One analytic form for four branches of the ABCD matrix

TL;DR

This paper introduces a unified analytic form for the four branches of the optical ABCD matrix, enabling easier analysis of complex optical systems by consolidating different matrix forms into a single framework.

Contribution

It presents a novel unified matrix form that combines the four Wigner matrices into one, applicable to various optical systems such as laser cavities and multilayer optics.

Findings

Unified matrix form simplifies optical system analysis

Applicable to optical activities, laser cavities, multilayer optics

Enhances understanding of ABCD matrix transformations

Abstract

It is not always possible to diagonalize the optical $ABCD$ matrix, but it can be brought into one of the four Wigner matrices by a similarity transformation. It is shown that the four Wigner matrices can be combined into one matrix with four branches. This result is illustrated in terms of optical activities, laser cavities, and multilayer optics.