Robust edge states in amorphous gyromagnetic photonic lattices

We numerically study amorphous analogs of a two-dimensional photonic Chern insulator. The amorphous lattices consist of gyromagnetic rods that break time-reversal symmetry, with the lattice sites generated by a close-packing algorithm. The level of short-range order is adjustable, and there is no long-range order. The topologically nontrivial gaps of the photonic Chern insulator are found to persist into the amorphous regime, so long as there is sufficient short-range order. Strongly nonreciprocal robust transmission occurs via edge states, which are shown to propagate ballistically despite the absence of long-range order, and to be exponentially localized along the lattice edge. Interestingly, there is an enhancement of nonreciprocal transmission even at very low levels of short-range order, where there are no discernible spectral gaps.

Figure 1

(a) Transverse-magnetic band structure for a 2D triangular photonic crystal. The Chern number $C$ for each topologically nontrivial band is indicated, and the topological band gaps are highlighted in green. The structure (inset) consists of gyromagnetic rods with radius $r=3.4\mathrm{mm}$, nearest-neighbor spacing $a=30.91\mathrm{mm}$, permittivity $\epsilon =15$, and permeability components ${\mu }_{xx}={\mu }_{yy}=14,{\mu }_{zz}=1,{\mu }_{xy}=-{\mu }_{yx}=10i$. The rods are surrounded by air. (b)–(d) Amorphous lattices generated with packing parameters $\mathrm{\Phi }=0.1,0.4$, and 0.7. (e) Spatial correlation function $C\left(\mathrm{\Delta }r\right)$ for different $\mathrm{\Phi }$. Each curve is averaged over 100 samples. Inset: close-up of the nearest-neighbor correlation peak.

Figure 2

(a) Spectral density of amorphous lattices with different values of the packing parameter $\mathrm{\Phi }$. These results are obtained from frequency-domain calculations for an amorphous lattice of 266 rods distributed within a square region, surrounded by empty space, with a point TM source of fixed current amplitude at the center. Each curve shows $|E_z{|}^2$ averaged over the lattice region, and over ten samples. For $\mathrm{\Phi }=0.7$, three spectral dips are observed, which diminish with greater disorder (smaller $\mathrm{\Phi }$). (b) Isolation ratio ${\mathcal{I}}_R=I_{AB}/I_{BA}$ versus frequency, for transmission along the lattice edge. Each curve is averaged over ten samples. There is no enhancement of ${\mathcal{I}}_R$ at the primary gap, but the two higher-order gaps show enhanced ${\mathcal{I}}_R$ corresponding to forward and backward transmission, respectively. (c) Schematic for the calculation in (b). The upper edge is a perfect electrical conductor (PEC), while perfectly matched layer (PML) boundary conditions are applied on the other three sides. The isolation ratio is calculated between the two points labeled $A$ and $B$, near the edge. Also shown is the steady-state real electric field $E_z$ emitted by a $4.448\mathrm{GHz}$ point source at $A$, with red and blue corresponding to positive and negative values of $E_z$; the lattice packing parameter is $\mathrm{\Phi }=0.7$. (d) Close-ups of (a) and (b) in the vicinity of the isolation ratio peaks.

Figure 3

Robust TM edge propagation in amorphous lattices of $\mathrm{\Phi }=0.5$ with (a) circular and (b) triangular shape, and PEC along all edges. The steady-state real electric field $E_z$ is indicated in red (positive) and blue (negative). Stars indicate point sources emitting at $4.437\mathrm{GHz}$.

Figure 4

(a) Dwell time $d\phi /d\omega$ versus path length $s$, for amorphous lattices with different packing parameters $\mathrm{\Phi }$ as well as the perfectly ordered triangular lattice. Each symbol is averaged over ten samples; error bars show the standard deviation. The calculation procedure is explained in the text. The legend indicates fitted values of the group velocity $v_g$, where $c$ is the speed of light. (b) Semilogarithmic plot of the intensity $I$ versus the distance from the edge $y$, where $I$ is the value of $|E_z{|}^2$ averaged over positions to the right of a point source, and over a frequency band and ten lattice samples. The results show that the waves are exponentially localized to the edge, $I\propto exp(-\beta y)$. Inset: plot of the fitted values of $\beta$ versus $\mathrm{\Phi }$ (including $\mathrm{\Phi }$ values omitted from the main plot for clarity).