Exceptional Points in the Scattering Resonances of a Sphere Dimer

TL;DR

This work develops a first-principles framework to predict, locate, and exploit exceptional points in the electromagnetic scattering of a sphere dimer, spanning from the electroquasistatic to full-wave regimes. By casting the problem as a two-port circuit and leveraging a coupled-dipole description, it derives analytic PT-symmetry conditions in the QS limit and synthesis conditions for real-frequency EPDs in the retarded regime via material-dispersion tuning. The study demonstrates that near an EPD, single-parameter perturbations induce a characteristic square-root splitting of eigenfrequencies, yielding highly sensitive spectral responses in scattering, extinction, and absorption, with strong agreement between the circuit model and full Mie theory for Drude spheres. The findings point to potential sensing applications based on minimal scattering units and offer a scalable framework extendable to higher multipoles and complex arrays.

Abstract

We investigate exceptional points of degeneracy (EPDs) in electromagnetic scattering of a sphere dimer from the electroquasistatic limit to the fully retarded regime. In the quasistatic limit, we prove that $\parity\trev$-symmetric configurations, realized by spheres with complex-conjugate susceptibilities, host EPDs. Beyond this limit, retardation breaks $\parity\trev$-symmetry; nevertheless, by jointly tuning the material dispersion of the two spheres, we derive analytic conditions for the existence of EPDs at \textit{real-frequencies}. Near an EPD, we show that single-parameter perturbations yield the characteristic square-root splitting of the eigenfrequencies, and we quantify its impact on scattering, extinction, and absorption, clarifying sensing implications.

Figures (7)

• Figure 1: (a) Dimer composed by two spheres of radius $a$ and inter-center distance $d$. (b) Two-port equivalent circuit of two interacting particles in the coupled dipole approximation and in the full-wave regime. (b) Two-port equivalent circuit of two interacting particles in the coupled dipole approximation and in the static limit. (c) Configuration for characterizing the resonant mode of the dimer (i.e., $Z_{\mathrm{in}}=0$). In panels (b),(c),(d) $h\in\{x,y,z\}$ denotes the polarization of the dipolar mode.
• Figure 2: (a) Normalized electric dipole resistance $\mathcal{R}_{\mathrm{ed}}/\zeta_0$ (blue curve) and (b) reactance $\mathcal{X}_\mathrm{ed}/\zeta_0$ (red curve) as a function of $k_0a$. The RLC approximations (dashed line) are provided for reference. (c),(e) Normalized mutual resistance $R_{M}^h/(\zeta_0 \rho^2)$ and (d),(f) reactance $X_{M}^h/(\zeta_0 \rho^2)$ of the longitudinal (c)-(d) and transverse (e)-(f) coupling as a function of the electric distance $k_0d$. The first zeros of the mutual resistance are marked with $\times$.
• Figure 3: (a) Normalized eigenfrequency deviation ${\delta\omega_{\mathrm{qs}}}/{\omega_{\mathrm{ep}}^{\mathrm{qs}}} = {(\omega_\pm - \omega_{\mathrm{ep}}^{\mathrm{qs}})}/{\omega_{\mathrm{ep}}^{\mathrm{qs}}}$ in the complex-frequency plane for varying $\gamma_{2} = \gamma = -\gamma_{1}$, comparing full-wave (FW, blue) and quasistatic (QS, red) models for a conjugate-symmetric ($\chi_{1} = \chi_{2}^*$) dimer of Drude spheres with $k_{\mathrm{p}1}a = k_{\mathrm{p}2}a = 0.1$ and $d = 5a$. (b) Normalized eigenfrequency deviation in the FW regime ${\delta\omega}/{\omega_{\mathrm{ep}}} = {(\omega_\pm - \omega_{\mathrm{ep}})}/{\omega_{\mathrm{ep}}}$ for varying $\gamma_{2}$ with $\gamma_{1} = -\gamma_{2} + \alpha\Gamma_\mathrm{ep}$ in a Drude sphere dimer with $k_{\mathrm{p}2}a = 0.1$, $k_{\mathrm{p}1}a = k_{\mathrm{p}1,\mathrm{ep}}a$, and $d = 5a$. Results for $\alpha \in \{0.95, 1.00, 1.05\}$ show eigenfrequency coalescence only at $\alpha = 1$. The EPD is marked with $\circ$.
• Figure 4: (a) Real and (b) imaginary parts of the normalized eigenfrequencies deviation $\delta \omega_{\pm}/\omega_{\mathrm{ep}}$ of the dimer ($k_{\mathrm{p}2} a = 0.1$, $d=5a$) as a function of the normalized damping ratio detuning $\delta \gamma_2/\gamma_{2,\mathrm{ep}} = (\gamma_2-\gamma_{2,\mathrm{ep}})/\gamma_{2,\mathrm{ep}}$. The gain is tuned as $\gamma_{1} = -\gamma_{2} + \Gamma_\mathrm{ep}$. The EPD is marked with $\circ$.
• Figure 5: Peak's frequency splitting $\Delta\omega_{\mathrm{pk}}/\omega_{0}$ versus normalized detuning $\delta d/d_{\mathrm{ep}}$ for the geometric control parameter $\vartheta_i=d/a$. Solid: CDA circuit model; dashed: multiparticle Mie theory. The value $d = 0.99d_{\mathrm{ep}}$ (red $\times$), $d = d_{\mathrm{ep}}$ (blue $\times$), and $d = d_{\mathrm{ep}}$ (yellow $\times$) are provided for reference.