We present numerical simulations of electromagnetic millimeter-wave propagation in a two-dimensional lattice of dielectric rods arranged in a tenfold Penrose tiling. We find (i) isotropic photonic band gap as expected for quasicrystals and (ii) localized states. We demonstrate that the high frequency edge of the second band gap is characterized by a very small refractive index (fast light). We study the transmission of electromagnetic waves in the frequency range corresponding to fast light and demonstrate that it is related to tunneling through localized states. We use the fast light phenomenon to design a focusing device—a planoconcave lens.

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Omni-directional negative refractive index is a desirable feature that may be hard to achieve by using conventional periodic structures. In 2D periodic photonic crystals, where rotational symmetry is four-or six-fold, negative refraction index can be achieved but it is highly anisotropic. Another way to achieve negative refraction in 2D arrays is by using photonic crystal superlattices consisting of rods with two different diameters. In this study we employ this method to build a 2D quasicrystal superlattice based on Penrose tiling. The omnidirectionality of the bandgaps that results from the quasicrystalline structure, while the effective negative index results from the superlattice. Together, these two features produce an omnidirectional negative refractive index. We demonstrate this effect by tracing the directions of phase and group velocities in the prism made of the photonic quasicrystal superlattice.

We present numerical studies of the electromagnetic wave propagation in a metamaterial built from dielectric rods. The rods were arranged in a quasicrystalline lattice based on 10-fold Penrose tiling. We find wide isotropic band gaps and localized states, as expected in a quasicrystalline lattice. Our most important finding is the presence of the band gap edge states that are characterized by a very small refractive index (fast light). We use this phenomenon to design a focusing device-a plano-concave lens.