1. Introduction

Dissipative Kerr solitons (DKSs) are self-organized optical pulses generated based on the double balance of the dissipation and gain, nonlinearity and dispersion in optical microresonators with the anomalous dispersion [1]. They were first observed in a $MgF_{2}$ whispering-gallery-mode (WGM) microresonator in 2014 [2], and have developed very rapidly in recent years. Because of the advantages of coherence in time domain and low noise in frequency domain, DKSs have shown potential application value in many fields, such as high precision optical ranging [3–5], optical clock [6,7], ultrahigh data rate communication [8], quantum key distribution [9], dual-comb spectroscopy [10–12], and astronomical calibration [13,14]. Especially with the progress of film preparation and fabrication technology, DKSs have been realized in a variety of complementary metal–oxide–semiconductor (CMOS) compatible materials, such as silicon nitride ($Si_{3}N_{4}$), lithium niobate, aluminum nitride ($AlN$), and aluminum gallium arsenide ($AlGaAs$) [15–20]. The integrated DKS generation system has also been demonstrated in recent two years [15,21,22], which indicates that optical frequency comb has begun to advance from experimental demonstration to practical application.

For the microresonator platforms supporting DKSs, multiple modes are usually supported in a microresonator due to the need of dispersion engineering. Consequently, avoided mode crossings (AMXs) caused by the interaction of different mode families are very common, which deform the smooth envelope of soliton spectrum and introduce additional losses [23–25]. Generally, AMX is considered to be detrimental to the formation of DKS [26], but in some cases, the presence of AMX can help the generation of DKS in normal dispersion microresonators and excite the dispersive wave to suppress the variation of repetition rate over the laser frequency drift [27–30]. In particular, the modulated intracavity background field via AMXs has been experimentally demonstrated to order the DKS pulses evenly on the microresonator circumference, in other words, to generate the perfect soliton crystal (PSC) with specific soliton number [31–33]. Compared to the single soliton state, the mode spacing of the PSC is not limited to a single free spectral range (FSR), but can be an integer multiple of the FSR, which makes the mode spacing adjustable for a microresonator with fixed geometry and enhances the conversion efficiency of DKS [34]. Previously, the mode spacing of the PSC is adjusted by delicately controlling the pump condition or adding a control laser whose frequency is an integer multiple of FSR apart from the pump laser frequency [33–35]. One direct way to adjust the mode spacing of the PSC is to change the location of the AMX, but for integrated devices, the location of the AMX is determined after finishing the fabrication processing. The process called post-trimming can be used to finely tune the location of the AMX in no cladding devices [36], but it is not universal and cannot be performed during experiments.

In this letter, we demonstrate the generation of single soliton state in our on-chip silicon nitride microring resonator by auxiliary-laser-assisted thermal response scheme [37–40]. By measuring the integrated dispersion via a fiber Mach-Zehnder interferometer (MZI) [41], we prove that the location and strength of the AMX can be changed with different input laser power, corresponding to the thermal effect in the microresonator. Finally, we demonstrate experimentally that by selecting the specific pumped resonance mode and appropriate input power, the PSC with desired soliton number can be generated. This scheme of thermal tuning the location and strength of the AMX is universal and can be extended to other on-chip platforms. Meanwhile, it is fast and convenient to adjust the mode spacing of the PSC based on this scheme, which is beneficial to the practical application of PSCs.

2. Experimental setup and soliton generation

For frequency comb generation, the $Si_{3}N_{4}$ integrated microring resonator is fabricated from the stoichiometric $Si_{3}N_{4}$ film of $800\,nm$ thickness, which is grown on a silicon substrate wafer with $500\,\mu m$ silicon and $3\,\mu m$ wet oxidation silicon dioxide ($SiO_{2}$), and more fabrication details are shown in Ref. [16]. The cross section of the microresonator is $800\,nm\times 1800\,nm$, which is designed to be anomalous around the pump wavelength. Figure 1 shows a scanning electron microscopy (SEM) picture of the microring with a radius of $100\,\mu m$. When the pump laser is tuned from blue-detuned side to red-detuned side relative to the pumped mode, the intracavity power changes too much to stably access the soliton state [32]. To overcome this challenge, the auxiliary laser is introduced in our experiment to balance the thermal effect and stabilize the soliton state, as shown in Fig. 1. The pump laser is amplified by erbium-doped fiber amplifier (EDFA) and then coupled into the chip through a circulator by the lensed fiber. The auxiliary laser is coupled to the chip by another circulator in opposite direction. The fiber polarization controllers (FPCs) are used to control the polarization of each laser. The coarse wavelength division multiplexer (CWDM) with the bandwidth of $20\,nm$ is for separating the different lasers: one channel is for recording the pump mode transmission, another one is for recording several comb lines power excluded the pump laser. For the $Si_{3}N_{4}$ microring with a radius of $100\,\mu m$, the corresponding FSR is about $231\:GHz$, which means the repetition rate signal is too large to be directly measured. The usual approach is to use the electro-optic modulator to down conversion the repetition rate to several gigahertz [42,43]. Therefore, we adopt the method of cascading the intensity modulator (IM) and the phase modulator (PM) to further expand the sideband spectrum width [44]. The radio frequency (RF) beat note signal is recorded by the electrical spectrum analyzer (ESA).

 

Fig. 1. The experimental setup for the soliton generation in a $Si_{3}N_{4}$ microresonator. EDFA: erbium-doped fiber amplifier. FPC: fiber polarization controller. MZI: Mach-Zehnder interferometer. CWDM: coarse wavelength division multiplexing. PM: phase modulator. IM: intensity modulator. DSO: digital oscilloscope. OSA: optical spectrum analyzer. ESA: electrical spectrum analyzer. And the inset scanning electron microscopy picture shows a $Si_{3}N_{4}$ microring with a radius of $100\mu m$.

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To realize the frequency comb, the input power is strong enough and heats the high-Q microresonator with obvious thermal effect [45,46]. Therefore, we characterize the dispersion of the microresonator by a probe laser through a fiber MZI with a strong pump laser [47]. The probe power is low enough to ignore the thermal effect and is modulated with the IM to avoid the backscattering from the pump laser. In our device, the TM modes have the greater Q factor than the TE modes. Therefore, we measure the transmission of fundamental TM modes from 1520nm to 1630nm, as shown in Fig. 2(a). Through the calibration of the MZI, Fig. 2(b) shows the integrated dispersion $D_{int}=\omega _{\mu }-\omega _{0}-\mu D_{1}$ of the fundamental TM mode versus the relative mode number $\mu$, which exhibits anomalous dispersion of $D_{2}/(2\pi )=2.98\:MHz$. The AMXs at $\mu =-4$ and $\mu =20$ are obviously changed with different launched power from $0\:mW-240\:mW$, indicated by the blue and gray strip, respectively. Figure 2(c) shows the detailed AMX at mode $\mu =-4$. A higher-order mode (indicated by the red arrow) is clearly observed around the fundamental mode. With increasing power, both of the two mode families exhibit red shift. Due to their different drift, the frequency interval between the fundamental mode and the higher-order mode gradually decreases, as depicted in Fig. 2(d). Therefore, the interaction between these two modes is getting stronger, making the change of the location and strength of the AMX more obvious, which is also observed around the mode of $\mu =20$. Especially, there is no obvious AMX around the mode of $\mu =20$ when there is no input power.

 

Fig. 2. (a) The typical tansmission spectrum of the $Si_{3}N_{4}$ microring. (b) The dispersion of the fundamental TM modes with various launched power around optical mode with $\mu =-2$. The red dots and blue lines are the experimental data and the theoretical fitting, respectively. (c) The detailed transmission of the fundamental TM mode ($\mu =-4$) with a high-order mode (indicated by the red arrow). The distance of these two modes is gradually decreasing with different launched power. (d) shows the frequency interval and AMX strength with various launched power.

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Figure 3(a) shows the typical transmission spectrum of the pump laser and the comb lines when the auxiliary laser is coupled into the resonance. The pump transmission is compressed accompanied by the appearance of the soliton steps. By applying such a thermal response control method, we eventually reached single soliton state, as shown in Fig. 3(b), which exhibits a spectral smooth $sech^{2}$ envelope. Here, the fundamental TM pump mode around $1566.98\,nm$ has a linewidth of $116\:\mathrm {MHz}$, as shown in the inset of Fig. 3(a). A loaded $Q$ of $1.65\times 10^{6}$ is estimated from the Lorentzian fitting (red line). The high extinction ratio of the resonance indicates an intrinsic $Q_{0}$ factor of $3.3\times 10^{6}$. Because of the Q factor of the auxiliary mode is one half of the pump mode, the power of pump laser and auxiliary laser are launched around $51\,mW$ and $214\,mW$, respectively [32,48]. The inset of Fig. 3(b) shows the RF beatnote $f_{b}\sim 183\:MHz$ with resolution bandwidth (RBW) of $10\:kHz$. For the method we adopt, the repetition rate can be calculated as $2N\Omega \pm f_{b}$, where $N=7$, $\Omega =16.5\,GHz$ and $f_{b}$ are the order of the generated sidebands, the RF signal frequency and the RF beatnote, respectively. The sign can be determined by slightly changing the modulation frequency. Consequently, the repetition rate is calculated as $231.183\:GHz$, which is agreed well with the size of the microring.

 

Fig. 3. (a) The typical transmission spectrum of the $Si_{3}N_{4}$ microresonator with an auxiliary laser. The blue trace is the transmission of the pump laser, and the orange one is the power of several comb lines. The inset is the transmission of the corresponding TM mode with a low power, which has a linewidth of $116\:\mathrm {MHz}$ according to the Lorentzian fitting (red line), corresponding to a loaded Q of $1.65\times 10^{6}$. (b) Optical spectrum of the single DKS. The pump laser wavelength is around $1567.59\:nm$, and the auxiliary laser wavelength is around $1563.36\:nm$, with the power of $51\,mW$ and $214\,mW$, respectively. The inset shows the RF beatnote, which resolution bandwidth (RBW) is $10\:kHz$.

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From Fig. 2(b), there are two obvious AMXs at $\mu =-4$ and $\mu =20$, which would be utilized for adjusting the soliton number $N$ of the PSCs [31–35]. For the generation of PSCs state, the power of the auxiliary laser and pump laser are strong enough to heat the microring to change the location and strength of the AMXs. To obtain the PSCs with number $N$, the pumped resonance mode should be chosen far away the AMXs with an integral multiple of $N\mathsf {x}FSR$ [31–33]. When the pumped mode is set at $\mu =2$, with appropriate pump power and laser-cavity detuning, the perfect soliton crystal state with $N=3$ is obtained in our experiment. Similarly, we obtained the PSCs with $N=2,4$ at $\mu =0$. We believed that the mode crossing of $\mu =-4$ and $\mu =20$ are contributing to the generation of the PSCs with $N=2,3,4$, together. Based on the mode crossing of $\mu =20$, the corresponding spacing are 3 times of $5\mathsf {x}FSR$ and $7\mathsf {x}FSR$, we obtain the PSC with $N=5,7$ with the pump mode of $\mu =5$ and $\mu =-1$, respectively. However, the PSC with number $N=6$ is achieved when the pumped resonance mode is at $\mu =0$, which has no the frequency spacing of an integral multiple of $6\mathsf {x}FSR$ for mode crossing of $\mu =-4$ or $\mu =20$. We believed that the strong thermal effect would induce the complex background to generate the PSC. Figure 4(a)-(f) are the spectra of the PSCs with different $N$ obtained in our experiment. The PSC number $N$, the index of the pumped mode and the avoided crossing mode are listed in the following Table 1.

 

Fig. 4. Optical spectra of the perfect soliton crystal (PSC) states with different soliton numbers in the $Si_{3}N_{4}$ microring. The pump wavelengths are $1567.598\:nm$ $(\mu =0)$, $1563.811\:nm$ $(\mu =2)$, $1567.609\:nm$ $(\mu =0)$, $1558.208\:nm$ $(\mu =5)$, $1567.670\:nm$ $(\mu =0)$, and $1569.598\:nm$ $(\mu =-1)$, respectively. The auxiliary mode is the fundamental TE mode around $1563.3\:nm$.

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Table 1. The PSC obtained in the experiment with the soliton number $N$, the mode number of the pumped mode and the refereed avoided mode crossing.

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3. Conclusion

In conclusion, by introducing an auxiliary laser to balance the thermal effect, we experimentally demonstrated the stable generation of single soliton state, with the repetition rate of $231.183\:GHz$ for the $Si_{3}N_{4}$ microring resonator. Furthermore, we characterized the integrated dispersion of the fundamental TM mode with varied intracavity power, and observed two apparent AMXs at $\mu =-4$ and $\mu =20$, clearly. Compared to our previous work [32], we proved that the local dispersion can be thermally tuned in the commonest single microresonator system by taking advantage of the frequency interval between fundamental and higher-order modes with different input power. Although, both the mode deviation in the crossing point and the mode interval can be easily controlled independently in the coupled microrings [28], it is convenient for generation of PSC with desired soliton number in the case of fixed pumped mode and in-depth exploration of the mechanism of AMX-induced PSC formation. The thermal effect induced by the strong input power is hard to avoid in such high-Q microresonators. Our results provide a precious and new way to effectively tune the local dispersion in the packaged microresonator. It not only enables formation of PSC with desired soliton number, which is convenient for future applications of soliton microcombs, i.e., a small mode spacing could be used for precision spectroscopy measurement, and a large mode spacing could be used for optical communication, but also shows the potential influences for other studies based on the high-Q microcavity, e.g. nonlinear optics [49].