Adaptive Optics with Adaptive Filtering and Control*

Yu-Tai Liu and Steve GibsonMechanical and Aerospace Engineering

University of CaliforniaLos Angeles, CA 90095-1597

yutai@ucla.edu, gibson@ucla.edu

Abstract— This paper presents a quasi adaptive controlmethod for adaptive optics. Adaptive compensation is neededin many adaptive optics applications because wind velocitiesand the strength of atmospheric turbulence can change rapidly,rendering any fixed-gain reconstruction algorithm far fromoptimal. The performance of the new method is illustrated byapplication to recently developed simulations of high energylaser propagation through extended turbulence.

I. I NTRODUCTION

Adaptive optics (AO) refers to the use of deformable mir-rors driven by active control loops that feedback wavefrontsensor (WFS) measurements to compensate for turbulence-induced phase distortion of optical waves propagatingthrough the atmosphere [1], [2], [3], [4], [5]. These controlloops reconstruct (i.e., estimate and predict) the phase profile,or wavefront, from the WFS data. The control loops inclassical AO systems are linear and time-invariant (LTI),having fixed gains based on assumed statistics of atmosphericturbulence. Such control loops are not themselves adaptive,in the sense in which the termadaptiveis used in the controland filtering community.

Adaptive compensation is needed in many AO applicationsbecause wind velocities and the strength of atmosphericturbulence can change rapidly, rendering any fixed-gain re-construction algorithm far from optimal. Recently, adaptivewavefront reconstruction algorithms based on recursive least-squares (RLS) estimation of optimal reconstructor matriceshave been proposed [6], [7], [8], [9]. In this approach, anadaptive control loop augments a classical AO feedback loop.Results in [7], [8], [9] have shown that the type of adaptiveloops used here are robust with respect to modeling errorsand sensor noise.

The real-time computational burden is a significant obsta-cle for adaptive wavefront reconstruction because the numberof actuators and the number of sensors each can be on theorder of 100 to 1000, while the digital control loops needto run at sample-and-hold rates of 1000 Hz and higher. It isa serious challenge to develop real-time adaptive algorithmswith RLS parameter estimation for a problem with so manyinput and output channels. For the adaptive control loopsproposed in [7], [8], a multichannel RLS lattice filter firstpresented in [10] has been reparameterized and embedded inan algorithm developed specifically for adaptive feedforward

* This research was supported by AFOSR Grant F49620-02-1-0319.

disturbance rejection in adaptive optics problems. This mul-tichannel lattice filter preserves the efficiency and numericalstability of simpler lattices, while accommodating very largenumbers of channels through a channel-orthogonalizationprocess. Although the problem formulation and much of thestructure of the adaptive control loops presented in [7], [8]and this paper do not require that a lattice filter be used foradaptive estimation of an optimal wavefront reconstructor,multichannel lattice filters do appear to be among the fewclasses of algorithms that can yield the speed and numericalstability required for real-time adaptive optics.

This paper presents three advances over previous publica-tions on the use of adaptive filtering and control in adaptiveoptics. First, the adaptive loop is designed to use the closed-loop wavefront sensor vector as the input to the adaptive loop,as opposed to an estimate of the open-loop wavefront sensorvector used in previous publications on this subject. Second,extensive simulations for varied scenarios have shown thatthe quasi adaptive loop, which updates gains periodicallyfrom short data sequences, is essentially as effective as thefully adaptive loop, which updates gains at every time step.Finally, the adaptive optics simulations presented here aremuch more realistic than those in [7], [8], [11] becausea recently developed adaptive optics simulation with high-fidelity wavefront propagation model and detailed sensorcharacteristics, including nonlinearities, is used. Furthermore,the application is more challenging than those in [7], [8]because the turbulence path is much longer.

II. A DAPTIVE OPTICSFORMULATED AS A CONTROL

PROBLEM

Figure 1 shows a schematic diagram for generic adaptiveoptics problem in directed energy weapons. Actuators aredistributed in a two-dimensional array over a deformablemirror. These actuators are driven to adjust the profile ofthe mirror surface and cancel the phase distortions inducedin a beam of light as it propagates through atmosphericturbulence. A wave front sensor (WFS) is used to measurethe residual phase profile, using an array of subapertures thatsense the spatial derivatives, or slopes, of the profile on a gridinterlaced with the locations of the actuators. The purposeofAO system is to compensate the outgoing high energy laserfor the wavefront error that will be induced by atmosphericturbulence, so that the laser forms a tight spot (image) onthe target. The control system uses a beacon created by

AO

HEL

WFS

FP

Beacon on Target

HEL Spot on Target

AO

HEL

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Beacon on Target

HEL Spot on Target

AO

HEL

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FP

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HEL

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FP

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HEL Spot on Target

Fig. 1. Diagram of an adaptive optics problem in directed energy weapons.Key components of the control system: adaptive optics algorithm (AO),deformable mirror (DM), wavefront sensor (WFS), high energy laser (HEL).

¾z−1¾d¾F1(z)

v - V -

− ¾E1¾d s¾r

z−1L(z)

u6

-6

G(z)

WOP Model

E0

-

s¾

s

FIR Loop

Fig. 2. Block diagram for adaptive optics.

illuminating the target with a low energy laser as the basis fordetermining the commands to the deformable mirror requiredto cancel turbulence-induced phase distortion. Because thebeacon is considered to be a distant point source, the wave-front propagating from the beacon would be very nearly aplane wave when it reached the mirror with no atmosphericturbulence. This plane wave is the desired set point forthe control algorithm. If the wavefronts propagating fromthe beacon and to the target travel through approximatelythe same atmosphere, then correcting the wavefront fromthe beacon should compensate for the turbulence effects onoutgoing beam.

For control purposes, the adaptive optics problem is rep-resented by the block diagram in Figure 2. The measuredwavefront slope vector is denoted bys. Since the wavefrontcannot be measured directly, the objective of the controlloops is to minimize the RMS value of the projection ofsonto a certain subspace. The top feedback loop is a classicalAO loop, so that is linear and time-invariant (LTI). The matrixV represents a parameterization of actuator space [11]. Theactuator command vectorc and the control command vectorv are related by

c = V v . (1)

The matrixΓ̃ is given by

Γ̃ = Γ V , (2)

whereΓ is the poke matrix. Thus,

Γ̃ v = Γ c . (3)

The reconstructor matrixE0 is equal to the pseudo inverseof Γ̃ , so that

E0Γ̃ = I . (4)

The objective of the control loops is to minimize the varianceof the part of the wavefront reconstructed byE0; i.e., the partof the wavefront in the range space ofE0.

The z−1 in each control loop represents a one-step delaydue to sensor read-out and computation. Hence, latency inthe adaptive optics problem considered in this paper is onetime step.

The block labeled WOP (wave optics propagation) Modelin Figure 2 is a high-fidelity simulation of a directed energyproblem like that represented in Figure 1. This model in-cludes the wave optics propagation of both the beacon andhigh energy laser beams, as well as models of the wavefrontsensor and deformable mirror, and focal plane imaging onboth the adaptive optics platform and the target. This waveoptics model is contained in the program WaveTrain, which isa product of MZA Associates Corporation. The model usedin this research is based on non-sensitive features of HELsystems.

III. T HE CONTROL LOOPS

A. Classical AO Loop

The linear time-invariant (LTI) feedback control loop (thetop loop in Figure 2) is a classical AO loop. It contains thereconstructor matrixE0 and the digital integrator

F1(z) = K1

z

z − 1, K1 = 0.5 , (5)

along with the one-step computation delay.

B. FIR Loop

The quasi-adaptive control loop is labeled “FIR Loop” inFigure 2 (enclosed in the dashed box). This control loopaugments the classical AO loop to enhance wavefront predic-tion and correction, particularly for higher-order wavefrontmodes. The main component of this loop is the FIR filterL(z). The gains in this filter are updated adaptively. Thisadaptation may be either fully adaptive (i.e., at each timestep) or quasi adaptive (i.e., periodically). In the simulationsfor this paper, the FIR gains were identified in the quasiadaptive fashion from data collected over one second (5000frames)e.

For identification of the FIR gains, the problem is for-mulated as a feedforward noise-cancellation problem withtuning signale and reference signalr. From Figure 2,

r = E1 s . (6)

In the FIR loop,E1 = E0 in this paper, although the methodsare extended easily without this condition. Thetuning signalfor the FIR filter is the sequence

e = E0s ; (7)

i.e., the FIR gains are identified to minimize the varianceof e. The transfer function from the control signalu to thesignale with only the classical AO loop closed is representedby G(z). Of course, only an estimate of this transfer functionis available in applications.

The filter L could be either FIR or IIR. While an IIRfilter theoretically would produce optimal steady-state per-formance for stationary disturbance statistics, an FIR filter ofsufficient order can approximate the steady-state performanceof an IIR filter, and the convergence of the adaptive algorithmfor an FIR filter is more robust with respect to modelingerrors. In most adaptive optics problems to which the currentmethods have been applied, FIR filter orders greater thanfour do not offer further performance improvement. Hence,an FIR filter is used in this paper. As the gains in the filterLare updated repeatedly, and they converge to optimal constantgains when the disturbances have constant statistics.

The least-squares objective for choosing the filter matrixL(z) is to minimize eT (t)e(t). An important result of theparameterization of actuator and sensor spaces in [7], [8],[11] is that each component ofe(t) is affected only by thecorresponding component of the control command vectorv. This means that the RLS problem reduces to a set ofindependent RLS problems for the gains in the individualrows of L(z).

A challenging feature of the problem here is the verylarge number of channels. Even though the parameterizationof actuator and sensor spaces causes the channels in thesignalsv, u, ande to remain uncoupled, all channels in thenoise referencer feed into each control command. Hence,the adaptive filtering problem to determine the optimal gainsfor L(z) is multichannel. An on-line algorithm for adaptivelydetermining the filter gains must be numerically robust in thepresence of many channels. The algorithm proposed here isa multichannel adaptive lattice filter, which is based on thealgorithm presented in [10].

IV. SIMULATION RESULTS

Simulation of a problem like that illustrated in Figure 1were performed. The high-fidelity wavefront propagationcode WaveTrain simulated the effect of atmospheric turbu-lence on the laser beams, was wells as the optics hardware(deformable mirror and wavefront sensor). The deformablemirror in this simulation has 196 master actuators, and theHartmann wavefront sensor had 156 subapertures. For thesimulation results presented here, the beacon for adaptiveoptics was a point source. In current research, the controlmethods used here are being applied to simulations withextended beacons.

The FIR filter had order 4. Initially, closed-loop wavefrontsensor data was generated with only the top AO loop closed.This data was used by the lattice filter to identify the FIRgains. Then the simulation was started, with a differentrandom seed for turbulence generation, with the FIR loopclosed and run for 5000 time steps to produce the resultsshown in the figures here.

Point spread functions were computed for simulations withno control, with the LTI feedback control loop only, andwith the feedback loop augmented by the adaptive loop.The point spread functions then were averaged over 50-point intervals to simulate a shutter speed that is1/100 ofthe control sample-and-hold rate of 5000 Hz. Typical PSFs,along the corresponding images of the beacon, are plotted inFigure 3. The diffraction-limited Strehl ratio was normalizedto 1. Figure 4 shows the on-target high-order Strehl ratiosfor the last 2000 time steps. The high-oder Strehl ratio is thenormalized intensity at the centroid of the image. The meanvalue of the high-order Strehl ratio is80% greater with theFIR loop than with the standard AO loop only.

Perhaps the single most important performance measurefor a directed energy system is the maximum accumulatedenergy on any single point on the target. Figure 3 showsthe maximum accumulated intensity with the FIR loop is69% greater with the FIR loop than with the standard AOloop only. the normalized accumulated intensities at all targetpixels, with and without the FIR augmentation to the standardAO loop. These distributions were computed by averagingthe on-target image over the 5000 frames (one second) ofeach simulation. The maximum accumulated intensity is69%greater with the FIR loop than with the standard AO looponly.

V. CONCLUSIONS

The quasi adaptive FIR control loop produces significantimprovement in the point spread function of the adaptiveoptics system. The results here have been achieved for asignificantly more challenging adaptive optics problem, anda more realistic simulation model, than those used in previousstudies to illustrate the effectiveness of augmenting a clas-sical adaptive optics loop with adaptive filtering and controlbecause the simulation here used an extended turbulencepath, rather than the relatively short turbulence paths commonin astronomy applications. Also, the new configuration ofthe adaptive loop here, which uses only the closed-loopwavefront sensor vector as input to the adaptive loop, hasadvantages for practical implementation.

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Fig. 3. Point spread functions (left) and images (right) for 50-point intervalending at time step 5000. Top: standard AO loop augmented by FIRloop(194 channels); Bottom: standard AO loop only (194 channels).

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Fig. 4. High-order Strehl ratio time histories for point spread functionsaveraged over 50-point intervals.

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Fig. 5. Long term averaged point spread functions (left) andlong termimages (right) for 5000 time steps. Top: feedback loop with FIRloop (194channels); Bottom: standard AO loop (194 channels).

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