Optical frequency combs (OFCs) have been attracting significant interest over the last two decades as excellent broadband coherent light sources, revolutionizing optical frequency metrology and spectroscopy [1–4]. Investigations of OFCs in a wide range of wavelengths from visible to mid-infrared (MIR) have been reported [5,6]. The MIR spectral region ($\sim 2\text{–}20\mathrm{\mu m}$) has been a hot topic of ongoing research for the following reasons. On the one hand, this wavelength range is known as the “molecular fingerprint” region and allows for direct exploration of the fundamental absorption bands of many gas molecules, where the absorption strengths of molecular transitions are typically 1 to 2 orders of magnitude higher than those in the near-infrared region [7,8]. On the other hand, it contains two important atmospheric transmission windows, i.e., 3–5 μm and 8–13 μm, that are related to free space communication and environmental monitoring [9]. Thus, MIR laser spectroscopy has numerous applications in biomedicine, defense, environment monitoring, and sensing [10,11]. MIR OFCs offer an attractive way for identifying and quantifying molecular species in a given environment [6,12]. Compared with previous MIR OFCs based on solid-state lasers, semiconductor lasers, and fiber lasers, microresonator-based combs that are free of bulky and complicated components are ideal light sources in practical applications [13–15].

Recently, there have been several demonstrations of MIR frequency comb generation based on microresonators, for example, silicon nitride (${\mathrm{Si}}_3{\mathrm{N}}_4$) [16], silicon (Si) [17], crystalline calcium fluoride (${\mathrm{CaF}}_2$) [18], and magnesium fluoride (${\mathrm{MgF}}_2$) microresonators [19]. However, those microresonator-based MIR frequency combs based on ${\mathrm{Si}}_3{\mathrm{N}}_4$ and Si platforms are difficult to extend beyond 5 μm because of the limitation of material property. It should be noted that fluoride crystalline resonators have not only ultra-high $Q$ factor in the broad transparent wavelength range from $\sim 0.1\mathrm{\mu m}$ to $\sim 10\mathrm{\mu m}$ [20] but also naturally anomalous material dispersion that can be further adjusted with resonator geometry [19]. In addition, with narrow linewidth, high output power, and broad spectral coverage, the quantum cascade laser (QCL) has attracted increasing attention as an excellent MIR laser source [21,22]. Therefore, a highly promising route to the broadband MIR frequency comb generation is pumping ${\mathrm{MgF}}_2$ crystalline microresonators by a QCL, and this will lead to the access of longer wavelengths in the MIR region.

In this paper, we demonstrate a microresonator-based broadband Kerr frequency comb centered at about 4.78 μm pumped by the QCL. The simulation results show that the intracavity power exhibits characteristic step-like patterns via scanning the frequency of the pump laser from the blue to red detuning, and the microcombs will undergo the modulation instability (MI), multi-soliton, and steady single-soliton states in this process. The temporal and spectral evolution dynamics of single soliton formation in the ${\mathrm{MgF}}_2$ resonator are also analyzed in detail. The MI state OFC over an ultra-wide bandwidth of 4380 nm generated in the ${\mathrm{MgF}}_2$ microresonator is experimentally realized by adjusting the pump power and pump-resonance detuning in the MIR region. These results provide a path toward a portable and compact broadband MIR comb source via combining crystalline microresonators with miniaturized QCLs and can offer exciting opportunities for molecular spectroscopy, frequency metrology, and trace-gas detection.

The ${\mathrm{MgF}}_2$ crystalline resonator with exceptional optical, mechanical, and chemical properties is chosen for MIR frequency comb generation. The unique combination of ${\mathrm{MgF}}_2$ material properties includes an excellent mechanical stability, small thermo-refractive constant, and ultra-wide transparency window from the ultraviolet to the MIR. Moreover, the anomalous group velocity dispersion (GVD) over the full MIR region of this material is particularly relevant for achieving the high efficiency broadband four-wave mixing (FWM) process. The ${\mathrm{MgF}}_2$ resonator is fabricated by using the ultra-precision machining technology [23], and a side-view optical micrograph is exhibited in Fig. 1(a). Diameters of resonators typically range from 0.5 mm to 10 mm, corresponding to the free spectral range (FSR) varying from 7 GHz to 140 GHz. Figure 1(b) presents the calculated intensity profile of the fundamental mode of the ${\mathrm{MgF}}_2$ microresonator at 4.78 μm. Using the mode solver based on the finite element method (COMSOL Multiphysics), an effective mode area $A_{\mathrm{eff}}\sim 120{\mathrm{\mu m}}^2$ and a nonlinearity coefficient $\gamma =2\pi n_2/(\lambda A_{\mathrm{eff}})=1\times {10}^{-4}{\mathrm{m}}^{-1}{\mathrm{W}}^{-1}$ are obtained in a ${\mathrm{MgF}}_2$ resonator with a diameter of 2 mm and wedge angle of 30°, where $n_2\sim 0.92\times {10}^{-20}{\mathrm{m}}^2{\mathrm{W}}^{-1}$ is the nonlinear Kerr coefficient of ${\mathrm{MgF}}_2$. Limited by the tuning rate and step size of MIR QCL lasers, the quality factor ($Q$ factor) of the microresonators cannot be directly measured in the MIR band. An alternative way is to utilize the NIR data to estimate the $Q$ factor of the MIR band, so the transmission spectrum of the ${\mathrm{MgF}}_2$ resonator is measured via a tunable narrow-linewidth laser (Toptica CTL 1550) at the communication band with a scanning rate of 0.2 nm/s, as illustrated in Fig. 1(c). By Lorentzian fitting of a resonance mode centered at 1555.038 nm, we measure the typical full width at half-maximum to be 0.88 MHz, giving a loaded $Q$ factor of $2.2\times {10}^8$. The $Q$ factor at 4.78 μm is reduced by about an order of magnitude due to the multiphoton absorption of the ${\mathrm{MgF}}_2$ crystalline microresonator at the MIR region [20,24]. To couple evanescently QCL light into the resonator, a tapered optical fiber made of chalcogenide (ChG) glass is employed because ChG is the superior material due to its broadband transparency and low loss in the MIR [25,26]. The uncoated ChG (${\mathrm{As}}_2{\mathrm{S}}_3$) tapered fibers are fabricated with high precision via an electrically heated taper-drawing process [27]. The taper waist diameter can be precisely controlled to phase match the fundamental mode of the resonator by feedback control of the tapering setup, and Fig. 1(d) shows the suitable taper waist diameter of approximately 1.5 μm for light coupling to a ${\mathrm{MgF}}_2$ microresonator.

As we all know, to generate an MIR Kerr frequency comb, engineering of the microresonator dispersion plays a critical role. The resonance frequency ${\omega }_{\mu }$ of the modes can be Taylor-expanded around the center frequency ${\omega }_0$ as [28] $${\omega }_{\mu }={\omega }_0+\mu D_1+\frac{1}{2}{\mu }^2{D}_2+\frac{1}{6}{\mu }^3{D}_3+\cdots ,$$(1)where $\mu$ is the mode number with respect to the pump mode ($\mu =0$), $D_1/2\pi$ denotes the FSR, and $D_2$ is related to the GVD through the relation $D_2=-c/n_0{D}_1^2{\beta }_2$, where $c$ is the speed of light, $n_0$ is the refractive index, and $D_3$ represents the third-order dispersion. Figure 2(a) plots the simulated integrated dispersion curve $D_{\mathrm{int}}={\omega }_{\mu }-{\omega }_0-\mu D_1$. It reveals that the resonator’s anomalous dispersion $D_2/2\pi$ is $\sim 0.67\mathrm{MHz}$ while higher-order terms are negligible. The nonlinear passive cavity dynamics in the ${\mathrm{MgF}}_2$ microresonator and the corresponding Kerr frequency comb generation can be described by a normalized Lugiato–Lefever equation (LLE) [29]: $$\frac{\partial \mathrm{\Psi }}{\partial \tau }=-(1+i{\zeta }_0)\mathrm{\Psi }+i{|\mathrm{\Psi }|}^2\mathrm{\Psi }-{\sum _{n=2}^N(-i)}^{n+1}\frac{{\beta }_n}{n!}\frac{{\partial }^n\mathrm{\Psi }}{\partial {\theta }^n}+f,$$(2)where $\mathrm{\Psi }(\tau ,\theta )$ is the slowly varying field amplitude, $\tau =\kappa t/2$ describes the normalized time, $\kappa ={\kappa }_0+{\kappa }_{\mathrm{ext}}$ is the cavity decay rate as the sum of intrinsic decay rate ${\kappa }_0$ and coupling rate to the waveguide ${\kappa }_{\mathrm{ext}}$, ${\zeta }_0=2({\omega }_0-{\omega }_p)/\kappa$ denotes the normalized detuning as the pump laser ${\omega }_p$ from the cold resonance frequency ${\omega }_0,{\beta }_n$ represents the $n$th order dispersion term, $\theta \in [-\pi ,\pi ]$ is the azimuthal angle along the rim of the resonator, $f=\sqrt{8g\eta P_{\mathrm{in}}/({\kappa }^2\hslash {\omega }_0)}$ is the external pump related to the pump power $P_{\mathrm{in}}$, $\eta ={\kappa }_{\mathrm{ext}}/\kappa$ is the coupling efficiency, $g=\hslash {\omega }_p^{2}c{n}_2/(n_0^2{V}_{\mathrm{eff}})$ describes the nonlinear gain coefficient, in which $\hslash$ is the Planck constant, ${\omega }_p$ is the pump frequency, $c$ is the speed of light, $V_{\mathrm{eff}}=A_{\mathrm{eff}}L$ is the effective cavity nonlinear volume with the effective mode area $A_{\mathrm{eff}}$ and circumference of the cavity $L$, and $n_0$ and $n_2$ denote the refractive index and nonlinear refractive index, respectively.

To understand and observe the detailed process associated with soliton formation in a ${\mathrm{MgF}}_2$ microresonator, numerical simulations are carried out based on the LLE. Similar to the numerical analysis in the near-infrared [29,30], the thermal effect is also neglected. In practice, it will lead to the resonance shift and, thus, affect the effective detuning. This could be solved by choosing an appropriate wavelength tuning speed of the pump laser such that the resonator could reach the soliton state in a thermal equilibrium. Figure 2(b) depicts the variation of the intracavity average power versus the pump-resonance detuning at pump power $P_{\mathrm{in}}=300\mathrm{mW}$, which has terraced soliton solution states [see the inset of Fig. 2(b)]. As the pump wavelength increases from the blue-detuned side to the effectively red-detuned side, the microcomb transitions from the MI state to multi-soliton state and eventually to the steady single-soliton state. It can be obviously found that the intracavity power has a large drop during the evolution of the MI state to the soliton regime in the cavity. The spectrum evolution of the cavity field versus pump-resonance detuning is also observed in Fig. 2(c), which shows the generation of the Kerr frequency comb. With the pump laser tuning across a cavity resonance from the blue to red detuning, MI and chaotic states of the intracavity field are first observed, where the temporal and spectral distribution is erratic [region (i) of Fig. 2(d)]. The chaotic region is clearly indicated by the large oscillation of the energy. Moreover, multi-soliton states could also be excited in a small range of ${\zeta }_0$ (e.g., 3.421–3.456), where the intracavity fields become relatively stable [region (ii) of Fig. 2(d)]. Finally, the ${\mathrm{sech}}^2$-shape spectrum can be obtained by further increasing ${\zeta }_0$ (e.g., $>3.473$), indicating that the comb reaches the steady single-soliton state [region (iii) of Fig. 2(d)].

To further reveal single-soliton-based frequency comb generation dynamics in the ${\mathrm{MgF}}_2$ resonator, the temporal and spectral evolutions of the cavity field are investigated with the normalized detuning ${\zeta }_0\sim 3.56$ at $P_{\mathrm{in}}=300\mathrm{mW}$, as shown in Figs. 3(a) and (b), respectively. Additionally, the corresponding variations of time-dependent intracavity power are given in Fig. 3(c). Note that the intracavity average power will undergo a relaxation oscillation in the initial several microseconds [inset of Fig. 3(c)], then increase slowly in the long thermal stabilization process, and eventually converge to a constant power. This indicates that a stabilized microresonator-based MIR OFC can be realized with appropriate pump-resonance detuning and pump power [30].

Schematic illustration of the experimental setup for MIR OFC generation using a crystalline ${\mathrm{MgF}}_2$ microresonator is shown in Fig. 4. A continuous-wave pump light is generated from a tunable QCL with a spectral linewidth less than 3.5 MHz, as shown in the inset of Fig. 4. The input light is polarization-adjusted by a half-wave plate and focused by an aspheric lens onto one end face of ChG tapered fiber, which is stably controlled by fixing it with a glass slide. The evanescent coupling is achieved by adjusting the distance between the ${\mathrm{MgF}}_2$ crystalline microresonator and the ChG tapered fiber. Furthermore, the phase matching can be finely tuned by translating the microresonator along the taper of the fiber relative to the waist. The signal output from the tapered fiber is collimated by another aspheric lens and is split by a beam splitter. The reflected light is recorded by a photodetector and an oscilloscope to monitor the cavity resonance. The transmitted light is injected into a Fourier transform infrared spectrometer (Bruker VERTEX 80) for the spectrum analysis.

The QCL module is operated at 4.78 μm as illustrated in Fig. 5(a) (black line), and its tunable wavelength range is approximately 10 nm by adjusting the working temperature. The MIR frequency comb is generated by pumping a cavity resonance of a ${\mathrm{MgF}}_2$ microresonator. From the theoretical simulation, we notice that the pump laser detuning is a critical parameter in Kerr OFC generation. To search the effective pump-resonance detuning, the pump laser is scanned with decreasing optical frequency ${\omega }_p$ over an FSR. A sudden growth of the power floor can be observed on the oscilloscope, and the parametric frequency conversion will occur when the pump frequency is tuned to one of the resonances of the microresonator. Figures 5(a) and (b) show the output spectra in the ${\mathrm{MgF}}_2$ resonators with different sizes via tuning the pump wavelength at a fixed pump power of approximately 150 mW (estimated as $\sim 70\mathrm{mW}$ inside the tapered fiber), which is processed by the min-max normalization method [31]. While continuously increasing the pump-resonance detuning, more and more new frequency components are generated by the cascade FWM [29]. This phenomenon coincides with the transition from the blue-detuned regime to the red-detuned regime of Kerr frequency combs [28]. The broadband MIR optical parametric oscillation with evenly spaced modes is also obtained in the ${\mathrm{MgF}}_2$ resonators with different diameters at a proper pump detuning, as illustrated in Fig. 5(c). The spectral distance of $\sim 1.59\mathrm{THz}$ is acquired in a $\sim 2\mathrm{mm}$ diameter ${\mathrm{MgF}}_2$ resonator, while it is $\sim 0.81\mathrm{THz}$ in the $\sim 9\mathrm{mm}$ diameter ${\mathrm{MgF}}_2$ resonator. This mainly results from the different whispering-gallery modes excited by pumping different radial modes in the microresonator [32]. Furthermore, the FSR and dispersion vary with the structure parameters and will also lead to different mode spacing in microresonators with different sizes [33,34].

Next, we investigate the generation of MIR frequency comb by optimizing pump-resonance detuning and pump power in a ${\mathrm{MgF}}_2$ resonator with a diameter of $\sim 2\mathrm{mm}$. Figure 6(a) shows the parametric oscillation spectrum operated at approximately 200 mW, where one can observe the onset of MIR frequency comb generation. Moreover, when the input pump power is about 240 mW, the spectral components of the comb are obviously broadened and increased based on the cascaded FWM process, as presented in Fig. 6(b). This can be interpreted that the broadband phase matching could be satisfied through tuning the pump power and pump-resonance detuning, giving rise to the increase of the parametric gain bandwidth [35]. Further increasing the pump power to $\sim 280\mathrm{mW}$, the comb lines fill ulteriorly the spectrum, and an ultra-broadband MI state MIR frequency comb over an octave from 3380 nm to 7760 nm is obtained, as elucidated in Fig. 6(c). Indeed, limited by the performance of the QCL and ${\mathrm{MgF}}_2$ microresonator, the generated frequency comb is in the MI states, similar to the spectrum in region (i) of Fig. 2(d). Compared with the MIR OFC reported by Refs. [18,19,36–38], our results demonstrate the potential of the ${\mathrm{MgF}}_2$ crystalline resonator in octave-spanning OFC generation, which makes the $f\text{-}2f$ self-referencing method feasible. Greater bandwidth is also very important for practical applications. It is anticipated that the fully stabilized MIR soliton frequency comb generation with low-phase-noise based on the ${\mathrm{MgF}}_2$ crystalline microresonator and QCL is feasible, triggered by the progress of the fabrication technique for high $Q$ factor crystalline microresonators or high-power laser sources in the future.

In conclusion, we report the broadband MIR OFC generation in a ${\mathrm{MgF}}_2$ crystalline microresonator pumped by a QCL. The ${\mathrm{MgF}}_2$ resonator with a high $Q$ factor is obtained by optimizing the resonator’s geometry parameters. A steep transition from the MI to stable soliton state via the pump frequency tuning approach is observed in numerical simulation. The MI state MIR OFC ranging from 3380 nm to 7760 nm is generated experimentally based on the cascaded FWM in the resonator. This work would then represent an essential building block paving the road for MIR soliton frequency combs generated in microresonators.

Acknowledgment. The authors thank Prof. Haitao Guo of Xi’an Institute of Optics and Precision Mechanics of CAS for the support of mid-infrared fibers, Prof. Fengqi Liu of Institute of Semiconductors of CAS for the support of QCLs, and Zhejiang University and Peking University for the support of tapered fiber fabrication.