The Forgotten Quantum Number: A short note on the radial modes of Laguerre-Gauss beams

TL;DR

This paper highlights the importance of the often overlooked radial quantum number in Laguerre-Gauss beams, deriving its formalism and exploring its implications for beam stability and quantum communication.

Contribution

It provides a new differential operator formalism for the radial quantum number and discusses its physical significance and potential applications.

Findings

Derived the formalism for the radial quantum number

Connected the radial number to beam stability and Gouy phase

Suggested applications in quantum communication

Abstract

The orbital angular momentum quantum number of Laguerre-Gauss beams has received an explosively increasing amount of attention over the past twenty years. However, often overlooked is the so-called radial number of these beams. We present a derivation of the differential operator formalism of this "forgotten" quantum number. We then draw some connections between this new formalism and the effect the radial number has on beam stability with possible application to quantum communication. We also briefly outline how the radial number is tied to other physical aspects of the beam (such as the Gouy phase, and radial confinement). These do not necessarily constitute finished results, but are instead meant to stimulate discussion of this interesting and often overlooked physical parameter.