About the Use of Real Dirac Matrices in 2-dimensional Coupled Linear Optics

TL;DR

This paper introduces a novel approach to two-dimensional coupled linear optics using real Dirac matrices, providing a comprehensive framework for symplectic transformations, decoupling methods, and analogies with relativistic physics.

Contribution

It presents a systematic use of real Dirac matrices for analyzing coupled linear optics, including a new decoupling method based on symplectic transformations and analogies with Minkowski space-time.

Findings

Unified treatment of symplectic transformations

Decoupling method for coupled oscillators

Analogies with relativistic physics and Dirac spinors

Abstract

The Courant-Snyder theory for two-dimensional coupled linear optics is presented, based on the systematic use of the real representation of the Dirac matrices. Since any real $4\times 4$-matrix can be expressed as a linear combination of these matrices, the presented Ansatz allows for a comprehensive and complete treatment of two-dim. linear coupling. A survey of symplectic transformations in two dimensions is presented. A subset of these transformations is shown to be identical to rotations and Lorentz boosts in Minkowski space-time. The transformation properties of the classical state vector are formulated and found to be analog to those of a Dirac spinor. The equations of motion for a relativistic charged particle - the Lorentz force equations - are shown to be isomorph to envelope equations of two-dimensional linear coupled optics. A universal and straightforward method to decouple two-dimensional harmonical oscillators with constant coefficients by symplectic transformations is presented, which is based on this isomorphism. The method yields the eigenvalues (i.e. tunes) and eigenvectors and can be applied to a one-turn transfer matrix or directly to the coefficient matrix of the linear differential equation.