Extreme anisotropy of wave propagation in two-dimensional photonic crystals

We demonstrate that electromagnetic waves propagating in square and hexagonal photonic crystals can have fundamentally different anisotropy properties. The wave frequency and group velocity can be functions of the propagation direction even for vanishingly small wave numbers (near the $\Gamma$-point). This anisotropy, present in square but absent in hexagonal lattices, can be so extreme that the group velocity can be either parallel or antiparallel to the phase velocity depending on the propagation direction. An analytic explanation of this effect based on the $\vec{k}\bullet \vec{p}$ perturbation theory and group-theoretical considerations is confirmed by electromagnetic simulations. One manifestation of the extreme anisotropy is the divergent van Hove singularity in the density of photonic states at the $\Gamma$-point. New applications, including surface-emitting quantum cascade lasers, are proposed.

Figure 1

(Color online) TE propagation bands $\omega \left(\vec{k}\right)$ for the PC consisting of air holes of diameter $d=0.8a$ in silicon: (a) square lattice, and (b) hexagonal lattice. Four unit cells of each lattice are shown in the insets. Bands are labelled by their symmetry at the $\Gamma$-point. Band frequencies are scanned along the following directions inside the irreducible Brillouin zones: $\Gamma \text{−}X$ $(0<\vec{k}<{\vec{e}}_x\pi ∕a)$ and $\Gamma \text{−}M$ $(0<\vec{k}<({\vec{e}}_x+{\vec{e}}_y)\pi ∕a)$ in (a), and $\Gamma \text{−}M$ $(0<\vec{k}<2{\vec{e}}_y\pi ∕\sqrt{3}a)$ and $\Gamma \text{−}K$ $(0<\vec{k}<4\pi {\vec{e}}_x∕3a)$ in (b).

Figure 2

(Color online) Equal frequency surfaces (EFS) for the lower-frequency branches of the degenerate at $\Gamma$-point bands: (a) the $E$-band of the square lattice, and (b) the $E_2$ band of the hexagonal lattice. Frequencies are labelled by $(\omega -{\omega }_0)a∕2\pi c$, where ${\omega }_0=\omega (\vec{k}=0)$. For (a) ${\omega }_{0}a∕2\pi c=0.4587$, for (b) ${\omega }_{0}a∕2\pi c=0.6156$. Photonic crystal parameters: same as in Fig. 1.

Figure 3

(Color online) Refractive effects caused by the hyperbolic shape of the EFS at the frequency below the $\Gamma$-point $(\omega =0.45802\pi c∕a)$ of a square PC. Crystal parameters: same as in Figs. 1, 2. Main axes: group velocity angle of the refracted into PC wave(s) vs incidence angle. Inset: Cell-averaged Poynting flux $⟨P_y⟩$ into the PC. Squares: small incidence angle ${\theta }_{\mathrm{in}}=0.1\mathrm{rad}$: two refracted waves are present and interfere (bi-refraction). Diamonds: large incidence angle ${\theta }_{\mathrm{in}}=0.4\mathrm{rad}$: single refracted wave.