Hermite and Laguerre polynomials and their associated Hermite-Gaussian and Laguerre-Gaussian functions - or modes - are widely used in physics and information processing. Schemes to convert Hermite-Gaussian into Laguerre-Gaussian modes by means of appropriate linear canonical integral transformations - or first-order optical systems - are well known. In some recent papers, generalizations of these polynomials and their associated Gaussian functions were proposed; we mention Wünsche's Hermite and Laguerre two-dimensional polynomials and functions, and Abramochkin's Hermite-Laguerre-Gaussian modes. For all these cases we have a generating function that has a Gaussian form.
In this paper we propose a unified approach for the description of all polynomials and functions that are characterized by a Gaussian-type generating function, leading to a general class of sets of Gaussian-type modes; the polynomials and functions mentioned above, then appear as special cases. The general class contains not only the sets of orthonormal Hermite-Gaussian-type modes [1] - with Hermite-Gaussian, Laguerre-Gaussian, and Hermite-Laguerre-Gaussian modes as examples - but includes also mode sets that are not orthonormal. It will be shown that, in the non-orthonormal case, any set of modes has an associated bi-orthonormal partner set from the same class; in the orthonormal case, this bi-orthonormal partner set is then simply identical to the original set.
From the generating function, we will construct derivative relations and recurrence relations between Gaussian-type modes, and from these we will derive a closed form expression for them. Furthermore, it is shown that the evolution of non-orthonormal Gaussian-type modes under linear canonical integral transformations can be described by the same mechanism as used for the evolution of orthonormal Hermite-Gaussian-type modes [1], when, simultaneously, the associated bio-orthonormal modes are taken into account.
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