Estimation of the adaptive optics long-exposure point-spread function using control loop data

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1 Véran et al. Vol. 14, No. 11/November 1997/J. Opt. Soc. Am. A 3057 Estimation of the adaptive optics long-exposure point-spread function using control loop data Jean-Pierre Véran Département Images, Télécom Paris, 46 Rue Barrault, Paris Cedex 13, France, and Département de Recherche Spatiale, Observatoire de Paris, 5 Place Jules Janssen, 9195 Meudon Cedex, France François Rigaut Canada-France-Hawaii Telescope, P.O. Box 1597, Kamuela, Hawaii Henri Maître Département Images, Télécom Paris, 46 Rue Barrault, Paris Cedex 13, France Daniel Rouan Département de Recherche Spatiale, Observatoire de Paris, 5 Place Jules Janssen, 9195 Meudon Cedex, France Received January 1, 1997; revised manuscript received April 17, 1997; accepted April 30, 1997 Astronomical images obtained with adaptive optics systems can be enhanced by using image restoration techniques. However, this usually requires an accurate knowledge of the system point-spread function (PSF) which is variable in time. We present a method to estimate the PSF related to each image, using data from the adaptive optics control computer, namely, the wave-front sensor measurements and the commands to the deformable mirror, accumulated in synchronization with the acquisition. This method requires no extra observing time and has been successfully tested on PUEO, the Canada France Hawaii Telescope adaptive optics system. With this system, accurate PSF estimations could be achieved for guide stars of magnitude 13 or brighter Optical Society of America [S (97)0011-1] 1. INTRODUCTION Optical aberrations in ground-based astronomical images result mainly from random phase distortions of the incoming wave fronts induced by the atmospheric turbulence above the telescope. 1 Adaptive optics (AO) is a technique now commonly used to overcome this effect: the wave fronts are corrected in real time by a deformable mirror whose shape is continuously updated to match the current state of the atmospheric turbulence. The correction, however, is never perfect and the long-exposure images acquired with an AO system are still affected by a residual blur that reduces the contrast of the fine details. This residual blur is, of course, much less severe than the aberrations in the uncorrected images and may be further reduced by mean of postdetection data processing. When the set of short-exposure images making up each long exposure is available, the compensated speckle holography method (also called compensated deconvolution from wave-front sensing) 3,4 can be used. This method makes use of the wave-front sensor (WFS) measurements to correct each short-exposure image and obtain an improved long-exposure image. In this paper we are interested in the case in which only the integrated, longexposure image is available. Then the previous method cannot be applied, and one must usually rely on conventional image restoration (deconvolution) algorithms. Image restoration techniques are used in a wide range of scientific applications, including astronomical imagery. In particular, an important effort was launched a few years ago, to design software able to overcome the flaw in the initial optics of the Hubble Space Telescope. 5 Although several blind methods have been proposed, 6 a good knowledge of the system s point-spread function (PSF) is always highly desirable for achieving an accurate restoration. For AO imaging, the shape of the PSF depends on how well the system is able to compensate for the wave-front distortions. For a given AO system, this degree of correction depends not only on the size and magnitude of the object used as a reference for wave-front sensing but also on the local state of the turbulence, which fluctuates in time. As a result, the AO PSF is highly variable. 7 Except in the rare cases where an isolated point source is available close enough to the observed object, the usual way around this problem involves dedicating a significant portion of the observing time to the sole acquisition of the image of a point source (star) that gives the current PSF. 8 In this paper we propose instead to estimate the current PSF using data from the AO /97/ $ Optical Society of America

2 3058 J. Opt. Soc. Am. A/Vol. 14, No. 11/November 1997 Véran et al. system, namely the WFS measurements and the commands applied to the deformable mirror, which can be saved for each acquisition. This approach has two major advantages: first, no extra observing time is required for the determination of the PSF; second, the temporal variability of the PSF is not a problem since the estimation is based on data exactly synchronous with the image acquisition. The long-exposure PSF can be expressed in terms of second-order statistics of the phase of the residual wave front. In Section we present an operational analysis of the correction of a turbulent wave front by means of adaptive optics and establish the exact relationship between the WFS measurements and the phase of the residual wave front. In Section 3 we express the long-exposure PSF in terms of second-order statistics of this residual phase and describe the actual estimation method. In Section 4 we discuss the practical application of this method to PUEO, the curvature-based AO system operated by the Canada France Hawaii Telescope at the top of Mauna Kea. Results obtained on the telescope are presented, and the precision of the estimated PSF is discussed.. ADAPTIVE OPTICS LOOP ANALYSIS We assume a telescope with a mirror of diameter D equipped with an AO system. We use the twodimensional vector x to reference the points of the pupil plane of the telescope. The telescope aperture P is characterized by a pupil function defined as Px 1 inside the aperture 0 outside the aperture. (1) A. Turbulent Phase We define the turbulent phase a (x, t) as the twodimensional function representing the phase of the incoming turbulent wave front on the telescope aperture P at instant t. Because the state of the atmosphere changes randomly in time, a (x, t) can be seen as a realization of a random process. A classical model for the statistics of this random process is the Kolmogorov model, 9 which has only one free parameter r 0 called the Fried parameter. r 0 represents the coherence length of the turbulence and quantifies its strength. We call E the set of all the possible realizations of a (x, t). In a classical algebraic framework, E is a vector space of infinite dimension. The scalar product of two elements of E may be defined as 1 E 1 S P 1 x xdx () where the normalization constant S is the area of the pupil given by S P dx. The Zernike polynomials Z i (x), as defined in Ref. 10, are a set of basis functions of E that have been found very suitable for the description of the turbulent phase. a (x, t) can then be decomposed as a x, t i1 z i tz i x. (3) In the Kolmogorov model, the covariance of the expansion coefficients (Zernike coefficients) can be expressed as z i z j K ij D 5/3 r 0. (4) The analytic expression of K ij is given in Refs. 10 and 11. The temporal power spectrum of the Zernike coefficients S zi z i (f ) can be computed 1 by invoking the Taylor hypothesis of a frozen turbulence propagating at the wind velocity V. S zi z i (f ) is shown to asymptotically follow a rapid decrease in f 17/3 after a cutoff frequency given by f c i 0.3n i 1V/D, (5) where n i is the radial degree of the polynomial Z i (x). This behavior has been experimentally verified at the Canada France Hawaii Telescope. 13 Moreover, most of the power has been found to be concentrated around f c (i). The validity of the Kolmogorov model has been discussed and even questioned several times. A major problem is that the model considers the outer scale of the turbulence to be infinite, which is not true in practice. The presence of a finite outer scale mostly affects the variance of the lowest Zernike coefficients, which is less than what Eq. (4) predicts. 14 Furthermore, z (t) and z 3 (t), the tip and tilt coefficients, are strongly affected by different nonatmospheric phenomena such as telescope tracking errors. B. Phase Correction Upon entering the AO system, the turbulent phase is corrected by action of the deformable mirror (DM). If m (x, t) is the phase configuration of the mirror at instant t, the residual (corrected) phase (x, t) is x,t a x,t m x,t. (6) If the DM has m degrees of freedom (actuators), the phase configurations it can produce usually define a vector space M of dimension m which is a subspace of E. Note that the definition of M will be slightly altered in Subsection.D but the discussion below will still hold. The elements of M are low spatial frequency functions: most of their energy is concentrated at frequencies lower than the inverse of the inter-actuator distance. We define M as the orthogonal complement to M in E. Then any phase function of E can be decomposed as the sum of a component on M (mirror or low-order component) and a component on M (high-order component). For instance, Since the DM may compensate only for the mirror component of the turbulent phase, the residual phase has the same high-order component as the turbulent phase. (x, t) may then be decomposed as (7) x, t x, t x, t. (8)

3 Véran et al. Vol. 14, No. 11/November 1997/J. Opt. Soc. Am. A 3059 The mirror component of the residual phase is given by Eq. (6): x, t a x, t m x, t. (9) The optimal instantaneous correction is achieved when the norm of (x, t) is minimal, that is, when m (x, t) a (x, t). We define the mirror modes M i (x) as a set of basis functions of M. Any phase function from M can then be decomposed on the mirror modes and we call modal coordinates the coefficients of this decomposition. The following decompositions will be used later: m a x, t i1 a i tm i x, (10) m x, t i1 m i tm i x, (11) m x, t i1 i tm i x. (1) We will refer to the m i as the modal commands to the DM. There is a bijective relationship between the modal commands and the actual voltages applied to the DM. Equation (9) translates into modal coordinates as i (t) a i (t) m i (t), or in vector notation, a m Each mirror mode is an element of E and can therefore be decomposed on the Zernike basis. Following Eq. (4), a i a j can thus be expressed in a Kolmogorov turbulence as a i a j K ij D 5/3 r 0 (13) and K ij can be computed from the K ij. There is an infinite number of possibilities for choosing a set of mirror modes and any of them could actually be used. However, since the deformations achieved by continuous deformable mirrors are mostly low-frequency functions, usually some of the first Zernike polynomials can be well approximated. It is then convenient to choose these Zernikelike deformations as mirror modes so that Eq. (5) can be directly used to estimate their temporal cutoff frequency. The remaining modes can be chosen arbitrarily and an upper bound for their cutoff frequency is given by the highest Zernike polynomial with a significant contribution in their decomposition. In Section 4 we will detail the choice of such a particular base for PUEO. For now, we just define f M as the maximal temporal cutoff frequency of the mirror modes in the turbulent phase. C. Measurement of the Residual Phase After correction, the wave front is split: one part is used to actually form the corrected image onto the detector; the other part is directed onto a WFS so that the residual phase can be measured. The WFS is a device able to sample the derivatives of the corrected phase (x, t), at a finite number of points (subapertures) on the pupil. Commonly used devices include Shack Hartmann WFS and curvature WFS. 15 In any case, the WFS measurement is an n-dimensional vector, where n is the number of subapertures in the case of the curvature WFS and twice this number in the case of a Shack Hartmann WFS. We denote W (x, t) the WFS measurement of (x, t), in the absence of any measurement error. The WFS is usually supposed to work in its linear domain so that W can be considered a linear operator. If some known voltages are applied to the DM in the absence of turbulence and the corresponding WFS measurements are recorded, W can be calibrated on the mirror space M. From this calibration, the WFS response to the mirror modes can be deduced and expressed as the columns of a n m matrix D, called modal interaction matrix. Usually, the number of subapertures on the WFS and the number of actuators on the DM are dimensioned so that n m. Using Eqs. (8) and (1) we can write W x, t W x, t W x, t Dt W x, t. (14) Another way to calibrate W is to use a physical model to simulate the WFS. In Ref. 16, for instance, Shack Hartmann and curvature WFS were simulated to compare their respective performances in an AO system. These simulation codes allow computation of the WFS response for any phase function of E (i.e., not only of M as above). As a test of the accuracy of the WFS model, a simulated modal interaction matrix can be computed and compared to the modal interaction matrix measured as above. For the curvature WFS used in PUEO, a very good agreement was found. 17 All the wave-front sensing techniques entail detecting photons coming from a guide star during a very short integration time (WFS integration time, typically 1 to 50 ms). The WFS measurements are therefore affected by detection noise that results from the statistical fluctuations of the incoming photon flux (photon noise) as well as from the imperfections of the photon detector (dark current, readout noise, etc.). The effect of all these undesirable contributions to the WFS measurements may be summarized as an additive random vector n w (t): the measurement error. The actual WFS measurement w(t) can then be expressed as wt Dt W x, t n w t. (15) D. Reconstruction of the Residual Phase In an AO system it is the task of a real-time computer (RTC) to update the shape of the deformable mirror so that the spatial variance of the residual phase remains minimal. However, in order to compute the actual commands to be applied to the DM, the RTC needs first to derive an estimate of (x, t), the mirror component of the residual phase, based on the WFS measurements. The modal approach 18 has proved to be very efficient to solve this problem and is therefore used in many current AO systems, including PUEO. This technique allows identification of the directions (modes) of M that give a zero measurement on the WFS. These invisible modes depend on the geometry of both the DM and the WFS, and extensive design studies are usually performed to reduce their numbers. They always include the piston mode,

4 3060 J. Opt. Soc. Am. A/Vol. 14, No. 11/November 1997 Véran et al. which has no effect on the image formation, and sometimes other modes. The invisible modes must be excluded from the correction loop so that the actual phase compensation is not performed on M but on M, a subspace of M that is supplementary to the subspace generated by the invisible modes. This is then equivalent to having a mirror M M, that is, with a number of degrees of freedom m reduced by the number of invisible modes. With this new definition of M the discussion in Subsections.B and.c is still valid, including the choice of a particular base of modes for M. The point is that now the modal interaction matrix D is well conditioned and therefore (x, t) may be efficiently estimated. This can be achieved by a number of different techniques. One of them, which appeals because of its simplicity and has been implemented in several AO systems including PUEO, is based on the computation of D, the generalized inverse of the modal interaction matrix, given by D D T D 1 D T, (16) where the superscript T denotes the transpose-matrix operation. This approach leads to a least-squares estimation of (t), the modal coordinates of the residual phase: ˆt D wt. (17) It is clear that ˆ(t) (t) only in the absence of detection noise and high-order component. In the general case, the estimation error may be assessed as where ˆt t nt rt, (18) n D n w t (19) is the propagation of the WFS measurement error onto the mirror modes and rt D W x, t (0) originates from the turbulent phase having a high-order component that gives a nonzero measurement on the WFS and is mistaken for a low-order component through the estimation process. This error is a consequence of the incomplete (finite) spatial sampling of the WFS and has been studied in Refs It has been shown to be the combination of two effects, namely, (spatial) aliasing and (spatial) cross coupling. As in Ref., we will refer to r(t) as the remaining error. E. Control of the Deformable Mirror Considering that the estimation of (x, t), the mirror component of the residual phase, is affected by estimation error, as discussed above, and that the commands to the DM are always affected by a time delay, it has been demonstrated 18 that to minimize the variance of the corrected phase, the best updates to the voltage applied to the DM are such that the modal commands m i are increased by g i ˆ i. The g i are called modal gains. This approach is known as optimized modal control and has been implemented in a number of AO systems, including PUEO. The temporal variations of the modal commands to the DM can then be expressed in the Fourier domain 18 : m if H cl g i, f ã i f H cl g i, f r if H n g i, f ñ i f. (1) H cl ( g i, f ) is a low-pass filter called closed-loop transfer function. We denote f cl ( g i ) its cutoff frequency at 3 db. H n ( g i, f ) is called noise transfer function and is very similar to H cl ( g i, f ). The analytical expression of H cl ( g i, f ) and H n ( g i, f ) can be found in Ref. 18. f cl ( g i ) represents the temporal bandwidth of the system, that is, its ability to respond to a quick change in the turbulent phase. Decreasing (resp. increasing) g i decreases (resp. increases) f cl ( g i ) but also decreases (resp. increases) the effect of the errors in the estimation of the residual phase. The g i are thus set to yield an optimal trade-off. From Eqs. (6) and (1) it can be deduced that i f H cor g i, f ã i f H cl g i, f r if H n g i, f ñ i f. () H cor ( g i, f ) is a high-pass filter called the correction transfer function and given by H cor g i, f 1 H cl g i, f. (3) 3. LONG-EXPOSURE POINT-SPREAD FUNCTION A. Theoretical Expression To derive the corrected long-exposure PSF, we apply a procedure similar to the method presented in Ref. 1 for the calculation of the uncorrected long-exposure PSF. In the near-field approximation, the instantaneous optical transfer function (OTF) B(, t) related to the monochromatic image at wavelength is defined as B /, t 1 S PPxPx expi x, t expi x, tdx. (4) The normalization by S, the area of the telescope aperture, ensures that the PSF has a unit energy. The longexposure OTF is the average of the instantaneous OTF over the integration time interval. Assuming first that as the uncorrected phase, the corrected phase at any position on the pupil has a Gaussian statistic and second that the integration time is long enough that the statistical average can be substituted to the temporal average, the expression of the long-exposure OTF is B / B /, t where 1 S PPxPx exp 1 D x, dx, (5)

5 Véran et al. Vol. 14, No. 11/November 1997/J. Opt. Soc. Am. A 3061 D x, x, t x, t (6) is the structure function of the corrected phase. The long-exposure PSF is then the Fourier transform of the long-exposure OTF: PSFu F B /. (7) We now address the problem of the actual computation of B( /). For a turbulent wave front with Kolmogorov statistics, the phase function is spatially stationary, 1 so that the structure function is only the function of the separation and not of the location x, which greatly simplifies its calculation. This is not the case when the phase is partially corrected, so that an exact calculation of D (x, ) requires averaging four dimensional functions, which is computationally very demanding. However, it has been suggested 3,4 that D (x, ) could be replaced by D (), its mean over the variable x given by: D so that Eq. (5) simplifies as PPxPx D x, dx PPxPx dx (8) B / exp 1 D PPxPx dx. (9) This approximation is equivalent to assuming that the dispersion along variable x is small enough that the exponential of the mean may be approximated by the mean of the exponential. Since the exponential is a convex function, this leads to an underestimation of the OTF. 3 However, extensive tests through simulations led us to conclude that for a system such as PUEO, only the lowest values of the OTF, corresponding to the highest spatial frequencies, are affected by the approximation. We have found that this does not result in a significant loss in the quality of the reconstructed PSF. With this approximation the computation of B( /) requires the averaging of only two-dimensional functions. Using Eq. (8), D can be developed as with obvious notations. In Eq. (30) the third term (cross term) is not rigorously zero because the and may be correlated through the remaining error. Here we assume that this cross term is negligible. Simulations show that this assumption is fully granted for systems such as PUEO. Then Eq. (9) can be conveniently expressed as the product of three contributions: B / B / B / B tel /. (31) B tel ( /) is the OTF of the telescope in the absence of turbulence: B tel / 1 S PPxPx dx. (3) B ( /) is the contribution of the high-order phase to the long-exposure OTF: where D () is given by D P B / exp 1 D, (33) PxPx x, t x, t dx PxPx dx P. (34) B ( /) is the contribution of the mirror component of the residual phase to the long-exposure OTF: B / exp 1 D. (35) With the modal expansion (1), D can be expressed as n D i, i j U ij, (36) j where the U ij () are functions of the mirror modes and need to be computed only once: PxPx x, t x, t x, t x, tdx P D D D PPxPx dx (30) U ij PPxPx M i x M i x M j x M j x dx PPxPx dx. (37)

6 306 J. Opt. Soc. Am. A/Vol. 14, No. 11/November 1997 Véran et al. For Zernike polynomials or Karhunen Loève functions with no central obscuration, an analytical expression of U ij () has been derived. 4 For any other set of modes, these functions can be computed numerically. One last important point is that since after correction by the DM the wave front is split, as discussed in Subsection.C, the wave front that forms the image and the wave front that goes to the WFS do not follow the same optical path. Because of this, uncommon path aberrations may result. These aberrations are not seen by the WFS but do affect the final image and therefore must be taken into account in the PSF. The uncommon path aberrations are of static nature and may be calibrated in the form of a static PSF denoted PSF s (u). PSF s (u) is obtained by imaging a point source in the absence of turbulence, with the deformable mirror flat, in the sense that all the WFS measurements are zero. The OTF of the telescope B tel ( /) should then be replaced by the static OTF B s ( /) F 1 (PSF s (u)) in Eq. (31). The longexposure OTF is thus finally given by B / B / B / B s /. (38) B. Estimation from the Control Loop Data The computation of the long-exposure OTF requires B and B (Eq. (38)). In this section we present a method to estimate these contributions from the control loop data, that is, the data that were used by the RTC to drive the DM during the exposure. These are the WFS measurements and the commands to the DM. We first introduce some notation. If x(t) and y(t) are two random vectors and x i (t) and y i (t) their respective components, we denote as C xx the covariance matrix of x(t) and C xy the cross-covariance matrix of x(t) and y(t). Also, we denote as S xi x i (f ) the temporal power spectrum (spectral density) of x i (t) and S xi y i (f ) the temporal cross-power spectrum of x i (t) and y i (t). The general definitions of these quantities are given in Appendix A. To simplify the notation, we also use xi for (C xx ) ii. 1. Estimation of B B, the contribution of the mirror modes to the longexposure OTF, can be computed from C, the covariance matrix of the modal coordinates of, the mirror component of the residual phase (Eqs. (35) and (36)). Our estimation method is based on ˆ(t), the RTC estimates of (t) obtained from the WFS measurements. The computation of ˆ(t) has been discussed in Subsection.D. Recalling Eq. (18), we have ˆt t nt rt (39) where n(t) is the WFS measurement error propagated onto the mirror modes and r (t) is the remaining error that is due to the high-order phase affecting the measurement of. Since the measurement error has no temporal correlation and may not propagate through the loop without a delay (loop delay), there is no correlation between n(t) and (t) and between n(t) and r(t). Expressing Eq. (39) in terms of covariance matrices gives C C ˆˆ C nn C rr C r. (40) C ˆ ˆ can be computed from C ww, the covariance matrix of the WFS measurements. Equation (17) gives C ˆ ˆ D C ww D T. (41) Similarly, C nn can be computed from C nw n w, the covariance matrix of the measurement error: C nn D C nw n w D T. (4) The computation of C nw n w is WFS specific and will be presented in Subsection 4.A. We now evaluate C rr C r. Using Eqs. (), (3) and Eqs. (A5) (A7) from Appendix A, each element of this matrix may be expressed as C rr ij C r ij C rr ij H* cor g i, f S ri r j f d f H* cor g i, f S ai r j f d f. (43) For a given AO system, the power spectra S ri r j ( f ) and S ai r j ( f ) depend only on the turbulent phase, but unfortunately, their exact computation does not seem feasible. However, we can infer that these spectra are decreasing at high frequencies, with a cutoff frequency around f M, the DM cut-off frequency defined in Subsection.B. This is because the remaining error r (t) originates from the high-order phase, and we know from the Kolmogorov model that most of the variance of the high-order phase is carried by the uncorrected modes of lowest spatial frequency, that is, those with a cutoff frequency close to f M. We also notice in Eq. (43) that before integration, both S ri r j ( f ) and S ai r j ( f ) are multiplied by H cor ( g i, f ), the correction transfer function, which is a high-pass filter. Let us now assume that the AO system is fast enough so that its temporal bandwidth f cl ( g i ) is higher than f M. Numerical values will be given in Section 4, and we will find that this assumption is generally valid for all the modes, except when the guide star is very faint. Then, we can consider that the correction transfer function H cor ( g i, f ) filters out most of the power in S ri r j ( f ) and S ai r j ( f ) so that the magnitude of the integral terms in Eq. (43) is negligible compared with (C rr ) ij. We then obtain the simplified expression: C C ˆˆ C nn C rr. (44) Relation (44) thus states that the estimate of C, which is required for computing B, can be obtained from C ˆ ˆ, the covariance matrix of the RTC estimates of (t) during the exposure, by subtraction of C nn, the covariance matrix of the measurement error propagated onto the mirror mode, and addition of C rr, the covariance matrix of the remaining error. The computation of C rr is discussed in Subsection 3.B... Estimation of B and C rr B, the contribution of the high-order phase to the long-exposure PSF, can be computed from D, the structure function of [Eq. (33)]. Likewise, for a given AO system, C rr, the covariance matrix of the remaining error, depends only on the statistics of [Eq. (0)]. Unfortunately, because of the limited spatial sampling of the

7 Véran et al. Vol. 14, No. 11/November 1997/J. Opt. Soc. Am. A 3063 WFS, no information on can be derived from the WFS measurements. However, since is not affected at all by the AO correction, we can assume that its statistics obey the Kolmogorov model. Then it follows that D and C rr depend only on the value of D/r 0 for the turbulence. A method for computing D/r 0 will be presented in Subsection 3.B.3. We now assume that D/r 0 is known and address the problem of the actual computation of D () and C rr. Although analytical methods exist,,4 they are restricted to the case of a mirror space generated by Zernike polynomials or Karhunen Loève functions with no central obscuration. In the general case, a Monte Carlo method must be used. Random phase screens with Kolmogorov statistics can be simulated, for instance, with the method described in Ref. 5. By extracting the high-order component of each of them, a large number of realizations of (x, t) can be generated. D () can then be estimated by replacing the statistical average in Eq. (34) with an average over all the realizations. The same process may be used to estimate C rr : from the realizations of (x, t), a WFS simulation code can be used to compute W (x, t), and, using Eq. (0) a large number of realizations of r(t) can be obtained and accumulated to give an estimate of C rr. D () and C rr need to be computed only once at D/r 0 1, for instance. Then these quantities can be scaled to any D/r 0 : D x, D x, D/r0 1D/r 0 5/3, (45) C rr C rr D/r0 1D/r 0 5/3. (46) 3. Estimation of D/r 0 In this section we address the problem of the estimation of the Fried parameter of the turbulence r 0, which is needed to scale the contribution of the high-order modes [Eqs. (45) and (46)]. For each mirror mode, an estimate of D/r 0 can be derived using Eq. 13: D/r 0 i ai /K ii 3/5, (47) where ai is the statistical variance of the mode coefficient in the turbulent phase. The different estimates of (D/r 0 ) i can be combined to give a statistically more accurate estimation of D/r 0. However, the tip and tilt modes should be excluded since they are strongly affected by the finite outer scale and different nonatmospheric effects, as discussed in Section.A. For the remaining modes, we found that the most robust method was to take for D/r 0 the median of the (D/r 0 ) i computed from Eq. (47). We now explain how to compute ai. The coefficients a i (t) are related to the modal commands applied to the DM c i (t) by Eq. (1). As in Subsection 3.B.1, we state that the measurement error has no temporal correlation, so that there is no correlation between n(t) and (t) and between n(t) and r(t). If we also consider that, since the remaining error originates from the high-order modes of the turbulent phase the magnitude of the covariance a i r i is much smaller than the variance ai, we find, using Eqs. (A5) (A7) from Appendix A, mi H cl g i, f S ai a i f d f H cl g i, f S ri r i f d f ni H n g i, f d f. (48) Equation (48) states that the variance of the modal command to the deformable mirror is the variance of the mirror modes in the turbulence, which are filtered by the closed-loop transfer function, plus the variance of the remaining error, also filtered by the closed-loop transfer function, plus the variance of the measurement error, filtered by the noise transfer function. Now if we assume as in Subsection 3.B.1 that the system bandwidth is high so that f cl (g i ) f M, then the filtering action of the closed-loop transfer function on the a i (t) and on the r i (t) is negligible. Again, this assumption will be numerically verified in Section 4. Then, from Eq. (48), we obtain an expression of ai : ai mi ri ni H n g i, f d f. (49) In Eq. (49), mi can be computed directly from the commands applied to the DM during the exposure. H n ( g i, f ) is known analytically (Ref. 18), and the computation of ni (C nn ) ii has already been discussed in Subsection 3.B.1. The problem is that ri, which usually has a significant contribution on the mirror modes of highest spatial frequency, is a function of D/r 0 [Eq. (46)] which is the quantity that we are trying to estimate. This problem can be solved by the following procedure, which allows an iterative bracketing of the correct D/r 0 : Set D/r 0 0; Repeat until convergence: Compute ri with Eq. (46); Compute ai with Eq. (49); Compute (D/r 0 ) i with Eqs. (47); Take as new estimate for D/r 0 the median of the (D/r 0 ) i for all the nonexcluded modes. In practice, this procedure has been found always to converge. 4. APPLICATION TO PUEO PUEO is a curvature-based AO system operated at the Canada France Hawaii telescope, situated on the top of Mauna Kea, Hawaii. The primary mirror of the telescope has a diameter of 3.6 m with an obscuration ratio of The deformable mirror is a bimorph with 19 actuators, and the WFS has 19 subapertures. 6 A. Noise on the Curvature Wave-Front Sensor We present here a method to estimate C nw n w, the covariance matrix of the measurement error, in the particular case of the curvature WFS. The measurement of the curvature WFS is the normalized difference (contrast) between the illumination in an intrafocal plane and the il-

8 3064 J. Opt. Soc. Am. A/Vol. 14, No. 11/November 1997 Véran et al. lumination in an extrafocal plane. In the absence of any measurement error, the WFS measurement for a given subaperture would be Wt N 1t N t N 1 t N t Dt St, (50) where N 1 (t) and N (t) are the number of photons detected in the sub-aperture, respectively, in the intrafocal plane and in the extrafocal plane. W(t) is directly related to the average local curvature of the measured wave front at the subaperture. W(t) is therefore a random variable with mean value 0. D(t) also has a mean value of 0 whereas S(t) has a mean value of N N 1 (t) N (t), the average number of photons in the subaperture during the WFS integration time. Since the measured quantity is the corrected wave front, we can expect the fluctuations of S(t) to be of small amplitude. If N is high enough that S N, then a first-order development can be used to express W : N. (51) In the presence of measurement errors, the WFS measurement for the given sub-aperture becomes: W D wt n 1t n t n 1 t n t dt st. (5) In a curvature sensor, photon detection is carried out by avalanche photodiodes (one per subaperture). These devices are characterized by a very low dark current and a negligible readout noise. Therefore, except when N is very small, the main origin of the measurement error is the photon noise, which has Poisson statistics. The measured intrafocal and extrafocal intensities n 1 (t) and n (t) can then be seen as realizations of two independent Poisson processes with respective means N 1 (t) and N (t). It follows that N st, (53) S s N, (54) D d N. (55) From Eq. (51) we can deduce the variance of the noiseless measurement: W d N 1 N. (56) The variance of the measurement noise is then nw w W. (57) s(t) and d(t) are available from the WFS so that nw may be estimated for each subaperture as follows: From s(t) and d(t) compute s, d, w and N s(t); Compute S [Eq. (54)] and verify that S N; Compute nw (Eqs. (56) and (57)). C nw n w, the covariance matrix of the measurement error is then the diagonal matrix whose diagonal elements are nw for each sub-aperture. As a remark, we point out that because of the photon noise s S. However, if the mean flux on the subaperture is so high that we still have s N, then w can be developed as w N. (58) Combining Eq. (58) with Eqs. (56) and (57), we find that the variance of the measurement error becomes nw d 1 N, (59) which is the expression commonly used to characterize the photon noise on any WFS. 15 However, the method we propose is much more accurate than Eq. (59) at low flux levels. B. System Calibrations In this section we discuss the various calibrations required for applying to PUEO the PSF retrieval method presented above. The 19 influence functions of the deformable mirror, corresponding to each of the 19 actuators, have been precisely measured by an interferometric technique. From these influence functions, we can construct an orthonormal set of 15 modes M 1 (x),..., M 15 (x) that approximate the 15 first Zernike polynomials with a good precision. Zernike 16, however, cannot be achieved by the mirror, so that the four remaining modes M 16 (x),..., M 19 (x) are just taken as an orthogonal complement to generate all M. We found that M 16 (x),..., M 19 (x) may be expanded as a sum of Zernike polynomial of radial degree 5, 6 and 7 with good accuracy. We thus conclude that, given a wind speed of 0 m/s (typical conditions at the top of Mauna Kea), the maximal temporal cutoff frequency of the mirror modes (as defined in Section.B) for PUEO is f M 13 Hz. (60) 3.6 From the mirror modes, the function U ij () can be computed by using Eq. (37). With 19 mirror modes, and excluding the piston mode, there is a total of 171 different functions to be computed. D () and C rr can be computed as explained in Subsection 3.B.1. We used a total of 10,000 random Kolmogorov screens in order to reduce statistical errors. The spatial variance of the high-order component of the turbulent phase was found to be 0.01D/r 0 5/3, (61) which is about % of the spatial variance of the uncorrected turbulent phase. For a curvature system, the WFS measurements depend on the actual defocalization distance used for the intrafocus and extrafocus imaging.

9 Véran et al. Vol. 14, No. 11/November 1997/J. Opt. Soc. Am. A 3065 This distance is usually expressed as an optical gain G opt, which is approximately inversely proportional to the defocalization distance. The diagonal elements of C rr are shown in Fig. 1 for a typical optical gain. For this optical gain, the remaining error contributes to the residual phase about as much as the high-order phase, that is, with a spatial variance of approximately. The control system of PUEO has been designed for a nominal WFS sampling frequency of 1 khz. The total loop delay was experimentally measured to be near 0.7 ms. Using the analytical expressions given in Ref. 18, the closed-loop transfer function H cl ( g i, f ) can be computed for different modal gains, as shown in Fig.. Except for the very low modal gains, we have indeed f cl ( g i ) f M, which justifies the assumptions made in Sections 3.B and 3.B.3. Very low modal gains do happen when the flux onto the WFS is very low. This limiting case will be discussed in Section 4.D. The static PSF, PSF s (u), must be measured for every run and for every imaging band. For PUEO, with the near-infrared camera MONICA from the University of Montreal, the static PSF was found to have a Strehl ratio ranging from 96% in the K band to 80% in the J band. From the static PSF, the static OTF B s ( /) can be computed by Fourier transform as discussed in Section 3.A. The interaction matrices are normally computed during the run setup for the range of optical gains available. For PUEO there is only one invisible mode (piston mode), which is filtered to compute the control matrices D. Finally, we point out as a practical note that Eqs. (34) and (37), which are used to compute D () and U ij (), respectively, can be expressed in terms of spatial autocorrelation functions and thus be efficiently evaluated on a discrete grid with the fast Fourier transform. Fig. 1. Solid curve: variance of the mirror modes in a turbulent Kolmogorov phase. Dotted curve: variance of the remaining error. Both graphs are for D/r 0 1. C. Summary of Operations In this section we summarize the different steps required to compute the estimation of the long-exposure PSF associated with a given acquisition. The following data must be available for the entire integration time: The WFS measurements w(t), The sum and the difference of the intrafocal and extrafocal flux on the WFS, s(t) and d(t), The modal commands to the deformable mirror m(t). With a WFS sampling frequency of 1 khz, this represents a huge quantity of data for a typical long-exposure acquisition. However, only the covariance matrices of these quantities are of interest. The real-time control system of PUEO has thus been modified so that these covariance matrices are computed in real time and saved at the end of the exposure; then it is no longer necessary to save each w(t), s(t), d(t), and m(t). The values of the optical gain G opt and the modal gains g for the exposure also need to be saved. The file generated is then very compact and still contains all the information required for computing the long-exposure PSF. The following steps must then be executed sequentially: Fig.. Closed-loop transfer function H cl ( g i, f ) for different modal gains g i and corresponding 3-dB cutoff frequencies f cl ( g i ). From g compute H n ( g i, f ), the noise transfer function for each mode (Ref. 18); From G opt compute the control matrix D [Eq. (16)]; From D, C ww, C ss, and C dd, compute C nw n w, the covariance matrix of the measurement error as explained in Section 4.A, and derive C nn using Eq. (4); From C mm, C nn, and H n ( g i, f ) compute the value of D/r 0 for the turbulence as explained in Subsection 3.B.3; From D/r 0 scale D () [Eq. (45)] and compute B ( /) [Eq. (33)]; From D/r 0 scale C rr the covariance matrix of the remaining error [Eq. (46)]; From D and C ww compute C ˆ ˆ [Eq. (41)];

10 3066 J. Opt. Soc. Am. A/Vol. 14, No. 11/November 1997 Véran et al. From C ˆ ˆ, C nn, and C rr compute C [Eq. (44)]; From C and U ij () compute D () [Eq. (36)] and B ( /) [Eq. (35)]; From B ( /), B (/), and B s ( /) compute B(/), the long-exposure OTF [Eq. (38)]; From B( /) compute PSF(u), the long-exposure PSF [Eq. (7)]. D. Results and Performance The method presented in this paper has been successfully implemented and tested during the first commissioning runs of PUEO on the telescope in Spring Three typical examples are presented in Figs The numerical values describing these acquisitions can be found in Table 1. The value of r 0 in the three examples is typical for the top of Mauna Kea. The object acquired was a point source (star), so that its image is the real PSF and thus can be directly compared with the estimated PSF. A numerical comparison in terms of the Strehl ratio (SR) and the FWHM is given in Table. The imagery band is H(.1 m) for Example 1 and H( 1.65 m) for Examples and 3. In example 1 the guide star was bright, so the measurement error was low and the correction was very good. The PSF is well estimated for all but the highest spatial frequencies. In example the guide star was fainter, with an average of ten photons per subaperture and per millisecond on the WFS. The PSF is still well estimated, although the error becomes more important at high spatial frequencies. With the data we acquired during the Spring 1996 runs, we have found that for guide stars of magnitude 13 or brighter, the method presented in this paper allows estimates of an accurate long-exposure PSF, which are potentially useful for image restoration: FWHM within 0.01 arc sec, Strehl ratio within 4%, accurate morphology, and precision on the OTF ranging from 10 at low spatial frequencies to 10 1 at high spatial frequencies. However, with guide stars fainter than magnitude 13 (less than eight photons per WFS subaperture and per millisecond), the accuracy of the estimated PSF degrades, as illustrated by example 3 (Fig. 5). In this example, the guide star was of magnitude 15.4, that is, only an average of one photon per subaperture and per millisecond. The estimated PSF shows significant deviations from the measured (real) PSF, with large errors even at low spatial frequencies. The reason the method fails on faint stars is twofold: first, the estimation of the contribution of the measurement error (Subsection 4.A) becomes less accurate, leading to a poor estimate of C ; second, decreasing the number of photons increases the measurement error, which results in the use of smaller control gains in the closed loop and leads to a smaller correction bandwidth f cl ( g i ). Then the assumption f cl ( g i ) f M is not valid and the approximations made in Subsections 3.B and 3.B.3 are no longer accurate. We believe that, most of Fig. 3. Example 1: bright guide star (magnitude 10.4). Comparison of the estimated PSF and the real PSF in the case of a bright guide star. The left column (plots a, b, and c) shows the image, displayed with a square-root vertical scale. The right column (plots d, e, and f ) shows the modulus of the OTF displayed with a logarithmic scale. In each column, the X cut (plots a and d), Y cut (plots b and e), and circular average (plots c and f ) are presented. The real PSF is shown by the solid curve and the plus signs. The estimated PSF is shown by the diamond signs. The dotted curve is the absolute difference of the two curves, plotted with the same vertical scale (square root in the left column, logarithmic in the right column). In the left column, for a better comparison, both the real and the estimated PSF have been scaled so that the maximum of the real PSF is 1.

11 Véran et al. Vol. 14, No. 11/November 1997/J. Opt. Soc. Am. A 3067 Fig. 4. Example : fainter guide star (magnitude 1.7). Same legend as for example 1. Fig. 5. Example 3: very faint guide star (magnitude 15.4). Same legend as for example 1. the time, the dominant error lies in the estimation of D/r 0 : because the bandwidth is small, the mirror follows only the low-frequency fluctuations of the atmosphere, and thus the variance of the modal commands to the DM is less than the variance of the corresponding modes in the turbulence. Then using the method described in Subsection 3.B.3 leads to an underestimation of D/r 0. This results in an estimated PSF with too high a Strehl ratio. This trend has been confirmed by observations and is illustrated by example 3. Correcting for this effect by using our knowledge of the AO system transfer functions would probably be feasible, although it would also require some knowledge of the temporal characteristics of the turbulence. Another possibility would be to estimate D/r 0 from an open-loop acquisition just before or just after the closed-loop acquisition, but then the main advantage of the method described in this paper (simultaneously with the actual observation) would be lost.

12 3068 J. Opt. Soc. Am. A/Vol. 14, No. 11/November 1997 Véran et al. Table 1. Numerical Atmospheric and AO Loop Values (Given at 0.5 m) Atmospheric and AO Loop Quantities Example 1 Example Example 3 Guide star magnitude r 0 (cm) Mirror component (rad ): Estimated uncorrected phase ai RTC estimated residual phase i Estimated WFS measurement error ni Estimated remaining error ri Estimated residual phase i Estimated high-order component (rad ) Table. Numerical Comparison of the PSF s Example 1 Example Example 3 PSF Strehl Ratio (%) FWHM (arc sec) Strehl Ratio (%) FWHM (arc sec) Strehl Ratio (%) FWHM (arc sec) Real Estimated CONCLUSION In this paper we have demonstrated the possibility of using control data, namely, the WFS measurements and the commands to the DM, to estimate the long-exposure PSF of an AO system. Only the second-order statistics of these data are actually needed. This results in a very compact file that can be saved by the control computer after each acquisition and from which the PSF can be estimated. We have shown that the long-exposure OTF may be expressed as the product of three different contributions [Eq. (38)]: a static contribution, due to uncommon path aberrations, that can be calibrated with an artificial source in the absence of turbulence; a contribution from the mirror component of the residual phase that is partially corrected and can be estimated from the WFS measurements; a contribution from the high-order component of the turbulent phase that is uncorrected and can be estimated with the Kolmogorov model. The value of D/r 0 is estimated from the commands to the DM. Also, there exists a coupling between the high-order component and the mirror component through the remaining error, a cross-talk term that can be taken into account with an accurate modeling of the WFS. The different estimations are performed in the framework of the optimized modal control of the DM presented in Ref. 18. The method has been successfully tested on PUEO, the Canada France Hawaii telescope AO system and is now available for regular observing runs. However, in order to ensure a high quality estimated PSF, the temporal bandwidth of the AO system must be higher than the cutoff frequency of the highest mirror mode in the turbulent phase. This requirement is not satisfied when the guide source is too faint. For PUEO, we have found the limiting magnitude to be close to 13. Thus, for guide sources of magnitude 13 or brighter, our method allows an accurate estimate of the PSF related to any long-exposure image without requiring any observing time for this task. Another important result of this study is the ability to estimate D/r 0 for every image, which is useful in assessing the AO system performance. It should be noted, however, that the estimated PSF is valid only within the isoplanetic angle from the reference source. At greater angular distances the performance of the system is further degraded by the anisoplanetic errors, which we do not consider here. These errors are due to the fact that the turbulence in the atmosphere is extended in altitude, and thus taking them into account would require estimating the vertical distribution of the turbulence, perhaps by using extra guide sources and WFS measurements in different angular directions. In our method, the contribution of the anisoplanetism errors would simply appear as an additional term in Eq. (38). It should be quite possible to adapt this method to other AO systems, provided that the requirement mentioned above can be satisfied. For a Shack Hartmann WFS however, a specific method to estimate the variance of the measurement error has to be designed. APPENDIX A The definitions and properties introduced here are taken from Ref. 7. The autocorrelation of a stationary random process x(t) is defined as R xx xt x*t. (A1) The Fourier transform of the autocorrelation function is the power spectrum: S xx f R xx expjfd. (A)

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