Electromagnetic Energy Flow in Photonic Crystals



Photonic crystals (PCs) are periodic dielectric structures that enable an efficient control of optical electromagnetic signals (light) in geometries with features on the sub-wavelength scale; a control that is not possible in classical optics based on total internal reflection and refraction of light rays. In photonic crystals a periodic photonic potential can induce a photonic bandgap, i.e.; a range of frequencies where radiation energy in certain polarisation states is not allowed to flow in specific directions. This is the photonic analogue of the forbidden energy band for electrons in an electric periodic potential in the lattice of atoms of a semiconductor material.

Photonic crystals enable new advanced all-optical signal processing in regions with dimensions of a few cubic wavelengths because radiation losses can be greatly reduced; something that is impossible in classical optical components without serious energy loss.

Electromagnetic phenomena are governed by Maxwell’s equations and a set of boundary conditions that the electric vector field E and the magnetic vector field H must satisfy at material interfaces.

Figure 1 shows a planar cut of a two-dimensional (2D) photonic crystal made of a background material of relative dielectric constant er1 in which a triangular lattice of holes is etched. For silicon nitride the relative dielectric constant is assumed to be 4 at the wavelengths of interest, and that of air is 1. The lattice constant is L and the hole radius is r0. The eigenstates of this 2D structure can be categorised as H states (the magnetic field vector is parallel to the 2D plane and the electric field vector is perpendicular to the same) or E states (the electric field vector is parallel to the 2D plane and the magnetic field vector is perpendicular to the same). Banddiagrams usually present eigenfrequencies, w=L/l, versus k vector values, where l is the free-space wavelength and k is the Bloch mode propagation vector. The chosen k vectors usually lie on the contour of the irreducible Brillouin zone in order to determine the existence of bandgaps (the smallest bandgap is determined by the highest photon energies). The preferred regime of operation is typically for w<1.

We assume that only E states are present in the configuration.


















Figure 1: Two-dimensional photonic crystal made of a background material in which a triangular lattice of holes is etched.



By creating line defects of width W along the nearest neighbour direction, we can introduce localised electromagnetic states wd(k) with frequencies in the bandgap of the perfect crystal (for any crystal configuration exhibiting a gap of course).

Some of the questions one wonders about:



Energy coupling:









Negative refraction:

The numerical investigations are often based on methods mentioned in the following publications:



Related literature: