Geometric phases in astigmatic optical modes of arbitrary order

TL;DR

This paper explores the geometric phases in higher-order astigmatic optical modes, generalizing the Gouy phase, and discusses their geometric and quantum-mechanical implications.

Contribution

It introduces a geometric framework for understanding phases in arbitrary-order optical modes, extending the Gouy phase concept and linking it to quantum wave packet transformations.

Findings

Generalized Gouy phase derived for arbitrary-order modes

Connection between geometric phase and Aharonov-Bohm effect

Application to quantum wave packet solutions

Abstract

The transverse spatial structure of a paraxial beam of light is fully characterized by a set of parameters that vary only slowly under free propagation. They specify bosonic ladder operators that connect modes of different order, in analogy to the ladder operators connecting harmonic-oscillator wave functions. The parameter spaces underlying sets of higher-order modes are isomorphic to the parameter space of the ladder operators. We study the geometry of this space and the geometric phase that arises from it. This phase constitutes the ultimate generalization of the Gouy phase in paraxial wave optics. It reduces to the ordinary Gouy phase and the geometric phase of non-astigmatic optical modes with orbital angular momentum states in limiting cases. We briefly discuss the well-known analogy between geometric phases and the Aharonov-Bohm effect, which provides some complementary insights in the geometric nature and origin of the generalized Gouy phase shift. Our method also applies to the quantum-mechanical description of wave packets. It allows for obtaining complete sets of normalized solutions of the Schr\"odinger equation. Cyclic transformations of such wave packets give rise to a phase shift, which has a geometric interpretation in terms of the other degrees of freedom involved.