Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen–Loève functions

Abstract

Atmospheric Karhunen–Loeve functions are the optimal set of basis functions for modal atmospheric compensation. They are seldom applied in practice on account of their nonanalytical nature. A pseudoanalytical set of these functions is constructed with a least-squares-fitting procedure. To produce an analytical expression for the optical resolution of modal atmospheric compensation, a modified form of structure functions is used and applied to the compensated wave front. This results in analytical residual phase structure functions for Zernike polynomials and pseudoanalytical residual phase structure functions for Karhunen–Loeve functions. With these structure functions it is found that the modulation transfer function (MTF) after modal compensation is the product of the telescope MTF and the uncompensated atmospheric MTF under the assumption of isotropic compensating phase. Comparison with an accurate numerical method shows that the approximated analytical method developed is much faster and gives reasonably accurate results, especially for high-order compensations.