Coexistence of Multiple Nonlinear States in a Tristable Passive Kerr Resonator

Passive Kerr cavities driven by coherent laser fields display a rich landscape of nonlinear physics, including bistability, pattern formation, and localized dissipative structures (solitons). Their conceptual simplicity has for several decades offered an unprecedented window into nonlinear cavity dynamics, providing insights into numerous systems and applications ranging from all-optical memory devices to microresonator frequency combs. Yet despite the decades of study, a recent theoretical work has surprisingly alluded to an entirely new and unexplored paradigm in the regime where nonlinearly tilted cavity resonances overlap with one another [T. Hansson and S. Wabnitz, J. Opt. Soc. Am. B 32, 1259 (2015)]. We use synchronously driven fiber ring resonators to experimentally access this regime and observe the rise of new nonlinear dissipative states. Specifically, we observe, for the first time to the best of our knowledge, the stable coexistence of temporal Kerr cavity solitons and extended modulation instability (Turing) patterns, and perform real-time measurements that unveil the dynamics of the ensuing nonlinear structure. When operating in the regime of continuous wave tristability, we further observe the coexistence of two distinct cavity soliton states, one of which can be identified as a “super” cavity soliton, as predicted by Hansson and Wabnitz. Our experimental findings are in excellent agreement with theoretical analyses and numerical simulations of the infinite-dimensional Ikeda map that governs the cavity dynamics. The results from our work reveal that experimental systems can support complex combinations of distinct nonlinear states, and they could have practical implications to future microresonator-based frequency comb sources.

Popular Summary

Patterns abound in nature, ranging from the stripes on a zebra to the ripple formations in windblown sand. Remarkably, the key ingredients that underlie their formation also manifest themselves in many optical systems, giving rise to the analogous formation of patterns in light. The simplest of such systems is a device known as a passive Kerr resonator, formed, for example, by joining together the two ends of a telecommunications optical fiber. While the pattern formation dynamics in passive Kerr resonators has been studied for many decades, recent theoretical work has alluded to an entirely new paradigm in the highly nonlinear regime where adjacent resonator modes overlap. We report on the first experiments that access this regime, demonstrating the emergence of new combinations of patterned and self-localized states.

Our experiments are performed in optical fiber ring resonators that are synchronously driven with nanosecond laser pulses. Thanks to the low resonator losses and the large peak power of the driving pulses, very large nonlinear phase shifts can be realized. Through a combination of comprehensive temporal and spectral measurements, we unequivocally observe new combinations of distinct nonlinear structures associated with adjacent resonator modes. Specifically, we have observed the stable coexistence between patterned and self-localized soliton states—self-reinforcing solitary wave packets—and studied their dynamical interactions in real time. We have also observed coexistence between two distinct dissipative soliton states, thereby confirming earlier theoretical predictions.

By experimentally demonstrating the coexistence between distinct nonlinear states, our work opens up a new and exciting paradigm of nonlinear dynamics and pattern formation in general. Our findings could also have practical implications for the design of future sources of laserlike light based on Kerr microresonators, whose many prospective applications range from spectroscopy to telecommunications.

Figure 1

Cavity resonances and examples of nonlinear structures. (a) Blue curves at the bottom show power levels of cw steady-state solutions, while red curves at the top show CS branches predicted individually for the different resonances by the mean-field Lugiato-Lefever equation (LLE) (see Appendix pp2). For clarity, the CS branches are plotted as $f({\delta }_0)=10\mathrm{W}+P_p({\delta }_0)/2$, where $P_p({\delta }_0)$ is the soliton peak power. (b)–(e) Examples of steady-state nonlinear structures obtained from numerical simulations of the Ikeda map at different detunings [highlighted in (a) as dash-dotted vertical lines]: (b) MI pattern, ${\delta }_0=1.75\pi -2\pi =-0.25\pi$, (c) CS on a cw background, ${\delta }_0=\pi$, (d) CS coexisting with a MI pattern, ${\delta }_0=1.75\pi$, (e) coexistence of two CSs associated with adjacent resonances, ${\delta }_0=3.2\pi$. In (e), the dashed red and blue curves correspond to CS profiles predicted individually for the different resonances by the LLE. The parameters used in all of the calculations are $\theta =0.1$, $P_{\mathrm{in}}=15\mathrm{W}$, $\rho =0.73$, ${\beta }_2=-22{\mathrm{ps}}^2/\mathrm{km}$, $\gamma =1.2{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$, $L=100\mathrm{m}$. The mean-field results use $\alpha =0.145$, corresponding approximately to half the total cavity losses per round-trip (see Appendix pp2). Note the different axes in (b)–(d) and (e), highlighting the much larger power and shorter duration of the “super” CS.

Figure 2

Experimental evidence of coexisting CSs and MI patterns. (a) Experimentally measured spectra at the cavity output before (blue curve) and after (red curve) the excitation of a CS, with the cavity detuning set below the MI threshold of the second resonance (${\delta }_1\approx 1.71\pi$; ${\delta }_2\approx -0.29\pi$). (b) Corresponding numerical simulations. (c) Same as (a) but with the detuning set above the MI threshold of the second resonance (${\delta }_1\approx 1.79\pi$; ${\delta }_2\approx -0.21\pi$). (d) Numerical simulations corresponding to (c). (e) Experimentally measured FROG trace corresponding to the spectrum shown in (c). SH: second-harmonic.

Figure 3

Real-time dynamics of a CS on top of a patterned background. (a) Vertically concatenated segments extracted from a single oscilloscope trace measured at the cavity output after an offset filter, showing the round-trip-by-round-trip evolution of the intracavity field after the detuning has settled to a constant value. (b) Corresponding results from numerical simulations, processed to take into account the experimental detection method. The oscillatory features arise from the CS drifting across the MI pattern. (c) Temporal profiles at four different round-trips (indicated on the right) extracted from the simulation before taking into account the experimental detection method. The profiles are vertically offset for clarity. For a full animation of the simulation, see Supplemental Material [80].

Figure 4

(a) Numerical simulation of the Ikeda map, showing the intracavity dynamics as the cavity detuning is scanned across two cavity resonances and then maintained at ${\delta }_0=2.7\pi$. Dashed horizontal lines highlight the boundaries of different dynamical regimes, as indicated. (b) Simulated temporal profile at round-trip 400, before the collision of the two CSs associated with different resonances. SCS is the “super” CS associated with the first resonance. For a full animation of the simulation, see Supplemental Material [83].

Figure 5

Experimental results, showing evidence of the coexistence of two different CS states. (a), (b) Vertically concatenated segments of oscilloscope traces measured as the driving laser frequency is first scanned across two cavity resonances (round-trips 1–350) and then maintained at a constant value (round-trips 351–750). Traces in (a) were measured directly at the cavity output while those in (b) were obtained after the output had passed through an offset filter. The different slopes and spectral bandwidths of the localized structures emerging around the 350th round-trip are indicative of two different CS states. The white arrow highlights a “super” CS associated with the first resonance.

Figure 6

Comparison of cw solutions of the Ikeda map (solid gray curves) and the LLE (dashed black curves). All parameters the same as in Fig. 1.