Bandwidth considerations for tracking through turbulence

Abstract

The bandwidth requirements for tracking through turbulence have been studied for the case in which a closed-loop transfer function of the form H(f) = (1 + if/f3dB)−1 is used. The results illustrate that the one-axis one-sigma (rms) jitter σθ is given by the expression σθ = (fT/f3dB)(λ/D), where λ is the wavelength of light used, D is the diameter of the tracking aperture, and fT is the fundamental turbulence-tracking frequency that is determined so that, when the servo bandwidth f3dB is equal to fT, σθ is equal to the diffraction angle λ/D. In practice a tracking system may measure one of various kinds of tilt. As a consequence both G tilt (obtained from a centroid measurement) and Z tilt (the direction that is defined by the normal to the plane that minimizes the mean-square wave-front distortion) have been evaluated in detail. The fundamental turbulence-tracking frequencies fTG and fTZ corresponding to the G tilt and the Z tilt, respectively, are found to be almost identical and are given by the expressions fTG = 0.331D−1/6λ−1[∫ dzCn2(z)V2(z)]1/2 and fTz = 0.368D−1/6λ−1[∫ dzCn2(z)V2(z)]1/2, where z is the range coordinate, Cn2() is the refractive-index structure function, and V() is the wind-velocity profile. For turbulence models that are applied to systems of interest the fundamental turbulence-tracking frequency that is defined by these expressions is about one ninth of fG, the Greenwood frequency associated with higher-order wave-front distortion. This illustrates the important point that the bandwidth that is necessary to control the turbulence-induced tilt is significantly less than the bandwidth that is necessary to control the turbulence-induced higher-order wave-front distortion.