Simple trace criterion for classification of multilayers

Abstract

The action of any lossless multilayer is described by a transfer matrix that can be factorized in terms of three basic matrices. We introduce a simple trace criterion that classifies multilayers in three classes with properties closely related with one (and only one) of these three basic matrices. PACS numbers:Any linear system with two input and two output channels can be described in terms of a 2 × 2 transfer matrix [1,2]. In many instances (e.g., in polarization optics [3]), we are interested in the transformation properties of quotients of variables rather than on the variables themselves. Then, the transfer matrix induces a bilinear transformation that represents the action of the system in the complex plane [4].For layered media the use of 2 × 2 matrix methods is standard, but the corresponding complex-plane representation is not a common tool. In fact, it has been recently established that for a lossless multilayer the transfer matrix is an element of the group SU(1,1) [5], which is the key for a deeper understanding of its behavior and simplifies exceedingly its description in terms of bilinear transformations.Moreover, as we have pointed out recently [6], the Iwasawa decomposition provides a remarkable factorization of the transfer matrix representing any multilayer (no matter how complicated it could be) as the product of three matrices of simple interpretation. At the geometrical level, such a decomposition translates directly into three actions that are the basic bricks from which any multilayer action is built.In this Letter we go one step further and show that the trace of the transfer matrix allows for a classification of multilayers in three disjoint classes with properties very close to those appearing in the Iwasawa decomposition.To maintain the discussion as self contained as possible we briefly summarize the essential ingredients of multilayer optics we shall need for our purposes. The configuration we consider is a stratified structure that consists of a stack of plane-parallel layers sandwiched between two semi-infinite ambient (a) and substrate (s) media, which we shall assume to be identical, since this is the common experimental case. Hereafter all the media are supposed to be lossless, homogeneous, and isotropic.We assume an incident monochromatic, linearly polarized plane wave from the ambient, which makes an angle θ 0 with the normal to the first interface and has amplitude E (+) a . The electric field is either in the plane of incidence (p polarization) or perpendicular to the plane of incidence (s polarization). We consider as well another plane wave of the same frequency and polarization, and with amplitude E (−) s , incident from the substrate at the same angle θ 0 [7].As a result of multiple reflections in all the interfaces, we have a backward-traveling plane wave in the ambient, denoted E (−) a , and a forward-traveling plane wave in the substrate, denoted E (+) s . It is useful to treat the field amplitudes as the vectorwhich applies to both ambient and substrate media. Then, ...

Analyzing light localization using Iwasawa-canonical transfer matrices

“…The reader should be aware that because the similarity transformation is applied to matrices drawn at random, the resulting transfer matrix induced by the similarity transformation will not, in general, be in Iwasawa-canonical form. Each resulting matrix, though, will be in SU (1,1).…”

“…This form, for transfer matrices in SU (1,1), renders a transfer matrix in a special diagonal form for frequencies in the passband. This basis is accomplished by a similarity transformation that moves the interior fixed point of the associated bilinear transformation to the origin.…”

“…(2)) has been exploited to better analyze and classify periodically layered media. As noted, such a transfer matrix is an element of SU (1,1) and it is known to induce a transformation of the complex plane, mapping z, the ratio of elements in the vector into another point in the complex plane. The field quotient z will be taken here as:…”

“…In a recent series of papers 1,2,3,4,5,6 Sánchez-Soto, Monzón, Cariñena and others have examined the transfer matrices used in multilayer 1-D photonic crystals in some detail. They have noted a particularly useful decomposition, the Iwasawa decomposition, in which each transfer matrix can be expressed, no matter its frequency.…”

Characterizing the reflectance of periodic layered media

“…Indeed, after Ref. [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.…”