Hyperuniform Disorder in Photonic Crystal Slabs with Intrinsic non-Hermiticity

Abstract

Hyperuniform disorder is a type of correlated disorder characterized by vanishing spectral density at small wavevectors, making the configuration effectively homogeneous on long length scales. In photonics, hyperuniform disorder is promising for generating isotropic photonic pseudogaps and engineering photonic crystal waveguides. However, these studies are largely restricted to idealized lossless settings, although all photonic systems necessarily have loss. In this work, light propagation in photonic crystal slabs with imposed hyperuniform disorder is investigated theoretically and numerically. The system is intrinsically non-Hermitian due to radiative loss, with non-Hermiticity appearing as a complex effective mass of a quadratic photonic band. A theoretical framework for disorder scattering is analytically derived in Hermitian and non-Hermitian quadratic bands with real and complex effective mass, respectively. In contrast to the power law behavior $|\mathbf{k}|^α$ observed in the Hermitian case (where $α$ is the hyperuniformity exponent), the scattering loss in the non-Hermitian band is given by $C_0+C_{β_2}\cdot|\mathbf{k}|^{β_2}$, where $C_0$ is a finite constant and the exponent $β_2\leq 2$. Our theoretical predictions are verified with tight-binding and Finite-Difference Time-Domain simulations with realistic photonic crystal parameters, based on recent experiments.

Figures (5)

• Figure 1: The non-Hermitian nature of photonic crystal slabs. (a) An illustration of the photonic crystal slab. The in-plane geometry contains circular air holes in a square lattice. (b) The simulated reflection spectrum of the structure in (a) along $k_y=0$. (c) The resonance frequency of the band extracted from (b). A quadratic fit is performed to obtain the real part of effective mass $\mathrm{Re}\left(\frac{1}{2m}\right)$. (d) The linewidth of the band extracted from (b) by fitting the reflection spectrum with a resonance lineshape. The linewidth also grows as $O(k^2)$, indicating the effective mass is complex.
• Figure 2: Hyperuniform disorder in photonic crystal slabs. (a) Local potential configuration with uncorrelated disorder. The color and the size of the holes represent the local potential change at each site with respect to the periodic (without disorder) case. Only an area of $30\times 30$ unit cells is shown. (b) The spectral density $\tilde{\rho}(\mathbf{q})$ corresponding to the disorder pattern in (a). The system size is $N_x\times N_y=500\times 500$. (c) Hyperuniform local potential configuration with hyperuniformity exponent $\alpha=1$. The potential is highly spatially correlated. (d) The spectral density $\tilde{\rho}(\mathbf{q})$ corresponding to the disorder pattern in (c). The spectral density scales as $\tilde{\rho}(\mathbf{q})\to O(k^1)$ when $k\to 0$. The Fourier components for $k>q_\mathrm{max}=0.3\ [2\pi a^{-1}]$ are also filtered out to make the spectral density $\tilde{\rho}(\mathbf{q})$ symmetric in all the directions.
• Figure 3: Effects of Hyperuniform disorder in a Hermitian quadratic band. (a) The tight-binding model used in the simulation. The on-site energy $t_0$, the nearest neighbor hopping $t_1$, and the third nearest neighbor hopping $t_3$ are considered. (b) The spectral function $A_\mathbf{k}(E)$ when an uncorrelated on-site disorder is imposed on the tight-binding model in (a) with a system size of $N_x\times N_y=500\times 500$. The red dashed line shows the unperturbed band structure. (c) The scattering loss $\mathrm{Im}\left(\Sigma_\mathbf{k}\right)$ along $k_y=0$ when a hyperuniform on-site disorder is imposed on the system. The simulation results for two different values of $\alpha$ are shown in the upper panel ($\alpha=1$) and the lower panel ($\alpha=8$). The scattering loss $\mathrm{Im}\left(\Sigma_\mathbf{k}\right)$ is fitted by $C_0+C_{\beta_0}\cdot k^{\beta_0}$ to obtain the power law behavior with respect to $k$. (d) The comparison between the actual hyperuniformity exponent $\alpha$ and the fitted exponent $\beta_0$.
• Figure 4: Effects of Hyperuniform disorder in a non-Hermitian quadratic band. (a) The simulated reflection spectrum of the same quadratic band as in Figure \ref{['figure1']}(b) when an uncorrelated disorder (with disorder strength $w=3.44a^{-2}$) is imposed on the photonic crystal slab. The band becomes blurry because the linewidth is broadened by scattering loss $\mathrm{Im}\left(\Sigma_k\right)$ introduced by disorder. The system size is $N_x\times N_y=100\times 100$. (b) The excess linewidth along $k_y=0$ when $\alpha=1$ (upper panel) and $\alpha=8$ (lower panel). The FDTD simulation is averaged over 10 disorder configurations each with system size $N_x\times N_y=100\times 100$. The TB simulations and SCBA numerics are obtained from a system size of $N_x\times N_y=500\times 500$. (c) The excess linewidth at $k_x=0.004\ [2\pi a^{-1}]$ and $k_y=0$ when $\alpha$ is swept. (d) The excess linewidth is fitted by $C_0+C_{\beta_2}\cdot k^{\beta_2}$ in the tight-binding simulation to obtain the exponent of the second leading order $\beta_2$.
• Figure S1: The method to add disorder into a photonic crystal slab. (a) The unit cell (in the $x-y$ plane, $z$ is the out-of-plane direction) of our photonic crystal slab. The structure contains a circular air hole ($\varepsilon=1.0$) with $r_0=0.45a$ in a square lattice, where $a=1290nm$ is the lattice constant. The slab is made of silicon ($\varepsilon=12.11$) with thickness $h=0.05a$. (b) The simulated band structure of (a) along the $k_y=0$ line. (c) The simulated band structure of (a), but with $r=0.43a$. (d) The tip energy ($E=\left(\frac{\omega}{c}\right)^2$) of the quadratic band as a function of the air hole radius $r$.