The transfer matrix: a geometrical perspective

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.

50 pages, 24 figures

References in corpus (10)

• Accuracy of transfer matrix approaches for solving the effective mass Schrödinger equation
• Scattering theory of cooling and heating in opto-mechanical systems
• Analytical solution of linear ordinary differential equations by differential transfer matrix method
• Transmission probabilities and the Miller-Good transformation
• Analytical solution to position dependent mass Schrödinger equation
• Metamaterials for Ballistic Electrons
• Generalized composition law from 2x2 matrices
• The interplay of the polar decomposition theorem and the Lorentz group
• Ground-state energy eigenvalue calculation of the quantum mechanical well $V(x)=1/2k{x}^2+\lambda x^4$ via analytical transfer matrix method
• Transfer matrices for piecewise constant potentials

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• Analogue Special and General Relativity by Optical Multilayer Thin Films: The Rindler Space Case
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