Physical meaning of the radial index of Laguerre-Gauss beams

TL;DR

This paper provides a physical interpretation of the radial index of Laguerre-Gauss beams by introducing the concept of intrinsic hyperbolic momentum charge, enhancing understanding of these optical modes.

Contribution

It develops a differential operator formalism for radial modes and defines their physical meaning as intrinsic hyperbolic momentum charge.

Findings

Radial index linked to intrinsic hyperbolic momentum charge

Formalism applicable in position and momentum representations

Clarifies physical significance of radial modes

Abstract

The Laguerre-Gauss modes are a class of fundamental and well-studied optical fields. These stable, shape-invariant photons - exhibiting circular-cylindrical symmetry - are familiar from laser optics, micro-mechanical manipulation, quantum optics, communication, and foundational studies in both classical optics and quantum physics. They are characterized, chiefly, by two modes numbers: the azimuthal index indicating the orbital angular momentum of the beam - which itself has spawned a burgeoning and vibrant sub-field - and the radial index, which up until recently, has largely been ignored. In this manuscript we develop a differential operator formalism for dealing with the radial modes in both the position and momentum representations, and - more importantly - give for the first time the meaning of this quantum number in terms of a well-defined physical parameter: the "intrinsic hyperbolic momentum charge".