Spin-vertical-cavity surface-emitting lasers with by-design-defined birefringence for high-speed modulation.

We propose a new, to the best of our knowledge, type of spin-vertical-cavity surface-emitting laser (VCSEL) with controlled by design birefringence. To this aim, we utilize the so-called columnar thin films (CTFs) in the VCSEL dielectric distributed Bragg mirror and/or in a second dielectric cavity. We design such CTF-VCSELs with pre-defined birefringence and calculate their polarization-resolved resonant longitudinal modes and the corresponding quantum-well confinement factors and threshold gains. Using the spin-flip VCSEL model, we show that such spin CTF-VCSELs can achieve small-signal modulation response with a 3 dB cutoff frequency of several hundreds of GHz.

Breathing of dissipative light bullets of nonlinear polarization mode in Kerr resonators.

We demonstrate the existence of breathing dissipative light bullets in a birefringent optical resonator filled with Kerr media. The propagation of light inside the cavity for each polarized component, which is coupled by cross-phase modulation, is described by the coupled Lugiato-Lefever equations. The space-time dynamics of breathing light bullets are described using Stokes parameters and frequency spectra.

Discrete light bullets in coupled optical resonators.

We consider arrays of coupled nonlinear optical cavities subject to coherent optical injection. These devices are described by the discrete generalized Lugiato-Lefever equation. We predict that stable three-dimensional localized structures, often called discrete light bullets, and clusters of them may form in the output of the coupled optical resonators. We consider both anomalous and normal dispersion and show that it results in the generation of, respectively, bright and dark discrete light bullets.

Dissipative Light Bullets in Kerr Cavities: Multistability, Clustering, and Rogue Waves.

We report the existence of stable dissipative light bullets in Kerr cavities. These three-dimensional (3D) localized structures consist of either an isolated light bullet (LB), bound together, or could occur in clusters forming well-defined 3D patterns. They can be seen as stationary states in the reference frame moving with the group velocity of light within the cavity. The number of LBs and their distribution in 3D settings are determined by the initial conditions, while their maximum peak power remains constant for a fixed value of the system parameters. Their bifurcation diagram allows us to explain this phenomenon as a manifestation of homoclinic snaking for dissipative light bullets. However, when the strength of the injected beam is increased, LBs lose their stability and the cavity field exhibits giant, short-living 3D pulses. The statistical characterization of pulse amplitude reveals a long tail probability distribution, indicating the occurrence of extreme events, often called rogue waves.

Stationary localized structures and the effect of the delayed feedback in the Brusselator model.

The Brusselator reaction-diffusion model is a paradigm for the understanding of dissipative structures in systems out of equilibrium. In the first part of this paper, we investigate the formation of stationary localized structures in the Brusselator model. By using numerical continuation methods in two spatial dimensions, we establish a bifurcation diagram showing the emergence of localized spots. We characterize the transition from a single spot to an extended pattern in the form of squares. In the second part, we incorporate delayed feedback control and show that delayed feedback can induce a spontaneous motion of both localized and periodic dissipative structures. We characterize this motion by estimating the threshold and the velocity of the moving dissipative structures.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.

Strain induced polarization chaos in a solitary VCSEL.

Physical curiosity at the beginning, optical chaos is now attracting increasing interest in various technological areas such as detection and ranging or secure communications, to name but a few. However, the complexity of optical chaos generators still significantly hinders their development. In this context, the generation of chaotic polarization fluctuations in a single laser diode has proven to be a significant step forward, despite being observed solely for quantum-dot vertical-cavity surface-emitting lasers (VCSELs). Here, we demonstrate experimentally that a similar polarization dynamics can be consistently obtained in quantum-well VCSELs. Indeed, by introducing anisotropic strain in the laser cavity, we successfully triggered the desired chaotic dynamics. The simplicity of the proposed approach, based on low-cost and easily available components including off-the-shelf VCSELs, paves the way to the wide spread use of solitary VCSELs for chaos-based applications.

Cavity solitons in vertical-cavity surface-emitting lasers.

We investigate a control of the motion of localized structures (LSs) of light by means of delay feedback in the transverse section of a broad area nonlinear optical system. The delayed feedback is found to induce a spontaneous motion of a solitary LS that is stationary and stable in the absence of feedback. We focus our analysis on an experimentally relevant system, namely the vertical-cavity surface-emitting laser (VCSEL). We first present an experimental demonstration of the appearance of LSs in a 80 µm aperture VCSEL. Then, we theoretically investigate the self-mobility properties of the LSs in the presence of a time-delayed optical feedback and analyse the effect of the feedback phase and the carrier lifetime on the delay-induced spontaneous drift instability of these structures. We show that these two parameters affect strongly the space-time dynamics of two-dimensional LSs. We derive an analytical formula for the threshold associated with drift instability of LSs and a normal form equation describing the slow time evolution of the speed of the moving structure.

Localized structures in dissipative media: from optics to plant ecology.

Localized structures (LSs) in dissipative media appear in various fields of natural science such as biology, chemistry, plant ecology, optics and laser physics. The proposal for this Theme Issue was to gather specialists from various fields of nonlinear science towards a cross-fertilization among active areas of research. This is a cross-disciplinary area of research dominated by nonlinear optics due to potential applications for all-optical control of light, optical storage and information processing. This Theme Issue contains contributions from 18 active groups involved in the LS field and have all made significant contributions in recent years.

Physical random bit generation from chaotic solitary laser diode.

We demonstrate the physical generation of random bits at high bit rates (> 100 Gb/s) using optical chaos from a solitary laser diode and therefore without the complex addition of either external optical feedback or injection. This striking result is obtained despite the low dimension and relatively small bandwidth of the laser chaos, i.e. two characteristics that have been so far considered as limiting the performances of optical chaos-based applications. We unambiguously attribute the successful randomness at high speed to the physics of the laser chaotic polarization dynamics and the resulting growth rate of the dynamical entropy.

Bistability of time-periodic polarization dynamics in a free-running VCSEL.

We report experimentally a bistability between two limit cycles (i.e. time-periodic dynamics) in a free-running vertical-cavity surface-emitting laser. The two limit cycles originate from a bifurcation on two elliptically polarized states which exhibit a small frequency difference and whose main axes are symmetrical with respect to the linear polarization eigenaxes at threshold. We demonstrate theoretically that this peculiar behavior can be explained in the framework of the spin-flip model model by taking into account a small misalignment between the phase and amplitude anisotropies.

Mapping of two-polarization-mode dynamics in vertical-cavity surface-emitting lasers with optical injection.

We report theoretically on the interplay between polarization switching and bifurcations to nonlinear dynamics in a vertical-cavity surface-emitting laser (VCSEL) subject to orthogonal optical injection. Qualitatively different bifurcation scenarios leading to polarization switching are found and mapped out in the plane of the injection parameters, i.e., the frequency detuning vs injection strength plane. A Hopf bifurcation mechanism on the two-polarization-mode solution determines the injection-locking boundaries and influences polarization switching induced by optical injection. We furthermore report on a torus bifurcation emerging from a two-linearly polarized (LP) mode time-periodic dynamics before polarization switching and injection locking appear. It corresponds to an interesting combination of relaxation oscillation dynamics in the x -LP mode together with wave mixing dynamics in the injected y -LP mode. In agreement with recent experiments, we unveil a period-doubling route to chaos that involves both VCSEL orthogonal LP modes. The corresponding region of chaotic dynamics coincides with abrupt changes in the polarization switching boundaries in the plane of the injection parameters.

Nonlinear dynamics of the polarization of multitransverse mode vertical-cavity surface-emitting lasers under current modulation.

In this paper we report on a theoretical investigation of the nonlinear dynamics of the polarization of multitransverse mode vertical-cavity surface-emitting lasers (VCSELs) under current modulation. Special attention is given to the comparison with a previously studied case of single-transverse mode VCSEL emitting in two orthogonal polarizations. The consideration of spatial effects in VCSEL modifies the polarization dynamics that accompanies the period doubling route to chaos for large modulation amplitudes. Depending on the modulation parameters, the excitation of a higher order transverse mode may either induce chaotic pulsing in an otherwise regularly pulsating VCSEL, or induce a time-periodic pulsing dynamics in an otherwise chaotic VCSEL. Bifurcation diagrams obtained for different modulation frequencies, several values of the dichroism, and different transverse mode characteristics allow us to identify the different scenarios of polarization dynamics in a directly modulated VCSEL. Temporal analysis of carrier number radial profile reveals considerable changes for the multitransverse mode case only constituting the physical origin of the reported changes in the temporal and polarization dynamics.

Polarization synchronization in unidirectionally coupled vertical-cavity surface-emitting lasers with orthogonal optical injection.

We analyze theoretically the polarization dynamics in unidirectionally coupled vertical-cavity surface-emitting lasers (VCSELs). The master VCSEL is subject to an isotropic optical feedback. The slave VCSEL is subject to an orthogonal optical injection from the master VCSEL, i.e., only the linearly polarized mode orthogonal to the dominant linearly polarized mode of the free-running slave VCSEL is injected into the slave VCSEL. This laser configuration may lead the slave VCSEL polarization to switch to that of the injected master laser field. The injected power required for polarization switching depends on the frequency detuning. We identify in the plane of the injection parameters two regions of enhanced synchronization between the injected LP mode and the corresponding slave LP mode. In the so-called region II the slave VCSEL exhibits anticorrelated dynamics in its two LP modes while in the so-called region I the slave VCSEL exhibits dynamics in only one LP mode, which corresponds to the polarization of the injected field. The two regions exhibit different synchronization properties in both the LP mode dynamics and total intensity dynamics. We furthermore analyze the dependency of the synchronization quality on the parameter mismatch between master and slave VCSELs and on the polarization switching properties of each VCSEL.

Two-mode injection locking in vertical-cavity surface-emitting lasers.

We report on the dynamics that accompany polarization switching induced by orthogonal optical injection in a vertical-cavity surface-emitting laser (VCSEL). As the injection strength increases, the VCSEL bifurcates to injection-locked steady states and time-periodic and possibly chaotic dynamics. Of particular interest is a two-mode injection-locking solution, i.e., locking of the two VCSEL polarization modes. A detailed stability analysis unveils new bistability mechanisms in optical injection problems.

Optical feedback induces polarization mode hopping in vertical-cavity surface-emitting lasers.

Vertical-cavity surface-emitting lasers subjected to weak polarization-insensitive optical feedback are studied experimentally and theoretically. We find that the feedback induces random anticorrelated hopping between the two orthogonal linearly polarized modes. This polarization mode hopping is accompanied by rapid anticorrelated oscillations in the linearly polarized intensities at the external-cavity frequency. The study of a simple stochastic delay differential equation suggests that these oscillations generated by the delay are typical of any hopping phenomenon between states.

Nonlinear polarization dynamics in directly modulated vertical-cavity surface-emitting lasers.

A current-modulated vertical-cavity surface-emitting laser is shown to exhibit interesting nonlinear dynamics in its two orthogonal linearly polarized (LP), fundamental transverse modes. The intensities of the two LP modes may exhibit in-phase time-periodic dynamics or chaotic regimes with combination of in-phase and antiphase dynamics at two different time scales. Chaotic dynamics are found in a large range of laser and modulation parameters.

Photorefractive beam-fanning effect and self-pulsations in coated LiNbO3 slabs.

We investigate the dynamical behavior of optical Fabry-Perot resonators consisting of LiNbO3 slabs (x and c cut) that are coated with different (absorbing or lossless) dielectric multilayers deposited on both sides of the slabs. Bistable switching is observed experimentally. The buildup of beam fanning with time leads to destructive interference for a portion of the incident beam, inducing a change in absorption and heating, hence to switching off. As a result, self-pulsations appear whose frequency depends strongly on the input light intensity, the spot size, and the focusing. Switching and self-pulsations are not observed in the case of lossless coatings or for bare LiNbO3 slabs, although strong photorefractive beam fanning is still present. We also study the influence of the incident-beam characteristics (width and focusing) on the beam-fanning process and the pulsating behavior.