Abstract
Microresonatorbased optical frequency combs emitted from highqualityfactor microresonators, also known as microcombs, have opened up new horizons to areas of optical frequency comb technology including frequency metrology, precision sensing, and optical communication. To extend the capability of microcombs for such applications, large and reliable tunability is of critical importance. Here, we show broad spectral tuning of Kerr soliton microcombs in a thermally controlled crystalline microresonator with pumpdetuning stabilization. The fundamental elements composing frequency combs, namely the center frequency, repetition frequency, and carrierenvelope offset frequency, are spectrally tuned by up to −48.8 GHz, −5.85 MHz, and −386 MHz, respectively, leveraging thermal effects in ultrahighQ crystalline magnesium fluoride resonators. We further demonstrate a 3.4fold enhancement of soliton comb power resulting from thermal expansion with a temperature change of only 28 K by employing quantitative analyses of the fibertoresonator coupling efficiency.
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Introduction
Highrepetition rate microresonatorbased frequency combs offer powerful and compact optical frequency comb (OFC) sources that are of great importance to various applications including a phasecoherent link for the optical to microwave domain^{1,2}, optical communication^{3,4}, spectroscopy^{5,6}, and lownoise photonic microwave generation^{7,8,9,10}. In particular, the soliton modelocked states of microresonator frequency combs provide highcoherence and smooth spectral profiles^{11}, thereby making them highly desirable for most applications^{1,2,3,4,5,6,7,8,9,12,13}. One significant difference between the microcombs and conventional OFC sources is their resonator configuration^{14,15}. Most modelocked lasers are based on composite systems with a gain medium, bulky freespace optics, active and passive optical fibers^{15}; moreover, lasercavity actuators such as temperature control systems, piezoelectric transducer devices, or high bandwidth modulators can be easily combined to fully exploit the performance of lasers^{16}. As a result, oscillation frequencies and repetition rates are controlled by using such actuator elements or by regulating the cavity length by moving end mirrors or cutting/fusionsplicing optical fibers. In contrast, optical microresonators are usually made of a single dielectric material so that their waveguides can strongly confine light in a tiny mode space^{17}. In return, this ultimately elemental configuration makes it a challenge to freely tune the OFC properties.
Mechanical^{18,19} and electrical tuning^{20,21} can allow for dynamically control and stabilize the microcombs, and they can readily act as fast actuators for the comb frequencies in return for the limited spectral tuning bandwidth. Similarly, resonance control via thermal effects has also been implemented with an auxiliary laser coupled to a different resonance mode^{22,23}, pump power control using an intensity modulator^{2,8,24}, or a Peltier element installed in a cavitypackaged module^{25}. Although these techniques are suitable for narrow range of resonance tuning and compensation for environmental perturbation, i.e., soliton initiation^{23,24} or longterm stabilization^{2,8,25}, these studies have paid little attention to the capability of the spectral tuning bandwidth. In this regard, temperature control of the entire resonator system using integrated microheaters has been adopted as the most effective and useful method of frequency tuning and resonance control for onchip resonators operating mainly with a comb mode spacing exceeding 200 GHz^{26,27,28,29,30,31}. Despite the rapid progress made on larger mode spacing combs, broadband tunability and flexibility of soliton microcomb sources with a repetition rate in the microwave X and Kbands (8–27 GHz) have not been demonstrated. These frequency bands are important because they constitute the mainstream method for ultralownoise photonic microwave generation with a direct opticaltoelectric conversion process, and the microwave synthesis of the lowestlevel phase noise signals has been achieved in millimeterscale whisperinggallerymode (WGM) microresonators^{2,7,9,10}. Thus, widely tunable microcombs yielding the microwaves at these bands require a better understanding of the dynamics of soliton combs generated in WGM microresonators under thermal activation over a wide dynamic range.
Here, we demonstrate and extend the spectral flexibility of a soliton microcomb with a 23 GHz repetition rate by exploiting temperature control and a detuning stabilization system. The center frequency, repetition frequency, and offset frequency of a soliton comb with a radiofrequency (rf) repetition rate are simultaneously tuned by thermal effects and stabilized by a feedback loop. We further show that the output power of a soliton comb is distinctly enhanced via thermallyinduced coupling variation owing to the thermal expansion of a resonator coupled to a tapered fiber that ensures coupling flexibility. A qualitative analysis allows us to explain this interesting result. Moreover, we address the complex dynamics underlying soliton tuning arising from birefringence of crystalline magnesium fluoride (MgF_{2}) resonators that gives rise to the difference in the thermal sensitivity of optical resonances belonging to different polarization modes. This feature leads to polarizationdependent tunability and has a unique effect on the soliton selffrequency shift via modecouplinginduced dispersive wave (DW) emission.
Results and discussion
System overview
The frequency tuning of cavity resonances in optical microresonators can be driven by two thermal effects, namely the thermooptic and thermal expansion effects. The former originates from the change in refractive index and the latter corresponds to the volume change of the material in response to temperature variation, thus altering the optical length of the resonator. In general, both effects alter the resonance frequencies following a simple relation, df/dT = −(α_{n} + α_{l})f, where α_{n} = (1/n)(dn/dT) and α_{l} = (1/l)(dl/dT) are the thermorefractive and thermal expansion coefficients, respectively. The thermal coefficients in most resonator materials including crystalline MgF_{2} are positive, whereas some of the fluoride crystals (e.g., calcium fluoride, barium fluoride) or polydimethylsiloxane possess negative thermorefractive coefficients. Although recent studies have demonstrated a potential advantage of the negative thermooptic effect for microcomb generation^{32,33}, it often results in resonance instability induced by thermooptical oscillations^{34}. As a result, a crystalline MgF_{2} resonator is widely used to obtain a stable soliton state in thermal equilibrium^{2,4,6,7,11}.
For uniaxial crystals such as MgF_{2}, optical anisotropy should be considered^{35,36}. The use of a resonator made from zcut MgF_{2} crystal results in slightly different refractive indices for the transverseelectric (TE) mode (extraordinary polarized, n_{e} = 1.382) and the transversemagnetic (TM) mode (ordinary polarized, n_{o} = 1.371), and similarly the thermal coefficients exhibit anisotropy. In this work, we adopted the following values: α_{n,e} = 0.23 × 10^{−6} K^{−1}, α_{n,o} = 0.64 × 10^{−6} K^{−1}, and α_{l} = 8.9 × 10^{−6} K^{−1} at room temperature^{35}. We also note that the thermal expansion can mainly be attributed to the ordinary direction α_{l,o}, which corresponds to the expansion of the radial direction of the resonator^{37}. Strictly speaking, thermal expansion coefficients are given as a function of temperature, but the impact is considered negligibly small in this study.
Figure 1a, b shows the experimental setup and concept of thermal resonance tuning in an MgF_{2} microresonator. While the tuning efficiency of resonance frequencies is independent of the resonator diameter, that of the modespacing, i.e., the freespectral range (FSR), depends on the native mode spacing: df_{FSR}/dT = −c(α_{n} + α_{l})/(2πn_{g}R), where c is the speed of light in a vacuum, n_{g} is the group index, and R is the effective radius of a resonator. Thus, the tuning efficiency of the FSR corresponds −213.2 kHz/K for the TE mode and −222.8 kHz/K for the TM mode for the 23.35 GHzFSR resonator that we used for soliton generation as described in the following section. The tuning efficiency of the resonant frequencies corresponds to −1.765 GHz/K for the TE mode and −1.845 GHz/K for the TM mode in the 1550 nm band. Figure 1c, d shows the transmission spectra and resonance shifts with the resonator temperature increasing from 300 K (ΔT = 0 K) to 325 K (ΔT = 25 K), respectively. The fitted slopes yield an average efficiency of −1.77 GHz/K, which indicates that these modes belong to the TE mode families. We also clearly observe the polarization dependence of the resonance tuning efficiency as shown in Fig. 1e, f. By offsetting the frequency shift of one of the TE modes, we can recognize the mode families exhibiting different tuning efficiencies as the TM modes, and the difference is Δf_{TM−TE} ≈ 80 MHz/K. It is sometimes difficult to assign the polarization for WGM resonators; however, a simple measurement of the amount of thermal shift readily distinguishes between the polarizations in MgF_{2} resonators. Interestingly, a few measured resonances show intermediate efficiency between the TE and TM modes due to the degeneracy of the effective indices, which forms a hybridized mode between orthogonal polarization modes^{38}.
Versatile tuning of soliton microcombs
We perform the spectral tuning of a soliton microcomb using a temperaturecontrolled MgF_{2} microresonator with an FSR of 23.35 GHz. A thermoelectric cooler element is used to control the resonator temperature with 0.01 K accuracy with the appropriate feedback control, which enables active thermal tuning. First, we generate a soliton comb at 25 °C (ΔT = 0 K) and measure the soliton beat note, which directly yields the soliton repetition rate. Here, the pump detuning is flexibly controlled by applying a modulation frequency to a phase modulator. We obtain temperaturedependent optical and electrical spectra as shown in Fig. 2, while pump detuning dependence is presented in Fig. 3 (see Methods and Supplementary Note 1 for the details of our experimental setup). The pump detuning is an essential parameter for determining both the fundamental properties of a soliton comb such as the soliton pulse width and the soliton noise limit, which involves the soliton spectral recoil induced by DWs and the Raman effect^{9,39,40}.
Figure 2a shows the measured optical spectra of singlesoliton states when the temperature is increased up to 53 °C with a fixed detuning of 10.5 MHz. The frequencies of each comb line are defined by the wellknown relation, f_{m} = f_{ceo} + m × f_{rep}, where f_{m} is the frequency of each comb mode and the index m is an integer. Since the repetition rate f_{rep} is directly obtained as the beat frequency as shown in Fig. 2b, c, it is easily possible to extract the carrierenvelope offset frequency f_{ceo} by subtracting the pump frequency from an integer multiple of f_{rep} (m = 6365). Figure 2d–f shows the variation in the pump (center) frequency f_{p}, f_{rep}, and f_{ceo}, where the fitting slope (solid line) yields the tuning efficiency, −1.74 GHz/K, −208 kHz/K, and −14.1 MHz/K, respectively. The total tuning ranges of f_{p}, f_{rep}, and f_{ceo} reach −48.8 GHz, −5.85 MHz, and −386 MHz, respectively, with 28 K temperature change. The measured tuning efficiencies agree well with the theoretical predictions discussed in the previous section, and the results confirm that the pump mode belongs to the TE modes. It should also be noted that the total tuning range of the pump frequency exceeds 2FSRs owing to the narrow mode spacing of our millimeterscale resonator. This feature makes it possible to search for a desired resonance mode by using a thermal heating method instead of broadband laser frequency control, whose tuning range depends on the performance of the pump laser and may be insufficient to access all resonance modes within 1FSR.
To investigate the effect of thermal tuning on soliton properties in detail, we then plot the recoil frequency and average soliton power as a function of temperature (Fig. 2g, h). We draw attention to the fact that temperature change leads to a variation in the soliton recoil frequencies even with the same detuning values (Figs. 2g and 3b). Since the Raman effect can be neglected in MgF_{2} crystals because of the very narrow gain bandwidth, DW emission, which results from higherorder dispersion and mode couplings, must critically affect the degree of soliton recoil^{9,31,40}. Even though the spectral positions of the DWs are inherent features in a single microresonator with fixed detuning, we find that the intensity and positions of strong peaks vary irregularly with temperature, and this indicates that changes in resonator temperature have a strong influence on resonator mode structure, i.e., thermallyinduced resonance shift. Specifically, birefringent properties, which affect the tuning efficiency for orthogonal polarization, induce considerable mode hybridization rather than linear frequency shifts of all the resonant modes via strong crosspolarization mode coupling^{38}. The design and prediction of the spectral position of the DWs are still challenges^{41}; nevertheless, we believe that thermal activation could be utilized for the active tuning of the DWs in a WGM resonator supporting many different transverse modes.
More surprisingly, the soliton power is significantly increased with the temperature change even though a tapered fiber is in contact with the resonator and the position is not adjusted during the series of measurements. This result is not intuitively anticipated because the changes in the pump frequency and repetition frequency have little impact on the average soliton power given by \({P}_{{{{{{{{\rm{sol}}}}}}}}}=\frac{2\eta {A}_{{{{{{{{\rm{eff}}}}}}}}}}{{n}_{2}Q}\sqrt{2nc{\beta }_{2}\delta \omega }\)^{4,40,42}, where η is the coupling efficiency, A_{eff} is the effective mode area, n is the refractive index, n_{2} is the nonlinear index, and Q is the total Qfactor. Although the pump detuning δω = (ω_{0} − ω_{p}) and resonator groupvelocity dispersion β_{2} contribute to the soliton power, we can decouple them because the measured pulse widths \(\tau =\sqrt{c{\beta }_{2}/2n\delta \omega }\) are almost constant (τ ≈ 178 fs) under a fixed detuning of 10.5 MHz, which is direct evidence that the overall dispersion is unchanged. In fact, the pulse widths follow the theoretical prediction even for different temperatures at the same detuning value (Fig. 3c). When we neglect the changes in the mode area, refractive index and nonlinear refractive index for a temperature increase of 28 K, another possibility is that the significant change in the soliton power is caused by the variation in the coupling rate \(\eta /Q\,(={n}_{2}{P}_{{{{{{{{\rm{sol}}}}}}}}}/2{A}_{{{{{{{{\rm{eff}}}}}}}}}\sqrt{2nc{\beta }_{2}\delta \omega })\). The green symbol in Fig. 2h represents the extracted coupling efficiency η = κ_{ext}/(κ_{int} + κ_{ext}) as a function of temperature change at a constant intrinsic cavity decay rate (κ_{int}/2π = 0.43 MHz). This result indicates that the coupling strength, namely κ_{ext}/2π, is eventually increased 3.4 times (0.38–1.3 MHz) by heating the resonator despite the fact that neither the position nor the diameter of the fiber, which could alter the coupling strength, is adjusted during the measurement. It should be noted that the accessible soliton step varies due to the change in the fibertoresonator coupling condition as the cavity temperature increases. We emphasize that the coupling strength is known to be determined by the mode overlap integral and phase mismatch between a resonant mode and a fiber mode. In this respect, we theoretically detail the mechanism of the thermallyinduced coupling variation.
Analysis of dynamic change in coupling strength
In light of the experimental results, we numerically analyze the effect of the thermal activation on the coupling strength between a WGM resonator and a silica tapered fiber coupler (The calculations are detailed in Methods and Supplementary Note 2–4). Figure 4a shows a schematic model of our simulation for the coupling strength in a millimeterscale MgF_{2} resonator. On the assumption of a weak coupling approximation (i.e., the coupling coefficient is sufficiently small compared with the propagation constant), the coupling coefficient can be expressed as^{43}:
where E_{f(r)} corresponds to the complex electric field amplitudes of the fiber (resonator), N_{f(r)} is the normalization constant for the fiber (resonator), and n_{r} and n_{0} are the refractive indices of the resonator and air. The difference between the propagation constants Δβ = β_{f} − β_{r} contributes to the phase mismatch between the fiber and resonator modes. The integration is performed over a volume of the resonator in a threedimensional Cartesian coordinate system, where the zaxis represents the length direction of a tapered fiber. The coupling strength is then obtained from the relation \({\kappa }_{{{{{{{{\rm{ext}}}}}}}}}\approx  \tilde{\kappa }{ }^{2}(c/2\pi {n}_{{{{{{{{\rm{r}}}}}}}}}R)\), which accordingly corresponds to ω/Q_{ext}. Importantly, Eq. (1) indicates that both the overlap integral between the fiber and resonator modes and the phasematching condition are central to the coupling strength. As a hypothesis, the thermal expansion effect can modify the coupling strength as a result of the variation in the mode overlap, which is determined by the degree of contact between the fiber and resonator, and the thermooptic effect changes the effective refractive indices, thus altering the phasematching condition.
To investigate the dynamics of coupling variation, we first begin with numerical simulation of the coupling strength with respect to fiber diameter for different transverse modes when the fiber and resonator are placed in contact at one point. Figure 4b, c presents the mode field distribution for featured five transverse modes and the calculation results. As mentioned earlier, MgF_{2} resonators exhibit different refractive indices for orthogonal polarized modes, and therefore the fiber diameter which minimizes the phase mismatch also differs for the TE and TM mode families. In addition, higherorder polar and radial modes exhibit different coupling strengths due to the difference of the propagation constant and mode profiles. The two modes (2nd and 4th) of the five WG modes are not shown in Fig. 4c because the coupling strengths are several orders of magnitude smaller, and coupling does not occur. This is because the fiber is positioned at y = 0 in this simulation, thus significantly reducing the mode overlap. The coupling strength for varying the fiber position, defined as polar angle φ, is presented in Fig. 4d, where the fiber position is moved along a resonator curvature. This result confirms that higherorder WG modes can also be efficiently excited by changing the contact position of the fiber. Indeed, the coupling strength can be controlled by carefully adjusting the contact position, fiber thickness, and effective coupling length, namely the degree of contact with a resonator, in practical experiments.
To consider the effect of the effective coupling length Δl instead of single point coupling at z = 0, we modified Eq. (1) by including an additional coupling term \({\tilde{\kappa }}_{{{\Delta }}l}\) supposing that the mode overlap is maximized at z = 0 and remains constant in the finite coupling regime Δl. Then, the additional coupling term is given by \({\tilde{\kappa }}_{{{\Delta }}l}={\tilde{\kappa }}_{z = 0}\sin ({{\Delta }}\beta {{\Delta }}l/2)/({{\Delta }}\beta /2)\)^{44}. Such an approach is referred to as a pulleystyle configuration and widely employed with integrated resonators to precisely engineer the coupling strength^{44,45,46}. The significant difference is that the tapered fiber used in our system provides a considerable degree of coupling flexibility in contrast to integrated resonators with a fixed waveguide as mentioned earlier. The dependence of the coupling strength on effective coupling length and fiber diameter is shown in Fig. 4e as a contour map. The coupling strength exhibits a periodic dependence on the effective coupling length, which originates from a relative phase shift between the fiber and resonator modes as seen in a directional coupler^{47}. When the fiber diameter is 3.3 μm (typical diameter of a tapered fiber is 2–4 μm in our experiments), the coupling strength for the fundamental (1st) TE mode is gradually increased from κ_{ext}/2π ≈ 0.56 to 2.7 MHz with ~ 10 μm coupling length change. The results in the fundamental TM mode are also presented in Supplementary Note 5.
Our interest here is whether or not the thermal effects indeed modify the coupling strength. Since the temperaturedependent resonator expansion is given by ΔR(ΔT) = α_{l}RΔT, the radius of the resonator (R = 1400 μm) is increased by ~0.5 μm with 30 K heating. It should be noted that the change in the effective coupling length is a difficult parameter to assess experimentally because it relies on the tensile strength of the tapered fiber and the actual tapered length. Nevertheless, we emphasize that the effective coupling length is likely to increase by several microns or several tens of microns in practical experiments (See Supplementary Note 3), and therefore, the thermal expansion effect (i.e., the change in the effective coupling length) could cause a dynamic, widerange coupling enhancement as seen in our experimental results in Fig. 2. We further mention that the thermooptic effect also contributes to the coupling strength via the phasemismatch condition Δβ due to the considerable difference in the thermooptic coefficients of silica and MgF_{2} crystal. In thermal equilibrium, the temperature of a silica tapered fiber is considered to be the comparable to that of the resonator in the region where the fiber is in contact with the resonator. However, we find that the phasemismatch induced by the thermooptic effect only alters the coupling strength of up to ~20% with a 50 K temperature change (Fig. 4f, g) and is thus less effective than the expansion effect.
Conclusions
In summary, we have experimentally demonstrated the versatile tuning of rf repetitionrate soliton combs in a crystalline MgF_{2} microresonator. The main properties of frequency combs, namely, the center frequency, repetition frequency, and carrierenvelope offset frequency are tuned by −48.8 GHz, −5.85 MHz, and −386 MHz, respectively, by using the temperature control of a fibercoupled resonator system. Additionally, we show the possibility of spectrally tuning the DWs accompanied by soliton recoil via the dynamic modification of cavity mode structures due to the birefringence of crystalline microresonators. Thermallyinduced coupling variation that leads to pronounced soliton power enhancement has also been demonstrated and studied theoretically to validate this effect. So far, optimization of a coupling position and a fiber thickness has been performed based on practical experiences and patient adjustment rather than a theoretical method. The quantitative analyses also provide helpful guideline for the rational optimization of this “rule of thumb” fiberresonator coupling to multimode WGM resonators.
Previous studies have not reported such a significant coupling change induced by thermal effects^{4,26,29,30,31}. In our case, the flexibility of the tapered fiber coupler and the relatively large thermal expansion coefficient of the crystalline material (e.g., MgF_{2}: 8.9 × 10^{−6} K^{−1}, fused SiO_{2}: 0.6 × 10^{−6} K^{−1}, Si: 2.6 × 10^{−6} K^{−1 }^{35}) allow modification of the coupling strength, and resulting in the enhancement of the soliton comb power. Although the implementation of thermallyinduced dynamic coupling control in integrated resonators is a challenge, it would be possible if the resonator and the waveguide have very different thermooptic coefficients. For future practical applications, broad and reliable soliton tuning could be employed in microcomb systems. In particular, photonic rf oscillators exploit the intrinsic low phasenoise properties of soliton combs in ultrahighQ crystalline resonators^{2}, whereas the thermal drift of optical resonances leads to the degradation of the phasenoise performance in the low Fourier frequency regime. Thus, temperature control and subsequent highlevel stabilization^{48,49} are essential for mitigating the thermal drift and pursuing the fundamental noise limit of soliton microcombs^{50}. This study provides a greater understanding regarding widerange tunability of the comb frequency and output power toward the implementation of both a highpurity photonic oscillator and a highcoherence laser source for optical communications and dualcomb spectroscopy that requires precise control of the difference between repetition frequencies.
Methods
Experimental setup
An MgF_{2} microresonator is fabricated from a zcut crystalline disk by shaping and hand polishing with diamond slurry. The FSR of 23.35 GHz corresponds to a major radius of 1.49 mm and typical loaded Qfactors are 10^{8}–10^{9} in the 1550 nm band. The curvature radius is ~50 μm. The groupvelocity dispersion of the pumped mode in our work is β_{2} = d^{2}β/dω^{2} ≈ − 6.3ps^{2}/km, yielding anomalous dispersion. The groupvelocity dispersion is defined as the derivative of the inverse group velocity with respect to the angular frequency^{51}. The crystalline resonator is fixed with adhesive to a metal rod a few centimeters long, so that the temperature of the resonator is controlled through the metal parts with a temperature transducer. For soliton generation, a continuouswave fiber laser (linewidth < 0.1 kHz) is amplified by an erbiumdoped fiber amplifier, and an optical power of ~400 mW is coupled via a silica tapered fiber. The length of the tapered region is ~40 mm, and the diameter of the tapered fiber and the contact position with the resonator is optimized to make it possible to observe distinct soliton steps. Typically, we employed fiber diameters of 2–4 μm, and brought the fiber into contact with the resonator to realize stable light coupling. A singlesoliton is generated via a forward laser frequency sweep with an adequate scan speed (0.1–10 MHz/ms), and then the soliton is captured by activating a servo controller to lock the effective detuning via a phasemodulation technique^{4,8,52}. The feedback control of the pump laser frequency enables us to stabilize the soliton state against the thermal drift of the cavity resonance and the longterm frequency fluctuation of the pump laser^{52}. The absolute frequency of the pump laser is monitored by using a wavelength meter, whose spectral resolution (0.1 pm) limits the absolute accuracy of f_{p} and f_{ceo}.
Numerical analysis of resonator and fiber modes
Mode profiles of the WGM resonator and tapered fiber are calculated by conducting a finite element simulation and a theoretical analysis, respectively. We perform a finite element method (FEM) simulation by using FEM software (COMSOL Multiphysics) to calculate the resonator modes. For the simulation, we model an axisymmetric WGM resonator with a resonator radius of 1400 μm and a curvature of 25 μm, respectively, with a calculation area of 55 μm by 55 μm. The refractive indices of MgF_{2} for the TE and TM modes are set independently for the simulation.
To calculate the mode overlap integral between the fiber and resonator modes, the threedimensional electrical components are mapped on Cartesiancoordinate grids and then normalized by a factor, given as:
The y and xaxes correspond to the dominant directions of the electrical field of the TE, and TM modes, respectively. The mode field of a tapered fiber can be analytically obtained by solving the wave equation in the cylindrical coordinates, where the propagation direction is taken to be the zdirection and the center of the contact position is defined as z = 0. A detailed derivation is presented in Supplementary Note 2. The equation gives several fiber modes with different propagation constants, whereas we only focus on the fundamental fiber mode, i.e., HE_{11} mode, which is linearly polarized singlemaximum mode for the simulation. The refractive indices for a fiber and surrounding air are set at n_{f} = 1.444 and n_{0} = 1.000 at 25 °C.
Calculation of coupling coefficient
The coupling coefficient is obtained by performing a threedimensional integration of Eq. (1). The integration for the x–y plane, which is transverse to the propagation direction, is performed over the crosssectional area of an MgF_{2} microresonator at each zpoint. Because of the resonator curvature r, the overlap integral gradually decreases in accordance with the light propagation in the zdirection. Then, the overlap integral is further integrated along the zaxis to take the phasematching condition into account. A sufficiently large integral interval is chosen so that the coupling coefficient converges to a constant value. When we calculate the additional coupling introduced by the effective coupling length, the section is considered to be a straight waveguide for simplicity, which can be justified because the resonator radius (R = 1400 μm) is much larger than the diameter of a tapered fiber (~2–4 μm). The effective refractive index for a resonator mode is calculated from the electric field distribution and temperaturedependent refractive indices for extraordinary and ordinary rays (see Supplementary Note 3 for details).
Data availability
The data that support the plots of this paper and other findings within this study are available from the corresponding author upon reasonable request.
Code availability
The codes used for this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work is supported in part by JSPS KAKENHI (JP19H00873, JP22K14625). S.F. acknowledges support from RIKEN Special Postdoctoral Researcher Program. The authors thank Dr. W. Yoshiki for supporting numerical simulation.
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S.F. and K.W. performed the experimental measurements. S.F. and R.S. conducted the numerical simulations. K.W. fabricated the crystalline microresonators, and S.F., K.W., H.K. and S.K. developed the experimental setups. S.F. and K.W. analyzed the data. S.F. wrote the manuscript with input from Y.K.K. and T.T. All authors had constructive discussions. S.F. and T.T. supervised the project.
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Communications Physics thanks Xiaoxiao Xue, Jonathan Silver and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
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Fujii, S., Wada, K., Sugano, R. et al. Versatile tuning of Kerr soliton microcombs in crystalline microresonators. Commun Phys 6, 1 (2023). https://doi.org/10.1038/s42005022011184
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DOI: https://doi.org/10.1038/s42005022011184
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