Rogue waves in injected semiconductor lasers with current modulation: role of the modulation phase.

Semiconductor lasers with continuous-wave optical injection display a rich variety of behaviors, including stable locking, periodic or chaotic oscillations, excitable pulses, etc. Within the chaotic regime it has been shown that the laser intensity can display extreme pulses, which have been identified as optical rogue waves (RWs), and it has also been shown that such extreme pulses can be completely suppressed via direct modulation of the laser current, with appropriated modulation amplitude and frequency. Here we perform a numerical analysis of the RW statistics and show that, when RWs are not suppressed by current modulation, their probability of occurrence strongly depends on the phase of the modulation. If the modulation is slow (the modulation frequency, fmod, is below the relaxation oscillation frequency, fro), 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. The most extreme RWs occur for modulation phases that are at the boundary of the safe window. When the modulation is fast (fmod > fro), there is no safe phase window; however, the RWs are likely to occur at particular values of the modulation phase. Our findings are of interest for the study of RWs in other systems, where a similar response to external forcing could be observed, and we hope that they will motivate experimental investigations to further elucidate the role of the modulation phase in the likelihood of the occurrence of RWs.