieved through the use of reduced-range single-bandwidth control.

Ellerbroek et al.


Open-loop sensormeasurements

y

Sensormeasurement

correction

Phase profilecorrection

Open-loopphase profile

Closed-loopsensor measurements

y - Gc

C ~ c) Recon-a~y - Istructor

Field-of-view-averaged- phase variance

C2

(a)

Closed-loopsensor measurements

y - Gc

Sensormeasurement

correction

(b)Fig. 1. Adaptive-optics system control-loop dynamics: (a) an adaptive-optics control system in which all wave-front modes arecompensated at the same servo bandwidth; (b) a system that uses different bandwidths and estimation algorithms for distinct, orthogonalsubspaces of wave-front modes. Only two separate bandwidths are illustrated for the sake of concreteness, but the approach maybe generalized to any number of distinct control bandwidths.

Section 4 discusses the implications of these results foradaptive control concepts designed to adjust adaptive-optics control bandwidths in real time in response tovariations in sensor noise level and wave-front-distortiondynamics. Appendix A summarizes the iterative algo-rithm that is used to maximize the functional f(U) inthe general case. Appendix B considers the special caseof exactly two control bandwidths.

2. ANALYTICAL RESULTS

Figure 1 illustrates the closed-loop adaptive-optics controlapproaches to be investigated in this paper. Figure 1(a)is a block diagram of the single-control-bandwidth ap-proach. The top two thirds of this figure represents theadaptive-optics control loop itself, and the bottom thirddescribes the effect of the control system on the optical

Open-loop sensormeasurements

y

Recon-structors

Phase profilecorrection

Open-loopphase profile

Ellerbroek et al.


performance of the telescope. Figure 1(b) illustrates anadaptive-optics control loop that employs multiple controlbandwidths for separate modes, or components, of theoverall wave-front-distortion profile. The control loopsand the notation introduced in these figures are describedfurther in the paragraphs below. Formulas for evaluat-ing and optimizing the optical performance of these sys-tems are then developed in the following subsections.

The input disturbance to be nulled by the control sys-tem is the time-varying m-dimensional vector y(t) ofwave-front-slope sensor measurements. This vector canbe composed of components obtained from one or sev-eral natural or artificial wave-front-sensing beacons thatmay be displaced from or coincident with the object to beimaged. The vector y(t) includes the effect of wave-front-sensor measurement noise but not the feedback applied tothe sensor measurements by deformable-mirror-actuatoradjustments. This second effect is included in the quan-tity y(t) - Gc(t), where c(t) is an n-dimensional vectorcomposed of the deformable-mirror-actuator position com-mands at time t, and G = ay/ac is the m X n Jacobianmatrix of first-order changes in wave-front measurementsy with respect to variations in the deformable-mirror-actuator command vector c. The actuator commandvector c may include commands for actuators on oneor several deformable mirrors that are conjugate to dif-ferent locations along the propagation path between thetelescope and the scene to be imaged. Each componentof the vector c may also correspond to a linear combina-tion of commands to multiple physical actuators. It isconvenient to assume that the null space of the matrix Gcontains only the zero vector, so that all nonzero actuatorcommand vectors result in some effect on the wave-front-slope sensor measurement vector. This condition can beobtained by restriction of attention to a linear subspaceof the space of all possible actuator command vectorsif necessary.

The single-bandwidth control law that is used to de-termine deformable-mirror-actuator commands from theclosed-loop wave-front-slope sensor measurement vectory - Gc is defined by the equations

e = M(y - Gc), (2.1)

by the notation k(r, 0) and is a function of both thepoint r in the telescope aperture plane at which it isevaluated and the direction in the field of view of thetelescope from which the phase profile has propagated.Since the phase profile (r, 0) and any modified profileof the form O (r, 0) + f (0) will have identical effects on theimaging performance of the telescope, it is convenient toassume that the function k(r, 0) has been summed witha function of 0 to satisfy the condition

J drWA(r)O(r, 6) =O (2.3)

where WA(r) is a {0, 1}-valued function describing theclear aperture of the telescope. The function 0 (r, 0) thatincludes this mathematical adjustment will be referredto as a piston-removed phase profile. The adjustmentAO(r, 0) applied to the phase profile by the adaptive-optics system is assumed to be a linear function ofthe actuator command vector c and is described by theequation

A4(r, 0) = Y. cihi(r, 0), (2.4)

where each function h(r, 6) is the adjustment to thephase profile that results when a unit command is ap-plied to actuator number i. This influence function willdepend on field angle 0 when the actuator is located ona deformable mirror that is not conjugate to the apertureplane of the telescope. We will assume that the over-all piston has been removed from each of these influencefunctions, so that Eq. (2.3) applies with O(r, 6) replacedwith h,(r, 6). Equation (2.4) will sometimes be abbrevi-ated with operator notation in the form

A = Hc. (2.5)

H is a linear operator by Eq. (2.4).The optical performance of the adaptive-optics system

is quantified by use of the expected mean-square valueof the residual phase-distortion profile 0 - AO. Thismean-square residual phase error is expressed in termsof an inner product [f, g] defined by the expression

dc-= ke. (2.2)

The n x m matrix M is referred to as the wave-front re-construction matrix, and the vector e is the estimate ofthe instantaneous closed-loop wave-front distortion rep-resented as a sum of deformable-mirror-actuator adjust-ments. Equation (2.2) relates the rate of change of thedeformable-mirror-actuator command vector to the in-stantaneous wave-front-distortion estimate e. Note thatall components of the command vector c are driven at thesame bandwidth by use of parallel single-input, single-output control laws. The particular form for this controllaw was selected for the sake of simplicity and concrete-ness. The methods to be developed below apply equallywell to any control law described by a linear ordinary dif-ferential equation with constant coefficients.

The wave-front-distortion profile to be corrected by theadaptive-optics control loop is represented in Fig. 1(a)

[f,g]=J dr f d0WA(r)WF(6)f(r, )g(r, )

f dr f d6WA(r)WF(6)(2.6)

where f and g are real-valued functions of r and 0 andWF is a nonnegative function of 0 that quantifies therelative weight attached to different points in the field ofview of the telescope. This inner product is symmetric([If, g] = [g, fl) and linear in both f and g. The expectedmean-square residual phase error o-2 is defined by theformula

2=([q > - AO, 4q5 - AOD (2.7)

where the angle brackets denote ensemble averaging overthe statistics of random sensor noise and phase-distortionprofiles O(r, 0). The details of how these averages arenumerically calculated for specific wave-front-sensor and

Ellerbroek et al.


deformable-mirror configurations are described in previ-ous studies.2 2

-2 4 For the present paper it suffices to note

that ensemble averaging is a linear operation.The following analysis is simplified if we assume that

deformable-mirror-actuator commands are expressed interms of a basis set satisfying the condition

[Hc, He'] = cT c, (2.8)

for any two actuator command vectors c and c'. This isequivalent to the requirement that

(2.9)

where ij is the Kronecker delta function. One mayalways obtain this condition by orthogonalizing anyprespecified basis set of deformable-mirror-actuatorcommands. After this orthogonalization procedure, thebasis elements of the deformable-mirror-actuator com-mand vector c will correspond to linear combinations ofmultiple physical deformable-mirror actuators.

The multiple-bandwidth adaptive-optics control systemillustrated in Fig. 1(b) replaces Eqs. (2.1) and (2.2) withthe estimation and control law

e(i) M(i)(y - Gc), (2.10)

dc_ = k(i)P(')e(L) (2.11)dt (.1

c =EcMi. (2.12)

A. Single-Bandwidth SystemsThe steady-state solution to the single-bandwidth controllaw described by Eqs. (2.1) and (2.2) takes the form

c(t) =f drk exp(-krMG)My(t - ). (2.17)

The deformable-mirror-actuator command vector c(t) isa highly nonlinear function of the reconstruction matrixM, and either optimizing or evaluating adaptive-opticsperformance appears to be difficult in this general case.We therefore introduce the constraint

MG = I (2.18)

on the coefficients of the matrix M. Equation (2.18)may be interpreted as requiring the reconstruction ma-trix to estimate precisely the current set of deformable-mirror-actuator commands in the absence of noise andatmospheric turbulence. Because the null space of thematrix G contains only the zero vector, one possible choiceof M that satisfies Eq. (2.18) is the matrix (GTG)-lGT.2

Equation (2.18) does not uniquely specify the reconstruc-tion matrix M, however, except in the highly unusual casein which there are equal numbers of wave-front-slope sen-sor measurements and deformable-mirror actuators.

Substituting Eq. (2.18) back into Eq. (2.17) gives theformula

c(t) = Ms(t),

The index i varies over the n different bandwidthsemployed by the adaptive-optics control loop. Onceagain, the particular loop compensation equation givenin Eq. (2.11) is employed for the sake of simplicity andconcreteness. The bandwidth 0) is used for deformable-mirror adjustments within the range space of the or-thogonal projection operator p(i), and the total correctionapplied to the deformable mirror is the sum of theseseparately computed components. The projection opera-tors p(i) must satisfy the conditions

p(i)p(i) = p(i), (2.13)

[p(i)]T - p(i) (2.14)

p(i)p(i) = 0 for i # j, (2.15)E p(i) = I. (2.16)

Equations (2.13) and (2.14) are necessary and sufficientconditions for each matrix p(i) to be an orthogonal pro-jection operator.2 25 Equations (2.15) and (2.16) implythat the range spaces of these projections decompose thespace of all deformable-mirror-actuator commands intomutually orthogonal linear subspaces. The multiple-bandwidth adaptive-optics system illustrated in Fig. 1(b)includes the previously described special case of modalcontrol with use of a Zernike polynomial basis fordeformable-mirror-actuator commands.5

With the above introduction it is now possible to evalu-ate and optimize the performance of these two adaptive-optics control approaches. Formulas for this purpose aredeveloped in the following subsections.

where the temporally filtered wave-front-slope measure-ment vector is defined as

s(t) =f d-k exp(-kr)y(t - r). (2.20)

The deformable-mirror-actuator command vector c(t) isnow a linear function of the coefficients of the wave-frontreconstruction matrix M. If Eq. (2.2) were replaced byanother linear differential equation with scalar-valued,constant coefficients, Eq. (2.20) for s(t) would becomethe corresponding steady-state solution with zero initialconditions.

A formula for the expected mean-square residual phaseerror a2 can now be obtained by substitution of Eqs. (2.5)and (2.19) back into Eq. (2.7). This yields the result

2 ([, - HMs, - HMs]). (2.21)

With use of the linearity and symmetry properties of theinner product operator, Eq. (2.21) may be expanded intothe form

2 = ([, 0]) - 2 E ([O h]sj)Miji'j

(2.22)+ Ei [hi, hi'iMijMit(SjSj)

We note that the term (, ]) is finite for the case ofturbulence-induced phase profiles with Kolmogorov sta-tistics, because the function has been defined as the

(2.19)

Ellerbroek et al.

[hi, hj] = .5ij,


piston-removed component of the wave-front-distortionprofile. Introducing the standard definitions2 32 4

,0 2 = (k, 0 ]), (2.23)

Aij = (k, hi]sj), (2.24)

Sjj = (SjSj) (2.25)

and applying the orthogonality condition for hi given byEq. (2.9) permit this expression for o- 2 to be simplified tothe form

2 = 0.o2-2 - AijMij + MijMiySjyi~j iljlj'

= oo2 - tr(AMT) - tr(MAT) + tr(MSMT ), (2.26)

where tr(V) denotes the trace of a square matrix V. Thisis the desired formula for the optical performance of asingle-bandwidth adaptive-optics control loop. Compu-tational formulas for numerically evaluating the statis-tical quantities oo2, A, and S for the case of Kolmogorovturbulence have been described previously.2 2

The formula for the expected mean-square residualphase distortion is quadratic in the coefficients of the re-construction matrix M, and the constraints imposed uponM by Eq. (2.18) are linear. The value of M that mini-mizes the error o.2 may therefore be determined with useof Lagrange multipliers. The result is the expression

to Eq. (2.19) in the single-bandwidth case.again assume a constraint of the form

M()G = I

We will once

(2.29)

for each of the reconstruction matrices M(i), although theparticular value of this matrix does not have to be derivedwith use of the results of Subsection 2.A. The derivationof a formula for c(t) begins with the identity

dt [P(i)c] =-pU) dc* ~dt dt

E p(i) dc(i)dt

(2.30)

where the second equality follows from Eq. (2.12). Withuse of Eqs. (2.10), (2.11), and (2.29), Eq. (2.30) becomes

d [P(e)c] = k(L)P()M(L)y - k(L)P(t)e, (2.31)

where all terms of the summation may be dropped ex-cept for i = j on account of Eq. (2.15). The steady-statesolution to this differential equation takes the form

POi~c(t) = f drk) exp[-k(i)r]P()M()y(t - r). (2.32)

Introducing the definition

S(i)(t) =f drk(') exp[-kP),r]y(t - r) (2.33)

permits this solution to be abbreviated as the formula

M = AS- + (I - AS-lG)(GTS-l G)-'lGTS-l. (2.27)

The first term in this formula for M is the usual minimum-variance reconstructor without constraints, 2 3 2 4 and thesecond term is the adjustment necessary for satisfyingthe constraint imposed by Eq. (2.18).26 The spatial andtemporal statistics of the phase-distortion profile to be cor-rected influence the value of the reconstruction matrix Mthrough the covariance matrices S and A. The minimumexpected mean-square residual phase error for a single-bandwidth adaptive-optics control loop at the bandwidthk may be determined by substitution of this value of Mback into Eq. (2.26). After algebraic simplifications, theformula for o,2 becomes

0, 2 =0,.2 tr(AS-'AT )

+ tr[(I - AS-lG)(GTS-lG)-l(I - GTS-lAT)].

(2.28)

In analogy to Eq. (2.27), the first two terms in thisformula can be recognized as the usual result for theperformance of an unconstrained minimum-variancereconstructor. 2 3 24

B. Evaluating Multiple-Bandwidth SystemsThe first step in evaluating the performance of a multiple-bandwidth adaptive-optics control loop is to derive an ex-pression for the actuator command vector c(t) equivalent

PM')C(t) = p MMSM(t). (2.34)

Summing Eq. (2.34) over i and applying Eq. (2.16) givethe result

c(t) = Z P(OMMS(i)(t) (2.35)

which is the desired formula for the deformable-mirror-actuator command vector c(t). This solution is simplythe sum of contributions from the individual wave-frontreconstruction matrices Mi on the range spaces of theprojection operators p(i), and there is no cross couplingamong the separate reconstructors. This simplificationwould not have been obtained without Eq. (2.15).

A formula for the expected mean-square residual phasedistortion of the multiple-bandwidth adaptive-optics sys-tem may now be obtained by substitution of Eqs. (2.5) and(2.35) into Eq. (2.7). This gives the result

0.2= ([PS, - ( E q HP(i)M()S(i)]),

(2.36)

which may be expanded to become the expression

0.2 O2 - 2 FI ([k, hj]sk )[P(i)M(i)]jki, j,k

+ Y [hj, hj,][P(i)M(')]j[P(i )M(i)Ijk,(sk S)ijki',j,k'

(2.37)

Ellerbroek et al.


Introducing the definitions

A(') = (, h]si ),s(ii') = (S(i)s(i'))

and applying the orthogonality conditionEq. (2.9) yield the result

for hi given in

2 = 0.2 2 (i)i,j,k

+ iI [PiMi)1ik[P(i )M(i )]ik1Skh, (2.40)i,il,j,k,kl

for the value of the expected mean-square residual phaseerror 0.2. The sums over j, k, and k' in this expressionmay be rewritten in the form

a.2 = 0.r2 - E tr{P(i)M(i)[A(i)]T}

- tr{A(i)[M(i)]T[p(i)]T}

+ EI tr{P(i)M(i)S(ii)[M(i')]T[p(i')]T}.iil

Because the identity

tr(MN) = tr(NM)

(2.41)

(2.42)

is valid whenever the product MN is a square matrix,Eqs. (2.13)-(2.15) may be substituted into Eq. (2.41) forthe quantity 0.2 to yield

0.2 = 0.2 - Y tr(p(i){M(i)[A(i)]Ti

+ A(i)[M(i)]T - M(i)S(ii)[M(i)]T}p(i)). (2.43)

The definition

NM (i) = M(i)[A(i)]T + A(i)[M(i)]T - M(i)S(ii)[M(i)]T (2.44)

allows this expression for 0.2 to be rewritten in the form

O'2 ='2 (2450.2 = 0.0t - E tr(P'J~1~i'P~').- (2.45)

This is the desired formula for the expected mean-square residual phase distortion for a multiple-bandwidthadaptive-optics control loop.

We note for use below that matrix XM(i) is symmetric.It follows from the definitions of matrices AM and (ii)that X(i) satisfies the relationship

N(k = ([hj, 0][hk, 0]) - ([hj, -HM()s()]

X [hk, 0 -HM(i)S(i)]) (2.46)

Matrix Jvl (i) describes the change to the second-order sta-tistics of the correctable component of the phase-distortionprofile that results when the reconstruction matrix M(i)and the control bandwidth P) are employed in a single-bandwidth adaptive-optics control loop. Equation (2.45)then illustrates the relationship between the overall per-formance of the multiple-bandwidth control loop and theperformance of the individual reconstruction matrices andbandwidths from which it is constructed. We note once

again that there is no cross coupling among different re-construction algorithms in this formula and that this con-dition would not be obtained without Eqs. (2.13)-(2.15).

C. Optimizing Multiple-Bandwidth SystemsSuppose now that the single-bandwidth reconstruction al-gorithms M(i) have been optimized with use of Eq. (2.27)for a range of control bandwidths P) and that the corre-sponding matrices 52(i) defined by Eq. (2.44) have beencomputed. We assume that the sampled values of P)represent the range of control bandwidths that are pos-sibly of interest for the given adaptive-optics parame-ters and operating conditions. How can we determineorthogonal projection operators p(i) that will optimizethe performance of the resulting multiple-bandwidthadaptive-optics control loop? More precisely, we areinterested in determining the value of the minimumexpected mean-square residual phase distortion 0.*2 de-fined by the formula

*2

= min0o2 - E tr[P(i)M(i)P(i)]: [p(l) p(nc)] E p

= 0o2 max{X tr[P(i) ()P(i)] [p('), .. , P(nc)] E P

(2.47)

where the set P is the collection of sequences of matrices[p(l). , p(nc)] satisfying Eqs. (2.13)-(2.16). The maxi-mum in Eq. (2.47) is taken over both the bandwidths atwhich modes are controlled and the control modes them-selves. This approach is more general than is selectingoptimal control bandwidths for each mode in a preselectedbasis set, such as Zernike polynomials. The value of 0.*2obtained will therefore be less than or equal to the bestperformance achievable with a prespecified basis set andthe same range of allowable control bandwidths.

The goal of this subsection is to obtain an equivalentformula for 0.*2 in which the maximization operator isapplied over a set that is more easily characterized than2P. Every sequence [P(l), .. , P(nc)] that is an element ofP' may be written in the form

p(i) = UA(i)UT, (2.48)

where the matrix U is a unitary matrix,

UUT = UTU = I, (2.49)

and the diagonal matrices A(') satisfy the conditions

A(-) diag[A), ... ,A()]A i) = 0 or 1,

E A(') = I.

(2.50)

(2.51)

(2.52)

One constructs the columns of matrix U by combiningorthonormal basis sets for the range spaces of eachp('). One defines the diagonal matrices A(') by settingA() equal to unity if column number j of U is withinthe range space of p(i) and equal to zero otherwise.Equations (2.13)-(2.16) ensure that U is unitary and

Ellerbroek et al.


that Eq. (2.52) is satisfied. Conversely, any sequence ofmatrices constructed with use of Eqs. (2.48)-(2.52) willsatisfy Eqs. (2.13)-(2.16) and therefore will be an elementof the set P. This equivalency and Eq. (2.42) permit thedefinition of 0*2 to be rewritten as

0.* 2 = 0.02 -max{ tr[A(i)UT(i)U): U unitary,

[A(),.. , A ] E I } (2.53)

where £ is the collection of sequences of diagonal matri-ces satisfying Eqs. (2.50)-(2.52).

For fixed U the maximum over L in Eq. (2.53) takesthe value

max{Z tr[A(M)UTM ~U] [A(1), .. Ane] C L Jn

= E max[U T M(i)U]jj, (2.54)j= i

where this maximum is achieved for [A('), ... , A(n,)],given by the prescription

A 1 if [UTlM(i)U]jj = maxk[UTN(k)U]jj

0 otherwise(2.55)

Substituting Eq. (2.54) back into Eq. (2.53) yields theformula

o*2=.O2_ max{Z max[UTNv(i)U]jj: U unitary}X

(2.56)

for the value of the minimum expected mean-squareresidual phase distortion achievable with a multiple-bandwidth adaptive-optics control loop. The columnsof the unitary matrix U that is associated with thisminimum become the control modes for the optimalcontrol algorithm. Mode number j is controlled byuse of bandwidth P) precisely when [UTMU jj =maxk[UTv (k)U]jj.

Several theoretical and practical considerations mustbe addressed before Eq. (2.56) can be viewed as a fea-sible approach to selecting control modes and controlbandwidths for a closed-loop adaptive-optics system.Does a unitary matrix U that minimizes o2 necessarilyexist, and is it unique? Is there a unitary U that willminimize 0.2 over the entire continuous range of controlbandwidths? Can the value of a-*2 so achieved be ap-proached to arbitrary precision by means of optimizationover sufficiently dense finite sets of bandwidths? Fi-nally, is there a practical algorithm for finding a valuefor U that minimizes Eq. (2.56) for any prespecified valueof n, and any set of matrices X(l), .. , .2(nc)? We con-sider the theoretical questions first.

The set of n X n unitary matrices is a compact set forthe standard topology defined by the Hilbert-Schmidtnorm. The right-hand side of Eq. (2.56) is a continuousfunction of U with respect to this topology for any fixed setof matrices N (l), ... , N (nc); and continuous, real-valuedfunctions defined on compact sets always achieve their

minimum and maximum values. Therefore a value ofU that actually minimizes o2 in Eq. (2.56) always ex-ists. This matrix U is not necessarily unique, since morethan one value of U will correspond to the same set ofprojection operators p(l), .. , p(%l) in Eq. (2.47) wheneverthe range space of at least one of these projections ismore than one dimensional. This condition appears tobe consistently obtained whenever matrices A, S, andG contain symmetries arising from x-y-symmetric de-formable mirrors and wave-front sensors. Multiple val-ues of U associated with a common set of projectionsp(l),... , p(%i) in Eq. (2.47) do, however, yield the samemultiple-bandwidth control algorithm. The uniquenessor nonuniqueness of the multiple-bandwidth control al-gorithm yielding the minimum value of 0.2 in Eq. (2.56)must depend on properties of matrices X(l), ... , ()

that we have not attempted to characterize.Consider now how the value of o0* 2 achieved in

Eq. (2.56) varies as the number and range of controlbandwidths included in the optimization increases towardinfinity. For any particular adaptive-optics system, theset of bandwidths k of interest is restricted to a boundedrange [0, K] by practical considerations (such as the fi-nite speed of light). From Eqs. (2.20), (2.24), and (2.25) itfollows that the covariance matrices S and A are continu-ous functions of the bandwidth k of a single-bandwidthadaptive-optics control loop. Owing to Eqs. (2.27) and(2.46), the matrices M and Nl (k) are similarly continuousin k, where (for the remainder of this subsection only) thematrices EM(k) are now parameterized directly in termsof the bandwidth k. It follows that the function gj (U, k)defined by the expression

gj(U, k) = [UT M(k)U]jj (2.57)

is continuous on the product space of pairs of n X nunitary matrices and bandwidths k from the inter-val [0, K]. This domain is compact, so the func-tions are in fact uniformly continuous. The functionsmaXok^sK[UT_%(k)U]jj and lj~. maxOk5kK[UTM (k)U]jj

are therefore continuous in U and must achieve theirmaximum value on the compact set of n X n unitary ma-trices. It follows that there exists a unitary matrix U*that minimizes the mean-square residual phase error,

nO2 = 102- max[UT.M(k)U]jj,

j=1 :k:SK(2.58)

over all possible definitions of wave-front control modes(or unitary matrices U) and control bandwidths selectedfrom the range [0, K].

The performance of the continuously optimized solutionobtained above may be approximated to arbitrary preci-sion by means of optimization over sufficiently dense fi-nite sets of bandwidths. Since each function gj(U, k) isuniformly continuous, for every positive e there exists apositive a such that IiU - U'11 < a and Ik - k'l < a implythat Igj(U, k) - gj(U', k')I <e/n forj= 1,..., n. Ifthefinite set of bandwidths {k1, ... , kn,} is selected so that allbandwidths k in the range [0, K] satisfy 1k - kiI < 8 forsome i, it follows that

max [UTM(k)U]jj c max[UTX(ki)U]jj + en (2.59)0:5k.-K

Ellerbroek et al.

Vol. 11, No. l1/November 1994/J. Opt. Soc. Am. A 2879

for all unitary matrices U and j, with 1 ' j c n. Sum-ming over j and maximizing over U yields the desiredresult, namely,

Nmax _ max [UTJ(k)U]jj

U j= 0k'KN

c max max[UTJM(ki)U]jj + . (2.60)j=1

Quantitative bounds on convergence rates (i.e., the re-quired values for and n, as functions of the desired E)are beyond the scope of this paper. Numerical results inSection 3 below will illustrate how we have selected n,and the bandwidth bound K for a sample problem.

We now consider the more practical question, findingthe optimal unitary U in Eq. (2.56) for a given value ofn, and prespecified matrices .(l), ... , N(nc) We havenot found a practical closed-form solution to this problem.The objective function 2 is not differentiable with re-spect to U, and the collection {[U, 0.2(U)]: U unitary} isnot a convex set. Our approach to approximating 0.*2 forsample numerical problems uses a pairwise, iterative op-timization procedure somewhat analogous to the Jacobimethod for diagonalizing a symmetric matrix. The de-tails of this hill-climbing search algorithm are describedin Appendix A. Although the convergence (termination)of the search is guaranteed, we do not know whether themaximum reached is global. Several reasons for confi-dence in the practical utility of this method for numericalcalculations are described below in Section 3.

D. Reduced-Range Single-Bandwidth SystemsThe simplest possible multiple-bandwidth control systememploys one nonzero and one zero control bandwidth.In the present notation this system is described by theparameters

n, = 2, (2.61)

) k, (2.62)k(2) . (2.63)

With k(2) = 0 it follows quickly from the definitions givenin Eqs. (2.33), (2.38), (2.39), and (2.44) that

s(2) = 0, (2.64)

A(2) = 0, (2.65)

s(22) = 0, (2.66)

N(2) = 0, (2.67)

so that Eq. (2.56) for the minimum expected mean-squareresidual phase distortion becomes

0.*2 = 002- max{Z max[(UTJ1 U)jj, 0]: U unitaryj.

(2.68)

Because matrix 1 is symmetric, it may be written withan eigenvalue/eigenvector decomposition of the form

N = UoDUOT

where U0 is unitary and D = diag(dl, .. , d,,) is diagonal.The formula for *2 consequently becomes

0.*2 =002

- maxiZ max[(UTUoDUoTU)jj, 0]: U unitary} j=l I (

(2.70)

It is intuitively plausible that the maximum in Eq. (2.70)is achieved for the value of U that diagonalizes M,namely, U = U0; this result is proven in Appendix B.The formula for 0.*2 therefore simplifies to the closed-formsolution

(2.71)n

0*2 = 02 _ max(dj, 0).j=1

The control modes associated with this minimum aresimply the eigenvectors of 14, and a given mode is cor-rected at nonzero bandwidth precisely when the associ-ated eigenvalue is positive. This expression recovers anearlier result for the optimal range space of a wave-frontreconstruction algorithm for systems that use a singlenonzero control bandwidth.2 2

3. NUMERICAL RESULTS

In this section we apply the above analytical results torepresentative cases of atmospheric-turbulence compen-sation. The atmospheric-turbulence profile consideredconsists of a single thin phase screen 0 (r, 0) (r) inthe aperture plane of the telescope, with a structure func-tion of the form 12) ( t )5/3

(1sk(r) - 0(r + r12) = 6.88 r' -\roj (3.1)

where r is the turbulence-induced atmospheric coher-ence diameter. This structure function corresponds to aKolmogorov turbulence spectrum with an infinite outerscale and zero inner scale. The temporal dynamics of thephase screen are given by the Taylor hypothesis,

(3.2)

where t represents time and v is a constant wind-velocityvector. The windspeed v is assumed to be known, andseparate results are presented for the two cases in whichthe direction of v is either known or unknown and isuniformly distributed. For the case of a single translat-ing phase screen, the definition for the Greenwood fre-quency fg that is used to describe the temporal bandwidthof turbulence-induced phase distortions simplifies to therelationship8

fg = 0.423v/ro. (3.3)

The adaptive-optics systems considered consist of aShack-Hartmann wave-front sensor and a continuous-face-sheet deformable mirror. The dimensionality of thesystem is parameterized by the ratio D/d, where D isthe diameter of the circular, unobscured telescope aper-ture and d is the width of an individual wave-front-sensor

Ellerbroek et al.

0(r, t = O(r - tv, 0),

(2.69)


Table 1. Dimensionality of theAdaptive-Optics Systems Considered

D/daSystem Component 8 12 16

Fully illuminated subapertures 32 88 164Partially illuminated subapertures 28 44 60

Total subapertures 60 132 224Actuators in pupil 49 113 197Edge actuators 28 44 60

Total actuators 77 157 257

aD, diameter of the circular, unobscured telescope aperture; d, widthof a single subaperture in the Shack-Hartmann wave-front sensor.Deformable-mirror actuators are located conjugate to the corners ofthe wave-front-sensor subapertures in the Fried actuator/subaperturegeometry.

subaperture. The cases Did = 8, 12, and 16 have beenevaluated, and details on the dimensionality of thesethree cases are summarized in Table 1. Partially illu-minated subapertures and actuators outside of but cou-pling into the telescope aperture have been included inthe analysis.

Shack-Hartmann-sensor measurements were modeledas x and y gradients of the wave-front-distortion profileaveraged over the illuminated portions of each subaper-ture. The wave-front-sensor temporal sampling rate wasassumed to be sufficiently large to support the highestcontrol bandwidth of interest. Measurement noise wastreated as uncorrelated, temporally white for each indi-vidual subaperture, and uncorrelated between separatesubapertures. Wave-front-sensor measurement noise isparameterized by the quantity 'n2 , defined as the mean-square phase-difference measurement error for a fully il-luminated subaperture and a sensor sampling rate equalto ten times the Greenwood frequency. The mean-squaresensor noise for partially illuminated subapertures andother sampling rates was assumed to vary linearly withthe sampling rate and inversely with the area of the sub-aperture. These sample dependencies imply that detec-tor noise can be neglected as a source of wave-front-sensormeasurement error and that d s ro. Different scalinglaws could be substituted to model the effects of noisy de-tectors or larger subapertures.

The deformable-mirror-actuator influence functionswere modeled as linear splines with their widths equalto twice the interactuator spacing. Actuators were lo-cated conjugate to the corners of each wave-front-sensorsubaperture in the so-called Fried geometry. 15

Given the above assumptions, the performance of asingle-bandwidth adaptive-optics system with use of anoptimized wave-front reconstruction algorithm is fully de-termined by the size of the aperture, the strength of theturbulence, the Greenwood frequency of the turbulence,the control-loop bandwidth f = k/(2vr), and the wave-front-sensor noise variance o.,2 at the nominal samplingrate of 10fg. By inspection of Eqs. (2.26) and (2.27), theminimized mean-square residual phase error 0.2 is a func-tion of the open-loop mean-square phase error o2, thecovariance matrices A and S, and the Jacobian matrixG of first-order derivatives of wave-front-sensor measure-ments with respect to deformable-mirror-actuator adjust-ments. Numerical values for matrices A and S may be

computed with Eqs. (3.18)-(3.28) of Ref. 22. Using theseformulas and the above assumptions, we may show thatthese four quantities take the functional form

2 ( d 5/3'. ro=

1.0299( D ) (3.4)

(3.5)

(3.6)

G G(D),

A= ( )Ao D' f°'rol . dfd,

S = ( d ) 1SO( d f- ) + 2n (fI) 1( d )

( dr) [ S( d fd)

+dlron2 (fdr S(D)]- (3.7)

By Eq. (3.3) the quantity (fgro)i(fd) is proportional to thefraction of a wave-front-sensor subaperture traversed bythe translating phase screen in one servo time constant.The normalized quantities S0 and Si are the separate,uncorrelated contributions of atmospheric-turbulence andwave-front-sensor noise to the covariance matrix S oftemporally filtered sensor measurements. With use ofEqs. (2.26) and (2.27) and the above functional forms,it follows that the mean-square residual phase error o2for an optimized single-bandwidth adaptive-optics systemsatisfies a scaling law of the form

.2 2 ( D fgro 0.n -(diro)513 d fd (diro )4 3

(3.8)

The normalized mean-square phase-error e2 is a functionof three quantities that may be interpreted as a normal-ized aperture diameter, a normalized servo lag, and anormalized wave-front-sensor noise level. When one isapplying this scaling law it is important to note that,owing to Eq. (3.3), both variables fg and 0.n2 depend onthe values of ro and d. These dependencies must be ac-counted for when one is comparing the performance ofsystems with different subaperture sizes.

Figure 2 plots the normalized mean-square residualphase error e2 as a function of the normalized servo lagand the normalized wave-front-sensor noise level for thecase Did = 8 and a single-bandwidth adaptive-optics sys-tem when an optimized wave-front reconstructor is used.The direction of the wind is random and unknown. Theresults obtained are qualitatively similar to those of previ-ous trade studies on this subject.'2 The best performanceis naturally obtained for the case of no wave-front-sensornoise, and in this case the residual mean-square phaseerror decreases monotonically with decreasing servo lagtoward an asymptote imposed by the effect of fitting error.For each of the four nonzero noise levels considered, thereis an optimum control bandwidth that minimizes the com-bined effect of servo lag and sensor noise. The curvesfor the residual phase variance do not follow the idealized(fg/f)

513 dependence for large values of (fgif) primarilybecause the telescope aperture diameter is finite.2 7

Ellerbroek et al.

Ellerbroek et al.

1 0

00

0

0

0.

a,

e8

0

z

0.1


0.1 1 10Normalized Servo Lag

Fig. 2. Performance of an adaptive-optics control system withuse of a single control bandwidth, as a function of normalizednoise level and normalized servo lag. These results were ob-tained with use of the results described in Subsection 2.A. Theresults in this figure assume D/d = 8 and an unknown andrandom direction for the wind. The normalized residual phasevariance is o-2/(d/ro)5/3, the normalized servo lag is (fgro)i(fd),and the normalized root-mean-square (RMS) wave-front-sensornoise is defined as crn/[2ir(dro)41 3]. The factor 2r is includedin this definition so that the noise level is expressed in terms ofwaves when diro = 1. WFS, wave-front-sensor.

1 0

00

a)

Ca

-a

:0

N

0E0z

0.1

0.1 1 10Normalized Servo Lag

Fig. 3. Performance of an adaptive-optics control system withuse of reduced-range single-bandwidth control as a function ofnormalized noise level and servo lag. These results correspondto those illustrated in Fig. 2, except that they assume the use ofa reduced-range single-bandwidth control system as described inSubsection 2.D. WES, wave-front-sensor.

Figure 3 plots comparable results for the performanceof an adaptive-optics system with use of reduced-rangesingle-bandwidth control. We computed these results byusing Eq. (2.71) of Subsection 2.D. Simply not attempt-ing to control those modes that are poorly sensed fora given level of wave-front-sensor noise and degree ofservo lag yields a significant reduction in the mean-squareresidual phase error. For a given control loop bandwidth,the RMS wave-front-sensor noise level corresponding to aspecified mean-square residual phase error is consistentlyat least doubled through the use of reduced-range single-

bandwidth control. Some improvement is obtained evenfor the case of no wave-front sensor noise or a very smallnormalized servo lag. This result occurs because thereare a small number of high-spatial-frequency modes thatare poorly sensed because of aperture edge effects andthat should not be controlled regardless of the controlbandwidth or wave-front-sensor noise level.

Table 2 lists corresponding results for a multiple-bandwidth control system optimized as described inAppendix A over a set of 18 different control bandwidthsyielding normalized servo lag values of 0.25-10. Theprecise bandwidths used are listed in the notes to Table 2and are identical to the values evaluated for the single-bandwidth and the reduced-range single-bandwidth re-sults plotted in Figs. 2 and 3, respectively. Becausethese sample numerical results are intended to be pri-marily illustrative, we performed no additional calcula-tions to quantify how the results obtained depend onthe value of n, that we used and the exact bandwidthsthat we considered. The authors think that the resultsin Table 2 are near the limit for optimization over thecontinuum of all possible bandwidths, judging from thesmoothness of the curves and location of the minima inFigs. 2 and 3.

Figure 4 compares the performance of the three differ-ent control options as a function of normalized noise level.The curve for the multiple-bandwidth control system istaken directly from Table 2. The curves for the single-bandwidth and the reduced-range single-bandwidth sys-tems derived from the minima of the curves plotted inFigs. 2 and 3, so that the single bandwidths used forthis comparison have been optimized as a function of thenormalized wave-front-sensor noise level. The multiple-bandwidth and the reduced-range single-bandwidthapproaches provide nearly identical performance im-provements over the single-bandwidth control system.The significance of this improvement depends on whichaxis of Fig. 4 is considered the independent variable.The reduction in the mean-square phase error for a fixedsensor noise level is never greater than a factor of 2, butthe relative change in the RMS sensor noise level per-mitted for a given mean-square residual phase error can

Table 2. Performance of a Multiple-BandwidthAdaptive-Optics Control System as a

Function of Wave-Front-Sensor Noise withD/d = 8 and a Random Wind Directiona

Normalized Wave-Front-Sensor Noise

0.00 0.05 0.10 0.20 0.40

Normalizedmean-square error 0.316 0.369 0.449 0.636 1.069

'The normalized wave-front-sensor noise level is defined ason/[27r(d/ro)4 3], and the normalized mean-square error is o-2 /(d/ro)513 .o-, is the RMS wave-front-sensor phase-difference measurement error inradians for a fully illuminated subaperture and a sampling rate equalto ten times the Greenwood frequency, d is the subaperture width, rois the turbulence-induced coherence diameter, and o2 is the residualmean-square phase error for the multiple-bandwidth adaptive-opticscontrol system computed with the methods described in Subsections 2.Band 2.C. A factor of 2vr is included in the definition of the normalizednoise level so that this quantity is expressed in terms of waves whendiro = 1. The 18 different bandwidths used for constructing themultiple-bandwidth control algorithm correspond to normalized servolags of 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, 2.50, 3.00, 3.50, 4.00, 4.50, 5.00,6.00, 7.00, 8.00, 9.00, 10.00, and X (i.e., a control bandwidth of 0).

___ ~~Normalized RSWFS Noise

___0.00

___ 0.05-0.10

0.20

l l l -l l l { l- - - - 0 .4

1

1


2a)

0Ca

a)

0

._co

:0

a)

cca3)

0E02

1.6

1.2

0.8

0.4

0.01 0.1Normalized RMS WFS Noise

Fig. 4. Performance comparison of single-bandwidth control,reduced-range single-bandwidth control, and multiple-bandwidthcontrol for Did = 8 and a random wind direction. The multiple-bandwidth control results are taken from Table 2. The single-bandwidth and the reduced-range single-bandwidth results arethe minima of the curves plotted in Figs. 2 and 3, respectively.The definitions for normalized residual phase variance and nor-malized RMS wave-front-sensor noise are as in Fig. 2. VVFS,wave-front-sensor.

Table 3. Distribution of Modal ControlBandwidths for the Multiple-Bandwidth

Control Results Given in Table 2a

Normalized Normalized Noise LevelServo Lag 0.00 0.05 0.10 0.20 0.40

0.25 72 11 - - -0.50 - 56 12 - -0.75 - 4 48 6 -1.00 - - 10 29 -1.50 - 1 1 29 82.00 - - - 4 242.50 --- - 153.00 1 - - - 73.50 - - - 2 34.00 - - - - -4.50 - - - - 25.00 - - - - -6.00 - - - - 2

X0 3 4 5 6 15

aThe definitions for normalized noise level and servo lag are as given inthe caption for Fig. 2. Each column of the table lists the number of modescontrolled at each bandwidth for the indicated normalized wave-front-sensor noise level. The row labeled - is the number of modes left en-tirely uncontrolled by the bandwidth optimization process. Bandwidthscorresponding to normalized servo lags of 7, 8, 9, and 10 were includedin the optimization but were never selected. The total number of modesis 76 for each case.

be a considerably larger factor. The intensity requiredfor the wave-front-sensing beacon would vary inverselywith the square of the allowable RMS noise level in thecase of a photon-limited wave-front sensor. For a sensorlimited by either background noise or detector read noise,the required intensity would vary with the inverse of theRMS wave-front-sensor noise level.

Because the performances of the multiple-bandwidthand the reduced-range single-bandwidth approaches areso similar, it might be expected that the two control al-

gorithms themselves would be approximately the same.One measure of their degree of similarity is given byTable 3, which lists the numerical distribution of band-widths selected for the multiple-bandwidth control ap-proach at the five different wave-front-sensor noise levelsevaluated for Table 2. The sets of nonzero control band-widths selected for the three lowest sensor noise levelsare clustered tightly around the optimal bandwidths forthe reduced-range single-bandwidth algorithm, as indi-cated by Fig. 3. The bandwidths utilized for the twohighest sensor noise levels are more widely distributed,and the optimal reduced-range single-bandwidth solutionis at most an engineering approximation to the multiple-bandwidth approach.

At this point it must be stressed that the multiple-bandwidth performance estimates presented in Table 2and Fig. 4 are the results of the iterative hill-climbingoptimization method described in Appendix A and thatthere is no mathematical certainty that these valuesare necessarily close to the global minima for the setof sample cases considered. It is at the least interest-ing to note, however, that (a) the performance of themultiple-bandwidth control algorithm as optimized bythis approach is consistently very slightly superior tothe reduced-range single-bandwidth approach for boththis sample problem and all others that have been con-sidered, (b) according to Table 3 the multiple-bandwidthcontrol algorithms themselves are relatively close to thereduced-range single-bandwidth solutions, and (c) themultiple-bandwidth performance estimates in Fig. 4 werecomputed with a different random starting point for theiterative optimization procedure for each separate wave-front-sensor noise level listed in Table 2. A number ofthese cases were evaluated several times beginning withdifferent random initial conditions, and the separate es-timates for the normalized mean-square residual phaseerror varied by amounts of the order of 1.0 x 10-6. Allthese points would certainly be grounds for additionalinvestigation if the results in Table 2 were found to besignificantly different from the actual global minima forthe multiple-bandwidth control approach.

The results in Figs. 2-4 and Table 2 all assume thatDid = 8 and that the direction of the wind is random andunknown. Figure 5 presents results for a range of nor-malized aperture diameters Did and both unknown andknown wind directions.2 8 The performance advantageafforded by either multiple-bandwidth control or reduced-range single-bandwidth control is smaller for Did = 12or 16 than for Did = 8 and appears to have reachedan asymptote. The results obtained are effectively in-dependent of the wind model for a fixed value of Did,although there is a very slight performance advantage as-sociated with the use of the multiple-bandwidth approachwhen the wind direction is known. In no case are theresults for multiple-bandwidth control and the reduced-range single-bandwidth control significantly different.

Because the present analysis has modeled the tem-poral dynamics of atmospheric turbulence with use ofthe Taylor hypothesis, it may be somewhat surprisingthat a priori knowledge of the wind direction has such asmall effect on adaptive-optics performance. Wave-frontmodes evolve into linear combinations of other modes asthe phase-distortion profile translates across the aper-

Solid: Single bandwidthDashed: Reduced range

single bandwidthDotted: Multiple bandwidths ........ ..... . -......

. -------------- - .1----

. . . . . . . . . . . . .

,............................. ............. ............................ ,_....... ..i._.._.._._....

Ellerbroek et al.

Ellerbroek et al. Vol. 11, No. 11/November 1994/J. Opt. Soc. Am. A 2883

2a)00

a)C0

a-0

ILa)en

a)

N0

co

02

1.6

1.2

0.8

0.4

0

0.0.1Normalized RMS WFS Noise

(a)


single bandwidthDotted: Multiple bandwidths .... -- ..... ......

._ .. . .. . .. . . .. .. . .. . . .

0 1 01I

Normalized RS WFS Noise

(b)

Soid Sngle bandwidthDashed: Reduced range

single bandwidthDotted: Multiple bandwidths .. -- -----------

, A//~~-

0.1Normalized RMS WFS Noise

(c)

2

C

0

0

coEL

0

0

0a)

1.6

1.2

0.8

0.4

0


single bandwidthDotted: Multiple bandwidths . ....................... ... .... . .. . ....... .. .

. _ .. . ...... .. .. .......... .. ..... .................... ................................... ............................. ........... ... .._ . ....- _

_..__ ..... __ .. . ._ . ....... ...... .. ..... . ._ .. _.. . __ . __ ._ ..... . .............. . .. ........... .... .__.......

01 0.1

Normalized RMS WFS Noise

(d)


single bandwidthDotted: Multiple bandwidths ...... .. .... ............ .. .

_ .... ...... ~~~~~~~~~~~~~~~~~~~. ... . ........... ........,_.......... . ....................................... ;.._

. _. . . ......... . ..... -. ....... . .... ....

0.01 0.1Normalized RS WFS Noise

(e)

Fig. 5. Performance comparison of single-bandwidth control,reduced-range single-bandwidth control, and multiple-bandwidthcontrol for different normalized aperture diameters and windmodels. These plots are similar to those of Fig. 4, except thateach subfigure describes results for a different normalized aper-ture diameter (Did) and wind condition (known or unknowndirection). (a) Did = 12, random wind direction; (b) Did = 16,random wind direction; (c) Did = 8, known wind direction; (d)Did = 12, known wind direction; (e) Did = 16, known winddirection. WFS, wave-front-sensor.

ture, and a control algorithm must account for this crosscoupling to exploit the predictability associated with theTaylor hypothesis. None of the three algorithms con-sidered here satisfies this requirement. Figure 6 plots

the mean-square phase distortion in each wave-front con-trol mode for the multiple-bandwidth control approachboth before and after correction and for unknown andknown wind directions. The normalized aperture di-

2Solid: Single bandwidthDashed: Reduced range

single bandwidthDotted: Multiple bandwidths . ............ .-............... -.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. . .. . .. . . .. .. .. ... . .. . .. ... .. . . . .. . . . . . -'--'-"-- .'--"--'''''-''--'---'1. "'-

_~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ . . .. . ._. . . .. ... . _ . . .. . ........ _

a,

00

a)

Ca

0M

a-0

E

S0

a)

N

E02

1.6

1.2

0.8

0.4

0.01

2a)00

C0

.X

a-

0.

'RV

Ca~0N

0E0z

1.6

1.2

0.8

0.4

0

0.

2aUC0

c

a

0co

a-

0

a

0

0

a)

c

0E0z

1.6

1.2

0.8

0.4

0

0.01


10 1 20 3 4010 20 30 40

Index

(a)

40Index i

50 60 70 8

50 60 70 80

control modes and their associated control bandwidths.A special case of this technique yields a control systemthat uses only a single bandwidth on a reduced range ofthe possible deformable-mirror degrees of freedom. Thetwo approaches yield nearly identical performance im-provements relative to an adaptive-optics control systemthat uses a single control bandwidth for all observabledeformable-mirror degrees of freedom with an optimizedwave-front reconstruction algorithm. These results ap-ply for either a known wind direction or a wind directionthat is both unknown and random. This lack of depen-dence on wind direction could simplify the implemen-tation of adaptive control systems that are intended toadjust automatically for varying atmospheric conditions.It may also prove simpler to adjust a single nonzero con-trol bandwidth in real time than to optimize an entire setof control bandwidths simultaneously.

Practical considerations may modify the above observa-tions. All deformable-mirror degrees of freedom must becontrolled by use of at least a low bandwidth to avoid driftresulting from hysteresis and roundoff error. Secondarymirror misalignments that are due to wind buffeting andline-of-sight acceleration introduce significant focus andastigmatism errors that may actually require higher con-trol bandwidths for these low-order Zernike modes. Itwould also be interesting to see whether the results ofthis study apply to higher-order control laws and less-classical control architectures.

APPENDIX A: MAXIMIZING F(U)(GENERAL CASE)

Given n n X n real symmetric matrices M (1),[M(2), ... , N (nC), we want to find a unitary matrix Uthat maximizes the function

nf(U) = max {[UT._(k)U1ii}-

i=1 15ksn(Al)

(b)Fig. 6. Modal distribution of wave-front errors before (x) andafter (+) compensation by the multiple-bandwidth control algo-rithm. These plots are for the normalized aperture diameterDid = 8 and a normalized wave-front-sensor noise level of 0.1wave at a sampling rate of ten times the Greenwood frequency.(a) Unknown wind direction, (b) known wind direction.

ameter is Did = 8, and the normalized wave-front-sensornoise level is 0.1 wave at a sampling rate of ten timesthe Greenwood frequency. The values of the residualmean-square phase errors frequently come in pairs forthe case of a random wind direction. This is consistentwith the x-y symmetry of the wave-front-sensor sub-aperture geometry and the deformable-mirror-actuatorgeometry. The values of the residual errors do not oc-cur in pairs and are much more scattered for a knownwind direction, but their range and average value remainapproximately the same.

4. DISCUSSION

This paper has described a technique for improving theperformance of a modal adaptive-optics control system bysimultaneously optimizing both the basis of wave-front

A. Two-Matrix AlgorithmLet n, = 2, F = [M(1), and G = J.(2). Suppose that theunitary matrix U with columns u1 2 ... Un is themaximizer. Without loss of generality the columns of Ucan be ordered in such a way that Eq. (Al) can be writtenas

r nf(U) = uFu + . ui TGu,

i=l i=r+l(A2)

where r is the number of the diagonal elements of theproduct UTFU that are larger than the correspondingdiagonal elements of UTGU. Let U = [U1 1U2 ], with U1 =[Ul U2 . .. Ur]. Since U is unitary, it follows that

UUT = U1U1T + U2U2T = I.

Using this expression and the trace properties

tr(M + N) = tr(M) + tr(N),

W0

++

+~H ++. + +++ + +

10

1 0t

lo'

10,4

0

100

10'

+:+++ + + + + + +

0 10 20 3

r

Ellerbroek et al.

. -.9

lo'

lo'

10,4

-r

u / 10 20 30

tr(MN = WNW


we can rewrite Eq. (A2) as follows:

f(U) = tr(UiTFUi) + tr(U2TGU2)

= tr(FU 1UJT) + tr(GU2 U2T)

= tr(FU1U1T ) + tr[G(I - U1UT)]

= tr(FUU1T) + tr(G - GUUJT)

= tr(FUUT - GU1U1T) + tr(G)

= tr[(F - G)U1 UT] + tr(G)

= tr[UiT(F - G)U1 ] + tr(G).

Therefore in maximizing f(U) the best we can do is tomaximize the term tr[UiT(F - G)U1] by taking as U1 theeigenvectors of F - G that correspond to positive eigen-values; for a proof see Appendix B. This computation in-volves the Schur decomposition of the symmetric matrixF - G. For a description of the Schur decomposition al-gorithm see Ref. 21.

B. General ApproachWith the two-matrix maximization algorithm we can solvethe general n-matrix problem with a Jacobi-like ap-proach. If M is an n X n matrix and s is a subset ofthe integers from 1 to n, we use the notation M(s, s) todenote the submatrix of M with rows and columns in s.M(:, s) denotes the submatrix with columns in s.

Let U = U0 (an initial guess).While the sequence is not converged, do the following:

Let B(k) be UTM(k)U for all k = 1 ... n,.Choose a pair of matrices B('), B(m).Let u be the set of indices i such that either B(')(i, i)

or B(m)(i, i) is the maximum element over allB(k)(i, i) for k = 1 ... n,.

Let U1 be the optimizer of the two-matrix subproblemB(1)(u, u), B(-)(u, ).

Update U(:, u) with U(:, u)Ul.

Since N (k) is symmetric, the submatrix v (k)(u, u)is symmetric for any set of indices u. The two-matrixsubproblem contains the current maximum elements forthe indices in u, so for any increase in the value of thesum of the maximum diagonal elements there will beat least as big an increase in the value of f(U). Thatmeans that the sequence of matrices {U(k)}, where U(k)is matrix U on the kth iteration of the main loop of thealgorithm, defines a nondecreasing sequence of values{f[U(k)]}. The algorithm terminates when the sequence{f[U(k)]} converges.

The strategy for choosing the pair of matrices to im-prove upon leads to different algorithms. The simplestone is a sweep of all possible pairs of matrices (hill climb-ing). An alternative strategy is to choose the pair thatgives the biggest increase in the value of f(U) (steepest-ascent hill climbing). Other techniques of heuristicsearch can be applied, too. We used the hill-climbingsearch for the numerical tests presented in Section 3.

We have no rigorous convergence proof, and we donot know whether the maximum reached is global orwhether the global maximum is unique. We plan to con-

sider some of these questions, along with related topics,in a separate paper.

APPENDIX B: MAXIMIZING F(U)(SINGLE-MATRIX CASE)Let A(A) be the vector of the eigenvalues of an n X n realsymmetric matrix A such that

Ai(A) ... -An(A).

Lemma 1. Let Q be an n n unitary matrix withQ =[qj, ... , qn]. For any r such that 1 r n defineQr = [qj, ... , qj. For any n X n symmetric matrix Athe following inequality holds:

r r7 qiTAqi = tr(QrTAQr) C Z A(A).i=1 i=1

Proof: Let A(r) = QrTAQr. From the interlacing prop-erty (corollary 8.1.4 in Ref. 21) for any k,

Ak+l[A(k+l)] ' Ak[Alk)] I Ak[A(k+l)] ' ... 5 A[A(k)]

' A[A(k+l)]

By induction we infer that

Ar [A(r)] C Ar [A(n)], ... AA(r)] - A[A(n)].

Therefore tr[A(r)] = )tIA(r)] + ... + Al[A(r)] ' = Ai(A).

Theorem 1. Given an n X n real symmetric matrix NM,

nmax f(U) = max >j max{(UTNU)M O}

U unitary U unitary i=l

= Ai ( ) -Aij(N)>°

(B1)

Proof: When U is the matrix of the eigenvectors of i,we obviously achieve the value defined in Eq. (Bi).

We will show that for any U

n

y max{(UT MU)i, O} Ai(NM).i~l Ai(Jv)>0

Without loss of generality assume that the first r di-agonal entries of UTi U are positive; then, usinglemma 1, we get

r rZ (UTN U)ii = tr(UrT N Ur) 5. Ai(M)i=l i=l

_ Z Ai(M),Ai (M)>O

where Ur is the matrix consisting of the first r columnsof U.

ACKNOWLEDGMENT

Brent L. Ellerbroek and Robert J. Plemmons acknowl-edge support from the U.S. Air Force Office of ScientificResearch.

Ellerbroek et al.


REFERENCES AND NOTES

1. R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L.Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M.Wopat, "Measurement of atmospheric wave-front distortionusing scattered light from a laser guide-star," Nature (Lon-don) 353, 144-146 (1991).

2. F. Roddier, M. Northcott, and J. E. Graves, "A simple low-order adaptive optics system for near-infrared applications,"Publ. Astron. Soc. Pac. 103, 131-149 (1991).

3. R. Q. Fugate, B. L. Ellerbroek, C. H. Higgins, M. P. Jelonek,W. J. Lange, A. C. Slavin, W. J. Wild, D. M. Winker, J. M.Wynia, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F.Moroney, M. D. Oliker, D. W. Swindle, and R. A. Cleis,"Two generations of laser-guide-star adaptive-optics experi-ments at the Starfire Optical Range," J. Opt. Soc. Am. A 11,310-324 (1994).

4. V. E. Zuez and V. P. Lukin, "Dynamic characteristics ofoptical adaptive systems," Appl. Opt. 26, 139-144 (1987).

5. C. Boyer, E. Gendron, and P. Y. Madec, "Adaptive optics forhigh resolution imagery: control algorithms for optimizedmodal corrections," in Lens and Optical Systems Design,H. Zuegge, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1780,943-957 (1992).

6. E. Gendron, "Modal control optimization in an adaptive op-tics system," presented at the International Commission forOptics-16 Satellite Conference, August 2-5, 1993, Garch-ing, Germany.

7. C. Schwartz, G. Baum, and E. N. Ribak, "Turbulence-degraded wave fronts as fractal surfaces," J. Opt. Soc. Am.A 11,444-451 (1994).

8. G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut,P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, and F. Merkle,"First diffraction limited astronomical images with adaptiveoptics," Astron. Astrophys. 230, L29-L32 (1990).

9. G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer,P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P.Gaffard, F. Rigaut, and P. Lena, "The Come-On-Plus adap-tive optics system: results and performance," presented atthe International Commission for Optics- 16 Satellite Con-ference, August 2-5, 1993, Garching, Germany.

10. F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P.Gaffard, F. Merkle, and P. Lena, "Adaptive optics on the3.6 m telescope: results and performance," Astron. Astro-phys. 250, 280-290 (1991).

11. G. Rousset, J. Fontanella, P. Kern, P. J. Lena, P. Gigan, F.Rigaut, J. Gaffard, C. Boyer, P. Jagourel, and F. Merkle,"Adaptive optics prototype system for infrared astronomy,"in Amplitude and Intensity Spatial Interferometry, J. B.Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1237,336-344, 1990.

12. E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella,G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot,F. Merkle, and N. Hubin, "Come-On-Plus project: an up-grade of the Come-On adaptive optics prototype system," inActive and Adaptive Optical Systems, M. A. Ealey, ed., Proc.Soc. Photo-Opt. Instrum. Eng. 1542,297-307 (1991).

13. F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna,and D. Roddier, "One-dimensional spectra of turbulence-

induced Zernike aberrations: time-delay and isoplanicityerror in partial adaptive compensation," J. Opt. Soc. Am.A 10, 957-965 (1993).

14. R. R. Parenti and R. J. Sasiela, "Laser-guide-star systems forastronomical applications," J. Opt. Soc. Am. A 11,288-3091994.

15. D. L. Fried, "Least-squares fitting a wave-front distortionestimate to an array of phase difference measurements,"J. Opt. Soc. Am. 67, 370-375 (1977).

16. R. H. Hudgin, "Wave-front reconstruction for compensatedimaging," J. Opt. Soc. Am. 67, 375-378 (1977).

17. J. Herrmann, "Least-squares wave-front errors of minimumnorm," J. Opt. Soc. Am. 70, 28-35 (1980).

18. D. P. Greenwood and D. L. Fried, "Power spectra require-ments for wave-front compensation systems," J. Opt. Soc.Am. 66, 193-206 (1976).

19. J. Y. Wang and J. K. Markey, "Modal compensation of at-mospheric turbulence phase distortion," J. Opt. Soc. Am. 68,78-87 (1978).

20. R. J. Noll, "Zernike polynomials and atmospheric turbu-lence," J. Opt. Soc. Am. 66, 207-211 (1976).

21. G. H. Golub and C. Van Loan, Matrix Computations, 2nd ed.(Johns Hopkins U. Press, Baltimore, Md., 1989).

22. B. L. Ellerbroek, "First-order performance evaluation ofadaptive-optics systems for atmospheric-turbulence com-pensation in extended-field-of-view astronomical telescopes,"J. Opt. Soc. Am. A 11,783-805 (1994).

23. E. P. Wallner, "Optimal wave-front correction using slopemeasurements," J. Opt. Soc. Am. 73, 1771-1776 (1983).

24. B. M. Welsh and C. S. Gardner, "Effects of turbulence-induced anisoplanatism on the imaging performance ofadaptive-astronomical telescopes using laser guide stars,"J. Opt. Soc. Am. A 8, 69-80 (1991).

25. R. C. Fisher, An Introduction to Linear Algebra (Dickenson,Encino, Calif., 1970).

26. Equation (2.27) requires that the matrix S be positive defi-nite. S is positive semidefinite by construction, and anyeigenvector of S with a zero eigenvalue defines a wave-front-sensor measurement mode that must be identically zero.Such measurement modes, in the cases in which they existat all, cannot contribute to the wave-front estimate. Whennecessary we may replace the measurement vector s by itsprojection onto the subspace orthogonal to all such modesand redefine M and G as their restrictions to this subspace.

27. G. A. Tyler, "Turbulence-induced adaptive-optics perfor-mance evaluation: degradation in the time domain,"J. Opt. Soc. Am. A 1, 251-262 (1984).

28. The case of a fixed, known wind direction requires a slightmodification to the formulas developed in Ref. 22 for the ma-trices A and S. Ensemble averaging over the direction ofthe wind in Eq. (3.22) of Ref. 22 now has no effect, and theray separation vector A must be replaced by A + ( - T2)V

for the remainder of the derivation. The velocity v of therandom wind must also be set to zero. The resulting for-mula for a general element of the matrix A or S is identicalto Eq. (3.27) of Ref. 22, except that the term f (28v/D, 2A/D)is replaced by f(0, 21,5v + A liD). The function f(a, b) re-mains as defined by Eq. (3.28) of Ref. 22.

Ellerbroek et al.

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Optimizing closed-loop adaptive-optics performance with ...users.auth.gr/pitsiani/reprints/J-004-AOJOSAA94.pdf · Fig. 1. Adaptive-optics system control-loop dynamics: (a) an adaptive-optics