The geometric phase of rotations and 3D coordinate transformations

TL;DR

This paper extends the wave-based explanation of geometric phase to circular polarization and rotating systems, demonstrating how phase shifts can be derived directly from electromagnetic vectors without advanced mathematics.

Contribution

It introduces a wave-based approach to geometric phase for circular polarization and rotating systems, avoiding differential geometry and solid angle calculations.

Findings

Phase shift can be derived directly from electromagnetic wave vectors.

The approach applies to systems like three-fold mirrors and helical fibers.

The method simplifies understanding of geometric phase in optical systems.

Abstract

The wave description of geometric phase uses the superposition of light waves to explain the geometric phase's origin. While our previous work focused on a basis of linearly polarized waves, here we show that the same concepts can be applied to circularly polarized waves, and to any case in which a rotator is itself subjected to rotation. As with a linear polarization basis, we show that the addition of two vectors (rotators) with different orientations and magnitudes causes the orientation of the resulting vector to shift towards the component vector of greater magnitude, i.e. it introduces a geometric phase. We illustrate this approach with two classic examples of the geometric phase of rotations in space: a system of three fold mirrors, and the helical coiled fiber. In both cases we show that it is possible to derive the phase shift directly from the electromagnetic wave vector without needing to resort to mathematical abstractions such as differential geometry, or calculating solid angles in the space of directions.