Enhanced Photonic Band-Gap Confinement via Van Hove Saddle Point Singularities

We show that a saddle point Van Hove singularity in a band adjacent to a photonic crystal band gap can lead to situations which defy the conventional wisdom that the strongest band-gap confinement is found at frequencies near the midgap. As an example, we present a two-dimensional square photonic crystal waveguide where the strongest confinement is close to the band edge. The underlying mechanism can also apply to any system that is described by a band structure with a gap. In general, the saddle point favors the appearance of a very flat band, which in turn results in an enhanced confinement at band-gap frequencies immediately above or below the flat band.

Figure 1

(a) A waveguide is formed by removing one row from a 2D square lattice of dielectric rods with refractive index 3.4 and radius $r=0.18a$. (b) The dispersion relation of the waveguide mode (thick black line) is shown on top of a color contour plot of the confinement strength of the bulk photonic crystal $\kappa (\omega ,k_x)$. The black regions include $(\omega ,k_x)$ pairs for which propagating modes exist in the bulk crystal, while the colors from dark red to bright yellow represent increasingly stronger confinement, as shown by the colorbar. (c) Confinement strength $\kappa$ along the dispersion relation of the guided mode. The gray areas delimit the complete band-gap range. The $y$ axis is the same as in part (b).

Figure 2

The projected TM band structure of the bulk crystal is shown by the blue areas and the black lines that represent $\omega (k_x)$ curves at constant $k_y$ values ranging from 0 to $0.5(2\pi /a)$ in steps of $0.05(2\pi /a)$. Note that the second band of bulk crystal has a saddle point singularity that is indicated by a thick black arrow. Also shown is the dispersion relation of the fundamental waveguide mode (red curve).

Figure 3

Dispersion relations $\omega (k_y^{(n)})$ at fixed $k_x$ for propagating and evanescent modes in the bulk crystal. The axial wave vector $k_x$ takes the values 0, 0.2, 0.3, and $0.05(2\pi /a)$ for parts (a) through (d). The imaginary part of $k_y^{(n)}$ is shown in red on the left side of each plot, while the real part is shown on the right, in blue for propagating modes and gray for evanescent and complex modes. In parts (a) and (c), the individual modes $k_y^{(n)}$ are labeled by the value of their $n$ index. The A and B labels track the range of frequencies for which the $n=0$ mode is evanescent, while the C label tracks the cutoff frequency of the $n=-1$ mode.

Figure 4

Contour plots of the second photonic crystal band $\omega (k_x,k_y)$ for varying radii of the dielectric rods: (a) $r=0$ (vacuum), (b) $r=0.06a$, (c) $r=0.12a$, and (d) $r=0.18a$. The same color scheme is used for all four parts of the figure. Constant-frequency contours are labeled by the corresponding value of the angular frequency $\omega$ in units of $(2\pi c/a)$. In parts (b), (c), and (d), a black dot indicates the position of the saddle point Van Hove singularity.