1 November 1998

Ž .Optics Communications 156 1998 10–15

Optimal Hartmann sensing at low light levels

D.L. Ash a, C.J. Solomon a, G. Loos b

a School of Physical Sciences, UniÕersity of Kent, Canterbury CT2 7NR, UKb Air-Force Research Laboratories, Kirtland Air-Force Base, NM 87117-6008, USA

Received 13 January 1998; revised 8 June 1998; accepted 14 July 1998

Abstract

A computer model has been used to investigate the performance of a Shack-Hartmann sensor operating under low lightlevel conditions. The trade-off between sub-aperture size and slope noise is examined in detail and optimal sizes aredetermined for a number of imaging scenarios. Our results are discussed in relation to related literature. q 1998 ElsevierScience B.V. All rights reserved.

1. Introduction

It is well known that atmospheric turbulence severelylimits the performance of ground based telescopes andthere now exists a considerable literature on both techno-logical and computational techniques to compensate for

w xatmospheric effects 1 . Accurate estimation of atmospheri-cally perturbed wavefronts is of critical importanceŽ .whether directly or implicitly in any attempt to correctincident wavefronts in real time using adaptive optics. It isalso a crucial aspect of the post-processing technique of

Ždeconvolution from wavefront sensing also termed self-.referenced speckle holography originally developed in

w xRef. 2 . Although wavefront sensing devices such as theshearing interferometer and increasingly the curvature sen-

w xsor 3 have been developed and used for astronomical andsurveillance imaging systems, the practical and conceptualsimplicity of the Shack-Hartmann device and its competi-tive performance still make it a first-choice in manysystems.

In this paper, we use a computer model employing amodal wavefront reconstructor to investigate Hartmannsensor performance at low signal levels and consider thecorresponding implications for optimal design. A key fac-tor in employing Hartmann sensors for adaptive opticsapplications is to optimise the pupil-plane sub-aperturesize. When the signal is limited, we see that by increasingthe Hartmann sub-aperture size, the signal–noise of theindividual slope measurements at the sub-apertures is in-

creased but at the expense of larger sampling intervals andhence larger modelling error. It is evident that a balancemust be achieved between these two competing factors ifthe best performance is to be attained.

Previous authors have addressed the question of Hart-mann sub-aperture size for adaptive optics applications.

w xGardner et al. 4 discuss laser-guide star adaptive opticssystems and present results for that sub-aperture size whichwill minimise the required sub-aperture photon count toachieve a number of target rms wavefront error values.Their analysis, however, does not include the effects of

w xread-noise. Ellerbroek 5 discusses the effect of varyingŽDrl i.e. varying the number of sub-apertures across the

telescope pupil plane but this effectively only studies the. w xfitting error . Gavel et al. 6 derived analytic results for

optimum sub-aperture size but under the restrictive as-sumption that r -1 and that CCD read-noise may be0

neglected. They also present results for the optimum sub-aperture size as a function of laser power for the Keck

w xtelescope system. Sechaud et al. 23 have also presented acomprehensive analysis which defines the limiting visualmagnitude as a function of sub-aperture size for theCOME-ON system.

In this paper, we use a computer model incorporatingturbulence, shot and CCD read-noise and employing amodal wavefront reconstructor to investigate Hartmannsensor performance at low signal levels. In certain scena-rios, both for astronomical and non-astronomical applica-tions, it may not be feasible to illuminate the object or sky

0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0030-4018 98 00398-8

( )D.L. Ash et al.rOptics Communications 156 1998 10–15 11

artificially. The primary goal of this study is to considerthe optimal sub-aperture size of systems operating in lowsignal–noise regimes without employing laser guide starsto assist with wavefront sensing.

2. Computer model

2.1. Generation of Hartmann images

Randomly distorted optical wavefronts obeying thew xKolmogorov power spectrum were generated 7 and con-

sidered incident at the pupil plane of the telescope. As iscommonly assumed in problems of imaging through turbu-lence, amplitude fluctuations were neglected. The Hart-mann sub-aperture images were calculated by considering

Ž .the appropriate segment of a high i.e. infinite light levelwavefront and using standard Fourier optics to generatethe image plane intensities as

2w xI j ,h A FT A x , y , 1Ž . Ž . Ž .Ž . Ž .where I j ,h is the intensity in the image plane, A x, y

is the complex amplitude of the incident wavefront portionand FT denotes the operation of Fourier transformation.

Photon limited images are generated in a standard wayfrom the infinite light level images via generation of acumulative distribution function which is then randomlysampled in order to determine the location of each photo-electron event in the image plane. Poisson sampling for thenumber of detected events in each sub-aperture is alsoincluded in the model. In our study, we have assumed aCCD camera with read-noise of 1.5 electrons rms per pixelper frame – this figure corresponds to an upper limit on

w xthe performance of current state-of-the-art devices 8 . Anumber of statistical models have been described for read

w xnoise in CCD arrays 8,9 . In our study, read-noise issimulated by adding an appropriately scaled Gaussian de-viate to each pixel in the CCD image frame.

Calculation of the sub-aperture centroid positionsŽ Ž . Ž ..C j ,h s C ,C was performed using the discretisedj h

version of the classical centre of massrfirst moment,

j I j ,h h I j ,hŽ . Ž .Ý Ý ÝÝh hj j

C s , C s . 2Ž .j hI j ,h I j ,hŽ . Ž .Ý Ý ÝÝh hj j

Initially centroiding was performed over a compara-tively large area to locate the approximate position of thecentroid. The centroiding envelope was then reduced insize and centroiding was repeated centred on the newlyestimated ‘‘spot’’ location. This process was then repeatediteratively until the envelope had approximately reached

w xthe optimal dimension 10 of 3=3 pixels and the centroidlocation had stabilised. This typically takes between 4 and7 iterations depending on the stability tolerances specifiedand the rate at which the centroiding envelope is reduced.

Conversion of the centroid estimate to a local, averagewavefront slope is straightforward and the slope data isthen available to form an estimate of the incident wave-front.

2.2. Modal waÕefront estimation

A substantial body of work has now appeared in theliterature on the subject of wavefront estimation and, be-ginning with Wallner’s paper, specifically on optimal

w xwavefront estimation 5,12–16 . Within this body of litera-Žture, due to variations in notation and approach i.e. inte-

.gral or linear algebraic formulation to the wavefrontŽ .model whether zonal or modal and the specific meaning

attached to optimal, the common thread amongst them isnot apparent. For clarity, we here present what is, to thebest of our knowledge, the most direct and simple deriva-tion of a minimum variance estimator for a linear model.Other common solutions used in wavefront estimation maybe considered either as alternative forms or as limitingcases of this estimator.

In the so-called modal approach, the random phasefunction is approximated by an expansion of the form

N

F x s a P x , 3Ž .Ž . Ž .Ý k kks1

Ž .where F x is the atmospherically distorted wavefront and� Ž .4P x are the chosen basis functions and the system isk

modelled as

Axqesm , 4Ž .

where x is the vector of modal coefficients, A is thesystem or design matrix, m is the vector of phase gradientsand e is the error vector on the slopes.

We make the following assumptions on this linearŽmodel all of which are reasonable for Hartmann and

.shearing sensors² :Ø Signal and noise vectors are both zero mean – x s

² :e s0.² T :Ø Noise and signal are uncorrelated x e s0.

² T:Ø Noise covariance Vs ee is assumed known.² T:Ø A priori covariance of the unknown vector C s xxx

is assumed known. This covariance was explicitly cal-culated for the Zernike basis and Kolmogorov spectrum

w xby Noll 11 . A suitable unitary transformation enablesŽit to be calculated for any other orthogonal basis e.g.

. w xthe Karhunen-Loeve basis 7 .Accordingly, we seek a linear estimator xsLm whichˆ

will minimise the error covariance matrix P defined by

T² :Ps xyx xyx . 5Ž .Ž . Ž .ˆ ˆ

Ž . Ž .We substitute xsL Axqe in Eq. 5 to obtainˆT Tw x w xPs IyLA C IyLA qLVL . 6Ž .x

( )D.L. Ash et al.rOptics Communications 156 1998 10–1512

Taking variations in matrix P with respect to L andw xsetting to zero 17 , we have

T Tw xdPsydL AC IyLA qVLŽ .x

w x T Ty IyLA C A qLV dL s0, 7Ž .Ž .x

which yields an optimal matrix L

y1T TL sLsC A AC A qV .M V x x

Ž .Substitution of the matrix L into Eq. 5 defining PM V

yields an error covariance matrix given by

y1T TP sC IyA AC A qV AC . 8Ž .M V x x x

w xIt is easy to show 19 that two equivalent forms forL and P areM V M V

y1T y1 y1 T y1xsL ms A V AqC A V m ,ˆ M V x

y1T y1 y1PsP s A V AqC . 9Ž .M V x

Wallner’s lengthy analysis may be considered as theWiener-Hopf integral for the optimum filter assumingcontinuous data and, in fact, formally reduces to the

Ž . Ž .minimum variance estimator defined by Eqs. 7 and 9Žunder the same assumptions that we have stated see Ref.

w x.16 . This solution may also be derived directly fromw xBayes theorem 14,15 . Further, since since both the prior

and likelihood laws on the modal coefficients may bew xwell-modelled as Gaussians 11 , the choice of a different

ŽBayesian criterion such as finding the mean or median of. w xthe posterior distribution rather than the mode 15 is

academic and produces the same result.In the limit that the a priori covariance tends to infinity,

the minimum variancerMAP estimator tends to the bestŽ . w xlinear unbiased estimator BLUE 18 in the mean-squared

sense

y1T y1 T y1w xxs A V A A V m. 10Ž .ˆ

If we assume that the noise on each measurement isuncorrelated and the noise associated with each measure-

2 Ž .ment is equal, we have Vss I and Eq. 10 gives

y1T Tw xxs A A A m , 11Ž .ˆ

which is recognisable as the standard least-squares solu-tion. The standard least-squares solution is thus the limit-ing form of the MAP estimator when no a priori knowl-edge of the parameter statistics and noise is imposed orknown.

A modal estimator was implemented in our computermodel using both the Zernike and Karhunen-Loeve bases.

Fig. 1. Fitting error: The maximum possible compensated Strehl ratio with number of basis functions. The sensor has 52 active sub-aperturesof size r . A least-squares reconstructor was used.0

( )D.L. Ash et al.rOptics Communications 156 1998 10–15 13

3. Simulation results

In practice, Hartmann wavefront sensing systems arelimited to a finite number of sub-apertures and a finitenumber of modes only can be estimated. We conducted anumber of studies simulating an 8=8 Hartmann sensorhaving 52 active subapertures. The primary goal was tostudy the variation in maximum possible compensatedStrehl ratio as a function of flux and sub-aperture size. Thestudies were conducted using the standard least-squares

Ž .modal reconstructor described by Eq. 11 . As this estima-tor does not include a priori knowledge, no constraints areplaced on the modal coefficients. It is therefore importantin this case to avoid modelling errors which may be due tothe use of either too few or too many modes. Accordingly,

w xwe first study modelling errors for our system. Dai 21 haspreviously examined this issue in detail. In Fig. 1 we showthe maximum compensated Strehl ratio as a function of thenumber of estimated modes with sub-aperture size fixed atr . The Strehl ratios given were generated from the resid-0

Žual wavefront known wavefront subtracted from the esti-.mated and do not include the effects of any real phase

compensating system. The incident wavefronts correspondto high light level and obey the Kolmogorov power spec-trum of phase fluctuations. The wavefront reconstructionwas performed using the standard least-square estimatorand each data point is averaged over 400 realisations. We

observe a peak at approximately 45 modes after whichover-fitting effects increase the errors significantly. Wealso note that although the peak occurs for approximatelythe same number of modes, there is a significant differencebetween the Karhunen-Loeve basis and the Zernike poly-

w xnomials 22 .Fig. 2 shows the effect of variations in the photo-

electron flux on the optimal value of lrr . Here, an 8=80

Hartmann sensor having 52 active subapertures was used.Read-noise was simulated at 1.5 electrons rms per pixelper frame. As intuitively expected, the peak moves to-wards higher values of lrr as the flux falls and lower0

values as the flux increases. It is clear from Fig. 2 that thecorrect choice of sub-aperture size lrr is of critical0

importance to any attempt to accurately estimate and com-pensate wavefronts under conditions of low incident flux.We also note that the pronounced skew in the figuresindicates that choosing lrr smaller than the optimal0

value has more serious consequences than slight over-estimation of the optimum value. Fig. 3 shows the resultsof a study to estimate the optimal values of lrr as a0

function of flux levels for a number of CCD read noisevalues. Fig. 3 was obtained using an 8=8 Hartmannsensor having 52 active sub-apertures and using a least-squares reconstructor employing 45 Karhunen-Loevemodes and thus, strictly, corresponds to a special case.However, in Fig. 4 we show that the absolute number of

Fig. 2. Maximum possible compensated Strehl ratio as a function of pupil-plane subaperture size for signal levels of 10, 15, 20, 25 and 30photoelectrons per r 2. The simulation used an 8=8 Hartmann sensor having 52 active sub-apertures and a least-squares reconstructor using0

45 Karhunen-Loeve modes. Read-noise of 1.5 electrons rms per pixel per frame was generated.

( )D.L. Ash et al.rOptics Communications 156 1998 10–1514

Fig. 3. Optimum subaperture size as a function of photoelectron flux for varying amounts of read-noise. Flux is in photoelectrons per r 2 and0

read-noise in electrons rms per pixel per frame. The studies were conducted using an 8=8 Hartmann sensor having 52 active sub-aperturesand a least-squares reconstructor using 45 Karhunen-Loeve modes.

sub-apertures does not significantly affect the achievableStrehl ratio and thus the dominant factor is the signal–noiseobtained in the individual sub-apertures. It is therefore areasonable basis from which to extrapolate the optimumsize to imaging scenarios not directly studied herein.

Consider a set of observational parameters, namely agiven telescope diameter, stellar magnitude, atmosphericcoherence size r and frame exposure time. In a particular0

imaging scenario, the temporal bandwidth required and themagnitude of the star used for wavefront sensing will

Ždetermine the frame exposure time assuming this can be. 2altered and hence the flux per r area per frame. The0

read-noise of a given CCD device for the chosen orspecified integration time can then be determined. Usingthese values for flux and read-noise, Fig. 3 may be used todirectly estimate the optimal sub-aperture size. The opti-mal number of sub-apertures is then simply calculatedfrom the telescope diameter. If the flux and read-noisevalues are not represented on the curves, we suggest thatthe optimum size may be approximated by treating thecurves as realisations of a family controlled by these twofree parameters and thus generate an empirical curve forthe given values of flux and read-noise. Clearly, the accu-

Fig. 4. The maximum compensated Strehl ratio as a function ofthe number of Hartmann sub-apertures. No significant variation isobserved. The sub-aperture size was set equal to r and an infinite0

signal–noise ratio assumed within the sub-apertures.

( )D.L. Ash et al.rOptics Communications 156 1998 10–15 15

racy of such an approach will depend on the particularfunctional form chosen and the discrepancy between thevalues of the flux and read-noise from those studieddirectly by simulation.

We point out that the Strehl ratios indicated on thegraphs are limiting Õalues for real adaptive optics systemswhich will not be achievable in practice since they entirelyneglect finite bandwidth effects and limitations in thedeformable mirrorrcorrection device. However, the trendobserved as lrr increases is of direct relevance to real0

AO systems.Since the optimal sub-aperture size clearly depends on

sub-aperture signal–noise, it follows that the optimal val-ues indicated in Fig. 3 are dependent on the specificcentroiding algorithm and the reconstructor used. We choseto use the least-squares wavefront reconstructor because,despite the significant theoretical studies that have beenmade on optimal reconstructors, implementation of realis-tic and reliable a priori knowledge is quite complex and

w xwith notable exceptions 5 has rarely been employed inpractical systems. We note that superior slope estimationbased on matched filtering techniques rather than centre ofmass should yield higher signal–noise at low light levelsw x20 . However, since the trade-off between sub-aperturesize and slope estimation noise is such that less slopemeasurement noise allows smaller sub-aperture sizes, wepredict that the main effect of such methods will be simplyto shift the optimal sub-aperture size to slightly smallervalues than those indicated. The precise amount will de-pend on the relative performance of the matched filter butcan reasonably be expected to be small.

4. Summary

We have shown that the optimal sub-aperture size in aHartmann sensor operating at low light levels is criticallydependent on the prevailing signal to noise level and havegiven the optimal value for a number of situations. Criti-cally, our results include the effect of CCD read-noise. Theoptimum value for a number of typical imaging scenariosmay be estimated empirically from the results presented inFig. 3. This may be )3r in very low signal–noise0

conditions. We note that the specific results we haveobtained in Fig. 3 for the optimal sub-aperture size for thezero read-noise case are in good agreement with those

Ž w x.obtained by Gardner et al. Table1 in Ref. 4 .

Acknowledgements

The authors would like to acknowledge the supportprovided for this work by the European Office of Aerospace

Ž .Research and Development SPC-96-4054 and the Engi-neering and Physical Sciences Research Council, UKŽ .GRrL26339 .

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