A set of orthogonal functions, convenient for the aberration analysis of the quasi-gaussian beams such as the laser TEM<sub>00</sub> mode is derived. It is a set of hypergeometric functions. We can expand the aberration function of a quasi-gaussian beam in terms of it, and can calculate the diffraction patterns of aberrations. This set of functions has some characteristics of orthogonality to each other and relations to the Bessel functions. It bears a resemblance to the Zernike’s circle polynomials, but has some different characteristics. It is useful in holography, optical communications, and optical measurement.
Akira Arimoto, "Aberration expansion and evaluation of the quasi-gaussian beam by a set of orthogonal functions," J. Opt. Soc. Am. 64, 850-856 (1974)