Bi-orthonormal sets of Gaussian-type modes

M.J. Bastiaans, T. Alieva

Research output: Contribution to journalArticleAcademicpeer-review

2 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)9931-9939
Number of pages9
JournalJournal of Physics A: Mathematical and General
Issue number46
Publication statusPublished - 2005


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