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Simple 2D photonic crystals are composed of periodic dielectric structures in a two-dimensional world. Electromagnetic waves in 2D are decomposed into two categories: TE and TM. Maxwell's equations are thus eigenvalue equations solving for TE and TM modes. A set of eigenfrequencies exist for each wavevector k. The plot of frequencies v.s. wavevectors is so-called band diagram.
We explore a set of rods arranged either in square or triangular lattice. The left part of the picture shows solid rods of square lattice in air background, while the right shows air rods of triangular lattice in solid background. The lattice constants are used as length units. We solve the band diagrams for both TE and TM modes. Try it out solving band diagrams for yourselves.