Skew ray tracing in a step-index optical fiber using Geometric Algebra

TL;DR

This paper employs Geometric Algebra to analytically trace skew rays in a step-index optical fiber, deriving invariants and a generalized numerical aperture formula for complex ray paths.

Contribution

It introduces a novel application of Geometric Algebra for skew ray tracing in optical fibers, deriving invariants and a generalized numerical aperture formula.

Findings

Rays follow a polygonal helical path with three key invariants.

Derived a generalized numerical aperture formula for skew rays.

Reproduced known formulas as special cases.

Abstract

We used Geometric Algebra to compute the paths of skew rays in a cylindrical, step-index multimode optical fiber. To do this, we used the vector addition form for the law of propagation, the exponential of an imaginary vector form for the law of refraction, and the juxtaposed vector product form for the law of reflection. In particular, the exponential forms of the vector rotations enables us to take advantage of the addition or subtraction of exponential arguments of two rotated vectors in the derivation of the ray tracing invariants in cylindrical and spherical coordinates. We showed that the light rays inside the optical fiber trace a polygonal helical path characterized by three invariants that relate successive reflections inside the fiber: the ray path distance, the difference in axial distances, and the difference in the azimuthal angles. We also rederived the known generalized formula for the numerical aperture for skew rays, which simplifies to the standard form for meridional rays.