May 1, 1995 / Vol. 20, No. 9 / OPTICS LETTERS 955

Sparse matrix wave-front estimators for adaptive-opticssystems for large ground-based telescopes

Walter J. Wild, Edward J. Kibblewhite, and Rodolphe Vuilleumier

Department of Astronomy and Astrophysics, The University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637

Received December 6, 1994

A closed-loop adaptive-optics servo system is an iterative optical–digital processor that can be characterized by afirst-order difference equation. This identification leads to a new class of wave-front estimators, some of whichare extremely sparse and permit real-time subaperture intensity weighting.

Adaptive-optics systems are often designed by con-sidering the performance of individual subsystemsthat are then combined in an overall error budget.1 – 5

This approach does not always recognize that thecharacteristics of the servo loop, principally the loopgain, affect the total system error in different ways.By considering the closed-loop adaptive-optics sys-tem specifically as an iterative optical–digital com-puter we are able to optimize the loop gain for givenobserving conditions and have discovered that ex-tremely sparse matrices can be used for wave-frontreconstruction. The use of these estimators dramati-cally reduces the computation required to control thesystem and can accommodate real-time subapertureintensity (scintillation) weighting.6 Both of theseproperties become important for high-order adaptive-optics systems proposed for visible light and plane-tary detection.7

The residual mean square corrected wave-frontphase error in units of radians squared, s2

w, can beexpressed as

s2w ­ gns2

ph 1 s2td 1 s2

fe 1 . . . , (1)

where gn is the noise propagator, s2ph is the photon

noise error, s2td is the time-delay error, and s2

fe is thedeformable mirror-fitting error. Other error sourcesmay exist, e.g., focus anisoplanatism for laser bea-cons. To minimize s2

w we make trade-offs betweenthe subaperture size, system sampling rate, or cycletime t and the servo loop gain k. Subaperture sizeaffects the deformable mirror-fitting error s2

fe and thenumber of detected photons per subaperture in t,hence in s2

ph, whereas the k and t influence the time-delay error s2

td, and k influences the noise propa-gator gn, which is a global measure of slope errorsin the reconstructed wave front.8 Clearly, these ef-fects are all coupled together and there is an opti-mum operating condition depending on atmosphericturbulence and winds, wavelength, DyL for telescopediameter D and subaperture dimension L, Dyr0 forseeing parameter r0,9 and various hardware and bea-con considerations.10

The primary components of the adaptive-opticsservo loop are the deformable mirror, the wave-frontsensor, a wave-front reconstructor, and the mirrordrivers, shown in Fig. 1. Let the wave-front sensor

0146-9592/95/090955-03$6.00/0

integration and readout, reconstruction, and actu-ator updates occur within t ­ ti11 2 ti, with ti andti11 the temporal interval boundaries. Systems mayhave detector integration and readouts beyond t inthe past; e.g., when a cw laser guide star is used.11

The shape on the mirror at time ti is fi ­ fstid,where boldface denotes a vector. The wave-frontsensor integrates the light over a duration G # t andthe subaperture slopes are averaged over G. If thesensor integration starts at ti, the mirror shape atti11 is that at ti plus a correction term proportionalto the reconstructed phases

fi11 ­ fi 1 kM1G

Z ti1G

ti

s0stddt , (2)

where M is the reconstruction matrix, s0std ­sstd 2 Afstd with sstd the atmospheric subapertureslopes and Afstd the slopes from the mirror shape,and A is the geometry matrix relating local slopesand phases.12,13 Typically A has ,2N2 elements, ofwhich ,4N are nonzero. Over the interval t themirror surface is fi and is static. The integral inEq. (2) can be approximated by the instantaneous

Fig. 1. Adaptive-optics servo loop. The residual wave-front phase from reflection off the deformable mirror iswsti11d 2 fstid, where wsti11d is the phase of the inci-dent wave front and f stid is the reconstructed phaseon the mirror’s surface. This residual is converted tosubaperture tilt information by the wave-front sensor andsubsequently reconstructed by the matrix M and thenscaled by k. An updated signal is then sent to the mirror,thus completing the loop.

1995 Optical Society of America

956 OPTICS LETTERS / Vol. 20, No. 9 / May 1, 1995

slopes at the beginning of t, whereby the servo ismodeled by the first-order difference equation

fi11 ­ fi 1 kMssi 2 Afid . (3)

For a natural star or cw laser beacon Eq. (3) can begeneralized to difference equations of higher orderbecause wave-front sensor integration occurs whileprevious data are read out and processed.

Equation (3) is the standard iterative approachused to solve systems of linear equations of theform sstd ­ Afstd 1 nstd, for noise nstd, though sis fixed in the mathematical paradigm.14 In thisoptical–digital processor the deformable mirror is ananalog memory device and the reconstructor a digitalcomponent.15 The adaptive-optics system performsonly one iteration on fi before si is updated becausethe atmosphere is undergoing continuous change.The wave-front phase decorrelation over several tdepends on the wind velocity v, assuming that theatmospheric turbulence satisfies the Taylor frozen-turbulence hypothesis16; ,Ly3tv iterations occur be-fore significant wave-front decorrelation occurs.

When viewed as an iterative algebraic computer,the matrix M can correspond to the Richardson M ­AT , Jacobi M ­ D21AT for D ­ diagsAT Ad, successiveoverrelaxation (SOR) M ­ fs1yvdD 2 Lg21AT , or sym-metric successive overrelaxation (SSOR) M ­ fs2 2vdyvg hfs1yvdD 2 LgD21fs1yvdD 2 LT gj21AT solvers(for AT A ­ D 1 L 1 LT and L the lower triangu-lar part of AT A); the latter two incorporate a relaxa-tion parameter v and generally converge faster.14

We refer to these as iterative estimators. The SORand SSOR matrices are 75% and 50% sparse (zeroelements) for large N, respectively, where N is thenumber of controlled actuators in the mirror, whereasRichardson and Jacobi are extremely (and equally)sparse matrices with a proportion sN 2 2dyN of zerosfor large N. These sparse estimators use the nearestneighbor slopes in the reconstruction. For compari-son the least-squares estimator M ; A1 is at most50% sparse, depending on the subaperture-actuatorgeometry.13

For subaperture intensity weighting the Richard-son matrix generalizes to M ­ W21AT and theJacobi to M ­ D21W21AT , where W is a diagonal ma-trix with elements proportional to the subaperturephoton flux and D ­ diagsAT W21Ad. For M ; A1

the reconstructor performs 2N2 operations per cycletime for the Fried geometry,12 whereas the Jacobirequire 4N operations. This is a significant com-putational reduction and an adaptive-optics systemwith large N based on sparse estimators can be con-trolled by averaging slopes—which can be weightedfor subaperture intensity in real time—around eachactuator, and these operations can be accomplishedby using digital signal processors rather than special-ized matrix–vector multipliers. With subapertureintensity weighting the Richardson and Jacobi ma-trix requires 8N for static D and 15N otherwise forthe Jacobi matrix. Intensity-weighted, least-squaresM ­ sAT W21Ad21W21AT requires ,N3 operations forthe matrix inversion and 2N2 for each matrix multi-plication. If sparse matrix techniques are used, suchas nested dissection of the A matrix before doing

a LU decomposition, ,N3/2 operations are neededfor the inversion and ,N log2 N for each matrixmultiplication.6

Equation (3) can be exactly solved for fi for anyA and arbitrary M as a function of k and the pasthistory of si0 for i0 ­ 0, 1, 2, . . . , i. From this we cancompute the first two error terms in Eq. (1). For thefirst term in Eq. (1)

gns2ph ­ s2

n ­s2

phk2

NTrsHMMT d , (4a)

where s2n is the noise variance and

H ­V21Xt0­0

sI 2 kAT MT dt0

sI 2 kMAdt0

, (4b)

for V ! ` cycles since the system was turnedon. When M ­ A1, gn , kys2 2 kd, so for k , 1,gn , ky2; for k , 1, gn , k; and the system is un-stable for k . 1. For the Richardson estimatorgn , k Trfs2I 2 kAT Ad21g; for this and the Jacobiestimators where gn diverges depends on the largesteigenvalue of AT A. For small k the sparse estima-tors have smaller gn than that for A1. Larger kincreases the system bandwidth, which is ,kyt fork , 1, though gn increases and the servo loop beginsto go unstable. With the iterative estimators higherbandwidth can be attained by decreasing t, thus di-minishing s2

td, though sph , 1ySNR grows so thatlow noise detectors are desired. In general, there isan optimum value of k, and this depends on the ratioof the photon-to-detector noise.

Low subaperture intensity levels creates unrealis-tic slope values that cause deformable mirror wafflingfor particular geometries with independent spaces13;the sparse estimators limit waffling to the nearestneighbor actuators, whereas the SOR and SSOR es-timators spread the waffling further out but withless amplitude. This arises because full convergenceis not attained in one iteration as is the case forM ­ A1. There is a trade-off between how well thewave front is reconstructed in one cycle time andthe distance that waffle propagates over the mir-ror’s surface from the noisy subaperture. Becausethe iterative estimators do not propagate waffle overthe entire surface of the mirror they are less sensi-tive to scintillation effects. Shorter t enables con-vergence before significant wave-front decorrelationoccurs. The iterative estimators possess a smallergn, for a given k, than A1 because slopes errors re-main localized rather than propagate over the entiregrid of reconstructed phase points.

The time-delay formula incorporates the covari-ances of the atmospheric wave fronts spaced at timest apart, which can be modeled assuming Kolmogorovturbulence and Taylor’s hypothesis. The elements ofthis matrix are

fXtgij ­ 23.44µ

LD

∂ 5/3

jui 2 uj 2 vtj5/3 , (5)

for ui the position of the ith actuator normalized toL. The piston and global tilt removed in Xt, i.e., forlaser guidestars, is X̃t; X̃0 is for zero time delay. Thetime-delay error, including computation error for M,is

May 1, 1995 / Vol. 20, No. 9 / OPTICS LETTERS 957

Fig. 2. Noise propagator for a 6 3 6 actuator array asa function of k. Solid curve, least-squares estimator;dashed curve, Richardson estimator; dotted–dashedcurve, Jacobi estimator.

Fig. 3. Time-delay error normalized to sDyr0d5/3 as afunction of k. The tip/tilt removed normalized varianceat k ­ 0 converges to s

2tdysDyr0d5/3 ­ 0.134 in the limit

of large N. Solid curve, M ­ A1; dashed curve, Jacobiestimator.

s2td ­

1N

TrfX̃0 2 2sI 2 kMAd21C`

1 k2HMAX̃0AT MT 1 2kHC`AT MT gµ

Dr0

∂5/3, (6a)

where

C` ­V21Xt00­1

sI 2 kMAdt00

kMAX̃t00 . (6b)

The cycle time t is implicit in v ­ jv j, which can becast in units of pupil space, i.e., v ­ fLyt for somefractional translation, f, of L over time interval t.

Figure 2 shows gnskd for a 6 3 6 square actu-ator arrangement with the actuators at the subaper-ture corners, V ­ 300. Here gn is smaller for theJacobi estimator for most k , 1 and diverges atk ­ 1; the Richardson estimator diverges for k , 0.56.Figure 3 shows s2

td normalized by sDyr0d5/3 for A1 andJacobi estimators for wind speed v ­ Ly10t. Thereis an optimal k, depending on Dyr0 and s2

ph, for bothestimators in the combined curves from Figs. 2 and 3.

Consider the case where k ­ 0.5. From Figs. 2and 3 s2

w ­ gns2ph 1 s2

td ø 0.3s2ph 1 0.015sDyr0d5/3

for the Jacobi estimator and s2w ø 0.45s2

ph 10.008sDyr0d5/3 for A1. If D ­ 2 m so L ­0.33 m and for r0 , 0.2 m at l ­ 0.5 mm, sothat at l , 2.0 mm r0 , 1.05 m because r0 , l6/5, sosDyr0d5/3 , 3, and for v ­ 10 mys, and with a photon-limited detector s2

ph , 1ynt ­ 33vyn, assumingn ­ 100, for n detected photons per t per subaper-ture, and a fitting error s2

fe ­ 0.3sLyr0d5/3, the Jacobiestimator has s2

w , 0.99 rad2, which A1 exceedswith s2

w , 1.43 rad2. For a D ­ 10 m telescopeat l , 2.0 mm, Dyr0 , 9.5, a similar calculationyields s2

w , 1.47 rad2 for the Jacobi estimator ands2

w , 1.24 rad2 for A1. Here each subaperture has25 times the collecting area therefore an identicalbeacon source enables t to be diminished wherebys2

w will diminish. For l , 0.5 mm it is essential todecrease L, hence increase N, to reduce s2

fe.Experiments using the iterative estimators with

bright stars were conducted with the 4-kHz Wave-front Control Experiment (WCE) adaptive-opticssystem13,17,18 on the 41-in. (104-cm) reflector atYerkes Observatory. Performance is generally supe-rior than with A1; this arises because scintillation-induced errors, especially from partially obscuredsubapertures, do not propagate over the entire wavefront, and because the WCE is such a high-speedsystem.

Funding for this work is furnished by NationalScience Foundation Cooperative Agreement AST-8921756.

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