Paraxial meridional ray tracing equations from the unified reflection-refraction law via geometric algebra

TL;DR

This paper derives paraxial meridional ray tracing equations from a unified reflection-refraction law using geometric algebra, explicitly incorporating sign conventions for clarity and generality.

Contribution

It introduces a unified derivation of ray tracing equations from geometric algebra, explicitly embedding sign conventions within the equations.

Findings

Derived finite meridional ray tracing equations.

Established paraxial limits using sign function identities.

Unified reflection-refraction law applied to geometric algebra.

Abstract

We derive the paraxial meridional ray tracing equations from the unified reflection-refraction law using geometric algebra. This unified law states that the normal vector to the interface is a rotation of the incident ray or of the refracted ray or of the reflected ray by an angle equal to the angle of incidence or of refraction. We obtain the finite meridional ray tracing equations by simply equating the arguments of the exponential rotation operators. We then derive the paraxial limits of these equations with the help of sign function identities. We show that by embedding the sign functions in the ray tracing equations, we explicitly declare our chosen sign conventions in symbols and not in prose.