Abstract

Driven Kerr nonlinear optical resonators can sustain localized structures known as dissipative Kerr cavity solitons, which have recently attracted significant attention as the temporal counterparts of microresonator optical frequency combs. While conventional wisdom asserts that bright cavity solitons can only exist when driving in the region of anomalous dispersion, recent theoretical studies have predicted that higher-order dispersion can fundamentally alter the situation, enabling bright localized structures even under conditions of normal dispersion driving. Here we demonstrate a flexible optical fiber ring resonator platform that offers unprecedented control over dispersion conditions, and we report on the first experimental observations of bright localized structures that are fundamentally enabled by higher-order dispersion. In broad agreement with past theoretical predictions, we find that several distinct bright structures can coexist for the same parameters, and we observe experimental evidence of their collapsed snaking bifurcation structure. Our results also elucidate the physical mechanisms that underpin the bright structures, highlighting the key role of spectral recoil due to dispersive wave emission. In addition to enabling direct experimental verifications of a number of theoretical predictions, we show that the ability to judiciously control the dispersion conditions offers a novel route for ultrashort pulse generation: the bright structures circulating in our resonator correspond to pulses of light as short as 230 fs—the record for a passive all-fiber ring resonator. We envisage that our work will stimulate further fundamental studies on the impact of higher-order dispersion on Kerr cavity dynamics, as well as guide the development of novel ultrashort pulse sources and dispersion-engineered microresonator frequency combs.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

In 1983, McLaughlin, Moloney, and Newell predicted the existence of transverse localized structures in a passive, coherently driven nonlinear optical cavity [1]. The subsequent two decades saw the dynamics and characteristics of such structures—today known as spatial cavity solitons (CSs)—extensively investigated, motivated by fundamental interest and application prospects alike [25]. These studies arguably culminated in the pioneering experiments of Barland et al., who in 2002 showed that a semiconductor microcavity could sustain CSs in the form of transversely self-localized (non-diffracting) beams that could be turned on and off at will [6].

In 1993, Wabnitz used the analogy between spatial diffraction and temporal dispersion to predict the existence of temporal analogs of spatial CSs in passive, coherently driven Kerr nonlinear ring resonators [7]. Such temporal structures correspond to “bright” pulses of light that sit atop a low-intensity continuous wave (cw) background, and that can circulate around the resonator indefinitely, without changing their shape or energy. They attracted scant attention until 2010, when they were first experimentally observed by Leo et al. in a macroscopic fiber ring resonator [8]. In homage to the extensive and pioneering studies performed in the context of spatially diffractive resonators, the authors coined the term temporal CSs to describe the structures. Following the first experimental observation of temporal CSs, numerous studies embraced the unprecedented degree of control afforded by fiber-based systems to explore fundamental CS physics: experimental evidence of oscillatory (Hopf) instabilities [9], emission of dispersive waves [10], control with phase gradients [11,12], coexistence of distinct CS states [13,14], and a number of other phenomena [1517] have been observed over the past decade.

In 2013, Herr et al. reported on the experimental observation of temporal CSs in a monolithic Kerr microresonator—a microscopic analogue of a macroscopic fiber ring resonator [18]. It is in such microresonators that the full application potential of (temporal) CSs has finally been unleashed—more than three decades after studies on localized structures in passive cavities first emerged [1]. Specifically, microresonator CSs (also known as dissipative Kerr solitons) correspond to coherent and (potentially) broadband optical frequency combs [1925], whose utility has now been demonstrated in a host of applications, including telecommunications [26], optical distance measurements [27,28], spectroscopy [29,30], and optical frequency synthesis [31]. Moreover, the unique characteristics of microresonators (e.g., strong thermo-refractive effects [32] and complex mode interactions [33]) have been shown to engender rich new physics, such as the emergence of periodic soliton crystals [34,35], soliton switching [36], and single-mode dispersive waves [37].

It is commonly held that “bright” CSs—intensity peaks atop a low-level background—only manifest themselves under conditions where the resonator is driven in the region of anomalous group-velocity dispersion. In stark contrast, the regime of normal dispersion driving can permit “dark” solitons—intensity dips on a high-level background—which were first observed in a Kerr resonator by Xue et al. in 2015 [38]. Such dark solitons have been explained to arise through a fundamentally different mechanism than their counterparts in the anomalous dispersion regime [39]. Specifically, whereas the latter can be understood as a single cycle of an underlying Turing (or modulation instability, MI) pattern, the former arise via the interlocking of switching waves (or fronts): nonlinear structures that connect the two homogeneous states of the bistable cavity system. This fundamental difference gives dark solitons particular advantages, such as enhanced spectral conversion efficiency [40], which has recently been leveraged for high-order coherent communications [41,42].

While bright CSs are not, under typical conditions, to be expected with normal dispersion driving, studies have shown that higher-order perturbations can fundamentally alter the situation. For example, interactions between different transverse mode families can permit the existence of bright localized structures with flattop profiles (“platicons”) [4346] even in the regime of normal dispersion. Likewise, similar to the formation of bright dissipative solitons in normally dispersive mode-locked lasers with strong intracavity spectral filtering [47,48], frequency-dependent cavity losses have been shown to permit a variety of bright soliton structures in passive Kerr resonators with normal dispersion [49,50]. Very recently, it has also been theoretically predicted that even the simplest of perturbations—third-order dispersion (TOD)—is sufficient to allow bright structures in Kerr cavities driven in the normal dispersion regime [51]. Like dark solitons, such TOD-enabled bright CSs are predicted to arise via interlocking of switching waves, and for given system parameters, several of them can coexist, arranged in a bifurcation structure referred to as collapsed snaking.

To the best of our knowledge, conclusive experimental observations of TOD-enabled CSs and their collapsed snaking have yet to be reported. This is arguably because existing experimental configurations are not suitable for systematic studies of TOD effects in isolation. Microresonator devices are hindered by thermo-refractive effects [32] and mode interactions [33], while fiber ring resonator systems typically operate under conditions where TOD is negligible [8,9,1317]. A handful of studies have used dispersion-managed fiber resonators to enhance the impact of TOD [10,5254], but in this case the dispersion management itself constitutes a significant perturbation that can obfuscate the cavity dynamics [55].

In this paper, we report on a flexible experimental platform that permits systematic studies of Kerr cavity dynamics in the presence of higher-order dispersion. Our system comprises a homogeneous (i.e., non-dispersion-managed) ring resonator made entirely out of dispersion-shifted fiber (DSF) with a zero-dispersion wavelength (ZDW) at 1565.4 nm; by driving the resonator with a widely tunable external cavity diode laser (ECDL), we are able to controllably explore dynamics around the ZDW, and hence control the impact of higher-order dispersion. We observe clear evidence of bright CSs that are fundamentally enabled by TOD, and we identify experimental signatures of their collapsed snaking bifurcation structure. Moreover, the ability to examine CS characteristics across the ZDW has also allowed us to observe—for the first time to the best of our knowledge—the emission of spectrally symmetric dispersive waves as predicted in earlier theoretical works [56]. Our experiments also reveal that the range of existence of the conventional CSs that manifest themselves in the anomalous dispersion regime can extend into the normal dispersion regime [56], and we discuss the relationship between the localized structures appearing in the different dispersion regimes. We highlight in particular the fact that, even though the resonator is driven in the normal dispersion regime, the resulting ultrashort bright structures in fact reside predominantly in the anomalous dispersion regime due to spectral recoil induced by dispersive wave emission [20,56].

Our experiments confirm a string of past theoretical predictions, and they unveil a rich range of novel dynamics in Kerr resonators operating close to the ZDW. From an applied perspective, the prospect of systematically generating localized structures in the normal dispersion regime (and arbitrarily close to the ZDW) could pave the way for novel sources of ultrashort pulses and broadband optical frequency combs. In this context, with durations as short as 230 fs, the bright structures observed in our experiments correspond—to the best of our knowledge—to the shortest CSs directly generated in a passive fiber ring resonator without any extra-cavity dispersion compensation.

2. EXPERIMENTAL SETUP

Figure 1 shows a schematic illustration of our experimental platform. At the heart of it lies a passive fiber ring resonator made of a 26-m-long segment of DSF. The resonator incorporates two couplers made out of the same DSF as the main cavity: a first coupler with 95/5 coupling ratio is used to inject the driving field into the cavity, while a second coupler with 99/1 coupling ratio is used to extract a small portion (1%) of the intracavity field for analysis. The cavity has a free-spectral range (FSR) of 8.4 MHz and a measured finesse of ${\cal F} = 45$, corresponding to 180 kHz resonance linewidth.

 

Fig. 1. Schematic illustration of the experimental setup. A passive fiber ring resonator made out of a 26-m-long segment of dispersion-shifted fiber (DSF, orange shaded region) with a zero-dispersion wavelength of 1565.4 nm (see inset) is driven with nanosecond pulses carved from an external-cavity diode laser (ECDL). The pump repetition rate is actively locked to the intracavity soliton repetition rate by a computer-based measurement and feedback system (green shaded region). The detuning between the ECDL frequency and a linear cavity resonance is actively stabilized using the Pound–Drever–Hall (PDH) technique (magenta shaded region); a digital laser locking module (Toptica Digilock) simultaneously provides the PDH modulation signal, demodulates the photodetector signal measured at the cavity output, and acts as the proportional-integral-derivative controller for the driving laser. AM, amplitude modulator; PPG, pulse-pattern generator; EDFA, erbium-doped fiber amplifier; BPF, bandpass filter; circ., circulator; PC, polarization controller; AOM, acousto-optic modulator; PM, phase modulator; OSA, optical spectrum analyzer; osc., oscilloscope.

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We drive the resonator with flattop nanosecond pulses carved from an ECDL whose wavelength can be continuously tuned from 1510 nm to 1630 nm (Toptica CTL). An erbium-doped fiber amplifier is used to amplify the driving field, thus allowing for the peak power of the nanosecond pulses to be adjusted over a wide range. An external clock signal generated by an RF signal generator is used to carefully adjust the repetition rate of the nanosecond pulse train to match the round trip time of the solitons in the resonator; a computer-automated measurement and feedback scheme (see green shaded area in Fig. 1) continuously monitors and corrects the clock signal so as to maintain the solitons close to the center of the driving pulse. In this context, we must emphasize that the nanosecond duration of our driving pulses is considerably longer than the sub-picosecond durations of the solitons under study. As a consequence, the solitons experience the driving field effectively as cw.

To stabilize the linear phase detuning ${\delta _0} = 2\pi ({f_0} - {f_{\rm P}})/{\rm FSR}$ between the driving laser (with carrier frequency ${f_{\rm P}}$) and a cavity resonance (with frequency ${f_0}$), we launch a low-power cw beam derived from the ECDL into the cavity such that it counterpropagates with respect to the main pump, and use the Pound–Drever–Hall (PDH) technique to lock the laser frequency to the peak of the probe’s Lorentzian resonance (magenta shaded region in Fig. 1). By using an acousto-optic modulator (AOM) to frequency shift the probe beam before its injection into the cavity, we are able to continuously tune the linear detuning experienced by the main pump [55].

Because dispersion plays a central role in our study, we have carefully characterized the dispersive properties of our cavity (see Fig. 1). This was achieved by measuring, for a wide range of pump wavelengths, the frequency shifts of spectral sidebands generated via phase-matched four-wave mixing with the detuning stabilized at zero (see Supplement 1). By fitting a theoretically predicted phase-matching curve on the experimentally measured tuning curve, we find that the cavity has a ZDW of 1565.4 nm and TOD and fourth-order dispersion coefficients of ${\beta _{3,{\rm ZDW}}} = 0.135\;{{\rm ps}^{3}}{\rm /km}$ and ${\beta _{4,{\rm ZDW}}} = - 9 \times {10^{- 4}} {{\rm ps}^{4}}{\rm /km}$ at the ZDW, respectively. In our experiments, we set the pump wavelength close to the ZDW, thus enhancing the impact of TOD (see Appendix A). This should be contrasted with an earlier study by Bessin et al., who used a uniform resonator made out of DSF to study the dynamics of MI in the normal dispersion regime [57]; no signatures of bright CSs were reported since the resonator was driven comparatively far from the ZDW where TOD is negligible.

 

Fig. 2. Observations of different bright structures with normal dispersion driving at 1563 nm. (a)–(d) The left panels show experimentally measured (blue solid curves) and numerically simulated (orange solid curves) optical spectra, while the right panels show corresponding temporal profiles extracted from numerical simulations. Dashed red vertical lines highlight the theoretically predicted dispersive wave wavelengths (see Appendix B), while dashed black vertical lines highlight the ZDW of 1565.4 nm. All results obtained for the same system parameters, including pump peak power (6.7 W) and linear cavity detuning (${\delta _0} = 1.05\;{\rm rad} $).

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Our experimental platform incorporates a number of advances compared to earlier studies on CSs, each crucial to obtaining the results presented in the following sections. First, the use of a homogeneous cavity with no dispersion management ensures that effects observed are representative of pure Kerr cavity physics. Second, the use of a widely tunable ECDL allows us to systematically explore the cavity dynamics in the vicinity of the ZDW, providing control over the sign of the group-velocity dispersion as well as the relative magnitude of higher-order dispersion. Third, in contrast to active stabilization schemes used in prior studies, our PDH-based technique permits the cavity detuning to be linearly and continuously tuned over a wide range and, together with our computer-automated measurement and feedback scheme, allows for the robust study of detuning-dependent soliton dynamics. Our experiments also suggest that the PDH-based scheme is more efficient than the side-of-fringe locking applied in previous studies. In fact, we find that a simple side-of-fringe lock used, e.g.,  in [55] is not sufficient to allow the pump ECDL, with a measured linewidth of 80 kHz, to coherently drive our cavity; in contrast, the PDH scheme allows the detuning to be more robustly stabilized, enabling us to observe and study coherent cavity dynamics over a wide range of parameters.

3. FIRST OBSERVATIONS OF TOD-ENABLED BRIGHT CAVITY SOLITONS

We first present illustrative experimental observations of bright CSs with normal dispersion driving. The results described below were obtained with the pump wavelength and power set to 1563 nm (2.4 nm below the ZDW) and 6.7 W (peak power of the flattop nanosecond pulses), respectively. While we have obtained similar observations for a range of parameter values, we must note that excitation of bright solitons in practice requires the pump wavelength to be comparatively close to the ZDW. This is because the solitons’ range of existence decreases as the relative impact of TOD reduces [51], which occurs as the pump is tuned deeper into the normal dispersion regime (see Appendix A).

Because our fiber cavity exhibits a negative fourth-order dispersion coefficient ${\beta _4}$, the upper state of the bistable cavity response is modulationally unstable even when pumping in the normal dispersion regime [57,58]. As a consequence, it is not possible to sustain dark solitons in our system when operating close to the ZDW. (While beyond the scope of our present study, dark solitons may be observable sufficiently far from the ZDW where the MI gain bandwidth is smaller than the cavity FSR, such that the resonator mode structure can quench the instability [58].) In stark contrast, we find that a negative ${\beta _4}$ does not prevent or even noticeably perturb the bright CSs that are fundamentally enabled by TOD; rather, the MI permitted by fourth-order dispersion in fact offers a convenient route for the solitons to be spontaneously excited simply by scanning the detuning over a cavity resonance. We must emphasize, however, that the soliton existence does not require a negative ${\beta _4}$ (or MI), nor do the solitons correspond to localized elements of any ${\beta _4}$-induced MI pattern; the MI only facilitates the solitons’ spontaneous formation (see Supplement 1).

In our experiments, we lock the detuning close to the point where the system becomes bistable (as identified by the collapse of the MI state to the lower homogeneous state). We then mechanically perturb the cavity, thus causing the detuning stabilization system to scan across a resonance; solitons emerge spontaneously as a result. In agreement with theoretical predictions [51], we find that several soliton states with distinct spectral and temporal profiles can coexist for the same cavity detuning. The solid blue curves in Figs. 2(a)–2(d) show the optical spectra measured for four such states, obtained by repeatedly perturbing the cavity with the detuning locked at $1.05\;{\rm rad} $. The orange curves show corresponding results from numerical simulations of a generalized mean-field model (see Supplement 1 for details regarding the model) with no free-running parameters.

The experimentally measured and numerically simulated spectra in Fig. 2 show very good overall agreement. The only discernible discrepancy is the presence of Gordon–Kelly sidebands in the experimental measurements [55,59,60]; these are not reproduced by a mean-field model of the cavity but arise when considering a more general lumped cavity model (not shown). Our simulations further show that the spectra shown in Fig. 2 correspond to bright structures with complex multi-peak profiles in the time domain. While limited detection bandwidth (12.5 GHz photodetector and 40 GSa/s oscilloscope) prohibits us from experimentally resolving the structures’ sub-picosecond temporal features, photodetector measurements at the 1% cavity output port provide evidence that our observations correspond to bright solitons (see Supplement 1). As such, these results demonstrate the existence of TOD-enabled bright CSs under conditions of normal dispersion driving.

 

Fig. 3. Signatures of collapsed snaking: (a) 704 optical spectra measured at different detunings for a three-peak soliton, concatenated vertically to form a single pseudo-color plot. (b) Experimentally measured bifurcation curve for the different bright structures considered in Fig. 2. (c) Corresponding theoretical bifurcation curve. Numbers next to the curves correspond to the number of peaks the different structures possess in their temporal profile.

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It is interesting to note that, while each of the four spectra shown in Fig. 2 are clearly distinct, some qualitative similarities can also be observed. First, a strong dispersive wave like feature at around 1540 nm is apparent in each case. As highlighted by the red dashed vertical lines, the position of the feature can be accurately predicted by the well-known dispersive wave resonance condition (see [10,56] and Appendix B). As for conventional CSs that exist under conditions of anomalous dispersion driving, this dispersive wave feature manifests itself in the time domain as a strong oscillatory tail, and it can give rise to robust binding between individual solitons [54]. Indeed, while the experimental results presented in Fig. 2 correspond to single isolated structures, our experiments often show signatures of multi-pulse states consisting of distinct structures (that may or may not be bound to each other). A second point of similarity is that each of the spectra display clear modulations that are particularly evident on the long-wavelength side. These modulations are linked to the solitons’ complex, multi-peak temporal profiles, and they offer a convenient fingerprint to identify the different states. In what follows, we will refer to the different states in terms of the number of peaks in their temporal profile ($\textit{N}$), deduced experimentally from the number of modulation minima on the long-wavelength edge of the spectrum ($N - 1$).

The third and final noticeable similarity in the spectra of Fig. 2 is that, even though the pump experiences normal dispersion, considerable portions of the spectral components generated inside the resonator in fact reside in the anomalous dispersion regime. This observation, together with the strong dispersive wave-like feature, hints at the possibility that the bright structures may in fact arise via the balancing of anomalous dispersion and focusing Kerr nonlinearity, enabled here by spectral recoil from the dispersive wave emission. The observations to be reported in Section 5 support this physical interpretation.

4. SIGNATURES OF COLLAPSED SNAKING

Theory predicts that CSs that manifest themselves under conditions of normal dispersion driving are organized in a bifurcation structure known as collapsed snaking (see [51] and Supplement 1). A defining feature of such a structure is that several distinct solitons may coexist for identical system parameters, but the range of parameters over which they exist is different. To systematically test this prediction, we have performed extensive experiments so as to map the range of existence of the different soliton states shown in Fig. 2. Since it can be straightforwardly adjusted in our experiments, we use the cavity detuning as a control parameter. Specifically, we first lock the cavity detuning at an appropriate value, and perturb the cavity until a given soliton is excited. A computer-automated algorithm then adiabatically increases the detuning by changing the frequency applied on the AOM in small discrete increments, recording the soliton spectrum at each step. Once the upper limit of the soliton existence is deduced in this manner, we repeat the procedure but now incrementally reducing the detuning until the lower limit of existence is identified.

The pseudo-color plot in Fig. 3(a) shows an example of the spectra measured for a soliton with three peaks in its temporal profile. Exactly 704 individually recorded spectra are shown over the entire range of the soliton’s existence, visualizing how the soliton spectrum broadens and the dispersive wave features shift away from the pump as the detuning increases. (Due to the spectral recoil effect, the latter phenomenon also causes the soliton to spectrally shift to longer wavelengths.) As the detuning is increased (decreased) beyond the values shown in Fig. 3(a), the intracavity field switches to the cw state (fluctuating MI state), thus causing the soliton to cease to exist.

To plot bifurcation-like diagrams, we integrate over each spectra (excluding the pump) so as to obtain a quantity proportional to the soliton energy that can be plotted as a function of detuning. Figure 3(b) shows the resulting curves for each of the soliton states shown in Fig. 2. As can be seen, the different states (i) coexist for a range of detunings and (ii) exhibit different ranges of existence, with the four- and five-peak states existing over a narrower range than the two- and three-peak states. These observations are consistent with the solitons’ hypothesized collapsed snaking bifurcation structure [51].

To compare the experimentally observed range of soliton existence with theoretical predictions, in Fig. 3(c) we show the energies of the different soliton states as derived from mean-field modeling (excluding the pump as in our experiments; see also Supplement 1). The theoretical results show very good qualitative agreement with our experimental observations, yet a number of quantitative discrepancies can be observed. We note in particular that, in our experiments, the lower boundary of soliton existence is delineated by an abrupt and spontaneous switching of the entire intracavity field to a fluctuating MI state, which is not captured by theory. Furthermore, in contrast to theory, our experiments suggest that the lower boundary of existence of the two-peak soliton is significantly higher than the other solitons. A detailed study is beyond the scope of our present work, but we speculate that these discrepancies stem from experimental imperfections (e.g.,  inhomogeneities on the nanosecond driving pulses) as well as from the inability of the theoretical model to account for the Gordon–Kelly sidebands. Regardless, the experimental results presented in Fig. 3(b) provide clear evidence of the collapsed snaking bifurcation structure of the bright CS structures under conditions of normal dispersion driving.

5. SINGLE-PEAK CAVITY SOLITONS IN THE NORMAL DISPERSION REGIME

The results presented above pertain to bright soliton structures characterized by multiple peaks in their temporal profile (see Fig. 2). Interestingly, it is only such multi-peak structures that were identified in the theoretical study of Parra-Rivas et al. [51], raising the natural question whether it is possible to sustain, in a Kerr resonator driven in the normal dispersion regime, single-peak bright CSs reminiscent of the conventional CSs in the anomalous dispersion regime.

The answer to the question posed above is yes. As a matter of fact, already in 2014, Milián and Skryabin discovered via numerical simulations that higher-order dispersion can extend the range of existence of a conventional CS into the regime of normal dispersion driving [56]. In our experiments, we find that such single-peak, bright CSs can be readily excited when increasing the driving power to about 20 W. However, once excited, it is possible to reduce the driving power level without destroying the soliton (provided that the detuning is simultaneously reduced).

Figure 4(a) shows an experimentally measured spectrum characteristic of a single-peak bright CS, obtained with the driving wavelength in the normal dispersion regime at 1563 nm (2.4 nm below the ZDW). Superimposed with the experimental data is the corresponding numerically simulated spectrum, while Fig. 4(b) shows the numerically simulated temporal profile. As can be seen, the spectral profile is highly reminiscent of a conventional CS generated when driving close to the ZDW in the anomalous dispersion regime [20,61]. The resemblance can be readily understood by noting that, similar to the multi-peak structures shown in Fig. 2, the soliton experiences predominantly anomalous dispersion, being centered at around 1572 nm (7 nm above the ZDW), despite the pump residing in the normal dispersion regime. This is made possible by the emission of a strong short-wavelength dispersive wave at about 1545 nm, which causes the soliton to spectrally recoil towards the long-wavelength anomalous dispersion regime [56,61]. Accordingly, the dispersive wave and the soliton rather remarkably correspond to a single symbiotic structure, where one cannot exist without the other. It should also be clear how the existence of such a structure is fundamentally enabled by TOD, which underpins the phase-matched energy flow between the soliton and the dispersive wave.

 

Fig. 4. Experimental observation of a single-peak CS with normal dispersion driving. (a) Experimentally measured (solid blue curve) and theoretically predicted (solid orange curve) spectra for a driving wavelength and peak power of 1563 nm and 6.7 W, respectively. Black dashed line indicates the zero-dispersion wavelength of 1565.4 nm, while red dashed lines indicate the theoretically predicted dispersive wave positions, with $\Delta\textit{f}$ the corresponding frequency shift from the pump frequency ${f_{\rm p}}$. Inset shows a zoom around the weak dispersive wave at 1578 nm. (b) Simulated temporal profile corresponding to the theoretical spectrum shown in (a). All parameters as in Fig. 3 except for the linear cavity detuning, which was set to ${\delta _0} = 0.8\;{\rm rad} $.

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In addition to the strong short-wavelength dispersive wave peak at about 1545 nm, both the experimentally recorded and numerically simulated spectra show a small but noticeable long-wavelength spectral feature at about 1580 nm [see inset of Fig. 4(a)]. This latter feature corresponds in fact to the symmetrically detuned (with respect to the pump frequency) counterpart of the strong dispersive wave at 1545 nm. It is a known theoretical result ([10,56]; see also Appendix B) that, in cavity systems, dispersive waves will always be generated in pairs, but until now, experimental observations have remained elusive due to the small amplitude of one the constituents of the pair. To the best of our knowledge, the results presented in Fig. 4(a) correspond to the first experimental observation of a dispersive wave pair as predicted by theory.

The experimental results described above corroborate the hypothesis that, by enabling dispersive wave emission and concomitant spectral recoil, TOD can extend the existence of conventional anomalous dispersion CSs into the regime of normal dispersion driving [56]. To gain more insights, we have repeated the experiment at various wavelengths across the ZDW, with Fig. 5(a) showing an assortment of the spectra recorded. As can be seen, the main consequence of tuning the pump wavelength is the shifting of the short-wavelength dispersive wave feature, which is well-predicted by theory (red diamonds). While not readily visible in the plot, we also remark that, as the pump wavelength increases beyond the ZDW, the long-wavelength dispersive wave feature discussed above [see also Fig. 4(a)] shifts to longer wavelengths and becomes essentially unobservable, arguably explaining why this feature has not been observed in previous experiments. (Note that the peaks beyond 1590 nm in Fig. 5 correspond to Gordon–Kelly sidebands.) It is, however, the complete absence of any qualitative change in the soliton spectrum as the driving field tunes across the ZDW that is the main result to be gleaned from Fig. 5. Indeed, these measurements unequivocally demonstrate that the single-peak structures manifesting themselves under conditions of normal and anomalous dispersion driving correspond to the very same CS, whose range of existence spans across the ZDW.

 

Fig. 5. (a) Experimental results, showing how the single-peak soliton spectrum changes as the driving wavelength is tuned across the ZDW. The different curves show spectra measured for driving wavelengths ranging from 1563 nm (top) to 1568 nm (bottom) in 0.5 nm intervals. The dashed black line indicates the ZDW of 1565.4 nm, while the red diamonds indicate the positions of the theoretically predicted (short-wavelength) dispersive waves. All measurements use a driving pulse peak power of 15 W and a linear cavity detuning ${\delta _0} = 1.5\;{\rm rad} $. (b) Blue solid curve shows an experimentally measured intensity autocorrelation (AC) trace for a single-peak soliton when driving in the normal dispersion regime at 1563 nm, while the red dashed curve shows the calculated autocorrelation of a CS predicted by theory.

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Finally, we must note that numerical modeling reveals the solitons whose spectra are shown in Fig. 5(a) to have a temporal duration as short as 220 fs (full width at half-maximum). This value is in very good agreement with the value of 230 fs measured using intensity autocorrelation and assuming a ${{\rm sech}^2}$ pulse shape [see Fig. 5(b)]. [Note that (i) the autocorrelation was measured at the 1% cavity output port after the CSs were amplified using an L-band EDFA and (ii) the solitons sit atop the quasi-cw background of the nanosecond driving pulse, which manifests itself both in the experimental and theoretical autocorrelation traces.) To the best of our knowledge, these results corresponds to the shortest CS durations ever achieved in a uniform (i.e., non-dispersion-managed) passive fiber ring resonator.

6. DISCUSSION AND CONCLUSION

The fact that single-peak CSs can exist under conditions of normal dispersion driving, together with the fact that a considerable proportion of the bright soliton spectra reside in the anomalous dispersion regime, raises interesting questions on the physical origins of the different structures. Specifically, it is fascinating to speculate whether the bright multi-peak structures reported in Fig. 2 (and theoretically studied in [51]) could be understood as bound states of individual single-peak solitons that extend from the anomalous dispersion regime. On the one hand, TOD is well-known to permit robust CS bound states [54], and we find that the single-peak structures are part of the same collapsed snaking bifurcation structure as the multi-peak structures (see Supplement 1). But on the other hand, the characteristically close proximity of the peaks as seen in Fig. 2 speaks against the presence of individual (albeit bound) structures, as does the fact that the multi-peak structures can exist over a wider range of parameters than their single-peak counterparts (see Supplement 1). Further study is required to fully understand where and how the different soliton states (and their physical origins) meet.

The results obtained in our work represent direct experimental observations of bright CSs enabled by TOD, and they provide clear signatures of the solitons’ collapsed snaking bifurcation structure. Moreover, our findings demonstrate that higher-order dispersion can extend the range of existence of conventional single-peak bright CSs into the regime of normal dispersion driving, as well as give rise to symmetric dispersive wave pairs. Taken together, our work confirms a string of theoretical predictions [51,56] that have remained unobserved in direct experiments until now.

Besides the verification of earlier theoretical predictions, another major outcome of our work is the demonstration of a robust experimental platform that allows for the systematic study of novel Kerr cavity dynamics that manifest themselves under the influence of higher-order dispersion. In this context, our experiments have revealed a rich diversity of phenomena that have hitherto not been studied neither in experiment nor theory. These include discontinuities in the soliton existence range at high pump powers, as well as persisting coexistence of incoherent MI and coherent soliton states. Future works will be dedicated for the comprehensive study of such dynamics.

The CSs achieved in our experiments are the shortest ever reported in uniform passive fiber ring resonators, yet we envisage that the soliton durations can be further reduced through judicious optimization of the cavity length and losses. When combined with demonstrated techniques to manipulate CSs in macroscopic fiber ring resonators [11,12], our work could pave the way for a novel source of ultrashort pulses with widely tunable repetition rate that can be easily locked to an external RF reference. Moreover, by showing that bright CSs can exist (and be spontaneously excited) under conditions of normal dispersion driving, our work may provide new degrees of freedom for the design of dispersion-engineered microresonator frequency combs. In fact, while finalizing our manuscript, encouraging signatures of bright CSs in microresonators driven with optical pulses in the normal dispersion regime have been observed [62].

Note: generation of stretched CSs that can be de-chirped down to 210 fs was reported in a dispersion-managed fiber ring resonator after the submission of our manuscript [63].

APPENDIX A: DISPERSION COEFFICIENTS

As mentioned in our main text, we measured the resonator to have a zero-dispersion wavelength of ${\lambda _{{\rm ZDW}}} = 1565.4\;{\rm nm} $ and third- and fourth-order dispersion coefficients of ${\beta _{3,{\rm ZDW}}} = 0.135\;{{\rm ps}^{3}}{\rm /km}$ and ${\beta _{4,{\rm ZDW}}} = - 9 \times {10^{- 4}} {{\rm ps}^{4}}{\rm /km}$ at the ZDW, respectively (see also Supplement 1). These coefficients readily allow us to evaluate the dispersion coefficients at all other wavelengths $\lambda$ using the following relations:

$$\begin{split}{\beta _2}& = {\beta _{3,{\rm ZDW}}}\Omega + \frac{{{\beta _{4,{\rm ZDW}}}}}{2}{\Omega ^2}, \\ {\beta _3}& = {\beta _{3,{\rm ZDW}}} + {\beta _{4,{\rm ZDW}}}\Omega , \\ {\beta _4} &= {\beta _{4,{\rm ZDW}}},\end{split}$$
where $\Omega = 2\pi\! c[{{\lambda ^{- 1}} - \lambda _{{\rm ZDW}}^{- 1}}]$ with $c$ the speed of light in vacuum.

The relative strength of TOD, which underpins the bright solitons studied in our work, can be estimated using the dimensionless quantity [10,51]

$${d_3} = \sqrt {\frac{{2\pi}}{{L{\cal F}}}} \left({\frac{{{\beta _3}}}{{{{(3|{\beta _2}|)}^{3/2}}}}} \right),$$
where $L$ and ${\cal F}$ are the resonator length and finesse, respectively. By allowing us to operate arbitrarily close to the ZDW, and hence control the value of $|{\beta _2}|$, our platform gives access to a wide range of ${d_3}$ coefficients. When driving at 1563 nm (see Figs. 24), we estimate ${d_3} \approx 0.9$; when scanning the pump wavelength across the ZDW to obtain data shown in Fig. 5, we estimate ${d_3}$ to vary from 0.7 to 67.

APPENDIX B: DISPERSIVE WAVE POSITIONS

The spectral positions of dispersive waves emitted by Kerr cavity solitons (CSs) can be found from the roots of the following characteristic equation [10,56]:

$$\begin{split}{- \frac{\pi}{{\cal F}}}&+ {i\frac{{{\beta _3}L}}{{3!}}{Q^3} - iVQ}\\ &{\pm i\sqrt {{{\left({2\gamma\! \textit{LP}_0 - {\delta _0} + \frac{{{\beta _2}L}}{{2!}}{Q^2} + \frac{{{\beta _4}L}}{{4!}}{Q^4}} \right)}^2} - {{(\gamma \textit{LP}_0)}^2}} = 0.}\end{split}$$

Here $V$ represents the group-delay accumulated by the CSs with respect to the driving field over one round trip, ${P_0}$ is the power level of the (quasi) cw background on top of which the solitons sit, ${\delta _0}$ is the phase detuning, and $Q$ is a complex frequency whose real part yields the dispersive wave frequency shift from the pump, $\Delta\! f = {\rm Re}[Q]/(2\pi)$.

The two signs in front of the square root in Eq. (B1) signal that, in a cavity geometry, dispersive waves come in pairs. It is straightforward to show that if ${Q_1}$ is a solution for one of the signs, then the solution for the second sign is given by ${Q_2} = - Q_1^*$. This implies that the corresponding frequency shifts $\Delta\! {f_1} = - \Delta\! {f_2}$, thus demonstrating that the two dispersive waves are symmetrically detuned with respect to the pump.

For parameters pertinent to our study, the second term inside the square root of Eq. (B1) is small. Focusing on the ${+}$ sign in front to of the square root, we may approximate the equation as a polynomial in $Q$ [10],

$$\begin{split}\frac{{{\beta _4}L}}{{4!}}{Q^4} & + \frac{{{\beta _3}L}}{{3!}}{Q^3} + \frac{{{\beta _2}L}}{{2!}}{Q^2} - \textit{VQ} \\& + \left[{(2\gamma\! \textit{LP}_0 - {\delta _0}) + i\frac{\pi}{{\cal F}}} \right] = 0.\end{split}$$

All the theoretically predicted dispersive wave frequency shifts shown in our work were obtained by finding the roots of this polynomial, with the background power level ${P_0}$ obtained from the usual cubic polynomial characteristic of dispersive bistability, and the CS group-delay $V$ extracted from numerical simulations that use experimental parameters.

Funding

Marsden Fund; Rutherford Discovery Fellowship of the Royal Society of New Zealand; James Cook Fellowship of the Royal Society of New Zealand.

Acknowledgment

We acknowledge useful discussions and help from Alexander Nielsen and Ian Hendry.

Disclosures

The authors declare no conflicts of interest.

 

See Supplement 1 for supporting content.

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References

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2020 (1)

X. Dong, Q. Yang, C. Spiess, V. G. Bucklew, and W. H. Renninger, “Stretched-pulse soliton Kerr resonators,” Phys. Rev. Lett. 125, 033902 (2020).
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2019 (5)

N. L. B. Sayson, T. Bi, V. Ng, H. Pham, L. S. Trainor, H. G. L. Schwefel, S. Coen, M. Erkintalo, and S. G. Murdoch, “Octave-spanning tunable parametric oscillation in crystalline Kerr microresonators,” Nat. Photonics 13, 701–706 (2019).
[Crossref]

Ó. B. Helgason, A. Fülöp, J. Schröder, P. A. Andrekson, A. M. Weiner, and V. Torres-Company, “Superchannel engineering of microcombs for optical communications,” J. Opt. Soc. Am. B 36, 2013–2022 (2019).
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A. U. Nielsen, B. Garbin, S. Coen, S. G. Murdoch, and M. Erkintalo, “Coexistence and interactions between nonlinear states with different polarizations in a monochromatically driven passive Kerr resonator,” Phys. Rev. Lett. 123, 013902 (2019).
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X. Xue, X. Zheng, and B. Zhou, “Super-efficient temporal solitons in mutually coupled optical cavities,” Nat. Photonics 13, 616–622 (2019).
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M. Karpov, M. H. P. Pfeiffer, H. Guo, W. Weng, J. Liu, and T. J. Kippenberg, “Dynamics of soliton crystals in optical microresonators,” Nat. Phys. 15, 1071–1077 (2019).
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2018 (10)

A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4, e1701858 (2018).
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D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
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D. C. Cole, J. R. Stone, M. Erkintalo, K. Y. Yang, X. Yi, K. J. Vahala, and S. B. Papp, “Kerr-microresonator solitons from a chirped background,” Optica 5, 1304–1310 (2018).
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E. Lucas, G. Lihachev, R. Bouchand, N. G. Pavlov, A. S. Raja, M. Karpov, M. L. Gorodetsky, and T. J. Kippenberg, “Spatial multiplexing of soliton microcombs,” Nat. Photonics 12, 699–705 (2018).
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J. K. Jang, A. Klenner, X. Ji, Y. Okawachi, M. Lipson, and A. L. Gaeta, “Synchronization of coupled optical microresonators,” Nat. Photonics 12, 688–693 (2018).
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T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. L. Gorodetsky, “Dissipative Kerr solitons in optical microresonators,” Science 361, eaan8083 (2018).
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P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359, 887–891 (2018).
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M.-G. Suh and K. J. Vahala, “Soliton microcomb range measurement,” Science 359, 884–887 (2018).
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A. Fülöp, M. Mazur, A. Lorences-Riesgo, Ó. B. Helgason, P.-H. Wang, Y. Xuan, D. E. Leaird, M. Qi, P. A. Andrekson, A. M. Weiner, and V. Torres-Company, “High-order coherent communications using mode-locked dark-pulse Kerr combs from microresonators,” Nat. Commun. 9, 1598 (2018).
[Crossref]

A. U. Nielsen, B. Garbin, S. Coen, S. G. Murdoch, and M. Erkintalo, “Invited article: emission of intense resonant radiation by dispersion-managed Kerr cavity solitons,” APL Photon. 3, 120804 (2018).
[Crossref]

2017 (12)

F. Copie, M. Conforti, A. Kudlinski, S. Trillo, and A. Mussot, “Dynamics of Turing and Faraday instabilities in a longitudinally modulated fiber-ring cavity,” Opt. Lett. 42, 435–438 (2017).
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Y. Wang, F. Leo, J. Fatome, M. Erkintalo, S. G. Murdoch, and S. Coen, “Universal mechanism for the binding of temporal cavity solitons,” Optica 4, 855–863 (2017).
[Crossref]

P. Parra-Rivas, D. Gomila, and L. Gelens, “Coexistence of stable dark- and bright-soliton Kerr combs in normal-dispersion resonators,” Phys. Rev. A 95, 053863 (2017).
[Crossref]

V. E. Lobanov, A. V. Cherenkov, A. E. Shitikov, I. A. Bilenko, and M. L. Gorodetsky, “Dynamics of platicons due to third-order dispersion,” Eur. Phys. J. D 71, 185 (2017).
[Crossref]

X. Xue, P.-H. Wang, Y. Xuan, M. Qi, and A. M. Weiner, “Microresonator Kerr frequency combs with high conversion efficiency,” Laser Photon. Rev. 11, 1600276 (2017).
[Crossref]

F. Bessin, F. Copie, M. Conforti, A. Kudlinski, and A. Mussot, “Modulation instability in the weak normal dispersion region of passive fiber ring cavities,” Opt. Lett. 42, 3730–3733 (2017).
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E. Obrzud, S. Lecomte, and T. Herr, “Temporal solitons in microresonators driven by optical pulses,” Nat. Photonics 11, 600–607 (2017).
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P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H. P. Pfeiffer, P. Trocha, S. Wolf, V. Brasch, M. H. Anderson, R. Rosenberger, K. Vijayan, W. Freude, T. J. Kippenberg, and C. Koos, “Microresonator-based solitons for massively parallel coherent optical communications,” Nature 546, 274–279 (2017).
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D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” Nat. Photonics 11, 671–676 (2017).
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H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
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X. Yi, Q.-F. Yang, X. Zhang, K. Y. Yang, X. Li, and K. Vahala, “Single-mode dispersive waves and soliton microcomb dynamics,” Nat. Commun. 8, 14869 (2017).
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M. Anderson, Y. Wang, F. Leo, S. Coen, M. Erkintalo, and S. G. Murdoch, “Coexistence of multiple nonlinear states in a tristable passive Kerr resonator,” Phys. Rev. X 7, 031031 (2017).
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2016 (7)

M. Anderson, F. Leo, S. Coen, M. Erkintalo, and S. G. Murdoch, “Observations of spatiotemporal instabilities of temporal cavity solitons,” Optica 3, 1071–1074 (2016).
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J. K. Jang, M. Erkintalo, J. Schröder, B. J. Eggleton, S. G. Murdoch, and S. Coen, “All-optical buffer based on temporal cavity solitons operating at 10  Gb/s,” Opt. Lett. 41, 4526–4529 (2016).
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V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip–based optical frequency comb using soliton Cherenkov radiation,” Science 351, 357–360 (2016).
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M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
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P. Parra-Rivas, D. Gomila, E. Knobloch, S. Coen, and L. Gelens, “Origin and stability of dark pulse Kerr combs in normal dispersion resonators,” Opt. Lett. 41, 2402–2405 (2016).
[Crossref]

J. K. Jang, Y. Okawachi, M. Yu, K. Luke, X. Ji, M. Lipson, and A. L. Gaeta, “Dynamics of mode-coupling-induced microresonator frequency combs in normal dispersion,” Opt. Express 24, 28794–28803 (2016).
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F. Copie, M. Conforti, A. Kudlinski, A. Mussot, and S. Trillo, “Competing Turing and Faraday instabilities in longitudinally modulated passive resonators,” Phys. Rev. Lett. 116, 143901 (2016).
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2015 (5)

S.-W. Huang, H. Zhou, J. Yang, J. F. McMillan, A. Matsko, M. Yu, D.-L. Kwong, L. Maleki, and C. W. Wong, “Mode-locked ultrashort pulse generation from on-chip normal dispersion microresonators,” Phys. Rev. Lett. 114, 053901 (2015).
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V. E. Lobanov, G. Lihachev, T. J. Kippenberg, and M. L. Gorodetsky, “Frequency combs and platicons in optical microresonators with normal GVD,” Opt. Express 23, 7713–7721 (2015).
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V. E. Lobanov, G. Lihachev, and M. L. Gorodetsky, “Generation of platicons and frequency combs in optical microresonators with normal GVD by modulated pump,” Europhys. Lett. 112, 54008 (2015).
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X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9, 594–600 (2015).
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J. K. Jang, M. Erkintalo, S. Coen, and S. G. Murdoch, “Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons,” Nat. Commun. 6, 7370 (2015).
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2014 (4)

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
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J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Observation of dispersive wave emission by temporal cavity solitons,” Opt. Lett. 39, 5503–5506 (2014).
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T. Herr, V. Brasch, J. D. Jost, I. Mirgorodskiy, G. Lihachev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode spectrum and temporal soliton formation in optical microresonators,” Phys. Rev. Lett. 113, 123901 (2014).
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C. Milián and D. Skryabin, “Soliton families and resonant radiation in a micro-ring resonator near zero group-velocity dispersion,” Opt. Express 22, 3732–3739 (2014).
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2012 (1)

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6, 84–92 (2012).
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F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photonics 4, 471–476 (2010).
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P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
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P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6, 84–92 (2012).
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M. Anderson, Y. Wang, F. Leo, S. Coen, M. Erkintalo, and S. G. Murdoch, “Coexistence of multiple nonlinear states in a tristable passive Kerr resonator,” Phys. Rev. X 7, 031031 (2017).
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M. Anderson, F. Leo, S. Coen, M. Erkintalo, and S. G. Murdoch, “Observations of spatiotemporal instabilities of temporal cavity solitons,” Optica 3, 1071–1074 (2016).
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Anderson, M. H.

P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H. P. Pfeiffer, P. Trocha, S. Wolf, V. Brasch, M. H. Anderson, R. Rosenberger, K. Vijayan, W. Freude, T. J. Kippenberg, and C. Koos, “Microresonator-based solitons for massively parallel coherent optical communications,” Nature 546, 274–279 (2017).
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Ó. B. Helgason, A. Fülöp, J. Schröder, P. A. Andrekson, A. M. Weiner, and V. Torres-Company, “Superchannel engineering of microcombs for optical communications,” J. Opt. Soc. Am. B 36, 2013–2022 (2019).
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A. Fülöp, M. Mazur, A. Lorences-Riesgo, Ó. B. Helgason, P.-H. Wang, Y. Xuan, D. E. Leaird, M. Qi, P. A. Andrekson, A. M. Weiner, and V. Torres-Company, “High-order coherent communications using mode-locked dark-pulse Kerr combs from microresonators,” Nat. Commun. 9, 1598 (2018).
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P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
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Bilenko, I. A.

V. E. Lobanov, A. V. Cherenkov, A. E. Shitikov, I. A. Bilenko, and M. L. Gorodetsky, “Dynamics of platicons due to third-order dispersion,” Eur. Phys. J. D 71, 185 (2017).
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D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. B. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
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Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1. Schematic illustration of the experimental setup. A passive fiber ring resonator made out of a 26-m-long segment of dispersion-shifted fiber (DSF, orange shaded region) with a zero-dispersion wavelength of 1565.4 nm (see inset) is driven with nanosecond pulses carved from an external-cavity diode laser (ECDL). The pump repetition rate is actively locked to the intracavity soliton repetition rate by a computer-based measurement and feedback system (green shaded region). The detuning between the ECDL frequency and a linear cavity resonance is actively stabilized using the Pound–Drever–Hall (PDH) technique (magenta shaded region); a digital laser locking module (Toptica Digilock) simultaneously provides the PDH modulation signal, demodulates the photodetector signal measured at the cavity output, and acts as the proportional-integral-derivative controller for the driving laser. AM, amplitude modulator; PPG, pulse-pattern generator; EDFA, erbium-doped fiber amplifier; BPF, bandpass filter; circ., circulator; PC, polarization controller; AOM, acousto-optic modulator; PM, phase modulator; OSA, optical spectrum analyzer; osc., oscilloscope.
Fig. 2.
Fig. 2. Observations of different bright structures with normal dispersion driving at 1563 nm. (a)–(d) The left panels show experimentally measured (blue solid curves) and numerically simulated (orange solid curves) optical spectra, while the right panels show corresponding temporal profiles extracted from numerical simulations. Dashed red vertical lines highlight the theoretically predicted dispersive wave wavelengths (see Appendix B), while dashed black vertical lines highlight the ZDW of 1565.4 nm. All results obtained for the same system parameters, including pump peak power (6.7 W) and linear cavity detuning ( ${\delta _0} = 1.05\;{\rm rad} $ ).
Fig. 3.
Fig. 3. Signatures of collapsed snaking: (a) 704 optical spectra measured at different detunings for a three-peak soliton, concatenated vertically to form a single pseudo-color plot. (b) Experimentally measured bifurcation curve for the different bright structures considered in Fig. 2. (c) Corresponding theoretical bifurcation curve. Numbers next to the curves correspond to the number of peaks the different structures possess in their temporal profile.
Fig. 4.
Fig. 4. Experimental observation of a single-peak CS with normal dispersion driving. (a) Experimentally measured (solid blue curve) and theoretically predicted (solid orange curve) spectra for a driving wavelength and peak power of 1563 nm and 6.7 W, respectively. Black dashed line indicates the zero-dispersion wavelength of 1565.4 nm, while red dashed lines indicate the theoretically predicted dispersive wave positions, with $\Delta\textit{f}$ the corresponding frequency shift from the pump frequency ${f_{\rm p}}$ . Inset shows a zoom around the weak dispersive wave at 1578 nm. (b) Simulated temporal profile corresponding to the theoretical spectrum shown in (a). All parameters as in Fig. 3 except for the linear cavity detuning, which was set to ${\delta _0} = 0.8\;{\rm rad} $ .
Fig. 5.
Fig. 5. (a) Experimental results, showing how the single-peak soliton spectrum changes as the driving wavelength is tuned across the ZDW. The different curves show spectra measured for driving wavelengths ranging from 1563 nm (top) to 1568 nm (bottom) in 0.5 nm intervals. The dashed black line indicates the ZDW of 1565.4 nm, while the red diamonds indicate the positions of the theoretically predicted (short-wavelength) dispersive waves. All measurements use a driving pulse peak power of 15 W and a linear cavity detuning ${\delta _0} = 1.5\;{\rm rad} $ . (b) Blue solid curve shows an experimentally measured intensity autocorrelation (AC) trace for a single-peak soliton when driving in the normal dispersion regime at 1563 nm, while the red dashed curve shows the calculated autocorrelation of a CS predicted by theory.

Equations (4)

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β 2 = β 3 , Z D W Ω + β 4 , Z D W 2 Ω 2 , β 3 = β 3 , Z D W + β 4 , Z D W Ω , β 4 = β 4 , Z D W ,
d 3 = 2 π L F ( β 3 ( 3 | β 2 | ) 3 / 2 ) ,
π F + i β 3 L 3 ! Q 3 i V Q ± i ( 2 γ LP 0 δ 0 + β 2 L 2 ! Q 2 + β 4 L 4 ! Q 4 ) 2 ( γ LP 0 ) 2 = 0.
β 4 L 4 ! Q 4 + β 3 L 3 ! Q 3 + β 2 L 2 ! Q 2 VQ + [ ( 2 γ LP 0 δ 0 ) + i π F ] = 0.

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