Light propagation in semi-infinite photonic crystals and related waveguide structures

A transfer-matrix method (TMM) that employs the plane-wave expansion of electromagnetic (EM) fields has been developed to handle EM wave propagation in semi-infinite photonic crystal and related waveguide structures. The major aim is to account for wave scattering only at the concerned boundary and to completely remove multiple reflections in the presence of other structural boundaries. It turns out that the scattering problem is closely connected to the eigenmodes of the transfer matrix for the unit cell of the crystal. A natural boundary condition is imposed to describe the asymptotic propagation behavior of scattered EM waves in a region far away from the interface. Theories for a variety of important structures have been systematically set up. These include wave propagation in a semi-infinite photonic crystal, a coated semi-infinite photonic crystal, a heterostructure formed by two different semi-infinite photonic crystals face to face, and a complex sandwiched structure formed by two semi-infinite photonic crystals separated by a general grating slab. In combination with a supercell technique, the developed formulations can also be used to handle photonic crystal waveguide structures. We have applied the developed TMM to two-dimensional photonic crystal and related waveguide structures. The first is the coupling of an external wave into a photonic crystal (photonic crystal waveguide) and the related inverse problems of coupling of a Bloch's wave (guided wave) out of the photonic crystal (photonic crystal waveguide). The second is scattering of a guided wave by a cavity introduced into a photonic crystal waveguide. The developed TMM can help to understand optical properties and design optimal structures of individual functional elements in an optical integrated circuit built in a photonic crystal environment.