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

TL;DR

The paper addresses how purely lossy defects in one- and two-dimensional non-Hermitian photonic crystals can induce transmission/reflection singularities not accessible with lossless defects, enabling dramatic gain enhancements. It develops a transfer-matrix and topological-analysis framework to show that exceptional-point degeneracies and branch-cut singularities in the reflection coefficient, characterized by nontrivial winding numbers, underlie these effects. In 1D, loss-optimized defects produce extreme gain peaks (up to >$10^8$) and high-Q resonances linked to topological transitions; in 2D, analogous phenomena yield quasi-BIC-like resonances with very high $Q$ at corresponding topological transitions. The findings highlight loss as a powerful design parameter to engineer singularities and topological protection in non-Hermitian photonics, offering routes to enhanced nonlinear responses in photonic devices.

Abstract

We show that purely lossy defects in one- and two-dimensional non-Hermitian photonic crystals can induce transmission matrix singularities not accessible with lossless defects. These singularities in turn can enable dramatic enhancement in overall system gain not accessible through conventional means. We further show that the underlying mechanism behind the loss-induced gain enhancement is due to the resonances being located specifically at topological branch cut singularities in the reflection coefficient with nontrivial winding numbers. The resulting resonances can exhibit exceptionally high quality factors in excess of $\sim 10^4$. Our work highlights the counterintuitive role of loss in engineering singularities in the gain response in non-Hermitian systems and its connection to topological phenomena in photonic systems.

Figures (6)

• Figure 1: (a) Schematics of the baseline periodic and defected 1D non-Hermitian photonic crystals on the top and bottom respectively. The baseline periodic system has 50 unit cells of length d of lossless $\varepsilon_a$ = 2 with length a = 0.8*d and gain layers of $\varepsilon_b$ = 2-0.1i with length b = 0.2*d. The defected structure is inversion symmetric about the center of the lossy defect layer of $\varepsilon_d = 2 + \varepsilon_d"*i$ with length d. 25 unit cells of the original baseline periodic system are located on the left of the defect. Those 25 unit cells on the left are then reflected across the center of the defect to create the right half of the structure. (b) Gain as measured by $|1-R-T|$ for the baseline periodic structure, defected system at a loss value of 0.2074.(c), (d) Plots of the real and imaginary parts of eigenvalues of the transmission matrix for the baseline periodic and lossy defect structures respectively.
• Figure 2: (a), (b) Surface plots of the real and imaginary parts of the eigenvealues of the 2-by-2 transmission matrix respectively in the defect loss and $Re(\omega)$ parameter space. An EP forms from a degeneracy in the real and imaginary parts coinciding at around a loss of 0.2074 and frequency of 0.3571. (c) Gain for the baseline periodic structure, defected system at a loss value of 0.01, defected system at the critical defect loss value of 0.14, and defected system at a defect loss value of 0.5.
• Figure 3: (a), (b), (c) Maximum gain achieved in the defect structure by sweeping defect loss vs defect size, complex reflection phase plotted over $Re(\omega)*a/(2\pi c)$ where a is the unit cell length and c is the speed of light vs defect loss, and cross sections of the reflection phase color plot at various defect loss values. A branch cut singularity appears starting at $\varepsilon_c" = 0$ and goes until $\varepsilon_c" = 0.14$ after which the complex reflection phase is no longer singular. The branch cut singularity also has a topological charge of C = +1. (d) Maximum gain achieved in the defect structure by sweeping $Re(\varepsilon)$ of the defect with no loss vs defect size. Complex reflection phase plotted over $Re(\omega) a/(2\pi c)$ vs $Re(\varepsilon_c)$ of the defect without any defect loss with color plot and cutlines in (e) and (f) respectively. The complex reflection phase exhibits a discontinuity for all values of $Re(\varepsilon_c)$. Maximum gain is achieved when the loss in the defect is around 0.14 at a reflection pole.
• Figure 4: (a) Examination of the role of various structural parameters on the gain response and reflection phase singularity in 3 ways: changing the real part of the permittivities in all the layers, changing the gain in the gain layers, and adjusting the thicknesses in all layers. Gain spectra and complex reflection phase cutlines corresponding to modification in (b), (c), and (d) respectively. The red curves show the maximum gain achieved by positioning the gain peak at a reflection pole through increasing the defect loss.
• Figure 5: (a) schematics of the baseline periodic and point-defected 2D non-Hermitian system. The baseline periodic system is shown on top with 12 unit cells of length a in the x-direction while infinitely periodic in the y-direction using Floquet periodic boundary conditions. Each unit cell is composed of air surrounding lossless rods of $\varepsilon_a = 2$ and gain rods of $\varepsilon_b = 2-0.1i$ with radii $0.17a$. The defected case is shown below which contains a lossy point defect composed of a rod with $\varepsilon_d$ and radius $r_d = 0.3*a$. The defect is surrounded by 6 unit cells of the baseline periodic system on the left and right extents and made to be $C_4$ symmetric about the defect. (b) Gain vs frequency plots of the baseline periodic crystal compared against the lossy line defected case at loss values of 0.2, 0.32, and 0.35.