2 Phase screen behaviour

This section describes the wavefront decomposition into Zernike polynomials, the phase structure function of the wavefront, the generalization of the Fried parameter and the phase correlation length behavior.

2.1 Wavefront description

Optical wavefronts are two dimensional functions that can be decomposed into Zernike polynomials which are separable in angle and radius and form an orthogonal basis. We will use the definition given by Noll (1976):

$$% \phi(r,\theta)=\sum_{i=1}^\infty a_i Z_i(r,\theta)$$ (1)

where a_i are the coefficients of the corresponding Zernike polynomials (Z_i). The effect of partial compensation on the wavefront is that some of the decomposition coefficients vanish. The residual distortion in the compensated wavefront may be estimated using the Noll expression for the average variance over the wavefront surface once the first j Zernike terms have been corrected:

$$% \Delta_j=\sum_{i=j+1}^\infty \left\langle \left\vert a_i^2 ... ...ight\rangle =\mbox{coef}(j) \left( \frac{D}{r_0}\right)^{5/3}$$ (2)

where $\langle$ $\rangle$ denotes an ensemble average and coef(j) is the corresponding coefficient given by Noll. The maximum number of modes that can be corrected is related with the number of actuators in the system.

2.2 Structure function

The phase structure function is defined as:

$$% D_\phi\left( \vec{r}-\vec{r'}\right)= \left\langle\left[ \phi(\vec{r})-\phi(\vec{r'})\right]^2\right\rangle.$$ (3)

For apertures over 1 m, even in the best seeing sites, the phase fluctuation has a Gaussian distribution because of the central limit theorem. Its mean value is zero and, hence, it is fully described by its second moment. The structure function being the second moment for differential phase fluctuation, completely specifies the spatial statistics of phase fluctuation. Based on the Kolmogorov (1961) theory of turbulence, it is possible to show that in the absence of compensation the phase structure function may be written as:

$$% D_\phi(r)=6.88 \left(\frac{r}{r_0}\right)^{5/3}$$ (4)

where r₀ is the Fried parameter (Fried 1965). It is known (Roddier 1990) that the shape of the structure function varies as a function of the compensation degree. As the separation between points becomes arbitrarily large the autocorrelation function tends to zero and the structure function saturates to $2\Delta_j$. However, for small separation distances the curve still follows the 5/3 power law (Rigaut et al. 1997; Hardy 1998) and can be fitted using the following expression:

$$% D_\phi(r)=6.88 \left(\frac{r}{\rho_0}\right)^{5/3}$$ (5)

where the value of the parameter $\rho_0$, the generalized Fried parameter, increases with the compensation degree. Figure 1 shows this behavior of the structure function for uncompensated wave fronts (short-dashed curve) obtained from simulation, and for 6 (solid curve), 21 (circles), 41 (long-dashed curve) and 81 (triangles) corrected modes. This behaviour of the structure function is not restricted to those systems that correct pure Zernike modes. A first region described by Eq. (5) that saturates to $2\Delta_j$ for distances greater than the correlation length has been found in noisy systems and in curvature-based systems (Cagigal & Canales 2000a).

2.3 Correlation length

The phase correlation length in the wavefront, $l_{\rm corr}$, can be obtained from the structure function (Valley & Wandzura 1979). It is defined as the distance value for which the structure function leaves the 5/3 power behavior and reaches the constant value $2\Delta_j$. In this point it is fulfilled that:

$$% 6.88 \left(\frac{l_{\rm corr}}{\rho_0}\right)^{5/3}=2\Delta_j.$$ (6)

It is interesting to note that the parameters $\Delta_j$ and $\rho_0$ are determined by the number of corrected polynomials and by the value of the ratio D/r₀. However, the correlation length is completely determined by the number of corrected polynomials (in D units) as it can be deduced from previous works (Valley & Wandzura 1979). The behaviour of $l_{\rm corr}$was described in a previous paper (Cagigal & Canales 2000a) by fitting a generic curve to the values obtained from the simulated structure functions. The fitted curve is given by:

$$% l_{\rm corr}\approx0.286j^{-0.362}D$$ (7)

where j is the number of corrected polynomials, that we relate to the number of system actuators. We have checked that this is a robust curve under deviations from an ideal compensation process.

Figure 1: Structure function of the wavefront phase for D/r0=38.4and 1 (short-dashed curve), 6 (solid curve), 21 (circles), 41 (long dashed curve) and 81 (triangles) corrected modes