Understanding multilayers from a geometrical viewpoint

Abstract

We reelaborate on the basic properties of lossless multilayers. We show that the transfer matrices for these multilayers have essentially the same algebraic properties as the Lorentz group SO(2, 1) in a (2 + 1)-dimensional space-time as well as the group SL(2, R) underlying the structure of the ABCD law in geometrical optics. By resorting to the Iwasawa decomposition, we represent the action of any multilayer as the product of three matrices of simple interpretation. This group-theoretical structure allows us to introduce bilinear transformations in the complex plane. The concept of multilayer transfer function naturally emerges, and its corresponding properties in the unit disk are studied. We show that the Iwasawa decomposition is reflected at this geometrical level in three simple actions that can be considered the basic pieces for a deeper understanding of the multilayer behavior. We use the method to analyze in detail a simple practical example.

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“…(12). This provides a full characterization of the reflectance of any periodic system, as we shall demonstrate in next Section.…”

“…In Ref. [12] we have provided simple examples of realistic systems that accomplish these requirements.…”

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“…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]:…”

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“…To choose the remaining two real quantities, we note 18,19 that M allows one to define a mapping in the complex plane, ͑i͒ The flux of a state written as in Eq. ͑1͒, with components (a,b), is jϭ͉a͉ 2 Ϫ͉b͉ 2 and is conserved by the action of M. Therefore, waves with net flux to the right, jϾ0, correspond to points ͉z͉Ͻ1 which are mapped onto points ͉w͉ Ͻ1.…”