Wave description of geometric phase

TL;DR

This paper presents a wave-based visualization of optical geometric phase, explaining its origin from wave superposition and shifts, contrasting with traditional algebraic and geometric methods.

Contribution

It introduces a wave description of geometric phase, linking it to wave superposition and providing a visual understanding of its physical mechanism.

Findings

Geometric phase arises from wave superposition and maximum shifts.

Derived relationship between wave geometric phase and interferogram phase.

Identified conditions where wave phase matches interferogram phase.

Abstract

Since Pancharatnam's 1956 discovery of optical geometric phase, and Berry's 1984 discovery of geometric phase in quantum systems, researchers analyzing geometric phase have focused almost exclusively on algebraic approaches using the Jones calculus, or on spherical trigonometry approaches using the Poincar\'e sphere. The abstracted mathematics of the former, and the abstracted geometry of the latter, obscure the physical mechanism that generates geometric phase. We show that optical geometric phase derives entirely from the superposition of waves and the resulting shift in the location of the wave maximum. This wave-based model provides a way to visualize how geometric phase arises from relationships between waves, and from the transformations induced by optical elements. We also derive the relationship between the geometric phase of a wave by itself and the phase exhibited by an interferogram, and provide the conditions under which the two match one another.