This chapter is dedicated to periodic structures that are geometrically more complex than 2D photonic crystals, but not as complex as full 3D photonic crystals. In particular, we consider the optical properties of photonic crystal fibers, optically induced photonic lattices, and photonic crystal slabs.
Photonic crystal fibers
First, we consider electromagnetic modes that propagate along the direction of continuous translational symmetry ẑ of a 2D photonic crystal (Fig. 7.1(a)). In this case, modes carry electromagnetic energy along the ẑ direction, and, therefore, can be considered as modes of an optical fiber extended in the ẑ direction and having a periodic dielectric profile in its cross-section. Fibers of this type are called photonic crystal fibers. From Section 2.4.5, it follows that fiber modes can be labeled with a conserved wave vector of the form k = kt + ẑkz, where kt is a transverse Bloch wave vector, and kz ≠ 0. To analyze the modes of a photonic crystal fiber with periodic cross-section, we will employ the general form of a plane-wave expansion method presented in Section 6.2.
Furthermore, if a defect that is continuous along the ẑ direction is introduced into a 2D photonic crystal lattice (Fig. 7.1(b)), such a defect could support a localized state (see Section 6.6), thus, effectively, becoming the core of a photonic crystal fiber.