Multinanoparticle scattering in a multimode microspheroid resonator

We theoretically investigate the scattering of multiple Rayleigh nanoparticles in a microspheroid resonator which supports multiple polar whispering gallery modes. Different from a standard microsphere resonator in which polar modes with the same radial and angular mode numbers are degenerate, a spheroid resonator supports spectrally nondegenerate polar modes. The mode splitting, mode shift, and linewidth broadening of polar modes are deduced at the presence of multiple Rayleigh scatterers following the Weisskopf-Wigner semi-QED treatment. It is found that the radial, polar, and azimuthal coordinates of the nanoparticle can be inferred from the transmission spectra of the polar modes. In particular, the changes of both the mode shift and linewidth broadening show cosinoidal dependence on the azimuthal angle. Our scheme provides an efficient platform to locate the three-dimensional positions of multiple nanoparticles in a microcavity, which may push the microcavity sensing to sizing and further quantitative measurements.

Figure 1

(a) Schematic of the proposed system: multiple Rayleigh nanoparticles are located in the evanescent field of a slightly prolate WGM microspheroid. The microcavity is coupled by a tapered fiber. ($r_n,{\theta }_n,{\phi }_n$) are the radial, polar, and azimuthal positions of the $n\mathrm{th}$ scatterer. $a_{\mathrm{CW}}^{\mathrm{in}}$ denotes the input field of the clockwise mode, while $a_{\mathrm{CW}}^{\mathrm{out}}$ and $a_{\mathrm{CCW}}^{\mathrm{out}}$ are the output fields of the clockwise and counterclockwise modes. (b),(c) Typical transmission spectra of the polar modes with $m=l$ and $m=l-2$ upon the attachment of one nanoparticle (black solid curve) and two nanoparticles (red dotted curve) to the microcavity. The radii of the first and second scatterers are $100\mathrm{nm}$ and $40\mathrm{nm}$, respectively.

Figure 2

(a) Transmission spectra under different numbers of target nanoparticles $N=10$, 50, 100, and 200. The quality factor of the $m$ cavity mode $Q=1\times {10}^8$. (b) Typical transmission spectra under different values of $Q$. $N$ is assumed to be 50, $\lambda =1550\mathrm{nm}$, and $f_m^2\left(\overrightarrow{r_n}\right)=0.2$. Radii of the cavity, the preset nanoparticle, and target nanoparticles are $R=50\mu \mathrm{m}$, $r_0=70\mathrm{nm}$, and $r_n=30\mathrm{nm}$.

Figure 3

Upper limit $N_1$ of the number of target scatterers that can be resolved through mode splitting under different values of radii for target scatterer $r_n$between the range of $10\mathrm{nm}$ to $30\mathrm{nm}$. The radius of the preset scatterer is set at $r_0=60\mathrm{nm}$, $r_0=70\mathrm{nm}$, and $r_0=80\mathrm{nm}$ for the black solid curve, the red dashed curve, and the blue dotted curve. Quality factor of the cavity mode $Q=1\times {10}^8$, $\lambda =1550\mathrm{nm}$, and cavity radius $R=50\mu \mathrm{m}.$

Figure 4

Upper limit $N_2$ of the number of target nanoparticles that can guarantee the coupling strength condition under different radii of the target scatters $r_n$ and preset scatterer $r_0$. Other parameters are the same as in Fig. 3.