Temporal cavity solitons (CSs) are pulses of light that can persist in coherently driven passive resonators, such as fiber ring resonators and monolithic microresonators. It has been theoretically predicted that they can exhibit rich instability dynamics, yet experimental observations have remained scarce. Here, we report on the observations of complex spatiotemporal instabilities of temporal CSs in a synchronously driven fiber ring resonator. Through continuous variation of a single control parameter, we observe a string of predicted instabilities, including irregular oscillations (breathing) and chaotic collapses. Beyond a critical point, we find behavior reminiscent of a phase transition: CSs trigger localized domains of spatiotemporal chaos that invade the surrounding continuous wave background. Our findings directly confirm a number of theoretical predictions, and they highlight that complex CS instabilities can play a role in experimental systems.
© 2016 Optical Society of America
The Lugiato–Lefever equation (LLE) is a prototypical model equation for pattern formation, and it has applications in many non-equilibrium systems . In particular, it provides for the canonical description of coherently driven Kerr nonlinear optical cavities [2,3], including fiber ring resonators [3–6] and monolithic microresonators [7–9].
The LLE has steady-state solutions known as cavity solitons (CSs) or dissipative Kerr solitons. These are wave packets that sit on top of a homogeneous background, with potential applications as coherent optical frequency combs [9–13] or as bits in all-optical buffers [4,5]. Besides stable stationary CSs, theoretical analyses have revealed several dynamic instabilities [1,6,14–16]. Periodically oscillating CSs represent the simplest examples but more complex instability behaviors, including chaotic oscillations and transient collapses, have also been predicted [1,6,14]. Unstable CSs can even trigger a chain reaction of proliferation, giving rise to dynamics analogous to a phase transition: nucleation of a domain of spatiotemporal chaos that invades the surrounding meta-stable homogeneous state .
Only stable [4,5,9–13] and periodically oscillating [6,17,18] CSs have so far been studied in experiments. Observations of more complex behaviors have remained elusive, arguably due to the lack of appropriate experimental configurations. In particular, while continuously driven optical fiber ring resonators allow for the investigation of stable CSs [4–6], they cannot easily access the strong driving regime where instabilities manifest themselves. The use of synchronous pulsed pumping [19–22] could alleviate that deficiency, yet no studies involving individual CSs have been reported. Sufficient driving strengths can also routinely be achieved in microresonator frequency comb experiments [9–12]; however, the microresonators’ small physical size obstructs direct (i.e., time-resolved) observations.
In this Letter, we experimentally investigate an optical fiber ring resonator that is synchronously driven with flat-top laser pulses. Through variation of a single control parameter, we observe several CS instabilities that have been predicted to manifest themselves in the strong-driving regime of the LLE; this includes chaotically oscillating and collapsing CSs as well as transitions to expanding domains of spatiotemporal chaos. Our experimental findings directly confirm theoretically predicted CS bifurcation characteristics, and they demonstrate that exotic instability behaviors can play a role in experimental systems.4], ; ; . Here, is the cavity roundtrip time, is equal to half of the percentage power dissipated over one round trip, is the length of the resonator, and and are the fiber group-velocity dispersion and nonlinearity coefficients, respectively. The normalized driving strength , where is the intensity transmission coefficient of the coupler used to inject the coherent driving laser, with power , into the resonator. Finally, characterizes the frequency detuning of the driving laser at from the closest cavity resonance at , with .
Complex CS instabilities have been predicted for . In Fig. 1, we illustrate generic bifurcation characteristics for a driving strength similar to the experiments that will follow (). Here, we show the peak amplitudes of the coexisting continuous wave (cw, black curves) and CS (red curves) solutions as a function of . (The solutions were obtained using standard techniques .) Considering first the cw solutions, the intermediate branch (dotted line) is unconditionally unstable while the top and bottom branches are stable against cw perturbations. Nevertheless, for , periodic perturbations can cause the top branch solutions (dashed line) to experience a modulation (or Turing) instability , leading to stationary patterns or full spatiotemporal chaos consisting of fluctuating structures. The lower branch is stable for all , and corresponds to the cw background on top of which the CSs sit. Accordingly, the up-switching point represents the lower boundary of CS existence, as seen in Fig. 1(a). For CS solutions, structures in the lower branch (dotted line) are metastable  and shall not be discussed further. On the upper branch, CSs are stable (solid line) for large but then become unstable (dashed line) through a Hopf bifurcation as decreases.
Referring to the inset in Fig. 1(a), CSs in the unstable regime first exhibit oscillatory breathing behavior close to the Hopf threshold [region I, Fig. 1(b)]. The oscillations are initially periodic but turn chaotic as is reduced. For sufficiently small , behavior not dissimilar to excitability  emerges [region II, Fig. 1(c)]: the CSs undergo a large excursion in phase space (chaotic oscillations), followed by collapse to the cw state. A narrow regime of oscillatory behavior resurfaces for even smaller until finally, for close to the up-switching point , the CS instability manifests itself as the triggering of a wake of spatiotemporal chaos [region III]. Figure 1(d) shows simulated dynamics in this regime: the CS proliferates into multiple copies that invade the surrounding cw state. The dynamics can be interpreted as coexistence between a coherent metastable state (the lower-branch cw solution) and chaotic patterns ensuing from the modulation instability of the upper branch. The CS triggers a (phase) transition from the former to the latter. We note that the resulting rise of a spatially localized (albeit expanding) domain of spatiotemporal chaos is different from the dynamics observed for driving frequencies , where the lack of a stable (or metastable) cw solution forces the entire field to simultaneously transform into delocalized spatiotemporal chaos. We also note that, although this discussion involves a single driving strength , the list of instabilities (periodic oscillations, chaotic oscillations, collapses, and transitions to spatiotemporal chaos) encompasses all dynamical behaviors identified for CSs in the strong-driving regime of the LLE .
To experimentally study CS instabilities, we use the setup shown in Fig. 2. It consists of a 100 m long fiber ring cavity (; ; ; ) that is similar to the one we have previously used for the study of stable temporal CSs . In these earlier studies, however, the cavity was driven with cw laser light, which does not readily allow access to the strong driving regime () required to study CS instabilities. To overcome this issue, here we drive the cavity with approximately 20 ns flat-top pulses (see inset in Fig. 2) at a duty cycle of . We obtain our pump pulses by intensity modulating a 1550 nm narrow linewidth cw laser. The modulator is driven with a frequency corresponding to an integer multiple of the cavity free-spectral range . Before the pump pulses are coupled into the cavity, they are amplified with an erbium-doped fiber amplifier to a quasi-cw power of about 6 W (normalized driving strength ) and passed through a BPF centered at 1550 nm to remove amplified spontaneous emission. It should be emphasized that the 20 ns width of our quasi-cw pump pulses is about 4 orders of magnitude larger than the sub-picosecond widths of the CSs that exist for our experimental parameters. The CSs thus experience the driving as effectively homogeneous. Compounded by the facts that (i) the cavity has high finesse of about 20 and (ii) the pump pulses are synchronized to the cavity roundtrip time, we expect the system to be well-modeled by Eq. (1) [4,22].
To investigate different CS regimes, we systematically control by using a function generator to adjust the carrier frequency of the pump laser. Moreover, we monitor by leveraging the non-ideal extinction of the modulator used to create the pump pulses: the linear resonance associated with the low-power cw background that exists in between the pump pulses provides for a reference. (We estimate the error in obtained in this way to be about .) As is varied, we continuously examine the intracavity field profile (extracted with a 99/1 tap coupler) using a 12.5 GHz photodetector and a real-time oscilloscope (12 GHz bandwidth). Before detection, the cavity output passes through a narrow (0.6 nm) bandpass filter (BPF) centered at 1551 nm. This removes the cw background at 1550 nm, thereby improving the signal-to-noise ratio of the measurement .
Referring to Fig. 1, we start the experiment with the pump laser blue-detuned from a cavity resonance () and then slowly increase to by reducing the laser frequency. This results in the spontaneous excitation of CSs [9,23]. We then reverse the direction of the frequency scan and continuously reduce at a rate of about . (Similar forward and backward tuning has recently been used to control the number of stable CSs in microresonators .) Because the cavity photon lifetime , the intracavity field reacts almost adiabatically to the frequency scan. Accordingly, by recording a long time trace at the cavity output, we are able to examine CS behavior as a function of . To facilitate visualization, we divide the experimentally measured time trace into segments spanning one roundtrip and concatenate the resulting sequences on top of each other. Figure 3 shows typical results obtained from such a measurement: the density maps depict the spatiotemporal field evolution (for clarity, we only show a 2 ns segment of the full 20 ns pump profile), while the line plots capture the evolution of the integrated energy around a single CS. The full measurement encompasses values continuously reducing from about 27 to 5, but we only show snapshots around four regions that highlight the main dynamical behaviors.
The results are in remarkable agreement with the predicted bifurcation characteristics [see Fig. 1(a)]. We first observe six stable CSs until passes the theoretically predicted Hopf threshold at about , beyond which simple oscillatory behavior ensues [region I in Fig. 1(a)]. For , we witness irregular oscillations that lead to the spontaneous collapse of particular structures [region II]. Owing to the chaotic nature of the dynamics and the fact that is continuously reduced, some CSs avoid a collapse and survive to the second regime of stable oscillations that manifests itself for below the collapse region. Finally, around , the few remaining CSs transform into localized chaotic domains [region III] that are quickly engulfed by the full destabilization of the cw background at about . This point coincides almost exactly with the theoretically predicted up-switching point .
Before proceeding, we comment on two aspects of our measurement. First, energy variations of oscillating CSs are exaggerated in our experiment. This is due to the offset BPF in the detection path of our setup (see Fig. 2): oscillating CSs exhibit variations in their spectral bandwidth that can impact strongly on the energy transmitted through the BPF. Second, due to the 50 ps response time of our detection electronics, we are not able to directly discern the sub-picosecond profiles of the CSs or the proliferation into numerous closely packed structures that are predicted in the chaotic regime [see Fig. 1(d)]. Nevertheless, the abrupt increase in integrated energy (and the spread of the CS envelope) around provides convincing evidence for such behavior.
The results in Fig. 3 show clear evidence of oscillating and collapsing CSs. However, due to the rate at which the driver frequency was varied, the transitions to spatiotemporal chaos are not satisfactorily captured. We have therefore performed another experiment where we significantly reduced the laser scan rate after reaching the region of interest (). Figure 4(a) shows the detected signal over the full 20 ns width of the pump pulses when is slowly reduced around over 1 ms (we estimate that reduces from about 6.2 to 5.8 during the measurement). We observe six oscillating CSs (highlighted with arrows), some of which abruptly transition into expanding fronts. Figure 4(b) shows a zoom on a particular event [white arrow in Fig. 4(a)], illustrating how the oscillating CS triggers a chaotic domain that invades the surrounding homogeneous state at an almost constant rate. These observations are in good agreement with corresponding simulations of Eq. (1), shown in Fig. 4(c). The simulations use experimental parameters aside from , which is reduced from 6.75 to 5.75 during the simulation (the values are nevertheless in reasonable agreement with experimental estimates). To facilitate comparison with experiments, the simulation results have been post-processed to mimic the effect of the offset filter and limited detection bandwidth in our experiment; the raw theoretical data is qualitatively similar to that shown in Fig. 1(d), consisting of chaotically oscillating, rapidly proliferating CSs. In this context, besides results shown in Fig. 4(c), we have carefully verified that all of our experimental observations are in good agreement with numerical simulations of both the LLE and a more rigorous Ikeda-like cavity map .
The slow expansion of the domains triggered by individual CSs is followed by an abrupt transition into fully distributed spatiotemporal chaos. This behavior is explained by the detuning falling below the up-switching point [see Fig. 1(a)]: the stable cw state ceases to exist, thus forcing the entire field to switch to the turbulent modulation instability pattern on the upper branch. This transition takes place irrespective of the presence of CSs. Indeed, we have performed additional experiments where no CSs are present and observed similar transition behavior. We also note that the transition clearly highlights how inhomogeneities in the pump pulse strength (see inset in Fig. 2) can lead to locally different threshold behavior. We suspect this explains why individual CSs are observed to trigger the expanding chaotic domain at different . Besides pump inhomogeneities, the intrinsically chaotic oscillations of the CSs can also contribute to the behavior.
Our experimental results are in excellent agreement with the theoretically predicted CS bifurcation characteristics. To the best of our knowledge, they represent the first experimental observations of CSs exhibiting irregular oscillations, collapses, and transitions into expanding domains of chaos. In this way, the findings directly confirm theoretical predictions that have been drawn over the past three decades [1,6,14], and they show that complex CS instabilities may play a role in experimental systems described by the LLE, including cw-driven microresonators. More generally, as CSs are a class of localized dissipative structures (or dissipative solitons) that are characterized by universal phenomenologies , we expect that the results and experimental techniques demonstrated in our work will be of significant interest in the wider study of dissipative systems. For example, we mention in closing that instabilities similar to those reported here have also been predicted for solitons of the parametrically driven nonlinear Schrödinger equation , whose applications range from vertically driven fluids  to optical parametric oscillators .
Royal Society of New Zealand.
The authors acknowledge support by the Marsden Fund and the Rutherford Discovery Fellowships from Government funding, administered by the Royal Society of New Zealand.
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