Spectral modification of whispering-gallery-mode resonances in spheroidal resonators due to interaction with ultra-small particles

Abstract

The recently developed general ab initio theory of nanoparticle-induced modifications of the spectrum of whispering gallery modes of optical spheroidal resonators is applied to the case in which distinct particle-induced resonances overlap and cannot be resolved. This situation occurs in the case of resonances with low Q-factors and/or ultra-small particles. The position of the single resonance observed in these situations depends on the strengths and widths of the overlapping resonances. We determine this position by considering the spectral characteristics of the total power scattered by the resonator. The obtained theoretical results are compared against the available experimental data and heuristic theories.

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

“…Our calculations showed that even smallest deviation of the resonator's shape from an ideal sphere renders spherical approximation for the shape of the resonator invalid and that the analytical resonant approximation for spheroidal resonances developed in Ref. [67,68] gives an accurate description of the spectral characteristics of the resonator-particle system. However, our calculations also showed that for the values of the ellipticity parameter of the order of 3-5%, which is typical for many nominally spherical resonators used in experiments, the corrections due to non-resonant elements of the T-matrix can become quite significant and result in up to 4-6% deviations of the resonant frequencies from the values predicted by the resonant approximation.…”

“…This work is an extension of our earlier efforts, where we used a similar idea to obtain a semianalytical description of the resonator-particle system in the so-called resonance approximation, which takes into account the interaction of the particle with only two remaining degenerate modes of a spheroidal resonator [67,68]. Now we are going outside of the resonant approximation and include into the consideration all modes of the system until convergence of the corresponding sums is achieved.…”

About size distribution of particles adsorbed to a spherical whispering-gallery-mode resonator in sensing experiments

“…This phenomenon was more fully investigated in subsequent work, Ref. [33], where the theory of Ref. [32] was applied to the case of ultra-small particles and the process of formation of the single shifted peak from two overlapping split peaks was explicitly demonstrated.…”

“…In this regard, it needs to be pointed out that the dipole approximation used in Refs. [28,30,32,33] is not equivalent to the uniform field approximation. While the latter requires that 𝑘 0 𝑛 𝑝 𝑅 𝑝 ≪ 1, where 𝑘 0 is the vacuum wave number at the resonance, 𝑛 𝑝 and 𝑅 𝑝 are refractive index and the radius of the particle respectively, the former only requires that the contributions from the multipoles with polar number 𝑙 > 1 and the magneto-dipole terms remain smaller than the electro-dipole contribution with 𝑙 = 1.…”

“…Therefore, the results for the frequency shift derived in Ref. [33] do not require any corrections in the form of a heuristic form factor used in Ref. [10] to extend the validity of RSM beyond the uniform field approximation.…”