Propagation-invariant beams with quantum pendulum spectra: from Bessel beams to Gaussian beam-beams

TL;DR

This paper introduces a new class of propagation-invariant light beams with spectra based on quantum pendulum eigenfunctions, unifying Bessel and Gaussian beams through a parameter-dependent spectrum.

Contribution

It presents a novel family of beams with spectra derived from quantum pendulum eigenfunctions, bridging Bessel and Gaussian beams, and explores their mathematical properties and physical implications.

Findings

Pendulum beams are eigenfunctions of an operator interpolating angular and linear momentum.

They generalize Bessel beams and resemble Gaussian wavepackets at large parameters.

Connections with Mathieu beams and insights into paraxial approximation are established.

Abstract

We describe a new class of propagation-invariant light beams with Fourier transform given by an eigenfunction of the quantum mechanical pendulum. These beams, whose spectra (restricted to a circle) are doubly-periodic Mathieu functions in azimuth, depend on a field strength parameter. When the parameter is zero, pendulum beams are Bessel beams, and as the parameter approaches infinity, they resemble transversely propagating one-dimensional Gaussian wavepackets (Gaussian beam-beams). Pendulum beams are the eigenfunctions of an operator which interpolates between the squared angular momentum operator and the linear momentum operator. The analysis reveals connections with Mathieu beams, and insight into the paraxial approximation.