Paraxial Hamiltonian for photons in two-dimensional photonic crystal microstructures

Abstract

New solid-state physics based approach is developed for analysis of the paraxial light propagation in two-dimensional (2D) photonic lattices of coupled dielectric waveguides or microcavities. In particular, using Maxwell's equations, a non-Hermitian Hamiltonian eigenproblem with respect to the spinor wave function of a photon is obtained for energy-dissipating photonic microstructures. The Hamiltonian is suitable for almost the entire subclass of 2D structures encompassing arrays of semiconductor microcavities and microstructured photonic crystal fibers, characterized by light propagating mostly normal to the periodic lattice plane. Methods of numerical solution are discussed and the formalism is applied to a square array of coupled semiconductor microcavities, revealing reach possibilities for tailoring photonic band structure both in the photon energy and photon lifetime energy broadening domains. In particular, a feasibility to open a double photonic crystal band gap simultaneously in the energy and lifetime energy broadening domains is demonstrated.