Temporal Characteristics and Modelling of Atmospherically-Distorted Wavefront Slopes

Abstract

The wavefronts distorted by a Kolmogorov turbulent atmosphere are fractal surfaces [1] and have the properties of fractional Brownian motion (FBM) with a self-similarity parameter of 5/6. Such a FBM process is nonstationary and has a power-law spectrum with spectral index -8/3. In an adaptive optics system a Shack-Hartmann (SH) wavefront sensor (WFS) delivers the time series of wavefront (WF) slopes measured at each lenslet subaperture. While the FBM wavefront exhibits persistence, and consequently has predictability, the derivatives or slopes of this FBM process are antipersistent [2] with spectral index -2/3. This means that the WF slopes would have limited predictability at least by conventional mean-square prediction methods. Thus the adaptive optics (AO) control strategy that treats WF slopes as an unpredictable random walk process would seem justified. Within a closed loop system the difference between the incident wavefront and the correction by the deformable mirror is equivalent to a differentiation or increments process because of the loop delay, in which case the wavefront sensor gives essentially the second derivative. The second derivative process of FBM is also antipersistent.