Multiple embedded eigenstates in nonlocal plasmonic nanostructures

Trapping light in open cavities is a long sought “holy grail” of nanophotonics. Plasmonic materials may offer a unique opportunity in this context, as they may fully suppress the radiation loss and enable the observation of spatially localized light states with an infinite lifetime in an open system. Here, we investigate how the spatial dispersion effects, e.g., caused by the electron-electron interactions in a metal, affect the trapped eigenstates. Heuristically, one may expect that the repulsive-type electron-electron interactions should act against light localization, and thereby that they should have a negative impact on the formation of the embedded eigenstates. Surprisingly, here we find that the nonlocality of the material response creates new degrees of freedom and relaxes the requirements for the observation of trapped light. In particular, a zero-permittivity condition is no longer mandatory and the same resonator shell can potentially suppress the radiation loss at multiple frequencies.

Figure 1

(a) Geometry of the open bounded bilayered spherical meta-atom. The shell is made of a plasmonic spatially dispersive material. (b) Characteristic function $D_{\mathrm{s}}$ as a function of frequency, for a meta-atom with $\beta /c=1/{10}^{3/2},R_{21}=1.1,R_1=0.973R_{10}$, and ${\omega }_{\mathrm{c}}=0$. The zeros indicate the values of the frequencies ${\omega }_{\mathrm{trap}}^{\left(j\right)}$. The insets give the values of the core permittivity ${\varepsilon }_1$ and of the shell transverse permittivity ${\varepsilon }_{2,\mathrm{T}}$ for the first three solutions. (c), (d) Embedded eigenstate frequency (blue lines) and susceptance at the core interface (green lines) as a function of the (c) locality strength $c/\beta$ and (d) normalized shell radius $R_{21}=R_2/R_1$. The susceptance is normalized to the free-space impedance ${\eta }_0$. The solid, dashed, and dotted-dashed curves correspond to the $j=1,2,3$ solution branches, respectively. The structural parameters are $\beta /c=1/{10}^{3/2},{\omega }_{\mathrm{c}}=0,R_{21}=1.1$, and $R_1/R_{1,0}=1$, except when one of the parameters is shown in the horizontal axis of a plot.

Figure 2

(a) Quality factor (of the dipolar mode) as a function of the normalized core radius $R_1/R_{1,\mathrm{trap}}$, for different values of the material loss in the plasmonic shell. (b) Variation of the real ${\omega }^{\prime }$ and imaginary ${\omega }^{″}$ parts of the eigenmode frequency $\omega ={\omega }^{\prime }+i{\omega }^{″}$ with $R_1/\phantom{R_1{R}_{1,\mathrm{trap}}}{R}_{1,\mathrm{trap}}$, for a lossless plasmonic shell $\left({\omega }_c/{\omega }_{\mathrm{p}}=0\right)$. The structural parameters are $\beta /c=1/{10}^{3/2},R_{21}=1.1,{\varepsilon }_1=1$. The quality factor diverges to infinity when $R_1=R_{1,\mathrm{trap}}=0.973R_{1,0}$ which yields $\omega ={\omega }_{\mathrm{trap}}=1.026{\omega }_{\mathrm{p}}$.

Figure 3

Electromagnetic fields of an embedded eigenstate. (a) Field amplitudes in the core-shell resonator (normalized to the electric field at the core center, $r=0)$ as a function of the radial distance in the xoy plane, calculated using the local (dashed lines) and the nonlocal models (solid lines). Local model results: $c/\beta \to \infty ,{\omega }_{\mathrm{trap}}={\omega }_{\mathrm{p}}$, and $R_{1,\mathrm{trap}}=R_{1,0}$. Nonlocal model results: $\beta /c=1/{10}^{3/2},{\omega }_{\mathrm{trap}}=1.026{\omega }_{\mathrm{p}}$, and $R_{1,\mathrm{trap}}=0.973R_{1,0}$. (b) Zoom of ${\eta }_0\left|H_z\right|$ (blue) and $|E_{\phi }|$ (green) in the shell for the nonlocal case. (c) ${\eta }_0|H_z{|}_{x=0}$ (blue) and $|E_r{|}_{y=0}$ (green) as a function of $c/\phantom{c\beta }\beta$ at center of the shell $\left(r=1.05R_1\right)$. (d) Time snapshots of the electric field in the nonlocal meta-atom: (i) $E_r\left(t=0\right)$ and (ii) $H_z\left(t=0\right)$ in the xoy plane. In all the panels, $R_{21}=1.1,{\varepsilon }_1=1$, and ${\omega }_{\mathrm{c}}=0$.

Figure 4

(a), (b) Spatial variation of the electromagnetic fields in the shell (cut in the xoy plane) for the modes labeled (i) and (ii) in Fig. 1, respectively. The right vertical axis is used for ${\eta }_0|H_z{|}_{x=0}$ and $|E_{\phi }{|}_{x=0}$, and the left vertical axis for $|E_r{|}_{y=0}$.

Figure 5

(a) Mie coefficient $|a_1^{\mathrm{TM}}|$ in the core region and (b) Mie coefficient $|c_1^{\mathrm{TM}}|$ in the air region as a function of the normalized frequency $\omega /{\omega }_{\mathrm{p}}$ and for three different values of the core radius. The meta-atom parameters are $\beta /c=1/{10}^{3/2},R_{21}=1.1,{\varepsilon }_1=1$, and ${\omega }_{\mathrm{c}}=0$. The embedded eigenstate is characterized by ${\omega }_{\mathrm{trap}}=1.026{\omega }_{\mathrm{p}}$ and $R_{1,\mathrm{trap}}=0.973R_{1,0}$.