Bohm approach to the Gouy phase shift

TL;DR

This paper uses the Bohm approach and Ermakov-Lewis techniques to relate the Gouy phase shift in Gaussian beams to a quantum invariant and an effective index of refraction, offering a new perspective on beam focusing.

Contribution

It introduces a quantum mechanical invariant and links the Gouy phase to an effective Bohm index of refraction within the paraxial wave framework.

Findings

Gouy phase related to phase form in Gaussian propagation

Quantum invariant is explicitly time-dependent

Effective Bohm index creates a GRIN medium for focusing

Abstract

By adapting the Madelung-Bohm formalism to paraxial wave propagation we show, by using Ermakov-Lewis techniques, that the Gouy phase is related to the form of the phase chosen in order to produce a Gaussian function as a propagated field. For this, we introduce a quantum mechanical invariant, that it is explicitly time dependent despite the fact that the Hamiltonian is itself time-independent. We finally show that the effective Bohm {\it index of refraction} generates a GRIN medium that produces the focusing needed for the Gouy phase.