Power series evaluation of covariances for turbulence-induced phase distortions including outer scale and servo lag effects

Power series formulas are developed to efficiently compute the covariance of the integrated turbulence-induced phase distortions along a pair of ray paths through the atmosphere from two points in a telescope aperture to a pair of sources at finite or infinite range. These covariances may be used to evaluate and optimize the predicted performance of adaptive optical (AO) systems. The power series formulas apply when one or both of the phase distortions is temporally filtered by the closed-loop impulse response function of the AO control loop, thereby allowing the effects of a finite servo bandwidth to be included in AO system performance modeling without the introduction of additional numerical integrations. Results are presented for the Kolomogorov turbulence spectrum with an infinite outer scale, as well as for the case of a finite outer scale with the von Karman turbulence spectrum. Amplitude scintillation effects are neglected. The Taylor, or frozen flow, hypothesis is used to model the temporal behavior of the turbulence, by using a fixed windspeed profile w(z) and a random wind direction profile theta(w)(z) for which the mean values of the quantities cos[k theta(w)(z)] and sin[k theta(w)(z)] can be computed. The resulting formulas for the covariances are weighted integrals of the refractive-index structure constant C-n(2)(z), where the weighting functions are power series in from one to three indices depending on the choices made regarding the atmospheric turbulence spectrum and the direction of the wind. The integral with respect to z may also be evaluated analytically, provided that (i) the C-n(2)(z) profile is a sum of terms of the form z(P) exp(-cz) and (ii) the phase distortion profiles are not temporally filtered. (C) 1999 Optical Society of America [S0740-3232(99)03603 -0].