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We present an analytical derivation of the on-axis performance of adaptive optics systems using a given number of guide stars of arbitrary altitude, distributed at arbitrary angular positions in the sky. The expressions of the residual error are given for cases of both continuous and discrete turbulent atmospheric profiles. Assuming Shack-Hartmann wavefront sensing with circular apertures, we demonstrate that the error is formally described by integrals of products of three Bessel functions. We compare the performance of adaptive optics correction when using natural, sodium, or Rayleigh laser guide stars. For small diameter class telescopes (≲5 m), we show that a small number of Rayleigh beacons can provide similar performance to that of a single sodium laser, for a lower overall cost of the instrument. For bigger apertures, using Rayleigh stars may not be such a suitable alternative because of the too severe cone effect that drastically degrades the quality of the correction.

This paper studies the effects on Zernike coefficients of aperture scaling, translation, and rotation, when a given aberrated wavefront is described on the Zernike polynomial basis. It proposes an analytical method for computing the matrix that enables the building of transformed Zernike coefficients from the original ones. The technique is based on the properties of Zernike polynomials and Fourier transform, and, in the case of a full aperture without central obstruction, the coefficients of the matrix are given in terms of integrals of Bessel functions. The integral formulas are exact and do not depend on any specific ordering of the polynomials.