Laser-localized structures have been observed in several experiments based on broad-area semiconductor lasers. They appear as bounded regions of laser light emission which can exist independently of each other and are expected to be commuted via external optical perturbations. In this work, we perform a statistical analysis of time-resolved commutation experiments in a system of coupled lasers and show the role of wavelength, polarization and pulse energy in the switching process. Furthermore, we also analyse the response of the system outside of the stability region of laser-localized states in search of an excitable response. We observe not only a threshold separating two types of responses, but also a strong variability in the system's trajectory when returning to the initial stable fixed point.
Spatially extended nonlinear systems often evolve towards non-uniform states in which some physical quantity organizes in spatial domains, either spontaneously or in response to external perturbations. Localized structures are one among the many types of spatial patterns which can result from this organization process . Their distinctive feature is their finite extension which naturally translates into the coexistence of zero, one or more of them in sufficiently large systems. As clearly shown by this Theme Issue and also earlier publications [2–4], localized structures are by far not a specific feature of optical systems and they may arise in many different contexts. However, the analysis of localized states (LSs) in the optical context has been remarkably prolific, from both the theoretical and experimental points of view, since they were discovered in the 1990s [5,6]. Even if most of the initial experiments were performed in systems with coherent forcing [7–9], LSs in systems with phase symmetry (such as lasers) were envisioned early on [10–12].
An appealing feature of LSs in optical systems is their bistability: the LS and the homogeneous background are stable for the same region of parameters. This feature is very generic and is observed for LSs which appear in the transverse dimension [13,14] or along the direction of propagation  and even in feedback systems . An external perturbation is expected to switch an LS on or off provided that the perturbation overcomes a certain threshold. The threshold is fixed by the unstable branch of the LS solution which connects the stable background and the stable LS branch. The control of optical LSs is of course fundamental for applications and the switch-on and switch-off duration determines the speed at which information can be written and processed. Because of their rapid response, semiconductor devices may allow for fast manipulations of LSs. For this reason, the switching of LSs in semiconductor devices has been the subject of several studies. The switching techniques that have been proposed may differ according to the system under consideration.
In semiconductor microcavities with coherent injection (holding beam), two kinds of switching techniques have been studied: coherent and incoherent. In the coherent switching, part of the holding beam is used as a control beam [9,17]. The control beam is locally injected in phase or out of phase in order to write or erase the LSs. In the incoherent one, the perturbation does not have a stable phase relation with the holding beam and it acts, as an additional localized pumping, on the population inversion. Experimentally, it was realized in optically pumped devices . From a numerical point of view, both methods were analysed [19,20].
In cavity-soliton lasers, phase, polarization and frequency are not fixed by any holding beam and therefore they are chosen freely by the system. The switching methods that have been proposed for cavity-soliton lasers have been called semi-coherent and incoherent. The principle of the incoherent switching is the same as that described above, whereas the semi-coherent consists of a perturbation of the electric field by the injection of an external laser beam whose phase is independent of the phase of the system.
The different techniques are described and compared numerically in the case of a vertical cavity surface-emitting laser (VCSEL) with an intracavity saturable absorber in . Experimentally, in an optically pumped monolithic VCSEL with an intracavity saturable absorber , solitons were repeatedly switched on and off at a repetition rate up to 80 MHz by incoherent switching. In a coupled amplifier/absorber microcavity system, it has been shown that the switch-on occurs with a long transient (hundreds of nanoseconds) [23,24] that decreases upon reduction of the external cavity length. However, no analysis of the properties that the perturbation should have to trigger the nucleation of LSs has been performed in any cavity-soliton laser system. We address experimentally this issue and report our findings on the impact of wavelength and polarization of the perturbation.
In addition, lasers with saturable absorbers without transverse degrees of freedom are known to be able to reach an excitable regime [25–28]. We analyse the effect of a localized perturbation outside the parameter range in which LSs are bistable in search of excitable LSs  which do not rely on convective effects [30,31].
2. Experimental set-up
In figure 1, a scheme of the set-up with the injected laser is depicted.
L1 and L2 are two nominally identical broad-area (200 μm) VCSELs, both of which are equipped with temperature and current control, respectively T1, I1 and T2, I2. They are placed face to face at a distance of about 30 cm in a self-imaging configuration via lenses and collimators. The coupling strength between the two devices is set by the beam splitter (BS) placed in the external cavity. Its transmission depends on the polarization (80% for P-polarization and 60% for S-polarization), and the polarization emitted by the system ends up being P-polarized. The output of the system is monitored by two 8 GHz fibre-coupled photodetectors D1 and D2. We remark that D1 receives the system output coming from the amplifier L1, whereas D2 detects the radiation coming from the absorber L2.
The laser (LD in figure 1) that we used to generate the perturbation is a 100 mW edge emitter. Its current is modulated (MOD in figure 1) with a pulse sequence provided by a waveform generator. The minimal duty cycle achieved is 0.1% of the sequence period plus 0.2% of the rise and fall time. Because of the electrical characteristics of the LD package, the minimal pulse duration that we could achieve is about 15 ns.
The wavelength emitted by the LD is λWB=982 nm. Because the usual wavelength of cavity solitons (CSs) (λCS) that switch on spontaneously in our system is around 976 nm, a diffraction grating is added in front of the LD in order to tune its laser frequency in the proximity of λCS. The switching process is very sensitive to the perturbation wavelength as we will see below. The half-wave plate highlighted in figure 1 allows us to change the polarization of the writing beam (WB) from parallel to orthogonal with respect to the CS's one.
The injection is done on the absorber L2 in such a way that the perturbation pulse could locally saturate the absorption and start the lasing process. The same method is used in [32,23]. The switching process is monitored through the fast detectors, D1 and D2, and the iris selects the area of interest.
The control beam is injected locally in the absorber through reflection on the BS. During the injection, D1 receives the part of the pulses that is transmitted (60% or 80% according to the polarization) by the BS and that did not enter the system. After one round-trip time from the beginning of the injection it also starts receiving the effects of the injection on the system: the beam reflected on the BS goes in L2, then it is transmitted to L1 and eventually it is reflected on the BS to go towards D1. Because the pulse duration is of the order of tens of nanoseconds and the round-trip time is 1.8 ns, the injected beam and its effects on the system overlap on D1. However, the system response is much weaker than the injected pulses because of the BS transmission–reflectivity ratio.
On the other hand, D2 receives only the system response to the injected pulses.
The signal in D2 has a delay of 3.5 ns compared with the signal in D1 because of the optical path difference from the output BS to the fibre-coupled detectors. Moreover, it is much weaker because of the two beam samplers which the output beam passes through before reaching D2.
3. Localized states commutation
Here, we investigate the influence on the switch-on dynamics of the WB parameters, such as wavelength and power and polarization. In particular, we focus our attention on the critical energy for the switch to occur and on the switching time, i.e. the delay between the rise of the injected pulse and the rise of the CS.
In order to take into account the effects of power fluctuations in the WB (caused by the feedback configuration), we perform several realizations for the same injection and system parameters and statistically analyse the results. We analyse a series of at least 500 switch-on events for different WB wavelengths and powers. The system is prepared in the non-lasing state in the bistability region. The WB is addressed where a CS is known to be stably pinned on a defect. The writing pulses are fired with a period of 2 ms. In order to bring back the system to the non-lasing state after a switch-on event, a short downward pulse is applied in the bias current of the amplifying microcavity I1 a few hundreds of microseconds after the application of the perturbation. The delay between this reset current pulse and following WB injection is chosen to be long enough to allow stabilization of the system parameters. An example of switching is shown in figure 2. In the following, we show only the smoothed traces which average out the multi-mode dynamics during the transient. The multi-mode dynamics is due to multiple longitudinal modes of the coupled cavity involved in the transient. Because the coupled system is about 30 cm long, the pulses are separated by about 2 ns (hardly resolved here; see also the transient analysed in §4). The multi-mode dynamics is progressively damped out and a stationary state is finally reached (close to 300 ns). Owing to the length of the coupled cavity, the electric field lifetime in this system can be expected to be of several nanoseconds, much longer than that of the semiconductor medium. As a consequence , we did not observe any signature of relaxation oscillation dynamics or Q-switched operation.
The strong overshoot observed in figure 2a between the vertical dashed lines is the result of part of the writing pulse reaching the detector. By construction of the experimental set-up, it is much more visible on detector D1 than on detector D2.
In figure 3, we measure switch-on events at the maximum pulse power for two different wavelengths in order to directly compare the system response. Owing to different reflectivities for S- and P-polarized light of the outcoupling beam splitter, CSs are always P-polarized, independently of the WB perturbation. In figure 3, the perturbation is S-polarized. As previously, the strong overshoot at the beginning of the trace is due to a part of the perturbation beam which reaches the detectors (especially D1) after reflection on the semiconductor microcavity.
We conclude that the blue detuned perturbation is more efficient, in agreement with observations reported in [34,35]. We interpret that this detuning corresponds to the detuning between the CS wavelength and the absorber cavity resonance. In fact, we have observed that the most efficient perturbation is at the wavelength of maximal absorption when applied on the absorber device in the absence of coupling with the amplifier device. In order to increase the success rate, the perturbation has been amplified via an optical tapered amplifier (TA in figure 1) for the following measurements. Other than the TA, the perturbation beam is so astigmatic that it has to be corrected. After a cylindrical and spherical lens, the perturbation beam has a diameter of about 10 μm and a circular geometry when it reaches the semiconductor microcavity. While the previous measurement shows that orthogonal polarization can be used, its efficiency is much lower, as shown by the comparison of figure 4a,c with figure 4b,d. Both experiments have been realized for the optimal wavelength (λWB=975.82, whereas λCS=976.02 nm). The power reaching the absorber (figure 4a,b) in the case of S-polarized perturbation (figure 4b) is four times larger than the power reaching the absorber in the P-polarized case (figure 4a). In spite of this large difference, the S-polarized perturbation has a success rate of 90%, whereas the P-polarized one has a success rate of 100% (figure 4c,d). In all cases, the LS which remains after the perturbation is P-polarized, owing to weaker losses of the system for this polarization (caused by the outcoupling beam splitter).
Finally, we measure the impact of the perturbation energy (figure 5). We apply an optimally tuned perturbation of constant duration (15.5 ns) and increasing intensity (controlled via the TA) and plot the corresponding success rate. The resulting threshold is found to be about 25 pJ. This energy appears to be in the power-dependent reflectivity regime (but not in the saturation regime) of the absorber device. This reflectivity has been measured to increase from 9% for 16 pJ pulses to 15% for 30 pJ. In correspondence with the increased efficiency of the perturbation, we also observe a very strong reduction of both the switching time average and the dispersion as shown on the histograms in figure 5b. We note that the residual dispersion in the switching time remains quite large, even when 99% of the perturbations are successful. One possible interpretation is that, in addition to (partial) bleaching of the absorption, the perturbation also has a thermal effect, which induces a slow displacement of the threshold, blurring its determination . Nevertheless, partial saturation of the absorption is the main mechanism behind the nucleation of CSs.
The annihilation of phase-locked LSs (in forced systems) can be achieved by applying a perturbation which is coherent and out of phase with the LS. In laser systems, there is no such possibility. Some previous observations of annihilation of CSs in laser systems were based on spatial degrees of freedom , but others have suggested the role of thermal processes  or only the local creation of additional carriers . In the following, we perform time-resolved measurements which show that the annihilation of a structure can be achieved within the 10 ns following the end of the perturbation, which indicates that thermal processes (which would be much slower) are not necessary for the annihilation of a structure.
As previously the perturbation wavelength is very important but, contrary to the nucleation case, the perturbation wavelength must be as close as possible to the CS wavelength in order to have the maximal efficiency. In this case, the CS is affected by the perturbation and its intensity drops down. For a given perturbation power, the effectiveness of CS switch-off increases with the distance, in terms of parameters, from the spontaneous switch-off. In the case reported in figure 6, the CS is bistable for values of 306.6 mA≤I1≤311.1 mA and the erasing beam polarization is orthogonal. If the bias current is not close enough to the lower boundary of the bistability region, the CS switches on again some time after the perturbation as shown in figure 6. The drop in intensity becomes deeper and the off-time is longer as the system gets closer to the spontaneous switch-off (I1=306.6 mA).
This observation indicates that the homogeneous stable solution was actually not reached by the system, which therefore returns to the CS state after a transient.
When the parameters are set close enough to the CS instability boundary (figure 6b), the annihilation can be achieved albeit with low efficiency (less than 10%), possibly related to the maximal power available in the perturbation beam. In spite of this low efficiency, the process is completed within the 10 ns following the end of the perturbation, a duration which is comparable to the electric field lifetime which can be estimated from the response of the system outside the bistable range (§4 and figure 7).
4. System response outside of the bistable range
In order to address the eventual existence of excitable LSs , which do not rely on convective effects [30,31], we analyse the response of the system out of the stability region of CSs. We observe two qualitatively different responses depending on the characteristics of the perturbation (wavelength or power). In the general case, the response of the system is so weak that it can only be observed by averaging over many realizations. In this case, the system seems to act essentially as a low-pass filter for the perturbation as illustrated in figure 7.
On the contrary, for adequately chosen perturbation (wavelength tuned to the absorber resonance as measured previously) and sufficient power, a much larger response takes place. Although this response cannot be detected in a single shot by standard CCD cameras, the averaged response is similar in shape and size to a stationary CS. The time-resolved response is essentially identical to the transient leading to the nucleation of a stable LS, with the exception that after this transient the system always returns back to the initial state. A typical example is shown in figure 8a. We define the ‘success rate’ as the ratio between the number of these large responses and the number of perturbations applied to the system. In figure 8b, we show the success rate obtained with 20 ns long pulses whose average power is 300 μW. The fact that the number of responses increases abruptly from 20% to 100% demonstrates the presence of a threshold separating different kinds of responses.
The existence of a threshold separating linear from excitable responses is a necessary condition for the system to be qualified as ‘excitable’. The second condition is that the excitable orbit should be strongly attractive, such that any above-threshold perturbation leads to the same response. In order to check the uniqueness of the trajectory following a perturbation, we statistically analysed the time taken by the system to relax back to its original stationary state. The results are shown in figure 9.
A deterministic time scale clearly appears in the histogram of the response durations, because no return trajectory has a duration shorter than 100 ns. Together with the threshold shown in figure 8, this confirms the existence of a separatrix in phase space and once this separatrix is overcome the system will settle via a much longer trajectory. However, there is also a strong dispersion in the response durations, even if the filtered response intensity is always identical to the one of a stationary CS. Besides this dispersion, one can also distinguish a secondary peak centred around 250 ns. The existence of several peaks in the return time may be related to the remnants of multiple coexisting attractors for CSs , and stochastic effects can also not be ruled out. In any case, the large variability of the response prevents an interpretation of the results in terms of excitable LSs.
We have analysed the response of a system of coupled semiconductor microcavities in an absorber/amplifier configuration and we have studied the optimal perturbation parameters to commute on and off localized structures. The optimal beam characteristics ended up to be different for the two processes. Switch-on is more probable when the perturbation pulses are P-polarized and blue detuned with respect to the CS. Switch-off, on the other hand, is favoured by S-polarization and a wavelength that matches the CS's one. The shortest switch-on process that has been observed takes 20 ns to start and about 200 ns to be completed owing to the complex multi-mode dynamical transient. On the contrary, the switch-off, when successful, occurs in about 20 ns. The switch-off efficiency, which we found to be very low for any of the parameter sets we could test, may have been affected by the rather low power available in the perturbation beam. However, the shape of the perturbation may also have a strong impact, as discussed in , and this parameter has not been tested at all yet.
Besides these results, we have also studied the response of the system below the bistability region and found some elements peculiar to excitability such as a threshold-like behaviour and a certain determinism in the return trajectory (the way by which the system returns to its initial state). Indeed, the majority of the responses have a duration of 125 ns and none lasts for a shorter time. Because the spatial profile of the time-averaged response and the emitted intensity matches that of the stable CS, these observations could indicate the existence of excitable LSs. However, the strong dispersion in the duration of the return trajectory prevents such an interpretation and may be related to the multi-stability of monochromatic solutions implied by the longitudinal geometry of this experimental system.
One contribution of 19 to a Theme Issue ‘Localized structures in dissipative media: from optics to plant ecology’.
- © 2014 The Author(s) Published by the Royal Society. All rights reserved.