Conical Dispersion and Effective Zero Refractive Index in Photonic Quasicrystals

It is recognized that for a certain class of periodic photonic crystals, conical dispersion can be related to a zero-refractive index. It is not obvious whether such a notion can be extended to a noncrystalline system. We show that certain photonic quasicrystalline approximants have conical dispersions at the zone center with a triply degenerate state at the Dirac frequency, which is the necessary condition to qualify as a zero-refractive-index medium. The states in the conical dispersions are extended and have a nearly constant phase. Experimental characterizations of finite-sized samples show evidence that the photonic quasicrystals do behave as a near zero-refractive-index material around the Dirac frequency.

Figure 1

Supercell crystalline approximants and conical band dispersion near $\mathbf{k}=\mathbf{0}$ for 12-fold photonic quasicrystals. (a) and (b) are for the first level, while (c) and (d) are for the second level. Conical dispersions (purple) with a Dirac point are evident, with a flat band (orange) intersecting the Dirac frequency. The lower cone is intersected by multiple flat sheets (green) of localized ring modes.

Figure 2

Normalized electric fields in the 12-fold quasicrystals with different number of rods. (a) and (b) Extended states with constant phase excited at the Dirac frequency of $0.5584c/a$. (c) and (d) Localized modes at the frequency of $0.448c/a$. Zero phase extended states persist as the size becomes larger. The extended states allow energy to be transported while the localized modes trap energy inside the quasicrystal. The patterns are the real part of the fields.

Figure 3

Demonstration of zero refraction and Snell’s law. (a)–(c) Field patterns at three different frequencies. The plane wave is incident to the right-angle prism (black arrows) and forms the wave front (thin black lines) traveling upper right. (d) The effective refractive index (black) and the term $\delta =\sin {\theta }_i-n_{\text{eff}}\sin {\theta }_o$ (red) as a function of frequency. The term $\delta$ being approximate to zero shows that light refraction is governed by Snell’s law.

Figure 4

(a) Schematic of the experimental setup for field measurement. The sample is made by alumina rods (yellow) arranged in a 12-fold quasicrystalline array, sandwiched between two aluminum plates. A PEC obstacle (blue) is placed in the channel. (b) Measured $E_z$ fields with quasicrystalline array at 10.542 GHz, and the wave front retrieval on the exit side. (c) Control experiment of an obstacle in the channel without quasicrystalline array. Note the shadow cast by the obstacle on the exit side and the interference due to reflection on the left side. All the fields are normalized.

Figure 5

(a) and (b) Normalized calculated field pattern in a multichannel waveguide containing a quasicrystalline prism, showing asymmetric transmission at the Dirac frequency of 10.42 GHz. (a) The left-going waves have high transmittance while (b) the right-going waves suffer from nearly total reflection. (c) Measured transmission ratio between the left and right incidence wave. The ratio is above 26 and can reach the maximum of 94 at the frequency of 10.67 GHz.