ABCD Matrices as Similarity Transformations of Wigner Matrices and Periodic Systems in Optics

TL;DR

This paper reveals that ABCD matrices in optics can be expressed as similarity transformations of Wigner matrices, facilitating the analysis of periodic optical systems like laser cavities and multilayer structures.

Contribution

It demonstrates that any ABCD matrix can be represented as a similarity transformation of Wigner matrices, enabling simplified computation of scattering in periodic optical systems.

Findings

ABCD matrices can be transformed into Wigner matrices via similarity transformations.

The method allows calculation of the transfer matrix for multiple cycles in periodic systems.

Application to laser cavities and multilayer optics shows practical utility.

Abstract

The beam transfer matrix, often called the $ABCD$ matrix, is a two-by-two matrix with unit determinant, and with three independent parameters. It is noted that this matrix cannot always be diagonalized. It can however be brought by rotation to a matrix with equal diagonal elements. This equi-diagonal matrix can then be squeeze-transformed to a rotation, to a squeeze, or to one of the two shear matrices. It is noted that these one-parameter matrices constitute the basic elements of the Wigner's little group for space-time symmetries of elementary particles. Thus every $ABCD$ matrix can be written as a similarity transformation of one of the Wigner matrices, while the transformation matrix is a rotation preceded by a squeeze. This mathematical property enables us to compute scattering processes in periodic systems. Laser cavities and multilayer optics are discussed in detail. For both cases, it is shown possible to write the one-cycle transfer matrix as a similarity transformation of one of the Wigner matrices. It is thus possible to calculate the $ABCD$ matrix for an arbitrary number of cycles.