4 Mean PSF and PSD estimation

The correction achieved by adaptive optics depends on the magnitude and angular size of the reference object, as well as on the strength of atmospheric turbulence. As a result, the PSF evolves both in time and space. Each image has a different PSF, which is difficult to determine.

One possibility to estimate the PSF, is to image a star near the observed object (Northcott 1996). This technique however presents some major problems: waste of observation time, small sky coverage, evolution of the turbulence between object and reference star acquisition.

Recently, a method has been proposed to accurately retrieve the mean PSF ($h_{\rm m}$) from the data accumulated by the AO control system during the acquisition (Veran et al. 1997). This mean PSF however corresponds to an image with an infinite integration time. $h_{\rm m}$ is obtained using the theoretical expression:  

$$\tilde{h}_{\rm m}(\vec{f}) = \exp{(-\frac{1}{2} D_{\phi_{\rm r}(\lambda \vec{f})})}\times OTF_{\rm free}(\vec{f})$$ (6)

where $D_{\phi_{\rm r}}(\lambda \vec{f}) = <[\phi_{\rm r}(\vec{\rho})-\phi_{\rm r}(\vec{\rho} + {\lambda \vec{ f}})]^2\gt$ is the residual phase structure function estimated from the AO data and $OTF_{\rm free}$ is the aberration-free OTF. Note that, in practice, the WFS data never allow the calculation of the true $\phi_{\rm r}$ but give a noisy measurement of this phase (Veran et al. 1997).

$h_{\rm m}$ differs from the actual finite time PSF because of the so-called turbulence noise, related to the residual speckles in the AO corrected images. The shorter the integration time, the larger the turbulence noise. The variance of this noise on the optical transfer function for an integration time $T_{\rm i}$can be estimated at each spatial frequency by (Conan 1994):  

$$\sigma^2_{\rm turb}(\vec{f}) = \frac{\tau_{\rm s}}{T_{\rm i}}(STF(\vec{f}) - \vert\tilde{h}_{\rm m}(\vec{f})\vert^2)$$ (7)

where $STF(\vec{f})$ is the Speckle Transfer Function (Roddier 1981) and $\tau_{\rm s}$ is the coherence time characterizing the turbulence evolution. Using some approximations, the STF can be expressed as a function of ${\tilde h}_{\rm m}$ (Conan 1994) and can therefore be also estimated from the AO loop data.

The difficulty lies in the estimation of the coherence time $\tau_{\rm s}$. It corresponds roughly to the turbulence evolution time. Without AO correction, $\tau_{\rm s}$ is the so-called speckle boiling time characterizing the speckle life time. An approximate value is given by Roddier (Roddier et al. 1982). It could be different in the case of AO correction and certainly depends on the correction degree. However we assume, in a first approximation, that the expression given by Roddier is valid here:

$$\tau_{\rm s} = 0.36\frac{r_0}{\bar{v}}$$ (8)

where r₀ is the Fried parameter of the turbulence and $\bar{v}$ is the mean wind speed across the different turbulence layers. r₀ can be computed from the AO loop data (Veran et al. 1997) as well as $\bar{v}$ (Conan et al. 1995).

With accurate models of the servo-loops, WFS, deformable mirror, a bright enough object, the error on the PSF estimated from the AO loop data is dominated by the turbulence noise (Veran 1997). Therefore, we have:  

$$PSD_{\rm h}(\vec{f}) = \sigma^2_{\rm turb}(\vec{f}).$$ (9)

However, if some other errors are introduced in the calculation of $h_{\rm m}$ (unexpected fixed aberrations for example), $PSD_{\rm h}(\vec{f})$ should be increased accordingly.