Theory of nanoparticle-induced frequency shifts of whispering-gallery-mode resonances in spheroidal optical resonators

Nanoparticle-induced modifications of the spectrum of whispering-gallery modes (WGMs) of optical spheroidal resonators are studied theoretically. Combining an ab initio solution of a single-resonator problem with a dipole approximation for the particle, we derive simple analytical expressions for frequencies and widths of the particle-modified resonances, which are valid for resonators with moderate deviations from the spherical shape. The derived expressions are used to analyze spectral properties of the resonator-particle system as functions of the particle's position, the size of the resonators, and the characteristics of WGMs. The obtained results are shown to agree well with available experimental data. It is also demonstrated that the particle-induced spectral effects can be significantly enhanced by careful selection of the resonator's size, refractive index, and other experimental parameters. The results presented in the paper can be useful for applications of WGM resonators in biosensing, cavity QED, optomechanics, and other areas.

Figure 1

Resonator-particle system with all geometric and material parameters. Also shown are the resonator's and particle's coordinate systems centered at the resonator and the particle. The magnified image of the particle in the upper right corner shows its size and refractive index.

Figure 2

Frequency dependence of the element of the $T$ matrix in the vicinity of the second-order radial resonance $(s=2)$ expressed in terms of the size parameter $x=k{R}_r$. The black curve shows this dependence for a spherical resonator $(e=0)$, for which peaks with the same $l$, but different $m$, coincide. The gray curves show the same graphs for a spheroidal resonator with $n_r=1.59$ ellipticity parameter $e=1-R_{s\mathrm{m}}/R_r=0.048$ for different values of $m$.

Figure 3

Dependence of the frequency shifts $\delta {\omega }_{L,M,S}^{(1,+)}\left(\pi /2\right)$ versus orbital number $L$ for the equatorial position of the particle. The two upper graphs correspond to $M=L$ mode and radial numbers $S=1$ and $S=2$, while the two lower curves represent $M=L-2$ mode with the same two radial numbers. The second shift $\delta {\omega }_{L,M,S}^{(1,-)}\left(\pi /2\right)$ vanishes for the equatorial position of the particle.

Figure 4

Dependence of $\delta {\omega }_{L,L,S}^{\left(1,+\right)}$ and $\delta {\omega }_{L,L-1,S}^{\left(1,-\right)}$ on the particle's angular coordinate for $L=630, M=L,L-1$, and $S=1,2$.

Figure 5

Dependence of the major, $\delta {\omega }_{40,40,1}^{(1,+)}$ (left vertical axis), and minor, $\delta {\omega }_{40,40,1}^{(1,-)}$ (right vertical axis), frequency shifts on the angular coordinate of the particle for a fundamental $L=40$ mode of the first radial order.

Figure 6

Angular dependence of the two particle-induced frequency shifts $\delta {\omega }_{40,M,1}^{\left(2,-\right)}$ (dashed lines) and $\delta {\omega }_{40,M,1}^{\left(2,+\right)}$ (solid lines) for the $L=40\mathrm{TM}$ modes of the first radial order. The black single-maximum lines present this dependence for the fundamental mode. The gray lines with two maxima present $M=39$ polar mode.

Figure 7

The particle-induced frequency shifts $\delta {\omega }_{630,M,1}^{\left(2,-\right)}$ (dashed lines) and $\delta {\omega }_{630,M,1}^{\left(2,+\right)}$ (solid lines) versus the particle's angular coordinate for the TM $L=630$ fundamental (black) and $M=L-1$ (gray) modes of the first radial order.