Polaritonic spectra of optical Mie voids

TL;DR

This paper addresses how to realize and characterize polaritons in spherical Mie void cavities. The authors study three configurations—empty voids in a non-dispersive background, voids filled with a Lorentz resonant medium, and empty voids in a dispersive background—to derive analytical and semi-analytical descriptions of resonant frequencies and $Q$-factors, and to map weak versus strong light-matter coupling regimes. A compact polaritonic model is derived from a $2\times2$ non-Hermitian Hamiltonian with coupling $g=(\omega_0/2)\sqrt{f/\varepsilon_{\infty}}$, yielding a polaritonic spectrum that obeys $(\omega-\omega_v+i\gamma_v/2)(\omega-\omega_0+i\gamma_{ex}/2)=\frac{f\omega_0^2}{4\varepsilon_{\infty}}$ and enabling the identification of coupling thresholds $f_{th1}$ and $f_{th2}$ for anticrossings. The study shows that increasing background permittivity enhances $Q$-factors and facilitates strong coupling, while in dispersive backgrounds polariton gaps can form with potential localization of quasinormal modes due to absorption, offering design strategies for all-dielectric polaritonic devices. Overall, Mie voids emerge as versatile platforms for polaritonic engineering in high-index environments.

Abstract

The progress in understanding the optical and microscopic properties of polaritons relies on various optical cavities to confine electromagnetic radiation, which causes a demand for new platforms with higher $Q$-factors and better fabrication robustness. In this context, so called Mie voids -- spherical cavities inside a dielectric medium, where the light confinement occurs due to refractive index contrast at the air-dielectric interface -- present a substantial interest. Here, we theoretically study the resonant characteristics and polaritonic spectra of spherical Mie cavities loaded with resonant media, as well as address the inverted problem, where a Mie void is formed inside a resonant dispersive medium. We establish approximate analytical expressions for the $Q$-factors of Mie void cavities, find the parameter ranges of spherical voids leading to the regimes of weak and strong light-matter coupling and analyze the concomitant effects, such as $Q$-factor enhancement and spatial field localization, from the polaritonic perspective. Our result could be valuable for the design of new polaritonic systems.

Figures (10)

• Figure 1: Schematics of the three classes of systems analyzed in this paper. (a) An empty Mie void inside a transparent dielectric; (b) a Mie void loaded with a resonant medium embedded in a transparent dielectric; (c) an empty Mie void embedded in a dispersive resonant medium. The resonant media are represented by the Lorentz model, Eq. \ref{['Eq_4']}.
• Figure 2: Resonant frequencies of the TM$_{\ell=1,2}$ and TE$_{\ell=1,2}$ quasinormal modes of an empty spherical Mie void surrounded by different transparent dielectrics as functions of the void inverse radius in double logarithmic scale. Gray areas are guides for the eye, approximately indicating the spectral ranges of each particular mode. For the electric field profiles of the presented modes, see Fig. S1 of the Supporting Information.
• Figure 3: $Q$-factors of (a) TM$_{\ell=1,2}$ and (b) TE$_{\ell=1,2}$ modes of a Mie void as functions of the dielectric permittivity of the background material. The black lines represent the analytical expressions for the obtained $Q$-factors, Eq. \ref{['Eq_3']}.
• Figure 4: Polaritonic spectra of a Mie void surrounded by a transparent dielectric ($\varepsilon_{ \mathrm{bg} } = 16$) and loaded with a resonant medium. (a), (b) Eigenfrequency spectra of the TM$_{\ell=1}$ and TE$_{\ell=1}$ modes of the dielectric void loaded with a resonant medium ($f = 0.1$, $Q_{ex} = 100$), revealing a set of anti-crossings. (c) Same as (a) but for $f = 0.01$, $Q_{ex} = 100$. $\lambda_0 = 2 \pi c/ \omega _0$ is the resonant wavelength of the Lorentz material. The colored curves represent the analytical polaritonic spectra, Eq. \ref{['polspec']}.
• Figure 5: (a) Coupling diagram in the parameter space of the background dielectric permittivity $\varepsilon_ \mathrm{bg}$ and reduced oscillator strength $f$ in logarithmic scale for various $Q$-factors of the resonant medium filling the void. The lines indicate the corresponding threshold values of reduced oscillator strength $f_{\mathrm{th}1}$; the colored areas depict strong coupling domains between the TM$_{\ell=1}$$N=2$ cavity mode and the resonant medium in accordance with Eq. \ref{['Eq_7']}. Insets are a graphical illustration of weak and strong coupling in Mie voids loaded with resonant media. Black dashed circle marks the point of $Q$-factors matching, $Q_{ex} = Q_{v}$, resulting in strong coupling for arbitrarily low values of $f$. (b) Same as (a) but for the stronger threshold $f_{\mathrm{th}2}$, Eq. \ref{['Eq_9']}.