Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps

An efficient numerical method based on the Dirichlet-to-Neumann (DtN) maps of the unit cells is developed for accurate simulations of two-dimensional photonic crystal (PhC) devices in the frequency domain. The DtN map of a unit cell is an operator that maps the wave field on the boundary of the cell to its normal derivative and it can be approximated by a small matrix. Using the DtN maps of the regular and defect unit cells, we can avoid computations in the interiors of the unit cells and calculate the wave field only on the edges. This gives rise to a significant reduction in the total number of unknowns. Reasonably accurate solutions can be obtained using 10 to 15 unknowns for each unit cell. In contrast, standard finite element, finite difference or plane wave expansion methods may require a few hundreds unknowns for each unit cell at the same level of accuracy. We illustrate our method by a number of examples, including waveguide bends, branches, microcavities coupled with waveguides, waveguides with stubs, etc.