An Exact Energy Conservation Law for Magneto-Optical Nanoparticles

Abstract

Energy conservation imposes fundamental bounds on the polarizabilities of nanoparticles (NPs). While such bounds are well established for isotropic and bianisotropic NPs, they remain unexplored for magneto-optical NPs. Here, we derive the exact energy-conservation law governing the electric and magnetic dipolar response of axially symmetric magneto-optical NPs under general illumination conditions and arbitrary external magnetic fields. Two central results follow from energy conservation: (i) purely magneto-optical scattering, where the non-magnetic polarizability vanishes, is fundamentally forbidden, and (ii) strong magneto-optical scattering regimes, in which the magneto-optical polarizability dominates, are intriguingly allowed.

Figures (3)

• Figure 1: A magneto-optical NP sustaining electric (${\bf p}$) and magnetic (${\bf m}$) dipolar modes is subjected to an external magnetic field ${\bf B}_{\rm{ext}}$.
• Figure 2: Optical response of a spherical magneto-optical nanoparticle (radius $R = 15~\mathrm{nm}$) described by a Drude model with $\varepsilon_{\rm inf} = 1$ and $\hbar \omega_p = 4~\mathrm{eV}$. Top row: Conservation of energy given by $(\mathcal{J}_{\rm{E}} - |\mathcal{O}_{\rm{E}}|)/V$ for different damping rates $\hbar \gamma$. Here $V$ denotes the volume of the NP. Bottom row: logarithmic ratio $|\alpha_{\rm{MO}, \rm{E}}|^2 / |\alpha_{\rm{OR}, \rm{E}}|^2$. The incident wavelength $\lambda$ is indicated in nm, and $\hbar\omega_c$ in eV. Subplots are labeled (a)–(f).
• Figure 3: $|\alpha_{\rm{MO}, \rm{E}}|^2 / |\alpha_{\rm{OR}, \rm{E}}|^2$ (log scale) vs the wavelength for several constant values of the damping rates $\hbar \gamma$. The parameters are fixed: $\omega_c = 0.01$ eV, $R = 15~\mathrm{nm}$, $\varepsilon_{\rm inf} = 1$ and $\hbar \omega_p = 4~\mathrm{eV}$.