Coupling between plane waves and Bloch waves in photonic crystals with negative refraction

A comprehensive analysis of the coupling coefficients between plane waves in conventional dielectric media and Bloch waves of photonic crystals with effective negative refractions is performed by the layer Korringa-Kohn-Rostoker method. Employing the infinite layers refraction operator, semi-infinite size photonic crystals are considered. Some special coupling properties are discussed. In particular, the strong angular dependence of coupling coefficients is found even for an interface between air $(n=1)$ and a photonic crystal with effective refractive index $(n_{\mathrm{eff}}=-1)$. Thus, a negative refractive index defined by the radius of a circular equal-frequency contour does not guarantee an isotropic behavior.

Figure 1

The semi-infinite PhC structure consists of the front $N$ layers and the rear semi-infinite layers. ${\mathbf{u}}_{N+1}^{\pm }$ denotes the field amplitude at the $N+1$ interface (between the front $N$ layers and the rear semi-infinity system).

Figure 2

Schematic diagram of light propagation: (a) from air to the negative index material with $n=-1$ (b) from the air to the PhC with an effective refractive index $n_{\mathrm{eff}}=-1$.

Figure 3

(Color online) For air-PhC interfaces, the coupling coefficients at the different interface (a1) normal to the $\mathrm{\Gamma }\text{−}\mathrm{M}$ direction, and (b1) normal to the $\mathrm{\Gamma }\text{−}\mathrm{K}$ direction, at the frequency $\omega =0.31(2\pi c∕a)$. The reflection coefficients are plotted in (a2) and (b2) for these two case, respectively. The total line in (a2) and (b2) denotes the sum of all propagation order reflectance. Here $k_x$ is the $x$ component of the incident wave vector and $k_0$ is the wave number in the vacuum.

Figure 4

(Color online) The stable field $\left(E_z\right)$ distribution of incident Gaussian beams at the frequency $\omega =0.31(2\pi c∕a)$. (a) and (b) are for the surface interfaces of the PhC slab normal to the $\mathrm{\Gamma }\text{−}\mathit{M}$ direction, and (c) and (d) are for the surface interfaces normal to the $\mathrm{\Gamma }\text{−}\mathit{K}$ direction. The arrows indicate the incident angle, $\varphi =0\mathrm{°}$ in (a) and (c) and $\varphi =30\mathrm{°}$ in (b) and (d).

Figure 5

(Color online) The $E_z$ field distribution of the bulk modes of the photonic crystal at the frequency $\omega =0.31(2\pi c∕a)$, where (a) corresponds to the Bloch wave vector $\mathbf{k}$ at the $\mathrm{\Gamma }\text{−}M$ direction, and (b) corresponds to $\mathbf{k}$ at the $\mathrm{\Gamma }\text{−}K$ direction.

Figure 6

The equal frequency surface plot for the frequency $\omega =0.31(2\pi c∕a)$. The plane wave $\mathbf{k}$ is incident on the interface normal to the $\mathrm{\Gamma }\text{−}\mathit{K}$ direction. For the larger incident angles (i.e., the wave vector $\mathbf{k}$ in the region of the dotted line), the plane wave can excite Bloch waves in PhC with the Bloch wave vectors, ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$. Here $\mathbf{G}$ is the reciprocal lattice vector.

Figure 7

(Color online) The calculated coupling coefficients (solid lines) and the reflection coefficients for interfaces between the dielectric medium $(n=3.24)$ and the photonic crystal $\left(n_{\mathrm{eff}}=-0.73\right)$ at the frequency $\omega =0.325\left(2\pi c∕a\right)$. The interfaces are (a1) normal to the $\mathrm{\Gamma }-M$ direction, and (b1) normal to the $\mathrm{\Gamma }-K$ direction. The dashed-dotted lines are the coupling coefficients between two conventional isotropic dielectric media with $n=3.24$ and $n=0.73$, respectively, calculated by the Fresnel transmission formulas. The reflection coefficients are plotted in (a2) and (b2) for these two cases, respectively. The total line in (a2) and (b2) denotes the sum of all propagation order reflectance.