Impedance of square and triangular lattice photonic crystals

We give a rigorous semianalytical definition of impedance for square and triangular (hexagonal) lattice two-dimensional photonic crystals (PCs). Our impedance is a small matrix, derived from transfer matrices, which stores the information required to calculate the reflection and the transmission between PCs. We apply our definition to design PC antireflection coatings efficiently. This task is O(n) with the number of candidate PCs as only one simulation per PC is required to find the impedances; the reflection and the transmission properties of a large number of coatings may then be evaluated quickly using the impedances in a simple matrix equation, in a way similar to the design of thin-film coatings. This is much faster than directly finding the reflections and the transmissions between pairs of candidate PCs, which requires one simulation per pair, a task that is O(n^2)