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

Abstract

Nanoparticle-induced modifications of the spectrum of whispering-gallery-modes (WGM) 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 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 others. PACS 42.25.Bs, 42.25.Fx, 42.60.Da

Impact of nanoparticle-induced scattering of an azimuthally propagating mode on the resonance of whispering gallery microcavities

“…This phenomenon earlier discussed in Ref. [19] in terms of field symmetries under different polarizations can be explained here with a symmetry relation [52] τ − 1 ≈ ρ of APM scattering coefficients. The symmetry relation derives from a mirror symmetry of the particle-induced scattered electric field about ϕ 0, which arises from the mirror symmetry of the incident APM electric field (with only E z 3, we simply obtain ν c;AS ≈ cm∕2πn eff for the AS-mode, which is exactly the complex resonance frequency of the unperturbed WGM in absence of the nanoparticle τ 1; ρ 0.…”

“…The symmetry relation derives from a mirror symmetry of the particle-induced scattered electric field about ϕ 0, which arises from the mirror symmetry of the incident APM electric field (with only E z 3, we simply obtain ν c;AS ≈ cm∕2πn eff for the AS-mode, which is exactly the complex resonance frequency of the unperturbed WGM in absence of the nanoparticle τ 1; ρ 0. Our interpretation here provides a new viewpoint to understand the preservation of resonance in terms of the scattering properties of APMs at the nanoparticle, in comparison with the viewpoint in terms of the location of the nanoparticle at the node of the anti-symmetric resonant mode [4,19,20]. For TE S-mode or TM modes, the resonance frequency and Q-factor change with the particle size because no similar symmetry relation works.…”

“…The refractive indices of the nanoparticle, the cavity, and the surrounding medium take values 1.59 (polystyrene), 1.45 (silica), and 1 (air), respectively. The cavity is assumed to be excited by an external source, and the interaction between the cavity and exciting device (such as tapered fiber [4] or prism [20]) is neglected [15,19].…”

“…Here, note that the fields of the two counterpropagating APMs matched to the splitting modes are almost identical to the fields of the two counterpropagating unperturbed WGMs except for some subtle difference: they correspond to slightly different complex resonance frequencies; within the deep subwavelength ϕ range of the nanoparticle, the matched APMs have no definition (Appendix A), while the unperturbed WGMs have [15,20,26,38]. The latter difference arises from the different nature of the APM and the WGM: the APM is a waveguide mode (corresponding to a dispersion curve describing the dependence of k 0 n eff on frequency, see more details in Appendix A) [31,[33][34][35][36]40], while the WGM is a resonant eigenmode (corresponding to a single complex eigenfrequency) [19,20,26]. To seek the solution, Eq.…”

“…A degenerate perturbation theory [15] is developed when the interaction between the particle and WGM is strong enough to resolve a mode splitting. To give a complete description, a rigorous ab initio analysis based on the standard multisphere Mie formalism is performed for a spherical [16,17] 2D disk [18] and spheroidal optical resonators [19]. An intuitive semi-quantum electrodynamics (semi-QED) model [20] is proposed to describe the perturbation-induced coupling between two counterpropagating quantized WGMs and the coupling between WGMs and free-space radiation modes by deriving the system's Heisenberg equations of motion, with a classical dipoleapproximation of perturbations.…”