The AO system corrects a limited number of aberration modes. In the case of AO76 the system is typically able to correct between 65 – 80 Karhunen–Loève (KL) or Zernike modes. Higher order modes remain uncorrected. Hence, the AO system acts as an imperfect high pass filter. The corrected modes are not corrected perfectly, as was discussed in Section 6, and a residual variance remains for each corrected mode. The magnitude of the residual variance is mode dependent.

Closely following Veran et al. (1997), Marino (2007), and Marino and Rimmele (2010), the residual phase after correction can then be expressed as:

where is the residual phase of the corrected modes and for the uncorrected, respectively. The phase structure function can then be written as: where cross terms have been assumed to be negligible (Veran et al., 1997). is a function of only since the uncorrected atmospheric phase errors in a statistical sense (long exposure) are homogeneous and isotropic. Hence, can be placed in front of the integral in Equation 11 resulting in a long exposure OTF that can be written as the product of three independent terms: where andis the contribution from the residual variance of the corrected modes and can be derived from the residual wavefront errors measured by the WFS with the AO in closed loop. derives from the high order modes that the AO system cannot correct. is simply the diffraction limited telescope OTF. is the main contributor to the seeing halo of the PSF, while the core of the PSF is formed by the other two terms.

can be extracted from the WFS measurements after accounting for various noise terms, such as the WFS noise described above and aliasing noise (Herrmann, 1981; Dai, 1996). A detailed description of this process and the calibration procedures involved is given by Veran et al. (1997) and Marino (2007).

The residual wavefront variance after correction can be expressed as:

where are the coefficients KL modes of the uncorrected phase, are the KL coefficients of the applied correction, and the are the noise free residual KL coefficients. The phase structure function of the residual corrected phase becomes: and using Equation (33):Due to the isotropy assumption mentioned above computing the mean phase structure function of the residuals averaged over the pupil is sufficient in this case:

where is the pupil function. Combining Equations (35) and (36) leads to: where the functions are defined as:The important conclusion is that the information, which the AO telemetry has to deliver is the covariance of the noise free residual KL coefficients: . The functions can be pre-computed and stored. The computation of the covariance for the exposure interval can be performed within the AO system, which reduces the telemetry data rates and storage requirements significantly. The computation of the covariance is typically initiated and terminated by an external trigger signal from the science camera.

The term in Equation 32 can be derived from the Kolmogorov model (Kolmogorov, 1941, 1991)
if the Fried parameter r_{0} is known. The residual uncorrected phase after correction of N KL modes can be
expressed in terms of KL modes:

Using the definition of the phase structure function the residual uncorrected phase structure function can be expressed as:

The covariances for Zernike modes was given by Noll (1976) and scales with , where
is the aperture size of the telescope. A similar scaling law for the KL covariances can be derived. Only
r_{0} needs to be determined, which can be estimated from the actuator commands. The modal
variance of the DM actuator commands is compared in a least squares sense to the modal variance
predicted by the Kolmogorov model in order to derive an estimation of the Fried parameter
(r_{0}).

Figure 24 summarizes the major steps of the PSF estimation method.

Living Rev. Solar Phys. 8, (2011), 2
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