Flat and self-trapping photonic bands through coupling of two unidirectional edge modes

We find a flat band and a self-trapping band through the coupling of two unidirectional edge modes, which was originally achieved by Wang et al. [Phys. Rev. Lett. 100, 013905 (2008); Nature 461, 772 (2009)] in two-dimensional magneto-optical photonic crystals. We break up a square-lattice yttrium-iron-garnet photonic crystal forming a waveguide and two edges. Given a proper interval of the two edges, two unidirectional edge modes with opposite group velocity directions can be coupled. The coupling leads to a wide flat band in which the group velocity is near zero, and a self-trapping band in which light is totally localized around the source. The position of the flat band and the group velocity can be adjusted by the external magnetic field. Numerical simulations and theoretical analysis both demonstrate the two interesting band structures.

Figure 1

Magneto-optical unidirectional edge modes. (a) 2D MO PC with its two edges in two opposite directions. (b) Projected band diagram for 2D MO PC with metallic cladding. Blue dashed and red solid lines represent the unidirectional edge modes for two different edges, respectively. (c) Steady-state field pattern for each unidirectional edge mode. Red and blue represent positive and negative field values. A source, indicated by a star, is located at the two edges and operates at a frequency $\omega =0.5426\times 2\pi c/a$.

Figure 2

2D MO PC with a line defect waveguide. The width of the waveguide is denoted as $d$. Two arrows denote the directions of two isolated unidirectional edge modes. The dashed rectangle denotes a supercell in the calculations.

Figure 3

The band diagrams for the structure as Fig. 2 with $d=1.5a$. There are two cases: without a magnetic field (left plot), and with applied magnetic field in the same directions (right plot) on the two PCs, respectively. $P_1,P_2$, and $P_3$ denote three frequencies at different mode dispersion curves.

Figure 4

The position change of the upper dashed band in Fig. 3 with the value of the nondiagonal element ${\mu }_2$.

Figure 5

The group velocity versus ${\mu }_2$ for $\omega =0.5426(2\pi c/a)$ which is at the position of the flat band in Fig. 3.

Figure 6

(a–c) The steady-state ${\mathbf{E}}_z$ field patterns correspond to three mode frequencies $P_1,P_2$, and $P_3$ at different mode dispersion curves in Fig. 3. $P_1$ corresponds to a typical waveguide mode. $P_2$ is at the flat band. Its field is along two edges which result from the coupling of two unidirectional edge modes. $P_3$ is at the lower dashed band in Fig. 3. Its field is self-trapped around the source because of the destructive interference of two degenerated modes with opposite group velocity directions. (d–f) The distributions of the Poynting vectors correspond to $P_1,P_2$, and $P_3$, respectively.