Reflection optics, an Einstein addition law for nonparallel boosts, and the geometric algebra of space–time

Abstract

The geometric algebra of space–time is used to write the addition of nonparallel boosts, usually a cumbersome and inelegant process, in a compact form similar to that of the familiar addition law for parallel boosts. This compact expression is related to that used by Vigoureux [ J. Opt. Soc. Am. A8, 1697 ( 1991);J. Opt. Soc. Am. A9, 1313 ( 1992);J. Phys. A26, 385 ( 1993);Am. J. Phys.61, 707 ( 1993)] for combining reflection coefficients for stratified media, thus generalizing and strengthening the links between the formalisms used in special relativity theory and in optics. The comparison is then shown to hold when the addition schemes are performed to an arbitrary number of iterations.