3 Comparison between theory and simulation

3.1 Turbulence simulation method

In order to validate the Eqs. (5) and (12) derived in Sect. 2, wide FOV AO images are simulated. We consider a three layer $C_n^2\$profile: one on the pupil, the second at one km and the third at ten km with respectively 20, 60 and 20% of the turbulence. Using the so-called near-field approximation [Roddier1981], the scintillation effects are neglected and the phase on the telescope pupil for a given direction $\text{\boldmath$\alpha$ }$is simply the sum of the corresponding part of each phase screen in each turbulent layer:

$$\Phi_{\text{\boldmath$\alpha$ }}(\text{\boldmath$\rho$ })=\su... ... (\text{\boldmath$\rho$ }+h_j\text{\boldmath$\alpha$ }\right )$$ (15)

where $\varphi_j$ and h_j are respectively the phase screen and the height of the $j^{\rm th}$ layer. Each phase screen is simulated by N. Roddier's method [Roddier1990] using the 861 first Zernike polynomials (radial order up to 40). The size of these phase screens corresponds to a 20 arcsecs FOV radius and a telescope diameter of 4 m. The overall D/r₀ is 10.

3.2 Long exposure PSF simulation

Let us consider an adaptive optics system which can perfectly correct the i₀ first Zernike polynomials, in our case, i₀=21. We have simulated a PSF at every arcsecond in the FOV (20 arcsec). Each PSF is the sum of 1000 time-decorrelated short exposures which are deduced from $\Phi_{{\rm res},\alpha}(\text{\boldmath$\rho$ })$. If we consider that the speckle lifetime is about 10 ms, the simulation are roughly equivalent to a 10 s long exposure. Figure 1 shows the PSF evolution as a function of angle. Note that the PSF does not vary significantly within 3 arcsec, which roughly corresponds to the isoplanatic angle given by Roddier Roddier-81a:

$$\theta_{0} = 0.314 \frac{r_0}{\bar{h}} = 1.64\hbox{$^{\prime\prime}$ }$$ (16)

with $\bar{h}$ given by [Fried1982]

$$\bar{h} = \left [\frac{\int_0^{\infty}h^{\frac{5}{3}}C_n^2(h)... ..._0^{\infty}C_n^2(h) {\rm d}h}\right ]^{\frac{3}{5}} = 3953 m .$$

In this field, the PSF variations are only due to the turbulence noise (finite number of short exposures in the PSF calculation).

Figure 1: PSF in the FOV versus angle. On the top: 2D images of the simulated PSF's for each arcsecond. On the bottom:X-cut of the images. The PSF's normalized by the diffraction (in percent)

Outside the isoplanatic field, the PSF degradation is important, the Strehl Ratio goes from 43% at the center of the field to less than 10% at the border of the field. Of course, a deconvolution scheme, assuming that the PSF is constant in the whole field of view, would lead to a poor restoration. Note that the non-circularity of the off-axis PSF's appears in Fig. 1.

  
3.3 Results

For each angle, we compare the theoretical OTF (see Eqs. 7 and 8) to the simulated one. A very good estimation for each OTF is obtained, even in the case of large values of $\alpha$ (see Fig. 2).

Figure 2: Y-cut of the simulated OTF (dashed line) and theoretical OTF given by Eq. (7) (solid line) for $\alpha = 5, 10, 15 \text{ and } 20 \hbox{$^{\prime\prime}$ }$. The OTF for $\alpha =0$ is shown for comparison

Because we only consider a Y-cut of the OTF's in Fig. 2, the anisotropic (elongation) effect is not visible. In Fig. 3 we only consider one angle ( $\alpha=20\hbox{$^{\prime\prime}$ }$) but we plot two cuts (X and Y axis) of the simulated and theoretical OTF. In that case, the anisotropic effect and its good restitution by the analytical ATF is shown.

A small under-estimation of the OTF can be seen in each case with the analytical method. The difference between the simulated and theoretical OTF's is due to the phase non-stationarity error. In Sect. 2 we have assumed the phase stationarity on the telescope pupil. This assumption corresponds to approximate the mean of the exponential of function by the exponential of the mean of the function. Even if this approximation is quite good [Véran1997], it leads, nevertheless, to a small under estimation of the theoretical OTF. This error gives the fundamental limitation of the method, but in experimental data processing, other error sources may also degrade the results, such as:

Figure 3: Y-cut and X-cut of the simulated OTF (dashed line) and the theoretical OTF (solid line) for $\alpha=20\hbox{$^{\prime\prime}$ }$

- atmospheric parameter estimation errors on: r₀, L₀, C_n² profile;

- AO full correction of the first Zernike modes which is only an approximation of a real system;

- turbulent noise: finite exposure time which leads to a residual speckle pattern in the long exposure image.

But we show in the next section, that in a real case, all these errors do not significantly influence the ATF estimation.