Cavity solitons (CSs) are solitary waves that can persist in driven nonlinear passive resonators [1]. Their tendency to broaden is balanced by nonlinear self-focusing, and all energy they lose is replenished by the coherent field driving the cavity. Several of them can coexist, and they can be independently excited, erased, and manipulated [2–5]. These properties have led to CSs being identified as promising candidates for bits in all-optical buffers and processing units [6–8].

Historically the focus has been on spatial CSs, i.e., nondiffracting localized beams of light trapped in planar cavities [2–4]. More recently experiments performed in optical fiber loops [5,8–10] have stimulated interest in their temporal counterparts: recirculating nondispersing pulses of light [7]. Such temporal CSs do not suffer from material defects that hinder the performance of spatial cavities [11], although somewhat analogous impairments can arise due to electrostriction-mediated interactions [9]. These interactions can, however, be overcome by phase modulating the continuous wave (cw) laser driving the cavity; the CSs are then trapped to the peaks of the ensuing intracavity phase profile [2,5]. In parallel with investigations in macroscopic fiber cavities, temporal CSs have also attracted attention in the context of monolithic microresonators [12,13]. In these devices, CSs can under certain conditions correspond to the temporal structures underlying broadband “Kerr” frequency combs [12,14,15], suggesting applications in metrology and high repetition rate pulse train generation. Finally, we note that temporal localized structures analogous to CSs have also been reported recently in semiconductor laser configurations [16].

Excitation of CSs requires that the resonator cw steady state is suitably perturbed [6,17–19]. For temporal CSs two techniques have been demonstrated. Most experiments in fiber resonators have used an incoherent “writing” scheme [5,8–10], where an optical “addressing” pulse perturbs the cw steady state via nonlinear cross-phase modulation (XPM). In microresonators, stable soliton states have been reached by carefully tuning the frequency of the cavity driving field [12,20], and similar spontaneous CS creation has recently been observed also in a macroscopic fiber cavity [21]. Both methods yield CSs, but also possess drawbacks: optical addressing increases the system complexity [8], while adjusting the driving laser frequency provides limited control over how many CSs are excited and at what temporal positions [15]. Moreover, neither of these techniques can easily be adapted to achieve controlled erasure of existing CSs. XPM has been numerically proposed for this purpose [8], but difficulties in synchronization have prevented experimental realization. This represents a major shortcoming, since the potential to be independently erased is widely regarded as a defining characteristic of CSs, and underpins many of their proposed functionalities [4,17,18].

Here, we implement another method for temporal CS excitation that alleviates these deficiencies. Our technique relies on the phase modulation that is typically applied to the cavity driving field to trap CSs into specific time slots [5]. By applying local boosts in the phase modulation, we are able to write CSs into corresponding empty slots. Significantly, similar boosts also allow us to demonstrate selective erasure of CSs already trapped in the cavity. This technique minimizes the complexity of CS systems, since the same components enable writing, trapping, and erasure. It could also represent a step toward controlled pulse generation in optical microresonators [22].

The ability to switch dispersive cw bistable devices through appropriate changes of the phase of the driving field was first proposed by Hopf et al. [23], and analyzed theoretically in the context of CSs by McDonald and Firth [19]. Phase modulation has also been used recently to excite localized topological phase structures in a laser with external forcing and feedback [24]. For a better understanding of our experiment, we first illustrate the phase-induced switching dynamics of temporal CSs by means of numerical simulations. To this end, we model the evolution of the field envelope $E(t,\tau )$ inside a high-finesse fiber resonator using the well-known mean-field equation [7,8,14]:

 

Fig. 1. Numerical results of CS writing and erasing by means of abrupt changes in the phase of the cavity driving field. The Gaussian phase modulation trap $\mathrm{\Gamma }(\tau )$ is shown as the curve on the back while its amplitude over subsequent roundtrips $A(t)$ is shown on the left. The numerical parameters are similar to the experiments that follow: $P_{\mathrm{in}}=908\mathrm{mW}$, $t_{\mathrm{R}}=0.48\mathrm{\mu s}$, $L=100\mathrm{m}$, $\theta =0.1$, $\alpha =0.146$, $\gamma =1.2{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$, ${\beta }_2=-21.4{\mathrm{ps}}^2/\mathrm{km}$, ${\delta }_0=0.4426\mathrm{rad}$, and ${\tau }_0=43\mathrm{ps}$. Note we only show a short 200 ps portion of the full cavity roundtrip.

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For simplicity, we consider here a one-bit pattern phase modulation, $\varphi (t,\tau )=A(t)\mathrm{\Gamma }(\tau )$, where $\mathrm{\Gamma }(\tau )=\exp (-{\tau }^2/{\tau }_0^2)$. For a temporal CS, $\mathrm{\Gamma }(\tau )$ represents a local trap in the fast-time frame [in our case with 70 ps full width at half-maximum (FWHM)] and mimics the modulation used in our experiments to overcome soliton interactions [5,9]. It also defines where writing/erasing takes place within the cavity. $A(t)$ is the phase modulation amplitude whose changes determine when CSs are written or erased. Figure 1 illustrates the process. Starting with an empty cavity, we turn on the driving field with a weak phase modulation, $A=0.52\mathrm{rad}$. As can be seen, this weak phase modulation alone does not lead to CS excitation. Instead, after some tens of roundtrips the field converges to a cw steady state. After 100 roundtrips we abruptly boost the Gaussian phase modulation amplitude to $A=2.17\mathrm{rad}$ for 10 roundtrips. This leads to a broad peak in the optical intensity which evolves and settles into a 2.6 ps wide (FWHM) CS in about 100 roundtrips (or 30 photon lifetimes), with transient oscillations similar to those observed with optical addressing pulses [8]. The CS then persists until we again boost the phase modulation amplitude to 2.17 rad for 15 roundtrips, which erases the CS after a brief transient lasting a couple of photon lifetimes only.

The ability to write and erase using phase modulation can be understood by noting that an abrupt step in the driving phase is equivalent to rotating the intracavity field in complex phase space around the origin [23]. Writing (erasing) occurs if the sequence of rotations, combined with the intervening nonlinear evolution, is such that the field falls within the basin of attraction of a single CS (lower cw state). This of course implies that not all phase operations lead to switching. To illustrate these concepts, Fig. 2 shows the simulated evolution of the amplitude $E(t,\tau =0)$ in the complex plane (with phase calculated relative to that of the driving field at $\tau =0$) for different writing and erasing sequences. The blue curve corresponds to the successful writing procedure in Fig. 1: starting from the cw solution, the phase modulation amplitude is first boosted to 2.17 rad (the corresponding rotation is labeled as step 1) and then reduced back to 0.52 rad after 10 roundtrips (step 2). As can be seen, the field eventually spirals to the stable CS solution (this spiraling represents the transient oscillations seen in Fig. 1). In contrast, if the phase modulation amplitude during the 10-roundtrip boost is slightly lower (2.00 rad, red curve), no writing takes place; the field remains in the basin of attraction of the original cw solution. Similar behavior is observed when the modulation amplitude is only boosted but never returned to its original value, and the green curve in Fig. 2 shows a trajectory for this case. The gray curve in Fig. 2 also illustrates the successful erasing of Fig. 1. Here little spiraling is observed: the approach to the cw state is fast and direct, and is actually disturbed when the phase modulation amplitude is lowered back to 0.52 rad. We note, however, that these dynamics are particular to our choice of parameters. For example, for some parameters writing can be achieved even with a single phase step [19]. In fact, the precise dynamics depends quite sensitively on the parameters involved. This may be linked to the metastable nature of the unstable CS solution (which exists for the same parameters [8]), to which the soliton field may temporarily be attracted (see blue curve in Fig. 2), as well as to the associated noncritical slowing [19]. We find that, for given parameters, some trial and error is required to determine phase sequences suitable for writing and erasing. Fortunately this task is quite effortless; while the precise dynamics may somewhat differ, switching nevertheless occurs over a wide range of conditions. We must remark that CS switching can also be realized with abrupt changes to other parameters in the presence of a non-evolving phase profile. Such changes can be induced by, e.g., mechanically perturbing the cavity.

 

Fig. 2. Simulated phase-space dynamics during attempts of CS writing/erasing with different phase-modulation sequences.

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To experimentally demonstrate CS writing and erasing, we use the setup schematically illustrated in Fig. 3. The optical cavity is similar to that used in [5,9], consisting of 100 m of single-mode fiber (SMF) closed on itself with a 90/10 fiber coupler. The cavity is driven with a narrow linewidth cw laser at 1550 nm whose output is phase-modulated with an electro-optic modulator (EOM) and then amplified with an erbium-doped fiber amplifier (EDFA). An electronic feedback control loop is used to actively lock the laser frequency near a cavity resonance and within the bistable region [8]. The cavity also hosts a fiber isolator, used to suppress stimulated Brillouin scattering, as well as a 99/1 tap coupler, through which the intracavity dynamics are monitored with a 40 GSa/s real-time oscilloscope. Before detection, the cavity output is filtered using a narrow (0.6 nm width) bandpass filter (BPF) centered at 1551 nm. This improves the signal-to-noise ratio by removing the cw component [8].

 

Fig. 3. Schematic illustration of the (a) optical and (b) electronic segments of the experimental setup.

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The abrupt phase changes that enable switching are achieved by manipulating the electronic signal that drives the EOM [see Fig. 3(b)]. Two programmable pattern generators, capable of producing approximately 70 ps FWHM electronic pulses, are synchronized by a single external clock, such that the repetition rate of their output patterns coincide with the cavity free-spectral range. One of them [top in Fig. 3(b)] is set to selectively produce pulses whose amplitude is $\sim 3$ times larger than those from the other. The low-amplitude pattern is fed to the EOM at all time; this gives rise to an intracavity phase profile that allows CSs to be trapped [5], but not written or erased. In contrast, the high-amplitude pattern is controlled by an electronically gated switch. When the switch is activated, the outputs of both pattern generators are combined for about 10 roundtrips. In this way the amplitudes of those electronic pulses that are contained in the high-amplitude pattern can be selectively boosted to $\sim 4$ times their original level.

For a particular experimental demonstration, we set the low-amplitude pattern generator to produce a sequence of five electronic pulses separated by 300 ps (the whole pattern repeats once per cavity roundtrip). We then explore complex writing and erasing sequences by selectively boosting some of the amplitudes as described above. Experimental results are shown in Fig. 4. The density map is a vertical concatenation (with slow/laboratory time $t$ running bottom to top) of successive oscilloscope traces recorded at the cavity output, illustrating how the optical intensity inside the resonator is affected by abrupt changes in phase modulation amplitude (the bit sequences on the right indicate which electronic pulses are boosted in amplitude). During the first 5 s of measurement no CSs are excited, confirming that the low-amplitude modulation itself is insufficient for this purpose. At $t=5\mathrm{s}$ all five electronic pulses are boosted for 10 roundtrips, resulting in the creation of five CSs. After excitation, the CSs remain trapped to the peaks of the low-amplitude phase modulation [2,5]. We then demonstrate erasure of individual CSs, by boosting selected phase pulses already trapping a CS, and highlight the flexibility of the scheme by performing complex simultaneous writing and erasing operations. In this context we remark that we are experimentally able to write and erase using the same operation (same phase amplitude applied over the same number of roundtrips). This should be contrasted with simulations of Fig. 1, where different numbers of roundtrips were required for writing and erasure. We believe this discrepancy arises due to environmental fluctuations that alter the system parameters during the experiment. This notion is supported by the fact that several attempts are occasionally required for successful switching, and that for each attempt, either all or none of the target CSs are affected.

 

Fig. 4. Experimental density map showing successive oscilloscope traces of the optical intensity at the cavity output. All the traces were acquired at 1 frame/s with a 40 GSa/s real-time oscilloscope. The bit sequences on the right indicate which of the five phase pulses are boosted to achieve writing and erasing.

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The above experiment demonstrates that direct phase modulation allows temporal CSs to be written and erased. However, the acquisition rate in this measurement was limited to 1 frame/s, which is too slow to capture the switching transients. To gain more insights, we have thus also taken real-time measurements that resolve the roundtrip-by-roundtrip dynamics of the optical intensity at the cavity output together with the electronic signal that drives the EOM. As above, we use the same phase operation for both writing and erasing (except that we use here only one electronic pulse per roundtrip), and Fig. 5(a) shows the corresponding electronic signal. Experimental results for transient writing and erasing dynamics are shown in Figs. 5(b) and 5(c), respectively. During writing, the field displays ringing similar to that observed during optical excitation [8] and numerical simulations (Fig. 1), and stabilizes in about 100 roundtrips after the moment of addressing. Erasure, on the other hand, appears to take place almost immediately. For comparison, we also plot in Figs. 5(b) and 5(c) results from numerical simulations (green curves), taking into account the effects of the BPF and the limited bandwidth of our photodetector. The simulations use parameters approximated from experimental measurements (see caption of Fig. 1), yet for erasure we allowed 5% increase in the cavity detuning ${\delta }_0$ so that both writing and erasure can be achieved with the same experimentally used phase sequence. We believe this is justified given the imperfections in the electronics used to stabilize the laser frequency. Regardless, the numerical results show good qualitative agreement with experiments. In this context, we also note that the writing (erasing) dynamics appear slower (faster) than they truly are since the optical BPF does not capture the broad temporal feature that appears during switching (see Fig. 1).

 

Fig. 5. Real-time dynamics of CS writing and erasing. (a) Electronic signal fed to the EOM. (b), (c) Experimentally measured roundtrip-by-roundtrip dynamics of (b) writing and (c) erasing. The green curves show results from numerical simulations.

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To conclude, we have experimentally demonstrated writing and erasing of temporal CSs by direct manipulation of the phase profile of the cavity driving field. The demonstrated technique offers a means to simplify the implementation of temporal CS-based all-optical buffers, and could enable controlled pulse train generation in monolithic microresonators. In this context, we must emphasize that writing is achieved with phase pulses that have a much longer duration than the CSs (70 ps versus 2.6 ps). Our work also reports the first experimental demonstration of temporal CS erasure. This highlights that temporal CSs are truly independent entities, and constitutes a step toward their use as fully addressable bits in optical buffers.