3 - One-dimensional photonic crystals – multilayer stacks

Summary

In this chapter, we will consider reflective properties of planar multilayers, and guidance by multilayer waveguides. We will first introduce a transfer-matrix method to find electromagnetic solutions for a system with an arbitrary number of planar dielectric layers. We will then investigate the reflection properties of a single dielectric interface. Next, we will solve the problems of reflection from a multilayer stack, guidance inside a dielectric stack (planar waveguides), and finally, propagation perpendicular to an infinitely periodic multilayer stack. We will then describe omnidirectional reflectors that reflect radiation completely for all angles of incidence and all states of polarization. Next, we will discuss bulk and surface defect states of a multilayer. We will conclude by describing guidance in the low-refractive-index core waveguides.

Figure 3.1 presents a schematic of a planar multilayer. Each stack j = [1 …N] is characterized by its thickness d_j and an index of refraction n_j. The indices of the first and last half spaces (claddings) are denoted n₀ and n_N+1. The positions of the interfaces (except for j = 0) along the axis ẑ are labeled z_j,j = [1 …N + 1], whereas z₀ can be chosen arbitrarily inside of a first half space. In the following, we assume that the incoming plane wave has a propagation vector k confined to the xz plane. The planar multilayer possesses mirror symmetry with respect to the mirror plane xz.