1. Introduction

Optical whispering gallery mode (WGM) resonators are known for outstanding Q-factors and small mode volumes, exploiting the strength and diversity of nonlinear-optical (NLO) phenomena, and for a small footprint together with mechanical robustness [1–3]. Their applications span from detection of single atoms and molecules to low-power frequency conversion and comb generation [4–9]. Although they differ in shape, almost all WGM resonators are axially symmetric. Here, light is strongly localized near the resonator rim which is generally characterized by the major and minor radii R and r, see Fig. 1. Many NLO applications, such as low-power second-harmonic generation, optical parametric oscillation, and comb generation, benefit from resonators made of wide band-gap anisotropic materials where WGMs can be regarded as ordinary (o) and extraordinary (e) modes [2,4–6].

 

Fig. 1 Geometry of an anisotropic WGM resonator, R and r are the major and minor radii, z is the optical axis, φ is the polar angle, while u and θ are the cross-sectional coordinates.

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The applications often require continuous tuning of discrete WGM resonances and imply regularity of the frequency spectrum. Tuning can be implemented by varying the resonator temperature, by applying an electric field or by inducing mechanical stress [2]. For anisotropic resonators, however, one runs, into difficulties caused by anticrossing (avoided crossing) of resonances. This general effect, known in atomic/molecular physics and optics [10, 12], has specific features in the context of WGMs. Owing to different temperature/field dependences of the ordinary (o) and extraordinary (e) refractive indices n_o and n_e, numerous crossings of primary o- and e-resonances are inevitable. Some of the crossings are robust, so that orthogonally polarized WGMs stay uncoupled. But in many cases o- and e-modes become coupled with each other leading to the anticrossing behavior and o-e hybridization [12–14]. While the anticrossings can be useful for polarization transformations, they introduce numerous irregularities into the WGM spectrum and ruin continuous tuning. The situation is exemplified by Fig. 2, heralding an experimental map of o- and e-resonances. While the temperature tuning does not reveal a disturbed mode spectrum in the o-case, it is disrupted by numerous strong and weak anticrossings for the e-polarization. Electro-optic tuning shows similar features.

 

Fig. 2 An experimental map of WGM resonances for thermal tuning in a lithium niobate resonator (R ≃ 1350 µm and r ≃ 900 µm) for o- and e-polarizations at 1040 nm pump wavelength. The maximum avoidance gap, shown in red, is ≃ 410 MHz. The minor anticrossings visible in the e-case are orange-highlighted.

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The origin of the interactions controlling the anticrossings and the values of the frequency splits (avoidance gaps) remain unclear. The anticrossings are often attributed to non-controllable remnant cross-polarization scattering processes [2, 14]. They should not be mixed up with the frequency splits for counterpropagating WGMs with the same modal number m caused by a weak violation of the azimuth symmetry [15, 16]. Uncertainty with the anticrossing behavior is greatly due to the dominating view on WGMs as on scalar modes obeying the scalar Helmholtz equation. In particular, a big work has been done on solution of this equation, both analytically and numerically, for different resonator geometries – spheroidal, toroidal, etc. [17–19]. When applied to the o- and e-modes, it enabled experimentalists to identify WGM resonances [20, 21]. It is known, however, that Maxwell’s equations allow the scalar reduction only for spherical resonators made of isotropic materials [22]. The existing approaches to the description of WGMs in anisotropic resonators are impractically complicated even for the spherical geometry [23, 24]. Avoiding the simplified scalar treatment and consideration of the vectorial Maxwell equations may provide new insights. Quantification of the impact of the vectorial o-e coupling can be the key for mode control in anisotropic resonators.

2. Theoretical analysis

The actual geometry is illustrated in Fig. 1. The rotational z-axis coincides with the optical axis of an uniaxial crystal (refractive indices n_o and n_e). The only nonzero components of the optical permittivity tensor $\widehat{\epsilon }$ are ${\epsilon }_{xx}={\epsilon }_{yy}=n_{\mathrm{o}}^2$ and ${\epsilon }_{zz}=n_{\mathrm{e}}^2$. The anisotropy is quantified by the parameter β = (n_o − n_e)/(n_o + n_e); it is always very small, |β| ≪ 1. The radii ratio η = R/r is expected to be of the order of 1. Also we employ the u, θ, φ orthogonal curvilinear coordinate system, where u is the distance to the surface, θ is the polar angle measured from the equator, and φ is the azimuth angle. The factor exp(imφ − 2πiνt), where m is the azimuth number and ν is the WGM frequency, is split off from all components of the light electric and magnetic fields E and H.

The theoretical part is original in two respects. First, we use the field components E_z, H_z as a pair of independent variables. Each of them obeys the scalar Helmholtz equation providing zero coupling in the bulk. This follows from Maxwell’s equations and is intuitively clear: the optical anisotropy can be accounted for by rescaling of Cartesian x, y, z coordinates. Coupling between E_z and H_z comes solely from the boundary conditions. Second, we instantly employ a perturbative approach benefiting from strong localization of light near the rim. It is known [19] that the localization scales in u and θ are u_m = R/(2m²)^1/3 and θ_m = η^3/4/m^1/2. Since m ≃ 2πRn_o,e/λ ≳ 10⁴ for typical values of wavelength λ, radius R and ratio η, we have u_m ≪ R and θ_m ≪ 1. This inequality enables us to strongly simplify the theory.

Within the perturbative approach, E_z and H_z are amplitudes of the e- and o-modes, respectively. With the vectorial coupling neglected, each of them is characterized by an azimuth number m, a radial number q = 1, 2, …, a polar number p = 0, 1, … and the frequency [18, 19]

The WGM solution of the scalar Helmholtz equation is [19]

Employment of boundary conditions (BCs) for E_z and H_z at u = 0 allows us to determine s_o,e and ν_o,e. For simplicity, we use the perfect-conductor BCs. They do not pretend to give exact values of ν, but are playable and sufficient to describe the intermode distances. In the leading approximation in 1/m and β (see Section 5.1) the BCs are

Accounting for the coupling terms in BCs (4) gives (see Section 5.2) a generic anticrossing relation for the frequencies

Importantly, the condition ν_o(m, q_o, p_o) = ν_e(m, q_e, p_e, necessary for the anticrossings, can be fulfilled only because of the anisotropy. Using Eq. (1), we represent it as ${\zeta }_{q_{\mathrm{o}}}-{\zeta }_{q_{\mathrm{e}}}\simeq \beta {(m/2)}^{2/3}\pm {(2/m)}^{1/3}$, where the sign ± corresponds to the cases p_o − p_e = 1. The effect of the specific tuning parameter (e.g., the temperature T) on the product $\overline{n}R$ has canceled out from this condition. For |β|m 1, which is practically always well fulfilled, the ± contribution is negligible and we have ${\zeta }_{q_{\mathrm{o}}}\simeq {\zeta }_{q_{\mathrm{e}}}+\beta {(m/2)}^{2/3}$. As follows from this relation and Eq. (6), the split δν does not only depend on the value, but also on the sign of β. At the same |β|, it is larger for β < 0.

2.1. Application to lithium niobate

Now we consider representative numerical estimates for lithium niobate (LN) resonators. Setting R = 1.5 mm and λ = 1 µm, we have at room temperature: m ≃ 2 × 10⁴, $\overline{n}\simeq 2.2$, β ≃ 0.075, β(m/2)^2/3 ≃ 35, and FSR ≃ 15 GHz. Correspondingly, we obtain for η = 1, q_e ∼ 1, and $p_{*}=1:{\zeta }_{q_{\mathrm{o}}}\approx 45$, q_o ≈ 50, δν/FSR ≈ 0.03, and δν ≈ 450 MHz. The predicted avoidance gap is several orders of magnitude larger than the linewidth. The o-modes with q_o ≈ 50 are still localized near the rim, but they cannot be efficiently excited with standard couplers. Furthermore, their localization depth is much larger than that (∼ u_m) for the e-modes. This enables us to get rid of the o-modes by a proper mode management and to eliminate the anticrossings, see below.

An important aspect is how frequent the anticrossings are. Again, we exemplarily assume thermal tuning. The thermo-optic coefficients are strongly different in LN crystals, $n_{\mathrm{o}}^{\prime }=d{n}_{\mathrm{o}}/dT\simeq 5.6\times {10}^{-6}$ and $n_{\mathrm{e}}^{\prime }=d{n}_{\mathrm{e}}/dT\simeq 44\times {10}^{-6}{\mathrm{K}}^{-1}$ [25]. This big difference is only partially compensated by the transverse thermal expansion with dR/RdT ≃ 14 × 10⁻⁶ K⁻¹. The dependences ν_o (T) and ν_e(T) are almost straight lines with two distinct slopes, and different combinations of the modal numbers give numerous intersections. Blue and orange lines in Fig. 3(a) give ν_e(T) and ν_o(T) for q_o = 49 and 50, p_o = 0 and 1, q_e = 1, and p_e = 0. Within the temperature interval of ≈ 45 °C, the lines show only two anticrossings. The frequency difference between them exceeds 16 FSRs. Figure 3(b) shows the area near the up-left anticrossing point in more detail. Here, we add lines corresponding to p_e = 1 and p_e = 2. This leads to two additional anticrossings. The frequency difference between them exceeds one FSR. Thus, the anticrossings with large avoidance gaps and |p_o − p_e| = 1, as predicted by our theory, are relatively rare events. It should be noted that higher-order perturbation terms in the BCs, which are outside our consideration, give much smaller avoidance gaps, but different selection rules for the polar numbers. In particular, θ² terms give the selection rule |p_o − p_e| = 0, 2. The number of weak anticrossings is expected to be substantially larger than that of strong ones.

 

Fig. 3 Calculated dependences ν_o,e(T) for m = 2 × 10⁴ and different combinations of the radial and polar numbers taking into account thermal expansion and the thermo-optic effect. Blue and orange lines correspond to e- and o-modes. Green spots indicate the anticrossings. (a) q_o = 49 and 50, p_o = 0 and 1; q_e = 1, p_e = 0. b) The shadowed upper-left anticrossing area of a) in some detail: q_o = 50, p_o = 0 and 1; q_e = 1, p_e = 0, 1, and 2. The vertical bar shows the FSR.

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3. Experimental results

Now we switch back to the observations. In our experiments, we used three different mm-sized WGM resonators made of MgO-doped LN crystals and shaped as shown schematically in Fig. 1. The radii ratio η = R/r ranged between ≃ 1 and 3. A fine-tunable diode laser emitting at 1040 nm served as the pump source. A TiO₂ prism allowed to couple light in and out of the resonator. The measured intrinsic Q-factors of o- and e-polarized WGMs were ∼ 10⁸. During the thermal tuning, the resonator temperature T was slowly increasing (some mK/s) while the pump frequency was sweeping forth and back within the range of ≈ 40 GHz exceeding the FSR ≈ 15 GHz. Recording of the transmitted power gives thus a two-dimensional map indicating the positions of the WGM resonances versus T. A similar procedure is applicable to the field tuning and leads to equal avoidance gaps. More details can be found in Section 5.3.

Figure 2 shows representative fragments of the frequency maps for o- and e-polarized pump light. In Fig. 2(a) we see an unperturbed, almost linear decrease of the frequencies ν_o(T) with a common slope. This behavior is now quite obvious: Only WGMs with small radial numbers q_o are excited from outside, so that fulfillment of the anticrossing condition ν_o(m, q_o, p_o) = ν_e(m, q_e, p_e) is impossible. Although the scheme with o-polarized pump is free of anticrossings, it is unfavorable for NL applications because of typically low values of the nonlinear coefficients. The situation is entirely different for e-polarized pump light, highly beneficial for NL applications, see Fig. 2(a). Here we see a single major anticrossing with an avoidance gap δν ≃ 410 MHz, consistent with the theoretical prediction, and several minor anticrossings with much smaller values of δν. The latest are obviously due to higher-order vectorial perturbation terms. The o-modes with very large values of q_o are excited internally in this case.

The big difference in the radial numbers q_o and q_e leads to the corresponding big difference in the scales of localization of the o- and e-modes near the resonator rim, as shown in Fig. 4(a). This distinctive feature has encouraged us to employ a mode management and to get rid of the anticrossings completely. This can be achieved by adding additional optical losses to the undesired modes, as described in Refs. 26, 27. To accomplish this task, we made a neat ≈ 100 µm-long through slit near the the resonator rim leaving only a small gap of ≈ 20 µm for light to travel, see Fig. 4(b). As the desired low-order e-mode is unaffected, the unwanted high-order o-mode is fully suppressed. The result of this mode management is presented in Fig. 4(c) for a representative resonance. It is evident that the resonator without the mode-control suffers from strong mode perturbations while the resonator with the mode-control is completely free of anticrossings.

 

Fig. 4 a) The radial dependence of light intensity for the o- (orange) and e-modes (blue) with q_o = 50 and q_e = 1. b) Schematic of the mode management suppressing the o-modes with q_o ≲ 1. c) Fine structure of a e-resonance with (right) and without (left) the mode management during temperature tuning. The resonance center of the mode has been kept a zero detuning. The color code represents the transmission of the pump light, normalized to the achievable coupling efficiency.

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4. Conclusion

In conclusion, vectorial coupling, inherent in any anisotropic axisymmetric resonator, is responsible for the anticrossings of the ordinary (o) and extraordinary (e) WGM resonances. The avoidance gap sizes are determined by the material birefringence and the resonator geometry. They are quantified using a new perturbative approach to the description of WGM structures beyond the scalar models. Strong difference in the radial numbers and in the degree of spatial localization near the rim is predicted for anticrossing o- and e-resonances in LN-based WGM resonators. Experimentally, we have detected the predicted large avoidance gaps and found a striking difference in the anticrossing behavior for the o- and e-excitation of the resonances. Based on the expected properties of the anticrossings, we have introduced a mode management technique to get rid of unwanted WGMs. As the result, we eliminated the devastating impact of the anticrossings on the frequency tuning.

5. Appendix

5.1. Boundary conditions for E_z and H_z

The inequalities |β| ≪ 1, u_m/R ≪ 1, and θ_m ≪ 1 lie in the basis of our perturbative approach. They allow to greatly simplify the differential operations in the curvilinear u, θ, φ coordinate system, see Fig. 1 of the main text. In particular, the spatial derivatives of the WGM amplitudes can be estimated as ∂/∂u ≈ 1/u_m and ∂/∂θ ≈ 1/θ_m. They are much larger than the derivatives of the Lame (scaling) coefficients h_u ≃ 1, h_θ ≃ r − u, and h_φ = ρ(u, θ) ≃ R. Thus, one can compare different contributions relevant to the same amplitude and chose the dominating one.

The initial perfect-metal boundary conditions (BCs) at u = 0 read E_θ = H_u = 0. Transferring to the cylindric ρ, φ, z coordinate system, we express geometrically E_θ by E_z, E_ρ and H_u by H_z, H_ρ. For θ ≪ 1, we have in the leading approximation with the help of Fig. 1 of the main text: E_θ = E_z − θE_ρ and H_u = − H_ρ − θH_z. Next, we obtain straightforwardly from Maxwell’s equations:

(7)$$\left(\frac{{\partial }^2}{\partial z^2}+n_{\mathrm{o}}^2{k}^2\right)E_{\rho }=\frac{{\partial }^2{E}_z}{\partial \rho \partial z}-\frac{km}{\rho }{H}_z$$

(8)$$\left(\frac{{\partial }^2}{\partial z^2}+n_{\mathrm{o}}^2{k}^2\right)H_{\rho }=\frac{{\partial }^2{H}_z}{\partial \rho \partial z}+\frac{k{n}_{\mathrm{o}}^{2}m}{\rho }{E}_z.$$

Since ∂/∂z ≈ (1/r)∂/∂θ ≈ 1/rθ_m, the differential terms in the left-hand sides are small as 1/m compared to the non-differential ones and can be omitted. After that E_ρ and H_ρ are algebraically expressed by E_z and H_z and their second-order derivatives. Substituting the corresponding expressions for E_ρ and H_ρ into the BCs, we obtain two relations for E_z and H_z at u = 0:

(9)$$E_z+\frac{\theta m{H}_z}{n_{\mathrm{o}}^{2}k\rho }-\frac{\theta }{n_{\mathrm{o}}^2{k}^2}\frac{{\partial }^2{E}_z}{\partial \rho \partial z}=0$$

(10)$$\theta H_z+\frac{m{E}_z}{k\rho }+\frac{1}{n_{\mathrm{o}}^2{k}^2}\frac{{\partial }^2{H}_z}{\partial \rho \partial z}=0.$$

We need to keep here only the leading terms for E_z and H_z. To make so, we replace ρ by R, n_o by $\overline{n}$, and k = ν/2πc by $m/\overline{n}R$. Furthermore, the second-order derivative can be estimated as ∂²/∂ρ∂z ≈ 1/(ru_mθ_m). Thus, the third term in each of the above relations is negligible compared to the first one, so that Eqs. (9) and (10) give independently the first of the boundary conditions (4) of the main text. The neglected contributions are relatively small either in 1/m or in β.

To derive the second BC for E_z and H_z, we start from the equality$\nabla \cdot \widehat{\epsilon }\mathit{E}=0$. With the birefringence neglected, it gives ∂E_u/∂u = 0 at u = 0 in the leading approximation in 1/m. Next, using the above relations of this section, we obtain: $E_u\simeq -E_{\rho }-\theta E_z\simeq H_z/\overline{n}-\theta E_z$. From here we get the second of Eqs. (4) of the main text. The birefringence term gives merely small (in β) corrections to the contributions already taken into account.

5.2. Derivation of Eq. (5) of the main text

The field components E_z and H_z represent the amplitudes of the e- and o-modes, respectively. They are given by Eqs. (2) and (3) of the main text when prescribing the subscripts o and e to Ψ, C, s, p, and n. Substituting the expressions for E_z and H_z into the BCs given by Eqs. (4) of the main text, we obtain two relations for the constants C_o and C_e and the frequency ν. These relations can be strongly simplified taking into account smallness of the vectorial coupling terms in the BCs. With the vectorial coupling neglected, we have $s_{\mathrm{e}}={\zeta }_{q_{\mathrm{e}}}{u}_m$ and $s_{\mathrm{o}}={\zeta }_{q_{\mathrm{o}}}{u}_m$, where ${\zeta }_{q_{\mathrm{e}}}$ is the q_e-th zero of the Airy function Ai(−x) and ${\zeta }_{q_{\mathrm{e}}}$ is the q_e zero of the derivative $\mathrm{A}{\mathrm{i}}^{\prime }(-x):{\zeta }_{q_{\mathrm{e}}}\simeq 2.338$, 4.088, 5.53, … and ${\zeta }_{q_{\mathrm{e}}}\simeq 1.01$, 3.248, 4.82, …. The frequencies ν_e(m, q_e, p_e) and ν_o(m, q_o, p_o) are given by Eq. (1) of the main text in this scalar approximation.

To account for the vectorial coupling terms, we set $s_{\mathrm{e}}/u_m={\zeta }_{q_{\mathrm{e}}}+{\tau }_{\mathrm{e}}(\nu -{\nu }_{\mathrm{e}})$ and $s_{\mathrm{o}}/u_m={\zeta }_{q_{\mathrm{o}}}+{\tau }_{\mathrm{o}}(\nu -{\nu }_{\mathrm{o}})$ with τ_o,e = 2πR²n_o,e/cmu_m. Assuming that the vectorial correction terms in these relations are small, we can employ the first-order expansions of Ai(−x) and Ai′(−x) near ${\zeta }_{q_{\mathrm{e}}}$ and ${\zeta }_{q_{\mathrm{o}}}$, respectively. Furthermore, we can neglect a small (in β) difference between τ_e and τ_o by setting ${\tau }_{\mathrm{o},\mathrm{e}}=\tau =2\pi R^2\overline{n}/cm{u}_m$ with $\overline{n}=(n_{\mathrm{o}}+n_{\mathrm{e}})/2$. After that we obtain:

(11)$$C_{\mathrm{e}}\overline{n}\tau {\mathrm{\Theta }}_{p_{\mathrm{e}}}\mathrm{A}{\mathrm{i}}^{\prime }(-{\zeta }_{q_{\mathrm{e}}})(\nu -{\nu }_{\mathrm{e}})-C_{\mathrm{o}}\theta {\mathrm{\Theta }}_{p_{\mathrm{o}}}\text{Ai}(-{\zeta }_{q_{\mathrm{o}}})=0$$

(12)$$C_{\mathrm{e}}\overline{n}\theta {\mathrm{\Theta }}_{p_{\mathrm{e}}}\mathrm{A}{\mathrm{i}}^{\prime }(-{\zeta }_{q_{\mathrm{e}}})+C_{\mathrm{o}}\tau {\mathrm{\Theta }}_{p_{\mathrm{o}}}\mathrm{A}{\mathrm{i}}^{″}(-{\zeta }_{q_{\mathrm{o}}})(\nu -{\nu }_{\mathrm{o}})=0.$$

Next, we multiply the first and second equations by ${\mathrm{\Theta }}_{p_{\mathrm{e}}}(\theta /{\theta }_m)$ and ${\mathrm{\Theta }}_{p_{\mathrm{o}}}(\theta /{\theta }_m)$, respectively, and integrate over θ. The integrations can be performed using the orthogonality relations for the wave functions of quantum harmonic oscillator Θ_p(x), the normalization condition ${\displaystyle \int {\mathrm{\Theta }}_p^2(x)dx=1}$, and the identity $x{\mathrm{\Theta }}_p(x)=\sqrt{(p+1)/2}{\mathrm{\Theta }}_{p+1}(x)+\sqrt{p/2}{\mathrm{\Theta }}_{p-1}(x)$, see Ref. 11 of the main text. Furthermore, we have the equality Ai″(−x) = −xAi(−x) for the Airy function. As a result, we obtain two linear algebraic equations for C_e,o:

(13)$$C_{\mathrm{e}}\overline{n}\tau \mathrm{A}{\mathrm{i}}^{\prime }(-{\zeta }_{q_{\mathrm{e}}})(\nu -{\nu }_{\mathrm{e}})-C_{\mathrm{o}}{\theta }_m\text{Ai}(-{\zeta }_{q_{\mathrm{o}}})\sqrt{p_{*}/2}=0$$

(14)$$C_{\mathrm{e}}\overline{n}{\theta }_m\mathrm{A}{\mathrm{i}}^{\prime }(-{\zeta }_{q_{\mathrm{e}}})\sqrt{p_{*}/2}-C_{\mathrm{o}}\tau {\zeta }_{q_{\mathrm{o}}}\text{Ai}(-{\zeta }_{q_{\mathrm{o}}})(\nu -{\nu }_{\mathrm{o}})=0$$

with p∗ = max{p_o, p_e}. The condition of their solvability (zero determinant condition) gives elementary Eq. (5) of the main text for the hybridized frequencies ν = ν_±.

In addition to the frequencies, Eqs. (13) and (14) describe also the polarization states of the hybridized modes. We have from them two equivalent relations for the ratio C_e/C_o:

(15)$$\frac{C_{\mathrm{e}}}{C_{\mathrm{o}}}=\frac{\alpha (\nu -{\nu }_{\mathrm{o}})}{\gamma }\equiv \frac{\alpha \gamma }{\nu -{\nu }_{\mathrm{e}}}$$

with $\alpha =\sqrt{{\zeta }_{q_{\mathrm{o}}}}\text{Ai}(-{\zeta }_{q_{\mathrm{o}}})/\overline{n}\mathrm{A}{\mathrm{i}}^{\prime }({\zeta }_{q_{\mathrm{e}}})$ and $\gamma =({\theta }_m/\tau )\sqrt{p_{*}/2{\zeta }_{q_{\mathrm{o}}}}$. They include parameter α which does not enter the expression for ν. The ratio C_e/C_o has to be considered separately for the (+) and (−) branches, as defined by Eq. (5) of the main text. Far from the anticrossing region, where |ν_o − ν_e| ≫ γ, we have o- and e-polarized modes. Within the anticrossing region, |ν_o − ν_e| ≲ γ, the polarizations of the ± modes are mixed and dependent on α.

5.3. Experimental details

All optical measurements were performed using a set-up shown schematically in Fig. 5. A tunable fiber-coupled diode-laser emitting at 1040 nm wavelength is coupled into the whispering-gallery resonator using a rutile prism. A fiber polarization-controller allows to freely choose the polarization direction. The transmitted light is collimated again and recorded with a Si-photodiode. The temperature of the resonator is stabilized on the mK level. The gap between the coupling prism and the resonator can be controlled using a piezo-actuator. During the measurements we tried to maintain a constant-coupling regime close to critical coupling.

 

Fig. 5 Illustration of the measurement setup. ECDL: tunable diode laser; PC: polarization controller; WGR: whispering-gallery resonator; L1: lens, D1: Si-photodiode.

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The experimental results presented in the main text were obtained by scanning the pump-laser frequency forth and back while recording the transmission spectra. At the same time, the temperature of the whispering-gallery resonator is increased with a constant heating rate of a few mK/s. In order to measure the data in Fig. 2, we automatically identified the position of the individual resonances using a peak finder and indicated the resonance position in the figure with a point. This was necessary as the resonance widths are very small compared to the y-axis and would otherwise appear as very thin lines.

Three different resonators made out of 5-mol.-%-MgO-doped lithium niobate with the crystal z-axis oriented parallel with the symmetry axis of the resonator have been used. The resonator used to measure the large split had a major radius of 1.35 mm and a minor radius of 0.9 mm and was equipped with top and bottom electrodes. These electrodes where short-circuited during measurement. Applying a voltage to the electrodes allowed us to verify that comparable results can be obtained with electric field tuning as well. To demonstrate the efficiency of the mode-control technique, two almost equal resonators with 1.5 mm major and 0.2 mm minor radius have been used. One of them has been equipped with a through slit perpendicular to the rim (see Fig. 6). The gap between the rim of the resonator and the slit was approximately 20 µm. Here, a femtosecond pulse laser (388 nm wavelength, 2 kHz repetition rate, 250 mW average power, 30 µm focal spot) was used to ablate the material. The generated hole penetrates the whole crystal and is approximately 100 µm long in radial direction. The same laser was also used to shape the resonators on a precision turning lathe while the focal spot was set to approximately 70 µm. Here, the laser was oriented in grazing incidence in order to ablate the excess material. Subsequently, the resonators were polished.

 

Fig. 6 Dark-field microscope image of the mode-control slit. The camera is looking perpendicularly on the rim of the resonator. The crystal z-axis is indicated with an arrow.

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Funding

German Federal Ministry of Education and Research (13N13648).

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