The performance of an AO system is usually described in term of Strehl ratio. The Strehl ratio is the ratio of the psf maximum intensity to the theoretical diffraction-limited psf one; the latter is the Airy pattern in the case of a clear circular aperture telescope of diameter D. We present the steps to compute an unbiased estimation of the Strehl ratio from any AO image of a point source provided that the Nyquist criterion is fulfilled for the effective spatial frequency cutoff.
For a theoretical diffraction-limited psf centered, sampled by a Shah function, and with energy normalized to one, the discrete peak value M0 at zero position is given by the formula:
where is the ratio of the Shah function sampling frequency to the Nyquist frequency (), and U is the linear central obstruction of the telescope. Let be an AO image sampled by a array of detectors. In order to compare with the previous psf, we first normalize the image energy to 1.0 and defilter from the pixel function (for a square pixel with a filling factor g, one divides the discrete Fourier transform of the image by the function sinc sinc). The defiltering operation is not needed in case of smooth peak or oversampled images (). The peak position in the image is then located from a spline function interpolation. The image is shifted to this position via the Fourier transform and not via a spline interpolation. Using Eq. (A1 (click here)), the Strehl ratio of the AO image is thus given by:
where M is the peak value in the shifted image. Of course, if we use a non point source image, the Sr value will be under-estimated all the more as the source is resolved. Assuming data without noise, the error source in this technique comes from the position estimated from the spline interpolation. For synthetic psfs which have been shifted by fractional pixel values in x and y, the maximum position error is always pixels which yields to a marginal relative error for the Strehl ratio .