2 Theoretical estimation of the PSF angular dependence

Let $o(\text{\boldmath$\alpha$ }')$ be the object of interest. For wide FOV, the AO corrected image is given by:

$$i(\text{\boldmath$\alpha$ })=\int o(\text{\boldmath$\alpha$ ... ...rm d}\text{\boldmath$\alpha$ }' + b(\text{\boldmath$\alpha$ })$$ (1)

where $\text{\boldmath$\alpha$ }$ is the angle in the FOV, $h(\text{\boldmath$\alpha$ },\text{\boldmath$\alpha$ }')$ the space variant long exposure PSF and b an additive zero mean noise.

Note that, within the isoplanatic patch, Eq. (1) becomes a convolution between the object and the PSF. In order to well estimate the object $o(\text{\boldmath$\alpha$ }')$ an accurate knowledge of the PSF $h(\text{\boldmath$\alpha$ },\text{\boldmath$\alpha$ }')$ in the whole FOV, is needed.

The computation of the PSF requires the use of the residual phase second order statistics. The long exposure Optical Transfer Function (OTF) (Fourier transform of the PSF) is given by:

$$\begin{align} \text{OTF}_{\alpha}({\bf f})=& \int \exp{\left \lbrace-\frac{1}{2}... ...ath$\rho$ }+\lambda {\bf f}){\rm d}\text{\boldmath$\rho$ } \nonumber \end{align}$$ (2)

where $P(\text{\boldmath$\rho$ })$ is the telescope pupil function, $\Phi_{\text{res},\alpha}(\text{\boldmath$\rho$ })$the residual phase in the pupil after AO correction and $\left \langle.\right \rangle$ denotes the expectation over turbulence realizations. The stationarity of the turbulent phase on the telescope pupil is well known. Conan Conan-t-94 and Véran Veran-97a have shown that the AO corrected phase is still quasi-stationary. Then, with this assumption, the OTF expression becomes:

$$\text{OTF}_{\alpha}({\bf f})\simeq T_0({\bf f}) \exp{\left \... ...ac{1}{2}D_{\alpha}\left (\lambda{\bf f}\right )\right \rbrace}$$ (3)

where $T_0({\bf f})$ is the telescope transfer function without atmospheric turbulence, and with $D_{\alpha}\left (\lambda {\bf f}\right )$ the spatially average residual phase structure function:

$$\begin{align} &D_{\alpha}\left (\lambda {\bf f} \right )= \notag \\ & \frac{\in... ...dmath$\rho$ }+\lambda {\bf f}){\rm d}\text{\boldmath$\rho$ }} \notag \end{align}$$ (4)

in a direction $\alpha$ and for a wavelength $\lambda$. Introducing the on-axis structure function Eq. (4) can be re-written as

$$D_{\alpha}\left (\lambda {\bf f}\right )= D_{0}\left (\lambd... ... f}\right )+D_{\rm ani}\left (\lambda {\bf f},\alpha \right ).$$ (5)

where $D_{\rm ani}$ is defined as follows:

$$\begin{align}& D_{\rm ani}\left (\lambda {\bf f},\alpha \right )= \nonumber \\ ... ...o$ }+\lambda {\bf f}){\rm d}\text{\boldmath$\rho$ }\bigg ) \nonumber \end{align}$$ (6)

OTF$_\alpha$ is the product of OTF₀, the OTF on the optical axis, with a term which can be called "anisoplanatic transfer function'' (ATF) and reads:

$$\text{OTF}_{\alpha}({\bf f})=\text{OTF}_{0}({\bf f})\ \text{ATF}({\bf f},\alpha)$$ (7)

with

$${ATF}({\bf f},\alpha) = \exp{\left \lbrace-\frac{1}{2}D_{\rm ani}(\lambda {\bf f},\alpha)\right \rbrace}\cdot$$ (8)

Note that, on-axis means here: on the wavefront sensor optical axis. $\text{OTF}_{0}({\bf f})$ can be estimated using the method proposed by Véran Veran-97b which is based on real-time data statistics accumulated by the AO control system during the observation. To obtain a theoretical expression of the ATF, $D_{\rm ani}(\lambda {\bf f},\alpha)$ must be computed.

Let us use the modal decomposition of the turbulent phase onto the Zernike polynomial basis:

$$\Phi(\text{\boldmath$\rho$ }) = \sum_{i=2}^{\infty} a_i Z_i(\text{\boldmath$\rho$ }).$$ (9)

The properties of turbulence in this particular basis are described by Noll Noll-76. In a first approximation, the AO correction can be seen as a high-pass filter which provides a full correction of all the Zernike polynomials up to a given number. With this assumption and for an AO correction up to the polynomial number i₀, we have:

$$\!\!\!\!\!\!\!\!\!\Phi_{{\rm res},0}(\text{\boldmath$\rho$ })... ...}^{\infty} a_i(0) Z_i(\text{\boldmath$\rho$ }) \text{\ \ and}$$ (10)

$$\!\!\!\!\!\!\!\!\!\Phi_{{\rm res},\alpha}(\text{\boldmath$\rh... ...{i=i_0{+}1}^{\infty}} a_i(\alpha) Z_i(\text{\boldmath$\rho$ })$$ (11)

where a_i(0) and $a_i(\alpha)$ are the Zernike coefficients of the phase expansion on the optical axis and for a direction $\alpha$ respectively. Using this expansion, a theoretical expression of $D_{\rm ani}(\lambda {\bf f},\alpha)$ can be obtained:

$$D_{\rm ani}(\lambda {\bf f},\alpha) = \sum_{i=1}^{i_0}\sum_{... ...t [C_{i,j}(0) - C_{i,j}(\alpha) \right ]U_{ij}(\lambda{\bf f})$$ (12)

where $C_{i,j}(\alpha)$ are the angular correlation of the Zernike coefficients a_i and a_j,

$$C_{i,j}(\alpha_2-\alpha_1) = C_{i,j}(\alpha_1-\alpha_2) = \le... ...left (\alpha_1\right )a_j\left (\alpha_2\right )\right \rangle$$ (13)

and U_ij are functions defined as:

$$\begin{align} & U_{ij} (\lambda{\bf f})=\nonumber \\ & \frac{\int \left [Z_i(\t... ... }{+}\lambda{\bf f}){\rm d}\text{\boldmath$\rho$ }}{\cdot} \nonumber \end{align}$$ (14)

The angular correlations $C_{i,j}(\alpha)$ can be theoretically computed [Chassat1989] assuming that the C_n² profile is known. Note that a crude estimation of the C_n² is enough because of the weak dependency of the angular correlation of the phase with the atmospheric profile [Chassat1989,Molodij & Rousset1997]. Nevertheless, a good estimation of the $C_n^2\$profile can be obtained by a SCIDAR measurement [Fuchs et al.1998], for example.

Equation (12) is a generalization of the expression given by Voitsekhovich & Bara Voits-99 who consider the case of a perfect correction ( $\Phi_{{\rm res},0}(\text{\boldmath$\rho$ }) = 0$) on the optical axis. Note that, because of the difference between the correlations $C_{2,2}(\alpha)$ and $C_{3,3}(\alpha)$ [Chassat1989], Eq. (12) gives an anisotropic ATF which leads to an elongated PSF [Voitsekhovich & Bara1999,Close et al.1998].

Using Eqs. (7), (8) and (12), the OTF in the whole field of view can be theoretically computed, assuming an infinite exposure time.