Dispersion relations and wave propagation in photonic hypercrystals

A photonic hypercrystal is a subwavelength periodic structure consisting of alternate layers of hyperbolic metamaterial and dielectric material. The structure can be treated as an effective medium as well as a periodic medium. Since two length scales are involved, the better treatment is to treat the hyperbolic metamaterial as an effective medium and the overall structure as a periodic medium. The dispersion relations are derived and plotted to show the appearance of propagating bands and gaps in the frequency and wave vector domains. Then using the transfer matrix approach, the transmissivity versus the frequency plot for propagating waves and grayscale plot of the transmission coefficient in the frequency versus wave vector plane for the evanescent waves are plotted and analyzed.