Nozaki-Bekki solitons in semiconductor lasers.

Optical frequency-comb sources, which emit perfectly periodic and coherent waveforms of light1, have recently rapidly progressed towards chip-scale integrated solutions. Among them, two classes are particularly significant-semiconductor Fabry-Perót lasers2-6 and passive ring Kerr microresonators7-9. Here we merge the two technologies in a ring semiconductor laser10,11 and demonstrate a paradigm for the formation of free-running solitons, called Nozaki-Bekki solitons. These dissipative waveforms emerge in a family of travelling localized dark pulses, known within the complex Ginzburg-Landau equation12-14. We show that Nozaki-Bekki solitons are structurally stable in a ring laser and form spontaneously with tuning of the laser bias, eliminating the need for an external optical pump. By combining conclusive experimental findings and a complementary elaborate theoretical model, we reveal the salient characteristics of these solitons and provide guidelines for their generation. Beyond the fundamental soliton circulating inside the ring laser, we demonstrate multisoliton states as well, verifying their localized nature and offering an insight into formation of soliton crystals15. Our results consolidate a monolithic electrically driven platform for direct soliton generation and open the door for a research field at the junction of laser multimode dynamics and Kerr parametric processes.

Control of single and multiple phase solitons in a ring cavity.

Phase solitons are localized structures characterized by phase jumps of 2π or multiples arising in forced ring lasers. Here, we show numerically that they can be created by superimposing to the constant driving field a suitable control beam matched in frequency with a different cavity mode for a time of the order of ten cavity round trip times. If the two beams are separated in frequency by n free spectral ranges of the cavity, a train of solitons like a perfect soliton crystal consisting of n equispaced phase solitons is generated. This may represent a simple way to produce frequency combs with flexible frequency spacing and high power per line.

Interaction of cavity solitons on an unstable background.

The interaction of two cavity solitons in a driven semiconductor laser above lasing threshold is investigated. We focus on the case in which the background field of the solitons is turbulent because the laser is below the injection locking point. We show that the solitons move spontaneously and either reach some equilibrium distance or merge. Different behaviors are found depending on how far from the injection locking point the laser is. The laser is modeled by a set of effective Maxwell-Bloch equations which include an equation for the macroscopic polarization that mimics the complex susceptibility of the semiconductor. In that way we avoid the emergence of an unphysical behavior of the background which instead appears when the polarization is adiabatically eliminated, which amounts to assuming infinite gain linewidth. The simulations are slow because the time scales of the different dynamical variables differ by four orders of magnitude. Yet, we show that the results of the complete set of equations can be accurately reproduced with a reduced set of equations where the polarization is adiabatically eliminated but a diffusion term is included in Maxwell equation, which accounts for the finiteness of the gain linewidth.

Coherent master equation for laser modelocking.

Modelocked lasers constitute the fundamental source of optically-coherent ultrashort-pulsed radiation, with huge impact in science and technology. Their modeling largely rests on the master equation (ME) approach introduced in 1975 by Hermann A. Haus. However, that description fails when the medium dynamics is fast and, ultimately, when light-matter quantum coherence is relevant. Here we set a rigorous and general ME framework, the coherent ME (CME), that overcomes both limitations. The CME predicts strong deviations from Haus ME, which we substantiate through an amplitude-modulated semiconductor laser experiment. Accounting for coherent effects, like the Risken-Nummedal-Graham-Haken multimode instability, we envisage the usefulness of the CME for describing self-modelocking and spontaneous frequency comb formation in quantum-cascade and quantum-dot lasers. Furthermore, the CME paves the way for exploiting the rich phenomenology of coherent effects in laser design, which has been hampered so far by the lack of a coherent ME formalism.

Exponentially decaying interaction potential of cavity solitons.

We analyze the interaction of two cavity solitons in an optically injected vertical cavity surface emitting laser above threshold. We show that they experience an attractive force even when their distance is much larger than their diameter, and eventually they merge. Since the merging time depends exponentially on the initial distance, we suggest that the attraction could be associated with an exponentially decaying interaction potential, similarly to what is found for hydrophobic materials. We also show that the merging time is simply related to the characteristic times of the laser, photon lifetime, and carrier lifetime.

Abnormal chiral events in a semiconductor laser with coherent injection.

We study experimentally and theoretically the dynamics of a spatially extended (along the propagation direction) oscillatory medium with coherent forcing. We observe abnormally high events, responsible for a different statistics of intensity and pulse height, in a regime where solitons and roll patterns are unstable. We focus on the formation of these high-peak events and their connection to the phase dynamics. Each abnormal event can be associated with a change in the slope of the phase time trace. Furthermore, the coexistence of ±2π phase rotations inside the cavity can be associated to the observation of abnormal events, similarly to recent predictions in bidimensional vortex turbulence.

Extreme events induced by collisions in a forced semiconductor laser.

We report on the experimental study of an optically driven multimode semiconductor laser with a 1 m cavity length. We observed a spatiotemporal regime where real-time measurements reveal very high-intensity peaks in the laser field. Such a regime, which coexists with the locked state and with stable phase solitons, is characterized by the emergence of extreme events that produce heavy tail statistics in the probability density function. We interpret the extreme events as collisions of spatiotemporal structures with opposite chirality. Numerical simulations of the semiconductor laser model, showing very similar dynamical behavior, substantiate our evidences and corroborate the description of interactions such as collisions between phase solitons and transient structures with different phase rotations.

Terahertz optically tunable dielectric metamaterials without microfabrication.

We theoretically investigate the terahertz (THz) dielectric response of a semiconductor slab hosting a tunable grating photogenerated by the interference of two tilted infrared (IR) plane waves. In the case where the grating period is much smaller than the THz wavelength, we numerically evaluate the ordinary and extraordinary component of the effective permittivity tensor by resorting to electromagnetic full-wave simulation coupled to the dynamics of charge carriers excited by IR radiation. We show that the photo-induced metamaterial optical response can be tailored by varying the grating and it ranges from birefringent to hyperbolic to anisotropic negative dielectric without resorting to microfabrication.

Controlling cavity solitons by means of photorefractive soliton electro-activation.

We consider a hybrid system consisting of a centrosymmetric photorefractive crystal in contact with a vertical-cavity surface-emitting laser. We numerically investigate the generation and control of cavity solitons (CSs) by propagating a plane wave through electro-activated solitonic waveguides in the crystal. In such a compound scheme, which couples a propagative/conservative field dynamics to a bistable/dissipative one, we show that by changing the electro-activation voltage of the crystal, the CSs can be turned on and shifted with controlled velocity across the device section, on the scale of tens of nanoseconds. The configuration can be exploited for applications to optical information encoding and processing.