Waveguiding at the Edge of a Three-Dimensional Photonic Crystal

We find that electromagnetic waves can be guided at the edge of a three-dimensional photonic crystal in air. When the waveguide is defined by the intersection of two surface planes, the edge modes are associated with the corresponding surface bands. A simple cell counting approach is presented to describe the periodic evolution of the system with interesting interplays among edge, surface, and bulk states.

Figure 1

(a) Illustration of a diamond lattice. The cubic unit cell of length $a$ is outlined in blue and the fcc primitive cell is outlined in red. Two $(1\overline{1}1)$ surfaces of equivalent termination parameters ${\tau }_1=0$ and ${\tau }_1=1$ are shown in yellow and green. They lie on the end faces of the fcc primitive cell connected by ${\overrightarrow{a}}_3$. The $(\overline{1}11)$ surface of ${\tau }_2=1$ is also shown in green. Any of the two intersecting planes define the [110] edge along ${\overrightarrow{a}}_1$. $\widehat{v}(\equiv \frac{\overrightarrow{v}}{|\overrightarrow{v}|})$ represents a unit vector along $\overrightarrow{v}$. (b) The projected band structure of the $\{111\}$ surfaces of the inverse diamond PhC in air. The $\{111\}$ surfaces, containing a $C_{3v}$ symmetry, have a triangular lattice of lattice constant $a/\sqrt{2}$. The brown area is the projected bulk modes. The light cone is shaded in transparent gray. The irreducible BZ is shown as an inset. $\Gamma \mathrm{\text{−}}{K}^{\prime }$ is the irreducible BZ of the edge waveguide along the $⟨110⟩$ directions.

Figure 2

(a) One unit cell of the edge waveguide along ${\overrightarrow{a}}_1$. The 3D PhC consists of air spheres of diamond lattice in a dielectric matrix rendered in gray solid. The cells and $\{111\}$ surfaces are the same as those in Fig. 1a. The top purple area in the cubic cell (001) plane is the region where the 2D cuts are made in Fig. 4 and in the Supplemental Material [22]. (b) Band structure of the edge waveguide formed by two $⟨110⟩$ surfaces, both of zero termination parameters.

Figure 3

(a) Schematic illustration of the periodic evolution of the edge unit cell. The cells of the outlined font indicate the increment from the original structure. By dashing the left and bottom boundaries of the structures, we mean the spatial extension of the cells beyond what is drawn. (b) Evolution of the edge dispersion diagrams of different surface terminations. The mode profiles of the circled edge dispersions at the zone boundary ($K^{\prime }$) are presented in Fig. 4 and in the Supplemental Material [22].

Figure 4

Mode profiles of the edge modes associated with the TM-like surface bands at the BZ boundary ($K^{\prime }$). The dielectric-air interfaces are drawn as transparent contour surfaces in gray. The arrows point to the directions of the electric fields; their sizes represent the field amplitudes. The colors of the arrows represent the field locations along the waveguide propagation direction (${\overrightarrow{a}}_1$). The fields of the amplitudes one order of magnitude less than the maximum field amplitude are not plotted. The insets on the sides of the vector plots are the plane cuts of the intensity square of the electric fields. The area and location of the plane (in air) is illustrated in Fig. 2a. Central lines are drawn in yellow dash to indicate the mode positions.

Figure 5

Effective mode areas and group indices of the edge modes of the (0.1, 0.1) edge are evaluated over their operational bandwidths. Field profiles are plotted for three representative modes with the gray outlines of the dielectric-air interfaces.