Analytic properties of photonic crystal superprism parameters

We study the analytic properties of the photonic crystal superprism resolution parameters $p$, $q$, and $r$ introduced previously by Baba and Matsumoto [Appl. Phys. Lett. 81, 2325 (2002)], which characterize the potential dispersive power of a superprism. We find closed form expressions for these quantities that greatly simplify their accurate evaluation and reveal significant insights about their behavior. The expressions imply general properties of the parameters which are true for all bands and all photonic crystals. In particular, we demonstrate that all photonic crystals exhibit infinite resolution as measured by the parameter $r$ along particular contours in any photonic band.

Figure 1

Construction for refraction at the interface between a uniform medium and a photonic crystal. Equifrequency curves for a uniform (thick line) and triangular photonic crystal (fine line) are shown. Light is incident from a uniform medium with wave vector $\mathbf{k}$ upon an interface with the photonic crystal (chain line). The tangential wave vector component at the interface is conserved (dotted line) identifying the allowed output wave vectors $\mathbf{\kappa }$. The direction of the group velocity $\mathbf{v}$ in the photonic crystal is given by the normal to the equifrequency curve at its intersection with $\mathbf{\kappa }$.

Figure 2

Geometry for the superprism analysis. The coordinates $(x,y)$ are the conventional coordinates of the Brillouin zone. The coordinates $(\eta ,\chi )$ are aligned with the cut plane of the crystal which lies along the direction $\widehat{\mathbf{t}}$. The angle $\psi =0$ if the crystal cut plane is along the $\Gamma \text{−}X$ direction. The wave vector $\mathbf{k}$ represents the incoming beam at an angle ${\theta }_i$ to the normal to the interface $\widehat{\mathbf{n}}$, and $\mathbf{\kappa }$ is the wave vector of the refracted beam inside the crystal. The angle between the normal $\widehat{\mathbf{n}}$ and the group velocity $\mathbf{v}$ inside the crystal is denoted ${\theta }_c$. The vectors $\mathbf{\kappa }$ and $\mathbf{v}$ are in general not collinear.

Figure 3

Equifrequency contours and Brillouin zone for the first band of the photonic crystal with parameters given in the text. The band exhibits a (nondifferentiable) minimum at $\Gamma$, a saddle point at $X$, and a maximum at $M$.

Figure 4

Contour plots over the entire Brillouin zone of ${\log }_{10}\mid 1∕p\mid$, ${\log }_{10}\mid q\mid$, and ${\log }_{10}\mid r\mid$ (left to right) for a photonic crystal with parameters given in the text. The rows correspond to different orientations of the crystal cut plane with respect to the $\Gamma \text{−}X$ direction. From the top, the angles are $\psi =0°$ (cut plane along $\Gamma \text{−}X$), 30°, 45° (cut plane along $\gamma \text{−}M$), and 70°. The broken white lines through the origin lie along the interface between the two media and thus indicate the direction of the $\eta$ axis.

Figure 5

Role of the curvature $G$ in the structure of $p$ for the lowest two bands at $\psi =0$. The peak of the diffuse white contour represents zeros of $G$. Thin white curves are the equifrequency contours of each band. For the lowest band, shown in (a) the black dotted lines indicate the direction of the $p=0$ curves obtained by quadratic expansion of the band at the saddle points at $X$. For the second band, shown in (b), there is a band maximum at $\Gamma$, minima at $X$ and $Y$, and a saddle point just before $M$ on $\Gamma \text{−}M$.

Figure 6

Role of the isocontours of ${\theta }_i$ (dashed white) and ${\theta }_c$ (dashed black) in the structure of $q$ for the lowest band at $\psi =0$ (left), 30° (center), and 45° (right). The diffuse black contour represents zeros of $G$.