SPATIO-TEMPORAL PREDICTION FOR ADAPTIVE OPTICS WAVEFRONT RECONSTRUCTORS

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1 SPATIO-TEMPORAL PREDICTION FOR ADAPTIVE OPTICS WAVEFRONT RECONSTRUCTORS Michael Lloyd-Hart and Patrick McGuire Center for Astronomical Adaptive Optics, Steward Observatory, University of Arizona, Tucson, AZ 85721, U.S.A. ABSTRACT By taking advantage of the spatial and temporal correlation of the phase of the atmospherically-aberrated optical wavefront, we show in extensive computer simulations that the effect of the time delay in the servo loop of an adaptive optics system can be greatly reduced. Further work based on open-loop Shack-Hartmann sensor data from a 1.6-m telescope confirms the results of the simulations. Most of the work so far has explored linear algorithms which predict the output from the wavefront sensor based on immediate past history, although an investigation recently begun into the use of artificial neural networks holds promise for greater robustness in the low signal-to-noise regime, and offers the possibility of continuous on-line training, which can keep the network up to date on changing atmospheric statistics. In addition, in the linear case, we have computed predictors which attempt to track the changing phase during an individual wavefront sensor integration. Substantial improvement in the residual phase error caused by temporal decorrelation is obtained, particularly as the wavefront sensor integration time approaches the atmospheric decorrelation time τ 0, which is encouraging for adaptive optics systems pushing towards shorter wavelengths. 1. INTRODUCTION One of the many sources of residual wavefront error in a system for adaptive compensation of atmospheric aberration arises because of the finite time required to sense the aberration and apply a correction. The trade-off between photon and read noise in the wavefront sensor signal, and error caused by temporal evolution of the wavefront is such that even for relatively bright guide stars, integration times of the order of 1 to 10 ms are required on the wavefront sensor s detector. A further source of residual error arises in the transformation from the data vector measured by the wavefront sensor to the pupil-plane phase: there is never a one-to-one correspondence. Information on the shape of the wavefront is limited in that the wavefront sensor does not sample all spatial frequencies which contain appreciable power. Several groups have addressed the second of these problems, and have begun to perform experimental work at the telescope 1,2. In addition, Jorgenson and Aitken 3 and Aitken and McGaughey 4 have shown that the behaviour of the phase difference between two points is a chaotic function of time, and is therefore amenable to short-term prediction. We report here on numerical simulations in which the effect of the delay in the servo loop is reduced by using the immediate past history of the data vector from the wavefront sensor to project the value of the vector some time into the future, when the phase correction will actually be applied by the deformable mirror. Furthermore, in the current scheme, the matrix elements used by the predictor algorithms would be derived from scratch using data taken entirely at the telescope. Thus, one may envision a system in which the first few minutes of each observing night are devoted to building the predictor/ reconstructor matrices, which would then be ideally suited to the prevailing atmospheric conditions. The present results have been derived entirely from open-loop wavefront sensor data from a point-like reference source at infinity; in the near future, we will begin an analysis of closed-loop data. Most of the present work has relied on numerical simulations, described in Section 2, using a model of the atmosphere derived from observations at the Multiple Mirror Telescope (MMT) 5,6. Two algorithms, discussed in Section 3, have been investigated for predicting the slope vector generated by a Shack-Hartmann wavefront sensor. In Section 4, we describe the results of the simulation runs, and compare them to results obtained from open-loop wavefront sensor data taken at the Steward Observatory 1.6-m telescope, using a Shack-Hartmann sensor with 31-cm subapertures. Both the simulations and the experimental arrangement have been chosen to match closely the measured operating conditions of the adaptive optics system under construction for the 6.5-m MMT DESCRIPTION OF THE SIMULATION AND TELESCOPE EXPERIMENTS 2.1 Simulation of the atmosphere The simulated atmosphere consisted of three separate turbulent layers with Kolmogorov power spectra. Temporal evolution was modelled by assuming the Taylor hypothesis, and propagating the three layers at different speeds, in random directions. Parameters are given in Table 1, for a wavelength of 2.2 µm; the structure function is shown in figure 1.

2 Table 1. Parameters of the simulated atmospheric phase screens Layer Inner Scale Outer Scale r 0 (2.2 µm) Wind Speed Wind Direction 1 1 mm 100 m 1.8 m 40 m/s mm 100 m 1.5 m 20 m/s mm 100 m 1.0 m 10 m/s 281 Figure 1. The structure function for phase perturbations computed from the simulated atmosphere with parameters as given in Table 1. The effective value of r 0 in the pupil plane is m, and τ 0 is 14 ms. These values represent conditions which are somewhat worse than median for the site (r 0 = 0.9 m, τ 0 = 30 ms at 2.2 µm); they were chosen to demonstrate that the improvements to be expected from temporal prediction are not limited to those times when conditions are average or better. 2.2 Simulation of the telescope, optics, and wavefront sensor Results have been obtained for a 2.0-m diameter primary mirror with a Cassegrain hole of 0.2-m diameter. The deformable mirror was represented as a continuous facesheet with 5 5 actuators spaced at intervals of 50 cm on a square grid. The central actuator, falling in the Cassegrain hole, was not used, but those falling outside the illuminated pupil were retained as a guard ring. The system thus had 24 degrees of freedom. A gaussian with FWHM equal to 1.5 times the actuator spacing was used for the influence function of all the actuators. This was not a realistic modelling of an actual deformable mirror, but was rather intended to provide a reasonable approximation in the spirit of a proof-ofprinciple. The phase over the pupil was calculated on a square grid of 312 points with 0.1 m spacing. The wavefront sensor was a Shack-Hartmann sensor with a 4 4 lenslet array, and square subapertures 50 cm on a side, providing 32 slope measurements per exposure. The detector was assumed to be a CCD with 3 electrons rms read noise. Throughput for the optical system was assumed to be 100%. Simulations have been carried out for wavefront sensor integration times between 2 and 10 ms. To simulate the effect of turbulence more closely, the phase screen was recomputed every 2 ms, and for integration times longer than 2 ms, the phase was accumulated throughout the integration. Slope vectors were computed from the mean phase value. 2.3 Experimental setup for 1.6-m telescope data To provide some grounding in reality for the simulations, we have collected data from a simple Shack-Hartmann sensor mounted at the Cassegrain focus of Steward Observatory s 1.6-m telescope. The lenslet array divided the pupil into 19 close-packed hexagons 31 cm from edge to edge. The central lenslet fell into the Cassegrain hole, and was not illuminated, so the slope vector consisted of 36 values. The wavefront sensor camera was run at its maximum read rate, which provided frames at 42.4 Hz. This is roughly an order of magnitude slower than the planned frame rate for the 6.5-m system, but serves as an adequate check of the simulations. Runs of 2520 consecutive frames on bright stars were collected for later analysis. Data sets were collected at a number of different values of the signal-to-noise ratio (SNR) by inserting neutral density filters in front of the camera.

3 3. RECONSTRUCTOR ALGORITHMS The wavefront reconstructor algorithm has been separated conceptually into two distinct portions. The first is the predictor, which computes an estimate of the future value of the data vector from the wavefront sensor, and the second is the algorithm normally thought of as the reconstructor - the inversion of the data vector into the pupil-plane phase. Let S denote this standard reconstructor matrix, and pr be the predictive algorithm which operates on a short time series of Shack-Hartmann slope vectors d. The estimate d' t at time t is given by d' t = pr ( d t 1, d t 2, ). Eq. 1 The desired phase φ t is given by S d, ' t φ t = S pr ( d t 1, d t 2, ). Eq. 2. In our simulations, we are primarily interested in the behaviour of the predictor, so we have not attempted anything clever with the inversion algorithm, but have simply relied on the Gaussian inverse of the deformable mirror s influence on the slope vector from the Shack-Hartmann sensor, with each slope value weighted by the illumination of the corresponding subaperture. The influence function F was determined by applying the first 24 Zernike modes after piston to the mirror, and measuring the corresponding slope vectors. The standard reconstructor matrix S which derives those modes is then given by S = ( NF) T NF 1 ( NF) T N, Eq. 3 where N = ni, I being the identity matrix, and n a vector containing the relative illumination of the subaperture associated with each slope value. 3.1 Linear predictor Two algorithms have been explored for the predictor pr. The first is a linear algorithm, implemented as a matrix P in the software. P is computed so as to minimise the error in the estimate of d in a global least squares sense. With the deformable mirror in its flattened state, and the adaptive loop open, a large number of slope vectors are collected with the same wavefront sensor integration time as in closed-loop operation. The vectors are arranged as column vectors in a matrix M 1, where each column contains a number of slope vectors equal to the lookback depth L of the reconstructor. A second matrix M 2 of slope vectors is also built up, where each column contains the slope vector immediately succeeding those in the corresponding column of M 1. Thus, if column n of M 1 contains d 0, d 1, and d 2, then column n of M 2 would contain d 3. The least-squares best fit predictor is then given by T T P = M 2 M 1 M1 M 1 1. Eq. 4 We have also computed predictors which attempt to track the changing phase during a single exposure on the wavefront sensor detector. For instance, if the wavefront sensor is to be used in closed loop with an integration time four times the minimum cycle time of the servo loop τ s, then we can compute four separate predictor matrices, which on the basis of the same input vectors d t-1, d t-2,... compute different phase corrections at intervals of τ s throughout the integration. We refer to this technique as phase interpolation. 3.2 Artificial neural network predictor The results of Jorgensen and Aitken 3 have prompted us also to investigate the use of artificial neural networks 8 for the predictive part of the wavefront reconstructor pr. The work is ongoing, but preliminary results are presented below. As input, the network accepts 160 slope values (the Shack-Hartmann vectors from the 5 most recent exposures), preprocesses them in a manner described below, and passes them to a layer of 100 hidden neurons. The hidden neurons each perform a weighted sum of all 160 inputs, and pass the result through a non-linear sigmoid (S-curve) function, to give an output between 0 and 1. The outputs of the hidden neurons are passed to 32 output neurons, which perform a second weighted sum. The network is trained using the back propagation algorithm to predict 5 ms into the future by showing it a large number of input vectors, together with the corresponding desired output. It has been found that the network trains much better if it is shown, and asked to predict, the time derivative of the slope values rather than the slopes themselves. We believe this is because over such a short period, the derivative shows much smaller excursions than the slopes. Training is also greatly improved, both in the time required and in the accuracy

4 of prediction, if the slope derivatives are first preprocessed in such a way that the distribution of values p(x) is transformed into a flat distribution ranging from 0 to 1. We require that p ( x)dx = dx*, which leads to the following expression for the transformed slope derivative ḋ * in terms of the original value ḋ, ḋ * = ḋ p ( x) dx Eq. 5 ḋ min where ḋ min is the minimum value in the distribution of ḋ. This preprocessing technique has the advantage that it demands high resolution learning for values in the center of the distribution while allowing low resolution learning for values on the tails of the distribution. The technique also takes advantage of the full dynamic range of the neurons, allowing more training to occur on the non-linear portions of the S-curve. 4. RESULTS 4.1 Linear predictor - simulated data The predictors described in this section were derived from Eq. 4, using 3000 consecutive slope vectors. Testing was done on a further 1000 slope vectors drawn from the same realization of the atmosphere, but which had not been included in the derivation. In the testing phase, we first computed an estimate d' of the wavefront sensor slope output at some time in the future. The estimate is multiplied by the standard reconstructor S to derive the modes which are then fitted to the mirror surface. Results have been obtained for three different wavefront sensor integration times τ int = 2, 4, and 10 ms. For all the simulations, the predictor computed independent corrections to the shape of the mirror every 2 ms; thus, for the longer integration times, more than one correction was made during the course of each integration. The simulation includes the effects of many different sources of residual wavefront error, but the effect of interest here is the error due to the servo delay. This was computed by measuring the actual noise-free value of the slope vector just before each predicted correction was applied to the mirror, and computing the mean square difference between the actual and predicted values. In addition to τ int, the parameters we have explored are the SNR and lookback depth. Figures 2 to 4 show the results, where for each value of τ int we compare the performance of predictors which use from 1 to 5 immediate past slope vectors, as functions of the SNR in the slope data. For the assumed geometry of the telescope and wavefront sensor, the plotted range of SNR corresponds roughly to stellar magnitudes from 0 to 14. Figure 2. Comparison of the temporal decorrelation portion of the residual wavefront error for an integration time τ int = 2 ms as a function of SNR in the slope values from the wavefront sensor. The performance of five predictors is shown, where the predictors used from 1 to 5 immediate past slope vectors as input. The dashed line shows the performance of the standard reconstructor alone, (the predictor is taken to be the identity matrix), operating on the last measured slope vector. In general, the results are not surprising. At high SNR, using the time history of the slope vector can dramatically reduce the temporal delay error, but as the SNR is reduced, the predictors with larger lookback depths compute noisier predictions. The improvement in the phase error at high SNR is quite strongly dependent on the ratio τ int /τ 0. Figure 4 also shows the effect of not using phase interpolation. Compared to the equivalent predictor with interpolation, the prediction is somewhat less good at high SNR, but again not surprisingly, in the low SNR regime, trying to extract more information from the same signal gives a worse result.

5 Figure 3. The same as figure 2, for a wavefront sensor integration time of 4 ms. Phase interpolation was used - updates to the deformable mirror shape were computed and applied every 2 ms, on the basis of past 4-ms integrations. Figure 4. Results for an integration time of 10 ms. The solid lines represent simulations where phase interpolation was used to update the mirror shape every 2 ms; dotted lines show the curves from identical simulations except that only one correction to the mirror shape was made per integration time. 4.2 Neural net predictor - simulated data So far, just one set of conditions has been investigated with the neural network - the case where τ int = 5 ms, and the lookback depth is 5. The network was first trained on 144,000 consecutive slope vectors, with zero noise. Figure 5 shows the result after training, which is compared to a linear predictor computed in the same way as those in the previous section, and also to a second linear predictor computed from slope derivatives preprocessed in the same way as the input to the neural net. Three things should be pointed out. Firstly, in the high SNR regime, the temporal phase error in the neural net s output is actually slightly negative - that is, the net has learned to do a better job of correcting the phase from the 5 past slope vectors than the standard reconstructor can do from the immediate slope vector. Secondly, at the other end of the scale, the net outperforms both linear predictors by well over an order of magnitude. Finally, for the linear predictors, although preprocessing helps by about a factor of two with low signal, it is a distinct disadvantage in the low noise case. We believe the remarkable durability of the neural net in the region of low SNR is due to the preprocessing technique applied to the slope derivatives, described in Section 3.2. The uniform distribution between 0 and 1 of the preprocessed derivatives in the zero-noise case for which the net was trained provides hard stops for the neural net s outputs. In the noisy case, when the inputs are biased towards the extremal values because of the long tails of the original noise distribution, the net is prevented from outputing correspondingly large signals, and so acts as a very effective noise filter, something the linear predictor is not flexible enough to do. 4.3 Linear predictor - real data To provide some verification that the predictors are not simply learning to rely on artifacts of the simulation which do not exactly mimic the real atmosphere, we have computed linear predictors from data taken with a Shack-Hartmann

6 Figure 5. Comparison of a linear predictor with an artificial neural network trained to predict the slope vector under the same conditions. The curve labeled Linear is the result of a linear predictor derived in the same way as those of Section 4.1. The curve labeled Linear preprocessed shows the behaviour of a second linear predictor derived from slope vectors treated in the same way as those shown to the net. Clearly, the neural net does very much better in the low SNR regime. The fourth plot shows results from the standard reconstructor with no prediction sensor on a 1.6-m telescope. The frame rate of the camera was limited to 42.4 Hz, and τ 0 was about 18 ms; nevertheless the results shown in figure 6 demonstrate that a very significant reduction in slope error can be achieved with a predictor compared to correction using just the most recent available slope vector. We have also computed predictors from simulated data designed to match the real data. The results are also shown in figure 6. The agreement between with the simulation is quite reasonable, which gives us confidence in the simulated results. In all cases, the predictors were tested on data similar to but different from the data used in the predictor s derivation. Figure 6. Comparison of the results of linear predictors trained on slope data taken at Steward Observatory s 1.6-m telescope at 42.4 Hz with similar results from simulated data. The parameters of the simulation were changed from those of Sections 4.1 and 4.2 only to match the values of r 0, τ 0, τ int, and the telescope diameter in the real data. The dashed lines show the mean square uncorrected slope values; the dotted lines show the improvement obtained by assuming the most recently measured slope vector for the correction, and the solid lines show results from predictors trained on the data using Eq. 4 CONCLUSIONS The work presented here extends the conventional approach to wavefront reconstruction by postmultiplying the usual reconstructor matrix by a second function which attempts to predict the shape of the wavefront at the time when the correction will actually be applied to the deformable mirror. The reconstruction is thus applied to up-to-date data, and much of the wavefront error introduced by the delay between sensing and correction can be eliminated. These results suggest that a reduction by a factor of between 2 and 10 in mean square phase error due to temporal decorrelation should be achievable, with good SNR. For low SNR, artificial neural networks hold great promise for maintaining the reduction. Further work is planned on the exploration of nets, and other non-linear algorithms with both open- and closed-loop data taken at various telescopes. In addition, it seems to be practical to build a reconstructor entirely at the telescope. At a wavefront sensor frame rate of 1 khz, it would take less than 10 minutes to derive a matrix for a system with 300 actuators, such as the adaptive

7 secondary mirror 9 planned for the 6.5-m MMT. In the case of the neural network, the net can continue to be trained during closed-loop operation, providing continuous adaptation to slowly-evolving atmospheric conditions. ACKNOWLEDGEMENTS This work has been supported by the U.S. Air Force Office of Scientific Research (grant #F ), the National Science Foundation (grant #AST ), and the Flintridge Foundation. We thank Steve Ridgway of NOAO for the loan of his lenslet array, Buddy Martin for thought-provoking discussion, and the computer users of Steward Observatory for the sacrifice of uncounted billions of CPU cycles. REFERENCES 1. Wild, W. J. et al., Investigation of wavefront estimators using the Wavefront Control Experiment at Yerkes Observatory, 1995, Proc. SPIE Conf. on Adaptive Optical Systems and Applications, Ed. R. K. Tyson & R. Q. Fugate, 2534, Rhoadarmer, T. A. and Ellerbroek, B. L., A method for optimizing closed-loop adaptive optics wavefront reconstruction algorithms on the basis of experimentally measured performance data, ibid, Jorgenson, M. B. and Aitken, G. J. M., Neural network prediction of turbulence induced wavefront degradations with applications to adaptive optics, 1992, Proc. SPIE Conf. on Adaptive and Learning Systems, Ed. F. A. Sadjadi, 1706, Aitken, G. J. M. and McGaughey, D., Predictability of atmospherically distorted wavefronts, these proceedings 5. Lloyd-Hart, M. et al., Adaptive optics experiments using sodium laser guide stars, 1995, Astrophys J., 439, Lloyd-Hart, M. et al., Progress in diffraction-limited imaging at the Multiple Mirror Telescope with adaptive optics, 1993, JOSA A, 11, Sandler, D. G. et al., The 6.5-m MMT infrared adaptive optics system detailed design and progress report, these proceedings 8. Rumelhart, D. E., Hinton, G. E., and William,s R. J., Learning internal representations by error propagation, in Parallel Distributed Processing, 1986, Vol 1, Ed. D. E. Rumelhart & J. L. McClelland, (Cambridge: MIT) 9. Martin, H. M. and Anderson, D. S., Techniques for Optical Fabrication of a 2-mm-thick Adaptive Secondary Mirror, these proceedings