Band Structure Computation of 1D Photonic Crystals

Summary

Chapter 4 is dedicated to the detailed description of the plane wave expansion method used for computation of the band structure of 1D photonic crystals. We describe all steps of the band structure computation applied for simplest photonic crystal. After the problem formulation, terms such as unit cell, lattice vector, reciprocal lattice and reciprocal lattice vectors as well as the first Brillouin zone are introduced and their physical and mathematical essence is discussed in detail. Peculiarities of the band structure computation are considered and such crucial problems as Bloch theorem, Fourier expansion of a dielectric function and general model formulation are analyzed. Finally, the eigen-value problem for the matrix is considered which gives an idea of the eigen-states computation of the differential operator presented in matrix form. The described model is also applied to off-axis radiation propagation. At the end of the chapter the example of original Matlabcode for band structure computation with additional comments is presented and several problems and questions for self-training are given.