Normalization of the Mathieu-Gauss optical beams

Abstract

A series scheme is discussed for the determination of the normalization constants of the even and odd Mathieu-Gauss (MG) optical beams. We apply a suitable expansion in terms of Bessel-Gauss (BG) beams and also answer the question of how many BG beams should be used to synthesize a MG beam within a tolerance. The structure of the normalization factors ensures that MG beams will always be normalized independently of the particular normalization adopted for the Mathieu functions. In this scheme, the normalization constants are expressed as rapidly convergent series that can be calculated to an arbitrary precision.

Quantifying momenta through the Fourier transform

“…As an example, the mean value of the orbital angular momentum of Elliptic-cylindrical waves, so called Mathieu waves, has been presented. With further efforts the schema can be extended from these ideal scalar wave fields to real world beams; by using the orthonormal bases given by Gauss, Bessel-, Mathieu- [31] and Weber-Gauss [32] beams it may be possible to implement computational procedures to calculate momenta of captured experimental images. An extension of these procedures to non-paraxial optical vector fields is plausible.…”

Normalization of optical Weber waves and Weber-Gauss beams

“…Normalization at the z = 0 plane yields complex integrals wich can be avoided using a normalization scheme based on the Bessel decomposition [18]. Under this scheme, normalization for Weber beams yield…”

“…In this work, the link between Weber waves and the well-known Bessel and Mathieu waves is provided by deriving the integral relations between parabolic, circular and elliptical waves following the phase space method proposed by Boyer, Kalnis and Miller [4,8]. With the Bessel and Mathieu wave decompositions of Weber waves, we present the normalization of Weber beams under a series scheme [18]. The efficiency to approximate Weber-Gauss beams as a finite superposition of Bessel-Gauss beams is also given.…”

Helical Mathieu and parabolic localized pulses

“…where A n ͑m͒ and B n ͑m͒ are the expansion coefficients whose explicit expressions can be found in [27,28]. Replacing into Eq.…”