Whispering Gallery Modes: Theory & Applications

• Whispering gallery modes are resonant eigenfunctions that confine energy near boundaries or interfaces through high angular momentum and curvature.
• Their modal quantization utilizes tools like Airy-type boundary layers and Schrödinger-type operators to achieve exponential field localization.
• Applications in sensing, nonlinear optics, quantum photonics, and energy harvesting leverage their extremely high quality factors and tunable resonance frequencies.

Whispering gallery modes (WGMs) are special eigenfunctions and resonances of wave (acoustic, optical, electromagnetic) equations that concentrate energy in a curvilinear layer near the boundary or an internal interface, such that wave amplitude decays rapidly away from this layer. Historically associated with smooth convex boundaries (e.g. circular disks, spheres), WGMs are now established as a general phenomenon in domains and media exhibiting strong refractive index contrast or curvature. Their essential properties include quantized angular momentum, exponential boundary or interface concentration, extremely high quality factors (Q), tunable resonance frequencies, and unique local field distributions exploitable for sensing, nonlinear optics, quantum photonics, and energy harvesting.

1. Mathematical and Physical Foundations

WGMs arise as eigenmodes of differential operators governing wave propagation—most characteristically, the Laplacian or Maxwell’s equations with suitable boundary or material conditions. In a canonical setting, one considers a compact domain $M\subset\mathbb{R}^2$ or $\mathbb{R}^3$, possibly with a smooth boundary $\partial M$, or an interior interface $\Sigma$, and a possibly nonuniform coefficient $c(x)$ representing material inhomogeneity.

Classical Boundary WGMs

For smooth convex boundaries, e.g., in microcavities, the eigenfunction $u$ concentrates in an $O(\lambda^{-1/3})$ layer adjacent to $\partial M$, as a consequence of high angular momentum and boundary curvature. The modal structure can be accessed via separation of variables, yielding radial and angular quantum numbers, and the field envelope exhibits an Airy-type turning point matched to the boundary.

Transmission WGMs Across Interfaces

For a Laplacian with a jump in the coefficient $c(x)$ across an interface $\Sigma$, WGMs emerge exponentially close to $\Sigma$. Explicitly, if $c(x) = c_-$ for $x \in \Omega_-$ and $c(x) = c_+$ for $x \in \Omega_+$, with $0 < c_- < c_+$, and $\Sigma = \partial \Omega_-$, the operator

$\Delta_c u := \mathrm{div}(c(x) \nabla u)$

admits eigenfunctions $u_n$ with mass exponentially concentrated near $\Sigma$, decaying as

$$\|u_n\|_{L^2(\omega)} \leq C e^{-d \sqrt{\lambda_n}}$$

for any open $\omega$ with $\mathrm{dist}(\overline{\omega},\Sigma)>0$ (Filippas, 2023). The exponential barrier arises due to the discontinuity in the effective potential $V_c(r) = c(r)/r^2$, which is minimized at the interface.

2. Modal Quantization and Asymptotic Structure

The quantization of WGMs depends on domain geometry, refractive index profile, and symmetry. Typical modal indices include:

• Azimuthal (angular) number $m$ or $l$: the number of oscillations around the boundary.
• Radial order $p$ or $q$: number of nodes in the radial direction.

The resonance condition is commonly written as

$m\lambda \approx 2\pi n R$

for a sphere/disc of radius $R$, refractive index $n$, or more generally, as a solution to transcendental boundary matching equations involving spherical Bessel and Hankel functions.

Asymptotic expansions for resonance frequencies and mode localization are available for

• Smooth disks/spheres: Airy-function (boundary layer width $O(m^{-2/3})$), with super-algebraic decay of amplitude away from the boundary (Balac et al., 2020).
• Disks with radially varying index: Depending on the sign of the effective curvature $\kappa_{\mathrm{eff}} = 1/R + n'(R)/n(R)$, three regimes occur—step-linear, step-harmonic, and full harmonic—each with distinct leading behavior for resonance frequencies (Balac et al., 2020).

For transmission problems, semiclassical reduction leads to a one-dimensional Schrödinger-type operator, where the Agmon distance quantifies exponential concentration (Filippas, 2023).

3. Quality Factors and Field Localization

WGMs are unique in combining extremely high quality factors with tight spatial confinement:

• Boundary WGMs: Q factors can reach $10^8 - 10^{10}$ in optimized crystalline microcavities. Losses are dominated by material absorption, surface scattering, and radiative leakage; for large spheres/disks, the latter is exponentially suppressed.
• Transmission WGMs: The discontinuity at the interface produces a genuine exponential barrier, yielding even stronger concentration and exponential decay than the Airy boundary-layer regime.

Quality factor for a WGM is defined as

$Q = \frac{\omega U}{P_{\mathrm{loss}}}$

where $U$ is the stored energy, and $P_{\mathrm{loss}}$ is the total power lost per cycle (Strekalov et al., 2016, Perin et al., 2022). As $\lambda_n \to \infty$, mass outside the layer vanishes as $e^{-d\sqrt{\lambda_n}}$.

4. The Agmon Estimate and Exponential Localization

The exponential decay away from the concentration layer is mathematically established via Agmon estimates. For the transmission Laplacian in revolution-surface coordinates (with interface at $r = R_1$), passage to angular momentum modes yields a separated operator

$$P_h = -h^2 \frac{1}{r}\partial_r(c(r)r\partial_r) + V_c(r),$$

with semiclassical parameter $h = 1/n$. The Agmon identity delivers

$$\int h^2 |\partial_r(e^{\phi/h}\psi_h)|^2 + \int e^{2\phi/h}|\psi_h|^2 \leq C e^{2\delta/h}$$

where $\phi(r)$ is a weight proportional to the Agmon distance from the classically allowed region near the interface; this yields the exponential localization (Filippas, 2023).

5. Applications and Implications

WGMs underpin a broad range of scientific and technological applications:

• Sensing: Exploiting high Q and tight localization near boundaries or interfaces for biochemical detection, with sensitivities demonstrated up to 500 nm per refraction index unit in plasmonic coated resonators (Xiao et al., 2010).
• Nonlinear optics: High-Q WGMs in microresonators boost interaction strengths, with implementations for second-harmonic generation, parametric down-conversion, and frequency combs (Strekalov et al., 2016).
• Quantum photonics: Spin–momentum–decay locking in spherical WGMs facilitates chiral quantum devices and unidirectional photon routing (Khosravi et al., 2019).
• Energy harvesting: Low-Q WGMs in metamaterial shells yield broad-band, strongly radiative interface states tunable by shell thickness, relevant for electromagnetic energy capture (Díaz-Rubio et al., 2013).
• Transmission uniqueness: Construction of transmission WGMs demonstrates the sharpness of quantitative unique continuation estimates for singular media, with observation norms decaying as $e^{-d\sqrt{\lambda_n}}$ (Filippas, 2023).

6. Comparison of Classical and Transmission WGMs

Classical WGMs depend critically on smooth boundary geometry for their existence, with boundary-layer thickness scaling as $O(\lambda^{-1/3})$ and exponential localization governed by Airy-type phenomena. Transmission WGMs are predicated on internal material discontinuity, with the effective potential having a discontinuous minimum at the interface, resulting in even tighter trapping and stronger exponential decay.

Table: Localization and Decay Regimes

Regime	Localization Layer Width	Decay Rate Outside Layer
Classical	$O(\lambda^{-1/3})$	$\exp(-\mathrm{const}\,\lambda^{1/3})$
Transmission	Arbitrarily thin	$\exp(-d\sqrt{\lambda})$

This distinction underlies both mathematical theory and experimental application: the interface discontinuity in transmission WGMs acts as a genuine exponential barrier rather than the algebraic decay of the Airy regime, yielding superior isolation of modal energy.

7. Outlook and Further Advances

Ongoing research expands the modal concept of WGMs to:

• Arbitrary curved surfaces, where geodesic ray tracing and differential geometry tools generalize classical WGM quantization to complex manifolds (Wang et al., 2020).
• Hybrid photonic architectures including photonic molecules, plasmonic coatings, metamaterial shells, and stretchable microstructures, where modal properties are engineered by controlling geometry, material composition, and symmetry (Li et al., 2017, Kudo et al., 2013, Madugani et al., 2012).
• Topological and chiral WGMs, notably gyromagnetic photonic crystals supporting unidirectional, topologically protected edge states quantized by loop topology (Chen et al., 2023).

The mathematical theory of WGMs continues to evolve alongside their experimental exploitation, providing precision tools for wave control at the nanoscale, quantum interfaces, and robust functionalities in complex media.