Generalized Wigner-Smith analysis of resonance perturbations in arbitrary $Q$ non-Hermitian systems

TL;DR

This work generalizes the Wigner-Smith framework to predict resonance shifts in open, non-Hermitian systems with arbitrary quality factors by linking pole movements to residues of a generalized Wigner-Smith operator. By reformulating the perturbation problem with unconjugated field products and suitable factorizations, the authors derive residue-based expressions that recover the conventional perturbation result for large $Q$ while remaining valid for any $Q$, including wavenumber perturbations and open geometries. They extend the theory to zeros and discuss alternate matrices $\mathbf{M}$ sharing the same pole structure, enabling flexible tracking of resonances and zeros. The theory is demonstrated on homogeneous and core-shell spheres, showing good agreement with direct perturbations and highlighting that zeros can exhibit much higher sensitivity than poles, which has implications for optimized sensing devices. The approach offers a versatile tool for analysis and inverse design of resonance-based systems, with potential impact on plasmonic sensors and photonic devices.

Abstract

Perturbing resonant systems causes shifts in their associated scattering poles in the complex plane. In a previous study [arXiv: 2408.11360], we demonstrated that these shifts can be calculated numerically by analyzing the residue of a generalized Wigner-Smith operator associated with the perturbation parameter. In this work, we extend this approach by connecting the Wigner-Smith formalism with results from standard electromagnetic perturbation theory applicable to open systems with resonances of arbitrary quality factors. We further demonstrate the utility of the method through several numerical examples, including the inverse design of a multi-layered nanoresonator sensor and an analysis of the enhanced sensitivity of scattering zeros to perturbations.

Figures (4)

• Figure 1: Extinction (blue, solid), scattering (orange, dashed) and absorption (green, dot-dashed) cross sections for a silica-gold nano-sphere in water with $r_c = 60\,\mathrm{nm}$ and $d_s = 10\,\mathrm{nm}$ as a function of $k$. A schematic diagram of the nanoparticle's multi-layer structure is also shown.
• Figure 2: Heat map of $\log(|a_1|)$ for different complex wavenumbers $k = \mathrm{Re}(k) + i\mathrm{Im}(k)$. Dark red regions correspond to poles and dark blue regions correspond to zeros. Cross sections of $|a_1|$ and $|a_2|$ for $\mathrm{Im}(k_p) =0$ are also shown.
• Figure 3: Heat maps of $|\mathrm{Re}(\eta_{b})|$ (left column) and $|\mathrm{Im}(\eta_{b})|$ (right column) for the LSPR indicated in Fig. \ref{['fig:poleszeros']} as a function of core radius $r_c$ and shell thickness $d_s$, calculated using both the Wigner-Smith based approach (top row, 'Wigner-Smith') and by perturbing the nanoparticle manually as described in the main text (bottom row, 'Direct').
• Figure 4: Heat maps of $|\mathrm{Re}(\eta_{b})|$ (left panel) and $|\mathrm{Im}(\eta_{b})|$ (right panel) for the lower frequency zero shown in Fig. \ref{['fig:poleszeros']} as a function of core radius $r_c$ and shell thickness $d_s$.