As a next-generation sensor, whispering gallery resonators (WGRs) that support optical whispering gallery modes (WGMs) with a high quality ($Q$) factor possess ultrahigh sensitivity, long photon lifetime, and strong light confinement, and are widely used in label-free detection ^[1–13]. Tiny perturbations, e.g., adsorption of a single molecule ^{[2,6,9,11,13]}, virus ^[3,5,7], or nanoparticle ^{[4,7,8,10,12]} onto the surface of the WGR can change the resonance remarkably. Via tuning the excited wavelength and monitoring the transmission or reflection spectrum of the WGR, the change of resonance can be detected in the form of mode shift ^{[1–3,6,9,11–13]}, mode splitting ^{[4,5,8,10,14]}, or mode broadening ^[7,9,12].

To understand mode shift ^[1] and mode broadening ^[9] for sensing applications, a reactive sensing principle is proposed based on a first-order perturbation theory, and the perturbed eigenfrequency is expressed as a ratio of the perturbed energy relative to the unperturbed energy of WGMs. 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 dipole-approximation of perturbations. The semi-QED is widely used to explain the mode splitting ^[4,20,21] and mode broadening ^[7,22]. QED formalisms with the perturbation of quantum dot ^[23] or plasmonic nanoparticle ^[24] quantized as a two-level system are reported. Such perturbation-induced coupling of modes also can be described by a set of time-domain coupling equations derived from classical electrodynamics ^[25,26].

In its physical essence, the WGM is formed by azimuthally propagating modes (APMs) that satisfy the resonance condition ^[27]. This understanding of WGMs has inspired a variety of exciting works. For instance, by introducing an azimuthally periodic perturbation of WGRs that causes out-of-plane scattering of the APM, optical vortex beam emitters ^[28], and orbital angular momentum microlaser ^[29] are realized. Lasing and coherent perfect absorption are achieved simultaneously by using azimuthally periodic PT-symmetric WGRs ^[30]. WGMs can be modeled numerically with the use of APMs under a cylindrical coordinate system ^[31,32]. The viewpoint of APMs also is used to describe the coupling between the circular microresonator and straight bus waveguide ^[33–36].

In this paper, we report an intuitive and quantitative model to clarify the impact of nanoparticle-induced scattering of APMs on the resonant modes of WGRs. The model is built up by considering a dynamical multiple-scattering process of APMs at the adsorbed nanoparticle and can comprehensively reproduce all the features of the resonant mode of WGRs such as the resonance frequency, the $Q$ factor, and the field distribution. Simple analytical expressions describing the dependence of the mode shift, mode splitting, and mode broadening on the scattering coefficients of APMs are derived. The model is based on a first-principle calculation of the APM scattering coefficients without using any fitting or artificial setting of model parameters, which ensures a quantitative prediction and a further unveiling of the contribution of cross coupling between different WGMs to forming the resonant mode. The proposed model provides deep theoretical insight into the crucial phenomena of mode shift, mode broadening, and mode splitting for sensing applications from the viewpoint of APMs and has the strength to further clarify the physical origin of the particle-induced couplings among WGMs and radiation modes that are widely adopted in current theoretical formalisms ^{[20–22,26,37]}.

For simplicity, we consider a 2D cylindrical microcavity ^[18,38] [invariant along the $z$-axis, as sketched in Fig. 1(a)] and the developed model applies as well 3D structures (such as microdisk ^[39] or microtoroid ^[4] resonators). A sectorial nanoparticle (centered at $\varphi =0$ with a side length $D$), whose geometry is chosen to facilitate the calculation, is adsorbed on the surface of the microcavity (with a radius $R$). 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].

Without the perturbation of the nanoparticle, each pair of degenerate WGMs are formed by two matched counterpropagating APMs with a phase shift of $2m\pi$ over a $2\pi$ azimuthal angle ^[31,32]. The WGMs with different complex resonance frequencies can be indexed as $\mathrm{TE}/{\mathrm{TM}}_{s,m}$, with TE/TM denoting polarization (electric/magnetic vectors along the invariant $z$ direction), $s$ being the radial number (corresponding to APMs with different propagation constants along the azimuthal direction) and $m$ being the azimuthal number. With the adsorption of a nanoparticle, the degeneracy of the WGMs is lifted, which results in a pair of splitting resonant modes ^[4,5,8,10]. Next we will try to build up an analytical model to reproduce the splitting resonant modes by considering an intuitive multiple-scattering picture that incorporates the elastic transmission and reflection of APMs at the particle. In the model only two matched counterpropagating APMs (corresponding to the unperturbed degenerate WGMs) are considered, and all other mismatched APMs (corresponding to other different WGMs) are neglected. As sketched in Fig. 1(a), we use $a$ and $b$ to denote the unknown complex amplitude coefficients of the APMs propagating in positive and negative $\varphi$ directions, respectively. To determine $a$ and $b$, a set of coupled-APM equations can be written: where $\rho$ and $\tau$ are defined as the reflection and transmission coefficients of the APM at the particle that is assumed to be mirror-symmetric about $\varphi =0$ [as sketched in Fig. 1(b)], respectively, $u=\exp (i{k}_0{n}_{\text{eff}}2\pi )$ is the phase shift factor of the APM traveling azimuthally over one round of the cavity, $k_0=2\pi \nu /c$ ($\nu$ and $c$ being the frequency and the speed of light in vacuum, respectively), and $n_{\text{eff}}$ is the complex effective index of the APM that is a leaky waveguide eigenmode (with leaky loss so that $n_{\text{eff}}$ is generally a complex number) ^[40]. Here $n_{\text{eff}}$ is obtained with a full-wave aperiodic Fourier modal method (a-FMM) ^[41–43], and $\rho$ and $\tau$ are obtained as the scattering-matrix elements with the a-FMM ^[44] (more details about the calculation of $n_{\text{eff}}$, $\rho$, and $\tau$ under cylindrical coordinate system can be found in Appendix A, and some earlier work on the calculation of $n_{\text{eff}}$ can be found in Ref. ^[31]). The first-principle calculation of all quantities used in the model ensures a solid electromagnetic foundation and thus quantitative prediction of the model. Equation (1a) is written in view that the APM propagating along the positive $\varphi$ direction (with coefficient $a$) results from two contributions, one contribution from the transmission ($\tau$) of itself (with coefficient $a$ and a one-round phase shift factor $u$) at the particle, and the other contribution from the reflection ($\rho$) of the counterpropagating APM (with coefficient $b$ and a phase shift factor $u$) at the particle. Equation (1b) is written in a similar way. Equation (1) can be understood in a rigorous sense under a cylindrical coordinate system, as illustrated in Fig. 5(b) in Appendix A. To obtain nontrivial solutions of Eq. (1), which represent the resonant modes, the determinant of the coefficient matrix should be zero, which yields Equation (2) can be used to determine the complex resonance frequency ${\nu }_c$ of the pair of splitting resonant modes. Substituting Eq. (2) into Eq. (1), one obtains $a=b$ [corresponding to $\tau +\rho$ in Eq. (2)] or $a=-b$ (corresponding to $\tau -\rho$), which is defined as a symmetric (S) or anti-symmetric (AS) mode ^[4,20] (with electrical vector being symmetric or anti-symmetric about $\varphi =0$). For the special case without the particle, i.e., $\rho =0$ and $\tau =1$, Eq. (2) reduces to $u=1$, indicating that the one-round phase shift of the APM is multiples of $2\pi$ ^[31,32], which forms the two unperturbed degenerate WGMs. For the case with the presence of the particle, similar physical meaning preserves for Eq. (2) by additionally incorporating the particle-induced phase change $\arg (\tau \pm \rho )$ of APMs ^[45]. Equations (1) and (2) explicitly show that the degeneracy of the WGMs preserves if $\rho =0$, meaning that the particle-induced coupling between the two unperturbed counterpropagating WGMs (formed by the two matched APMs), which are widely adopted in current theoretical formalisms ^{[20–22,26,37]}, is realized via the reflection of APMs ($\rho \ne 0$) ^[35]. 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 $\varphi$ 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 $k_0{n}_{\text{eff}}$ on frequency, see more details in Appendix A) ^{[31,33–36,40]}, while the WGM is a resonant eigenmode (corresponding to a single complex eigenfrequency) ^[19,20,26]. To seek the solution, Eq. (2) can be rewritten as where $p=-\arg (\tau \pm \rho )+2\pi m$, $q=\ln |\tau \pm \rho |$. Details for deriving the approximate equality in Eq. (3) can be found in Appendix B. Thanks to the weak dependence of $n_{\text{eff}}$, $\rho$, and $\tau$ on frequency $\nu$ [and thus the weak dependence of the right side of Eq. (3) denoted as $f(\nu )$], the transcendental Eq. (3) can be solved with the contractive mapping method ^[46] with the iteration formula ${\nu }_{N+1}=f({\nu }_N)$, which converges fast and is fairly insensitive to the initial value ${\nu }_0$ [more details on solving Eq. (3) can be found in Appendix B]. In fact, a crude evaluation of the complex resonance frequency ${\nu }_c$ can be obtained by calculating $f(\nu )$ at a certain frequency.

With the complex eigenfrequency ${\nu }_c$ obtained, we can then determine the several key parameters in sensing applications ^[1–13], the frequency shift $\delta =\mathrm{Re}({\nu }_c)-\mathrm{Re}({\nu }_{c,0})$ (${\nu }_{c,0}$ being the complex resonance frequency of the unperturbed WGM), the $Q$-factor $Q=-\mathrm{Re}({\nu }_c)/[2\mathrm{Im}({\nu }_c)]$ and the frequency splitting $\mathrm{\Delta }=|\mathrm{Re}({\nu }_{c,S})-\mathrm{Re}({\nu }_{c,\mathrm{AS}})|$ (${\nu }_{c,S}$ and ${\nu }_{c,\mathrm{AS}}$ being the complex resonance frequencies of the S-mode and the AS-mode, respectively). The mode splitting can be resolved only if $\mathrm{\Delta }$ is greater than the sum $w=-[\mathrm{Im}({\nu }_{c,S})+\mathrm{Im}({\nu }_{c,\mathrm{AS}})]$ of the linewidths of the S-mode and the AS-mode ^[20,37]. Note that the experimentally resolvable splitting also depends on the external $Q$-factor ^[4] related to the fiber WGR coupling.

Now we check the validity of the model. We first consider the pair of splitting resonant modes arising from the unperturbed ${\mathrm{TM}}_{1,42}$ WGM for cavity radius $R=8\mathrm{\mu m}$. The quantities $\delta$, $Q$, and $\mathrm{\Delta }-w$ as a function of nanoparticle size $D$ are plotted in Figs. 2(a), 2(b), and 2(f), respectively, which are obtained with the finite element method (FEM) using COMSOL Multiphysics Software (circles), the original model (solid curves), and the simplified model (dashed curves). Here the original model and the simplified model refer to the equality and the approximate equality in Eq. (3), respectively. It is seen that the prediction of the original model and that of the simplified model have no observable difference (the dashed curves are in fact completely superimposed by the solid curves), and both predictions are quite accurate compared with the FEM numerical results, except for slight deviations for large particles ($D>600\mathrm{nm}$), which will be explained later. Similar accuracy of the model can be observed for ${\mathrm{TM}}_{1,59}$ ($Q$ close to ${10}^8$) and ${\mathrm{TE}}_{3,100}$ ($Q$ over ${10}^8$) modes with $R=11\mathrm{\mu m}$ and 20 μm, respectively (Fig. 7 in Appendix B).

As shown in Figs. 2(a) and 2(b), the redshift ($\delta <0$) of the resonance frequency increases and the $Q$-factor decreases with the increase of nanoparticle size $D$ ^[15,47]. To achieve an understanding, analytical expressions of $\delta$ and $Q$ can be obtained from Eq. (3): where $\mathrm{Re}({\nu }_{c,0})\approx cm/[2\pi \mathrm{Re}(n_{\text{eff}})]$ is the resonance frequency of the unperturbed WGM. In Eq. (4), $\tau$ and $\rho$ are dependent on $D$ and frequency $\nu$ [expressed as $\tau =\tau (D,\nu )$, $\rho =\rho (D,\nu )$], while $n_{\text{eff}}$ is independent of $D[n_{\text{eff}}=n_{\text{eff}}(\nu )]$, with $\nu$ taking the value of eigenfrequency ${\nu }_c$, which is further dependent on $D[{\nu }_c={\nu }_c(D)]$. In view of the weak dependence of $n_{\text{eff}}$, $\rho$, and $\tau$ on frequency $\nu$, we obtain $\tau \approx \tau (D)$, $\rho \approx \rho (D)$, and $n_{\text{eff}}$ approximately independent of $D$ [as confirmed by Figs. 2(c)–2(e)]. Therefore, Eq. (4a) indicates that the frequency shift $\delta$ is simply proportional to $-\arg (\tau \pm \rho )$. Physically, $\arg (\tau +\rho )$ represents the phase change of the two counterpropagating coherently incident APMs when scattered at the nanoparticle ^[45] for the S-mode and $\arg (\tau -\rho )$ for the AS-mode. As shown by the numerical results in Fig. 2(c), such scattering-induced phase change remains positive and increases monotonously with the increase of particle size. Blueshift ($\delta >0$) may happen ^[4,24,47] if the refractive index of the particle is smaller than that of its surrounding medium (Fig. 8 in Appendix B). The positive (resp. negative) $\arg (\tau \pm \rho )$ and their monotonous dependence on $D$ can be understood by considering that the incident APM hops over the nanoparticle via another APM with a higher (resp. lower) effective index (Appendix B), which provides insight into the increase of the effective optical path length ^[11,48] or average index ^[47] that causes the mode shift.

Concerning Eq. (4b), the first and the second terms correspond to $Q$-factors related to the propagation loss ($1/Q_{\text{prop}}$, composed of leaky loss and absorption loss of APMs, the latter being in fact excluded by considering lossless material here) and the particle-scattering induced loss ($1/Q_{\text{scat}}$) ^[9,49,50], which are shown in the inset of Fig. 2(b) and exhibit a crossing point $D_c$. It is seen that $1/Q_{\text{prop}}$ dominates over $1/Q_{\text{scat}}$ for $D<D_c$ and vice versa for $D>D_c$. Comparison between the insets of Figs. 2(b) and 7(b) (with the same radial number 1 of APMs) shows that $D_c$ becomes smaller for higher $Q$ factors (with larger azimuthal number), which can be understood in view of the decrease of $1/Q_{\text{prop}}$ and the approximate invariance of $1/Q_{\text{scat}}$. A lower $Q$-factor for a larger $D$ can be understood in view that $|\tau \pm \rho |<1$ holds due to the energy conservation ^[45] and $|\tau \pm \rho |$ decreases with the increase of the scattering-induced energy loss for larger $D$ [as confirmed by Fig. 2(d)]. This analysis shows that the particle-induced coupling between the unperturbed WGMs and the free-space radiation modes widely adopted in current theoretical formalisms ^{[20–22,26,37]} is realized via the scattering-induced free-space radiation of APMs. Note that if we define $n_{\text{eff}}^{\prime }=n_{\text{eff}}/R$ so that the phase shift factor of APM can be expressed as $\exp (i{k}_0{n}_{\text{eff}}\varphi )=\exp (i{k}_0{n}_{\text{eff}}^{\prime }L)$ with $L=\varphi R$ denoting the arc length, then $\mathrm{Re}(n_{\text{eff}}^{\prime })=\mathrm{Re}(n_{\text{eff}})/R\approx 1.27$ is reasonably between the refractive indices 1 and 1.45 of air and silica.

Due to the different amounts of frequency shift between the S-mode and AS-mode, the phenomenon of mode splitting ^[4,20] emerges if $\mathrm{\Delta }-w>0$. Figure 2(f) shows that, with the increase of particle size $D$, $\mathrm{\Delta }-w$ first increases from negative (no mode splitting but mode shift ^[1,9] and broadening ^[7,9,12] with high $Q$-factors for $D<40\mathrm{nm}$, below the red dashed–dot line) to positive values (with mode splitting), and then decreases to large negative values (no mode splitting with low $Q$-factors for $D>300\mathrm{nm}$). This observation is consistent with earlier results ^[15,51]. Simple analytical expressions of $\mathrm{\Delta }$ and $w$ can be obtained from Eq. (3):

Now we look at the case of TE resonant modes. Numerical results fully parallel with those in Fig. 2 but corresponding to unperturbed ${\mathrm{TE}}_{1,42}$ WGM are shown in Fig. 3. In sharp contrast with TM modes, it is seen that the frequency shift $\delta$ and $Q$-factor of the TE AS-mode (red curves) are almost unchanged until the nanoparticle size increases to large values ($D>200\mathrm{nm}$). This phenomenon earlier discussed in Ref. ^[19] in terms of field symmetries under different polarizations can be explained here with a symmetry relation ^[52] $\tau -1\approx \rho$ of APM scattering coefficients. The symmetry relation derives from a mirror symmetry of the particle-induced scattered electric field about $\varphi =0$, which arises from the mirror symmetry of the incident APM electric field (with only $E_z$ component under TE polarization) within the deep subwavelength $\varphi$ range of the nanoparticle. As numerically confirmed by the red curves in Figs. 3(c) and 3(d), $\tau -\rho \approx 1$ holds for small values of $D$ but becomes less accurate for large values of $D$ due to the vanishing of the symmetry. Inserting $\tau -\rho \approx 1$ into the right side of Eq. (3), we simply obtain ${\nu }_{c,\mathrm{AS}}\approx cm/(2\pi n_{\text{eff}})$ for the AS-mode, which is exactly the complex resonance frequency of the unperturbed WGM in absence of the nanoparticle $(\tau =1,\rho =0)$. 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 conclusions here are fully consistent with those from the rigorous ab initio analysis ^[18,19] but are only partially consistent with those from the semi-QED theory ^[20] for which the preservation of one resonance is independent of mode polarization.

The previous results show that, for considerably large particle size ($D>600\mathrm{nm}$), the model exhibits observable deviation from the numerical FEM results. This deviation unveils a fact that, besides the matched APMs considered in the model, other mismatched APMs also have contributions to forming the resonant mode, especially for large particles. This could be surprising because, in current theoretical formalisms ^{[4,20–22,26,38]}, the resonant mode is commonly treated as a superposition of two unperturbed counterpropagating WGMs (formed by the matched APMs). The contribution of mismatched APMs implies the existence of the cross coupling between the unperturbed WGMs and other different WGMs. To see this contribution directly, we show the residual field (denoted by ${\mathbf{E}}_{\mathrm{res}}$) that excludes the matched APMs’ field (${\mathbf{E}}_{\mathrm{APM}}$) contained in the total field (${\mathbf{E}}_{\text{tot}}$) of the resonant mode, i.e., ${\mathbf{E}}_{\mathrm{res}}={\mathbf{E}}_{\text{tot}}-{\mathbf{E}}_{\mathrm{APM}}$. Details for calculating the residual field with the mode orthogonality can be found in Appendix C. As shown by the numerical results in Fig. 4 for two particle sizes ($D=100$ and 500 nm), the residual field is much weaker than the matched APM field for both particle sizes, which explains the high accuracy of the model. On the other hand, the residual field for $D=500\mathrm{nm}$ is much stronger than that for $D=100\mathrm{nm}$. In more detail, the mean electric-field intensities of the residual field at the outer surface of the microcavity (excluding the $\varphi$ range of the particle where the APM has no definition) are 0.0018 and 0.0458 for $D=100$ and 500 nm, respectively, which explains the lower accuracy of the model for larger particle sizes.

An intuitive and quantitative model for the pair of particle-induced splitting resonant modes of WGRs is built up by considering a dynamical multiple-scattering process of two counterpropagating APMs matched to the resonant modes. Simple analytical expressions of the resonance frequency and the $Q$-factor of the splitting modes in terms of the APM scattering coefficients at the nanoparticle are derived. The model unveils explicit relations between the particle-induced phase change and energy loss of the APM and the crucial phenomena of mode shift, mode broadening, and mode splitting for ultrahigh-sensitivity sensing applications; the model further explains the polarization-dependent preservation of one resonance and the particle-dependent redshift or blueshift with the symmetry and phase change in the scattering of APMs, respectively. The particle-induced coupling between the two counterpropagating unperturbed WGMs and the coupling between the WGMs and free-space radiation modes are shown to arise from the particle-induced reflection and free-space radiation of the APM, respectively. Contribution of mismatched APMs (corresponding to WGMs other than the unperturbed WGMs) to forming the splitting resonant modes especially for large particles is identified. The present model can be readily extended to more complex cases such as asymmetric particles ^[17], resonant plasmonic particles ^[12,24], or particle ensembles ^[21,38,53].