Resonances in light scattering from nonequilibrium dipoles pairs

Abstract

We consider the light scattering from a pair of point-like electrical dipoles. Whenever the polarizability of each dipole violates the optical theorem, the response of the pair (both in far-field and near-field) exhibits exact resonances as a function of the frequency and the inter-dipole distance. This polarizability is consistent with causality and the crossing condition (i.e., a real field generates a real response). Hence, the emergence of the resonances requires nonequilibrium conditions, e.g., corresponding to active dipoles. Within our approach (classical optics, monochromatic incident field, point-like dipoles), the exact resonances can be infinite. The resonances also appear in the equilibrium domain, where the optical theorem is valid. In that domain, they are finite, but can produce large amplification factors; e.g., for a pair of metallic nanoparticles under Drude's model, the single-particle plasmonic resonance can be amplified $\sim 10^{2}$ times. But the global maximization of the scattering can still be achieved by violating the optical theorem. Our results for one electric and one magnetic dipole show how resonances can amplify a weak magnetic response of a single dipole to the incident field. We also discuss an anti-resonance (dark-state) effect present in the two-dipole scattering.

Figures (2)

• Figure 1: The absolute value of the effective polarizability $|\hat{\Theta}_{33}|$ versus $\omega r=\omega|\bm{r}|$. $|\hat{\Theta}_{33}|$ is defined from (\ref{['motik']}) with $\xi=\zeta=1$, i.e., $\bm{k}\perp \bm{r}$ and $\hat{\bm{x}}\perp \bm{r}=(r,0,0)$, where $\bm{r}$ is the inter-particle radius-vector, $\hat{\bm{x}}$ is the (far-field) observation direction, and $\bm{k}$ is the wave-vector of the incident wave. The dimensionless polarizability $\hat{\alpha}$ holds Drude's form (\ref{['cato4']}) for two identical gold nanoparticles with size $\ell$; see (\ref{['au']}). Red curve: $\omega=2\times 10^{15}$ Hz, $\ell=10^{-6}$ cm, $\hat{\alpha}=-1.25\times 10^{-3}+i 2.16\times 10^{-5}$. The red curve reaches $110.7$ amplifying $\approx 10^2$ times $\alpha$ in (\ref{['kosh']}). Green curve: $\omega=10^{15}$ Hz, $\ell=10^{-6}$ cm, $\hat{\alpha}=-6.25\times 10^{-4}+i 2.15\times 10^{-5}$. Black curve: $\omega=10^{15}$ Hz, $\ell=5\times 10^{-6}$ cm, $\hat{\alpha}=-7.81\times 10^{-2}+i2.69\times 10^{-3}$. Brown curve: $\omega=10^{15}$ Hz, $\ell=8\times 10^{-6}$ cm. Magenta curve: $\omega=10^{14}$ Hz, $\ell=11\times 10^{-6}$ cm.
• Figure 2: The absolute value of the effective polarizability $|\hat{\Theta}_{33}|$ versus $\omega r=\omega|\bm{r}|$ for two gold particles with $\ell=6\times 10^{-5}$ cm holding Drude's model (\ref{['cato4']}). We set: $\xi=\zeta=1$, $\bm{k}\perp \bm{r}$ and $\hat{\bm{x}}\perp \bm{r}=(r,0,0)$, as in Fig. \ref{['fig1']}. For all 3 curves, $\omega$ changes in the vicinity of $\gamma$; see (\ref{['au']}). Red curve: $\omega=3.74532\times 10^{13}$ Hz, $\hat{\alpha}=-2.73929+i2.52204$. The red curve reaches $\approx 1600$ amplifying $\alpha$ in (\ref{['kosh']}). Black: $\omega=3.57143\times 10^{13}$ Hz, $\hat{\alpha}=-2.49781+i2.41167$. Magenta: $\omega=3.44828\times 10^{13}$ Hz, $\hat{\alpha}=2.32995(-1+i)$. The resonance approaches infinity by properly tuning both the inter-particle distance $r$ and $\omega$; e.g. $|\hat{\Theta}_{33}|=1.12\times 10^6$, when (for $\ell=6\times 10^{-6}$ cm) $\omega r=3.59224$ and $\omega=3.7484\times 10^{13}$ Hz.