1. Introduction

Whispering gallery modes (WGMs) are of manifold application in state-of-the-art photonics. These high-quality optical modes are utilized in devices such as filters/switches [1–5] or low-threshold micro lasers [6–8], but also in topical studies of, e.g., cavity-quantum electro dynamics [9–11] or topological and exceptional point photonics [12–19]. The shapes of these cavities are nearly as numerous as their applications: WGMs have for example been demonstrated in spheres [6,20], rings [1,21], disks [22–24], toroids [5,9,25], goblets [26,27], and bottles [28–30].

Whispering gallery modes in spheres have been demonstrated first [31], show very high quality factors [20], and even allow for the sensing of single molecules [32–34]. Regarding such spherical WGM cavities, analytical solutions of the modes’ field distributions are well established [35]. But although the different possible solutions of WGMs are calculated to be orthogonal, interactions between these modes have been found experimentally. Such couplings have been identified via mode splittings or avoided crossings [36–39]. These mode couplings have been utilized to demonstrate various effects as narrow-band high reflectivity [37], coupled-mode induced transparency and absorption [39], or Kerr frequency comb generation [38]. Most often, such mode splittings are attributed to the clockwise and counter-clockwise mode being coupled due to sub-wavelength scatterers [37]. But also a coupling of other pairs of WGMs (including modes of differing polarization) due to scattering [38], strain-induced birefringence [36] or small and unintended asymmetries of the spherical cavity [39] has been shown.

Within recent years however, the majority of research activities concerns substrate-bound resonators suitable for lab-on-a-chip systems as well as upscaling to large-scale fabrication processes [40–42]. Most of these cavities are either ring-, disk-, or toroid-shaped and can therefore be condensed as rotationally symmetric structures including a mirror symmetry with respect to the plane of the whispering gallery mode. Just as in spherical cavities, splittings and avoided crossings of modes within such mirror-symmetric resonators have been experimentally demonstrated [23,43–45]. We also have experimentally found avoided crossings of WGMs in a polymeric disk-shaped cavity similar to that presented in [46] under thermal tuning. The avoided crossing has been accompanied by a gradual exchange of the polarization of the two modes. Often, such mode interaction is again attributed to birefringence or sub-wavelength scatterers as well as other cavity imperfections like angled sidewalls [23,44,47]. In contrast to the spherical case however, these perturbations are no fundamental condition for such mode couplings. Due to the reduced symmetry of these cavities, modes of different spatial field distribution and/or polarization within a perfect non-spherical cavity from isotropic media can be intrinsically coupled via the vectorial character of Maxwell’s equations [45,48]. In this case, these modes are non-orthogonal [35,49] and thus can interact. This mode coupling can also be understood from the close analogy to the interaction of electron terms in molecules with two or more atoms [50].

An analytical description of WGMs in such cavities is not directly possible but can be done in an approximating manner based on solutions of the scalar Helmholtz equations [51,52]. Recently, the intrinsic avoided crossing of modes in isotropic and birefringent cavities has been derived analytically by Werner et al. and Sturman et al. by treating the vectorial properties of Maxwell’s equations as a perturbation [45,48]. The analysis has been done for spheroidal cavities on the mm-scale and includes selection rules of the modal coupling. In case of birefringent resonators from lithium niobate, the calculated spectral distances of the avoided crossings have been shown to be in good agreement with experiment.

Another case, where this intrinsic coupling is of special interest is that of mirror-symmetric resonators on the $\mathrm {\mu }$m-scale. E.g. in the context of state-of-the-art integrated photonics, where such cavities are widely applied [53,54], the intrinsic coupling might enable novel applications like a WGM-based polarization converter. As the dimensions (especially the thickness) of such cavities is often in the order of the wavelength, a larger influence of the reduced symmetry and thus stronger coupling effects compared to those in mm-scale resonators can be expected. Due to the reduced size of the micro-scale resonators, numerical methods can be used efficiently and are often applied to investigate the mode structure. Carmon et al. have investigated the coupling of two WGMs of different polarization within an isotropic toroidal cavity using finite element method (FEM) simulations [25]. An avoided crossing, including a continuous exchange of mode properties, has been found. Regarding possible future applications of such mode couplings, a detailed understanding of this intrinsic effect is indispensable.

In this paper, we conduct a qualitative analysis of the intrinsic coupling of WGMs in rotationally symmetric $\mathrm {\mu }$m-scale resonators with an additional mirror symmetry on the basis of extensive FEM simulations of WGMs in an isotropic, cylindrical disk-shaped cavity. We especially analyze the systematics of this mode interaction as well as its impact on the mode structure of such cavities. At first, we recap the established analytical solutions of WGMs in cylindrical cavities and determine the range of their validity from a detailed study of intra-cavity mode couplings within the FEM simulations. We propose novel conserved mode properties solely based on true symmetries of the underlying refractive-index distribution. Thereof, we introduce an alternative classification scheme of WGMs in mirror-symmetric cavities. Although all calculations have been conducted for disk-shaped cavities, the main results of this analysis should apply to all non-spherical but rotationally symmetric resonators with an additional mirror symmetry [55].

2. Analytical description of whispering gallery modes

In the following, we shortly review first the perfect analytical solution of WGMs in spherical cavities and afterwards the approximate analytical solution of modes in cylindrical disk-shaped cavities as they can both be found in literature.

Lets assume an isotropic dielectric sphere of radius $R$. Starting from the Helmholtz equation in spherical coordinates ($\rho,\theta,\varphi$), a separation of variables can be performed. This approach leads to the following solution: For both transverse-electric (TE, electric field in the equatorial plane) and transverse-magnetic (TM, magnetic field in the equatorial plane) modes, the radial part of the field distribution is given by Bessel functions of the first kind inside the cavity ($\rho\; < \;R$) and Hankel functions of the second kind outside ($\rho\; > \;R$). The field distribution in the polar ($0\leq \theta \leq \pi$) and azimuthal ($0\leq \varphi \leq 2\pi$) directions is given by vector spherical harmonics $\textbf {Y}_{l,m}\left (\theta,\varphi \right )$ of degree $l$ and order $m$. The sign of $m$ defines the azimuthal propagation direction. As their resonance frequency does not depend on $m$, several of these modes are degenerate. As a more convenient approach to characterize these possible solutions, so-called mode numbers are introduced, which are counting the intensity antinodes along each direction of the coordinate system. In radial direction, the respective mode number $N_\rho$ is given by the number of extrema of the Bessel function within the cavity ($\rho\; <\;R$). Regarding the angular field distribution, the mode numbers can be drawn from the degree and order of the vector spherical harmonics. The number of intensity antinodes along the azimuthal(polar) direction is given by $N_\varphi = 2m$ ($N_\theta = l - \left |m\right | + 1$). The term of so-called fundamental modes refers to the case of $N_\rho = 1$ as well as $l = m$ and thus $N_\theta = 1$. In combination with the modes’ polarization state TE/TM, these three mode numbers describe a complete basis of orthogonal whispering gallery modes. [35]

The most suitable coordinate system to describe an isotropic and cylindrical disk-shaped cavity is that of cylindrical coordinates, in which a disk of radius $R$ and thickness $d$ is given by $\rho\; <\;R$ and $-d/2\;<\;z\;<\;d/2$. Such a resonator disk and the respective coordinate system are illustrated in Fig. 1(a). In line with the solutions of spherical cavities, it seems natural to also characterize the modes in these cavities based on their polarization state TE/TM as well as their number of intensity antinodes along the radial ($N_\rho$), azimuthal ($N_\varphi$), and axial ($N_z$) direction of the coordinate system. This approach is backed up by respective analytical solutions that are obtained in a similar manner as described above via a separation of the scalar Helmholtz equation [23,35,49,56]. In azimuthal direction, one obtains a field distribution following $\exp \left (\pm \text {i} N_\varphi \varphi \right )$ with $N_\varphi \in \mathbb {N}$ and $2N_\varphi$ intensity antinodes along the circumference of the disk. Along the radial direction, again Bessel functions of the first kind ($\rho\; <\;R$) and Hankel functions of the second kind ($\rho\; >\;R$) describe the field distribution. Also these solutions are characterized by their number of antinodes $N_\rho$ within the cavity. Regarding the axial direction, cosinusoidal field distributions with $N_z$ extrema for $-d/2\;<\;z\;<\;d/2$ similar to those of modes in slab waveguides are found. In combination, the latter two solutions of the separated differential equations lead to a two-dimensional array of $N_\rho \times N_z$ intensity antinodes in the $(\rho,z)$-plane. Again, the cases of $\left (N_\rho,N_z\right ) = (1,1)$ are referred to as fundamental modes. (We’d like to comment that often the mode numbers $N_\rho$ and $N_z$ are defined as number of intensity antinodes - 1, giving the fundamental mode the values $\left (N_\rho,N_z\right ) = (0,0)$. In contrast to that, we here use the definition given above for convenience.) In Fig. 1(b) and (c), ring- and toroid-shaped cavities are illustrated, that yield the same symmetries as resonator disks. For such cavities, comparable analytical solutions are found [35,49,57].

 

Fig. 1. Illustration of mirror-symmetric WGM cavities. (a) In this article, modes in disk-shaped cavities of radius $R$ and thickness $d$ (orange) are investigated in detail. The applied cylindrical coordinate system is shown in red. Aside the rotational symmetry, the cavity disks yield an additional mirror symmetry. The respective plane of symmetry is indicated in blue. (b)/(c) Illustration of ring-shaped/toroid-shaped WGM cavities in the $(\rho,z)$-plane. These cavities have the same symmetries as the disk in (a). For all resonator shapes, the area where the field distribution of high-quality WGMs is typically localized is shaded.

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As commonly known, these analytical solutions of WGMs in cylindrical disk-shaped (and other mirror-symmetric) cavities are only approximations due to two underlying assumptions that do not generally hold [23,35,49,56]: the existence of independent sets of TE and TM modes, which is adapted from the analytical description of slab waveguides, as well as the separability of the $\rho$- and $z$-components of the Helmholtz equation in cylindrical coordinates. These assumptions are incorporated into the analytical calculations via the so-called effective index method (EIM) [49,58] and are directly transferred onto the obtained solutions: independent TE and TM modes with separable field distributions of two-dimensional arrays of $N_\rho \times N_z$ intensity antinodes in the $(\rho,z)$-plane. Despite their limited validity, these analytical solutions are useful in numerous scenarios. For instance, they generally hold in cases where there is a large index contrast between the cavity of interest and its surrounding as well as if its thickness is very small or very large compared to the wavelength of the light. Thus, they are extensively used to effectively describe the mode dynamics in experimentally realized mirror-symmetric WGM resonators [23,57,59–66].

In this paper, we investigate the limitations of the validity of these analytical solutions in detail based on finite element method simulations.

3. Finite element method simulation

We have performed FEM simulations using the commercially available simulation software package JCMsuite (JCMwave GmbH, Berlin, Germany) in a so-called 2.5-dimensional configuration by utilizing the rotational symmetry of the cavities [67]. Such simulations are widely established and details can be found elsewhere [68,69]. The assumption of perfect rotational symmetry implies the conservation of the number of intensity antinodes along the azimuthal direction [25,45]. All modes within this paper have been calculated with an equal azimuthal mode number of $N_\varphi =139$. Thus, the presented investigations do not cover interactions between modes of different $N_\varphi$ due to a possible lack of rotational symmetry.

In comparison to the analytical EIM, the only approximations within the FEM methodology are the triangulation of the $(\rho,z)$-plane into finite area elements as well as the numerical solution of Maxwell’s equations within these elements. Especially, FEM simulations do not imply any pre-defined orientation of the polarization or a separability of the refractive-index distribution. Thus, such FEM calculations are well-suited to test the limitation of the validity of the EIM-based analytical description outlined earlier. We expect all deviations of the results of these two approaches from one another to be induced by the invalid assumptions underlying the analytical effective index method.

We have calculated all modes within this paper for a perfect resonator disk of radius $R = {25}\,\mathrm{\mu}\textrm{m}$, an isotropic and real refractive index of $n_\text {res}=1.481$ in a surrounding of index $n_\text {sur}=1.000\,275$ and an imaginary permittivity of the resonator medium of $\epsilon _\text {i}=10^{-4}$, leading to quality factors in the order of $10^{4}$. We have chosen these parameters in accordance with experimentally realized polymeric cavities operated near the infrared c-band [46]. The resonance frequencies of the WGMs are calculated with a typical relative accuracy of $10^{-6}-10^{-7}$. The computational domain is restricted using so-called perfectly matched layers.

An interaction between different WGMs is expected to depend on their spectral distance, which can for example be controlled via the disk’s aspect ratio. Here, we control this relative tuning of the modes by changing the cavity thickness. To hit the spot of interest where the cavity is neither too thin nor too thick, we choose the tunable thickness to be in the order of the wavelength of the light. In combination with the relatively small refractive index contrast, this thickness range should lead to a significant influence of the reduced cavity symmetry onto the investigated WGMs.

4. Non-separable mode profiles and avoided crossings

To test the limitations of the validity of the effective-index method, we compare field distributions of WGMs calculated via FEM simulations with the solutions derived from EIM. We have calculated exemplary modes with $N_\varphi =139$ in disk-shaped cavities of different thicknesses via FEM simulations. Figure 2 shows contour plots of the intensity distributions of nine of these modes in the $(\rho,z)$-plane of the cavities. Herein, only the section close to the resonator edge where the WGMs are located is depicted (compare Fig. 1(a)).

 

Fig. 2. WGMs in disk-shaped cavities of different thicknesses obtained from FEM simulations. The contour plots show the intensity distribution of different modes in the $(\rho,z)$-plane close to the resonator edge (compare Fig. 1(a)). Five of the nine mode profiles depicted meet the EIM-based expectation of a two-dimensional array of intensity antinodes (no background). For these separable modes, the respective mode numbers $N_\rho$ and $N_z$ are given. These modes also show a strong preferential field orientation. The four other modes (red background) strongly differ in their field distribution. Under close inspection, these modes can be understood as a combination of separable field distributions. Some of them also do not show a preferential field orientation (see main text for details). Adapted from [55].

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Most of these WGMs (no red background) match the solutions based on the analytical EIM calculations introduced earlier. The intensity distributions are given by a two-dimensional array of $N_\rho \times N_z$ intensity antinodes. In the following, we refer to such modes obtained from FEM as separable and still use their mode numbers $N_\rho$ and $N_z$ (see Fig. 2), although they are generally not conserved. We have found all these separable modes in Fig. 2 with a strong preferential field orientation corresponding to either a TE or a TM polarization. We refer to such separable modes to be TE-like or TM-like.

The remaining four depicted modes (red background) do not fit into the classification scheme of a combination of the analytical solutions in $\rho$- and $z$-direction. Instead, they show a more complex field distribution. Under close inspection, these intensity distributions can be understood as a combination of the respective separable $(1,N_z)$-mode on the left-hand side and the separable $(N_\rho,1)$-mode above. Herein, the upper two of the non-separable modes correspond to a simple superposition of the field distributions of the (1,2)- and the respective $(N_\rho,1)$-mode. The field distributions of these two non-separable modes do not show a preferential field orientation. The lower two modes on the other hand are given by a combination of the (1,3)- and the respective $(N_\rho,1)$-mode including constructive and destructive interference of their field extrema. Furthermore, these modes again do have a preferential field orientation. These differences can be understood in the context of the underlying selection rules of the mode coupling, which we introduce in Sec. 5.

To gain a thorough understanding of the nature of these non-separable mode profiles, we have performed detailed investigations of these modes under small changes of the cavity thickness. To that end, we have conducted additional FEM simulations in the thickness and frequency regimes where non-separable mode profiles have been first found. The result of one of these investigations is exemplarily shown in Fig. 3 for the case of the upper-left non-separable mode profile in Fig. 2. Within the chosen thickness and frequency regime, we have found four WGMs undergoing two avoided crossings. For all those found modes, their angular resonance frequency $\omega$ is depicted versus the disk’s thickness $d$ in Fig. 3(a).

 

Fig. 3. FEM simulations of WGMs in a disk-shaped cavity at varying thickness. (a) shows the angular resonance frequency of four different modes versus the thickness of the cavity disk. For small/large thicknesses, the modes of similar color and slope show the same separable mode profile and only differ by their polarization being either TE- or TM-like. The respective mode numbers $(N_\rho,N_z)$ are given. Pairs of modes (indicated by equal symbols) are found to undergo an avoided crossing when the disk’s thickness is changed. For the coupling-modes pair represented by full circles, the insets present contour plots of the mode profiles at exemplary thicknesses. Comparing the profiles of the two interacting modes on the left- and right-hand side of the avoided crossing, their properties have been interchanged. Investigating the mode profile of each single mode along the avoided crossing, a continuous transition between these different mode profiles is found. Regarding modes from different coupling-modes pairs, a spectral crossing and thus no interaction is apparent. (b) For the mode pair represented by full circles in (a), the TE polarization ratio $\gamma _\text {TE}$ is calculated (see main text). For small/large values of $d$, both modes yield a nearly perfect TE- or TM-like polarization. Within the thickness regime of the avoided crossing, a continuous exchange of these polarization properties takes place.

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At thicknesses far from the avoided crossings, all four modes show approximately separable field distributions with a preferential field orientation. Herein, two each of the four modes share the same mode numbers $(N_\rho,N_z)$ and only differ in being either TE- or TM-like. These modes also yield similar slopes of their angular resonance frequencies and are indicated via similar colors. The respective mode numbers are given. The avoided crossings however take place between pairs of modes of different field distribution. These coupling-modes pairs are indicated by equal symbols (open squares or full circles). The following discussion can be conducted in a similar manner for both of these coupling-modes pairs and is exemplarily done here for the one represented by full circles. For these modes, additional contour plots of their mode profiles comparable to those in Fig. 2 are shown as insets for exemplary thicknesses.

The investigated coupling-modes pair undergoes an avoided crossing of their resonance frequency within the depicted thickness regime. As usual for such an avoided crossing, the slope of the two curves is exchanged. For sufficiently small/large values of $d$, the two modes show different but both approximately separable mode profiles with mode numbers $(N_\rho,N_z)$. Comparing the mode profiles at very small and large thicknesses, an interchange of these mode profiles is apparent. As evident from contour plots of the mode profiles, this transition takes place in a continuous manner. (Herein, the upper-left non-separable mode profile in Fig. 2 corresponds to the mode represented by full blue circles at a thickness of $d = {2.45}\,\mathrm{\mu}\textrm{m}$.) This also corresponds to an exchange of the respective mode numbers. During this continuous exchange however, the mode profiles are given as superpositions of the two separable modes and are thus not separable anymore. This renders the mode numbers $(N_\rho,N_z)$ meaningless for these mode profiles. As evident from the presented plot, modes of the different coupling-modes pairs do not seem to interact in any way, but show a simple crossing behavior.

Within this analysis, we have not yet considered the polarization of the WGMs. To also get an understanding of this influence, we introduce the polarization ratio $\gamma _\text {TE}$, which is defined as the ratio of the electric energy density corresponding to the TE direction (meaning electric field along the $\rho$-direction) and the total electric energy density:

Based on coupled mode theory (CMT), we have derived the coupling strength of the coupling-modes pairs from the minimal spectral distance within the avoided crossing [70]. For the modes represented by full circles in Fig. 3, this leads to a coupling strength of around 1.25×10¹² s⁻¹ (1.21×10¹² s⁻¹ for open squares). These coupling strengths are in the order of $10^{-3}$ of the angular resonance frequency of the respective modes and are therefore in coarse agreement with those of other WGM interactions as, e.g., an inter-cavity coupling of modes in evanescently coupled resonators [71–73].

In combination, the presented results demonstrate an interaction between the separable TE-like mode with $\left (N_\rho,N_z\right )=(1,2)$ and the TM-like mode with $\left (N_\rho,N_z\right )=(2,1)$ (for open squares: TM-like mode with $\left (N_\rho,N_z\right )=(1,2)$ and the TE-like mode with $\left (N_\rho,N_z\right )=(2,1)$). All other possible combinations of these separable modes do not seem to interact. Based on these findings we suggest the following conclusion:

Separable mode profiles as analytically approximated via EIM are a valid description of WGMs in cylindrical (and similar) cavities under most conditions. However, these modes are non-orthogonal in the general sense. Despite the lack of a mathematical proof, this conclusion can be drawn from the fact that the EIM modes couple within the FEM simulations. As these calculations show, some of these modes couple to form new, non-separable eigenmodes. As only some of the separable mode pairs undergo such interactions, additional selection rules have to apply. The extent of this coupling depends on the spectral distance of the resonance frequency of the respective modes, which is suspect to, e.g., a disk’s aspect ratio. Far from any points of avoided crossing, the analytical solutions and the respective mode numbers offer a useful characterization of the WGMs. For certain aspect ratios close to an avoided crossing as depicted in Fig. 3 however, the established mode numbers are not useful anymore. To be able to still identify WGMs within such a parameter range in a reliable manner, an alternative mode classification scheme is indispensable. In the following, we introduce a comprehensive analysis of WGMs in cylindrical cavities and propose a respective classification scheme solely based on true symmetries of the underlying refractive index distribution.

5. Mode characterization of WGMs based on true symmetries

As demonstrated, the analytically obtained field distributions of WGMs that can be characterized by their three mode numbers $N_\varphi$, $N_\rho$, and $N_z$ as well as their polarization state can significantly differ from the modes actually propagating in mirror-symmetric cavities under certain conditions. We allocate this mismatch to an unjustified assumption of high symmetry of the underlying refractive-index distribution that entails a separability of the respective field distributions. In the following, we present a comprehensive analysis of such WGMs solely based on true symmetries of the underlying system. Finally, we propose an alternative mode classification scheme, that is more reliable for parameter ranges close to avoided crossings. We base this analysis on extensive FEM simulations.

The most prominent symmetry, that most WGM cavities share, is the rotational one. This symmetry is also valid in good approximation for most experimentally realized WGM resonators. Thus, we have also presumed a rotational symmetry within the FEM calculations presented in this paper. Consequently, the azimuthal mode number $N_\varphi$ is still expected to be a conserved quantity and no interactions between WGMs of different $N_\varphi$ are considered [25,45].

After taking the rotational symmetry into account, the only remaining true symmetry of the refractive-index distribution of the cavities of interest, e.g. cylindrical disks, is the mirror symmetry with respect to the plane of the WGM (compare Fig. 1). As this symmetry has to be transferred onto the intensity distributions of the WGMs, one can expect the field distributions of the modes to be defined eigenstates of the mirror transformation $z\mapsto -z$ with an eigenvalue of $\pm 1$. We here define the eigenvalue of the electric-field distribution of a WGM under this mirror transformation as the mode’s parity $P_z$. This property is expected to be generally conserved. Under close examination, it is possible to connect this parity $P_z$ to the mode number $N_z$ as well as the preferential field orientation derived from analytical EIM calculations: Separable modes with a TE-like polarization and an even $N_z$ as well as those with a TM-like polarization and an odd $N_z$ are anti-symmetric with respect to the mirror transformation $z\mapsto -z$, corresponding to a parity of $P_z=-1$. Respectively, TE-like WGMs with odd $N_z$ and TM-like modes with even $N_z$ correspond to $P_z=1$.

Based on the symmetry of the problem, we can expect no conserved mode properties other than $N_\varphi$ and $P_z$. Consequently, a coupling between all modes sharing these two characteristics might be possible, while WGMs that differ in at least one of these numbers should not show any interaction. We have checked this hypothesis by again employing FEM simulations in cylindrical cavities. Herein, we have again varied the thickness of the resonator disk (compare Fig. 3(a)). To investigate a large number of potential mode crossings, we have chosen a larger thickness and frequency regime compared to Fig. 3(a). Again, we have only included modes of equal azimuthal mode number $N_\varphi =139$. Furthermore, we have also considered the parity of the modes: The obtained WGMs are subdivided into two plots of parity $P_z=-1$ in Fig. 4(a) and $P_z=1$ in Fig. 4(b). Modes of similar color in (a) and (b) are connected in the same way as in Fig. 3(a): Far from any avoided crossing, these WGMs yield the same mode numbers $N_\rho$ and $N_z$ and only differ by their polarization state. Versions of these plots including the mode numbers and polarization state are given in Sec. S1 in Supplement 1.

 

Fig. 4. Angular resonance frequency of the first twelve (sorting by increasing frequency) WGMs obtained from FEM simulations versus the thickness of the disk-shaped cavity. All investigated modes share the same azimuthal mode number $N_\varphi = 139$ and are divided via their parity $P_z=-1$ (a) and $P_z=1$ (b) under a $z\mapsto -z$ mirror transformation. In each of the plots, the eight potential mode crossings are marked with circles (white/gray: interaction between separable modes of different/equal preferential field orientation; dotted: data from Fig. 3). Under close inspection, an avoided crossing is observed at each of these points. This demonstrates an interaction between all modes within each of the two plots. The actual coupling strength between the modes strongly differs. For the mode of highest frequency of each plot (cyan), the respective high-frequency coupling partner has not been plotted and is hence missing. For all other modes, an avoided crossing (indicated by a kink of the curve) takes place if and only if a mode with same $N_\varphi$ and $P_z$ comes spectrally close. For all modes, the spectral order number $N_\text {s}$ is given in respective colors. Further versions of these plots including separable mode numbers $\left (N_\rho,N_z\right )$ and the polarization state TE/TM as well as enlarged depictions of the avoided crossings of modes with equal preferential field orientation (gray circles) are given in Sec. S1 and Sec. S2 in Supplement 1. (Note: Although the data has been calculated for discrete disk thicknesses $d$ only, it is presented as continuous lines for convenience. For the mode of highest frequency in plot (b), some data points are missing due to numerical issues.) Adapted from [55].

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Within each of the plots Fig. 4(a) and (b), a total of eight potential mode crossings arises. These points are marked with circles. Under close investigation however, each of these mode pairs shows coupling indicated by an avoided-crossing behavior. The dotted circles highlight the avoided crossings depicted in detail in Fig. 3(a). The minimal spectral distance and thus the coupling strength of the different avoided couplings strongly varies. The majority of these couplings (white circle background) takes place between separable modes of different preferential field orientation (compare Fig. 3(b)). In these cases of cross-polarization coupling, the modes’ profiles can be understood as a simple superposition of the two separable profiles of differing field orientation (compare the upper two non-separable mode profiles in Fig. 2).

For the avoided crossings with gray circle background, an interaction between separable modes of similar preferential field orientation is found. These avoided crossings correspond to the lower two non-separable mode profiles in Fig. 2. As the interacting separable modes yield a similar polarization, the non-separable mode profiles are built up via constructive and destructive interference. Just as in case of the cross-polarization coupling demonstrated in Fig. 3, a gradual exchange of all mode properties takes place. For these coupling-modes pairs of similar preferential field orientation, the smallest coupling strengths are found. Nevertheless, the minimal spectral distance of the avoided crossings is at least around two orders of magnitude larger than the calculation accuracy of the respective resonance frequencies (compare Sec. 3.). The four avoided crossings are depicted in detail in Sec. S2 in Supplement 1.

Based on the FEM data presented in Fig. 4, we have checked our symmetry-based expectations of the conservation of certain mode properties. Within each of the two plots, the $\omega (d)$ curves of all modes undergo a kink, if and only if another mode within the same plot comes spectrally close and, consequently, an avoided crossing is found. (For the cases in both plots where this is not true regarding the mode of highest frequency (cyan line), we just have not plotted the respective high-frequency coupling partner and it is hence missing.) Thus, all separable WGMs under investigation within the mirror-symmetric cavity sharing the same $N_\varphi$ and $P_z$ interact. If two of these modes are spectrally close, they form new eigenmodes and thereby lift the separability of their mode profiles. These observations confirm our symmetry-based expectation of mode coupling stated above. As all characteristics of modes are exchanged within an avoided crossing, no other mode property than $N_\varphi$ and $P_z$ is conserved. If two WGMs differ in at least one of these properties, they do not undergo any interaction. Hence, these two quantities are always conserved and can be used as mode numbers within a reliable WGM classification scheme.

Although we expect $N_\varphi$ and $P_z$ to be strictly conserved in all mirror-symmetric cavities, they are not sufficient to fully characterize the respective WGMs. Based on the analysis presented above however, we have defined an additional mode number. Due to the absence of actual mode crossings within one mode subset of equal $N_\varphi$ and $P_z$, the spectral order of these modes is unchanged. Thus, by defining a spectral order number $N_\text {s}\in \mathbb {N}$ with $N_\text {s}=1$ describing the fundamental mode, an additional conserved mode number is found. The spectral order number is given in Fig. 4 for each of the calculated modes.

In summary, whispering gallery modes within mirror-symmetric cavities can be unambiguously characterized via their azimuthal mode number $N_\varphi$, their parity under a $z\mapsto -z$ mirror transformation $P_z$ and their spectral order $N_\text {s}$. In comparison to the established EIM-based mode numbers $N_\varphi$, $N_\rho$ and $N_z$, this novel scheme does not give full information on the field distribution or preferential field orientation of the modes. Instead, it is independent from any geometrical or material parameters of the cavity and can therefore be applied to characterize coupled WGMs undergoing avoided crossings.

6. Discussion

In the following, we shortly assess the impact of intrinsic mode coupling on WGMs applied on related photonic devices. The above analysis does generally presume a perfect rotational symmetry of the whispering gallery resonators that entails the conservation of the azimuthal mode number $N_\varphi$ [25,45]. As the rotational symmetry is actually given in good approximation for most experimentally realized cavities, this assumption should not limit the general applicability of our presented analysis. We also have not taken into consideration a possible coupling and splitting of clockwise and counter-clockwise propagating modes as demonstrated in [37]. We however expect this effect to be independent of the coupling effects introduced here and to therefore just lead to an additional splitting of the introduced modes.

If the thickness of a WGM cavity is small compared to the wavelength, only separable modes with a single antinode along the axial direction $N_z=1$ can propagate. Although the above analysis still holds for this scenario, WGMs of equal $N_\varphi$ and $N_z=1$ do not come spectrally close and their coupling is therefore negligible. If the cavity thickness is on the other hand much larger than the wavelength, the field distributions are localized far from the cavity’s upper and lower edge and are only weakly impacted by its limited symmetry. This leads to weak coupling effects and thus small avoided crossings [45,48]. The same applies to cavities of very large refractive index contrast to their surrounding, in which the modes are tightly confined. Hence, the intrinsic coupling analyzed here is mainly relevant for cavities with a medium thickness in the order of the wavelength and a rather small refractive index contrast. For such resonators, the actual strength of the intrinsic coupling strongly depends on the spectral distance of the single WGMs and thus all parameters, that can induce a relative spectral tuning of different WGMs: thickness/aspect ratio, refractive index, ratio of small/large radius of toroidal cavities, etc.. As it is obvious from the presented FEM data, a specific mode of interest is separable in good approximation for large parameter ranges. Due to typical limitations of applied fabrication processes however, a simultaneous and precise control over all these resonator parameters can be difficult. It therefore should be analyzed for each single kind of cavity and application, if there is a significant impact of the intrinsic coupling behavior and whether the application of the established WGM classification scheme of separable modes is justified.

For a large number of current photonic applications of WGMs, only fundamental modes $(N_\rho,N_z)=(1,1)$ are employed [41]. Although these fundamental modes are not orthogonal to other modes of equal $N_\varphi$ and $P_z$, no points of potential crossing with these other modes arise. Therefore, all respective spectral distances are large, and the coupling effects are negligible (compare Fig. 4). This observation should generally hold for cylindrical cavity disks from isotropic, dielectric media. In case of birefringent cavities however, a spectral overlap of fundamental and non-fundamental modes and thus a strong coupling is possible.

Due to limitations of state-of-the-art lithographic methods, a lot of experimentally realized WGM cavities do not show a perfect mirror symmetry [24,74]. Such a deviation leads to an interaction of modes of different parity under $z\mapsto -z$ mirror transformation and thus lifts the conservation of the respective symmetry of the mode and the mode number $P_z$. The actual coupling strength of this interaction strongly depends on the degree of deviation from a perfect mirror symmetry. As in this case all modes of equal $N_\varphi$ couple, no mode crossings are found in such a subset of WGMs and a conserved spectral order number could again be defined to fully characterize the modes. Furthermore, experimentally realized resonators always contain scatterers in form of defects, surface roughness, or other sub-wavelength features. Generally, these scatterers can also initiate a coupling between various kinds of WGMs. Hereby, the actual degree of this coupling should strongly depend on the kind and amount of scatterers. It nevertheless is usually difficult to allocate mode couplings in experimentally realized cavities to either unintended deformations and scattering processes, or the intrinsic coupling behavior analyzed in this paper.

In summary, the intrinsic coupling of WGMs in cavities of reduced symmetry is negligible for most modes and parameter ranges and the established classification scheme of WGMs is valid in good approximation. It nevertheless can be hard to distinguish, whether these separable solutions can still be applied. Thus, the intrinsic coupling behavior has to be generally taken into consideration, especially regarding applications based on an intra-cavity coupling of WGMs.

7. Conclusion

We have analyzed the systematics of whispering gallery modes in rotationally symmetric but non-spherical cavities with an additional mirror symmetry. We have introduced the established classification scheme of such modes based on their number of intensity antinodes along the different directions of a respective coordinate system and discussed the limitation of the validity of these solutions. Based on extensive FEM calculations, we have found various interactions of WGMs and have analyzed the underlying selection rules in detail. Furthermore, we have identified novel conserved mode properties based solely on true symmetries of the underlying refractive-index distribution. From these properties, we have deduced an alternative classification scheme of WGMs in mirror-symmetric cavities. Therein, we characterize the modes by their azimuthal mode number $N_\varphi$, their parity under a $z\mapsto -z$ mirror transformation $P_z$ and their spectral order number $N_\text {s}$. In contrast to the established mode numbers, this novel classification scheme does not give full information on the spatial field distributions, but is applicable even to WGMs undergoing avoided crossings. Finally, we have assessed the impact of the intrinsic coupling behavior on WGM-based photonic devices. Although such coupling is often negligible, it should generally be taken into consideration as it is an intrinsic property of all WGM resonators with reduced symmetry.