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 $|\mathbf{k}|<sup>α$ observed in the Hermitian case (where $α$ is the hyperuniformity exponent), the scattering loss in the non-Hermitian band is given by $C_0+C_{β_2}\cdot|\mathbf{k}|<sup>{β_2}$, 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.

• The paper shows that intrinsic non-Hermiticity changes disorder-induced loss from the Hermitian k^α law to a finite k-independent term plus a subleading contribution capped at k².
• The authors combine Born-approximation analysis with tight-binding and FDTD simulations of silicon photonic crystal slabs, confirming the predicted scaling and identifying SCBA as necessary for strong low-exponent disorder.
• The results demonstrate that radiative loss enables scattering between modes of different energies, limiting hyperuniform suppression and providing design guidance for low-loss photonic devices with realistic open-system dynamics.

Overview

This paper investigates how hyperuniform disorder scatters light in photonic crystal slabs whose quadratic band is intrinsically non-Hermitian due to out-of-plane radiative loss. The central result is a qualitative change in the momentum dependence of disorder-induced scattering loss: in a Hermitian quadratic band, the scattering loss scales as $\mathrm{Im}(\Sigma_{\mathbf{k}})\propto k^{\alpha}$, where $\alpha$ is the hyperuniformity exponent of the imposed disorder, whereas in the non-Hermitian case the leading-order scattering loss is a finite $k$-independent constant proportional to $\mathrm{Im}(m)$, with a subleading term $C_{\beta_2}k^{\beta_2}$ where $\beta_2\leq 2$ regardless of how large $\alpha$ is. The authors derive these results analytically within the Born approximation and validate them against tight-binding (TB) and full-wave Finite-Difference Time-Domain (FDTD) simulations using realistic photonic crystal parameters.

Physical platform and intrinsic non-Hermiticity

The system is a silicon photonic crystal slab ($\varepsilon=12.11$) patterned with circular air holes on a square lattice of period $a=1290$ nm and thickness $h=0.05a$. The slab hosts an isolated TE-like quadratic band near $\alpha$0, described by

$\alpha$1

with tip energy $\alpha$2 (wavelength 1529 nm) extracted from FDTD reflection spectra. Because the mode lies above the light line, it couples to the radiation continuum; symmetry-protected bound states in the continuum (BICs) at $\alpha$3 enforce that both the resonance frequency and the linewidth vary quadratically with $\alpha$4. Consequently, the effective mass is complex: $\alpha$5. The imaginary part encodes radiative loss, which enters the theory as a non-Hermitian perturbation even in the absence of material absorption. The excess linewidth observable, related to the self-energy by $\alpha$6, provides a direct experimental handle on the scattering loss.

Disorder generation

Hyperuniform disorder configurations are produced via Fourier filtering: starting from uncorrelated on-site potentials drawn uniformly from $\alpha$7, each Fourier component below a cutoff $\alpha$8 is rescaled so that the spectral density obeys $\alpha$9 for $k$0 and vanishes above the cutoff. A key design choice is that the real-space potential variance, $k$1, is independent of $k$2, enabling fair comparisons across different hyperuniformity exponents. Disorder is mapped onto hole-radius variations through a calibrated relation between hole radius and band tip energy.

Hermitian quadratic band

For real effective mass, evaluating the Born-approximation self-energy over the iso-frequency contour yields, with no free parameters,

$k$3

The exponent carries over directly from the spectral density because both the group velocity and the iso-frequency contour radius grow as $k$4 in a quadratic band. TB simulations on a $k$5 lattice, with hoppings tuned to reproduce the quadratic dispersion up to $k$6, confirm this scaling quantitatively for $k$7 ranging from 0 (uncorrelated, giving a $k$8-independent loss) to large values; fitted exponents $k$9 track $\mathrm{Im}(m)$0 closely, with small deviations attributed to finite-size effects from the $\mathrm{Im}(m)$1 prescription.

Non-Hermitian quadratic band

When the effective mass is complex, the structure of the self-energy changes fundamentally. The denominator acquires an extra term $\mathrm{Im}(m)$2, which regularizes the small-$\mathrm{Im}(m)$3 divergence of the integral and produces a finite constant contribution:

$\mathrm{Im}(m)$4

where the subleading exponent satisfies $\mathrm{Im}(m)$5 for all $\mathrm{Im}(m)$6, with logarithmic corrections $\mathrm{Im}(m)$7 exactly at $\mathrm{Im}(m)$8. This is the paper's most striking claim: arbitrarily weak non-Hermiticity qualitatively rewrites the scaling law, replacing $\mathrm{Im}(m)$9 by a constant plus a term capped at $C_{\beta_2}k^{\beta_2}$0. Physically, the constant arises because loss permits scattering between modes of different energy, removing the energy-conservation constraint that restricted single-scattering events to the iso-frequency contour. The result reduces continuously to the Hermitian power law when $C_{\beta_2}k^{\beta_2}$1. An additional feature is a discontinuous jump in $C_{\beta_2}k^{\beta_2}$2 at $C_{\beta_2}k^{\beta_2}$3, where the coefficient of the $C_{\beta_2}k^{\beta_2}$4 term accidentally vanishes.

FDTD simulations on $C_{\beta_2}k^{\beta_2}$5 slabs (averaged over ten disorder realizations) and non-Hermitian TB simulations confirm the theory for moderate and large $C_{\beta_2}k^{\beta_2}$6. For small $C_{\beta_2}k^{\beta_2}$7, however, both simulations fall below the analytical prediction: the leading-order self-energy becomes large enough to shift the spectral peak appreciably, invalidating the assumption that $C_{\beta_2}k^{\beta_2}$8 can be evaluated at the unperturbed band energy. The authors resolve this discrepancy with the self-consistent Born approximation (SCBA), which accounts for multiple scattering and restores quantitative agreement — at the cost of losing closed-form expressions.

Limitations and open questions

Several caveats qualify the results. The analytical formulas are perturbative in the disorder strength ($C_{\beta_2}k^{\beta_2}$9), and their breakdown at small $\beta_2\leq 2$0 — where the SCBA is required but no analytic form exists — marks a genuine gap between the transparent theory and the accurate numerics. The derivation also assumes $\beta_2\leq 2$1, $\beta_2\leq 2$2, and neglects $\beta_2\leq 2$3 relative to $\beta_2\leq 2$4 near $\beta_2\leq 2$5; the regime near the BIC itself, where the linewidth extraction is unreliable, remains inaccessible in FDTD. Finite-size effects limit observation of the predicted jump in $\beta_2\leq 2$6, and fitted exponents at large $\beta_2\leq 2$7 slightly exceed the theoretical bound due to higher-order terms entering the fitting window. Whether the constant-loss behavior persists for other dispersions (e.g., linear or quartic bands) or under strong disorder approaching localization is not addressed here.

Conclusion

This work establishes that intrinsic radiative loss, modeled as a complex effective mass, fundamentally alters the wave dynamics of hyperuniform disorder in photonic crystal slabs: the characteristic $\beta_2\leq 2$8 suppression of scattering gives way to a finite, $\beta_2\leq 2$9-independent loss plus a subleading term bounded by $\alpha$0. The agreement among analytics, TB, and realistic FDTD simulations — including the SCBA resolution of the small-$\alpha$1 discrepancy — provides a quantitative benchmark for non-Hermitian hyperuniform wave transport and indicates that loss must be incorporated into any accurate description of disorder effects in realistic photonic devices.