Both laser guide stars and natural guide stars may be used to sample atmospheric turbulence for adaptive optics. The phase representing the turbulence can be expanded as Zernike polynomials across a circular aperture. Assuming a Kolmogorov phase spectrum and C2n profile the error induced in using off-axis source to correct for atmospheric phase can be calculated for Zernike modes. Small-angle approximations of analytic results are presented and compared with the analytic expressions for model C2n profiles. A numerical simulation of the analagous errors for the models using an on-axis sodium layer laser guide star is also presented.
Steven M. Stahl, Steven M. Stahl,
"Zernike decomposition of anisoplanatism for laser beacons and natural guide stars", Proc. SPIE 2201, Adaptive Optics in Astronomy, (31 May 1994); doi: 10.1117/12.176099; https://doi.org/10.1117/12.176099