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Isoplanatism in a multiconjugate adaptive optics system Andrei Tokovinin and Miska Le Louarn European Southern Observatory, Karl-Schwarzschild-Strasse, 2 D-85748 Garching bei Mu¨nchen, Germany Observatoire de Lyon, 9 avenue Charles Andre´, F-69561 Saint Genis Laval, France

Marc Sarazin European Southern Observatory, Karl-Schwarzschild-Strasse, 2 D-85748 Garching bei Mu¨nchen, Germany Received November 18, 1999; revised manuscript received June 14, 2000; accepted June 26, 2000 Turbulence correction in a large field of view by use of an adaptive optics imaging system with several deformable mirrors (DM’s) conjugated to various heights is considered. The residual phase variance is computed for an optimized linear algorithm in which a correction of each turbulent layer is achieved by applying a combination of suitably smoothed and scaled input phase screens to all DM’s. Finite turbulence outer scale and finite spatial resolution of the DM’s are taken into account. A general expression for the isoplanatic angle ␪ M of a system with M mirrors is derived in the limiting case of infinitely large apertures and Kolmogorov turbulence. Like Fried’s isoplanatic angle ␪ 0 , ␪ M is a function only of the turbulence vertical profile, is scalable with wavelength, and is independent of the telescope diameter. Use of angle ␪ M permits the gain in the field of view due to the increased number of DM’s to be quantified and their optimal conjugate heights to be found. Calculations with real turbulence profiles show that with three DM’s a gain of 7–10⫻ is possible, giving the typical and best isoplanatic field-of-view radii of 16 and 30 arcseconds, respectively, at ␭ ⫽ 0.5 ␮ m. It is shown that in the actual systems the isoplanatic field will be somewhat larger than ␪ M owing to the combined effects of finite aperture diameter, finite outer scale, and optimized wave-front spatial filtering. However, this additional gain is not dramatic; it is less than 1.5⫻ for large-aperture telescopes. © 2000 Optical Society of America [S0740-3232(00)02010-X] OCIS codes: 010.0010, 010.1080, 010.1330.

1. INTRODUCTION Adaptive optics imaging (AOI) systems find everincreasing use in astronomy, delivering diffractionlimited images at large ground-based telescopes. However, the small field of view (FOV) characterized by the atmospheric isoplanatic angle remains a serious limitation, mostly because it is difficult to find a suitably bright reference source for wave-front measurements within the isoplanatic patch. The use of artificial laser guide stars solves this problem only partially, since a natural guide star is still needed for tilt correction. Sky coverage is improved with laser guide stars and may attain reasonable fractions of the sky in the near infrared but still remains limited in the visible band. It was suggested some time ago that the atmospheric turbulence could be compensated with several deformable mirrors (DM’s) conjugated to different heights. Such multiconjugate adaptive optics (MCAO) systems (see the historical review of the concept in Refs. 1 and 2) offer a possibility of widening FOV and, consequently, of improving the sky coverage of AOI systems. The purpose of this paper is to quantify the FOV gain brought about by the increased number of DM’s. Despite the existence of theoretical calculations of MCAO systems performance,1 the issue of the FOV gain is still not clear. Owing to the inherent complexity of the problem, the off-axis Strehl ratio depends on many system parameters. A case study of a particular MCAO per0740-3232/2000/101819-09$15.00

formance can be found in Ref. 3. A more general attempt to calculate the MCAO FOV gain by segmenting the atmospheric refractive-index structure’s constant profile C n2 (h) into several slabs, placing DM at the center of each slab, and summing up the remaining anisoplanatic effects was made by Yan et al.4 In the analysis of the MCAO problem and the associated three-dimensional wave front reconstruction by tomographic techniques, however, it is sometimes assumed that turbulence is concentrated in a few thin layers that are exactly conjugated to the DM planes.5 This assumption leads to unlimited FOV; hence the problem of calculating the realistic FOV gain still remains unsolved. The FOV of classical (single-DM) AOI systems is usually characterized by the isoplanatic angle ␪ 0 defined by Fried.6 It is clear from this work that ␪ 0 is only an approximate measure of the FOV, which is correct in the limiting case of infinite aperture size. For finite apertures the differential piston terms must be subtracted from the pupil-averaged phase variance, with the resulting increase of the FOV for a not-too-large D/r 0 case.7 An AOI system with partial wave-front correction also has a larger FOV.7–9 Nevertheless, the parameter ␪ 0 is still widely used because it is system independent and permits easy comparisons of various sites. Another advantage of ␪ 0 is its known scaling with wavelength and zenith distance. Below, we propose a generalized parameter ␪ M for an M-mirror MCAO system that shares the de© 2000 Optical Society of America

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sirable properties of ␪ 0 : It depends only on the C n2 (h) profile and DM conjugate heights and permits wavelength and zenith distance scaling. The problem is defined in Section 2. Then the residual phase variance is calculated in Section 3, and the optimal control algorithm is found in Section 4. The simplest case of a two-DM MCAO is discussed in Section 5 as an illustration. In Section 6 the definition of isoplanatic angle ␪ M is given, followed by its calculations for real turbulence profiles in Section 7. In Section 8 we investigate the gain in the FOV size resulting from the optimized spatial filtering and finite pupil size. Our conclusions are summarized in Section 9. The details of the phase variance calculation are given in Appendix A.

2. PROBLEM DEFINITION Our purpose is to study the dependence of the quality of wave-front compensation on the viewing direction. We consider here an idealized situation in which the phase perturbations produced by each atmospheric layer are perfectly known at any instant and can be corrected by means of M DM’s with infinite temporal resolution. (Of course, in any real MCAO system the amount of information on the instantaneous phase disturbances and the ability of DM’s to correct them are limited.) Thus, the number of DM’s is the only limiting factor in turbulence correction. The resulting FOV will depend only on the C n2 (h) turbulence profile and on the altitudes H 1 , H 2 ,... H M of the DM conjugate planes. It will be a fundamental MCAO performance characteristic in the sense of Roggemann and Welsh,10 that is, when performance degradation due to only one factor is considered. As usual, we use the approximations of geometric propagation and near-field turbulence. The turbulent atmosphere is considered to be a collection of mutually independent thin phase screens located at different heights. Each screen introduces a phase ␾ (x), where x is a twodimensional coordinate in the wave-front plane. The contributions of individual layers are simply added to yield a wave front for an on-axis object. For an object viewed at an angle ␪ (a two-dimensional vector, ␪ ⫽ 兩 ␪兩 ), the individual contributions are summed with space shifts of ␪h, where h is the altitude of the layers. In an MCAO system each of the M DM’s introduces a phase ␺ m (x) to correct for atmospheric disturbances. These corrections are subtracted from the wave front (with appropriate space shifts for off-axis viewing). It is important to note that every turbulent layer is corrected by all DM’s. Thus the residual wave-front phase after compensation can be written as M

⑀ 共 x, ␪兲 ⫽ ␾ 共 x ⫺ ␪ h 兲 ⫺

兺

␺ m 共 x ⫺ ␪H m 兲 .

(1)

to be known. The problem is thus to find a set of corrections ␺ m (x) that would give the best-compensated image quality. A further analysis will be made in the frequency domain. We use the spatial frequency f [f ⫽ (period) ⫺1 , not a wave number] and denote by f its modulus. Fourier transforms are denoted by tildes. Dependence on ␪ is dropped from the following equations. Thus the Fourier transform of the residual phase is ˜ 共 f兲 exp关 ⫺2 ␲ i 共 f ␪h 兲兴 ˜⑀ 共 f 兲 ⫽ ␾ M

⫺

兺

(2)

A model for a control algorithm must now be specified. The phase corrections ␺ m are sought as a spatially filtered and properly scaled input perturbation ␾, which means that the control is linear. We consider only spatial filters g m ( f ) with radial symmetry that depend on the modulus of f. This restricts us to MCAO systems that have symmetrical imaging properties about the center of the FOV. The spatial resolution of any real DM is limited by the finite actuator spacing. To take this limitation into account, we introduce the low-pass spatial filter r( f ) (the DM response function) into the correction. The response function may be used to model AOI systems that correct only a limited number of wave-front modes. The response is presumed to be rotationally symmetric. It would be straightforward to generalize to DM’s with different response functions, but here a common response is assumed, so the correction looks like ˜␺ 共 f 兲 ⫽ g 共 f 兲 r 共 f 兲 ␾ ˜ 共 f 兲. m m

(3)

The optimization of the control algorithm consists in finding the filters g m ( f ) that would give the minimum residual phase variance of the corrected wave front (this is called a minimum-variance approach).10 As we shall see, the algorithm must be optimized for a particular viewing direction ␪⬘. A simplified version of the control algorithm consists in applying to each DM just a linear combination of unfiltered input phase screens. Thus, for a given screen, the control signal applied to the Mth mirror is defined by a constant g m ( f ) ⫽ c m . Taken together with other simplifications, this restricted proportional-control model leads us to the system-independent definition of isoplanatic angle ␪ M .

3. RESIDUAL PHASE VARIANCE The adopted model of the control algorithm leads to the following expression for the aperture-averaged residual phase produced by a single turbulent layer:

m⫽1

This equation defines our basic model for an MCAO system. It corresponds to only one turbulent layer. When the phase variance of the residual wave front is computed, the contributions of all independent layers are added quadratically. The perturbation ␾ (x) is assumed

˜␺ 共 f 兲 exp关 ⫺2 ␲ i 共 f␪H 兲兴 . m m

m⫽1

具 ⑀ p2 典 ⫽ 2 ␲

冕

⬁

0

¯ 共 f 兲 兩 2 p 共 f 兲 df. fW ␾ 共 f 兲 兩 G

(4)

A derivation of Eq. (4) is given in Appendix A. Here ⑀ p is the so-called piston-removed residual variance, i.e., its aperture average is zero, and W ␾ ( f) is the power spec-

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Vol. 17, No. 10 / October 2000 / J. Opt. Soc. Am. A

trum of atmospheric phase disturbances. The high-pass piston filter p( f ) for a circular aperture of radius R is given as p共 f 兲 ⫽ 1 ⫺

冋

J 1 共 2 ␲ Rf 兲

␲ Rf

册

2

.

(5)

¯ ( f ) 兩 2 depends on ␪, conjugate altiThe spatial filter 兩 G tudes of DM H m , layer altitude h, DM response r( f ), and the spatial filters g m ( f ): ¯ 共 f 兲 兩 2 ⫽ 1 ⫺ 2bT g ⫹ gT Ag, 兩G

(6)

where the vector b and the matrix A are b m ⫽ r 共 f 兲 J 0 共 2 ␲ fx m 兲 , a mm ⬘ ⫽ r 2 共 f 兲 J 0 关 2 ␲ f 共 x m ⫺ x m ⬘ 兲兴 ,

For the atmospheric phase power spectrum W ␾ ( f ) we adopt the von Ka´rma´n model with finite outer scale L 0 . It depends only on f ⫽ 兩 f兩 and for a single layer (altitude h, thickness dh) can be expressed as11 W ␾ 共 f 兲 ⫽ 0.38␭ ⫺2 共 f 2 ⫹ L 0⫺2 兲 ⫺11/6C n2 共 h 兲 dh.

(8)

The combined effect of all atmospheric layers is finally found as an integral over altitude:

冕

具 ⑀ p2 典 ⫽

h max

0

C n2 共 h 兲 F ⬘ 共 h 兲 dh,

(9)

where F ⬘ 共 h 兲 ⫽ 2.40␭ ⫺2

冕

⬁

0

4. MINIMUM VARIANCE CONTROL OF MULTICONJUGATE ADAPTIVE OPTICS All quantities in integral equation (4) are nonnegative. This means that integral minimization is reduced to the minimization of the integrand at each f by a suitable choice of spatial filters g m (f ). This, in turn, requires ¯ ( f ) 兩 2 because other factors do not deminimization of 兩 G pend on the filters. It is noteworthy that G( f) contains only geometric factors and DM response but is independent of the turbulence model used. We seek a column vector g with elements g m ( f ) that minimizes Eq. (6). At minimum the partial derivatives ¯ ( f ) 兩 2 over g are equal to zero, which yields the set of 兩 G m of equations that is necessary for finding the optimum filters g m . The result is simply

xm ⫽ ␪共 Hm ⫺ h 兲. (7)

¯ 共 f 兲 兩 2 df. f 共 f 2 ⫹ L 0⫺2 兲 ⫺11/6p 共 f 兲 兩 G (10)

The function F ⬘ (h) depends on the viewing angle ␪ ¯ ( f ) 兩 2 ] and is expressed in units of [through the factor 兩 G m⫺1/3. It is assumed for simplicity that turbulence outer scale does not depend on the altitude. For observations at zenith angle z, altitude h must be replaced by h sec z. If we use a control algorithm without spatial filtering and assume in addition that DM’s have infinite spatial resolution 关 r( f ) ⫽ 1 兴 , that the telescope has an infinite aperture diameter and that the turbulence outer scale is infinite, then the piston’s removal becomes irrelevant 关 p( f ) ⫽ 1 兴 , and the phase residual variance can be written as

具 ⑀ 2 典 ⫽ 2.905共 2 ␲ /␭ 兲 2 兩 ␪兩 5/3C n2 共 h 兲 F M 共 h 兲 dh,

(11)

ˆ c. F M 共 h 兲 ⫽ bˆT c ⫺ 0.5cT A

(12)

1821

g ⫽ A⫺1 b.

(14)

In the limit ␪ → 0 the determinant of matrix A tends to zero, and it becomes difficult to compute its inverse. This reflects the fact that for on-axis viewing the turbulence can be corrected with any of DM; hence the corrections applied to individual DM’s become indeterminate. In the limit f ␪ h → 0 the Bessel functions in Eqs. (7) can be replaced by their Taylor approximations, and the optimum filters g m are then just coefficients independent of f. The phase variance with optimum filters can be computed from Eqs. (9) and (10). However, the optimum spatial filters must be found for some particular ␪⬘ and then kept fixed for other viewing directions. So, we have to use the general expression [Eq. (6)] to find the phase variance as a function of ␪. In the restricted case of proportional control and Kolmogorov turbulence we achieve the minimum residual phase variance by minimizing F M (h) [Eq. (12)] by a suitable choice of the control vector c with the constraint described by Eq. (A14) of Appendix A. We introduce the Lagrange multiplier ␦ and search for a minimum of F M (h) ⫹ ␦ (cT c ⫺ 1). At minimum the partial derivatives over c m must be equal to zero, which determines a vector c( ␦ ). Then parameter ␦ is found to satisfy the constraint of Eq. (A14), and this is the solution to our problem. The results of minimization can be written in compact matrix form. By 1 we denote a column vector of the size of M with all elements equal to 1. Three auxiliary scalar quantities ␣, ␤, and ␥ are defined as ˆ ⫺1 1, ␣ ⫽ 1T A

ˆ ⫺1 bˆ, ␤ ⫽ 1T A

ˆ ⫺1bˆ. ␥ ⫽ bˆT A

(15)

The minimization results are

␦ ⫽ 共 1 ⫺ ␤ 兲/␣,

where

The vector c with elements c m is the control algorithm ˆ are and the vector bˆ and the matrix A bˆ m ⫽ 兩 h ⫺ H m 兩 5/3,

aˆ mm ⬘ ⫽ 兩 H m ⫺ H m ⬘ 兩 5/3. (13)

We use this restricted result below to calculate systemindependent isoplanatic angle.

ˆ ⫺1 共 bˆ ⫹ ␦ 1兲 , c⫽A F m 共 h 兲 ⫽ 0.5关 ␥ ⫺ 共 1 ⫺ ␤ 兲 2 / ␣ 兴 .

(16)

5. TWO-MIRROR SYSTEM To get some insight into the operation of the optimum control algorithm derived above, let us consider a simple MCAO system with only two DM’s. To simplify further,

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we assume that the DM’s have infinite spatial resolution by putting r( f ) ⫽ 1. The matrix A and vector b in this case are

冋

A⫽

冏

1

J 0 关 2 ␲ f 共 x 1 ⫺ x 2 兲兴

J 0 关 2 ␲ f 共 x 1 ⫺ x 2 兲兴

1

册

c 1 ⫽ 0.5兩 H 2 ⫺ H 1 兩 ⫺5/3共 兩 h ⫺ H 2 兩 5/3

,

冏

J 0 共 2 ␲ fx 1 兲 b⫽ . J 0 共 2 ␲ fx 2 兲

⫺ 兩 h ⫺ H 1 兩 5/3 ⫹ 兩 H 2 ⫺ H 1 兩 5/3兲 . (17)

The calculation of the matrix inverse and multiplication yield the optimum spatial filters. Because of the system’s symmetry with respect to subscripts 1 and 2 it is sufficient to write only the first filter: g 1共 f 兲 ⫽

J 0 共 2 ␲ fx 1 兲 ⫺ J 0 关 2 ␲ f 共 x 1 ⫺ x 2 兲兴 J 0 共 2 ␲ fx 2 兲 1 ⫺ J 02 关 2 ␲ f 共 x 1 ⫺ x 2 兲兴

. (18)

It is nice to see that this filter has the desired behavior. Assuming that the turbulence layer is located at the same height as DM1, h ⫽ H 1 , leads to x 1 ⫽ 0 and hence to g 1 ( f ) ⫽ 1 and g 2 ( f ) ⫽ 0. This layer will be completely compensated by DM1, as one would expect. Suppose now that a layer is located exactly between the two DM’s. Now the filters g 1 and g 2 are identical: g 1共 f 兲 ⫽ g 2共 f 兲 ⫽

J 0共 ␲ f ␪ ⬘H 兲 1 ⫹ J 0共 2 ␲ f ␪ ⬘H 兲

,

Let us consider now the restricted problem with proportional control. In the two-mirror case the optimal control vector contains only two elements, their sum is equal to 1, and c 1 is given by

(19)

where H ⫽ H 1 ⫺ H 2 is the difference of DM conjugation heights. We see that the optimum phase correction involves low-pass filtering, because the filter starts to decrease at frequencies above ( ␪ ⬘ H) ⫺1 . It does not make sense to include high-frequency components in the corrections because they will be decorrelated and will degrade the off-axis imaging performance. At low frequencies g 1 ( f ) is close to 1/2, which means that the correction is shared equally between the two DM’s. A salient feature of this example is that optimum filtering depends on the angle ␪ and on the distance between the turbulent layers and the DM conjugate heights. In a wider FOV the optimal corrections for some layers will be of lower order, and on-axis image quality will be degraded in exchange for an increased field.9 However, for layers that are close to DM conjugate heights a high-order correction is still required. Thus, if DM’s are conjugated with strong turbulent layers, a high-spatial-frequency correction (i.e., a large number of actuators) is desirable. It has been assumed throughout this paper that phase screens in all layers are perfectly known. Now it becomes clear that the amount of information that is required depends on the distance between a given layer and a DM conjugation plane. If this distance is large, high spatial frequencies are not needed, because they will anyway be damped by the optimum filter. In any tomographic scheme for three-dimensional turbulence measurements the number of reconstructed layers is always limited. We see that what is actually needed for good MCAO operation are suitable projections of layers onto DM conjugate planes with a spatial resolution that is progressively degraded with increasing distance from the DM.

(20)

The behavior of this coefficient is in agreement with simple logic. For example, the coefficient is equal to 1 when h ⫽ H 1 and the turbulence can be entirely compensated by the first DM. A layer located exactly between the DM’s, h ⫽ (H 1 ⫹ H 2 )/2, will have c 1 ⫽ 0.5. It is an easy matter to show that such redistribution of the correction gives larger FOV than the full correction applied to only one of the two DM’s, although this result is probably not intuitively clear. What is still less evident is that, for layers outside the interval (H 1 , H 2 ), either c 1 or c 2 will be negative. This means that for optimum off-axis imaging the turbulence must be slightly overcorrected by the DM that is nearest to the layer, and a corresponding negative correction must be supplied to the second DM.

6. ISOPLANATIC ANGLE FOR LARGE APERTURES AND THE KOLMOGOROV TURBULENCE MODEL The residual phase variance depends on many system parameters. To obtain a system-independent characteristic of the isoplanatic FOV size we make several simplifying assumptions and adopt the proportional-control model. In this restricted framework the optimized control vector for each layer and the resulting function F M (h) are given by Eqs. (16). Now the residual phase variance summed over the whole atmosphere takes the form

具 ⑀ 2 典 ⫽ 共 兩 ␪ 兩 / ␪ M 兲 5/3,

(21) 6

which is similar to Fried’s expression for a classic AOI. The generalized isoplanatic angle ␪ M is an angular radius of a field where the residual phase variance reaches 1 rad2. It is calculated as

␪ ⫺5/3 ⫽ 2.905共 2 ␲ /␭ 兲 2 共 sec z 兲 8/3 M

冕

h max

0

C n2 共 h 兲 F M 共 h 兲 dh, (22)

Fig. 1. Weighting functions F 0 (dotted curve), F 1 (dashed curve) and F 2 (solid curve) plotted against altitude. Conjugation altitudes are 5 km for F 1 , 1 and 8 km for F 2 .

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Vol. 17, No. 10 / October 2000 / J. Opt. Soc. Am. A

where the dependence on the source zenith angle z was introduced in an obvious way. The function F M (h) is given above [Eqs. (16)]. For any given turbulence profile C n2 (h) the optimum altitudes of the DM’s may be found by numerical minimization of Eq. (22). The corresponding value of ␪ M gives a quantitative measure of the FOV widening that results from the use of several DM’s. Alternatively, one may find a loss of FOV size when DM’s are conjugated to some fixed but not optimum heights. For an AOI system with a single DM conjugated to the altitude H 1 we obtain F 1 (h) ⫽ 兩 h ⫺ H 1 兩 5/3. If H 1 ⫽ 0, as in classic AOI systems with the DM in the pupil plane, the appropriate function is F 0 (h) ⫽ h 5/3, and Fried’s isoplanatic angle ␪ 0 is obtained from Eq. (22). An explicit formula for a two-mirror MCAO looks like F 2 共 h 兲 ⫽ 0.5关 兩 h ⫺ H 1 兩 5/3 ⫹ 兩 h ⫺ H 2 兩 5/3 ⫺ 0.5兩 H 2 ⫺ H 1 兩 5/3 ⫺ 0.5兩 H 2 ⫺ H 1 兩 ⫺5/3共 兩 h ⫺ H 1 兩 5/3 ⫺ 兩 h ⫺ H 2 兩 5/3兲 2 兴 .

(23)

It is easy to verify that F 2 is symmetric with respect to H 1 and H 2 and that it reduces to F 1 if H 1 ⫽ H 2 . In Fig. 1 the functions F 0 , F 1 , and F 2 are plotted as an example. The gain in the FOV of the two-mirror MCAO system is directly related to the lower weighting of the C n2 (h) profile with F 2 , compared with that in a one-mirror AOI system. Even the layers that lie between H 1 and H 2 have low weights in comparison with F 0 weighting.

7. EXPECTED FIELD-OF-VIEW GAIN We evaluated the isoplanatic angles for two- and three-DM MCAO, using different turbulence profiles. The conjugate altitudes of the DM’s were optimized for the largest FOV. The results of these calculations are given in Table 1. All quantities refer to a 500-nm wavelength for observations at zenith. We also provide the classical isoplanatic angle ␪ 0 and the coherence radius r 0 for these profiles. The gain in the FOV size is expressed by the ratio ␪ M / ␪ 0 . Profile 1 in Table 1 is a constant, C n2 (h) ⫽ 8 ⫻ 10⫺18 m⫺2/3 distributed from 0 to 10 km. The result of this calculation can be easily scaled to any other constant

1823

turbulence profile. In particular, the FOV gain will be the same, independently of the maximum altitude and of the C n2 value. It has often been assumed that a constant turbulence distribution is probably the worst for MCAO. In fact, in this case the FOV gain from use of multiple DM’s is close to maximum compared with other, more realistic, profiles. The second case represents the well-known Hufnagel turbulence profile as used by Valley and Wandzura.12 It may be typical for most ground-based observatories. The gain in FOV is somewhat less than for a constant distribution simply because the strong turbulence layer near ground is already well corrected by the DM in the pupil plane. We note, however, that the optimal conjugate height of the lowest DM is not at the pupil but somewhere near 1 km. The third profile corresponds to good seeing conditions at the Mauna Kea Observatory and was taken from Ref. 3. It is not much different from constant, so the FOV gain obtained is quite similar to that of constant C n2 . Profile 4 and 5 in Table 1 were taken at the Paranal Observatory in Chile.13,14 From the total of 12 profiles we selected Paranal-10 as typical of the whole series. Paranal-01 represents the best observing conditions and the widest FOV. The median value of ␪ 3 for the 12 profiles is 16.2⬙. The profiles are plotted in Fig. 2. Examining Table 1, we note that the gain in the FOV size from use of several DM’s is 5–8 for two DM’s and 8–13 for three DM’s. In other words, the widening of the FOV does not depend very much on the particular turbulence distribution. Of course, if all turbulence were indeed concentrated in two or three thin layers, we would obtain infinite gain, but this is never the case in practice. One may argue that with a larger number of DM’s the layered turbulence structure as seen at Paranal would be compensated much better. We calculated the isoplanatic angles for all Paranal profiles with increasing number of DM’s, from 2 to 12. The results are shown in Fig. 3. For a 12-DM MCAO the median value of ␪ 12 is 67.3⬙, with a maximum of 108.4⬙. ␪ M scales roughly linearly with the number of DM’s, with a slope of 5.8. The calculations of ␪ M for Paranal have also been performed with fixed heights of DM conjugation for two and three DM’s, which may be attractive from the technical point of view. When these heights are chosen as the

Table 1. Isoplanatic Angles for ␭ Ä 500 nm r0 (m)

␪0 (arc sec)

1. Constant

0.365

4.21

2. Hufnagel

0.107

1.08

3. Mauna Kea

0.341

2.68

4. Paranal-10

0.207

2.22

5. Paranal-01

0.318

3.81

Profile

M

␪M (arc sec)

␪M /␪0

H1 (km)

H2 (km)

H3 (km)

2 3 2 3 2 3 2 3 2 3

27.52 44.98 5.54 8.39 17.16 30.0 9.80 15.84 17.97 25.57

6.5 10.7 5.1 7.8 6.4 11.2 4.45 7.2 4.71 6.71

1.96 1.20 1.11 0.91 4.65 3.71 0.02 0.02 1.12 1.04

8.04 5.00 12.25 8.64 14.25 9.91 12.83 8.10 12.13 9.80

– 8.80 – 15.05 – 15.30 – 16.15 – 16.01

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Fig. 2. Paranal-01 (the best) and Paranal-10 (typical) turbulence profiles. Heights are in kilometers above the observatory; C 2n is in units of 10⫺17 m⫺2/3. For clarity, the profiles have been convolved with a 500-m standard deviation Gaussian. The arrows show the optimum heights for a three-DM MCAO configuration.

Fig. 3. Isoplanatic angle (in arc sec) as a function of the number of deformable mirrors. Solid curve, median ␪ M obtained from the Cerro Paranal data; dashed curve, linear fit with a slope of 5.8. This slope depends on a turbulence profile.

median of the optimized heights, the loss in the size of the FOV is not very pronounced (two fixed DM’s yield a median ␪ 2 ⬃ 9.1⬙ , whereas optimized heights provide ␪ 2 ⬃ 9.8⬙ . For three DM’s, these numbers are, respectively, 14.5⬙ and 16.2⬙). In the near-infrared spectral region the FOV size will be larger. For example, in the K band (␭ ⫽ 2.2 ␮ m) the FOV size is increased to 5.9 times its value to 0.5 ␮m. Thus the median ␪ 3 at Paranal corresponds to a FOV diameter of 3.2 arcmin in the K band. In the case considered by Fusco et al.,3 the FOV diameter of a two-DM system in the K band would be 3.4 arcmin. These authors made calculations only within a 1-arcmin FOV and found that with two DM’s the Strehl ratio remains practically constant in the field, a finding that is consistent with our results.

Tokovinin et al.

simplest two-mirror MCAO system and a Hufnagel– Valley turbulence profile are taken for this test. We adopt turbulence outer scale L 0 of 26 m, close to the median outer scale reported by Martin et al.15 for the La Silla observatory in Chile. A telescope diameter D ⫽ 8 m and a wavelength ␭ ⫽ 0.5 ␮ m are assumed. The two DM’s are conjugated to the optimized heights, as given in Table 1. The corresponding isoplanatic angle is ␪ 2 ⫽ 5.54⬙ . The residual phase variance is calculated from Eq. (4) by numeric integration. In the case of proportional control, infinite pupil size, and infinite outer scale, we must obtain the variance equal to ( ␪ / ␪ M ) 5/3, as predicted by Eq. (21). This is the limiting 5/3 case. In fact, the results of integration are within 2% from the 5/3 curve—a precision that we consider acceptable. In Fig. 4 the phase variance is plotted as a function of viewing angle ␪. The three solid curves correspond to spatial filtering optimized for angles ␪⬘ of 1⬙, 4⬙, and 8⬙. The dashed curve is the limiting 5/3 curve. The intersection of these curves with the dotted line at 1 rad2 gives the FOV size according to the adopted definition. Inspecting Fig. 4, we note that the FOV gain that is due to the optimized filtering is noticeable but not dramatic as would be expected for large D/r 0 . The lower envelope of the optimized curves is approximately one half that of the 5/3 curve, and the corresponding FOV gain is hence not greater than 2 3/5 ⬇ 1.5 times. A price to pay is the reduced on-axis phase variance. The phase variance with optimized filtering is much more nearly uniform over the whole FOV than is the limiting case of proportional control. In Fig. 5 the results of similar calculations for a 4-m telescope working at 2.2 ␮m are presented. Here the isoplanatic angle ␪ 2 ⫽ 33⬙ , and D/r 0 ⫽ 5.4. The curves are qualitatively similar to those in Fig. 3. The results are, however, different with respect to the rate of the phase variance growth at large ␪. In Fig. 4 the curves follow the 5/3 power law, whereas in Fig. 5 they increase much less steeply. This difference is related to the piston removal. It is evident that for very large angles ( ␪ Ⰷ D/h max) the differential piston-removed phase variance will saturate at the value 2 ⫻ 1.03(D/r 0 ) 5/3, be-

8. EFFECTS OF OPTIMIZED FILTERING, FINITE PUPIL, AND FINITE OUTER SCALE In any realistic system the pupil size is finite, and wave fronts are compensated with only limited spatial resolution. As mentioned above, this results in reduced off-axis phase variance. In this section we investigate the importance of these effects for a specific MCAO system. The

Fig. 4. Residual phase variance as a function of off-axis angle for the 8-m telescope, the two-DM MCAO system, and Hufnagel– Valley turbulence profile. Wavelength, 0.5 ␮m. Solid curves, spatial filtering optimized for 1⬙, 4⬙, and 8⬙; dashed curve, limiting curve ( ␪ / ␪ 2 ) 5/3.

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9. CONCLUSIONS

Fig. 5. Same as in Fig. 4 but for a 4-m telescope working at ␭ ⫽ 2.2 ␮ m. The spatial filtering (solid curves) is optimized for 6⬙, 24⬙, and 60⬙ FOV size.

The concept of the isoplanatic angle has been generalized for an adaptive optics system with several deformable mirrors. The typical FOV gain with three DM’s is 7–10⫻, and a FOV diameter as large as 1 arc min in the visible can be attained under the best observing conditions. It would correspond to a FOV of 6 arc min in the K band. We must repeat that ␪ M is a pessimistic parameter in the sense that in any real AOI system the FOV will be wider, as was shown by the numerical examples in Section 8. For large-aperture telescopes, however, this additional gain is small, and ␪ M is thus a valid estimator of the FOV size.

APPENDIX A: CALCULATION OF THE RESIDUAL PHASE VARIANCE A direct consequence of the linear control algorithm [Eq. (3)] is that the phase residual ⑀ is just a linearly filtered input signal: ˜ 共 f 兲, ˜⑀ 共 f 兲 ⫽ G 共 f 兲 ␾

(A1)

where we obtain the filter G( f ) by putting Eq. (3) into Eq. (2): G 共 f 兲 ⫽ exp关 ⫺2 ␲ i 共 f␪h 兲兴 ⫻ Fig. 6. Residual phase variance as a function of off-axis angle for the 8-m telescope, the two-DM MCAO system, and the Hufnagel-Valley turbulence profile. Wavelength, 2.2 ␮m. Solid curves, turbulence outer scale L 0 ⫽ 26 m; dashed curves, infinite outer scale. The curves are for proportional control and for the spatial filtering optimized for a 40⬙ field.

再

冎

M

1⫺

兺

g m 共 f 兲 r 共 f 兲 exp关 ⫺2 ␲ i 共 fxm 兲兴 .

m⫽1

(A2)

Here the spatial shifts xm ⫽ ␪ (H m ⫺ h) are introduced for clarity, with x m ⫽ 兩 xm 兩 . The square modulus of the spatial filter G( f ) is M

兩 G共 f 兲兩 2 ⫽ 1 ⫺ r共 f 兲

兺

g m共 f 兲

m⫽1

cause the on-axis and off-axis phase screens will be completely independent. For the case presented in Fig. 5, D/33⬙ ⫽ 25 km, and piston removal becomes important. It seems to be still more important for classic singlemirror AOI systems.7 Figure 6 illustrates the effects of the finite turbulence outer scale. It corresponds to the 8-m telescope working at ␭ ⫽ 2.2 ␮ m. Solid curves are for L 0 ⫽ 26 m; dashed curves are for L 0 ⫽ ⬁. Both the 5/3 curve (proportional control) and the curve with filtering optimized for ␪ ⬘ ⫽ 40⬙ are plotted. It is evident that even for a large 8-m aperture the influence of a finite outer scale on size of the isoplanatic patch is almost negligible. This is quite understandable, because the finite outer scale reduces the variance of only the lowest-order modes, which are, anyway, well correlated over the FOV. The high-order modes, which contribute mostly to the differential phase variance, remain unaffected by the finite outer scale. It is thus clear that the optimized spatial filtering, finite pupil size, and finite outer scale reduce the residual phase variance and increase the FOV size. However, for large-aperture telescopes this reduction is not dramatic, and the FOV gain is always less than 1.5⫻ that of angle ␪M .

⫻ 关 exp共 ⫺2 ␲ ifxm 兲 ⫹ exp共 2 ␲ ifxm 兲兴 M

⫹ r 共f兲 2

M

兺 兺

m⫽1 m ⬘ ⫽1

g m共 f 兲 g m ⬘共 f 兲

⫻ exp关 2 ␲ if 共 xm ⫺ xm ⬘ 兲兴 .

(A3)

In the computation of the phase variance, the double integration over df can be reduced to the one-dimensional integral over f by transformation to polar coordinates in the frequency plane and by radial averaging of 兩 G 兩 2 . Here we use the integral formula of a zero-order Bessel function. With the angular average denoted by a bar, M

¯ 共 f 兲 兩 2 ⫽ 1 ⫺ 2r 共 f 兲 兩G

兺

g m 共 f 兲 J 0 共 2 ␲ fx m 兲 ⫹ r 2 共 f 兲

m⫽1 M

⫻

M

兺 兺

m⫽1 m ⬘ ⫽1

g m 共 f 兲 g m ⬘ 共 f 兲 J 0 关 2 ␲ f 共 x m ⫺ x m ⬘ 兲兴 . (A4)

Before computing the residual phase variance, we must subtract its mean value over the telescope aperture (so-

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Tokovinin et al.

called piston), because it has no effect on the image quality. The piston-removed residual ⑀ p (x) ⫽ ⑀ (x) ⫺ ⑀ 0 , where

冕

⑀0 ⫽

冓冕

具⑀ 典 ⫽

␴ ␾2

(A5)

冔

⫹

冕

W ⑀ 共 f 兲 df.

(A7)

We note that random process ⑀ 0 is just the first Zernike coefficient. It is the low-pass linearly filtered process ⑀ (x); the filter function is the Fourier transform of the first Zernike mode.7,8 It follows from Eq. (A6) that

具 ⑀ p2 典 ⫽

冕

W ⑀ 共 f 兲 p 共 f 兲 df,

(A8)

where p共 f 兲 ⫽ 1 ⫺

冋

J 1共 2 ␲ R f 兲

␲ Rf

册

具⑀ 典 ⫽ 2

␴ ␾2

冉

冕

⬁

0

.

(A9)

(A10)

which is just the result in Eq. (4). Now the phase variance in the restricted problem will be estimated. We put r(f ) ⫽ 1, neglect piston removal 关 p(f ) ⫽ 1 兴 , and set L 0 ⫽ ⬁. Within this restricted framework the integral of the phase power spectrum can be expressed through the phase covariance function ⌫ ␾ (x) by noting that

冕

c mc m⬘⌫ ␾共 x m ⫺ x m⬘ 兲 ,

M

1⫺

兺

cm

m⫽1

冊

2

W ␾ 共 f 兲 exp关 ⫺2 ␲ i 共 fx兲兴 df.

(A12)

M

⫹

兺

m⫽1

c mD ␾共 x m 兲

M

兺 兺

m⫽1 m ⬘ ⫽1

c mc m⬘D ␾共 x m ⫺ x m⬘ 兲 .

(A13)

One could derived this result directly from Eq. (1) by putting into it the proportional control model. We proceed further by using the Kolmogorov phase structure function, which corresponds to the infinite outer scale L 0 in Eq. (8). The phase dispersion ␴ ␾2 is then infinite, and to cancel out the first term in Eq. (A13) we have to impose the condition that M

兺

c m ⫽ 1.

(A14)

m⫽1

This condition is naturally needed for a perfect on-axis turbulence compensation. The phase structure function for a given turbulent layer is related to its strength and to the wavelength, ␭: D ␾ 共 x兲 ⫽ 2.905共 2 ␲ /␭ 兲 2 兩 x兩 5/3C n2 共 h 兲 dh.

¯ 共 f 兲 兩 2 p 共 f 兲 df, fW ␾ 共 f 兲 兩 G

⌫ ␾ 共 x兲 ⫽ 具 ␾ 共 y兲 ␾ 共 y ⫹ x兲 典 ⫽

M

兺 兺

M

⫺

2

From Eq. (A1) it follows that the power spectrum W ⑀ (f) is obtained as a product of the phase power spectrum W ␾ (f ) and the filter function 兩 G(f) 兩 2 . We put W ⑀ (f) into Eq. (A8), go to polar coordinates, perform the angular integration (which adds a 2␲ factor), and arrive at the formula

具 ⑀ p2 典 ⫽ 2 ␲

c m⌫ ␾共 x m 兲

where ␴ ␾2 is the total phase dispersion. The phase covariance can be expressed through the phase structure function D ␾ , ⌫ ␾ ⫽ ␴ ␾2 ⫺ 0.5D ␾ , to give

(A6)

The variance of the random process ⑀ (x) can be computed as the integral of its power spectrum W ⑀ (f) over frequencies:

兺

m⫽1

m⫽1 m ⬘ ⫽1

P 共 x兲关 ⑀ 共 x兲 ⫺ ⑀ 0 兴 2 dx ⫽ 具 ⑀ 2 典 ⫺ 2 具 ⑀ 02 典 ⫹ 具 ⑀ 02 典 .

具 ⑀ 2典 ⫽

⫺2 M

P 共 x兲 ⑀ 共 x兲 dx.

For a circular aperture of the radius R, the pupilweighting function P(x) is equal to 1/( ␲ R 2 ) for 兩 x兩 ⬍ R and 0 otherwise. If necessary, the overall tilt of the reconstructed wave front may be subtracted in the same manner.7 We need the aperture-averaged variance:

具 ⑀ p2 典 ⫽

M

2

(A15)

Now the phase variance produced by a single layer takes the form

具 ⑀ 2 典 ⫽ 2.905共 2 ␲ /␭ 兲 2 兩 ␪ 兩 5/3C n2 共 h 兲 F M 共 h 兲 dh,

(A16)

ˆ c, F M 共 h 兲 ⫽ bˆT c ⫺ 0.5cT A

(A17)

where

ˆ written as with vector bˆ and matrix A bˆ m ⫽ 兩 h ⫺ H m 兩 5/3,

aˆ mm ⬘ ⫽ 兩 H m ⫺ H m ⬘ 兩 5/3. This is the desired result.

(A18)

ACKNOWLEDGMENTS This research has benefited from the support of the European Training and Mobility of Researchers program, ‘‘Laser guide stars for 8 m class telescopes,’’ contract ERBFMRXCT960094. We are grateful to E. Viard for her comments on the manuscript and to N. Hubin for fruitful discussions.

(A11)

Again we write the residual phase variance as an integral of the filtered phase power spectrum, take the square modulus of the filter from Eq. (A3) with g m replaced by c m , and make use of Eq. (A11) to obtain

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