It can be written as an infinite series involving the one dimensional probability densities and Hermite polynomials of x and y (see the link to Slepian).
Accurate description of such a beam involves expansion over some complete, orthogonal set of functions (over two-dimensions) such as Hermite polynomials or the Ince polynomials.
orthogonal polynomials | Hermite polynomials | Tristan l'Hermite | Macdonald polynomials | François Tristan l'Hermite | Chihara–Ismail polynomials | Charles Hermite | Brenke–Chihara polynomials | Wilson polynomials | Sister Celine's polynomials | Koornwinder polynomials | Jean-Marthe-Adrien L'Hermite | Jacques l'Hermite | Jacobi polynomials | Hermite | Chihara-Ismail polynomials | Charlier polynomials | Boas–Buck polynomials | Angelescu polynomials | Al-Salam–Ismail polynomials |