Transmission in the vicinity of the Dirac point in hexagonal photonic crystals

We use a scattering matrix approach to simulate the transmission through a hexagonal photonic crystal in the vicinity of the Dirac point. If the crystal is oriented so that the propagation direction perpendicular to the surface corresponds to the ΓKΓK direction, no oblique transmission is possible for a very long (infinite) structure. For a finite structure with width, W, and length, L  , the length dependence of the transmission is given by Ttotal=Γ0W/LTtotal=Γ0W/L. For TtotalTtotal all waves with a wavevector parallel to the surface, k∥=n2π/Wk∥=n2π/W, described by a channel number, n, must be considered. We show the transmission at the Dirac point follows the given scaling law and this scaling law is related to the behavior of the individual channels. This leads to the establishment of a criterion for the maximum length for this scaling behavior when the total transmission reaches a constant value. We also compare this scaling behavior to the results in other frequency regions.