Statistical model of an adaptive-optics system for phase conjugation with measurement noise

Algorithms for reconstructing the wavefront and for cancelling phase distortions of the field of an adaptive-optics phase-conjugating system have been developed. It is assumed that there is a measurement noise and that there are errors in the control by the phase corrector. A statistical approach based on calculus of variations and matrix calculus is used. A priori information about the statistical characteristics of the measured wavefront, the measurement noise, and the control errors is used to calculate the phase distribution to be reconstructed and the correcting phase distribution. In addition, equations are derived for orthogonal modes of the Karhunen–Loéve type which minimize the error in the correction of the phase distortions of the field for a given number of modes. These orthogonal modes are realized without approximation error by a phase corrector with prespecified response functions. Also presented and discussed are the results of calculations of the residual reproduction and correction errors for the cases of 3, 7, and 19 channels of measurement by a Hartmann receiver and for the same numbers of actuators, as a function of the amplitude of the measurement noise and the control errors.