In optics, Zernike orthogonal polynomials are used as a basis set for the expansion of wavefront phases. Due to the atmospheric turbulence-induced stochastic nature of the underlying process involved, the spatial-temporal correlation functions of the Zernike polynomial phase expansion coefficients must be evaluated if adaptive optics is to be employed. In this paper, these correlation functions are developed using a layered atmospheric model and calculations for the first few low-order Zernike modes are performed. Using these correlation functions, an underlying linear dynamical system is identified. Within an acceptable error bound, the correlation functions of this system are representative of the calculated correlation functions. This syste, in turn, provides the basis for the employment of advanced control and estimation concepts in an adaptive optics system that is being developed for the ABL.
Michael W. Oppenheimer,
"Algorithm development for online control of the ABL", Proc. SPIE 3706, Airborne Laser Advanced Technology II, (3 August 1999); doi: 10.1117/12.356975; https://doi.org/10.1117/12.356975