Reflection and refraction of skew rays in optical fibers: A geometric algebra approach

Abstract

This paper presents a geometric algebra formalism for the reflection and refraction of skew rays in optical fibers. Ray directions are expressed in terms of spherical coordinates in exponential form. These expressions are readily converted into cylindrical and rectangular coordinates by expanding the exponentials using Euler's theorem in Complex Analysis. The exponential form of the laws of refection and refraction in geometric algebra simplify the derivation of the relations between the polar and azimuthal angles of the incident, refracted, and reflected rays.