Based on the recently introduced orthonormal Hermite-Gaussian-type modes, a general class of sets of nonorthonormal 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 Gaussiantype modes is derived from their derivative and recurrence relations, which result from the generating function. It is shown that the evolution of non-orthonormal Gaussiantype 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.
|Title of host publication||Proceedings of the 16th ProRISC, Annual Workshop on Circuits, Systems and Signal Processing (ProRISC 2005) 17 - 18 November 2006, Veldhoven, the Netherlands|
|Place of Publication||Utrecht, the Netherlands|
|Publication status||Published - 2005|
|Event||2005 Annual Workshop on Circuits, Systems and Signal Processing (ProRISC 2005) - Veldhoven, Netherlands|
Duration: 17 Nov 2005 → 18 Nov 2005
Conference number: 16
|Conference||2005 Annual Workshop on Circuits, Systems and Signal Processing (ProRISC 2005)|
|Abbreviated title||ProRISC 2005|
|Period||17/11/05 → 18/11/05|
Bastiaans, M. J., & Alieva, T. (2005). A general class of bi-orthonormal sets of Gaussian-type modes. In Proceedings of the 16th ProRISC, Annual Workshop on Circuits, Systems and Signal Processing (ProRISC 2005) 17 - 18 November 2006, Veldhoven, the Netherlands (pp. 589-593). Utrecht, the Netherlands: Technology Foundation.