1. Introduction

Semiconductor lasers (SCLs) provide an inexpensive and controllable setup for studying nonlinear dynamics and optical instabilities. While stand-alone SCLs emit a stable output intensity, they are easily perturbed, either by optical injection (OI) from another laser [1, 2], or by self optical feedback (OF) from an external reflector [3, 4]. OI and OF can be used to improve the laser performance (for example, OI can be used to increase the modulation bandwidth and OF, to narrow the laser linewidth), but under uncontrolled conditions both OI and OF can lead to detrimental instabilities, including broad-band chaos; however, this dynamical behavior has also found applications, for example, for ultra-fast random number generation [5].

When the laser emits a chaotic output, a particular behavior that is attracting attention is the generation of occasional ultra-high intensity pulses, i.e., pulses which belong to the extreme tail of the distribution of pulse amplitudes. Extreme events, which appear in a variety of different contexts, is an active research field and laser systems provide controllable experimental setups for their study [6, 7]. In semiconductor lasers, ultra-high pulses have been observed when the laser is subject to phase-conjugated OF [8], to conventional OF from a short external cavity [9] and to continuous-wave (cw) OI, when the polarization of the injected light is orthogonal [10] or parallel [11] to the natural lasing mode.

In the latter case the extreme pulses have been identified as optical rogue waves (RWs [12]), and simulations of a simple model have shown good agreement with the observations [13]. In a follow up work several of us studied numerically the statistics of RWs when the laser current is sinusoidally modulated [14]. We found that the modulation is capable of controlling the probability of occurrence of RWs: with appropriated modulation amplitude, A_mod, and frequency, f_mod, RWs can be completely suppressed. However, for values of A_mod and f_mod outside the “safe” parameter region, current modulation can induce RWs, and the resonant interplay with spontaneous emission noise resembled the phenomenon of stochastic resonance.

Our goal here is twofold: on one hand we aim at investigating whether current modulation induces some regularity in the occurrence of RWs that can give some predictive power; on the other hand, we aim at gaining a better understanding of the interplay of noise and modulation. We show that when RWs are not suppressed by the modulation, they tend to occur in a precise range of values of the modulation phase (for slow modulation frequency), or at specific values of the modulation phase (for fast modulation frequency). We also show that the highest RWs often occur at a specific phase of the modulation cycle: just before the pump current starts growing from its minimum value.

2. Model

The dynamics induced in a single-mode SCL with cw OI can be modeled by a set of two rate equations for the slow envelope of the complex electric field E and the carrier density N as [14]

In order to study the effect of sinusoidal current modulation, the parameter μ varies as μ(t) = μ₀[1 + A_mod sin(ω_modt)], where μ₀ is the dc bias current, A_mod = μ_mod/μ₀ is the normalized modulation amplitude and f_mod = ω_mod/2π is the modulation frequency.

The model equations were numerically solved using, unless specifically indicated, the same parameters as in [14]: κ = 300 ns⁻¹, α = 3, μ₀ = 2.4, γ_N = 1 ns⁻¹, P_inj = 60 ns⁻². Time traces of 250 μs of the laser intensity, I = |E²|, were generated from random initial conditions. The noise strength and the modulation parameters are considered control parameters. As in [14] we chose two values of the detuning: Δν = 0.22 GHz (Point A) and Δν = −0.24 GHz (Point B). In Point A RWs can be observed even in the absence of noise or modulation, and thus, they are referred as deterministic RWs. In Point B, without modulation RWs are induced by noise (there are no deterministic RWs), and in the presence of current modulation, they can be either induced or suppressed by modulation, depending on the modulation amplitude and frequency (as seen in Fig. 5 of Ref. [14]).

As RWs are ultra-high pulses when compared with the average height of all the intensity pulses, they are characteristic of the tail of a “L-shaped” probability distribution function (PDF). Thus, a suitable method to detect them is based on the analysis of the first and second order moments of the PDF of the intensity pulse heights. Following [11, 13, 14] we consider that a pulse is a RW if its height is higher than a threshold, I_th, that is a function of the mean value of the intensity, 〈I〉, and its standard deviation, σ_I. Here we use I_th = 〈I〉 + 6σ_I. Similar results were obtained with a higher threshold (8σ_I instead of 6σ_I) even though the number of detected RWs was considerably smaller. In order to properly choose the threshold we have taken into account that too high thresholds have the drawback of detecting a small number of extreme pulses, requiring long simulation times, while too small thresholds detect pulses that do not belong to the tail of the distribution of pulse heights. For our system the use of I_th = 〈I〉 + 6σ_I is an adequate compromise between these two facts.

3. Results

3.1. Dynamics in Point A: deterministic RWs and interplay of noise and modulation

Figure 1(a) displays the number of detected RWs as a function of the phase of the current modulation, Φ = ω_mod t mod(2π), for parameters corresponding to Point A, i.e., such that, in the absence of current modulation and noise, occasional RWs occur.

 

Fig. 1 (a) Number of detected RWs as a function of the phase of the modulation, Φ. (b) Mean RW amplitude as a function of Φ. The error bars indicate the standard deviation of the distribution of RW amplitudes. The parameters are: Δν = 0.22 GHz (Point A), A_mod = 0.2, f_mod = 3.5 GHz, D = 10⁻³ ns⁻¹, other parameters are as indicated in the text.

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One can notice that RWs occur in a well-defined range of modulation phases, roughly speaking, during the first 3/4 of the modulation period. On the contrary, during the last 1/4 of the period (corresponding to a current range from μ₀(1 − A_mod) to μ₀), RWs are very unlikely to occur. In other words, in each modulation cycle there is a ”safe window” of time, when the pump current is increasing from its minimum value, where the likelihood of RWs is very small.

Figure 1(b) displays the mean RW amplitude as a function of the modulation phase and one can notice that it is rather independent on the modulation phase in most of the modulation cycle; however, the highest -most extreme- RWs are emitted just before the secure window. This behavior is typically observed in systems where the phase space is organized by a saddle point that separates two different behaviors.

The variation of the number of RWs can be due, at least in part, to different current values (regardless of the modulation phase). As one can see in Fig. 1 of [14], where points A and B are labeled in the parameter space (pump current, frequency detuning) and the color-code indicates the number of RWs without modulation, there is indeed an influence of the value of the current in the number of RWs: as one can note, moving along an horizontal line in this plot, in between the current values 2.1 and 2.5: in point A, for lower pump current the number of RWs is lower (it increases with the pump current), while there are no RWs for higher currents; in point B, there is no clear trend and no –or few– RWs occur in this current interval.

However, we remark that what Fig. 1 shows is indeed an effect of the modulation phase, because, for two different modulation phases, but which correspond to the same current value, the RW amplitude and probability can be very different. For example, in Fig. 1 (a), the value of the current is the same when the modulation phase is π/4 and when it is 3π/4, but the number of RWs is very different (being almost 0 in the first case, and very large in the second case).

Next we analyze the influence of noise. Figure 2(a) displays the number of RWs vs the modulation phase for different noise levels. The existence of a secure window where no RWs occur is a robust observation (at least for the range of noise strengths considered here), but when the noise is strong, RWs are detected in a broader range of phases. Figure 2(b) shows that the total number of RWs is rather independent of the noise strength (triangles); in contrast, in the absence of modulation (circles) noise substantially reduces the number of RWs, in good agreement with previous findings [13].

 

Fig. 2 (a) Number of detected RWs vs the modulation phase when the noise strength is D = 10⁻⁴ ns⁻¹ (black) and D = 5 × 10⁻³ ns⁻¹ (red). (b) Total number of RWs vs the noise strength without modulation (A_mod = 0, circles) and with modulation (A_mod = 0.2, triangles). Other parameters are as in Fig. 1.

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We now consider the influence of the modulation frequency. Figure 3 plots the number of RWs vs the modulation phase for f_mod in the range 2–5 GHz (with the same modulation amplitude and noise level as Fig. 1). For slow modulation [Figs. 3(a), (b)] we note that the “safe window” remains, but the size and shape of the distribution depends on f_mod.

 

Fig. 3 Influence of the modulation frequency. Number of detected RWs vs the modulation phase for (a) f_mod = 2 GHz, (b) 3 GHz and (c) 5 GHz. Panel (d) displays a detail of (c). Note that the vertical scale in (a) and (b) is logarithmic, but a linear scale is used in (c) and (d) in order to better display the “quantized” character of the modulation phases when RWs occur. All other parameters are as in Fig. 1.

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As discussed in [14], for modulation frequencies near the relaxation oscillation frequency (f_ro = 4.5 GHz), the modulation fully suppresses the RWs, i.e., the “safe window” extends to all modulation phases. For faster modulation, Fig. 3(c), there is an abrupt change in the dependence of the number of RWs on the modulation phase: we observe that the RWs occur in well-defined phase intervals, and within each interval, the number of RWs is maximum at highest modulation phase, as seen in Fig. 3(d).

3.2. Dynamics in Point B: RWs induced by the interplay of noise and modulation

In Point B, in the absence of current modulation and noise, no RWs occur. So, any RW is caused by perturbations induced by the modulation, the noise, or the interplay of both. Figure 4 shows that the observations done for Point A are also valid in Point B: for slow modulation there is a secure window in the range from μ₀(1 − A) to μ₀, and the RW amplitude increases with the modulation phase until the border of the secure window is reached; for fast modulation the RWs occur in well-defined phase intervals, and within each interval, the number of RWs is maximum at highest modulation phase.

 

Fig. 4 (a) Number of detected RWs and (b) mean RW height as a function of the modulation phase. Δν = −0.24 GHz (Point B), f_mod = 3.5 GHz and other parameters are as in Fig. 1. (c) as panel (a) but for f_mod = 5 GHz.

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4. Conclusions

We studied numerically the generation of RWs in semiconductor lasers with continuous-wave optical injection, when they are sinusoidally forced via direct pump current modulation. As it was shown in [14] RWs can be completely suppressed when the modulation frequency is close to the relaxation oscillation frequency, f_ro. We have performed a detailed analysis of the RW statistics and found that, when RWs are not suppressed by the modulation, their probability of occurrence strongly depends on the phase of the modulation. For slow modulation (f_mod < f_ro) the RWs occur within a well-defined interval of values of the modulation phase, i.e., there is a “safe” window of phases where no RWs occur, which corresponds to the last 1/4 of the modulation period. The most extreme RWs occur for modulation phases that are at the boundary of this safe window. For fast modulation (f_mod > f_ro), there is no safe phase window; however, the RWs are likely to occur at well-defined intervals of the modulation phase. Our findings can be relevant for the study of RWs in other systems, where a similar response to external forcing could be observed.

We also hope that the results reported here will motivate experimental investigations to further elucidate the role of the modulation phase in the occurrence of RWs. For example, it would be very interesting to investigate experimentally the transition from very slow modulation (when the variation of the laser current is adiabatic and the modulation phase plays no role), to fast modulation. We expect that, when the phase of the modulation starts affecting the RW probability, the shape of the distribution of the number of RWs will become asymmetric under the transformation Φ → Φ − π (that leaves the value of the pump current unchanged). This hypothesis could be tested experimentally.