Lorentz Group in Ray and Polarization Optics

TL;DR

This paper reveals that the Lorentz group, fundamental in relativity, also underpins the mathematical framework of ray and polarization optics, unifying various optical matrices as Lorentz group representations.

Contribution

It demonstrates that key optical matrices like ABCD, Jones, and Mueller matrices are representations of the Lorentz group, bridging relativity and optical sciences.

Findings

ABCD matrices are 2x2 Lorentz group representations in 3D space-time.

Jones matrices are 2x2 Lorentz group representations in 4D space-time.

Mueller matrices and Poincaré sphere are Lorentz group representations.

Abstract

While the Lorentz group serves as the basic language for Einstein's special theory of relativity, it is turning out to be the basic mathematical instrument in optical sciences, particularly in ray optics and polarization optics. The beam transfer matrix, commonly called the $ABCD$ matrix, is shown to be a two-by-two representation of the Lorentz group applicable to the three-dimensional space-time consisting of two space and one time dimensions. The Jones matrix applicable to polarization states turns out to be the two-by-two representations of the Lorentz group applicable to the four-dimensional space-time consisting of three space and one time dimensions. The four-by-four Mueller matrix applicable to the Stokes parameters as well as the Poincar\'e sphere are both shown to be the representations of the Lorentz group.