We review photonic applications of dielectric whispering-gallery mode (WGM) resonators-tracing the growth of the technology from experiments with levitating droplets of aerosols to ultrahigh-Q solid state crystalline and integrated on-chip microresonators.

Optical resonators with whispering-gallery modes-part II: applications

Explicit asymptotic formulas are derived for the positions, widths, and strengths of the morphology-dependent resonances in Mie scattering. These formulas are compared with numerical data and found to be highly accurate, especially for the low-order resonances most relevant to nonlinear processes. They permit the interpretation of experimental data on light scattering from microdroplets without resorting to the full apparatus of the Mie scattering formalism.

Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering

We have observed that very high-Q Mie resonances in silica microspheres are split into doublets. This splitting is attributed to internal backscattering that couples the two degenerate whispering-gallery modes propagating in opposite directions along the sphere equator. We have studied this doublet structure by high-resolution spectroscopy. Time-decay measurements have also been performed and show a beat note corresponding to the coupling rate between the clockwise and counterclockwise modes. A simple model of coupled oscillators describes our data well, and the backscattering efficiency that we measure is consistent with what is observed in optical fibers.

Splitting of high-Q Mie modes induced by light backscattering in silica microspheres

The study of resonant dielectric nanostructures with a high refractive index is a new research direction in the nanoscale optics and metamaterial-inspired nanophotonics. Because of the unique optically induced electric and magnetic Mie resonances, high-index nanoscale structures are expected to complement or even replace different plasmonic components in a range of potential applications. We study a strong coupling between modes of a single subwavelength high-index dielectric resonator and analyze the mode transformation and Fano resonances when the resonator’s aspect ratio varies. We demonstrate that strong mode coupling results in resonances with high-quality factors, which are related to the physics of bound states in the continuum when the radiative losses are almost suppressed due to the Friedrich–Wintgen scenario of destructive interference. We explain the physics of these states in terms of multipole decomposition, and show that their appearance is accompanied by a drastic change in the far-field radiation pattern. We reveal a fundamental link between the formation of the high-quality resonances and peculiarities of the Fano parameter in the scattering cross-section spectra. Our theoretical findings are confirmed by microwave experiments for the scattering of high-index cylindrical resonators with a tunable aspect ratio. The proposed mechanism of the strong mode coupling in single subwavelength high-index resonators accompanied by resonances with high-quality factors helps to extend substantially functionalities of all-dielectric nanophotonics, which opens horizons for active and passive nanoscale metadevices.

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Bound states in the continuum and Fano resonances in the strong mode coupling regime

Electromagnetic cavity modes in photonic and plasmonic resonators offer rich and attractive regimes for tailoring the properties of light-matter interactions. Yet there is a disturbing lack of a precise definition for what constitutes a cavity mode, and as a result their mathematical properties remain largely unspecified. The lack of a definition is evidenced in part by the diverse nomenclature at use - "resonance", "leaky mode", "quasimode", to name but a few - suggesting that the dissipative nature of cavity modes somehow makes them different from other modes, but an explicit distinction is rarely made. This perspective article aims to introduce the reader to some of the subtleties and working definitions that can be rigorously applied when describing the modal properties of leaky optical cavities and plasmonic nanoresonators. We describe some recent development in the field, including {calculation methods for quasinormal modes of both photonic and plasmonic resonators and the concept of a generalized effective mode volume, and we} illustrate the theory with several representative cavity structures from the fields of photonic crystals and nanoplasmonics.

Modes and Mode Volumes of Leaky Optical Cavities and Plasmonic Nanoresonators

The magnetic (or electric) fields of morphology-dependent resonances of a dielectric sphere are shown to form an orthogonal complete set for expanding divergence-free vectorial functions inside the dielectric sphere, provided that there is a spatial discontinuity in its refractive index, say, at the edge of the sphere. A transverse projection dyad that picks up the divergence-free part (or its generalization) of a vector is defined and shown to be expandable in terms of the magnetic (or electric) fields of these morphology-dependent resonances. Moreover, the transverse dyadic Green’s function in these dielectric spheres is in turn expressed as a sum of tensor products of relevant morphology-dependent resonance fields. Each term in the sum manifests itself as a resonant response to external perturbations. Thus the morphology-dependent resonance expansion provides a powerful tool to analyze various optical phenomena in dielectric spheres.

Dyadic formulation of morphology-dependent resonances. I. Completeness relation

The extremely narrow morphology-dependent resonances of a microdroplet seen in Mie scattering are important for the nonlinear optical phenomena in these microdroplets, the large storage time being responsible for optical feedback. A formalism is developed within the scalar wave analog to calculate the change in the widths caused by a small perturbation, such as a distortion of the microdroplet shape from perfect sphericity. This formalism may be viewed as the generalization of the usual Rayleigh–Schrodinger perturbation scheme to the imaginary part of the energy. The results are checked against T-matrix calculations. In particular, it is proved that any perturbation can only increase the widths of these resonances, provided that terms of O(1/Q0) can be ignored in the first- and second-order corrections, where Q0 is the original quality factor of the resonance. The result can be interpreted in terms of a generalized golden rule and is relevant to similar problems involving quasi-normal modes in quantum mechanics. The full theory beyond the scalar wave approximation can be developed, and it is expected that all qualitative features will survive.

Effect of perturbations on the widths of narrow morphology-dependent resonances in Mie scattering

We present a method for obtaining a simple relation between bulk and cavity-modified Raman gains for microcavities with arbitrary geometric shapes. The analytical expression for the microcavity-enhanced Raman gain quantitatively accounts quite well for all the main features of the related experiments.

Theory of microcavity-enhanced Raman gain.

Morphology-dependent resonances commonly observed in both the elastic and the inelastic scattering of light waves from dielectric spheres are in fact direct manifestations of complex frequency poles of the scattering matrix. Time-independent solutions to the wave equation at these poles are termed quasi-normal modes, which are characterized by the outgoing wave boundary condition at infinity and cannot be normalized in the usual sense. These resonances (or quasi-normal modes) are shown to form a complete set inside the dielectric sphere, provided that there is a spatial discontinuity in the refractive index, say, at the edge of the sphere. Novel definitions of norm and inner product are introduced. In addition, a time-independent perturbation method based on this completeness relation is developed to evaluate shifts in resonance frequencies when the refractive index is changed.

Completeness and time-independent perturbation of morphology-dependent resonances in dielectric spheres

The distribution of the internal field energy in nonresonant Mie scattering is shown to exhibit certain regularities not previously noticed. The distribution, when suitably expressed, is independent of the size of the spherical scatterer and obtainable from geometric optics, with the average over any spherical surface calculable in closed form. A discontinuity at a radius r = a/n is found and described, where a is the radius of the sphere and n the refractive index.

P. T. Leung

Papers

Dyadic formulation of morphology-dependent resonances. I. Completeness relation

Effect of perturbations on the widths of narrow morphology-dependent resonances in Mie scattering

Theory of microcavity-enhanced Raman gain.

Completeness and time-independent perturbation of morphology-dependent resonances in dielectric spheres

Characterization of the internal energy density in Mie scattering