Analytic design of graded photonic crystals in the metamaterial regime

We propose an analytical approach to the study of graded photonic crystals operating in the metamaterial regime. Relationships are given to predict the optical index map and hole-drilling distribution required to make light follow a prescribed path. The method is applied to proof-of-concept structures based on silicon-on-insulator technology. Light propagation is studied using FDTD simulation to verify the light trajectory, study the influence of extended light beams, and evaluate the robustness of the semiclassical approach based on the equations of Hamiltonian optics. The overall approach can be used for the straightforward design of new optical functionalities within the photonic metamaterial regime.