Heteronuclear soliton molecules in optical microresonators

TL;DR

This paper reports the discovery of heteronuclear dissipative Kerr soliton molecules in optical microresonators, revealing complex bound states of dissimilar solitons with potential applications in metrology and spectroscopy.

Contribution

It introduces heteronuclear soliton molecules in Kerr microresonators, a novel state of bound solitons with dissimilar properties, expanding understanding of soliton interactions in nonlinear systems.

Findings

Heteronuclear soliton molecules are tightly bound despite differences in soliton properties.

Ultrafast sampling reveals short-range binding of dissimilar solitons.

Generated frequency combs have unusual mode structures useful for applications.

Abstract

Optical soliton molecules are bound states of solitons that arise from the balance between attractive and repulsive effects. Having been observed in systems ranging from optical fibers to mode-locked lasers, they provide insights into the fundamental interactions between solitons and the underlying dynamics of the nonlinear systems. Here, we enter the multistability regime of a Kerr microresonator to generate superpositions of distinct soliton states that are pumped at the same optical resonance, and report the discovery of heteronuclear dissipative Kerr soliton molecules. Ultrafast electrooptical sampling reveals the tightly short-range bound nature of such soliton molecules, despite comprising dissipative Kerr solitons of dissimilar amplitudes, durations and carrier frequencies. Besides the significance they hold in resolving soliton dynamics in complex nonlinear systems, such heteronuclear soliton molecules yield coherent frequency combs whose unusual mode structure may find applications in metrology and spectroscopy.

Equations

$$\begin{gathered} \frac{\partial A}{\partial t}+i\sum _{j=2}\frac{D_j}{j!}(\frac{\partial }{i\partial \varphi }{)}^{j}A-ig|A{|}^{2}A= \\ \left(-\frac{\kappa }{2}+i({\omega }_0-{\omega }_p)\right)A+\sqrt{{\kappa }_{\mathrm{ex}}}\cdot s_{\mathrm{in}}(1+\frac{ϵ}{2}{e}^{i\Omega t}) \end{gathered}$$

Full text

††preprint: APS/123-QED

I Introduction

Solitons are one of the most fascinating phenomena in nonlinear dynamics due to their universal spatial or temporal localization of wave forms and the resulting particle-like behavior Akhmediev and Ankiewicz (1997); Stegeman and Segev (1999a). The ubiquity of solitons has been manifested by the numerous observations in hydraulics Yuen and Lake (1975), plasmas Bailung et al. (2011), lasers Ryczkowski et al. (2018) and Bose-Einstein condensates Khaykovich et al. (2002); Strecker et al. (2002), despite the drastic difference in the governing physics. First discovered in optical fibers Kivshar and Agrawal (2003), in which case losses play a negligible role, optical temporal solitons have also been observed in “open systems”, i.e. systems which exhibit dissipation Akhmediev and Ankiewicz (2005). In such dissipative systems solitons result from non-equilibrium driving, i.e. from a double balance of loss and gain, as well as dispersion and nonlinearity and correspond to specific solutions of spatiotemporal self-organization. One specific example is dissipative Kerr solitons (DKS), which can form in a continuous wave (CW) driven Kerr nonlinear cavity Leo et al. (2010); Herr et al. (2014), as mathematically described by the Lugiato-Lefever equation (LLE) Lugiato and Lefever (1987). With the frequency combs (also referred to as microcombs Kippenberg et al. (2018)) they generate, DKS have been successfully applied in spectroscopy, ranging and telecommunication Dutt et al. (2018); Obrzud et al. (2019); Suh et al. (2019); Marin-Palomo et al. (2017); Suh and Vahala (2018); Trocha et al. (2018).

Like their spatial counterparts Stegeman and Segev (1999b); Crasovan et al. (2003); Desyatnikov et al. (2005), temporal solitons can form bound pairs or groups, akin to molecules. Temporal soliton molecules have been observed in conservative systems such as optical fibers Stratmann et al. (2005); Hause et al. (2008); Hause and Mitschke (2010, 2013), and have also been theoretically and experimentally investigated in dissipative systems Malomed (1991); Tang et al. (2001). Moreover, recent advances in dispersive-Fourier-transformation-based imaging techniques have revealed the formation of soliton molecules in a variety of mode-locked lasers Herink et al. (2017); Krupa et al. (2017); Liu et al. (2018); He et al. (2017). Investigation on soliton molecules provides a direct route to study the interactions between solitary waves, and the formation and dissociation of soliton molecules are closely related to subjects such as soliton collision Roy et al. (2005), soliton splashing Yi et al. (2018), soliton rains Chouli and Grelu (2010) and the trapping of solitons Jang et al. (2015). Besides the significance they bring to the fundamental understanding of soliton physics, soliton molecules also present the possibility of transferring optical data surpassing the limitation of binary coding Rohrmann et al. (2012).

To date, binding of DKS has only been possible when dispersive waves interlock multiple identical solitons Wang et al. (2017); Cole et al. (2017); Parra-Rivas et al. (2017), which leads to the formation of “homonuclear” soliton molecules with long-range binding (i. e., the distances between solitons are much larger than the widths of solitons). In this work, for the first time we generate heteronuclear DKS molecules, which are stable bound states comprised of solitons with distinct carrier frequencies, pump detunings, temporal widths and soliton energies. This is achieved by pumping one resonance with a laser that is phase-modulated at a frequency that is only one thousandth of the cavity free spectral range (FSR). This pumping scheme allows us to access a multistability regime Hansson and Wabnitz (2015); Anderson et al. (2017) where multiple dynamical microcomb states coexist. Theoretically, besides the usual dispersive-wave-mediated long-range binding, we predict the unusual binding mechanism that results in the direct interaction between dissimilar solitons in close proximity. Experimentally, we apply a dual-comb sampling technique to measure the inter-soliton separation, revealing that distinct solitons can indeed form stable bound structures in systems with instantaneous nonlinear response, despite the fact that the relative phase between constituent solitons is rotating.

II Discretely pumped Kerr microresonators

In contrast to a conventional monochromatic pumping scheme, here we drive a single resonance with two laser fields, in order to simultaneously generate two distinct soliton states that are each triggered by the bistability of Kerr cavities. Such complex dynamics of multi-valued stationary states, known as multistability, was recently investigated in fiber ring resonators with a single driving laser pumping two resonances Anderson et al. (2017). Naturally, Kerr microresonators appear to be an ideal platform for studying the multistability since they are more robust against environmental perturbations and their strong nonlinearity allows for wider soliton existence range in the frequency domain. However, the small sizes of microresonators leads to very large FSRs that significantly surpass the bistable range, thus forbidding the multistability regime to be entered. Here we demonstrate that such obstacles can be circumvented by driving one resonance with two laser fields whose frequencies differ by less than the soliton existence range. Since there is only one resonance being pumped, the LLE model is adequate to describe the dynamics. Including a second pump in the driving term, we express the discrete pumping scheme as:

[TABLE]

where ${{A}}$ is the envelope of the intracavity field, $\omega_{0}$ and $\omega_{p}$ are the angular frequencies of the pumped resonance and the laser respectively, $D_{j}$ is the $j$th-order dispersion, $\phi$ is the co-rotating angular coordinate that is related to the round-trip fast time coordinate $\tau$ by $\phi=\tau\times D_{1}$ (where $\frac{D_{1}}{2\pi}$ is the FSR), ${\kappa}$ is the cavity decay rate, ${\kappa_{\rm ex}}$ is the external coupling rate and ${|s_{\rm in}|^{2}}=\frac{P_{\rm in}}{\hbar\omega_{0}}$ is the driving photon flux, where $P_{\rm in}$ denotes the power of the main pump. Here $g=\frac{\hbar\omega_{0}^{2}cn_{2}}{n^{2}V_{0}}$ is the single photon induced Kerr frequency shift, where $n$ and $n_{2}$ are the refractive and nonlinear optical indices, $V_{0}$ is the effective mode volume, and $c$ is the speed of light. Moreover, $\epsilon$ is the modulation index for generating the blue-shifted sideband from a phase modulator, e.g. an electrooptic modulator (EOM), and $\frac{\Omega}{2\pi}$ represents the sideband offset frequency, which is set to be positive to reflect the blue-shifted frequency.

Our theoretical analysis shows that the discrete pumping can simultaneously generate two different soliton states, which can be approximated by the superpositions of solitons excited by only the main pump and by only the sideband, respectively (see Supplemental Material SM for details). Here we note that our discrete pumping scheme is fundamentally different from bichromatic pumping methods investigated in previous studies Okawachi et al. (2015); Wang et al. (2016); Ceoldo et al. (2016); Bao et al. (2017). First, while the second pumps in previous works were all offset from the main pump laser by approximately one or multiple FSRs, the offset we apply here is only 12 - 30 MHz, i. e. approximately a thousandth of the FSR (14.09 GHz). Second, the bichromatic pumping scheme was used for modulating the intracavity CW background to manipulate the spatiotemporal characteristics of the otherwise ordinary monochromatically pumped microcombs. In contrast, the modulated laser in this work generates two dissimilar sets of microcombs, each of which would still exist in the absence of the other’s drive.

The simulated formation of heteronuclear soliton molecules is presented in Fig. 1 (b). As the two pumps are swept towards lower frequencies across a resonance, the major pump excites modulational instability (MI) at first, followed by DKS formation. Next, the minor pump scans across the same resonance to generate its microcombs while the major pump is supporting the major DKS. After the minor solitons are formed, due to self-frequency-shifting effects such as high-order dispersion (in this study) and Raman scattering, the major and the minor DKS travel with different group velocities, until the two solitons are close, with a separation where an equilibrium is achieved. Such equilibrium is obtained when the soliton group velocity difference is balanced with the inter-soliton “repulsion” caused by cross phase modulation (XPM) (see SM for details). Intuitively, in Kerr microresonators where the nonlinear response is instantaneous and local, one may expect that discretely driven solitons can not bind because the Kerr-nonlinearity-mediated effect Hause et al. (2008) averages out as the relative phase between two distinct solitons is constantly rotating, given that the solitons have different carrier frequencies. Indeed, a fixed phase difference between similar solitons (and their oscillatory tails) is essential to the formation of multi-soliton long-range bound states Wang et al. (2017); Parra-Rivas et al. (2017). Even in a broader perspective that includes spatial solitons, when the relative phase between driving fields is not fixed (e. g. incoherent solitons and vector solitons), bound states can be formed only when the system’s nonlinearity has a non-instantaneous or nonlocal nature Mitchell et al. (1996, 1998); Rotschild et al. (2008). In our work the XPM effect creates a refractive index barrier to stop solitons of different carrier frequencies (and hence difference group velocities) from colliding. As a result, heteronuclear soliton molecules are formed, with a final group velocity that lies in between the native velocities of the two DKS respectively, according to the conservation of momentum (see SM for details).

The corresponding frequency comb spectra are presented in Fig. 1 (d). Because the major and the minor solitons are of different carrier frequencies, the coarsely resolved comb spectra of heteronuclear molecules acquired by an optical spectrum analyzer (OSA) do not show interference patterns that are typical of monochromatically pumped multi-soliton states Brasch et al. (2016). Rather, the averaged comb spectrum of the heteronuclear DKS molecules is the linear superpositions of the spectra of the major and the minor DKS.

III Experimental generation of heteronuclear soliton molecules

We generate heteronuclear DKS molecules in a magnesium fluoride (MgF2) whispering-gallery-mode resonator (WGMR). The experimental setup is depicted in Fig. 2 (a) (see SM for more details). Fig. 2 (b) shows the generated microcomb power as the laser frequency sweeps across a resonance. By scanning the laser frequency into the DKS molecule regime, we observe the spectrum of the superposed microcomb of major-single-DKS and minor-single-DKS states (Fig. 2 (c)), while Fig. 2 (e) and (f) show the spectra of the monochromatically pumped single-DKS states driven by the major and the minor pumps respectively. Dynamical probing with vector network analyzer (VNA) Guo et al. (2016) measures the radio-frequency (RF) spectrum of the transfer function in Fig. 2 (d). We observe two sets of the typical double-resonance features that are induced by the soliton resonance (“$\mathrm{\mathcal{S}}$-resonance”) and the CW resonance (“$\mathrm{\mathcal{C}}$-resonance”) Guo et al. (2016), showing that the molecule spectrum is indeed the superposition of two monochromatically driven DKS spectra.

We excite a variety of comb patterns of different compositions, which are presented in Fig. 3 with the corresponding simulated spectra, showing remarkable agreement. For all the comb patterns we observe only one repetition rate of the out-coupled soliton trains, indicating that the coexistences of solitons are truly bound states. Again, we note that the superposed comb spectra do not give information on the separations between the major solitons and the minor solitons in a bound state.

IV Structure of soliton molecules

Since the observed comb spectrum indicates that the generated solitons are inevitably impacted by high-order dispersion and mode-crossing-induced dispersive waves, the DKS repetition rate ($f_{\rm{rep}}$) depends on the effective pump-resonance detuning Cherenkov et al. (2017); Yi et al. (2017). Consequently, the major and the minor DKS should have different $f_{\rm{rep}}$ because of their different detunings (major detuning $\delta_{1}=\frac{\omega_{0}-\omega_{p}}{2\pi}$ and minor detuning $\delta_{2}=\delta_{1}-\frac{\Omega}{2\pi}$) when one assumes that there is no interaction between them. Such assumption, however, is false. Because the major and the minor solitons share the same optical mode family (i. e., whispering gallery modes with the same polarization and identical radial and polar numbers), they would either have different $f_{\rm{rep}}$ and show soliton collisions, or have the same $f_{\rm{rep}}$ by forming bound states, i. e., heteronuclear DKS molecules. As predicted by our simulation and supported by the single repetition rate of superposed microcombs, the dissimilar solitons travel with same group velocity, likely with a small inter-soliton separation.

To verify this prediction, we adopt the imaging technique Yi et al. (2018) with an electrooptic comb (EOC) to examine the bound structures. The setup is displayed in Fig. 4 (a). For DKS generation the pump-resonance detuning is stabilized by implementing the Pound-Drever-Hall (PDH) laser locking technique with an EOM as a phase modulator. The major pump frequency is locked to the high-frequency PDH sideband, thus locking the detuning ($\frac{\omega_{0}-\omega_{\rm{p}}}{2\pi}$) to be equal to the PDH modulation frequency of 25 MHz. The carrier frequency of the EOC is different from the soliton pump laser frequency by $\sim 5.5$ GHz. The repetition rate difference ($\Delta f_{\rm{rep}}$) between the EOC and the soliton microcombs is set to be $\sim 30$ MHz. As shown in Fig. 4 (b), (c), (e) and (f), the sampled interferograms show only one repetition period, once again showing that the two constituent microcombs have the same $f_{\rm{rep}}$. However, due to the limited spectral span of the EOC and the chirping of DKS which is introduced by the notch filter, the interferogram duration is typically above 2 ns, which leads to a temporal resolution that is much broader than the inter-soliton separations between dissimilar DKS, thus potentially forbidding us from uncovering the bound structures. Nevertheless, we use fast Fourier transform (FFT) to transform the sampled interferogram streams of the major-single-with-minor-dual DKS microcomb into the RF spectrum (Fig. 4 (h)). Because the major DKS and the minor DKS have different $f_{\rm{ceo}}$, we are able to decompose the RF spectrum into the major-DKS components (Fig. 4 (i)) and the minor-DKS components (Fig. 4 (j)). Then we apply inverse FFT to transform the separated RF components back into temporal interferograms to infer the temporal delay between the two soliton streams with resolution that is much shorter than the pulse width of the EOC (see SM for details).

Fig. 4 (k) shows the separated interferograms of the solitons whose spectrum is presented in Fig. 4 (e). The envelopes of the interferograms are shown in Fig. 4 (l), which also presents the inferred real-time separation between the major and the minor DKS. The same method is also applied to the major-single-with-minor-single soliton streams shown in Fig. 4 (b), and the separated envelopes are plotted in Fig. 4 (m). The derived separations of $\sim 500-800$ fs are on the same scale of the temporal pulse width of the individual solitons, providing evidence of extremely short-range binding of distinct solitons.

V Coexistence of short- and long-range binding mechanisms

To fully understand how complex heteronuclear soliton molecules form, instead of using a single-LLE model, we apply a model of coupled LLEs SM to confirm that the XPM effect is responsible for the binding between dissimilar solitons. In Fig. 4 (d) we present the simulated profile of the molecule that is comprised on a single major soliton and a single minor soliton. By including the third-order dispersion in the single-LLE model, we obtain the sub-ps inter-soliton separation, showing good agreement with the experimental result.

For heteronuclear soliton molecules that are of multiple major or minor solitons, the situation is more complicated as the binding between similar solitons relies on interlocking via dispersive waves. Indeed our simulations reveal that the coexistence of the short-range binding due to the XPM effect and the long-range binding caused by the dispersive-wave-mediated effect is essential for the formation of complex molecules beyond the basic form of single major soliton with single minor soliton. Fig. 5 (a) depicts the concept of such combined binding mechanism. Fig. 5 (b) shows the resonator dispersion used in the qualitative simulation. In order to introduce dispersive wave effect we add large frequency deviation to the resonance with mode number of 60. As can be seen from the inset of Fig. 5 (b), such resonance frequency deviation generally leads to enhancement of the comb power in the mode, e. g. a single-mode dispersive wave. The simulation is started with a major soliton and two minor solitons seeded in the intracavity field. Fig. 5 (c) shows the evolution of the solitons. After the major soliton is bound with one of the minor solitons, the other minor soliton that is well separated from the bound pair also changes its soliton group velocity due to the long-range binding connecting the minor solitons. Consequently the three solitons travel with the same velocity, becoming a complex bound group. To stress the unique role of the dispersive wave, we repeat the simulation without the single mode resonance frequency deviation. As displayed in Fig. 5 (d), the simulation for comparison shows that without the long-range binding, the second minor soliton moves with its original velocity after the first minor soliton changes its velocity due to the binding with the major soliton. After some time, the two minor solitons collide into each other, leading to the annihilation of both.

In Fig. 4 (g) we plot the simulated profile of the complex molecule that is corresponding to the experimental observations shown in Fig. 4 (g) and (f). Again, the inter-soliton separation between the major and minor solitons is in excellent agreement with the measured value.

VI Frequency coherence measurement

Despite the frequency offset imposed by the driving lasers, the binding of the solitons mutually locks the repetition rates, thus potentially giving rise to a high frequency coherence between the major and the minor DKS microcombs. To test the coherence, we use a 1553-nm laser whose frequency is stabilized to an ultrastable Fabry-Perot cavity to measure the frequencies of a pair of major and minor comb teeth that is 20 FSRs ($\sim 2.3$ nm) apart from the pumped resonance. The experimental scheme is illustrated in Fig. 6 (a). The RF spectrum of the beat signals (see Fig. 6 (b)) shows two frequencies that differ by the exact value of the EOM modulation frequency which was set to be 22 MHz in this experiment. We also fully stabilize the microcomb (see SI for details) and then count the two down-mixed beat signals ($f_{1}$ and $f_{2}$) at the same time and the recorded frequencies allow us to confirm unambiguously that the frequency of the minor comb is offset from the frequency of the major comb by $\frac{\Omega}{2\pi}$. The Allan deviations of the two frequencies are displayed in Fig. 6 (c), showing almost identical instabilities. We attribute the imperfect overlap of the Allan deviations to the imperfect synchronization of the counter gating, as well as the fluctuation of the pulse separation in soliton molecules Shi et al. (2018) and the internal motion of soliton molecules Krupa et al. (2017). The in-depth analysis of the fluctuations of inter-soliton separations is beyond the scope of this work. Nevertheless, we emphasize here that the frequency coherence the DKS molecule comb exhibited is already sufficient for a wide range of applications in frequency metrology.

VII Conclusion and Discussion

We use modulated light to enter a novel multistability regime in a Kerr microresonator to generate heteronulcear soliton molecules. The structures of the soliton molecules, as well as the underlying mechanisms that enable the formation of such new DKS bound states are analyzed experimentally and numerically. The mutual frequency coherence of the generated combs is verified with both spectral analysis and frequency counting.

For practical applications, comb-based sensing and metrology may benefit from heteronuclear DKS molecules that provide an additional coherent comb. In particular, with the feature that the major comb and the minor comb are highly coherent despite the fact that they share no frequency components, the heteronuclear solitons can be used with the interlocking of counter-propagating solitons Yang et al. (2017) to generate ultrahigh coherent dual-comb spectrometer without the overlapping of comb teeth (thus no RF spectrum folding). Furthermore, soliton bound states have been used for optical data buffers Leo et al. (2010), and it has been proposed to use soliton molecules in optical telecommunication to break the restriction of binary coding Rohrmann et al. (2012). Naturally one would expect the DKS molecules to be capable of storing and buffering soliton-molecule-based data.

Authors contributions. W.W. conceived the concept and the experimental setup, developed theoretical analysis, and performed numerical simulations. W.W. and R.B. performed the experiments and analyzed the data with assistance from E.L.. E.O. and T.H. constructed the electro-optic comb. W.W. wrote the manuscript, with input from other authors. T.J.K. supervised the project.

Acknowledgments. W.W. thanks Hairun Guo for assistance and discussion on numerical simulation. This publication was supported by Contract No. D18AC00032 (DRINQS) from the Defense Advanced Research Projects Agency (DARPA), Defense Sciences Office (DSO), and funding from the Swiss National Science Foundation under grant agreement No. 163864, No. 165933 and No. 176563 (BRIDGE). W.W. acknowledges support by funding from the European Union’s Horizon 2020 research and innovation programme under Marie Sklodowska-Curie IF Grant Agreement No. 753749 (SOLISYNTH).