A solid state theoretical approach to the optical properties of photonic crystals

Abstract

We outline a theoretical framework that allows qualitative as well as quantitative analysis of the optical properties of Photonic Crystals (PCs) and which is derived from solid state theoretical concepts. Starting from photonic bandstructure computations which allow us to obtain dispersion relations and associated Bloch functions, we show how related physical quantities such as densities of states and group velocities can be calculated. In addition, defect structures embedded in PCs can be efficiently treated with the help of photonic Wannier functions that are derived from photonic Bloch functions by means of a lattice Fourier transform. Nonlinear PCs may be investigated by an appropriate multi‐scale analysis utilizing Bloch functions as carrier waves together with an adaptation of k · p‐perturbation theory. This leads to a natural generalization of the slowly varying envelope approximation to the case of nonlinear wave propagation in PCs.

Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects

“…1(a), we calculated the group velocity vector for the 5 th band (band with negative curvature) and the 4 th band (band with positive curvature) for a range of frequencies where the EFS contours are "almost" isotropic. We used the k· p perturbation method [42,43,44] result given in expression (82) of App. II a (which is equivalent to expression (31)).…”

“…Busch et al [43,44] implemented this method to determine the group velocity and "photon mass" of a two-dimensional E-polarized photonic crystal. In the following, we provide the basic steps of such calculation and derive simple expressions in terms of the FB wave's coefficients.…”

Surface plasmon resonance in Ag nanoparticles deposited inside porous GaP templates

“…The photon density of states is determined by "counting" all allowed states with a given wavelength [14]: ρ(λ) = (С/λ²)Σ∫δ(ω -ω(j,k))d³k (2) where C is a constant, ω = 2π/λ is the frequency of the electromagnetic radiation, k is the wave vector, and ω(j,k) are the branches of the photonic band structure. The integral is over the first BZ of a Dirac-δ function, and the sum is over all branches of the photonic band structure.…”

Two-Dimensional Ferroelectric Photonic Crystals: Optics and Band Structure

“…Over the past ten years, there has been great interest in photonic crystal lattices (PCLs) due to their ability to manipulate light and great potential for technological applications in nanophotonics and integrated optoelectronic devices [1][2][3]. PCLs are artificial materials; optical and electronic properties of two (2D)-and three (3D)-dimensional PCLs are intensively studied with the goal of achieving control of electromagnetic propogation, and, especially, a complete photonic band gap in all directions [2].…”