Uncovering hidden resonances in non-Hermitian systems with scattering thresholds

TL;DR

The paper addresses how scattering thresholds in periodic non-Hermitian systems produce Wood's anomalies and how resonances near branch points can strongly modify the optical response. It introduces a multi-valued rational approximation built on a coordinate transform $\tilde{k}$ and the AAA algorithm to extract resonances from real-frequency data, including hidden resonances on different Riemann sheets. Applied to a plasmonic line grating, the method reveals hidden resonances near scattering thresholds and shows how a few resonances suffice to reproduce sharp spectral features and the derivative discontinuities. The work provides a practical, efficient framework for interpreting and engineering spectral responses in nanophotonic devices, with potential relevance to metasurfaces, BICs, and sensing.

Abstract

The points where diffraction orders emerge or vanish in the propagating spectrum of periodic non-Hermitian systems are referred to as scattering thresholds. Close to these branch points, resonances from different Riemann sheets can tremendously impact the optical response. However, these resonances are so far elusive for two reasons. First, their contribution to the signal is partially obscured, and second, they are inaccessible for standard computational methods. Here, the interplay of scattering thresholds with resonances is explored and a multi-valued rational approximation is introduced to access the hidden resonances. The theoretical and numerical approach is used to analyze the resonances of a plasmonic line grating. This work elegantly explains the occurrence of pronounced spectral features at scattering thresholds applicable to many nanophotonic systems of contemporary and future interest.

Figures (4)

• Figure 1: Outline of the approach following the panels in indicated order: The specular reflection spectrum $R_0(k)$ exhibits a branch point where the $z$-component $k_{z,1}$ of the first order reflection $R_1$ transitions from purely imaginary (evanescent) to purely real (propagating). Samples of the the zero order Fourier coefficient $f_j$ are evaluated at various real valued $k$ in the vicinity of the branch point. The transformation, used to map the sampling points $k_j$ to $\tilde{k}_j$, locally provides a map of the Riemann surface. The AAA algorithm uses a subset $I$ of the samples to construct a rational approximation in barycentric form interpolating the $f_j$ with $j\in I$ and approximating all other data points in a least square sense. The rational function provides an analytic continuation of the data points to the full $\tilde{k}$-space and its poles correspond to resonances of the physical system. In the physical domain a branch cut is introduced, that cuts the Riemann surface in two distinct sheets, each defined over the entire complex plane. We choose a branch cut parallel to the negative imaginary axis, which in the example at hand approximately corresponds to the line $\mathrm{Im}\bigl(\tilde{k}\bigr)+\mathrm{Re}\bigl(\tilde{k}\bigr)=0$. We refer to poles as hidden resonances when they do not reside on the Riemann sheet that contains the real axis, as their influence on the spectrum is substantially concealed by the branch point.
• Figure 2: The specular reflection spectrum $R_0(k)$ across the three branch points $k_{\mathrm{b},1} = 0.5 \, k_{\mathrm{b},2}$, $k_{\mathrm{b},2} = \qty{1.074e7}{\per \meter}$ and $k_{\mathrm{b},3} = 1.5 \, k_{\mathrm{b},2}$. The gold grating, sketched in the circular inset, is illuminated by a TM-polarized plane wave at an incidence angle of $\Theta=30^\circ$. (a) The approximated spectrum together with reference points, a decomposition into six modal contributions and the residual contribution, which remains after subtracting the modal contributions from the full spectrum. (b) Resonances of the system: six resonances (blue crosses) are found at identical positions with and without the coordinate transformation $\tilde{k}$. In the single-valued approach (top), additional resonances cluster at wave numbers close to the branch points. Conversely, the multi-valued approach (bottom) reveals three additional resonances. The labels of the resonances within the circle refer to Fig. \ref{['fig3']}.
• Figure 3: Resonances passing a branch point. (a) The three resonances closest to the 50 support points (black dots) are displayed as a function of the groove width $w$, which is varied in steps of 1 from $w_1 = \qty{42}{\nano \meter}$ to $w_5 = \qty{60}{\nano \meter}$. The resonances inside the green boxes are not directly connected to the real axis above them, i.e., they are hidden resonances. (b) Specular reflection spectra corresponding to the labeled subset of resonances. (c) The magnetic field patterns corresponding to the three relevant resonances at selected widths.
• Figure 4: Multi-valued specular reflection. (a) The contributions of the resonances $b_3$ and $c_3$ explain the shape of the specular reflection (black line) at the respective sides of the branch point. (b) The Riemann surface of the double-valued specular reflection. The positions of the resonances are projected to the complex plane in the bottom part of the graph. The colors refer to the argument of $k-k_{\mathrm{b},1}$ that ranges from $-2\pi$ to $2 \pi$ to uniquely define every point of the double-valued function.