Electromagnetic density of modes for a finite-size three-dimensional structure

The concept of the density of modes has been lacking a precise mathematical definition for a finite-size structure. With the explosive growth in the fabrication of photonic crystals and nanostructures, which are inherently finite in size, a workable definition is imperative. We give a simple and physically intuitive definition of the electromagnetic density of modes based on the Green’s function for a generic three-dimensional open cavity filled with a linear, isotropic, dielectric material.

Figure 1

Schematic representation of a point dipole embedded in a cavity of volume $V$ and surface $S$ filled with a generic linear and isotropic dielectric material.