The transfer matrix: a geometrical perspective

TL;DR

This paper offers a geometric perspective on the transfer matrix method for analyzing wave propagation in one-dimensional lossless systems, linking algebraic properties to spacetime symmetries and classifying systems via a trace criterion.

Contribution

It introduces a geometric interpretation of transfer matrices as mappings on the unit disk and applies this to classify and analyze various physical systems.

Findings

Transfer matrices relate to the Lorentz group and unimodular matrices.

A trace criterion classifies systems into three geometrical types.

The approach provides an alternative framework for periodic and quasiperiodic systems.

Abstract

We present a comprehensive and self-contained discussion of the use of the transfer matrix to study propagation in one-dimensional lossless systems, including a variety of examples, such as superlattices, photonic crystals, and optical resonators. In all these cases, the transfer matrix has the same algebraic properties as the Lorentz group in a (2+1)-dimensional spacetime, as well as the group of unimodular real matrices underlying the structure of the abcd law, which explains many subtle details. We elaborate on the geometrical interpretation of the transfer-matrix action as a mapping on the unit disk and apply a simple trace criterion to classify the systems into three types with very different geometrical and physical properties. This approach is applied to some practical examples and, in particular, an alternative framework to deal with periodic (and quasiperiodic) systems is proposed.