Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal

We point out that electromagnetic one-way edge modes analogous to quantum Hall edge states, originally predicted by Raghu and Haldane in 2D photonic crystals possessing Dirac point-derived band gaps, can appear in more general settings. We show that the TM modes in a gyromagnetic photonic crystal can be formally mapped to electronic wave functions in a periodic electromagnetic field, so that the only requirement for the existence of one-way edge modes is that the Chern number for all bands below a gap is nonzero. In a square-lattice yttrium-iron-garnet crystal operating at microwave frequencies, which lacks Dirac points, time-reversal breaking is strong enough that the effect should be easily observable. For realistic material parameters, the edge modes occupy a 10% band gap. Numerical simulations of a one-way waveguide incorporating this crystal show 100% transmission across strong defects.

Figure 1

Magneto-optical one-way waveguide. (a) Projected band diagram. The dispersion curve (red) for the edge modes spans the band gaps for the cladding crystals. (b) Steady-state field pattern for $E_z$, the out-of-plane electric field component. Blue and red represent positive and negative field values. A source, indicated by double arrows, is located at the interface between a magnetized ($+z$) YIG crystal (lower half plane) and an alumina crystal (upper half plane), and operates at a midgap frequency ($0.552\times 2\pi c/a$). The excited edge mode propagates to the right and undergoes evanescent decay to the left. (c) The time-averaged electric field intensity along the midline of the waveguide [dashed line in (b)].

Figure 2

Construction of a MO photonic crystal supporting one-way edge modes. The crystal consists of a square lattice of YIG rods [inset in (b), with $ϵ=15{ϵ}_0$ and $r=0.11a$] in air. (a) Band diagram with zero dc magnetic field ($\mu ={\mu }_0$, $\kappa =0$). The relevant quadratic degeneracy point is indicated. (b) Band diagram with a 1600 Gauss $+z$ dc magnetic field ($\mu =14{\mu }_0$, $\kappa =12.4{\mu }_0$). The degeneracies are lifted, resulting in the given nonzero Chern numbers (red numbers). (c) Contour plot for the second band. Although the MO crystal has only $C_4$ symmetry, the band structure retains an accidental $C_{4v}$ symmetry, with an irreducible Brillouin zone identical to that of a non-MO crystal. The Chern numbers are calculated by integrating Eq. (5) along the boundary of the first Brillouin zone. (d) The corrected band gap with material dispersion.

Figure 3

Backscattering suppression in the one-way waveguide. Inserting a slab of perfect electrical conductor with thickness $0.2a$ (black rectangle) causes the propagating modes to circumvent the defect, maintaining complete transmission. $E_z$ is plotted for (a) $t=0$, (b) $t=0.25T_0$, and (c) $t=0.5T_0$, where $T_0$ is the optical period. (d) Time-domain simulation results for a temporal Gaussian pulse with spectrum contained in the band gap. The electric field amplitude is plotted at the source point (black), at the same transverse position 13 lattice constants downstream along the waveguide in the absence of a defect (red), and at the same point with an intervening defect (blue). The pulse is completely transmitted regardless of the existence of the defect. (e) Transmission and phase shift plots for the time-domain simulation.