Mode splitting induced by an arbitrarily shaped Rayleigh scatterer in a whispering-gallery microcavity

We investigate theoretically the mode splitting induced by an arbitrarily shaped Rayleigh scatterer attached to a whispering-gallery microcavity. The information including polar position, orientation, and polarizability tensor of the nanoparticle can be obtained through the mode-splitting signal. It is found that when the electric field of a pair of counterpropagating modes have different orientations and are not linearly polarized, both of the split modes experience frequency shift and linewidth broadening. The polar angle, at which a nanoparticle binds on the microsphere cavity, can be extracted by comparing frequency splittings of two fundamental TE modes. Moreover, the orientation and polarizability of the particle can be acquired by combining frequency splittings and linewidth broadenings of three modes which polarize the particle in different orientations. This work provides a comprehensive theory base for microcavity sensing of nanoparticles of arbitrary shape instead of nanospheres and could push nanoparticle detection towards quantitative characterization.

Figure 1

(a) Schematic diagram of an anisotropic nanoparticle coupled to a WGM microcavity. The scattering of the particle induces the coupling between the counterpropagating modes with the strength $g$, and the coupling of modes to the reservoir with the damping rate $\mathrm{\Gamma }$. (b) The detection signal of anisotropic and isotropic nanoparticles for a TE mode of a microsphere cavity, as a function of the orientation $\mathrm{\Psi }$ of the particle. Here, the anisotropic particle is an ellipsoid with three semimajor axes of length (9, 3, 1). The orientation $\mathrm{\Psi }$ is the angle between the orientation of electric field and the longest semimajor axis of the ellipsoid, and the shortest semimajor axis of the ellipsoid is vertical to the cavity (Inset). The frequency splitting and the linewidth broadening induced by an isotropic particle are not relevant to the orientation $\mathrm{\Psi }$ of particle, which are denoted by the blue dashed curve. The frequency splitting (black solid curve) and the linewidth broadening (red solid curve) induced by an anisotropic particle are normalized by those induced by an isotropic particle with the same volume.

Figure 2

Ratio between the two effective polarizabilities of frequency splitting and linewidth broadening for an anisotropic nanoparticle as a function of the orientation of the TE electric field $\mathbf{E}$ of a microsphere cavity. The orientation of the electric field rotates in the (a) ${\alpha }_x-{\alpha }_y$, (b) ${\alpha }_y-{\alpha }_z$, and (c) ${\alpha }_x-{\alpha }_z$ planes. (d) The orientation of the electric field is arbitrary. The insets show the orientation of the electric field $\mathbf{E}$. The three coordinate axes are parallel to the three principle axes of the polarizability tensor of the particle. The polar angle $\theta$ and the azimuthal angle $\varphi$ are used to represent the orientation of the electric field $\mathbf{E}$. Here, the three principle polarizabilities of the particle are taken as (10, 3, 1).

Figure 3

Ratio of frequency splitting $\left|2g\right|$ to intrinsic linewidth ${\mathrm{\Gamma }}_0$ for ellipsoid nanoparticles with different geometrical shape factors ($L_x, L_y, L_z=1$). All curves are symmetric about the angular bisector of the $x$ axis and $y$ axis (the dashed black line). The inset shows that the lengths of three semimajor axes of the nanoparticle are $L_x, L_y$, and the given $L_z=1$.

Figure 4

Detectable region division under different particle volume $V$ and intrinsic linewidth ${\mathrm{\Gamma }}_0$ of cavity modes. Blue (purple) regions stand for totally (partially) detectable region where particles with any eccentricity (some eccentricity) in this region can be detected. Gray regions represent undetectable region where no particles in this region can be detected. The red (green) division line represents the detection limit of spherical (strip-shaped) nanoparticles. Here, $f\left(r\right)=0.26, V_c=500\mu \text{m}, \omega =1.216\times {10}^{15}\text{Hz}$.