Coherent superposition of orthogonal modes

TL;DR

This paper derives relations for the beam properties of superposed orthogonal modes, especially Hermite-Gauss modes, and applies the Courant-Snyder formalism to analyze their propagation through optical systems.

Contribution

It introduces a formalism to analyze the propagation and properties of superposed orthogonal modes using the Courant-Snyder approach, providing new insights into beam quality and mode decomposition.

Findings

Relations for beam moments and quality factors are derived.

The Courant-Snyder formalism is successfully applied to optical mode superpositions.

Hermite-Gauss solutions are interpreted in terms of generating and observable functions.

Abstract

The coherent superposition of orthogonal modes can result in transverse offsets, variations of the Rayleigh length and a reduction of the beam quality factor of the coherent sum of modes in comparison to the incoherent sum. Relations for first and second order moments, the beam quality and the Rayleigh length for the superposition of Hermite-Gauss modes are derived. The Courant-Snyder formalism, which was originally developed in the context of charged particle optics, is applied to propagate an arbitrary coherent sum of orthogonal modes through a lens system. Relations of generating and observable optical functions are highlighted. In the last part of the report the elegant Hermite-Gauss solution is interpreted in terms of generating and observable functions and the solution is decomposed into a sum of standard Hermite-Gauss modes.