Non-Hermitian Fabry-Pérot Resonances

TL;DR

This work develops a continuous-model framework for non-Hermitian Fabry–Pérot resonances in high-contrast 1D resonator chains using propagation-matrix methods and a generalized capacitance matrix. It systematically characterises exceptional points—both approximate (from leading-order non-Hermiticity and radiation losses) and exact (via parity-time symmetric radiation conditions)—and links EPs to higher-order poles of the Green’s function and to zeros of a characteristic determinant $f(\omega;\delta)$. It further proves a non-Hermitian skin effect in the non-reciprocal, continuous setting by employing a gauge transformation to a self-adjoint problem, yielding broadband edge localization across subwavelength and non-subwavelength modes. The results extend subwavelength insights to the full frequency range, provide a concrete, computable framework for resonances in non-Hermitian media, and point toward three-dimensional generalizations of these phenomena.

Abstract

We characterise non-Hermitian Fabry-Pérot resonances in high-contrast resonator systems and study the properties of their associated resonant modes from continuous differential models. We consider two non-Hermitian effects: the exceptional point degeneracy and the skin effect induced by imaginary gauge potentials. Using the propagation matrix formalism, we characterise these two non-Hermitian effects beyond the subwavelength regime. This analysis allows us to (i) establish the existence of exceptional points purely from radiation conditions and to (ii) prove that the non-Hermitian skin effect applies uniformly across resonant modes, yielding broadband edge localisation.

Figures (7)

• Figure 1: The transmission coefficient $T(\omega)$ for a dimer system ($N=2$, $\ell_i = v_i = v = s = 1$, $\delta = 10^{-1}$) exhibits transmission peaks forming a characteristic Fabry-Pérot-type comb-like structure.
• Figure 2: Convergence of resonant frequencies as $\delta\to 0$. We consider a dimer $N=2$ with equal wave speeds $v_1=v_2=v = s=\ell_1=1$ and $\ell_2=1.1$. The mismatch in resonator length leads to different asymptotic convergence rates around $0$ and $\pi$.
• Figure 3: Coalescence of resonant frequencies as $\delta\to 0$. For the trimer system with complex material parameters from \ref{['ssec:complex_ep']}, $\omega_\pm$ denote the two resonances converging to $0$ with positive real part as $\delta\to 0$. We compare the cases at and away from the exceptional point by tuning $\theta$ to $\theta=\pi/4$ and $\theta=0$. For the dimer system from \ref{['ssec:rad_ep']}, $\omega_\pm$ denote the two resonances converging to $\pi$. At exceptional points, we observe higher-order coalescence of the corresponding resonances.
• Figure 4: Characteristic square-root sensitivity to parameter perturbations around the approximate and exact exceptional points for the systems from \ref{['ssec:complex_ep', 'ssec:rad_ep', 'ssec:prc_ep']}. For all systems, we choose $\delta=10^{-6}$ and observe that the exceptional points closely match the square-root sensitivity profile with respect to $\theta$.
• Figure 5: Resonances, marked by black dots, as characterised by solutions of $f(\omega;\delta)=0$ for the standard outgoing radiation conditions and solutions of $\Tilde{f}(\omega;\delta)=0$ for the perfect transmission radiation conditions. We consider the modified dimer with $N=2, \ell_i = 1, v_i=1$ and $s_1 = 3/2$ and $\delta=10^{-2}$.

Theorems & Definitions (26)

• Definition 2.1: Characteristic determinant
• Lemma 2.2
• Lemma 2.3
• Definition 3.1: Exceptional point
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• Remark 3.5
• Remark 3.6
• Definition 3.7: Parity-time symmetry