1. Introduction

Frequency comb sources in the mid-infrared (mid-IR) wavelength range present themselves as a promising tool for spectroscopy, with several works already showing improved features with respect to state-of-the-art spectrometers [1]. In particular, dual-comb spectrometry attracts currently most interest, as it provides fast and accurate measurements without any moving parts [2]. In sight of these applications, there is also increasing interest in microresonator and (interband) quantum cascade laser (QCL) combs to achieve a miniaturized mid-IR frequency comb source in the fingerprint region, where most molecules exhibit many strong absorption lines. Such technologies may enable, in the long-term, the realization of comb-based spectrometers fully on a chip-scale [3,4].

At the present time, however, the availability of compact and broad mid-IR comb sources is still highly limited despite the progress in these fields. With microresonators, the direct generation of Kerr combs in the mid-IR is typically realized by using a bulk optical parametric oscillator as a pump source [5–9], which is not suitable for a fully integrated device. Using a QCL as a pump source is the immediate choice towards a reduced footprint [10], but demonstrations so far show THz range comb line spacings [11,12], which is not suitable for spectroscopy. Another possibility is converting near-infrared combs into the mid-infrared via difference frequency generation [13,14] or optical parametric oscillation [15]. Regarding QCL combs, mode-locking in the mid-IR has been demonstrated with high optical powers and appear as a promising candidate, but are still limited in bandwidth [16–20].

Here, we make use of cascaded three-wave-mixing in order to generate frequency combs in the mid-infrared region. By pumping a resonator with a $\chi ^{(2)}$ nonlinearity, phase-matched for optical parametric oscillation (OPO) at degeneracy, a series of cascaded three-wave-mixing processes will lead to the generation of combs around the pump frequency and around half the pump frequency [21]. This scheme was first realized in bulk mirror cavities [22–24] and more recently in bulk [25] and chip-integrated [26] microresonators, as well as in waveguide cavities [27]. In this work, we apply this concept with a CdSiP$_2$ mm-sized resonator in order to generate a frequency comb at 3.1 µm by pumping at 1.55 µm wavelength. This way, we generate 30 nm wide frequency combs, with mW pump powers and different repetition rates of 27.7 GHz and multiples. This scheme could potentially be fully integrated, as it only requires a low power telecom continuous wave laser as a pump source and the integration of a single $\chi ^{(2)}$ material. Furthermore, it has been recently predicted by us theoretically [28] that the non-solitonic modelocked frequency combs, i.e., Turing pattern combs, in optical parametric oscillators (OPOs) can be spectrally staggered, see an illustration in the middle column Fig. 1. Here, we provide the experimental confirmation of the existence of such combs. Staggering is a unique feature of the OPO combs related to the fact that the central mode of the half-harmonic field is allowed to have zero power, which is forbidden in the sister case of the $\chi ^{(2)}$ microcombs due to second-harmonic generation [29,30]. The resulting comb pattern has an alternating sideband number between the pump and half-harmonic. We also report here combs with asymmetric staggering, sketched in the right column of Fig. 1, emerging when the pump couples to a resonance with an absolute odd longitudinal mode number instead of even.

 

Fig. 1. The left column illustrates an ordinary comb with a spatial period $2\pi /\nu$, where $\nu$ is an integer, and $\mu$ is the relative mode number. The middle column shows a symmetric staggered comb with a spatial period $2\pi /2\nu$. The right column is an example of an asymmetric staggering corresponding to a spatial period $2\pi /3$. Red and blue colours mark the signal and pump combs, respectively.

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2. Methods

2.1 Numerical model

In order to simulate the resonator intracavity fields for OPO and frequency comb generation, we follow here the model, which detailed derivation from Maxwell equations can be found in [31], and which has been previously studied in, e.g., [25,28,32]. We assume that $\omega _p$ is the pump laser frequency and $\omega _{0p}$ is the frequency of the resonator mode with the number $2M$. $2M$ equals the number of wavelengths fitting along the ring circumference. We express the multimode intra-resonator electric fields of the half-harmonic signal, $\tfrac {1}{2}\omega _p$, and pump, $\omega _p$, fields as

Here, $\vartheta =(0,2\pi ]$ is the angular coordinate and $\mu =0,\pm 1,\pm 2,\dots$ is the relative mode number.

Coupled-mode equations governing the evolution of $a_\mu (t)$, $b_\mu (t)$ are [31]

The frequency matching, i.e. phase matching, parameter for the non-degenerate parametric process initiated by, e.g., the $\mu =0$ mode in the pump field is defined as

Here, $n_m$ is the effective refractive index taken for the frequencies of the modes with the absolute numbers $m=M\pm \mu$ (signal and idler) and $2M$ (pump), $c$ is the vacuum speed of light and $R$ is the resonator radius. Arranging for

Values of the dispersion and linewidth parameters matching our experimental setup are $D_{1p}/2\pi =27.06$ GHz, $D_{1s}/2\pi =27.57$ GHz, $D_{2p}/2\pi =-378$ kHz, $D_{2s}/2\pi =-164$ kHz, $\kappa _p/2\pi =55$ MHz, and $\kappa _s/2\pi =64$ MHz. The nonlinear parameters are $\gamma _p/2\pi =1~\text {GHz}~\text {W}^{-1/2}$, $\gamma _s/2\pi =2~\text {GHz}~\text {W}^{-1/2}$. The values for $\kappa _p$, $\kappa _s$ and $D_{1p}$ were determined experimentally. The remaining parameters, $D_{1s}$, $D_{2p}$ and $D_{2s}$ are estimated by approximating the resonance frequencies as in Ref. [33], including the Sellmeier equation [34] and thermal expansion coefficients [35] for CdSiP$_2$ and assuming the fundamental transverse mode numbers for the pump and signal waves. The nonlinear parameters $\gamma _p$ and $\gamma _s$, defined as in Ref. [31], are proportional to the second-order susceptibility and inversely proportional to the mode area, which we estimate also with the model from Ref. [33]. The transmittance is calculated as the power dissipated over the roundtrip,

2.2 Resonator and experimental setup

We manufactured a CdSiP$_2$ resonator with the procedure described in Ref. [36]. The resonator has a major radius $R = 560$ µm and it is shown in Fig. 2(a). We design it with a small transverse area in order to reduce the number of higher order transverse modes. The raw material was grown and provided by BAE Systems. During the manufacturing process we positioned the optic axis of the crystal parallel to the azimuthal symmetry axis in order to use birefringent phase-matching for the OPO. For 1550 nm wavelength and extraordinary polarization, we obtain an intrinsic quality factor $Q_\text {p}^\text {(i)} = 6 \times 10^6$, which is in agreement with the upper limit calculated from material absorption ($\alpha \approx 0.02~\text {cm}^{-1}$) [37]. This indicates that scattering losses due to surface roughness are negligible with our fabrication method. During the experiments, we work with a loaded quality factor $Q_\text {p}^\text {(l)} = 3.2 \times 10^6$, leading to a coupling coefficient $r_\text {p}=0.88$. The coupling efficiency is then given by $K=4r_\text {p}/(1+r_\text {p})^2$ [38]. We determine a free spectral range (FSR) of 27.0 GHz at 1550 nm for extraordinary polarization by scanning the laser frequency in a wider range and monitoring the transmission.

 

Fig. 2. a) Sketch of the experimental setup. OSA: Optical spectrum analyzer, o.a.: optic axis. b) OPO tuning curve towards degeneracy, where the output wavelengths depend on the relative pump detuning from resonance. The points correspond to the experimentally measured minimum and maximum wavelengths for a fixed temperature, while the curves are calculated. Panel c) shows the pump transmission and d) the excited resonator mode as a function of pump detuning for the enclosed region in panel b). The blue line and red dots correspond to experimental measurements, while the gray lines and dots correspond to the numerical simulation.

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The experimental setup is sketched in Fig. 2. The resonator is placed inside of a holder which is temperature stabilized down to mK. We pump the resonator with a continuous wave 1550 nm laser (Toptica DL Pro) and couple the light with a silicon prism by frustrated total internal reflection. In order to achieve optical parametric oscillation, we rely on birefringent phase-matching, with the pump wave extraordinarily polarized and the signal/idler waves ordinarily polarized. The near-IR and mid-IR beams are separated with a dielectric mirror. Ordinarily polarized generated light in the mid-infrared is detected with an optical spectrum analyzer (Yokogawa AQ6376). In order to study the possible OPO processes supported by different transverse modes, we scan the laser frequency over the FSR of the resonator while monitoring the transmission. In the transmission, we can identify different resonances corresponding to different transverse mode families. Then, we reduce the scan range to in order to sweep the laser frequency over only one resonance that we want to study [36]. The OPO output wavelengths are controlled by tuning the temperature of the resonator in order to achieve operation close to degeneracy, or by varying the pump wavelength for faster coarse tuning [39]. In order to access the frequency comb states, we then reduce the laser frequency manually across the resonance and then keep it fixed for a certain detuning. This way, the thermal broadening of the resonance allows us to passively lock the pump frequency to the cavity for the desired state [40,41], which remains stable for more than an hour. The frequency comb spectra are recorded simultaneously with two optical spectrum analyzers (Yokogawa AQ6370D and AQ6376).

3. Results and discussion

We employ the experimental setup shown in Fig. 2(a) and observe mid-IR light generation ranging from 2.3 $\mathrm {\mu }$m to 5 $\mathrm {\mu }$m depending on the transverse mode selected. We determine a pump threshold of about 100 $\mathrm {\mu }$W of in-coupled power. By initially setting the temperature to $87^{\circ }$C we observe OPO operation close the point of degeneracy around 3112 nm, which agrees with our predicted phase-matching temperature considering fundamental transverse modes for all waves. We then proceed to tune the OPO output further to the point of degeneracy by reducing the temperature, and increasing the laser frequency to keep the pump light close to resonance.

As the laser frequency is scanned across the resonance for a given temperature, the frequency of the output signal/idler waves can vary by hundreds of GHz. This occurs as the signal and idler waves oscillate in different longitudinal modes of the resonator depending on the pump detuning. The tuning curve in Fig. 2(b) displays the measured minimum and maximum signal/idler wavelengths for different temperatures and in-coupled pump powers of 350 $\mathrm {\mu }$W. This behaviour was studied theoretically in Ref. [28] and the curves can be fitted with the equation,

Values for signal and idler wavelengths inside the shaded area Fig. 2(b) can be selected in steps of one FSR by thermally locking the pump laser to the resonance, as shown for the enclosed region at $T=86.2 ^{\circ }$C. For this temperature, Fig. 2(c) shows the pump transmission (blue is experiment and gray is numerical modelling). A series of dips corresponds to the excitation of different consecutive longitudinal modes of the resonator as the laser frequency is scanned across resonance. Figure 2(d) shows a sequence of the measured (red circles) and numerically modelled (small gray dots) resonator mode numbers excited by tuning the pump frequency. The experimentally determined mode numbers are calculated from the measured output wavelengths with $D_{1p}/2\pi$, and the detuning is aligned to the transmission dips. The numerical data reveal the existence of the locking intervals for every signal-idler pair, i.e., $\pm \mu$, and the respective ladder-like sequence of transitions from one interval of $\omega _p$ to the next. The matching of the experimental and numerical data in Fig. 2(d) should be considered as the experimental evidence of the theoretically predicted equivalence between the microresonator OPO tuning and the so-called Eckhaus instabilities, see Ref. [28].

In order to generate the frequency combs, we set the temperature and pump wavelength to operate in between the tips of the tuning curves from Fig. 2(b). Then, as we slowly reduce the pump frequency across the resonance (increasing detuning), we observe a transition from comb states to non-degenerate OPO with the signal and idler frequencies getting further apart. For an in-coupled pump power of 3 mW, we first observe a dense frequency comb around 3.1 µm, shown in Fig. 3(a), with a width of about 30 nm. At the same time, we observe a weak comb around 1.55 µm, shown in Fig. 3(b), approximately 60 dB below the pump power. Both the pump and signal comb lines are separated by one FSR and, therefore, provide an example of the ordinary comb structure, like the one illustrated in the left column of Fig. 1. Our numerical modelling has reproduced such combs, Figs. 3(e) and 3(f), and revealed that they correspond to a variety of complex wave forms which could be either stationary or weakly breathing.

 

Fig. 3. Ordinary and symmetrically staggered frequency combs for the pump coupled to the even mode number. The comb spectra plotted in blue are centered at the pump wavelength, while the spectra in red are centered at the degenerate OPO wavelength. In the experiment, the comb state changes from the one FSR comb line spacing in a,b) to the four FSR spacing in c,d) as the pump frequency is reduced across the resonance. e-i) Numerically simulated spectra for different pump detunings. e,f) and g,h) show the spectra for $- 216$ and $- 144$ MHz detunings, respectively. The background in i) is at −70 dB and $\varepsilon _0=0$. c,d) and g,h) show the staggered combs.

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By further reducing the laser frequency we observed a different type of comb, see Figs. 3(c) and 3(d) and 3(g) and 3(h) for the experimental and numerical data, respectively. Now, the line spacing equals four FSRs for both pump and signal combs, but the signal sidebands are located at $\mu =\dots -6,-2,2,6,\dots$ and the pump ones are at $\mu =\dots,-8,-4,0,4,8,\dots$. To the best of our knowledge, this is the first experimental evidence of the staggered OPO combs after their recent theoretical prediction [28]. We shall note, that in this case, we are pumping in an even absolute mode number of the resonator, which is evident, e.g., from Fig. 3(c) where the spectral center of the practically symmetric comb occurs in the middle between the two lines. Our modelling shows that this state is a phase locked one. It belongs to the general category of the two-colour Turing pattern combs, see [30] and references therein.

Conversion efficiency of the sidebands from the signal comb (red spectra) back to the pump (blue spectra) is hampered by the large group velocity (repetition rate) difference, $(D_{1p}-D_{1s})/2\pi =0.51$GHz, which strongly dominates over dispersion and, therefore, makes all the three wave mixing processes, excluding the $\mu =0$ (pump) to $\mu =\pm 4$ (signal), to be significantly mismatched [28,30]. However, the reason for the lower sideband power observed experimentally compared to the simulations still requires further studies.

The near-IR (blue coloured) and mid-IR (red coloured) combs share the same repetition rate, which is a multiple of $27.7\pm 0.1$ GHz. This value lies closer to the linear FSR at 3.1 µm ($D_{1s}/2\pi =27.57$ GHz), than to the pump FSR ($D_{1p}/2\pi =27.06$ GHz). The selection of the nonlinear repetition rate reflects on the fact that the relative sideband power in the mid-IR comb strongly dominates over the near-IR one. Figure 3(i) shows the full extent of the values of the pump detunings that we have explored numerically. Within this and other sets of data and before the signal-idler regime kicks in, see detunings greater than $230$ MHz Fig. 3(i), we have also observed staggered Turing pattern combs with different line spacings (two, six, etc. FSRs). However, experimentally, we only captured the one with four. One possible reason for this selectivity could be an avoided crossing with another mode family, which preferentially triggers a certain pair of sidebands and is not accounted for in our theory [42].

We have also investigated the case of pumping the resonator mode with an odd number, $2M+1$, by shifting the pump frequency to $\mu =1$. In this case, the degenerate conversion is not possible, as well as the described above staggering of the two combs, since $(2M+1)/2$ is not integer. However, the spectral transformations induced by the pump detuning scan looked similar to the even mode number case, cf. Figure 4 and Fig. 3. One difference was that instead of the perfect staggering and the even order Turing patterns we have now seen the Turing combs with an odd number of FSRs between the lines corresponding to the asymmetric staggering, see Figs. 4(c), (d) and 4(g), (h), and compare the illustrations in the middle and right columns in Fig. 1. The experimental and modelling data in Figs. 4(c) and 4(d) and Figs. 4(g) and 4(h) show examples of the of the combs with the three FSRs between the lines, but the excited modes for the pump (blue) and signal (red) are located at $\mu =\dots,-5,-2,1,4,7\dots$ and $\mu =\dots,-4,-1,2,5,8\dots$, respectively.

 

Fig. 4. Ordinary and asymmetrically staggered frequency combs for the pump coupled to the odd mode number. The comb spectra plotted in blue are centered at the pump wavelength, while the spectra in red are centered at the degenerate OPO wavelength. In the experiment, the comb state changes from the one FSR comb line spacing in a,b) to the three FSR spacing in c,d) as the pump frequency is reduced across the resonance. e-i) Numerically simulated spectra for different pump detunings. e-f) and g-h) show the spectra for the $- 50$ and $- 44$ MHz detunings, respectively. The background in i) is at −70 dB and $\varepsilon _0=-0.75$ GHz. c,d) and g,h) show the asymmetric staggering.

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As further possibilities, the comb center wavelength, defined by the degenerate OPO phase-matching conditions, can be widely selected by changing the resonator size. Our predictions indicate that it is possible, for example, to generate a comb up to 4.3 µm with a resonator size of $R=0.25$ mm. Furthermore, theory and modelling suggest that a variety of the two-colour parametric solitons in microresonators can be expected in future experimental studies [43–45], in addition to the already observed ones [26]. One property of our resonator that could facilitate the soliton formation is that, the point of zero walk-off, i.e., $D_{1p}=D_{1s}$, is achieved at 2.16 $\mathrm {\mu }$m and 4.32 $\mathrm {\mu }$m wavelengths for the pump and half-harmonic respectively. Furthermore, dispersion is also reduced for these wavelengths which is important to increase the comb span, and the zero dispersion point for this resonator is found around 5 $\mathrm {\mu }$m.

4. Conclusion

Overall, this work demonstrates the generation of mid-IR microresonator frequency combs based on cascaded three-wave mixing and the existence of the staggered two-colour frequency combs. The comb lines around the pump and half-harmonic frequencies can be staggered either symmetrically (pump line is located in the middle between the two signal sidebands) or asymmetrically (pump line is shifted away from the mid point between the two signal sidebands), see Figs. 1, 3, 4. The used scheme is compatible with integrated a telecom wavelength lasers and therefore might enable the realization of the setup fully integrated on a chip, with a simple single material architecture. Moreover, control of dispersion and the reduction of the repetition rate mismatch will unlock the full potential of microcomb generation based on the $\chi ^{(2)}$ nonlinearity. Advances in this and other directions will be facilitated by the predictive capabilities of the coupled-mode theory for $\chi ^{(2)}$ microresonators, see, e.g., [31] and references therein.