Geometrical setting for the classification of multilayers

Abstract

We elaborate on the consequences of the factorization of the transfer matrix of any lossless multilayer in terms of three basic matrices of simple interpretation. By considering the bilinear transformation that this transfer matrix induces in the complex plane, we introduce the concept of multilayer transfer function and study its properties in the unit disk. In this geometrical setting, our factorization translates into three actions that can be viewed as the basic components for understanding the multilayer behavior. Additionally, we introduce a simple trace criterion that allows us to classify multilayers into three types with properties closely related to one (and only one) of these three basic matrices. We apply this approach to analyze some practical examples that are typical of these types of matrices.

Photon generation from vacuum in nondegenerate cavities with regular and random periodic displacements of boundaries

“…, n: see, e.g., [45,46] and references therein. We use the method of transfer matrix [6,[47][48][49][50][51][52][53] (see also [54] for historical review and [55][56][57] for the most recent applications).…”

Characterizing the reflectance of periodic layered media

“…[20] one is led to introduce the following criterion: a matrix is of type K when [Tr(M as )] 2 < 4, is of type A when [Tr(M as )] 2 > 4, and finally is of type N when [Tr(M as )] 2 = 4. Although this trace criterion has an elegant geometrical interpretation [13] and coincides with the one giving the stop bands in Eq. (8), let us remark that if a multilayer has a transfer matrix M as of type K, A, or N, one can always find a family of matrices C [also in SU(1,1)] such that…”

Geometrical aspects of first-order optical systems

“…In consequence, given any ray-transfer matrix M one can always find a C such that M C takes one of the following canonical forms [40,41]:…”