We present a comprehensive and self-contained discussion of the use of the transfer matrix to study propagation in one-dimensional lossless systems, including a variety of examples, such as superlattices, photonic crystals, and optical resonators. In all these cases, the transfer matrix has the same algebraic properties as the Lorentz group in a (2 + 1)-dimensional spacetime, as well as the group of unimodular real matrices underlying the structure of the abcd law, which explains many subtle details. We elaborate on the geometrical interpretation of the transfer-matrix action as a mapping on the unit disk and apply a simple trace criterion to classify the systems into three types with very different geometrical and physical properties. This approach is applied to some practical examples and, in particular, an alternative framework to deal with periodic (and quasiperiodic) systems is proposed.

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.

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.

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We determine the range of thicknesses and refractive indices for which omnidirectional reflection from quasiperiodic multilayers occurs. By resorting to the notion of area under the transmittance curve, we assess in a systematic way the performance of the different quasiperiodic Fibonacci multilayers.