Optics in Stratified and Anisotropic Media: 4×4-Matrix Formulation

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Abstract
A 4×4-matrix technique was recently introduced by Teitler and Henvis for finding propagation and reflection by stratified anisotropic media. It is more general than the 2×2-matrix technique developed by Jones and by Abelès and is applicable to problems involving media of low optical symmetry. A little later, we developed a 4×4 differential-matrix technique in order to solve the problem of reflection and transmission by cholesteric liquid crystals and other liquid crystals with continuously varying but planar ordering. Our technique is mathematically equivalent to that of Teitler and Henvis, but we used a somewhat different approach. We start with a 6×6-matrix representation of Maxwell’s equations that can include Faraday rotation and optical activity. From this, we derive expressions for 16 differential-matrix elements so that a wide variety of specific problems can be attacked without repeating a large amount of tedious algebra. The 4×4-matrix technique is particularly well suited for solving complicated reflection and transmission problems on a computer. It also serves as an illuminating alternative way to rederive closed solutions to a number of less-complicated classical problems. Teitler and Henvis described a method of solving some of these problems, briefly in their paper. We give solutions to several such problems and add a solution to the Oseen–DeVries optical model of a cholesteric liquid crystal, to illustrate the power and simplicity of the 4×4-matrix technique.