Bi-orthonormal sets of Gaussian-type modes

M.J. Bastiaans, T. Alieva

Research output: Contribution to journalArticleAcademicpeer-review

2 Citations (Scopus)

Abstract

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
Volume38
Issue number46
DOIs
Publication statusPublished - 2005

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Orthonormal
Hermite
Linear transformations
Generating Function
Polynomials
Canonical Transformation
Hermite Polynomials
Derivatives
Recurrence relation
Closed-form
Derivative
polynomials

Cite this

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abstract = "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.",
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Bi-orthonormal sets of Gaussian-type modes. / Bastiaans, M.J.; Alieva, T.

In: Journal of Physics A: Mathematical and General, Vol. 38, No. 46, 2005, p. 9931-9939.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Bi-orthonormal sets of Gaussian-type modes

AU - Bastiaans, M.J.

AU - Alieva, T.

PY - 2005

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AB - 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.

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