Synthesis of Taylor Phase Screens with Karhunen-Loeve Basis Functions

Phase screens above a telescope pupil represent the variation of the phase of the electromagnetic field induced by atmospheric turbulence. Instances drawn from such statistics are represented by a vector of random phase amplitudes which are coefficients of a linear superposition of two-dimensional basis functions across the pupil. This work shortly reviews Fried's analysis of this modal decomposition for the case of Kolmogorov statistics of the phase covariance as a function of separation in the pupil plane. We focus on the numerical synthesis of phase screens. The statistically independent modes are transformed into the eigen-modes of a gradient matrix as time-dependence is introduced such that on short time scales the instances of the phase screens are rigidly shifted into a direction imposed by some wind velocity - known as the Taylor frozen screen approximation. This simple technique factorizes spatial and temporal variables and aims at binding the time dependence of the phase screens to the few expansion coefficients of the basis functions that obey a stochastic time-dependent differential equation.