1. Introduction

Cavity solitons (CS) are bistable spatially self-localized waves which exist in the transverse aperture of broad-area nonlinear optical resonators (see, e.g., [1, 2, 3]). They are discussed as basis for future all-optical and potentially massively processing schemes [4, 5, 6, 1], especially if realized in quite fast, compact and robust systems as semiconductor microcavities [6, 7, 8]. For applications response times to external inputs are obviously of outmost importance, but the dynamics of bistable systems was also studied because of the principals interest in transition scenarios in bistable systems [9, 10, 11, 12, 13, 14]. In particular, the phenomena of critical [9, 10, 11, 15] and non-critical [12, 13] slowing down were analyzed in detail.

Critical slowing down describes the divergence of the switching time, if one approaches the limit point of the bistable loop. It is encountered, if the control parameter is suddenly switched from a continuous-wave (CW) value below the limit point to a CW value above the limit point. For solitons, it was investigated in [16] in a feedback scheme using sodium vapor. The switch-on of semiconductor CS in a situation close to this limit (using pulses of a writing beam (WB) longer than the response time of the system) was studied in [17].

Probably more important for applications is the case of excitation by short pulses. In the limit of switching pulses that are short compared to the relaxation of the medium, only the pulse area, i.e. the product F=I_a·τ_a in the case of a rectangular pulse, was found to be the decisive factor deciding whether switch-on occurs and determining the delay time of switch-on [12, 13]. This so-called non–critical slowing down was discussed for CS numerically in a 1D nonlinear cavity [18] and was experimentally observed and quantitatively characterized for CS-like localized states in a liquid crystal light-valve [19] and in sodium vapor with optical feedback [16].

We recently demonstrated a semiconductor-based cavity soliton laser (CSL; see, e.g., [20, 21, 22, 23] for other laser systems with CS) based on a vertical-cavity surface-emitting laser (VCSEL) structure with frequency-selective feedback [24]. The particular advantage of using a laser configuration is that no holding beam of high spatial and temporal coherence is necessary to support CS [6, 7, 8] and all energy is drawn from a simple incoherent source. This also implies that before switch-on there is no frequency and phase reference for the WB and hence the WB is nearly inevitably incoherent with respect to the developing CS. In contrast, in systems with a holding beam, usually the WB is split-of from the holding beam and the phase relationship between the two arms is critical [17]. There are some reports of incoherent switching in systems with a holding beam [25, 8] but they do not provide a systematic investigation of switching times. Interestingly, in [8] a strong asymmetry is found between switch-on (delay times in the hundreds of ns range) and switch-off (delay times in the few ns range). Hence, it appears to be interesting as well as important for possible applications to investigate the characteristics of switching times in our CSL.

In this paper we present experimental investigations on the effect of the WB detuning versus the CS frequency, show the temporal dynamics during switching, and also examine the required WB power for successful switching versus the pulse width.

2. Experimental set-up

In our experiment, we use a broad-area bottom-emitting VCSEL with a structure similar to that in [26], emitting at 980 nm, and electrically pumped through a 200 µm circular oxide aperture. The experimental set-up is shown in Fig. 1. Frequency-selective feedback is provided by a diffraction grating in a Littrow configuration [27, 24], using a 616 cm long external cavity and a self-imaging 2-lens system. The VCSEL is operated below its solitary threshold in the amplifying regime. The diffraction efficiency of the grating is strongly anisotropic, the horizontal polarization being twenty times stronger than the vertical, causing the VCSEL to lase in the horizontal polarization. For the same reason the external cavity finesse differs in both polarizations, being 55 MHz in the horizontal, and 134 MHz in the vertical. Two beam samplers are positioned in the external cavity: one is used to couple out part of the beam for detection purposes, while the other allows an external field from a tunable laser source to be injected into the VCSEL. Its beam is focused to a narrow 12 µm waist, and is used as a WB for switch-on and off purposes. An acousto-optic modulator (AOM) is used to pick up pulses from this beam, with a limited rise and fall time around 15 ns.

In this configuration, bistable lasing spots with a size of about 10 µm are obtained (see inset in Fig. 2), when the grating is tuned to enhance on-axis emission [24]. It was demonstrated in [24] that several of them can coexist within the aperture of the device and can be independently manipulated (switched-on and -off) by the WB. They also show indications for motility, i.e., they are easily relocated in response to an external perturbation. Hence, they are interpreted as self-localized nonlinear solutions independent from boundary conditions, i.e. cavity solitons (CS, e.g. [6, 2, 3]).

As reported previously [27, 24], the limited frequency selectivity of the system gives rise to an off-axis optical background, which hinders the formation of CS. In order to reduce this background, we positioned a comb-like filter (with several horizontal apertures, 16 µm wide) in the external cavity, as close as possible to the grating, to spatially filter the re-imaged near-field. The resulting near-1D confinement limits the spreading of this detrimental background, and also improves the sensitivity to the frequency dispersion from the diffraction grating, which occurs along the horizontal axis. In this present experiment only one aperture from the comb filter was used, with conditions (current, aperture position) adjusted such that only one CS appeared at a time. This was done to reduce the potential influence of surrounding CS or remaining optical patterns on our data.

 

Fig. 1. Experimental apparatus. The self-imaging external cavity consists of a 8 mm focal length aspherical lens (C), a 300 mm focal length lens and a holographic grating (1800 lines/mm). OI: Optical Isolator, M: mirror, BS: beam sampler, HWP: half wave-plate, AOM: Acousto optic modulator. The tunable laser is used for injection of the WB. The detection branch contains CCD cameras for imaging of the VCSEL near and far-field, an avalanche photodiode to record fast temporal dynamics, and a scanning Fabry-Perot interferometer to monitor frequency spectra.

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3. Experimental results and discussions

A light-intensity (LI) curve of one CS is shown in Fig. 2. This displays bistability with a bistable loop whose width usually is in the range of 4–5 mA for the studied CS. In subsequent experiments the electrical current is kept within this bistability range, and the switchings are performed via the injection of the WB directly on top of the CS location. We first evaluated the effect of the frequency detuning, defined as the difference between the WB and CS frequencies. The CS frequency does not necessarily develop at the frequency of the WB, but remains within a range of a few GHz around a central frequency determined by the tuning of the grating and the injection current. Our experimental set-up does not allow measurements of the CS frequency during the fast switching transients, therefore we use as a reference the asymptotic frequency of the CS, measured no earlier than some hundred milliseconds after the switch-on.

 

Fig. 2. Power versus current for a single CS. The rugged appearance of the upper bistability branch is due to competition between different external cavity modes. The inset on the right shows the near-field intensity distribution of the CS, the transverse size is 34 µm.

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Fig. 3. Minimal pulse width needed to switch on a CS in dependence on the detuning between WB frequency and the CS frequency for a fixed power in the WB (220 µW). The WB is in the perpendicular polarization to the CS, and its frequency was fine tuned for each data point such that it was maximally resonant in the external cavity. At higher detunings than 25 GHz the WB is increasingly distorted.

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Figure 3 displays the pulse width necessary for CS switch-on versus detuning, at a constant WB power. The WB is in the polarization orthogonal to the one of the CS. The necessary pulse width greatly varies with detuning, from 40 ns to above 5 µs. The curve is strongly asymmetric and reaches a broad minimum for small positive detunings (about 3–10 GHz). Data beyond a positive detuning of 25 GHz were not recorded, as the amplified WB is distorted. This is the result of the excitation of off-axis waves which fulfill the resonance conditions in the VCSEL cavity better than on-axis waves [28].

This amplification of the WB due to the VCSEL microcavity resonance is thought to be the most influential mechanism on the required pulse width for switch-on, as this affects the WB power depending on its detuning. (We will see below that higher power leads to a lower minimal pulse width.) The resonance bandwidth of the VCSEL was measured experimentally to be in the order a few tens of GHz, and amplifies the WB power by a factor of two at its peak. We note that the external cavity is also enhancing the WB, but over a smaller bandwidth (in the order of 130 MHz in this vertical polarization). So each point in the curve was measured when the WB experiences maximum enhancement in the external cavity, using a fine adjustment of the WB frequency (in the tens of MHz range).

In order to investigate how the CS frequency is related to the peak resonance of the VCSEL, we performed measurements of the peak amplification frequency of the WB (with a blocked external cavity). Different power levels were used to estimate the change of the local resonance due to thermal or carrier density induced refractive index changes (not shown). This established that the CS frequency is red detuned versus the VCSEL peak resonance by approximately 10 GHz. Hence this probably explains that the best excitation condition is for a blue-detuning to the CS, where the resonance enhancement of the WB power is maximum. We previously showed that the CS frequency is slightly blue detuned with respect to the frequency retroreflected exactly on-axis by the grating [24]. The red detuning versus the VCSEL resonance should therefore be a consequence of frequency pulling, where the CS chooses a compromise between the VCSEL resonance frequency and the on-axis grating frequency. We note that the WB in this experiment is in the perpendicular polarization to the CS, and we measured an anisotropy of 5 GHz between the two polarizations. This implies that the asymmetry of the curve in Fig. 3 is further increased when the WB is in the parallel polarization. However in this case the data are more affected by the increased finesse of the external cavity.

 

Fig. 4. Transient dynamics during a) switch-on with parallel polarization, b) switch-on with orthogonal polarization and c) switch-off with parallel polarization. The time traces ofWB pulses are in brown, while the response of the system is in black. Time traces from the WB pulses were recorded separately, with the pulse trigger ensuring proper synchronisation. The detector is AC-coupled with a lower cut-off of 16 kHz.

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Figure 4 shows time traces of CS switch-on and switch-off, when conditions are optimal for fast switches (i.e. choosing an appropriate detuning, but also current, cf. below). For a WB in the same horizontal polarization as the CS the switch-on (off) pulses can be as short as 20 ns (40 ns), and 40 ns for switch-on in the vertical polarization. These fast time scales, clearly limited by the AOM risetime in some cases, indicate that the mechanism of switching is mediated by the carriers, because thermal effects occur typically on longer (a few hundreds of ns) timescales.

Figure 4(a) shows a delay of about 6–7 ns between the input pulse and the response of the system. After the peak of the WB pulse, the response of the system follows the pulse envelope until it detaches at some point and reaches a steady state after some small amplitude transient oscillations. Since in this case both the WB and CS power are recorded, with a similar magnitude in their power, we cannot distinguish between the two in the time traces.

The delay is partly due to the build up time of the WB resonance in the external cavity, and possibly a remaining lethargic time necessary to reach the critical energy for the CS ignition [17]. A minimal lethargic time of some hundreds of ps was found for CS in vertical-cavity amplifiers [17] (for critical slowing down) and indicates probably a limit for this part of the delay for conventional semiconductor media with a carrier lifetime of about one nanosecond. The former contribution to the delay can be reduced by using shorter cavities.

On the other hand, for the switch-on in the orthogonal polarization (Fig. 4(b)) the two polarization components can be easily distinguished in the measurement. In this case, the response of the CS is clearly slower than the rise-time from the WB pulse, and the initial delay is about 20 ns.

The switch-off (Fig. 4(c)) is quite fast (18 ns transition time between the 90% and 10% points of the difference between peak power level and the asymptote). It is followed however by strong pulsing dynamics, with pulses separated by the cavity round trip time (4 ns). Such a behavior is not unusual for transients in external cavity systems [29, 30]. It is possible that it is not the CS which is pulsing but that the pulsing state is actually related to the transient appearance of low amplitude, slightly off-axis patterns. These can be observed sometimes in a non-transient way (so that the spatial structure can be investigated by slow CCD-cameras) on the lower branch of the hysteresis loop just before the spontaneous switch-on [24]. Pulsing states are sometimes also found during the spontaneous switch-on transient and probably to be interpreted as a transient via these intermediate off-axis patterns.

After the transients died out, amplitude and shape of the spots remain constant in time and space for up to several hours (the longest time span over which the experiment was conducted), which is in accordance with the solitonic character of the spots. This is evidenced by observation with a standard CCD-camera and monitoring the power of a CS by a DC-coupled photodetector.

 

Fig. 5. Necessary WB pulse widths for switch-on (left) and switch-off (right) of the CS, versus the WB power. The curves are recorded for three different currents, at one, two and three mA below the limit point. Two curves are recorded for each current, displaying experimental uncertainties. The detuning between the WB and CS is +6 GHz (+4 GHz) for the switch-on (switch-off), and is readjusted for each current to take into account temperature change from Joule effect.

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Fig. 6. Same data as in Fig. 5, but the WB energy is plotted instead, versus the WB power.

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The dependence of the necessary ignition/erasure WB pulse width versus its power was also analyzed for different locations in the CS bistability range. We characterize the latter by the current difference to the threshold for spontaneous up-switching. Figure 5 presents the data for both the switch-on and switch-off with a WB in the parallel polarization. For specific currents, we varied the WB power and measured the minimum pulse width required to the CS switch-on/off. Each curve was measured twice in order to access the variations due to experimental uncertainties and fluctuations. All curves show qualitatively the expected behavior for non-critical slowing-down that the necessary pulse width decreases with increasing peak power. This necessary pulse width for both the CS ignition and erasure also varies greatly with the current difference to the bistability threshold current. For example, the minimal pulse width for switch-on is less than 20 ns (for 50 µWpeak power) when the current is 1 mA below threshold, but nearly 700 ns at 3 mA below threshold. For the switch-off, the trend is reversed, with increasing pulse widths required when approaching the threshold. An increase in the necessary pulse width/power is expected with increasing distance to the limit point because the distance to the separatrix between the on and off states increases. We note that for a certain range of current the same input pulse can induce both a switch-on and a switch-off of a CS.

Figure 6 presents the same measurements, but with the pulse energy plotted versus the WB power. The resulting curves are approximately linear, but do not follow an area law, as their slope is different from zero. Instead, for both the ignition and the erasure, the required pulse area increases with increasing power. The slopes differ for different currents, and decrease if one is closer to the limit point. The mechanism for these phenomena is not clear. It is also apparent that the pulse energy generally varies considerably less than the input power. This is reminiscent of the area law. A strict validity is not expected here because our input pulses are not short compared to the relaxation times of the system.

The fact that the use of an increased pulse width requires less energy might hint to a contribution of thermal effects. Phenomena on a long time scale (more than a few hundred ns) are usually associated with temperature. The injection of the WB could locally affect the temperature (via radiative cooling, i.e. a local decrease of carrier density in the active zone, or via background absorption in the passive parts), and hence the local VCSEL resonance and the bistability range. However this would affect the switch-on and switch-off in opposite ways, while it is clear in our curves that both the switch-on and switch-off necessitate less energy with increasing pulse width. Additionally, we did not find strong evidence of a local change of temperature with injection of the WB, in the range of power used in the experiment. In measuring the WB reflection off the VCSEL (without external cavity) for a power of 1.7 µW (considerably greater than power levels used in Fig. 6), the pulse amplitude varied only by 5% during the transient to the stationary state (reached after about 1 µs). This indicates that the local resonance does not seem to be significantly changed by temperature variations induced by the WB.

4. Conclusion

In summary, we characterized the switch-on and switch-off dynamics of CS in a CSL. Switch-on is possible in a wide range of frequencies different from the CS but the best results (i.e. the shortest response time) are achieved in an interval of a few GHz width, which is still comfortably wide for applications. This is attributed to the resonance condition in the VCSEL cavity. Our investigations show a clear trade-off between necessary peak power and pulse width but no exact area law. The latter cannot be expected though because the pulses used were not short in comparison to the characteristic time scales. So it appears to be interesting to repeat these experiments with short pulses from a mode-locked laser.

The minimum response time demonstrated is about 35 ns but this seems to be limited by our addressing system. We anticipate that the dynamics is currently limited to the 10 ns range by the length of the external cavity but can be speed up by using shorter cavities, i.e. by replacing the diffraction gratings by a volume Bragg grating, which should enable sub-nanosecond round-trip times. In that case, limitations due to carrier lifetime (typically one ns) will become an issue (e.g. [17]). This might improved by using faster semiconductor media, e.g. quantum dots [31].