Slide-rule-like property of Wigner’s little groups and cyclic S matrices for multilayer optics

It is noted that 2×2 $“S”$ matrices in multilayer optics can be represented by the Sp(2) group whose algebraic property is the same as the group of Lorentz transformations applicable to two spacelike and one timelike dimensions. It is also noted that Wigner’s little groups have a slide-rule-like property that allows us to perform multiplications by additions. It is shown that these two mathematical properties lead to a cyclic representation of the S matrix for multilayer optics, as in the case of $\mathrm{ABCD}$ matrices for laser cavities. It is therefore possible to write the N-layer S matrix as a multiplication of the N single-layer S matrices resulting in the same mathematical expression with one of the parameters multiplied by N. In addition, it is noted, as in the case of lens optics, that multilayer optics can serve as an analog computer for the contraction of Wigner’s little groups for internal space-time symmetries of relativistic particles.