Theory and Mathematical Formulation of Electromagnetic Wave Propagation in One-Dimensional Photonic Crystals

Abstract

Photonic crystals (PCs) are periodic dielectric (and sometimes metal–dielectric or metamaterialbased) structures that enable strong control over electromagnetic wave (EMW) propagation through the formation of photonic band gaps (PBGs). The theoretical foundation of PCs lies in Maxwell’s equations combined with appropriate constitutive relations and boundary conditions at material interfaces. In this paper, we present a concise theory and mathematical formulation for optical wave propagation in layered media with special emphasis on one-dimensional (1-D) periodic structures. The refractive index (optical density) is introduced as a key phenomenological parameter connecting macroscopic material response to microscopic polarization mechanisms. For isotropic media, the electromagnetic fields satisfy decoupled Helmholtz-type wave equations, while for nonuniform periodic media the fields obey a master equation that leads to Bloch-mode solutions and dispersion relations within the first Brillouin zone. The transfer matrix method (TMM) is described as an efficient technique for evaluating band structure, reflectance, transmittance, and defect-mode behavior in 1-D photonic crystals. The formulation is extended conceptually to anisotropic and composite media where permittivity and permeability may become tensor quantities, enabling hyperbolic dispersion and tunable optical responses. This theory provides a rigorous basis for designing photonic devices such as filters, mirrors, sensors, and waveguiding structures across optical and terahertz regimes.