Interpretación del factor geométrico de la fase en una fibra óptica, a partir del Teorema de Gauss-Bonnet

Abstract

The Gauss–Bonnet Theorem (GBT) is one of the deepest and most significant results in the theory of surfaces. Its most important formulation relates the integral (or average) of the Gaussian curvature over a surface to a property known as the Euler characteristic, which is purely topological, meaning that it remains invariant under any continuous deformation of the surface. Thus, the GBT establishes a fundamental connection between geometric and topological properties.

In the context of physics, particularly in optics, the GBT emerges through phenomena that, at first glance, appear to be unrelated to geometry. It is precisely these "hidden" connections among optics, differential geometry, and topology that make the geometric phase a paradigmatic example of the interdisciplinary nature of these three fields. This work illustrates the relationship between the GBT and the rotation of the plane of polarization of light propagating through a helical optical fiber as a particular example of the geometric phase. In this case, the polarization vector traces a closed loop on the surface of the unit sphere, in accordance with the Theory of Parallel Transport (TPT).

As a further perspective of this work, the polarization angle was measured experimentally. Specifically, the geometric phase factor of a helical optical fiber was determined by applying the Gauss–Bonnet Theorem together with the Theory of Parallel Transport to the polarization vector as it propagated along the optical fiber.