Non-orthonormal Gaussian-type modes and their bi-orthonormal partner sets

M.J. Bastiaans, T. Alieva

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Abstract

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 - 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, when, simultaneously, the associated bio-orthonormal modes are taken into account.
Original languageEnglish
Title of host publicationProc. ICOL 2005, the International Conference on Optics and Optoelectronics, Dehradun, India, Paper PP.OIP.2
Place of PublicationDehradun, India
PublisherInstrument Research & Development Establishment
PagesPP.OIP.2.1-6
Publication statusPublished - 2005

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    Bastiaans, M. J., & Alieva, T. (2005). Non-orthonormal Gaussian-type modes and their bi-orthonormal partner sets. In Proc. ICOL 2005, the International Conference on Optics and Optoelectronics, Dehradun, India, Paper PP.OIP.2 (pp. PP.OIP.2.1-6). Instrument Research & Development Establishment.