Loss-driven gain enhancements driven by topological singularities in non-Hermitian photonic crystals defects

Technical Summary: Loss-driven gain enhancements driven by topological singularities in non-Hermitian photonic crystals defects

Problem Statement
While photonic crystals are widely utilized to enhance nonlinear optical responses and achieve high quality factors through lossless defects (such as bound states in the continuum or bandgap localization), the specific role of lossy defects in non-Hermitian photonic systems remains unexplored. Conventional wisdom often treats material loss as a detrimental factor that degrades performance. Furthermore, while frameworks exist to describe non-Hermitian phenomena—such as exceptional points (EPs) arising from eigenvalue coalescence and the analysis of poles and zeros in scattering matrices—the connection between lossy defects, topological singularities in the reflection coefficient, and dramatic gain enhancement has not been established. The authors investigate whether purely lossy defects can induce transmission matrix singularities and gain enhancements inaccessible to lossless systems.

Methodology
The authors employ a theoretical approach utilizing the Transfer Matrix Method (TMM) to model one-dimensional (1D) and two-dimensional (2D) non-Hermitian photonic crystals.

• System Design: The baseline 1D system consists of 50 unit cells alternating between lossless layers ($\epsilon_a = 2$) and gain layers ($\epsilon_b = 2 - 0.1i$). A point defect is introduced at the center, creating an inversion-symmetric structure with 25 unit cells on either side. The defect is characterized by a complex permittivity $\epsilon_d = 2 + i\epsilon''_d$, where $\epsilon''_d$ represents the magnitude of loss. A similar geometry is extended to a 2D system with rods arranged in a lattice, utilizing Floquet periodic boundary conditions in the transverse direction.
• Analysis Framework: The study analyzes the eigenvalues of the transmission matrix ($T$) to identify exceptional point degeneracies. Crucially, the authors examine the complex reflection coefficient ($r$) to calculate the integer winding number ($C$) of the reflection phase ($\phi = \text{Arg}(r)$) in the parameter space of frequency and defect loss. This topological invariant is used to characterize branch cut singularities and phase transitions.
• Parameter Sweeps: The gain response ($|1 - R - T|$) is evaluated while sweeping the defect loss ($\epsilon''_d$), defect thickness, and real permittivity components to map the conditions for maximum gain.

Key Contributions and Results

1. Loss-Induced Gain Enhancement: The authors demonstrate that introducing a purely lossy defect into a non-Hermitian photonic crystal can induce transmission matrix singularities that lead to dramatic gain enhancement. In the 1D system, tuning the defect loss to a critical value ($\epsilon''_d \approx 0.14$) results in a gain peak exceeding $10^8$, representing an enhancement of multiple orders of magnitude compared to the baseline periodic system or systems with lossless defects.
2. Exceptional Points and Singularities: While the system exhibits gain peaks, the authors clarify that the peak gain does not strictly coincide with the formation of an exceptional point (EP) in the transmission matrix eigenvalues (where $\text{Re}(\lambda_1) = \text{Re}(\lambda_2)$ and $\text{Im}(\lambda_1) = \text{Im}(\lambda_2)$). Instead, the optimal gain occurs when the system is positioned near reflection poles that possess nontrivial topology. The EP condition ($|t_{22}| = 2$) is approached but the maximum gain is achieved at a specific loss value where the system undergoes a topological transition.
3. Topological Phase Transitions: The core mechanism identified is a topological phase transition in the complex reflection phase.
   • As the defect loss ($\epsilon''_d$) is increased, the system transitions from a region with a trivial winding number ($C=0$) to a region with a nontrivial winding number ($C=+1$).
   • This transition is mediated by a branch cut singularity in the reflection coefficient.
   • The maximum gain is achieved precisely at the critical loss value where this topological phase transition occurs (the point where the branch cut singularity disappears or the winding number changes).
   • In contrast, varying the real part of the defect permittivity ($\text{Re}(\epsilon_d)$) in lossless defects results in discontinuous phase jumps but does not induce a topological phase transition or a comparable gain enhancement.
4. High Quality Factors: The resonances induced by these lossy defects exhibit exceptionally high quality factors ($Q > 10^4$). The authors characterize these states as symmetry-protected quasi-bound states in the continuum (quasi-BICs).
5. Generalizability: The phenomenon is shown to be robust across different structural parameters. Changing the real permittivity of the layers, the gain magnitude, or the layer thicknesses shifts the resonance frequency but maintains the requirement for a specific critical loss value in the defect to trigger the topological transition and maximize gain. The effect is successfully generalized from 1D to 2D non-Hermitian photonic crystals.

Significance
The paper highlights the counterintuitive role of loss in engineering singularities within non-Hermitian systems. It establishes that loss is not merely a dissipative mechanism but can be a design parameter to induce topological phase transitions in the reflection response. By positioning resonances at topological branch cut singularities with nontrivial winding numbers, the system achieves gain enhancements and quality factors unattainable through conventional lossless defect engineering. This work connects topological phenomena directly to gain response optimization, suggesting new pathways for engineering singularities to maximize performance in nonlinear photonic systems.