Wave-front temporal spectra in high-resolution imaging through turbulence

Abstract

A general formulation is given for the derivation of theoretical temporal power spectra of quantities related to turbulent wave-front phase. These temporal power spectra and their asymptotic power laws and cutoff frequencies are presented for various quantities of interest in the field of interferometry (differential piston), wave-front sensing (Shack–Hartmann and curvature sensor), adaptive optics (Zernike polynomials), and seeing monitoring (differential angle of arrival). We show that the differential piston spectrum has two cutoff frequencies and exhibits a very steep decrease at high frequencies. The curvature sensor is shown to be much less sensitive than the Shack–Hartmann sensor to the low temporal frequencies. A study of the Zernike temporal power spectra shows that their cutoff frequencies increase with the polynomial radial degree. Both single-layer and multilayer plane and spherical waves are considered. The effect of wind direction is also taken into account. We point out the influence of the cone effect on the temporal power spectra when Rayleigh or sodium laser guide stars are used for wave-front sensing. The cone effect results in a temporal decorrelation between natural and laser guide star wave fronts. Finally, we demonstrate that in adaptive optics systems low-order modes require higher servoloop bandwidths than do high-order modes in order for the residual variance to be balanced between the corrected modes. The same conclusion applies to fringe tracking in large telescope interferometers equipped with adaptive optics systems.