Optimizing the performance of closed-loop adaptive-optics control systems on the basis of experimentally measured performance data

B. L. Ellerbroek and T. A. Rhoadarmer Vol. 14, No. 8 /August 1997 /J. Opt. Soc. Am. A 1975

Optimizing the performance of closed-loopadaptive-optics control systems on the basisof experimentally measured performance data

Brent L. Ellerbroek

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

Troy A. Rhoadarmer

Center for Astronomical Adaptive Optics, University of Arizona, Tucson, Arizona 85721

Received October 7, 1996; revised manuscript received February 4, 1997; accepted February 4, 1997

An experimental method is presented to optimize the control algorithm for a closed-loop adaptive-optics systememployed with an astronomical telescope. The technique uses wave-front sensor measurements from an in-dependent scoring sensor to calculate adjustments to the wave-front reconstruction algorithm and the band-width of the adaptive-optics control loop that will minimize the residual mean-square phase distortion as mea-sured by this sensor. Specifying the range of possible adjustments defines the class of control algorithms overwhich system performance will be optimized. In particular, the technique can be used to compute an opti-mized wave-front reconstruction matrix for use with a prespecified adaptive-optics control-loop bandwidth, op-timize the control-loop bandwidth for a given reconstruction matrix, optimize the individual modal controlbandwidths for a fixed modal reconstructor, or simultaneously optimize both the wave-front modes and theirassociated control bandwidths for a fully optimized modal control algorithm. The method applies to closed-loop adaptive-optics systems that incorporate one or more natural or laser guide stars and one or more de-formable mirrors that are optically conjugate to distinct ranges along the propagation path. Initial experi-mental results are reported for the case of a hybrid adaptive-optics system incorporating one natural guidestar, one laser guide star, and one deformable mirror. These results represent what is to the authors’ knowl-edge the first stable closed-loop operation of an adaptive-optics system using multiple guide stars. © 1997Optical Society of America [S0740-3232(97)03008-1]

1. INTRODUCTIONExperiments have shown the reward that adaptive opticsprovides in improving the resolution of ground-based as-tronomical telescopes.1–4 A critical contributor toadaptive-optics system performance is the control algo-rithm that converts wave-front sensor (WFS) measure-ments into the deformable-mirror (DM) actuator com-mands. For the adaptive-optics systems in use today thiscontrol algorithm consists of a wave-front reconstructionstep to estimate the instantaneous phase distortion to becompensated,5–8 followed by a servo control law to tempo-rally filter this instantaneous estimate before it is appliedto the deformable mirror.9 So-called modal adaptive-optics systems can apply different temporal filters toseparate spatial components, or modes, of the overallphase distortion.10 Extensive analysis has been per-formed to evaluate and optimize the performance of theseadaptive-optics control systems,11–19 but the results ob-tained depend on atmospheric parameters that are sel-dom known exactly and are constantly fluctuating. Theuncertainty and variability of atmospheric conditions im-plies that an optimal degree of turbulence compensationcannot be achieved or maintained for long time intervalswith a fixed control algorithm. A need exists for methodsto update adaptive-optics control algorithms based on ac-tual system performance at the time of operation or dur-ing operation. Encouraging results have already been

0740-3232/97/0801975-13$10.00 ©

obtained, demonstrating the value of empirically optimiz-ing the control bandwidths for a modal adaptive-opticssystem.20 In comparison, the subject of real-time adjust-ments to reconstruction matrices on the basis of mea-sured system performance has received little attention.Any method for empirically optimizing the performance

of an adaptive-optics control system must account for thelimited linear dynamic range of many WFS’s actually inuse today. Given sensors with sufficiently large lineardynamic range, measurements of open-loop phase distor-tions could be used to determine the spatial and temporalstatistics of uncompensated atmospheric turbulence.Optimal control bandwidths and wave-front reconstruc-tion matrices could then be computed from these statis-tics, by using any of the approaches already developed foruse with theoretical atmospheric turbulence models.12–19

Unfortunately, concerns regarding cost and detector noiseeffects led to Shack–Hartmann WFS designs that mini-mize the number of detector pixels per subaperture andlimit the sensor’s linear dynamic range.21 In this case,accurate phase measurements are possible only for the re-sidual phase distortions that remain after the adaptive-optics loop has been closed. Existing optimization meth-ods developed for use with theoretical atmosphericmodels are not applicable to such closed-loop data.In this paper we present a technique that uses experi-

mentally measured performance data from a closed-loop

1997 Optical Society of America

1976 J. Opt. Soc. Am. A/Vol. 14, No. 8 /August 1997 B. L. Ellerbroek and T. A. Rhoadarmer

adaptive-optics system to empirically optimize the sys-tem’s control algorithm. These performance data aresupplied by an auxiliary scoring WFS that provides high-resolution, low-noise closed-loop measurements of abright star recorded simultaneously with additional mea-surements obtained from the primary WFS used to closethe adaptive-optics control loop. Empirical correlationsbetween the scoring and the primary WFS measurementsare used to calculate adjustments to the control algorithmthat will minimize the residual mean-square error for thereconstructed scoring WFS phase profile. Depending onthe particular parameters of the control algorithm thatare adjusted, this general process specializes to one offour specific optimization problems:

1. Optimizing the wave-front reconstruction matrixfor a specified control-loop bandwidth,2. Optimizing the control-loop bandwidth for a fixed

wave-front reconstruction matrix,3. Selecting optimal modal control bandwidths for a

predefined modal algorithm with an a priori set of modesand a wave-front reconstruction matrix, and4. Determining both the modes and the associated

control bandwidths for a fully optimized modal control al-gorithm.

The method is applicable to single-guide-star–single-deformable-mirror adaptive-optics systems as well as tomore-complex systems incorporating multiple guide starsand multiple deformable mirrors. Because the scoringWFS requires a bright on-axis natural guide star (NGS),the method as currently formulated is most applicable foroptimizing the adaptive-optics control algorithm once be-fore an extended observation of an actual target field. Itcould also be used to iteratively update the control algo-rithm in the special case in which the target field includesa bright NGS. The more difficult problem of iterativelyupdating the control algorithm for a laser guide star(LGS) adaptive-optics system while observing a field lack-ing a bright NGS is not addressed directly by the presenttechnique.Section 2 of this paper describes the basic formulas

used to model a closed-loop adaptive-optics system duringnormal operation and while collecting scoring sensordata. Section 3 develops the theory used to calculate anoptimized wave-front reconstruction matrix for a fixedadaptive-optics control bandwidth. The formulas ob-tained are highly analogous to previous expressions forcomputing optimized reconstructors in terms of theoreti-cal turbulence models,12,13,17,19 but all the statisticalquantities necessary for these calculations can be mea-sured during closed-loop system operation. The problemof selecting an optimal control bandwidth for a fixed re-construction matrix is addressed in Section 4, and Section5 combines the results of Sections 3 and 4 to develop em-pirical methods for optimizing modal control algorithms.Section 6 presents initial experimental results of testingthe theory described in Section 3 with an adaptive-opticssystem consisting of a NGS, a LGS, separate Shack–Hartmann WFS’s for the two guide stars, and a single de-formable mirror (Sections 4 and 5 are not prerequisitesfor the description of this experiment). These resultswere obtained on the 1.5-m telescope at the U.S. Air Force

Starfire Optical Range. In addition to illustrating howmeasured system performance data can be used to updatewave-front reconstruction matrices, we believe that theseresults also represent the first stable closed-loop opera-tion of an adaptive-optics system that uses multiple guidestars.

2. ADAPTIVE-OPTICS SYSTEM MODELSFigure 1 illustrates the adaptive-optics control systems tobe studied in this paper. Figure 1(a) is a block diagramof the system’s operation while it is collecting the scoringWFS data used to update the control algorithm, and Fig.1(b) describes the control system during normal systemoperation. Figures 1(a) and 1(b) represent zonal andmodal control algorithms, respectively. These conceptsand the notation introduced in Fig. 1 are more fully de-scribed in the following paragraphs.The inputs s(t) and f(t) to this diagram are the open-

loop vectors of WFS measurements recorded at time t.The vector s(t) is the measurement from the primaryWFS used to close the adaptive-optics control loop, andthe vector f(t) is the auxiliary scoring WFS measure-ment used to evaluate system performance and updatethe adaptive-optics control algorithm. These vectors in-clude WFS measurement noise and the effects of LGS po-sition uncertainty but not the adjustment to the sensormeasurements that results from the current commands tothe DM actuators. The closed-loop WFS measurementvectors that account for this effect are

sc0~t ! 5 s~t ! 2 Gc0~t !, (2.1)

sc~t ! 5 s~t ! 2 Gc~t !, (2.2)

fc0~t ! 5 f~t ! 2 Hc0~t !, (2.3)

fc~t ! 5 f~t ! 2 Hc~t !, (2.4)

where c0(t) is the DM actuator command vector at timet in Fig. 1(a), c(t) is the corresponding quantity for Fig.1(b), and G and H are the first-order derivatives of thesensor measurements s(t) and f(t), respectively, with re-spect to these DM commands. Separate notation is usedfor Figs. 1(a) and 1(b) to distinguish between the actuatorcommands and the corresponding closed-loop sensor mea-surements associated with two different control algo-rithms. The vectors s(t) and f(t) may be composed ofcomponents measured by either one or multiple WFS’s.The actuator command vectors c(t) and c0(t) may includecommands either for a single DM or for multiple DM’sconjugate to different ranges along the propagation path.In Fig. 1(a) the wave-front control algorithm operates

on the closed-loop WFS measurement vector sc0(t) andtakes the form

e0~t ! 5 E0sc0~t !, (2.5)

c0~t ! 5 ~d0 * e0!~t !

5 E0

`

d0~t!e0~t 2 t!dt, (2.6)

where E0 is the wave-front reconstruction matrix, e0(t) isthe instantaneous estimate of the required DM actuator


Fig. 1. Adaptive-optics control systems studied in this paper. (a) Block diagram describing system operation while the scoring WFSdata used to update the control algorithm are collected. (b) Control system used during normal system operation. For concreteness (b)illustrates a modal control algorithm using two different control bandwidths for two separate subspaces of the overall wave-front dis-tortion profile, but partitions into an arbitrary number of subspaces are allowed.


adjustments output by the reconstructor at time t, d0(t)is the servo control system’s open-loop impulse responsefunction, and the symbol * denotes the convolution opera-tor. In Fig. 1(b) the most general adaptive-optics controlalgorithm considered for normal system operation isgiven by

e j~t ! 5 Ejsc~t ! j 5 1, ..., n, (2.7)

c j~t ! 5 Pj~dj * e j!~t ! j 5 1, ..., n, (2.8)

c~t ! 5 (j51

n

c j~t !. (2.9)

Here the actuator command vector c(t) is computed as asum of orthogonal components c j(t), each of which hasbeen obtained by using a separate wave-front reconstruc-tion matrix Ej and open-loop impulse response functiondj(t). The matrix Pj is a projection operator defining thesubspace of DM actuator commands to be controlled byusing the reconstructor Ej and the loop compensationdj(t). If n 5 1, the modal control algorithm illustratedby Fig. 1(b) reduces to the simpler zonal algorithm in Fig.1(a). These two options encompass the range of controlalgorithms actually used in adaptive-optics systems oper-ating today, although modal algorithms are usuallyimplemented by using an equivalent set of equations withreduced real-time computation requirements.14 Also, inexisting systems the impulse response functions di(t) arefully determined by a small number (usually four orfewer) of scalar coefficients that define a finite-orderfinite-difference equation. Constraints on the allowablevalues of the matrices E0 , Ej , and Pj required for stableclosed-loop system operation are summarized in the finalparagraph of this section.The bottom portions of Figs. 1(a) and 1(b) describe the

metric used to evaluate adaptive-optics system perfor-mance and to update the control algorithm. This metricis defined by the equations

s02 5 ^~fc

0!TWfc0&, (2.10)

s 2 5 ^fcTWfc&, (2.11)

where W is a positive-definite symmetric matrix used tocalculate the performance metric, the superscript T de-notes the transpose operator, and the angle brackets rep-resent an ensemble average over the set of experimentalmeasurements. The quantity s0

2 is the value of thismean-square error metric for the nominal adaptive-opticscontrol algorithm illustrated in Fig. 1(a), and s 2 denotesthe value of the metric for the more general control sys-tem to be optimized in Fig. 1(b). Formulas for s 2 can bedeveloped that represent the overall mean-square wave-front error as a sum of contributions from error sourcesincluding fitting error and time delay,22 but such resultsare more useful for the initial design of an adaptive-opticssystem than the experimental optimization problem con-sidered here. Although the results presented below ap-ply for any positive-definite matrix W, this matrix willnormally be specified such that s 2 is the mean-squarephase error for the phase distortion profile reconstructedfrom the closed-loop scoring WFS measurement vectorfc(t).

In terms of the above notation, the purpose of this pa-per is to describe how synchronized measurements ofsc0(t) and fc

0(t) can be used to compute values of Ej andPj that will minimize the value of s 2 in Fig. 1(b), givenknowledge of the quantities G, H, E0 , d0 , and dj . Re-stricting the ranges of permissable values for Ej and Pj invarious ways gives rise to the four specific algorithm op-timization problems listed in Section 1.The following derivations can be somewhat shortened

by the introduction of three simplifying assumptions.Without loss of generality, the vectors f(t) and c(t) canalways be expressed in terms of bases, so the matricesW and H satisfy the conditions

W 5 I, (2.12)

HTH 5 I. (2.13)

In such bases, each component of these vectors will gen-erally correspond to a linear combination of multiple WFSmeasurements or DM actuator commands. We also as-sume that the vector space of DM actuator commands hasbeen restricted to the range space of the nominal recon-structor E0 , so DM modes not controlled by this recon-structor are not included in the optimization process.More fundamentally, the results of this paper requirethat the matrices E0 , Ej , and Pj observe a set of con-straints that greatly simplify the evaluation of closed-loopadaptive-optics system performance. Given the abovethree assumptions, these constraints take the form

EjG 5 I j 5 0, ..., n, (2.14)

PjT 5 Pj j 5 1, ..., n, (2.15)

Pj2 5 Pj j 5 1, ..., n, (2.16)

PjPk 5 0 j Þ k, (2.17)

(j51

n

Pj 5 I. (2.18)

Motivation for these constraints is contained in earlierwork.17,19 The simplifying assumption EjG 5 I is vitalfor all of the techniques developed in this paper. Theevaluation and optimization of closed-loop adaptive-opticssystems without this simplification is a more difficultproblem, and current efforts in this direction still requirecertain approximations to yield numerical results.22

3. RECONSTRUCTOR OPTIMIZATION FORA FIXED CONTROL BANDWIDTHWe first consider the special case of using measuredadaptive-optics system performance data to optimize thecontrol algorithm’s reconstruction matrix while leavingthe servo loop compensation fixed. In the notation ofSection 2 this corresponds to specifying n 5 1, P1 5 I,and d1(t) 5 d0(t) and then determining a new recon-struction matrix E1 5 E0 1 E8 that will minimize thesystem performance metric s 2 defined in Eq. (2.11). Thefirst step in this process is to determine an explicit for-mula for the actuator command vector c(t) in terms of theWFS measurement vector s(t). This relationship canthen be used to express the value of s 2 in terms of the


second-order statistics of measured system performancedata and the coefficients of the reconstructor adjustmentE8 5 E1 2 E0 . Because the dependence of s 2 on E8 isquadratic, Lagrange multiplier methods can be used tosolve for the value of E8 that minimizes s 2. We concludethis section by deriving a formula for the full second-orderstatistics of the residual scoring WFS measurement vec-tor fc(t), a result that will be required for the evaluationof modal control algorithms in Section 5 below.With use of Eqs. (2.7)–(2.9), Eq. (2.2), and the assump-

tions on the values of n, P1 , and d1(t) listed at the begin-ning of this section, it follows that the DM actuator com-mand vector c(t) and the WFS measurement vector s(t)are related by the expression

c~t ! 5 E0

`

d0~t!$E1@s~t 2 t! 2 Gc~t 2 t!#%dt.

(3.1)

Since E1G 5 I by Eq. (2.14), this relationship can be re-arranged into the form

E0

`

@d ~t! 1 d0~t!#c~t 2 t!dt

5 E1F E0

`

d0~t!s~t 2 t!dtG , (3.2)

where d (t) denotes the Dirac delta function. Transform-ing both sides of Eq. (3.2) to the frequency domain and ap-plying the convolution theorem yield the expression

@1 1 d0~n!#c~n! 5 E1@ d0~n!s~n!#, (3.3)

where the Fourier transform h(n) of a function h(t) is de-fined by

h~n! 5 E2`

`

h~t !exp~22pitn!dt, (3.4)

and the Fourier transform of a vector-valued function isdefined component by component. Equation (3.3) can berearranged into the form

c~n! 5 E1F d0~n!s~n!

1 1 d0~n!G . (3.5)

Transforming this expression back to the time domaingives the result that

c~t ! 5 E1y~t !, (3.6)

where the temporally filtered WFS measurement vectory(t) is defined by the formulas

y~t ! 5 ~f * s!~t !, (3.7)

f~t ! 5 E2`

` F d0~n!

1 1 d0~n!Gexp~2pint !dn. (3.8)

An analogous derivation yields the result that c0(t)5 E0y(t). The function f(t) is just the closed-loop im-pulse response function corresponding to the open-loopimpulse response d0(t), and Eq. (3.6) is the desired ex-pression for the DM actuator command vector c(t) interms of the WFS measurement vector s(t). For lateruse we define the temporally filtered closed-loop WFSmeasurement vector yc0(t) as

yc0~t ! 5 ~ f * sc0!~t !. (3.9)

From the definition for sc0(t) given by Eq. (2.1) it followsthat the functions y(t) and yc0(t) are related by the ex-pression

yc0~t ! 5 y~t ! 2 G@~ f * c0!~t !#. (3.10)

Given the above formula for c(t), it is now possible toderive an expression for s 2 explicitly in terms of the co-efficients of the reconstructor adjustment E8 5 E1 2 E0and the second-order statistics of measured system per-formance. Substituting Eqs. (2.12), (2.4), and (3.6) intoEq. (2.11) gives the result

s 2 5 ^@f 2 H~E0 1 E8!y#T@f 2 H~E0 1 E8!y#&.

(3.11)

Since fc0(t) 5 f(t) 2 Hc0(t) 5 f(t) 2 HE0y(t), Eq.

(3.11) can be rewritten in the form

s 2 5 ^@fc0 2 HE8y#T@fc

0 2 HE8y#&

5 ^~fc0!Tfc

0& 2 2^~HE8y!Tfc0&

1 ^~HE8y!T~HE8y!&. (3.12)

Because E0G 5 E1G 5 I by Eq. (2.14), it follows that

E8G 5 0 (3.13)

and therefore by Eq. (3.10) that

E8yc0~t ! 5 E8y~t !. (3.14)

Substituting Eqs. (3.14) and (2.13) into Eq. (3.12) and ex-panding the matrix operations into summation notationgive the desired result:

s 2 5 ^~fc0!Tfc

0& 2 2^~HE8yc0!Tfc0&

1 ^~E8yc0!T~E8yc0!&

5 ^~fc0!Tfc

0& 2 2(j,k,l

Ejk8Hlj^~fc0!l~yc0!k&

1 (j,k,l

Ejk8Ejl8^~yc0!k~yc0!l&. (3.15)

This formula expresses the value of s 2 in terms of the co-efficients of the reconstructor adjustment E8 and thesecond-order statistics of vectors that are either measured@fc

0(t)# or directly computed from measured quantities@ yc

0(t)#.At this point it is convenient to express the value of

s 2 in the form

s 2 5 s02 2 2(

j,kEjk8Ajk 1 (

j,k,lEjk8Ejl8Skl ,

(3.16)

where the following definitions have been used:

s02 5 ^~fc

0!Tfc0&, (3.17)

A 5 HT^fc0~yc0!T&, (3.18)

S 5 ^yc0~yc0!T&. (3.19)

The value of E8 that minimizes s 2 subject to the con-straint that E8G 5 0 can now be determined by


Lagrange multiplier techniques. With these techniquesthe reconstructor adjustment must satisfy the equation

2Ajk 1 (lEjl8Skl 2 (

lDjlGkl 5 0, (3.20)

where Djl are the Lagrange coefficients. In matrix nota-tion this expression becomes

2A 1 E8S 2 DGT 5 0. (3.21)

Solving Eq. (3.21) with the constraint that E8G 5 0yields

E8 5 AS21 2 AS21G~GTS21G !21GTS21 (3.22)

for the wave-front reconstructor adjustment E8. Substi-tuting this expression for E8 into Eq. (3.16) and using thenotation tr(M) for the trace of the square matrix M givethe result that

s 2 5 s02 2 2 tr~E8AT! 1 tr @E8S~E8!T#

5 s02 2 tr @AS21AT 2 AS21G

3 ~GTS21G !21GTS21AT# (3.23)

for the optimized value of the mean-square error metrics 2. Equations (3.22) and (3.23) are both similar to theanalogous formulas for the case of an unconstrainedminimum-variance reconstructor, with the final term ineach expression being the change necessary to guaranteethe constraint E8G 5 0. It has been shown that Eq.(3.22) simplifies to the conventional noise-weighted least-squares reconstructor in the special case when theadaptive-optics system’s primary WFS and scoring WFSare identical.23

The quantity s 2 is a scalar description of adaptive-optics system performance. The evaluation and optimi-zation of modal control systems in Section 5 will require amore complete characterization of system performance,one that is closely related to the full second-order statis-tics of the residual scoring WFS measurements obtainedwith the adaptive-optics loop closed. For this purposethe matrices S and S0 are defined by the expressions

S0 5 HT^fc0~fc

0!T&H, (3.24)

S 5 HT^fcfcT&H. (3.25)

The appearance of the matrix H in these definitions mayrequire some explanation. Because of the assumptionsthat W 5 I and HTH 5 I it follows that(HTW21H)21HTW21 5 HT, and the quantity HTfc(t) isthe actuator command vector that yields the least-squares fit to the residual scoring WFS measurementfc(t). In other words, the matrices S0 and S describethe second-order statistics of the DM actuator commandvector that yields the best possible fit to the residual scor-ing WFS measurement obtained with the adaptive-opticsloop closed.The value of s 2 is related to the trace of the matrix S

by the expression

s 2 5 tr ~^fcfcT&!

5 tr @~I 2 HHT!^fcfcT&# 1 tr ~HHT^fcfc

T&!

5 tr @~I 2 HHT!^~f 2 Hc!~f 2 Hc!T&#

1 tr ~HT^fcfcT&H !

5 tr @~I 2 HHT!^ffT&# 1 tr ~S!. (3.26)

Here the third equality follows by the definition of fc(t)and the identity tr (MN) 5 tr (NM) for any two matricesM and N for which the product MN is a square matrix,and the final equality is obtained using the assumptionthat HTH 5 I and the definition of S. The first term onthe final right-hand side of Eq. (3.26) is completely inde-pendent of the wave-front reconstruction algorithm. Andwe note, for later use in Section 5, that minimizing s 2 isequivalent to minimizing tr(S).Finally, for the optimized wave-front reconstruction

matrix considered in this section the formula for S takesthe form

S 5 HT^@f 2 H~E0 1 E8!y#@f 2 H~E0 1 E8!y#T&H.(3.27)

Substituting the relationships fc0(t) 5 f(t) 2 HE0y(t)

and E8yc0(t) 5 E8y(t) into Eq. (3.27) yields the resultthat

S 5 HT^@fc0 2 HE8yc0#@fc

0 2 HE8yc0#T&H

5 S0 2 E8AT 2 A~E8!T 1 E8S~E8!T, (3.28)

where the second equality is obtained by using the defini-tions for the matrices A and S given by Eqs. (3.18) and(3.19) and the assumption that HTH 5 I. SubstitutingEq. (3.22) for E8 into Eq. (3.28) gives the final result forthe value of S, namely,

S 5 S0 2 AS21AT 1 AS21G~GTS21G !21GTS21AT.(3.29)

4. CONTROL BANDWIDTH OPTIMIZATIONFOR A FIXED RECONSTRUCTORIn the notation of Section 2, the task of selecting the op-timal control bandwidth for a fixed reconstruction matrixcorresponds to assigning the parameter values n 5 1,E1 5 E0 , and P1 5 I and then determining the open-loopimpulse response function d1(t) that will minimize s 2,the mean-square value of the residual scoring WFS mea-surement. Since the impulse response function d1(t) inan actual adaptive-optics system is implemented as afinite-difference equation specified by a small number ofscalar coefficients (typically four or less), this problem re-duces to a finite-dimensional optimization problem if thevalue of s 2 can be evaluated for an arbitrary choice ofd1(t). For practical applications it should in fact only benecessary to evaluate s 2 for a prespecified, finite set ofvalues of d1(t) that adequately samples the range of fea-sible control bandwidths. The first purpose of this sec-tion is to describe how the measured values of the vectorssc0(t) and fc

0(t) can be used to compute functions sc1(t)and fc

1(t) that describe the closed-loop WFS sensor mea-surements that would have been recorded with d0(t) re-placed with a candidate value of d1(t) and the same val-


ues for the input disturbance vectors s(t) and f(t). Afterthese simulated values of sc1(t) and fc

1(t) have beencomputed, the corresponding value of s 2 can be deter-mined from the formula s 2 5 ^(fc

1)Tfc1&. Additionally,

the wave-front reconstruction matrix that minimizes s 2

for the given value of d1(t) can also be computed by theapproach described in Section 3 above.The development of the desired formulas for sc1(t) and

fc1(t) begins with the identity

sc1~t ! 5 sc0~t ! 1 @sc1~t ! 2 sc0~t !#. (4.1)

Substituting Eqs. (3.6) and (3.7) and the assumption thatE1 5 E0 into Eq. (2.1) gives the result that

sc0~t ! 5 s~t ! 2 GE0@~f * s!~t !#, (4.2)

where the closed-loop impulse response function f(t) cor-responding to d0(t) is defined in Eq. (3.8). If the open-loop response function d0(t) had been replaced by d1(t)while the input disturbance s(t) and the wave-front re-construction matrix E0 were held constant, the corre-sponding formulas would become

sc1~t ! 5 s~t ! 2 GE0@~f1 * s!~t !#, (4.3)

f1~t ! 5 E2`

` F d1~n!

1 1 d1~n!Gexp~2pint !dn.

(4.4)

Substituting Eqs. (4.2) and (4.3) within the square brack-ets in Eq. (4.1) gives the expression

sc1~t ! 5 sc0~t ! 1 GE0$@~f 2 f1! * s#~t !%. (4.5)

If the term @(f 2 f1)* s#(t) in Eq. (4.5) can be expressed interms of sc

0(t), the vector sc1(t) will be described entirely

in terms of measured quantities. Multiplying both sidesof Eq. (4.2) by the reconstruction matrix E0 yields

E0sc0~t ! 5 E0s~t ! 2 E0GE0@~f * s!~t !#

5 E0@s~t ! 2 ~f * s!~t !#, (4.6)

where the second equality follows from the assumptionthat E0G 5 I. In the frequency domain, Eqs. (4.5) and(4.6) take the form

sc1~t ! 5 sc0~t ! 1 @ f~n! 2 f1~n!#GE0s~n!, (4.7)

E0sc0~n! 5 @1 2 f~n!#E0s~n!. (4.8)

Solving Eq. (4.8) for E0s(n) and substituting this formulainto Eq. (4.7) yield the result

sc1~n! 5 sc0~n! 1 GE0H F f~n! 2 f1~n!

1 2 f~n!G sc0~n!J .

(4.9)

If the function g1(t) is defined by the expression

g1~t ! 5 E2`

` F f~n! 2 f1~n!

1 2 f~n!Gexp~2pint !dn, (4.10)

the inverse Fourier transform of Eq. (4.9) can be writtenin the form

sc1~t ! 5 sc0~t ! 1 GE0@~g1 * sc0!~t !#. (4.11)

This is the desired expression for sc1(t) in terms of the

measured quantity sc0(t). An analogous derivation yields

the result

fc1~t ! 5 fc

0~t ! 1 HE0@~g1 * sc0!~t !# (4.12)

for the auxiliary scoring WFS measurement fc1(t). This

expression can be used to compute the mean-square re-sidual scoring sensor error s 2 5 ^(fc

1)T(fc1)& for a given

value of the impulse response function d1(t), and one canrepeat the calculation for different values of d1(t) to se-lect a value that minimizes s 2.The function g1(n) can be written in a more explicit

form if the impulse response function d0(t) is a pure in-tegrator control law. In this case d0(t) is described bythe expression

d0~t ! 5 H k if t > 0

0 otherwise, (4.13)

where k is the bandwidth of the control loop in radiansper unit time. In this case the function f (n) takes thevalue

f~n! 5k

k 1 2pin, (4.14)

and the function g1(n) can therefore be written in theform

g1~n! 5f~n! 2 f1~n!

1 2 f~n!

5 F k

k 1 2pin2 f1~n!GF1 2

k

k 1 2pinG21

5k

2pin2

k

2pinf1~n! 2 f1~n!. (4.15)

If the function d1(t) also corresponds to a simple integra-tor control law or, equivalently,

d1~t ! 5 H k1 if t > 0

0 otherwise, (4.16)

it follows that f1(n) 5 k1 /(k1 1 2pin), and Eq. (4.15)simplifies further to the expression

g1~n! 5k 2 k1

k1 1 2pin. (4.17)

In the time domain, the formula for the function g1(t) it-self becomes

g1~t ! 5 H ~k 2 k1!exp~2k1t ! if t > 0,

0 otherwise. (4.18)

The bandwidth parameter k1 determines the decay rateof g1(t) and therefore by Eq. (4.11) determines the timeinterval over which one must measure sc0(t) to computean accurate value of sc1(t). The value k1 5 0 impliesthat one must measure sc0(t) for all prior times to com-pute sc1(t), an obvious impossibility.Finally, we note that the methods described in Section

3 can be used to compute an optimal reconstruction ma-trix for use with the impulse response function d1(t) oncethe functions sc1(t) and fc

1(t) have been calculated. The


formulas for this purpose are listed below to introduce thenotation that will be used in Section 5 for the evaluationand optimization of modal control algorithms. For theseequations the specific function d1(t) has been replaced bya generic dj(t):

scj~t ! 5 sc0~t ! 1 GE0@~gj * sc0!~t !#, (4.19)

fcj~t ! 5 fc

0~t ! 1 HE0@~gj * sc0!~t !#, (4.20)

gj~t ! 5 E2`

` F f~n! 2 f j~n!

1 2 f~n!Gexp~2pint !dn,

(4.21)

ycj~t ! 5 ~ fj * scj!~t !, (4.22)

fj~t ! 5 E2`

` F d j~n!

1 1 d j~n!Gexp~2pint !dn, (4.23)

Aj 5 HT^fcj~ycj!T&, (4.24)

Sj 5 ^ycj~ycj!T&, (4.25)

Ej8 5 AjSj21 2 AjSj

21G~GTSj21G !21GTSj

21,(4.26)

f j,02 5 ^~fc

j!Tfcj&, (4.27)

s j2 5 s j,0

2 2 tr @AjSj21Aj

T

2 AjSj21G~GTSj

21G !21GTSj21Aj

T#, (4.28)

S j,0 5 HT^fcj~fc

j!T&H, (4.29)

S j 5 HT^@fcj 2 HEj8y j#@fc

j 2 HEj8y j#T&H

5 S j,0 2 AjSj21Aj

T

1 AjSj21G~GTSj

21G !21GTSj21Aj

T. (4.30)

5. OPTIMIZING MODAL CONTROLALGORITHMSBeyond their use for optimizing an adaptive-optics controlalgorithm with a single control bandwidth, the methodsdescribed in Sections 3 and 4 can be applied to optimizinga modal control algorithm that employs a distinct and in-dividually optimized bandwidth for each mode of the over-all wave-front distortion profile. Suppose that this opti-mization is to be performed with n discrete controlbandwidths that adequately sample the range of band-widths of interest and that these values correspond to theopen-loop impulse response functions d1(t), ..., dn(t).First, we apply Eqs. (4.19)–(4.21) to closed-loop systemperformance measurements to compute simulated timehistories of WFS measurements for each impulse re-sponse function. We then use Eqs. (4.22)–(4.26) to com-pute an optimal reconstructor Ej 5 E0 1 Ej8 for eachtime history; the level of adaptive-optics system perfor-mance that would be achieved by using either this opti-mized reconstructor or the nominal reconstructor E0 canbe computed with Eqs. (4.27)–(4.30). The final step indefining a modal control algorithm is the selection of theorthogonal projections P1 , ..., Pn that appear in Eq.(2.8). The values of these matrices determine the basis ofwave-front modes to be used and each mode’s associated

control bandwidth. How these projections can be deter-mined optimally is described in the following paragraphs.The first step in this development is to obtain a formula

for the performance of an arbitrary modal control algo-rithm described by Eqs. (2.7)–(2.9) subject to the con-straints and assumptions given by Eqs. (2.12)–(2.18).Substituting Eqs. (2.2), (2.7), and (2.8) into Eq. (2.9) yields

c~t ! 5 (j51

n

Pj($dj * @Ej~s 2 Gc!#%~t !). (5.1)

Multiplying this expression by the operator Pk and apply-ing Eqs. (2.14), (2.16), and (2.17) give the result

Pkc~t ! 5 @dk * ~PkEks!#~t ! 2 @dk * ~Pkc!#~t !.(5.2)

The corresponding relationship in the frequency domainis

Pkc~n! 5 dk~n!PkEks~n! 2 dk~n!Pkc~n!, (5.3)

and solving for the quantity Pkc(n) yields the expression

Pkc~n! 5 F dk~n!

1 1 dk~n!GPkEks~n!. (5.4)

A derivation identical to that which appears in Section 3for the special case k 5 0 gives the result

Pkc~t ! 5 PkEkyk~t !, (5.5)

where

yk~t ! 5 ~fk * s!~t !, (5.6)

and the closed-loop impulse response function fk(t) is de-fined in Eq. (4.23). Using Eq. (2.18) yields

c~t ! 5 S (j51

n

PjD c~t !

5 (j51

n

PjEjyj~t ! (5.7)

for the performance of a general modal control algorithm.We recall from Eq. (3.26) that minimizing s 2, the

mean-square value of the closed-loop auxilliary WFSmeasurement, is equivalent to minimizing the trace of thematrix S 5 HT^fcfc

T&H defined in Eq. (3.25). To com-pute the value of this matrix for the modal control algo-rithm, we first express the vector HTfc(t) in terms ofquantities that have already been evaluated in Section 4.Using Eqs. (2.4), (2.13), and (5.7), we can write this vectorin form

HTfc~t ! 5 HT@f~t ! 2 Hc~t !#

5 HTf~t ! 2 (j51

n

PjEjyj~t !. (5.8)

Substituting the identities ( j51n Pj 5 I and HTH 5 I into

the first and second terms, respectively, of the right-hand-side of Eq. (5.8) yields


HTfc~t ! 5 S (j51

n

PjDHTf~t ! 2 (j51

n

PjHTHEjyj~t !

5 (j51

n

Pj HT@f~t ! 2 HEjyj~t !#. (5.9)

From Section 4, the term within square brackets on theright-hand side of Eq. (5.9) can be recognized as theclosed-loop scoring WFS measurement vector that corre-sponds to the input disturbances s(t) and f(t) and asingle-bandwidth adaptive-optics control algorithm de-fined by the reconstructor Ej and the open-loop impulseresponse function dj(t). Applying the identity Ej 5 E01 (Ej 2 E0) and rearranging now give the desired result

HTfc~t ! 5 (j51

n

PjHT@f~t ! 2 HE0yj~t !

1 H~Ej 2 E0!yj~t !#

5 (j51

n

PjHT@fc

j~t ! 1 H~Ej 2 E0!yj~t !#,

(5.10)

for the value of the vector HTfc(t).By substituting the above result into Eq. (3.25) we can

write the trace of the matrix S in the form

tr ~S! 5 tr XK H(j51

n

PjHT@fc

j 2 H~Ej 2 E0!yj#C3 X(

k51

n

PkHT@fc

k 2 H~Ek 2 E0!yk#J TL C.(5.11)

Writing the product of summations as a double sum anddistributing the trace and expected value operators yieldthe result that

tr ~S! 5 (j51

n

(k51

n

tr $PjHT^@fc

j 2 H~Ej 2 E0!y j#

3 @fck 2 H~Ek 2 E0!yk#T&HPk

T%. (5.12)

Since tr (MN) 5 tr (NM) for any two matrices M and Nfor which the product MN is a square matrix, we can usethe assumptions listed in Eqs. (2.15)–(2.17) to obtain

tr ~PjMPkT! 5 tr ~PkPjM !

5 H tr ~PjPjM ! if k 5 j

0 otherwise

5 H tr ~PjMPj! if k 5 j

0 otherwise(5.13)

for any square matrixM for which the product PjMPkT is

defined. Substituting Eq. (5.13) into Eq. (5.12) and rear-ranging slightly yield the result that

tr ~S! 5 tr H(j51

n

PjHT^@fc

j 2 H~Ej 2 E0!y j#

3 @fcj 2 H~Ej 2 E0!y j#T&HPjJ (5.14)

for the value of the trace of S.The final evaluation of tr (S) now breaks into two spe-

cial cases. First, assume that the reconstruction matri-ces Ej and the projection operators Pj are restricted onlyby the assumptions listed in Eqs. (2.14)–(2.18) and are tobe chosen to globally minimize tr (S) and optimize theperformance of the adaptive-optics control system. Inthis case it follows that the best results will be achieved ifeach Ej 5 E0 1 Ej8 is the minimum variance reconstruc-tor derived according to Eqs. (4.19)–(4.26). Equation(5.14) now simplifies to the form

tr ~S! 5 tr S (j51

n

PjS jPjD . (5.15)

The matrix S j is defined by Eq. (4.30). The minimumvalue of tr (S) can now be written as

min@tr ~S!# 5 2maxH tr F(j51

n

Pj~2S j!PjG J , (5.16)

where the unsubscripted min and max operators are per-formed over all sets of values for the matrices Pj that sat-isfy Eqs. (2.15)–(2.18) and the minuses have been intro-duced to yield a maximization problem that has alreadybeen studied.19 The right-hand-side of Eq. (5.16) hasbeen shown to be equivalent to a more simply formulatedmaximization problem, namely,

maxH tr F(j51

n

Pj~2S j!PjG J5 maxH (

k51

m

max1<j<n

@UT~2S j!U#kk: U unitaryJ . (5.17)

Here m is the dimension of each matrix S j . Numericalmethods to search for the value of U that maximizes Eq.(5.17) have also been described.19 Substituting Eq. (5.17)into Eq. (5.16) and canceling the minuses yield the finalresult,

min@tr ~S!# 5 minH (k51

m

min1<j<n

~UT S jU !kk: U unitaryJ ,(5.18)

for the minimum value of tr(S). The value of U thatachieves this minimum also defines the parameters forthe associated modal adaptive-optics control algorithm.The columns of U describe the wave-front control modesin terms of DM actuator commands. Mode numberk is controlled by using open-loop impulse responsefunction di(t) precisely when (U T S iU)kk5 min1<j<n(U

TSjU)kk .For the second special case, assume that each recon-

struction matrix Ej takes a single, prespecified valueE0 , that the basis of the wave-front control modes has


also been prespecified, and that the only free parameterthat remains for optimizing adaptive-optics performanceis the choice of which open-loop impulse response functiondj(t) to employ with each mode. Since Ej 2 E0 5 0, Eq.(5.14) simplifies to the form

tr ~S! 5 tr S (j51

n

PjS j,0PjD . (5.19)

The matrix S j,0 is defined by Eq. (4.29). The minimumvalue of tr(S) possible with a free choice of wave-front con-trol modes could now be computed by using Eq. (5.18)with each S j replaced by S j,0 . Fixing the basis of thewave-front control modes at a prespecified value elimi-nates the minimization over the value of U in this expres-sion, yielding the result that

min@tr ~S!# 5 (k51

m

min1<j<n

~U0TS j,0U0!kk , (5.20)

where the columns of the matrix U0 are given by the pre-specified wave-front control modes. Once again, modenumber k is controlled by using the open-loop impulse re-sponse function di(t) precisely when (U0

TS i,0U0)kk5 min1<j<n(U0

TSj,0U0)kk .

6. EXPERIMENTAL RESULTSWe performed experiments using the U.S. Air Force Star-fire Optical Range 1.5-m telescope to test the theory de-veloped in Section 3. The objective of these experimentswas to empirically determine a wave-front reconstructionmatrix that combines data from a high-spatial-resolution,low-altitude LGS and a low-spatial-resolution, high-altitude guide star. The motivation for these experi-ments grew out of the current push toward the develop-ment of adaptive-optics systems for larger telescopes.24–26

Because of the inadequate availability of bright naturalstars, the use of LGS’s is required for full-sky coverage.A major issue associated with the use of LGS’s is focusanisoplanatism.13,27 The light returned from a LGS doesnot sample the same atmospheric turbulence as lightfrom an astronomical object because of its relatively lowaltitude of 5–120 km. For a fixed LGS altitude the mea-surement error that is due to focus anisoplanatism in-creases as the telescope aperture becomes larger. To off-set this increase in measurement error it becomesnecessary to use higher-altitude LGS’s or multiple LGS’s.So-called Rayleigh LGS’s rely on Rayleigh and Mie

scattering of laser light from molecules in the atmo-sphere. This scattering mechanism can create brightguide stars that can be sensed by using large numbers ofWFS subapertures and thus provide high-resolutionwave-front information. However, focus anisoplanatismeffects are large because the altitude of this type of LGS isrestricted to less than 20 km owing to atmospheric den-sity. Resonant excitation of sodium atoms in the mesos-phere allows the creation of LGS’s at altitudes of 90–120km. Focus anisoplanatism is consequently less of a con-cern for sodium LGS’s, but their brightness is limited bycurrent laser technology and ultimately by saturation ef-fects. WFS signal-to-noise ratio considerations indicate

that, for now, sodium LGS’s can support only a limitednumber of WFS subapertures.28

Previous studies have shown that a hybrid adaptive-optics system augmenting a bright Rayleigh LGS with asodium LGS sensed by using a WFS with a smaller num-ber of subapertures can reduce the effects of focusanisoplanatism experienced by the Rayleigh LGS WFS.17

The experiments described in this section were designedto test this result on an actual adaptive-optics system.However, for the experiments a NGS was used in place ofa sodium LGS.The adaptive-optics system used in the experiments

consisted of a Rayleigh LGS at a range of 14 km, a natu-ral star that served as the NGS and the object to be im-aged, Shack–Hartmann LGS and NGS WFS’s, each with16 3 16 subapertures, and a continuous face sheet DMwith a 17 3 17 grid of actuators optically conjugate to thecorners of the WFS subapertures. Full aperture tip–tiltwas corrected with a separate tracker and a fast steeringmirror. Because of processing constraints on the size ofthe reconstruction matrix the NGS WFS was binneddown electronically to 8 3 8 subapertures by averagingthe measurements of 2 3 2 blocks of subapertures. As athird WFS was not available to serve as the auxiliaryscoring sensor, the measurements from the 16 3 16 LGSWFS and the 8 3 8 NGS WFS were combined to simulatea 16 3 16 scoring sensor. The combination was accom-plished by subtracting the average measurement of theLGS subapertures contained within a NGS subaperturefrom each of those LGS measurements and adding theNGS measurement. For each 2 3 2 block of LGS WFSsubapertures the transformation is given by

fci 5 F ~sc!LGS

i 21

4 (j51

4

~sc! LGSj G 1 ~sc!NGS ,

i 5 1, ..., 4, (6.1)

where i and j indicate the subaperture of the 2 3 2 blockand (sc)NGS is the spatially overlapping binned NGS WFSmeasurement. The transformation was applied sepa-rately to the x and y WFS gradients. This transforma-tion was chosen because high-altitude turbulence tends tocontain only low spatial frequencies, whereas high spatialfrequencies are found closer to the ground. The LGSWFS measurements will accurately represent the high-frequency content of the turbulence but not the low-frequency content. The NGS WFS measurements areused to represent the low-frequency content of the turbu-lence.The adaptive-optics control loop operated at a sampling

rate of 1667 Hz, but WFS measurements could be ac-quired only at a 50-Hz rate to optimize the wave-front re-construction matrix. Therefore the closed-loop impulseresponse function f (t) was modeled by a delta functiond (t) so that yc(t) equaled sc(t). This assumption corre-sponds to adjusting the reconstructor to yield the bestpossible instantaneous estimate of the wave front, not thebest possible correction over time.The experiments were performed on the morning of

July 20, 1995. A bright star, HR8308, was chosen for theobject and NGS so that the optimization calculation


would not be limited by the signal-to-noise ratio of theWFS’s. This star is a widely spaced K2 binary, but onlythe brighter star, with magnitude 2.4, fell inside the fieldof view of the science camera. The science camera had a64 3 64 pixel field with a field of view of 285 nrad/pixeland an imaging wavelength of 850 nm. Frame times of1–2 ms were used to integrate as much light as possiblewhile avoiding saturation.The initial reconstruction matrix E0 consisted of a

least-squares reconstructor for the LGS WFS measure-ments and zero coefficients for the NGS measurements.The adaptive-optics loop was closed with this reconstruc-tor, and 1024 frames of closed-loop LGS and NGS WFSdata were collected for the optimization calculation. Theactual form of the optimization calculation used in the ex-periments is given in Ref. 29, which with coordinatetransformations can be shown to be equivalent to Eq.

Fig. 2. Short-exposure OTF results for the images obtainedwith the various reconstructors. The transfer functions wereazimuthally averaged to create the plots.

Fig. 3. Short-exposure PSF results for the images obtained withthe various reconstructors. Cubic splines were fitted to the azi-muthally averaged PSF’s to create the plots.

(3.22). The new reconstructor E1 obtained from the op-timization calculation was then used to drive theadaptive-optics system, using measurements from bothWFS’s. We believe that this experiment is the firststable closed-loop operation of a higher-order adaptive-optics control system using multiple guide stars.Images obtained with E1 were then compared with im-

ages obtained with the initial reconstructor E0 and athird reconstructor ENGS , which consisted of a least-squares reconstructor for the 8 3 8 NGS WFS measure-ments and zero coefficients for the LGS measurements.The comparisons were obtained with 1024 images, and astandard shift-and-add routine was used to align thepeaks of the images in each group and calculate the aver-age images. Optical transfer functions (OTF’s) andpoint-spread functions (PSF’s) were calculated from theseaverage images. Figure 2 presents a comparison of theaverage short-exposure OTF’s for the three reconstruc-tors, and Fig. 3 presents a comparison of the averageshort-exposure PSF’s. Table 1 provides information onshort-exposure Strehl ratios and PSF half-widths at half-maximum. During the collection of the images that weused to calculate the OTF’s and PSF’s the atmospheric co-herence length r0 was 6.0 cm and the isoplanatic angleu0 was 7.8 mrad.From Figs. 2 and 3 and Table 1 it is seen that the new

reconstructor E1 produced significantly improved OTF,PSF, and Strehl ratio compared with those obtained fromthe nominal reconstructor E0 . The factor of improve-ment for the short-exposure Strehl ratio is 0.208/0.1235 1.69. This improvement is still somewhat inferior tothat achieved with the reconstructor ENGS , indicatingthat the empirically derived reconstructor was not yet op-timal for the given operating conditions. There are sev-eral possible explanations for this result. First, the num-ber of WFS data frames used to optimize thereconstructor may not have been large enough to charac-terize accurately the atmospheric turbulence statistics.Second, completion of the optimization calculation re-quired ;45 min on the available computers, and the at-mospheric conditions may have changed sufficiently toappreciably alter the optimal reconstructor. Third, theeffective range of the LGS varied during the experimentas the telescope’s elevation angle changed. This varia-tion would have caused a shift in the nominal focus of theLGS wave front that was not recalibrated before the opti-mal reconstructor was used to collect images. Finally,further analysis of the WFS data has shown that mea-

Table 1. Short-Exposure Strehl Ratios and PSFHalf-Widths at Half-Maxima for the ImagesObtained with Various Reconstructorsa

Reconstructor WFS’s Used Strehl Ratio PSF HWHM (mrad)

E0 16 3 16 LGS 0.123 0.360ENGS 8 3 8 NGS 0.253 0.323E1 16 3 16 LGS

and8 3 8 NGS

0.208 0.325

aThe half-widths at half-maxima were determined by fitting cubicsplines to the PSF data in Fig. 3. The diffraction-limited half-width athalf-maximum is 0.292 mrad.


surements from the partially illuminated edge subaper-tures included in the reconstructor computation werevery noisy. This noise may have further degraded the ac-curacy of the reconstructor E1 .

7. SUMMARYWe have described how adaptive-optics control algo-rithms can be updated on the basis of closed-loop WFSmeasurements to optimize system performance as evalu-ated by an auxilliary scoring WFS observing a bright star.The method can be applied to optimize a wave-front re-construction matrix for a fixed control-loop bandwidth, toselect an optimal control bandwidth for a fixed recon-struction matrix, or more generally to optimize both ofthese variables simultaneously for a modal control algo-rithm employing a distinct control bandwidth for eachmode of the overall wave-front distortion profile. The re-sults obtained are highly analogous to previous formulasfor a priori control system optimization for theoretical at-mospheric turbulence models, but the new formulas areexpressed entirely in terms of the second-order statisticsof closed-loop WFS measurements. The simplifying as-sumption EjG 5 I is vital for all of the techniques devel-oped in this paper. Heuristically, this assumption re-quires that each reconstruction matrix Ej obtain a correctestimate of the current DM actuator command vector (ig-noring intentionally uncontrolled modes) in the ideal casein which WFS noise and atmospheric turbulence effectsare neglected. Similarly, the results for modal control al-gorithms depend heavily on the use of a complete, or-thogonal basis of wave-front modes, as specified by the as-sumptions on the matrices Pj given by Eqs. (2.15)–(2.18).The technique described here for updating a recon-

struction matrix with a fixed control bandwidth has beenapplied to a dual natural–laser guide star experimentperformed with the 1.5-m telescope at the U.S. Air ForceStarfire Optical Range. Beyond demonstrating this opti-mization method, these experiments also represent whatwe believe is the first stable closed-loop operation of ahigher-order adaptive-optics system employing multipleguide stars. In these experiments a nominal least-squares reconstruction matrix using only the LGS WFSmeasurements was empirically updated to operate by us-ing both the LGS and NGS WFS’s. This optimizationprocedure resulted in an improved OTF and PSF and inan improvement factor of 1.69 for the short-exposureStrehl ratio. These results are encouraging even thoughthe empirically calculated reconstructor did not performquite as well as a least-squares reconstructor that usesonly the NGS WFS measurements. Further work is inprogress to resolve this discrepancy.

ACKNOWLEDGMENTSThe authors gratefully acknowledge the work performedby numerous Rockwell Power Systems and Adaptive Op-tics Associates employees, particularly James Spinhirne,David Swindle, and Mike Oliker, to implement and oper-ate a dual-guide star capability for the Starfire OpticalRange 1.5-m telescope adaptive-optics system. Part of

this research was funded by the U.S. Air Force Office ofScientific Research. The authors gratefully acknowledgeits support.

REFERENCES1. 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 adap-tive optics,’’ Astron. Astrophys. 230, L29–L32 (1990).

2. C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars,and H. T. Barclay, ‘‘Compensation of atmospheric opticaldistortion using a synthetic beacon,’’ Nature (London) 353,141–143 (1991).

3. F. Roddier, M. Northcott, and J. E. Graves, ‘‘A simple low-order adaptive optics system for near-infrared applica-tions,’’ Publ. Astron. Soc. Pac. 103, 131–149 (1991).

4. 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. Mo-roney, M. D. Oliker, D. W. Swindle, and R. A. Cleis, ‘‘Twogenerations of laser guide star adaptive optics experimentsat the Starfire Optical Range,’’ J. Opt. Soc. Am. A 11, 310–324 (1994).

5. 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).

6. R. H. Hudgin, ‘‘Wave-front reconstruction for compensatedimaging,’’ J. Opt. Soc. Am. 67, 375–378 (1977).

7. J. Herrmann, ‘‘Least-squares wave-front errors of mini-mum norm,’’ J. Opt. Soc. Am. 70, 28–35 (1980).

8. W. H. Southwell, ‘‘Wave-front estimation from wave-frontslope measurements,’’ J. Opt. Soc. Am. 70, 998–1005(1980).

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

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

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

12. E. P. Wallner, ‘‘Optimal wave-front correction using slopemeasurements,’’ J. Opt. Soc. Am. 73, 1771–1776 (1983).

13. 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).

14. C. Boyer, E. Gendron, and P. Y. Madec, ‘‘Adaptive optics forhigh resolution imagery: control algorithms for optimizedmodal corrections,’’ in Lens and Optical System Design, H.Zugge, ed., Proc. SPIE 1780, 943–957 (1992).

15. E. Gendron, ‘‘Modal control optimization in an adaptive op-tics system,’’ in ESO Proceedings of the ICO-16 SatelliteConference on Active and Adaptive Optics, F. Merkle, ed.(European Southern Observatory, Garching, Germany,1994).

16. 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 isoplanicityerrors in partial adaptive correction,’’ J. Opt. Soc. Am. A 10,957–965 (1993).

17. B. L. Ellerbroek, ‘‘First-order performance evaluation ofadaptive optics systems for atmospheric turbulence com-pensation in extended field-of-view astronomical tele-scopes,’’ J. Opt. Soc. Am. A 11, 783–805 (1994).

18. D. C. Johnston and B. M. Welsh, ‘‘Analysis of multiconju-gate adaptive optics,’’ J. Opt. Soc. Am. A 11, 394–408(1994).

19. B. L. Ellerbroek, C. Van Loan, N. P. Pitsianis, and R. J.Plemmons, ‘‘Optimizing closed-loop adaptive-optics perfor-mance with use of multiple control bandwidths,’’ J. Opt.Soc. Am. A 11, 2871–2886 (1994).


20. 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 adaptive opticssystem: results and performance,’’ in ESO Proceedings ofthe ICO-16 Satellite Conference on Active and Adaptive Op-tics, F. Merkle, ed. (European Southern Observatory,Garching, Germany, 1994), pp. 65–70.

21. T. L. Pennington, D. W. Swindle, M. D. Oliker, B. L. Eller-broek, and J. M. Spinhirne, ‘‘Performance measurements ofgeneration III wavefront sensors at the Starfire OpticalRange,’’ in Adaptive Optical Systems and Applications, R.K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 327–337(1995).

22. W. J. Wild, ‘‘Predictive optimal estimators for adaptive op-tics systems,’’ Opt. Lett. 21, 1433–1435 (1996).

23. W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F.Shi, and N. Farmiga, ‘‘Investigation of wavefront estima-tors using the wavefront control experiment at Yerkes Ob-servatory,’’ in Adaptive Optical Systems and Applications,R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 194–205 (1995).

24. B. L. Ellerbroek, S. M. Pompea, D. J. Robertson, and C. M.Mountain, ‘‘Adaptive optics performance analysis for theGemini 8-meter telescopes project,’’ in Adaptive Optics in

Astronomy, M. A. Ealey and F. Merkle, eds., Proc. SPIE2201, 421–436 (1994).

25. D. G. Sandler, M. Lloyd-Hart, T. Martinez, P. M. Gray, J. R.P. Angel, T. K. Barrett, D. G. Bruns, and S. M. Stahl, ‘‘6.5-meter MMT infrared adaptive optics system: detailed de-sign and progress report,’’ in Adaptive Optical Systems andApplications, R. K. Tyson and R. Q. Fugate, eds., Proc.SPIE 2534, 372–377 (1995).

26. A. D. Gleckler and P. L. Wizinowich, ‘‘W. M. Keck Observa-tory adaptive optics program,’’ in Adaptive Optical Systemsand Applications, R. K. Tyson and R. Q. Fugate, eds., Proc.SPIE 2534, 386–400 (1995).

27. D. L. Fried, ‘‘Anisoplanatism in adaptive optics,’’ J. Opt.Soc. Am. 72, 52–61 (1982).

28. M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. C. Slavin, R. E.Ruane, and R. A. Cleis, ‘‘Characterization of artificial guidestars generated in the mesospheric sodium layer with asum-frequency laser,’’ J. Opt. Soc. Am. A 11, 806–812(1994).

29. T. A. Rhoadarmer and B. L. Ellerbroek, ‘‘A method for op-timizing closed-loop adaptive optics wavefront reconstruc-tion algorithms on the basis of experimentally measuredperformance data,’’ in Adaptive Optical Systems and Appli-cations, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE2534, 213–225 (1995).

Documents

Optimizing the performance of closed-loop adaptive-optics control systems on the basis of experimentally measured performance data