A fitting formula for the effective error of angular anisoplanatism in adaptive optics

In adaptive optics (AO) systems, when the paths of the beacon source and the object source separate at an angle theta, a so-called angular anisoplanatism is produced. The angular anisoplanatic error described by isoplanatic angle (theta(0)) overestimates the degradation in AO correction, for it includes the piston component which has no effect on optical quality. So, it is necessary to study the effective error of angular anisoplanatism which has actual influence on AO performance. Define the variance reduction factor (V-rf) as the effective anisoplanatic error in which the piston is removed, divided by the anisoplantic error described by theta(0). Through theoretical derivation and numerical evaluation, it is found that the V-rf is primarily influenced by a normalized coefficient b (b = theta L/D, where L is the path length in turbulence and D is the receiver aperture diameter), and mildly influenced by the turbulence distribution. The major influence factor for V-rf with fixed b is turbulence strength in the high altitude while the influence from turbulence near the ground is negligible. Finally, a fitting formula of the V-rf is proposed, which can be used easily in practical applications to evaluate the performance of AO systems.