Ab initio computational analysis of spectral properties of dielectric spheroidal resonators interacting with a subwavelength nanoparticle

An efficient numerical method for determining the spectral characteristics and spatial distribution of the field of a spheroidal whispering-gallery-mode (WGM) resonator interacting with a dielectric nanoparticle is presented. The developed approach is based on a combination of $T$-matrix formalism applied to a single resonator with a dipole approximation for the field of the nanoparticle. The method is illustrated by computation of the scattered field of the resonator-particle system illuminated by an incident field in the form of a single WGM mode of TE or TM polarization mimicking the excitation of the resonances by a tapered fiber. Our calculations show that even a very small (less than 0.1%) deviation of the resonator's shape from an ideal sphere renders spherical approximation invalid. They also confirm that analytical resonant approximation for spheroidal resonators developed previously gives a reasonable qualitative description of the spectral characteristics of the resonator-particle system. It was found, however, that corrections to the resonant approximation are significant enough for realistic nominally spherical resonators to be taken into account for accurate analysis of the experimental data.

Figure 1

The resonator-particle system with all its geometric and material parameters: refractive indices of the resonator $n_r$, particle $n_p$, and the surrounding medium $n_1$; resonator's equatorial radius $R_r$, its smallest half axis $R_{sm}$, and particle's radius $R_p$. Also shown are four coordinate systems used in the paper: resonator-centered and particle-centered resonator's and particle's systems. Resonator's systems are characterized by the polar ($Z$) axis directed parallel to the axis of rotation of the resonator, while the particle's systems have their polar axes directed along a line connecting the centers of the particle and the resonator.

Figure 2

Real part of various diagonal and nondiagonal elements of the $T$ matrix ${\tilde{T}}_{10,l;10,\nu }^{(1,1)}$ (a) and ${\tilde{T}}_{39,l;39,\nu }^{(1,1)}$ (b) computed for the same frequency corresponding to a second radial order resonance of ${\tilde{T}}_{39,l;39,\nu }^{(1,1)}$ and $e=0.048$.

Figure 3

Real part of a diagonal element of the $T$ matrix ${\tilde{T}}_{m,39;m,39}^{(1,1)}$ as a function of the size parameter $x$. Black solid line corresponds to the spherical resonator with $2l+1$ degenerate polar modes ($e=0$). Colored dashed and dotted lines depict several resonances of a spheroidal resonator ($e=0.048$) with the same orbital number $l=39$ and different polar numbers $m$. Note that at the resonance the real part of the diagonal element of the $T$ matrix is very close to $-1$ in both spherical and spheroidal cases.

Figure 4

Dependence of the frequency splitting versus ellipticity of the resonator in case of TE excitation for two resonator-particle distances obtained from coefficients $p_{\pm 1}^{\left(1\right)}$. Gray lines (solid and broken) represent results of numerical computation, and black solid and broken lines show results obtained from analytical expressions derived using the resonance approximation of Ref. [67].

Figure 5

Dependence of the frequency splitting versus ellipticity of the resonator in the case of TM excitation for a single resonator-particle distance obtained from coefficients $p_{\mu }^{\left(2\right)}$. Gray lines (solid and broken) represent results of numerical computation, and black solid and broken lines show results obtained from analytical expressions derived using the resonance approximation of Ref. [67].

Figure 6

Angular dependence of the shift of the particle-induced resonance frequencies excited by a TM type ($p_{+}^{\left(2\right)},p_{-,0}^{\left(2\right)}$—left vertical axis) and TE type ($p_{-}^{\left(1\right)}$—left vertical axis and $p_{+}^{\left(1\right)}$—right vertical axis) excitation with $L=39$ and $M=L$ (a) and $M=L-1$ (b) obtained by observing maximums of $p_{\pm ,0}^{(1,2)}$ coefficients for $e=0.0005$. The coefficients $p_{+}^{\left(1\right)}$ and $p_{-}^{\left(2\right)}$ vanish at the equatorial position of the particle on the (a) graph, as well as coefficients $p_{-}^{\left(1\right)}, p_0^{\left(2\right)}$, and $p_{+}^{\left(2\right)}$ on the (b) graph.

Figure 7

Angular dependence of the shift of the particle-induced resonance frequencies excited by a TM type ($p_{+}^{\left(2\right)},p_{-,0}^{\left(2\right)}$—left vertical axis) and TE type ($p_{-}^{\left(1\right)}$—left vertical axis and $p_{+}^{\left(1\right)}$—right vertical axis) excitation with $L=39$ and $M=L$ (a) and $M=L-1$ (b) obtained by observing maximums of $p_{\pm ,0}^{(1,2)}$ coefficients for $e=0.048$. At the equatorial position of the particle the coefficients $p_{+}^{\left(1\right)}$ and $p_{-}^{\left(2\right)}$ vanish on the (a) graph, as well as coefficients $p_{-}^{\left(1\right)}, p_0^{\left(2\right)}$, and $p_{+}^{\left(2\right)}$ on the (b) graph.

Figure 8

Dependence of the $p_{-}^{\left(2\right)}$ coefficient on the offset from the resonator's frequency ($\delta \omega =0$) for a set of the polar angles ${\theta }_p, M=L-1$, and $e=0.048$.

Figure 9

Dependence of the scattered power on the frequency close to the resonator's frequency ($\delta \omega =0$) for $e=0.0005$. Graph (a) shows results for the fundamental resonator's mode $M=L=39$ and the particle located at the equator, while graph (b) represents results for $M=L-1=38$ and the particle located at polar angle ${\theta }_p={81}^{\circ }$. Solid lines correspond to analytical results and dashed lines correspond to numerical results obtained from Eq. (29).

Figure 10

Dependence of the resonator scattered power on the frequency close to the resonator's frequency ($\delta \omega =0$) for $e=0.048$. Graph (a) shows results for the fundamental resonator's mode $M=L=39$ and the particle located at the equator, while graph (b) represents results for $M=L-1=38$ and the particle located at polar angles ${\theta }_p={81}^{\circ }$ and ${77}^{\circ }$. Solid lines correspond to analytical results and dashed lines correspond to numerical results obtained from Eq. (29).

Figure 11

Dependence of the difference of peak positions of scattered power between the results of analytical approximation and numerical simulation on the angular position of the particle. Frequency difference is normalized on the modulus of the shift of the left peak from resonator's frequency at ${\theta }_p={90}^{\circ }$ [see Figs. 9 and 10] for fundamental mode $M=L=39$. Graph (a) corresponds to $e=0.0005$ and graph (b) is for $e=0.048$. Curves for TM mode are calculated down to the angle of ∼85° where both peaks merge together, while TE peaks are still separated even at 82°.

Figure 12

Azimuthal intensity pattern in the equatorial plane for the sphere (a) and spheroid with $e=0.0005$ (b) at radius $r=5\mu \mathrm{m}$ from the center. The angle is defined with respect to $Z_p$ axis with the particle located on the equator.