Optical microresonators with a high quality ($Q$) factor possess the ability to confine light in a small mode volume and obtain a photon lifetime of up to several milliseconds. As light-matter interactions are greatly enhanced within [1], optical microresonators have emerged as an ideal platform for studying nonlinear phenomena [2]. Over the last decade, microresonator-based Kerr frequency combs [3,4] (“microcombs”) have set up a research upsurge in the field of Kerr nonlinear photonics and led to significant advances in several applications including high-capacity telecommunications [5], spectroscopy [6–9], ranging [10–12], low-noise microwave/THz generation [13–17], and optical neural networks [18–20]. Since mode-locked microcombs operated in the regime of temporal dissipative Kerr solitons (DKSs) were observed in crystalline microresonators [21], the research focuses have rapidly transited from modulation-instability microcombs that exist in the effectively blue-detuned regime [22] to the soliton microcombs which exist in the effectively red-detuned regime [23]. It was shown that different forms of DKSs, including traditional single/multiple soliton states and bound states such as soliton crystals (SCs) [24,25] and perfect SCs (PSCs) [26,27], exhibit a rich panel of dynamic instabilities [28,29]. The breathing state [30–34], specifically a kind of localized pattern that undergoes periodic oscillations in both duration and amplitude, is one of the most widely studied dynamic instabilities in microresonators, which can be well described by the Lugiato–Lefever equation (LLE) [35,36]. The intrinsic breathing instabilities under the monochromatic pumping scheme only exist in the detuning region between the modulation instability (MI) and stable DKS state. In the effectively blue-detuned regime, the stable Turing rolls are directly connected with chaotic MI states [29].

In addition to the common monochromatic pumping scheme, bichromatic pumping schemes have been employed as an important complement. This pumping scheme [37] can be implemented either by launching two independent continuous wave (CW) lasers [38] or modulating a single CW laser [39]. The bichromatic pumping scheme was previously adopted to pump two different resonances apart from each other with one or multiple free spectral ranges (FSRs) to generate MI microcombs with a much lower pump power threshold [37,40,41] or tunable comb line spacing (CLS) [42]. The auxiliary pump is commonly used to balance the unwanted thermal effect for reliable DKS generation [38,43]. Recently, it was demonstrated that driving one resonance with two red-detuned monochromatic lasers can lead to the formation of heteronuclear soliton molecules by accessing the multistability regime [44]. When soliton microcombs are operated at the breathing state, the introduction of a secondary pump can realize the stabilizing and tuning of the breathing frequency [34].

In this work, we demonstrate a novel bichromatic pumping scheme where a single resonance is driven by two pump laser fields that are respectively located at the effectively blue- and red-detuned sides. The employment of two pumps provides additional degrees of freedom including the frequency difference and the power difference, which are experimentally tunable. Even though the stronger pump is working at the effectively blue-detuned side, an analogue of a PSC breather [26] is produced due to the existence of the red-detuned pump, which possesses significantly larger bandwidth than that of monochromatically pumped Turing rolls. The microcomb states with CLSs ranging from 10 to 17 times the FSR are observed experimentally. Given that Turing rolls and PSCs are both periodic solutions of the LLE, they are usually bracketed as cnoidal waves in the nonlinear wave community [45–47], and here we refer to the novel breathing states in our work as artificial cnoidal wave breathers (ACWBs). The breathing behavior of ACWBs is compulsive, and the corresponding breathing frequency directly depends on the frequency difference of the two pump lasers. Specially, breathing microcomb states with CLSs ranging from 2 to 9 times of FSR—referred to as molecular crystal-like breathers—are also generated, which are featured with peculiar optical spectra. By verifying with numerical simulations, these novel breathing states exhibit intriguing breathing behaviors which are significantly different from monochromatically pumped DKS breathers. High periodicity breathing and irregular breathing phenomena are also observed experimentally. The demonstration of ACWBs and molecular crystal-like breathers can extend the research of nonlinear wave dynamics, and the CLS reconfigurability may find applications in diverse photonic and optic applications.

In contrast to the common monochromatic pumping scheme [21,22,48] and the previous bichromatic pumping scheme working on different resonances [8,37,40,41], we drive a single resonance by two laser fields with distinct power, as shown in Fig. 1. The laser field with higher power referred to as the primary pump is located at the effectively blue-detuned side while the other with lower power referred to as secondary pump is located at the effectively red-detuned side. The frequency difference is set to be several times larger than the half-width-half-maximum (HWHM) of the pumped resonance. Novel breathing states possessing periodic waveforms can be generated in the optical microresonator under this bichromatic pumping scheme.

Since only one resonance is pumped, the LLE model is sufficient to describe the dynamics of ACWBs by appending a secondary pump in the driving term. The bichromatic pumping LLE can be expressed as $$\frac{\partial A(\varphi ,t)}{\partial t}=(-\frac{\kappa }{2}+i(2\pi \delta ))A+i\sum _{j=2}\frac{D_j}{j!}{\left(\frac{\partial }{i\partial \varphi }\right)}^{j}A-ig{|A|}^{2}A+\sqrt{{\kappa }_{\mathrm{ex}}}(s^{(b)}+s^{(r)}{e}^{i2\pi \mathrm{\Delta }ft}),$$(1)where $A(\varphi ,t)$ represents the envelope of the intracavity field, $\kappa$ is the cavity decay rate, $2\pi \delta ={\omega }_0-{\omega }_p^{(b)}<0$ is the frequency detuning of the effectively blue-detuned primary pump from the pumped resonance, $D_j$ is the $j$th order dispersion ($D_j=0$ for $j>2$), and $\varphi$ is the co-rotating angular coordinate, which is related to the fast time coordinate $\tau$ by $\varphi =2\pi \tau \times \mathrm{FSR}$. $g=\hslash {\omega }_0^{2}c{n}_2/n_0^2{V}_{\mathrm{eff}}$ is the single photon-induced Kerr frequency shift with the refractive index $n_0$, the nonlinear refractive index $n_2$, and the effective mode volume $V_{\mathrm{eff}}$. ${\kappa }_{\mathrm{ex}}$ is the external coupling rate, and ${|s^{(b,r)}|}^2=P_{\mathrm{in}}^{(b,r)}/\hslash {\omega }_0$ is the driving photon flux, where $P_{\mathrm{in}}^{(b,r)}$ respectively denotes the power of the primary and secondary pump. $2\pi \mathrm{\Delta }f={\omega }_p^{(r)}-{\omega }_p^{(b)}$ is the frequency difference between two pump lasers, and $\mathrm{\Delta }f<\delta <0$ means that the secondary pump is effectively red detuned. For broader applicability of our study, we introduce the following normalized parameters for $P_{\mathrm{in}}^{(b,r)}$, $\delta$, and $\mathrm{\Delta }f$, which are, respectively, defined as $$f^{(b,r)}=\sqrt{\frac{8g{\kappa }_{\mathrm{ex}}{P}_{\mathrm{in}}^{(b,r)}}{{\kappa }^3\hslash {\omega }_0}},{\zeta }_0=\frac{4\pi \delta }{\kappa },\mathrm{\Delta }{\zeta }_0=\frac{4\pi \mathrm{\Delta }f}{\kappa }.$$(2)

Figure 2 shows the simulated dynamics of a typical ACWB state (see Visualization 1 for the detailed evolution process), which is based on the realistic parameters of a silicon nitride (${\mathrm{Si}}_3{\mathrm{N}}_4$) resonator used in the following experiment (see Section 4.A). The simulated intracavity power evolution and the temporal envelope evolution of the ACWB are, respectively, displayed in Figs. 2(b) and 2(c). Similar to monochromatic pumping DKS breathers [31,32], ACWBs exhibit a periodic oscillation in their amplitude as well as a periodic compression (CP) and stretching (SP) in their duration. When discovered in the moving coordinate $\varphi$ with the speed of $D_1$, the low-intensity pedestals and the high-intensity pulses exchange their positions in the co-rotating angular coordinate rather than returning to its initial state after a power breathing period [see Figs. 2(a)–2(c)], which is different from intrinsic breathing instabilities. Figure 2(d) displays the corresponding optical spectra sampled at seven moments over a power breathing period. During the process of power decline, the reduced intensity difference between the pedestals and the pulses results in an approximate frequency-doubled temporal waveform [A in Figs. 2(a) and 2(b)] and a specific instantaneous optical spectrum [A in Fig. 2(d)], which shows a complex envelope due to the interference between the ACWB pulses. The averaged spectrum features a typical triangle-shaped envelope of breathers but with some enhanced comb lines [see red arrows in Fig. 3(a)] distinguished from the PSC breathers [26]. We note that although it is hard to directly measure the detailed evolution of the intracavity waveforms as shown in Fig. 2(c) in the experiment, these enhanced comb lines are important characteristics of the ACWB’s averaged optical spectrum acquired by the optical spectrum analyzer. This can be attributed to the special breathing dynamics of the intracavity pulses that exhibit an overall frequency-doubled temporal waveform as shown in Fig. 2(c).

Traditional DKS breathers in microresonators exhibit periodic energy exchange between comb lines around the center and the wings, which is related to the Fermi–Pasta–Ulam recurrence [30]. Different from DKS breathers, ACWB microcombs show almost in-phase oscillation as displayed in Fig. 3(b), indicating that the breathing phenomenon is directly caused by the periodically oscillating pump power under our bichromatic pumping scheme. Thus, this kind of breathing state can be referred to as the artificial breathing state distinguished from the intrinsic breathing state. Another important characteristic of ACWB microcombs is that the enhanced comb lines exhibit much smaller breathing depth, which is defined as $(P_{\max }-P_{\min })/(P_{\max }+P_{\min })$, with $P_{\max (\min )}$ being the maximum (minimum) power of each comb line, resulting from the specific evolution dynamics of the intracavity waveform. For example, as shown in Fig. 3(a), the comb lines with mode index $\mu =\pm 13$ possess almost the same averaged power as comb lines with mode index $\mu =\pm 26$, which makes it convenient to directly compare their breathing depth. The simulated breathing depth of mode $\mu =26$ is around five times smaller than that of mode $\mu =13$ [see Fig. 3(c)].

Figure 4(a) illustrates our experimental setup. A silicon nitride (${\mathrm{Si}}_3{\mathrm{N}}_4$) microring resonator with a cross-section of $1550\mathrm{nm}\times 800\mathrm{nm}$ and a radius of 240 μm is utilized for ACWB generation. Figure 4(b) shows the measured laser-scanned transmission spectrum of the pumped resonance, which exhibits a loaded optical quality ($Q$) factor of 1.2 million. The dispersion characteristic of the microresonator shown in Fig. 3(c) is measured through the Mach–Zehnder interferometer (MZI)-based optical sampling technique [17], and the measured dispersion matches well with finite difference eigenmode (FDE) simulated result, indicating an anomalous dispersion of $D_2/2\pi \sim 1.35\mathrm{MHz}$. In order to achieve bichromatic pumping with flexibly controlled power and frequency difference, a dual-parallel Mach–Zehnder modulator operating in the carrier-suppressed single sideband (CS-SSB) mode is employed to generate two laser fields via simultaneously modulating the continuous wave (CW) laser by two microwave signals from broadband voltage-controlled oscillators (VCOs). The two laser fields are then amplified by an erbium-doped fiber amplifier (EDFA) and then coupled to the optical microresonator via a lensed fiber.

We design a two-step tuning method for reliably accessing the ACWB states as shown in Fig. 5(a). First, the primary and secondary pumps are separated far from each other, and both are located at the effectively blue-detuned side. By manually scanning the CW laser, the two laser fields are simultaneously tuned towards the resonance from the short wavelength, which is referred to as “forward tuning” [stage I in Fig. 5(a)]. The secondary pump then scans into the resonance, exciting the microcombs and red shifting the resonance due to the strong thermal effect [stage II in Fig. 5(a)]. The resonance shifts back after the secondary pump crossing the effectively zero-detuned point and then is thermally locked to the high-power primary pump in the effectively blue-detuned regime [49] thanks to the large frequency difference between the two pumps, accompanied by the emergency of chaotic microcombs [stage III in Fig. 5(a)]. Afterwards, we tune the secondary pump backwards from the red-detuned side of the resonance by changing the voltage applied to the VCO. ACWBs can be generated at an appropriate detune value [stage IV in Fig. 5(a)]. Although the frequency detuning is considered unstable when the pump is working at the red-detuned regime [49], the stronger primary pump at the blue-detuned side stabilizes the resonance and makes it possible to freely tune the secondary pump at the red-detuned side without greatly impacting the resonance. A numerical simulation of this tuning process is seen in Appendix C.2.

Taking the generation of an ACWB state with $\mathrm{CLS}=13\times \mathrm{FSR}$ as an example, we launch a total pump power of 26.5 dBm onto the coupled waveguide of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ microring resonator with a power difference between the primary and secondary pump around 6.5 dB. While the two laser fields with an initial frequency difference approximately 1.5 GHz are simultaneously scanned from blue to red, the secondary pump first couples into the resonance and excites the microcombs [II in Fig. 5(b)]. After the secondary pump crosses the resonance and reaches the red-detuned regime, the resonance shifts backwards due to the thermal instability and is blocked by the primary pump. We further tune the primary pump into the resonance, and relatively noisy microcombs are excited as shown in Fig. 5(b) (subgraph III). Then the secondary pump is tuned backwards while fixing the primary pump until the frequency difference is close to 500 MHz. At this time, the ACWB microcomb is generated after the single-FSR components between the super modes with $\mathrm{CLS}=13\times \mathrm{FSR}$ decay rapidly [see IV in Fig. 5(b)]. The detailed optical spectrum is displayed in Fig. 6(a), which shows a quasi-triangle envelope with enhanced comb lines at mode index $\mu =\pm 26$. In order to further characterize the breathing behavior of the ACWB, the comb lines with mode index $\mu =-13$ and $\mu =-26$ are, respectively, filtered out and then detected by fast photodetectors (PDs). The power evolution traces of two comb lines recorded by the oscilloscope are shown in Fig. 6(c). The breathing depth of the comb mode $\mu =-26$ is much smaller than the comb mode $\mu =-13$, although they have almost the same averaged power. The enhanced comb lines and the difference of the breathing depth agree well with the simulated result in Section 3, which corroborates the intriguing breathing dynamics of the intracavity waveform as shown in Fig. 2(c). Figure 6(b) shows the corresponding RF spectrum of the ACWB microcomb, where the low-noise spectrum background indicates the high coherence of the ACWB state (see Appendix F for more coherence characterization). We note that the broad linewidth of the breathing frequency results from the high phase noise of the broadband VCO controlled by the AWG, and a pure breathing frequency can be expected when using a stable modulation signal source.

Moreover, ACWB states with different CLSs can be generated when changing the total pump power, frequency difference, power difference, and relative pump-resonance frequency detuning of the two laser fields. The ACWB microcombs with CLSs ranging from 10 to 17 times the FSR are observed in the same microresonator by bichromatically pumping the same resonance, as shown in Fig. 7. The transition between these different ACWB states is intermediated by chaotic microcombs. Those ACWB spectra are all characterized by the typical quasi-triangle envelopes with two enhanced comb lines (see Fig. 7), which is consistent with the simulated characteristics. In some cases, two ACWB states can be switched between each other by simply changing the relative pump-resonance frequency detuning without modifying other parameters of the two pump lasers (i.e., only need to tune the wavelength of the CW laser), such as switching from the 12-FSR CLS state to the 13-FSR CLS state (see Fig. 7). Different from the Turing rolls (or primary combs) whose CLSs are determined by the well-defined parametric gain [22,29,46] or perfect soliton crystals (PSCs) whose CLSs are determined by the spectral position of the avoided mode crossings (AMXs) [24,26,27], we have not yet summed up the general rules for generating the ACWB microcomb with a specific CLS. From the perspective of LLE simulations, we find that ACWB states with several different CLSs could be generated under exactly the same pump parameters with different initial states. For example, ACWB states with CLSs ranging from 13 to 17 times the FSR can be generated under the same parameters used in Fig. 2 by only changing the initial state of the simulation. However, in the experiment, the reproducibility of a specific ACWB state is pretty good, which may be attributed to the effects that are not considered in the simulation (e.g., the avoided mode crossing).

We note that the generated ACWB states are very robust, which exist for a long time in our free-running system even with compromised amplitude and frequency noise originating from the broadband VCOs driven by the AWG. This robustness may be attributed to the strong primary pump which works at the effectively blue-detuned regime and thermally locks the resonance. Moreover, the frequency difference of the two pump lasers needs not to be at a fixed value for maintaining an ACWB state and can be tuned for tens of megahertz (MHz) without losing the specific ACWB. If the ACWB state is losing due to the change of the secondary pump, we can recover it by simply restoring the original state of the second pump.

Specially, except for typical ACWB microcombs, novel breathing states with CLSs ranging from 2 to 9 times the FSR are also generated when we continuously scan the two pump lasers in their power-detuning phase plane. Some of the experimentally measured optical spectra are displayed in Fig. 8 (see Appendix D for more optical spectra under different pump parameters).

These novel breathing microcombs exhibit complex spectral envelopes, which indicate that complex periodical waveforms are generated in the microresonator. Figure 9(b) shows the simulated averaged spectrum of the breathing state with $\mathrm{CLS}=6\times \mathrm{FSR}$, which shows remarkable agreement with the experimentally measured result [see Fig. 9(a)]. The corresponding intracavity waveform evolution process is displayed in Fig. 9(c) (see Visualization 2 for detail evolution process), which shows significantly different dynamics with typical ACWB states. There are several pulses oscillating as a group within one period of the intracavity waveform [see dashed region in Fig. 9(d)]. The complex envelope of the optical spectrum exactly results from the interference between these multiple pulses. Here, we name these novel breathing states as “molecular crystal-like breathers” in analogue to “soliton molecules” [44] and “soliton crystals” [24]. It should be noted that not all of the experimentally measured optical spectra displayed in Fig. 8 could be well reproduced in the general LLE simulations as deduced in Eq. (1). We surmise that the wavelength-dependent quality factor, avoided mode crossings, and high-order dispersion will affect the specific intracavity waveforms and the corresponding optical spectra.

Similar to the intrinsic breathing instability of monochromatically pumped DKS breathers [28,32], we also observed high periodicity breathing and irregular breathing phenomena in the ACWB and molecular crystal-like breather states. Some experimental results are displayed in Fig. 10. The period-2 breathing, period-4 breathing, and irregular breathing phenomena are discovered in the microcomb states with 9-FSR, 16-FSR, and 4-FSR CLSs, respectively, which are characterized by the recorded frequency-domain RF spectra and time-domain power evolution of the corresponding microcombs [see Figs. 10(d)–10(f)]. Experimentally, the high periodicity breathing and irregular breathing phenomena tend to appear when the primary pump is tuned further into the chaotic regime (i.e., increasing the power or decreasing the frequency detuning of the primary pump). We find that the period-2 breathing phenomenon is frequently observed in the molecular crystal-like breathers but relatively rarely found in typical ACWB states, which could be attributed to the special evolution dynamics of the intracavity waveform as shown in Fig. 9(d). In most of the ACWB states, tuning the pump parameters sometimes directly led to chaotic states without the appearance of high periodicity breathing or irregular breathing. Numerically, these states can be well reproduced in the bichromatic pumping LLE simulations (see Appendix C.3 for details).

In conclusion, we have demonstrated a novel bichromatic pumping scheme and discovered artificial cnoidal wave breathers and molecular crystal-like breathers in a ${\mathrm{Si}}_3{\mathrm{N}}_4$ microresonator. The unique pumping scheme bridges the gap between the effectively blue- and red-detuned regimes of a single resonance and excites the compulsive breathing behavior of microcombs. Especially, the ACWB states with CLSs ranging from 10 to 17 times the FSR and the molecular crystal-like breathers with CLSs ranging from 2 to 9 times the FSR are produced by pumping the same resonance of a single resonator, which indicates the remarkable flexibility of this pumping scheme for generating microcombs with reconfigurable CLSs. Moreover, high periodicity breathing and irregular breathing phenomena were also observed in our experiment. Numerical simulation results agree well with the experiment results and reveal the intriguing breathing dynamics of the intracavity waveform, which are significantly different from the intrinsic breathing instabilities of monochromatically pumped DKS breathers. On the one hand, the mechanism of these novel breathing phenomena is universal for any kind of Kerr nonlinear resonators pumped bichromatically, which may broaden the research scope of nonlinear wave dynamics both theoretically and experimentally. On the other hand, the discovery of ACWBs and molecular crystal-like breathers will help develop reconfigurable microcombs to meet different demands of applications such as optical communication, spectroscopy, and microwave photonics.

Acknowledgment. J.L. acknowledges support from the National Natural Science Foundation of China, Innovation Program for Quantum Science and Technology, and Shenzhen–Hong Kong Cooperation Zone for Technology and Innovation. The authors acknowledge Shichang Li and Jiaxuan Wang for assisting with the paper preparation.