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Abstract
Mid-infrared (MIR) microcomb provides a new way into the “molecular fingerprint” region. However, it remains rather a challenge to realize the broadband mode-locked soliton microcomb, which is often limited by the performance of available MIR pump sources and coupling devices. Here, we propose an effective approach towards broadband MIR soliton microcombs generation via a direct pump in the near-infrared (NIR) region, through full utilization of the second- and third-order nonlinearities in a thin-film lithium niobate microresonator. The optical parametric oscillation process contributes to conversion from the pump at 1550 nm to the signal around 3100 nm, and the four-wave mixing effect promotes spectrum expansion and mode-locking process. While the second-harmonic and sum-frequency generation effects facilitate simultaneous emission of the NIR comb teeth. Both the continuous wave and pulse pump sources with relatively low power can support a MIR soliton with a bandwidth over 600 nm and a concomitant NIR microcomb with a bandwidth of 100 nm. This work can provide a promising solution for broadband MIR microcombs by breaking through the limitation of available MIR pump sources, and can deepen the understanding of the physical mechanism of the quadratic soliton assisted by the Kerr effect.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Mid-infrared (MIR) optical frequency comb (OFC) is significant for the spectroscopy of characteristic molecular vibration and biochemical sensing [1–5]. In past decades, several kinds of MIR combs have been implemented via mode-locked lasers [6], difference frequency generators [7], or optical parameter oscillators [8]. However, these OFCs usually have large size, weight and power consumption (SWaP), which hinders their practical applications outside the laboratory. In contrast, the microcombs, optical frequency combs recently developed based on microresonators with the high-quality factor (Q) and small mode volume, have unique advantages of compact size and low threshold power [9–11]. So far, the MIR microcombs have been preliminarily realized in material platforms such as Si [12], Si3N4 [13], MgF2 [14] and CaF2 [15]. However, due to the poor stability and tunability of MIR pump sources and the lack of high-efficiency coupling devices, the reported MIR microcombs still have a huge gap in mode-locking stability and system integration level compared with their counterparts in the near-infrared (NIR) region, making the compact MIR mode-locked soliton microcomb system difficult to implement.
Most of the reported MIR microcombs depend on the third-order (${\chi ^{(3)}}$) nonlinear effect [12–15]. However, the third-order nonlinear coefficients of most materials are relatively low, which leads to high threshold of microcombs and requirement of high-power pumps. Since the second-order (${\chi ^{(2)}}$) nonlinearity is usually stronger than ${\chi ^{(3)}}$ nonlinearity (typically 1-2 orders of magnitude higher), the generation of a microcomb via ${\chi ^{(2)}}$ process has intrinsic advantages in efficiency, which has attracted extensive attentions [16,19,20]. Up to now, microcombs in the visible and NIR regions have been realized in Si3N4 [18], AlN [19] and LiNbO3 [17] microresonators based on the sum-frequency generation (SFG) and second-harmonic generation (SHG). In these studies, the pump generates microcombs through ${\chi ^{(3)}}$ process in NIR region, and the ${\chi ^{(2)}}$ acts as an auxiliary to obtain frequencies in visible region. Conversely, by pumping the visible region, researchers recently obtain visible and NIR microcombs in an AlN microresonator, achieving energy transfer from shorter to longer wavelengths via the ${\chi ^{(2)}}$ process assisted by ${\chi ^{(3)}}$ nonlinearity (namely the quadratic soliton microcomb) [20]. Here, primary sidebands are generated by the quadratic optical parametric oscillation (OPO) process, and the spectrum expansion is realized by cascaded ${\chi ^{(2)}}$ and four-wave mixing (FWM) processes. Similarly, it can be expected to use the NIR lasers with better performance as the pump source to generate MIR soliton microcombs. For microcavity materials, LiNbO3 and Main III-V materials (e.g., AlGaAs and GaP [21]) are promising candidates. However, due to the large spectral span from NIR to MIR regions, it is relatively difficult to achieve phase-matching for second- and third-order nonlinearities simultaneously. Besides, the dynamics by combining ${\chi ^{(2)}}$ and ${\chi ^{(3)}}$ nonlinearities in the quadratic soliton microcomb are complex and not so clear as that in pure Kerr systems.
Generally, whether Kerr- or quadratic-soliton microcombs, the continuous wave (CW) laser is the most commonly used pump source [19,20,22]. Although the CW pump scheme is simple to operate, it can only provide limited controllable parameters (like detuning and power). Since the soliton duration is shorter compared to the cavity round-trip time (hundreds of femtoseconds vs several picoseconds), energy can only be extracted from their overlapping parts, resulting in low pump-to-soliton conversion efficiency (3∼5% for the bright type). Indeed, the pulse pump scheme can achieve higher conversion efficiency [23,24], as well as provide more tunable parameters such as the repetition rate, pulse duration and chirp. Nevertheless, whether the pulse pump can be used to generate MIR microcombs by the quadratic soliton scheme remains to be further investigated.
In this paper, the MIR soliton microcomb generation via a direct NIR pump is investigated. Under the condition of simultaneous phase-matching for second- and third-order nonlinearities, a LiNbO3 microresonator pumped at 1550 nm can simultaneously support the MIR soliton and NIR microcomb centered at 3100 nm and 1550 nm, respectively. The spectral coverage can span over 700 nm (2784-3498 nm) and 100 nm (1495-1601 nm) for the MIR and NIR regions under a low CW pump power of 200 mW, respectively. Meanwhile, the critical role of ${\chi ^{(3)}}$ nonlinearity in spectrum expansion and mode-locking process is explored, and it is found that reducing the group velocity mismatch (GVM) can enhance the microcomb stability. Additionally, the advantage of using the low peak power pulse as the pump is explored. It is proved that the spectral bandwidth and peak power of the MIR soliton can be increased by improving the NIR pulse energy. Our work suggests the potential of the cooperation of ${\chi ^{(2)}}$ and ${\chi ^{(3)}}$ nonlinearities in low power consumption and system-integratable MIR microcombs with more accessible and advanced NIR laser sources, and it can also deepen the understanding of quadratic solitons assisted by the Kerr effect for the realization of broadband mode-locked frequency combs.
2. Theoretical model
Basically, the pure Kerr microcomb generates new frequencies through the cascaded FWM process, as depicted in Fig. 1(a). The detuning is scanned from blue to red, making the spectral evolution process from the primary sidebands and chaotic combs until it reaches the soliton state. The pump laser needs to have the technical performance of narrow linewidth, wide tuning range and high frequency stability, which is accessible in the NIR region but rather a challenge in the MIR region. Based on the cooperation of ${\chi ^{(2)}}$ and ${\chi ^{(3)}}$ nonlinear effects, the problem above can be bypassed by pumping the NIR region to generate MIR microcombs. As shown in Fig. 1(b), with the power reaching the threshold of OPO, the NIR pump frequency $2{\omega _0}$ generates a pair of primary MIR frequencies on both sides of ${\omega _0}$. Meanwhile, new NIR frequencies are generated by the SHG process. Compared with the Kerr microcomb whose primary sidebands are generated by the degenerate FWM, the initial MIR comb teeth are obtained by the NIR pump through the ${\chi ^{(2)}}$ OPO process. This is meaningful for reducing high power fluctuations in the initial stage of microcomb formation caused by the Kerr effect because the threshold power of ${\chi ^{(3)}}$ nonlinearity is higher than that of ${\chi ^{(2)}}$. As indicated in Fig. 1(c), with the accumulation of energy, the MIR frequencies further expand through the OPO process, and the NIR spectrum also widens through the SFG and SHG effects. The anomalous dispersion optimization at the MIR center frequency ${\omega _0}$ can realize efficient phase matching for the FWM process. When the Kerr threshold power is reached, the FWM will promote the expansion of MIR frequencies.
Fig. 1. (a) The pure Kerr microcomb is extended by the pump frequency ω0 through degenerate four-wave mixing (DFWM) and non-degenerate four-wave mixing (ND-FWM). (b)-(c) Formation of the MIR quadratic soliton microcomb assisted by the Kerr effect. (b) The NIR pump produces the MIR primary sidebands by the OPO process, meanwhile the NIR initial frequencies are obtained by the SHG process. The solid and dotted black arrows indicate OPO and SHG processes, respectively. (c) On the basis of SHG and OPO processes, the SFG process is involved to produce more NIR frequencies, and the FWM process will promote the expansion of MIR frequencies when the Kerr threshold power is reached. The dotted red and blue arrows represent the SFG process.
The microcombs generation dynamics described above can be modeled by the coupled modified Lugiato-Lefever equations (LLEs) taking both the ${\chi ^{(2)}}$ and ${\chi ^{(3)}}$ nonlinearities into consideration as [25]
3. Results and discussion
3.1. Simultaneous phase-matching for second- and third-order nonlinearities in a LiNbO3 microresonator
Generating MIR microcombs by combining ${\chi ^{(2)}}$ and ${\chi ^{(3)}}$ nonlinearities can be achieved on non-centrosymmetric material platforms with transparent windows covering the NIR and MIR regions. Among them, LiNbO3 possessing both high second- and third-order nonlinearities (d33= 27 pm/V, n2= 1.8 × 10−19 m2/W) and low absorption loss covering 0.4-5 µm, is an ideal candidate. Therefore, a Z-cut thin-film LiNbO3 microresonator is optimized by using the finite element method (COMSOL Multiphysics), and the modal phase-matching (${\chi ^{(2)}}$) can be satisfied between the 3100 nm TE00 mode and the 1550 nm TE60 mode, as indicated in Fig. 2(a). The mode effective refractive index and second-order dispersion in the MIR and NIR ranges are shown in Figs. 2(b) and (c), respectively. Mode effective refractive indexes at 1550 nm and 3100 nm are equal so that the perfect phase-matching is achieved i.e., Δk = 0. Here, the second-order dispersion around 3100 nm is optimized for efficient FWM phase-matching (${\chi ^{(3)}}$). The microresonator radius, width and height are 400 µm, 5000 nm and 795 nm, respectively, and the other parameters are shown in Table 1. Recently, low-loss fully etched LiNbO3 with a depth of 700 nm has been reported [26]. With the optimization of the etching process, the etching depth of 795 nm is possible. The waveguide loss is reasonably estimated based on current reports [27]. Moreover, due to the difference of group refractive index, the free spectral ranges corresponding to 1550 nm and 3100 nm are 45 GHz and 51 GHz, respectively.
Fig. 2. (a) Mode field of 1550 nm and 3100 nm for the intermodal phase-matching. Effective refractive index (b) and second-order dispersion (c) in the MIR and NIR region.
Table 1. Part of the LLEs-related parameters for the optimized LiNbO3 microresonator.
To search for the optimal waveguide structure, geometric parameters like thickness and width are scanned for phase matching engineering and dispersion engineering. The ideal mode effective index curve of TE00 mode in the MIR and that of the TE60 mode in the corresponding NIR second harmonics should have small slope and cross when the second harmonics frequency equals the pump frequency. For the ideal dispersion curve, both MIR and NIR should show relatively flat and near-zero dispersion distributions. Currently, the dispersion of the NIR region is uneven, because it is difficult to control the dispersion of the MIR and NIR independently, and we mainly focus on the anomalous dispersion of the MIR region to match the FWM process. The device manufacturing only requires a relatively conventional process which can be applied to not only the lithium niobate thin-film platform but also other quadratic photonic materials. Higher pump power can be used to overcome larger phase-mismatch, and the mismatch parameter ξ = ΔkL/2 can reach 0.75π by this method according to Ref. [25]. In addition, the phase-mismatch can be compensated by microresonator tuning techniques like temperature control. Combining with quasi-phase-matching (QPM), a wider potential comb bandwidth might be achieved. However, the QPM scheme is only applicable to ferroelectric crystals, and domain engineering has strict requirements for domain uniformity, adding challenges for the on-chip microcavity that require small domain periods.
3.2. Simultaneous generation of MIR and NIR microcombs via direct NIR CW pump
Figure 3 depicts the detailed generation process of the MIR soliton microcomb and NIR microcomb in the designed LiNbO3 microresonator above by solving Eqs. (1) and (2) with the split-step Fourier method. In the simulation, a 1550 nm CW laser is adopted as the pump source. The results show that the obtained MIR bandwidth is larger than 700 nm (2784-3498 nm) and the NIR coverage exceeds 100 nm (1495-1601 nm) under a low pump power of 200 mW. As shown in Figs. 3(a) and (d), the stable MIR single soliton is generated, and the corresponding spectral profile is illustrated in Fig. 3(b), which has a characteristic sech2-shape. For Figs. 3(d-f), the pump-cavity detuning is tuned linearly from blue to red side in 900 ns and then held constant to stabilize soliton. Figure 3(c) displays the temporal envelope with large amplitude fluctuations and the spectral envelope with multiple spikes in the NIR region caused by the GVM, which is discussed later. As depicted in Figs. 3(d) and (e), unlike the pure Kerr system, the multi-soliton state of the quadratic microcomb is seeded directly, reducing high power loss from chaotic MI to multi-soliton state in the Kerr microcomb formation, which is consistent with the conclusion of the previous work [20]. As can be seen from the intracavity power variation in Fig. 3(f), the overall power fluctuation is small, which has advantages in thermal stability. Besides, distinct from typical Kerr microcombs with an obvious pump spike in the center, here the spectral profile in Fig. 3(b) exhibits a CW-background-free envelope. That is, initial comb lines stemming from the OPO process instead of the FWM process can lead to better flatness. More importantly, the ${\chi ^{(2)}}$ nonlinearity provides a low pump threshold and allows operation in spectral regions where the Kerr comb is difficult to achieve (such as lacking the available pump source).
Fig. 3. Simultaneous generation of MIR and NIR microcombs via a direct NIR CW pump. (a) Temporal and (b) spectral profile of the stable MIR single soliton. (c) Spectral profile for the NIR comb. Inset: the NIR temporal profile. (d) Temporal and (e) spectral evolution of the MIR intracavity field from multiple- to single-soliton. Two dashed lines separate the multi-soliton region from the single-soliton region. (f) The evolution of MIR and NIR intracavity power.
Meanwhile, the critical role of the Kerr effect in the formation of quadratic soliton microcomb cannot be ignored. Actually, the second- and third-order nonlinear cooperation effect in the microcomb generation has more complex dynamics than the pure ${\chi ^{(2)}}$ or ${\chi ^{(3)}}$ scheme. By setting Kerr-related parameters (${\gamma _1}$, ${\gamma _2}$, ${\gamma _{12}}$ and ${\gamma _{21}}$) to zero while keeping other parameters unchanged, results are obtained to illustrate the effect of ${\chi ^{(3)}}$ nonlinearity. As demonstrated in Figs. 4(a) and (c), in the absence of the Kerr effect, a single MIR pulse can be generated from the multi-pulse state. As shown in Fig. 4(b), the corresponding spectral profile is not a typical sech2-shape, showing the features of a narrow bandwidth and irregular profile. In contrast, when the Kerr effect is considered, the smooth MIR spectral envelope is shown by the red dashed curve in Fig. 4(b). The comb bandwidths with and without the Kerr effect are 714 nm and 352 nm, respectively. With the third-order nonlinearity, the bandwidth increases by a factor of 1.03. The spectral results without the Kerr effect are not ideal, and the deficiencies are also reflected in the phase locking. The spectral phase in the two cases above is shown in Fig. 4(d). It can be seen that in the absence of the Kerr effect, irregular and obvious phase fluctuations occur. On the other hand, the spectrum corresponds to a consistent phase (phase-locking state) when the Kerr effect is considered, which indicates ${\chi ^{(3)}}$ nonlinearity plays a key role in phase locking. For pure OPO-soliton, dispersion and detuning disturb the phase locking process. Achieving phase locking requires frequency generation at precise location and intensity distribution conforming to the spectral envelope of the soliton. Especially for the effect of dispersion, frequencies away from the comb center gradually experience phase-mismatch [20]. For these mismatched frequencies, the efficiency of the pure quadratic process is very low and it is difficult to achieve phase locking. Therefore, a suitable dispersion design is required, and the pump frequency and power need to be properly adjusted. When the third-order nonlinear effect is at work, self-phase modulation and cross-phase modulation effects can help compensate the phase-mismatch and promote the formation of stable phase distribution to achieve phase locking. The researches above prove that third-order nonlinearity is indispensable in this scheme, which has important significance in broadening spectrum bandwidth and promoting mode-locking process.
Fig. 4. MIR microcomb without the Kerr effect. (a) Temporal and (b) spectral profile of the stable MIR single pulse. (c) Temporal evolution of MIR intracavity field. (d) Spectral phase. For comparison, when the Kerr effect is considered, the spectral envelope and phase correspond to the red dashed curve in (b) and the red line in (d), respectively.
Due to the presence of the GVM, as shown in Fig. 3(d), the MIR soliton drifts slowly in time domain. In the NIR region, it appears as multi-peak comb teeth in frequency domain and amplitude fluctuation pulse envelope in time domain, as depicted in Fig. 3(c). To confirm the impact of the GVM, we set $\Delta k^{\prime} = 0$ while keeping all the other parameters unchanged. Figures 5(a) and (b) show temporal profiles of the stable MIR single soliton with no drifting and the accompanying NIR single pulse, respectively, when the GVM is eliminated. The spectral profiles corresponding to temporal pulses above are illustrated in Figs. 5(c) and (d). Compared with the irregular envelope in Fig. 3(c), the NIR temporal envelope evolves into a stable single pulse, which corresponds to a smooth spectral profile. In Eq. (2), when $\Delta k^{\prime} \ne 0$, it will change the first-order differentiation of the intracavity field, resulting in a slight variation of the round-trip time and free spectral range, which causes the drift of the pulse. Thus, in the quadratic soliton scheme, the GVM affects the performance in time- and frequency-domain, and can be mitigated by dispersion engineering to further improve the microcomb stability.
Fig. 5. MIR and NIR microcombs when the GVM is eliminated. (a) Temporal and (c) spectral profile of the MIR single soliton. The inset shows the stable transport of the temporal soliton with no drifting. (b) Temporal and (d) spectral profile of the NIR single pulse.
3.3 Quadratic soliton microcomb generation via pulse pump
Except for the CW pump, the pulse pump can also be used as an excitation source in the quadratic soliton scheme. In fact, pulse sources synchronized with the round-trip time have been shown to provide an attractive way to improve the efficiency of pump-to-soliton conversion [23]. Moreover, the pulse pump scheme can provide additional tuning parameters to manipulate soliton states, and can explore interesting fundamental phenomena like spontaneous symmetry breaking [28] and soliton tweezers [29]. Here, the simultaneous generation of MIR soliton and NIR microcomb is obtained by the NIR pulse pump using the synergistic effect of the ${\chi ^{(2)}}$ and ${\chi ^{(3)}}$ nonlinearities. In the simulation, a Gaussian pulse source centered at 1550 nm is used as the excitation. Its peak power and duration are 25 W and 2 ps, respectively, corresponding to a single pulse energy of 29 pJ.
In the pulse pump scheme, the detuning refers to the difference between the central spectral mode of the driving pulse ${\omega _P}$ and the nearest microresonator mode ${\omega _M}$ [30]. By adjusting the detuning from blue to red, a stable MIR soliton microcomb centered at 3100 nm with a spectrum spanning from 2819 to 3443 nm is obtained, with a concomitant NIR microcomb from 1495 to 1595 nm, as shown in Fig. 6(b). Comparing the input Gaussian pulse (blue curve) with the stable MIR soliton (red curve) to demonstrate the advantages of the pulse scheme, and the full width at half maximum is significantly compressed from 1.17 to 0.2 ps, while the peak power remains at a high level, from 25 to 22 W, as depicted in Fig. 6(a). The corresponding soliton formation and spectrum broadening processes are shown in Figs. 6(c) and (d), respectively. Unlike the CW scheme in Fig. 3(d), the pulse scheme exhibits fewer pulse numbers in the multi-soliton state because the initial optical field is already a single pulse, accelerating the capture of the steady single-soliton state. In addition, the temporal drift of the MIR soliton induced by the GVM can be suppressed by the pulse pump. However, the influence of the GVM cannot be completely eliminated, and it still causes a temporal delay between the stable MIR soliton and the pump pulse (Fig. 6(a)). For the inhibition of the soliton drift, it may be that the pulse pump suppresses the impact of GVM effect on the first-order differentiation of the intracavity field. Therefore, the pulse pump can exhibit special advantages in the quadratic soliton scheme and improve microcomb performance.
Fig. 6. MIR quadratic soliton microcomb generation via a NIR pulse pump. (a) The blue and red curves are the temporal profile of the NIR pump pulse and the stable MIR soliton, respectively. FWHM: full width at half maximum. (b) The spectral profile of the stable MIR single soliton. The inset shows the simultaneous NIR spectral profile. The MIR temporal profile (c) and spectrum (d) evolution.
Further studies show that increasing the input pulse duration or peak power i.e., pulse energy can significantly increase the peak power and spectral bandwidth of the MIR soliton. As depicted in Fig. 7(a), with the input NIR pulse duration widening from 1.8 to 2.4 ps, the spectral bandwidth, peak power and corresponding detuning of the MIR soliton raise from 424 to 789 nm, 7 to 39 W, and 0.01 to 0.0165, respectively. Similarly, when the input pulse peak power improves from 20 to 50 W, the spectral bandwidth and peak power of the MIR soliton show the same rising trend, and the corresponding detuning increases from 0.0094 to 0.0209, as seen in Fig. 7(b). It can be inferred that the rises of the NIR pulse duration or peak power require larger detuning to maintain the steady-state MIR soliton. Basically, when a stable soliton is formed, there are relations below [31]:
where $A$ and ${\tau _{width}}$ represent the soliton amplitude and pulse width, respectively. $\delta \omega$ describes cavity-pump detuning. It can be inferred from Eqs. (3) and (4) that both soliton peak power (${A^2}$) and spectral bandwidth (inversely proportional to the soliton pulse width) are positively correlated with the detuning, which is consistent with the numerical results in Fig. 7. It is worth noting that because there exists a maximum allowable detuning when the input peak power is fixed, the increase of input pulse duration is limited. Compared with the CW pump scheme, the pulse pump scheme is more flexible, which is convenient for better control of the soliton dynamic process and improvement of the conversion efficiency.Fig. 7. Influence of the input NIR pulse (a) duration and (b) peak power on the MIR soliton microcomb. The insets show the variation of the corresponding detune value.
4. Conclusion
We numerically demonstrate an effective MIR soliton microcomb generation method and explore its formation dynamics based on the collaboration of ${\chi ^{(2)}}$ and ${\chi ^{(3)}}$ nonlinearities. LiNbO3 and Main III-V materials such as AlGaAs and GaP are promising candidates for this scheme. An exemplary LiNbO3 microresonator is designed for simultaneous quadratic and cubic phase-matching. With a NIR pump power as low as 200 mW, the spectral bandwidth of MIR soliton is more than 700 nm (2784-3498 nm), while the accompanied NIR spectrum spans above 100 nm (1495-1601 nm). It is revealed that the ${\chi ^{(3)}}$ nonlinearity plays an important role in spectrum broadening and mode-locking processes, and the GVM should be reduced to alleviate the impact on the microcomb stability. In addition, the advantages of pulse sources for this scheme are explored. With a low peak power of 25 W, the pulse pump can stably excite a MIR soliton microcomb spanning above 600 nm, and simultaneously generate a NIR microcomb covering 100 nm. It is also verified that improving the NIR pulse energy can increase the spectral bandwidth and peak power of the MIR soliton. These results not only contribute to a deeper understanding of the complex nonlinear processes in microresonators based on quadratic photonic materials, but also provide a feasible method to realize MIR soliton microcombs by better-performance NIR pump sources.
Funding
National Key Research and Development Program of China (2019YFA0308200); CAS Project for Young Scientists in Basic Research (YSBR-069); CAS Project for Special Research Associate (E2291114); Natural Science Basic Research Program of Shaanxi Province (2022JC-43, 2023-JC-QN-0701); West Light Foundation of the Chinese Academy of Sciences (XAB2022YN11); .
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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