The transfer matrix: A geometrical perspective

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.

The transfer matrix: a geometrical perspective

A wealth of different expressions for the frequency of the Thomas precession (TP) can be found in the literature, with the consequence that this issue has been discussed over a long period of time. It is shown that the correct result was obtained in the works of several authors, which were published more than forty years ago but remained unnoticed against the background of numerous erroneous works. Several TP-related physical paradoxes formulated primarily to disprove the special relativity theory are shown to be fallacious. Different techniques for deriving the correct expression are considered and the reasons for the emergence of the main incorrect expressions for the TP frequency are analyzed.

http://iopscience.iop.org/article/10.1070/PU2006v049n08ABEH005870/pdf

Thomas precession: correct and incorrect solutions

It is shown that the matrix describing a general (transparent or absorbing) multilayer is an element of the group SL(2, C), which is locally isomorphic to the (3+1)-dimensional restricted Lorentz group SO(3, 1). In consequence, a natural identification of the multilayer reflection and transmission coefficients with the parameters of a Lorentz transformation is performed. The correspondence between electric field and space–time variables is discussed as well.

Fully relativisticlike formulation of multilayer optics

Nonlocal approach to scattering in a one-dimensional problem.

A close comparison is presented between special relativity and optics of stratified planar media. This presentation, which may provide some useful tools in optics, can also suggest interesting thoughts and questions both in optics, in special relativity, and in quantum theory.

Ph. Grossel

Papers

A relativistic‐like presentation of optics in stratified planar media