Bi-orthonormal sets of Gaussian-type modes

Based on the recently introduced orthonormal Hermite-Gaussian-type modes, a general class of sets of non-orthonormal Gaussian-type modes is introduced, along with their associated bi-orthonormal partner sets. The conditions between these two bi-orthonormal sets of modes have been derived, expressed in terms of their generating functions, and the relations with W\"unsche's Hermite two-dimensional functions and the two-variable Hermite polynomials have been established. A closed-form expression for Gaussian-type modes is derived from their derivative and recurrence relations, which result from the generating function. It is shown that the evolution of non-orthonormal Gaussian-type modes under linear canonical transformations can be described by the same mechanism as used for the evolution of orthonormal Hermite-Gaussian-type modes, when, simultaneously, the associated bio-orthonormal modes are taken into account.