Dissipative solitons, mode locking and phase symmetry

Since the seminal studies on solitary waves in dissipative nonlinear systems [Nozaki1984,Fauve1990] considerable attention has been given on a broader class of spatial structures called localized states, especially in optics. Indeed, some time after optics entered the world of pattern formation with the introduction of the so-called Lugiato-Lefever equation [Lugiato1987], the first theoretical predictions of localized structures in optical cavities [Tlidi1994] mentioned the potential of these structures in optical data processing applications. Indeed, due to their nature of bistable and independently controllable optical beams, they were immediately perceived as potential optical bits of information, and their prediction in semiconductor microcavities [Brambilla1997] gave a clearer insight to their potential due to the microscopic size of the devices and their fast time scales. The first experimental demonstration of existence and control of these structures in semiconductor was given in 2002 [Barland2002] by several of the participants to the present project and opened the way to several proof-of-concept applications of localized structures in semiconductor systems.

localized structures can be organized into reconfigurable matrix of bits
Among them, one can cite the coherent and incoherent commutation of pixels [barbay2006], an all-optical delay line [Pedaci2008], the realization of an optically reconfigurable matrix of bistable pixels [Pedaci2006] or even the implementation of a solitonic force microscope [Pedaci2008b]. In spite of these remarkable achievements, the localized structures considered in these cases are localized only in the transverse dimension of the system, which is in general described in a uniform-field approximation.

Along the propagation direction, dissipative solitons have also been very successfully studied, in particular in fiber systems. For instance, mode locked fiber lasers have proven to be a very fruitful system and many theoretical and experimental observations have been reported about solitons and soliton complexes (see [Grelu2008] for a review). In semiconductor systems, in spite of the maturity of mode locking techniques, most experiments have been concerned with reaching higher power or shorter pulses [Keller2006,Rafailov2007]. Indeed, more complex regimes differing from the periodic emission of one single pulse per cavity roundtrip are generally avoided and only very few works are directed towards more complex regimes [Saarinen2008]. Only very recent experiments in a Kerr fiber cavity [Leo2010] have shown the possibility to control temporal cavity solitons in almost arbitrary number along the propagation direction. It is important to remark that the dissipative solitons observed in this case are actually closer conceptually from the ones observed in [Barland2002] than from mode-locked lasers since the system described in [Leo2010] does not possess the phase symmetry of laser systems and is actually modelled by a slightly modified version of the Lugiato-Lefever equation. In [Leo2010], the authors strongly emphasize the need of a purely unidirectional cavity and model it with an instantaneous nonlinearity (as compared to the optical cavity lifetime). It is important to notice that numerical studies geared towards three-dimensional light localization phenomena have already underlined the detrimental role that can be played by slower material variables [Columbo2006]. A fascinating parallel can be done between the observations reported in [Leo2010] and the results about an optical frequency comb generation from a monolithic microresonator presented in [Del'Haye2007]. In this last case, a coherent beam is applied onto a nonlinear (Kerr) microresonator and results in the emission of a periodic train of pulses at the roundtrip frequency of the ring resonator, which is strinkingly similar to the observation of a single temporal cavity soliton. Finally, it is worth mentioning that mode-locking of semiconductor laser via optical injection has been claimed in [Kasuya1999], again calling for an interesting parallel with the experiment reported in [Leo2010].

The MOLOSSE project aims at bridging the notion of temporal cavity solitons in a nonlinear optical cavity with semiconductor laser mode-locking. We will build not only on our experience on light localization phenomena, but also on the preliminary results we have obtained on a suitably designed semiconductor ring laser.

Power emitted in clockwise and counterclockwise directions by the ring laser
Although in the longer term a monolithic ring cavity could be used provided suitable ratios for carrier and field lifetimes can be achieved, we have built a macroscopic semiconductor ring laser which as several advantages among which the possibility to control the cavity lifetime and the possibility to resolve in real time strongly multimode dynamics that we are expecting to control. We have already demonstrated that such a device can actually work as a laser and lock together [Tierno2012] the many longitudinal modes which are required for the observation of temporal cavity solitons, which are of course combs in Fourier space (see eg [Firth2010]).A very peculiar feature of this experimental system is that the ratio of field to carrier lifetime can be tuned by varying the losses of the cavity, and very importantly, we have observed that it is possible to make this system operate in a so-called "Class-A" regime, in which the material variables are actually faster than the optical one, which addresses the numerical observations made in [Columbo2006].

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