Analytic partial wave expansion and integral representation of Bessel beam

TL;DR

This paper develops partial wave expansion and integral representation techniques for Bessel beams, including their wavepacket expansion and relationships with Legendre polynomials, applicable in free space and dispersive media.

Contribution

It introduces a novel partial wave expansion and integral representation for Bessel beams, linking these to Legendre polynomial products and providing a unified framework.

Findings

Derived the partial wave expansion of Bessel beams.

Established the integral representation and its series form.

Connected the expansion with Legendre polynomial products.

Abstract

This paper describes the partial wave expansion and integral representation of Bessel beams in free space and in the presence of dispersion. The expansion of the Bessel beam wavepacket with constant spectrum is obtained as well. Furthermore, the sum of a triple Legendre polynomial product of same order but different argument follows naturally from the partial wave expansion. The integration of all Bessel beams over all conical angles is shown to have a simple series representation, which confirms the equivalence between the results for both expansion and integral representation.