1. Introduction

Cavity solitons (CS) are localized structures whose existence is now experimentally well established in semiconductor systems [2, 3, 1, 4, 5]. They appear in spatially extended systems in the presence of bistability and of a modulational instability. These conditions can be obtained in semiconductor microcavities with an injected signal (holding beam, HB). Although CS can appear in passive or active systems, active systems have proved more favorable from an experimental point of view, especially to get rid of a global, thermally driven instability [6]. CS currently attract much interest because of the possible applications to all-optical processing of information. In particular, CS can be written or erased at any transverse location of the microresonator, and can be manipulated by phase gradients [7].

In experiments, CS can appear either spontaneously, by sweeping a parameter (holding beam power or pump power) or by local addressing with an external beam. If the addressing beam is coherent with the holding beam, it is sufficient to control the phase mismatch between the local addressing beam and the holding beam to switch-on or switch-off a CS. This process has been studied in [8] and is now well understood. However, CS are composite objects since they have an electric-field component and a carrier-density component, and it has been recently demonstrated that CS can indeed be switched on and off with a local injection of carriers [1]. This property has interesting consequences, in that it is possible to create or erase CS with almost any wavelength able to generate carriers inside the semiconductor material filling the cavity. There is no need for a precise control of the phase of the addressing beam. Moreover, the creation of the CS is accompanied with a wavelength conversion process, since the CS created has the wavelength of the holding beam. However, until recently the precise mechanism leading to the incoherent writing/erasure of CS was not fully understood and important discrepancies remained between the theory presented in [9] and the observations. In particular, the directions of the hysteresis cycles when the pump is adiabatically ramped are different in the experimental observations (see Fig. 6 in [4]) and in the previous theory. Hence, if the experimental observations seem to predict an incoherent switch-on with a local addition of carriers, the expectation from the theory is instead an erasure of CS. In each cases, an explanation is missing for both the incoherent writing and erasure of CS. In this Letter we propose a mechanism that explains all the qualitative features experimentally observed. It allows also to understand the observed delay in the switch-on process and the absence of it in the switch-off process. We can then make interesting predictions on the tunability of the delay which could be exploited in real systems.

2. Model

The basic argument for the modeling of the incoherent switch-on and off relies on the existence of thermally-induced local effects. When carriers are injected at higher energy in the bands, the excess energy with respect to the semiconductor band-gap (the quantum defect) is released into the system as heat. Heating has the well known property of shifting the cavity resonance towards lower energy, thus introducing a local decrease of the detuning in the system. The addition of carriers and the local decrease of detuning have opposite effects on the plane-wave hysteresis curves (see Fig. 1). On addition of carriers, the hysteresis curve is shifted towards higher holding beam energies whereas the following local decrease of detuning shifts the hysteresis curve towards lower HB intensities, thus giving the possibility to act in two directions with the local carrier injection. Then, if the system starts close to the upper turning point A, a sudden addition of carriers makes the system fall back to the lower branch (A’) and stay there even after the subsequent local detuning. Conversely, if the system starts on the lower branch on B, addition of carriers lets the system on the lower branch whereas the local detuning brings it to the upper branch in B’. An important difference resides in the timescales at which the two processes operates, since the carrier relaxation dynamics takes place on a fast time scale (a few nanoseconds), whereas the local detuning happens on a slow one (several hundreds of nanoseconds). The previous arguments illustrated the physical processes at stake in a zero dimensional system (without spatial effects). In an extended system, considering the spatial dimensions, a similar process is expected to take place with the CS branches (as computed e.g. in [10]) instead of the plane-wave ones since they have a similar topology. We will now focus on what happens in a one dimensional system. This choice is made for practical reasons since it is computationally much faster than a 2D one and we do not expect any dramatic change in the general physical processes. Our model is based on models developed in Refs [9, 11, 12, 13] which have already proved to describe the system quite accurately, and includes an equation for the spatially dependant detuning. It consists in three nonlinear, coupled partial differential equations for the intracavity field E, the carrier density N and the detuning θ :

 

Fig. 1. Plane-wave, bistability curves (Eqs.1 without spatial terms and without the equation for θ) with the parameters α=5,C=0.2, θ=-2 and Λ=3.2 except blue curve (Λ=3.3) and red curve (θ=-2.2). The portions of the curves with negative slopes are unstable.

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The equations are in rescaled form. Time is expressed in cavity lifetime units, and the spatial dimension is in units of the diffraction length. E_I is the holding beam field amplitude, C the bistability parameter, α is the Henry enhancement factor. The carrier recombination rate is γ and the thermal relaxation rate towards the steady state value θ ₀ is γ_T. The pumping term for the carriers Λ also provides a source term in the detuning equation through a linear function f. Diffusion and diffraction processes are also included through second derivative terms. A small noise term ξ in the intracavity field equation has been added to mimic spontaneous emission such that <ξ(t)ξ(t^′)>=2D_Eδ (t-t^′) where D_E=10^-6. Local carrier injection by an external address beam occurs on a very fast time scale (~100fs) such that we consider that it amounts to a perturbation at x_i of duration τ on the pump term Λ=Λ₀[1+δΛΠ(t-t₀,τ)Π(x-x_i,δ x)], where Π(x,δ x) is a boxcar function centered on x of width δ x. The corresponding source term f is such that f (Λ₀)=0 and is given by f(Λ)=(Λ-3.2)/0.2. We choose standard parameters for semiconductor materials that give bistability and a modulational instability : C=0.2, θ ₀=-2, Λ₀=3.2, E_I=0.75, γ=0.01, γ_T=10^-4. The laser threshold is given by Λ_th=1+1/2C while transparency is at Λ₀=1, hence we are in an amplifying medium. The diffusion of carriers and of detuning are taken to be equal. Although this may not be true in general, this is a reasonable starting point since DT may vary a lot according to the design details of the microresonator, and in particular how thermal management is taken care of [14]. The same holds for the relaxation time of the detuning γ_T, and the ratio γ/γ_T=100 is chosen to be large as expected in the experiment. Larger ratios may be required to fit the actual parameters in the experiments at the expense of much longer computation times but without significant qualitative impact on the dynamics.

3. Results

Results of the simulation are shown in Fig. 2. We use a numerical split-step method with a stochastic, second order Runge-Kutta scheme [15] for the temporal part and a Fast Fourier Transform scheme for the Laplacian operators part. The boundary conditions are periodic. The carrier injection time is taken to be τ=20.

As can be seen on Fig. 2, the qualitative behavior observed in experiments is well reproduced here featuring a delay in the switch-on process and a fast switch-off. Moreover, the transition itself is fast in both cases. Inspection of the variable θ shows a decrease of the detuning after the pump pulse followed by a slow relaxation towards the steady-state value. Once the writing pulse energy has been released, the system is brought to an unstable state and a slow dynamics governed by the detuning takes place. CS switch-on is triggered by this slow dynamics. On the contrary, CS switch-off takes place immediately after the beginning of the injection of carriers and the following detuning has little effect if not a small but visible relaxation of the system to the steady state.

The steady-state response of the system when ramping the pump intensity is shown on Fig. 3. It is obtained by monitoring the steady-state response of the system at a CS location while ramping adiabatically the pump. The maximum of the field is plotted in each branch. The lower branch corresponds to the homogeneous steady state (the dashed line and full line curves are superimposed) and is obtained by starting the system with a uniform state. When the system jumps to the higher branch, a pattern solution is selected. The noise visible on the curve when further increasing the pump is due to the finite size of the simulation box that constrains the available transverse wavevectors. When ramping down the pump, the solution evolves from the patterned solution to a solution when several CS can coexist. In the latter region, we can check that the single CS solution is stable by starting the simulation on the upper branch near the upper turning point with a single CS as initial condition. This single CS solution becomes unstable when leaving the bistable region and transforms into an extended pattern. The sense of the hysteresis is now in accordance with the one observed in experiments.

 

Fig. 2. Time traces of the intensity at a CS peak |E|² vs time (blue line) and local injection of carrier (Λ, red line). Upper figure : experimental results obtained in [1]. Lower figure : incoherent switch-on and off of a CS, with respectively E_I=0.76, δΛ=3.6, δx=20 and E_I=0.75, δΛ=1.37 and δx=20 (in inset is a zoom of the initial part of the switch-off process).

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Fig. 3. Adiabatic response |E|²when ramping the pump Λ: 1D numerical simulation (full line, the maximum of the field is plotted) and plane wave hysteresis (dashed line). The injected field amplitude is E_I=0.76. The arrows indicate the sense followed by the system on the hysteresis cycle.

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Fig. 4. Scaling of the switch-on delay versus writing power for a fixed duration of the writing pulse τ=20. The critical writing power is δΛ_c≃3.50.

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Let us now focus on the switching process itself. Switching in bistable systems has already been the subject of many theoretical and experimental studies, especially in zero-dimensional systems [16, 17]. When a bistable system is perturbed by a pulsed excitation and brought into an unstable state and if the perturbation area A(product of the perturbation time and perturbation amplitude) is larger than a critical value A_c, the system switches to the other state with a delay (non-critical slowing down) which has a logarithmic scaling law δτ~-ln(A-A_c). When the perturbation area is close to its critical value, infinite delay can be observed whereas, on the other side, the larger the perturbation, the smaller the delay. This behavior is reproduced here in our 1D system as shown on Fig. 4.

The critical writing intensity perturbation has been obtained by a fitting procedure. The time-jitter on each delay point was negligible at the noise level used in the simulation, hence only one realization of the switch-on has been done for each writing power. The logarithmic scaling seems to hold very well, especially at writing powers close to the critical value. At much higher writing levels, a small deviation is observed. This may be due to the fact that at high perturbation levels, the lasing threshold is transitorily crossed thus changing the subsequent scaling law.

The switch-off process is also characterized by a critical erasing power δΛ_c≃1.0225. Below this value the switch-off fails and above this value the system evolves towards the homogeneous background (see Fig. 5). On this figure, we have prepared the system with a cavity soliton and after a time t=100 launched an erasing pulse of a given amplitude. When the switch-off is effective, for an erasure power larger than the critical value, the erasure of the cavity soliton is very fast. For erasing powers slightly below the critical value, the dynamics is slowed down around the unstable point and the system returns back to its on-state. If the erasing power is high enough however, the switch down is followed by a switch-on if the local heat-induced detuning is large. We note that precise rules for the writing/erasing conditions can in principle be found, at least numerically, e.g. using the numerical method introduced in [10] to find the bistability curves for the CS, but are out of the scope of this paper. With the particular parameter values used in the simulations, switch-on and off are possible for the same parameters (in particular same HB amplitude E_I) over a range of input fields E_I and writing/erasing powers.

 

Fig. 5. Switch-off dynamics with succeeded (blue) and failed (red) switch-offs. From bottom to top, the erasing powers are δΛ_c=1.10,1.03,1.0225,1.022,1.02. The erasing pulse in launched at t=100 and has a duration δτ=20.

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4. Conclusion

In conclusion we have proposed a model to describe the incoherent writing and erasure process of CS in a semiconductor optical amplifier. This model includes a thermally-induced detuning making it possible to write and erase CS incoherently. It is worth pointing out that this detuning is always present when injecting carriers, because it is impossible not to produce some heat in the system when doing so (be it by releasing energy through the quantum defect or by the non-radiative recombination of carriers). However, dissipation of this energy can be somehow controlled by a proper thermal management of the device. Indeed in [1], heat dissipation has been carefully taken care of in order to obtain a sample with a small thermal resistance and a fast relaxation time, which is all the more critical when dealing with a broad area system. Despite this, and because the system is very sensitive to small detunings, there remains a small effect which is responsible for the incoherent processes presented here. In addition, the thermal timescale involved is rather fast (some hundreds of nanoseconds) since carrier injection for writing a CS is made over a rather small area. In the case where the thermally driven detuning would become too small to be able to induce the CS switch-on, then other methods could be employed like the one presented in [18] in a zero-dimensional system, using two control beams with different wavelengths. Our model is in qualitative agreement with what has been observed in experiments [1] : it explains why it is possible to write and erase CS with a local addition of carriers, reconciles the observed sense of the hysteresis cycle with the predicted one and explains the observed delay in the CS switch-on. Although this delay represents a disadvantage for high bit-rate applications, it could prove interesting in applications requiring a tunable delay if the noise in the system is not too high.