Anatomy of the slope discrepancy structure function: characterization of turbulence

Abstract

A new approach, for estimating <i>r</i>₀ and the Greenwood frequency from wavefront sensor measurements, is introduced. Unlike other techniques, which use phase statistics calculated from reconstructed wavefront sensor measurements, this approach makes use of the slope discrepancy, the component of the measurement that is not reconstructed by a least squares reconstructor. It is shown that the <i>r</i>₀ estimator based on the slope discrepancy is a high degree-of-freedom estimator with a much smaller estimation variance than estimates based on the phase variance. An new temporal atmospheric sample rate, <i>f</i>₀(&epsilon;²), is introduced in this work. This quantity is calculated from the slope discrepancy structure function and sets the minimum required sampling rate to maintain the residual wavefront error below &epsilon;²/<i>K</i> for a system dependent gain, <i>K</i>. For Kolmogorov turbulence statistics it is shown that the Greenwood frequency is related to <i>f</i>₀ by <i>f</i>_G=<i>f</i>₀(<i>K</i>(2&pi;/20)²)/40. In non-Kolmogorov turbulence, <i>f</i>₀ continues to be a legitimate specification of sampling requirements. These results are all illustrated with wave optics simulations.