Soliton microcombs based on optical microresonators have triggered the rapid development of miniature and chip-scale optical frequency combs in recent years [1–4] and have resulted in the emergence of fully integrated frequency comb chips, opening an access to high-performance, high-compactness, and high-volume laser sources for advanced optical metrology [5–9]. Indeed, a number of proof-of-concept applications have been demonstrated with soliton microcombs, such as massive parallel optical communications [10,11], optical ranging [12–14], massive parallel LIDAR [15], low-noise microwave synthesis [16–18], astronomical spectral calibration [19,20], and optical neuromorphic computing [21,22].

While recent focus is on photonic integrated platforms where wafer-scale, high-quality, and highly nonlinear optical microresonators are accessible [23], a parallel platform is whispering gallery mode (WGM)-based crystalline resonators [24,25]. In particular, crystalline fluoride resonators could have a record-high finesse beyond 107 [26], which is suitable for the generation of ultra-narrow linewidth lasers [27–30] as well as soliton microcombs [1,11,31–33]. Given a weak thermo-refractive noise, such resonators could support solitons with low-noise repetition frequencies, serving as a photonic microwave synthesizer [16,17]. Moreover, on the study of soliton physics, they represent an ideal platform which is almost free from high-order dispersive or nonlinear effects. A number of dissipative soliton dynamics have been demonstrated with high-level of agreement with the theory, such as the soliton double resonance [34], soliton pulse scaling [35], soliton breathers [36], soliton molecules [37], soliton crystals [38], and soliton behaviors to intermode interactions [39].

Indeed, effects of intermode interactions have been widely observed and studied in soliton microcombs, which in most cases are reflected as localized power enhancement on certain comb modes [40]. These modes were also referred as single-mode dispersive waves, which would help to stabilize the soliton state if located in a particular “quiet point” [41]. In the strong coupling condition, energy exchange between mode families was also observed, which leads to instability on the soliton state, including a breathing-like dynamic [39]. Efforts were also made to engineer intermode interactions in microresonators. In particular in coupled resonator systems, lithographically engineering the coupling strength and the coupling frequency has evoked new features of cavity solitons, including the soliton comb with tunable dispersive waves [42], dark pulse excitation with tailored dispersion profile [43–45], and solitons with super efficiency [46,47].

Recent studies have also shown that with an induced auxiliary mode, an intermode pumping wave may cause the thermal equilibrium in the cavity and recover the access of soliton microcombs with reduced thermal instability [48–50].

Usually, such coupled mode or coupled resonator systems require a dedicated design to apply proper intermode interactions. In this regard, crystalline resonators with a high number of mode families would equally support a high volume of intermode interactions of different kinds and serve as an alternative platform in the study of emergent features of cavity solitons. Nevertheless, most reported effects of intermode interactions in crystalline resonators were narrowband and involving few comb modes, given a large difference in the free spectral range (FSR) between the coupled mode families. Effects of group velocity dispersion (GVD) in the second mode family, namely dispersive intermode interactions, were usually hindered and unexplored.

In this paper, we demonstrate soliton microcombs in crystalline magnesium fluoride (MgF2) resonators. Effects of broadband and dispersive intermode interactions are experimentally evidenced, when an intermode pumping scheme is applied, which lead to tailored comb spectrum dependent on the dispersion in the second mode family. We also demonstrate a soliton comb in company with the Raman lasing, including both narrowband lasers and Raman–Kerr combs [51–54]. A suspect Raman-soliton comb [55,56] is also observed around the anti-Stokes mode.

First, we prepared ultra-high-Q crystalline MgF2 resonators, by means of mechanical machining [57–59]. The preform of the cavity is diamond turned from a cylindrical bulk material, followed by surface polishing processes. Diamond slurries with three particle sizes (i.e., 1300, 250, and 50 nm) are used in the polishing in an order to improve the surface smoothness; see Table 1. During the polishing, the surface profile as well as the roughness is also monitored by a commercial optical interferometric profiler (Sensofar Metrology, system’s noise <1nm). Figures 1(a)–1(c) showcase a measure of the cavity surface. In particular, by having the detailed geometric structure, the transverse mode field of the cavity’s WGMs is numerically calculated, by which an axial range of care is defined to cover an area where the absolute mode field amplitude is greater than 1% of its central peak; see Fig. 1(a). The measured profile is also fitted by a 2D polynomial function, and the surface roughness is extracted as the residual of the fit; see Figs. 1(b) and 1(c). Remarkably, the estimated RMS roughness can be as low as a few nanometers, which indicates a high-quality surface polishing and an ultra-high-Q factor (typ.>109) of the resonator. In practice, such a surface assessment is carried out at 18 positions along the circumference of the resonator (spaced by every 20°), indicating a variation of 5% on the cavity diameter (i.e., 0.25 mm over a diameter of ca. 5 mm). The curvature on the edge of the cross section is ca. 35 μm.Table 1.

Polishing Slurry Granule Size, Polishing Time, and Corresponding Roughness

The MgF2 resonator is then coupled with a tapered fiber for the assessment of the Q-factor and for soliton microcomb generation, using an experimental setup shown in Fig. 2(a). The transmission of the cavity resonance is measured by scanning the cw diode laser over it. For ultra-high-Q resonators, the resonance linewidth is already comparable with that of the probe laser [O(10kHz)], and the laser frequency tuning within the linewidth could be faster than the intracavity photon lifetime. This leaves a ringdown profile at the ending edge of the resonance [see Fig. 1(d)], which is also theoretically derived in Ref. [60]. In detail, the time evolution of the intracavity field of one resonant mode could be described as a simple harmonic oscillator model, i.e., ∂A∂t=iω0A−κ2A+κex·sin,(1)where ω0 is the angular frequency of the resonance; κ is the overall loss rate of the system which consists of both the intrinsic loss rate κ0 and the coupling loss rate κex, i.e., κ=κ0+κex; sin stands for the external source, which in the frequency tunable cw mode can have the variation sin=s0·exp[iφ(t)]. In the stationary form, i.e., the laser frequency tuning with respect to its initial value ωi is slowly changed compared with the intracavity photon lifetime, we have φ(t)=ωit, and the solution of Eq. (1) gives a standard Lorentz profile as a function of the laser detuning δω=ω0−ωi. In contrast, in the condition that the laser frequency tuning is comparable with the photon lifetime, we have φ(t)=(ωi+Vst2)t, where Vs is the tuning speed of the laser frequency. Hence, a variation of Eq. (1) could be ∂a∂t=i(δω−Vst)a−κ2a+κex·s0,(2)where A=a·exp[iφ(t)]. The integration of Eq. (2) further gives a=κex·s0·exp(iδωt−κ2t)[f(t)−1iδω−κ2],(3)with $$\begin{gathered} f(t)=−iπ2Vsexp[i(iδω−κ/2)22Vs] \\ ×{erf[−ii(δω−Vst)−κ/22iVs]−erf[−iiδω−κ/22iVs]}, \end{gathered}$$where erf(z) is the complex error function, i.e., z∈C. The cavity transmission is then calculated as T=|s0−κex·as0|2.(4)

Note that fitting with Eq. (4) would deterministically return the coupling rate κex as part of the overall loss rate κ, such that the coupling regime of the cavity (over- or under-coupling) can be distinguished. As a result, an intrinsic Q-factor Q0=ω0/κ0=8.44×109 is obtained with respect to the transmission profile shown in Fig. 1(d).

In contrast to the theoretical fitting, reading out the decay rate of the ringdown profile would also reveal the photon lifetime of the resonator, as well as the overall loaded Q-factor (i.e., Q=ω0/κ) [26,61]; see Fig. 1(e). As a comparison, the loaded Q-factor via the theoretical fitting is 4.75×109 and that from the decay rate is 4.95×109.

Therefore, we have demonstrated ultra-high-QMgF2 resonators, with a convincing Q-factor above 109. In addition, the coupling ideality with respect to selected resonant mode of the resonator is characterized [see Fig. 1(f)], which is on a high-level indicating almost no loss at the coupling junction [62,63].

We next carried out experiments for soliton microcomb generation in MgF2 resonators. By a conventional laser tuning scheme with respect to an isolated high-Q cavity resonance, we could reproduce the classic transmittance of the system as a function of the laser-cavity detuning, which reveals soliton steps on the red-detuned side of the resonance [1]; see Fig. 2(c). In addition, the detuning regions corresponding to the modulation instability (MI) regime and to the soliton breathers are also noted. Tuning the laser from the blue-detuned side and landing on the soliton step, we could observe soliton microcomb generation, especially the single soliton state that has an overall smooth sech2-shape spectral envelope; see Fig. 2(b). Moreover, the stability of the soliton comb is assessed by measuring its low-frequency RF spectrum [Fig. 2(d)] as well as the repetition frequency [Fig. 2(e)], after the residual pump wave was removed by a fiber notch filter before the spectrum was photo-detected. As a result, while the comb in the MI regime shows a noise figure around DC and a broad line shape around the repetition frequency, the soliton comb shows a low noise state with a narrow linewidth repetition frequency. The linewidth of the soliton repetition frequency is measured to be O(10kHz), similar to that of the pumping wave.

In fact, there is a large number of mode families in our crystalline resonators, each corresponding to a transverse eigen-mode field [in the (r, z) plane]. As a consequence, there is a high probability to observe resonances that are in proximity, such that the intermode pumping scheme can be investigated. Figure 3(a) showcases a transmission of two resonances in proximity, via the laser tuning at the power of 3.8 mW. When the pump power is increased to ∼280mW, which is sufficient to excite the comb generation, a complex transmittance of the system is detected [see Fig. 3(b)], and a soliton step situated in between the two resonances is observed. Indeed, tuning the laser frequency from the blue-detuned side to stop on the soliton step, a soliton comb spectrum is observed; see comb 1 in Fig. 3(d). Intuitively, the soliton state is stemming from pumping the primary (left) resonance and is on the slope of the auxiliary (right) resonance, such that it may feature interactions in between. As previously reported, one effect is that the pumping wave being coupled to the second resonant mode would cause the thermal equilibrium of the system [49,50]. This effect also applies to our system, where we noted that the soliton comb would be running for hours free from additional feedback control (despite that the laser drift would surpass the soliton existence range during the time).

Moreover, we noted that the generated comb spectrum via the intermode pumping scheme usually features a strongly tailored spectral envelope. This includes not only two comb spectra [i.e., comb 1 and comb 2 in Fig. 3(d)] featuring certain spectral profiles at the center, but also a comb spectrum (comb 3) with broadband enhancement such that the overall envelope is beyond the standard sech2 profile.

In addition, the Raman lasing is observed in our MgF2 resonators, in company with the microcomb generation; see Fig. 3(d). The noted frequency shift for both the Stokes and anti-Stokes modes is ca. 7.35 THz (comb 1) and ca. 5.61 THz (comb 2 with a different polarization), which are much smaller compared with that of reported Raman bonds of fluoride materials, and are attributed to Raman bonds of MgF2 oligomers on the surface of the resonator [64]. We observed not only isolated Raman lasing and narrowband Raman–Kerr combs as usual states [51], but also a suspect Raman-soliton comb [55] on the anti-Stokes band, which has a wide spectral span and has a similar sech2-shape envelope to conventional soliton microcombs.

The stability of microcombs via the intermode pumping is also assessed by measuring the low-frequency RF spectrum; see Fig. 3(c). It can be noted that the soliton comb 1 accompanied with narrowband Raman–Kerr combs could feature a low-noise figure, indicating that the comb is overall in a stable state. The comb 2 features an outstanding RF tone at ca. 6 MHz, together with certain noise, which indicates that the comb is in a transition from the MI to the breathing state.

We also calculated the conversion efficiency of the observed microcombs, using the following equation [65]: η=∑μPμ−P0Pin,(5)where Pμ indicates the power of the comb mode and μ is the mode index, P0 is the power of the central mode which mostly contains the residual pump power, Pin is the pumping power in the tapered fiber before being coupled into the resonator. As a result, the soliton comb generated with the conventional pumping scheme [Fig. 2(b)] has an efficiency of 1.28%. The soliton comb 1 in Fig. 3(d) has an efficiency of 0.99% (taking the central comb) with certain pump power converted to the Raman lasing. The efficiency for the comb 2 is 9.54% (also taking the central comb), yet it is not a fully stabilized comb state. The comb 3 is calculated to have an outstanding efficiency, ca. 31.1%. However, there lacks essential evidence to show if this comb is in the low-noise or soliton state.

Physically, the efficiency of soliton combs in a single and anomalous mode family is governed by the cavity nonlinearity as well as affected by the coupling scheme, which in MgF2 microresonators, with the nonlinear coefficient ∼10−4Hz per photon and in the strongly over-coupled regime, could only reach a few percentages. As such, we notice that the efficiency of >30% in the comb 3 is truly outstanding and can be attributed to effects beyond the single-mode operation. That is, the pump power partially coupled into the auxiliary mode family would via intermode interactions transfer back to the comb mode and increase the comb power, and with constructive interference, the power efficiency is increased. In practice, microcombs with close to the 0 dBm power level (excluding the residual pump) would be highly desired in applications including telecommunications [10,11,66] and photonic microwave synthesis [16–18].

For a better understanding of the comb spectral tailoring effect with the intermode pumping, we referred to numerical simulations based on a pair of coupled Lugiato–Lefever equations (LLE) [39]: $$\begin{gathered} ∂A˜p(μ,t)∂t=(−κp2+iδω+iDint(μ))A˜p+iκcA˜aux \\ −igpF[|Ap|2Ap]μ−igcF[2|Aaux|2Ap]μ \\ +κex,p1−αsin, \end{gathered}$$(6)$$\begin{gathered} ∂A˜aux(μ,t)∂t=(−κaux2+iDaux(μ))A˜aux+iκcA˜p \\ −igauxF[|Aaux|2Aaux]μ−igcF[2|Ap|2Aaux]μ \\ +κex,auxαsin, \end{gathered}$$(7)where |A˜p(μ,t)|2 stands for the photon number in the primary mode family (where the soliton formation is expected), and |A˜aux(μ,t)|2 is that in the auxiliary mode family; Ap(θ,t) and Aaux(θ,t) are amplitudes in the Fourier domain; κp and κaux are overall loss rates of cavity mode families, including both the intrinsic loss rate and the coupling loss rate. In particular, the coupling loss rates are expressed as κex,p(1−α) and κex,auxα, where the subparameters α, 1−α are introduced revealing fundamental modal overlap coefficients between the bus waveguide mode (in the tapered fiber) and the cavity modes (and in a close system with two cavity modes, the sum of the coefficients is forced to be unity), and subparameters κex,p and κex,aux are more specifically referred as the coupling strength determined by the gap distance between the fiber and the cavity. This enables us to separate and monitor the transmitted light fields from the primary and the auxiliary mode families. The coefficient κc is the linear coupling strength between the two mode families. The laser-cavity detuning is δω=ωp(0)−ωi, where ωp is the angular frequency of resonances in the primary mode family, and ωi is that of the pumping wave. The resonant mode μ=0 is the central pumped mode and sin is the pump amplitude. The integrated dispersion of the primary mode family is Dint(μ)=ωp(μ)−ωp(0)−μD1,p, where D1,p/(2π) is the FSR around the pumped mode. In the same reference frame, the dispersion profile of the auxiliary mode family is Dint(μ)=ωaux(μ)−ωp(0)−μD1,p. Nonlinear coupling coefficients gp and gaux indicate the single-photon induced nonlinear frequency shift by means of the self-phase modulation, and gc is that by the cross-phase modulation. Indeed, gp(μ)∝n2/Veff,p(μ) and gc(μ)∝n2/Veff,c(μ), where n2 is the nonlinear refractive index of the material, Veff,p(μ) is the effective mode volume of the primary mode family and Veff,c(μ) is the effective overlapped mode volume between the primary and the auxiliary mode families.

Then, the transmitted field pattern of each mode family is sout,p=1−αsin−κex,pA˜p,(8)sout,aux=αsin−κex,auxA˜aux,(9)and the overall transmitted field is $$\begin{gathered} sout=1−αsout,p+αsout,aux \\ =sin−κex,p1−αA˜p−κex,auxαA˜aux, \end{gathered}$$(10)which contains wave interference between two field patterns, both transmitted and propagated through the tapered fiber.

Before LLE simulations, we first performed mode simulations based on the finite element method (COMSOL). The simulations take the geometry shown in Fig. 1(a) and cover 20 transverse mode families, and could confirm certain information of the MgF2 resonators, such as the FSR and the GVD of mode families. The simulated D2/2π component is between 1 and 2 kHz for most of the mode families, while few higher-order mode families would feature normal group velocity dispersion. Moreover, a mode map of all 20 mode families is illustrated (see Fig. 4), which reveals a large number of mode crossings in the resonator. Physically, a mode crossing indicates that there is phase matching between different mode families, which under perturbations (e.g., via imperfection induced light scattering in the resonator) would induce light coupling in between, efficiently at the crossing modes. In the map, we noticed that some of the mode crossings are sharp such that the light coupling is expected among few modes, namely narrowband intermode interactions. However, there are also mode families featuring slow crossings, such that the light coupling can be supported in a broader range. In particular, six mode families are highlighted in the map, forming three types of crossings (marked as case-I, -II, and -III) where dispersion components could make a decisive role in changing the crossing profile. As such, there could be a relative normal, anomalous, or close-to-zero dispersion profiles in the auxiliary mode family, with respect to the primary one.

Then, we carried out simulations with qualitatively same dispersion relations. In details, the dispersion component (D2,p) is always fixed to support the soliton comb generation in the primary mode family. The absolute value of D2,aux is also fixed but with inverse signs to model normal and anomalous dispersion profiles, and with a higher-order component D3,aux to model the close-to-zero dispersion profile. In simulations, the nonlinear self-phase modulation and cross-phase modulation regarding Aaux are neglected, i.e., gaux=0 and gc=0, provided that the field pattern in the auxiliary mode family is usually much weaker compared with the soliton pattern in the primary mode family. A complete list of parameters used in the simulations is shown in Table 2.Table 2.

Parameters for Numerical Simulations

As a result, the simulated soliton combs show close spectra to our experimentally observed comb spectra in Fig. 3(d). In particular, if the sign of the dispersion component D2 is flipped in the auxiliary mode family, we observed different transmitted field patterns, as the result of wave interference. Physically, if D2,aux>0 (case-I), the generated comb modes in the primary mode family would be on the red-detuned side of the auxiliary modes upon certain detuning (δω>0), and each mode would feature a phase shift that leads to destructive interference in the transmitted power; see Figs. 5(a) and 5(d). This gives rise to a narrowed power peak on the comb spectrum, which is close to the observation of the comb 2 in Fig. 3(d). In contrast, if D2,aux<0 (case-II), the comb modes would be on the blue-detuned side featuring constructive interference in the transmitted power; see Figs. 5(b) and 5(e). This gives rise to a smooth comb spectrum beyond the standard sech2 profile, which is close to the observation of the comb 3 in Fig. 3(d). In the third dispersion profile, the comb modes feature both the constructive and destructive interference, resulting in a quick change of comb power when the phase shift is reversed; see Figs. 5(c) and 5(f). The transmitted comb is similar to the observation of the soliton comb 1 in Fig. 3(d).

Therefore, numerical simulations could confirm the comb spectral tailoring effect via intermode interactions. Moreover, we realized that the missing nonlinear effects in the simulation could be critical, which may lead to instability of the soliton state if the laser frequency is tuned into the MI region of the coupled auxiliary mode in the anomalous dispersion regime. This would explain, e.g., the comb 2 in Fig. 3(d), as an unstable state. Nevertheless, the simulation could not capture a high-efficiency soliton comb, nor the Raman lasing (as Raman effects are switched off).

In conclusion, we have demonstrated soliton comb generation in ultra-high-Q crystalline fluoride resonators. The home-developed MgF2 resonators have an intrinsic Q-factor approaching 1010, which is on par with the best performances of the WGM-based microresonators. Cavity dissipative soliton state can be excited in these resonators, both with the conventional pumping scheme regarding an isolated cavity resonance, and via the intermode pumping scheme. In the latter, comb spectral tailoring effect has been widely observed and studied, and found dependent on the dispersion regime in the coupled auxiliary mode family. Indeed, given a large number of transverse mode families in the resonator, the presence of slow crossings between mode families further reveals structured dispersion profiles in the auxiliary mode families, including a relatively normal, anomalous, or close-to-zero dispersion, which are decisive in tailoring the comb spectral envelope. In addition, the Raman lasing was observed in company with the comb generation in our crystalline resonators, which in contrast to previous understandings could be in a low-noise and stabilized state. A suspect Raman-soliton comb was also observed around the anti-Stokes side, rather than being on the Stokes side as previously reported in high-Q silica microresonators [55,56].

Overall, our work reveals certain insights of cavity dissipative structures in nonlinear microresonator systems and would contribute to the understanding of even complex soliton dynamics in the presence of dense mode families. It also opens an alternative approach to access stable soliton microcombs in crystalline resonators where dispersion engineering is challenging. Moreover, the results would be supplementary to recent advances in the engineering of intermode interactions in photonic integrated platforms, which has enabled properly tailored soliton comb spectra [42] and super-efficient soliton combs [47].

Acknowledgment. Preforms of the crystalline resonators were fabricated in the Engineering Technology Training Center at Shanghai University.