-An overview of chaos in laser diodes is provided which surveys experimental achievements in the area and explains the theory behind the phenomenon. The fundamental physics underpinning this behaviour and also the opportunities for harnessing laser diode chaos for potential applications are discussed. The availability and ease of operation of laser diodes, in a wide range of configurations, make them a convenient test-bed for exploring basic aspects of nonlinear and chaotic dynamics. It also makes them attractive for practical tasks, such as chaos-based secure communications and random number generation. Avenues for future research and development of chaotic laser diodes are also identified.The emergence of irregular pulsations and dynamical instabilities from a laser were first noted in the very early stages of the laser development. Pulses whose amplitude "vary in an erratic manner" were reported in the output of the ruby solid-state laser 1 [ Fig. 1 a] and then found in numerical simulations 2 . However the lack of knowledge of what would later be termed "chaos" resulted in these initial observations being either left unexplained or wrongly attributed to noise.The situation changed in the late 1960s with the discovery of sensitivity to initial conditions by Lorenz 3 , later popularized as the "butterfly effect". As illustrated in Fig. 1 (b), numerical simulations of a deterministic model of only three nonlinear equations showed an irregular pulsing with a remarkable feature: the state variables evolve along very different trajectories despite starting from approximately the same initial values [ Fig. 1 b]. The distance δ(t) between nearby trajectories diverges exponentially : δ(t) = δ(0) exp(λt) provided that λ, the effective Lyapunov exponent of the dynamical system, is positive. Consequently such systems are unpredictable in the long term. Plotted in the x-y-z phase space of the state variables, the trajectories converge to an "attractor" which has the geometric property of being bounded in space despite the exponential divergence of nearby trajectories [ Fig. 1 c]. Such attractors are found to have a fractional dimension 4 and are thus termed strange. Aperiodicity, sensitive dependency to initial conditions and strangeness are commonly considered as the main properties for "chaos" The fields of laser physics and chaos theory developed independently until 1975 8 when Haken discovered a striking analogy between the Lorenz equations that model fluid convection and the MaxwellBloch equations modelling light-matter interaction in single-mode lasers. The nonlinear interaction between the wave propagation in the laser cavity (represented by the electric field E) and radiative recombination producing macroscopic polarization (encapsulated in the polarization P and carrier inversion N) yield similar dynamical instabilities to those found in the Lorenz equations. More specifically, in addition to the conventional laser threshold, Haken suggested a second threshold would exist above which "spiking occurs ...

Fifty years after the invention of the laser diode and fourty years after the report of the butterfly effect -i.e. the unpredictability of deterministic chaos, it is said that a laser diode behaves like a damped nonlinear oscillator. Hence no chaos can be generated unless with additional forcing or parameter modulation. Here we report the first counter-example of a free-running laser diode generating chaos. The underlying physics is a nonlinear coupling between two elliptically polarized modes in a vertical-cavity surface-emitting laser. We identify chaos in experimental time-series and show theoretically the bifurcations leading to single-and double-scroll attractors with characteristics similar to Lorenz chaos. The reported polarization chaos resembles at first sight a noise-driven mode hopping but shows opposite statistical properties. Our findings open up new research areas that combine the high speed performances of microcavity lasers with controllable and integrated sources of optical chaos.The discovery of deterministic chaos -i.e. the aperiodic deterministic dynamics of a nonlinear system showing sensitivity to initial conditions -has been a major paradigm shift overthrowing two centuries of Laplacian viewpoint of dynamical systems [1][2][3][4][5] . Looking into a system behavior with the use of chaos theory has helped to interpret and control many of such ordered or disordered behaviors in our present day life, such as the bifurcations leading to epilepsy and cancer 6 , the stabilization of cardiac arrhythmias 7 , and the improvement of complex behavioral patterns in robotics 8 .Soon after the invention of the laser, the possibility to observe light chaos raised attention. In 1975, Haken discovered an analogy between the Maxwell-Bloch equations for lasers and the Lorenz equations showing chaos 9 . The Maxwell-Bloch equations are three equations for the field E, the polarization P and the carrier inversion N, each with its own relaxation time. However, while in Lorenz equations the relaxation times of the dynamical variables are of similar order of magnitude, they may take very different values in lasers. If one variable relaxes much faster than the others, this variable is adiabatically eliminated, hence resulting in a reduced number of dynamical equations. Therefore, so-called class A (ex: He-Ne, Ar and Dye), class B (ex: Nd:YAG, CO2 and semiconductor) or class C (ex: NH3) lasers have dynamics governed either by a single equation for the field, two equations for the field and population inversion or the full set of equations, respectively. In class A or class B laser systems chaos cannot be observed unless one adds one or several independent control parameters 10 . Chaos has then been reported in, for example, free-running NH3 lasers 11 , He-Ne lasers with modulation of the external field 12 , CO2

We investigate theoretically the possibility of retrieving the value of the time delay of a semiconductor laser with an external optical feedback from the analysis of its intensity time series. When the feedback rate is moderate and the injection current set such that the laser relaxation-oscillation period is close to the delay, then the time-delay identification becomes extremely difficult, thus improving the security of chaos-based communications using external-cavity lasers.

International audienceWe perform a theoretical investigation of the polarization dynamics in a vertical-cavity surface-emitting laser (VCSEL) subject to orthogonal optical injection, i.e., the injected field has a linear polarization (LP) orthogonal to that of the free-running VCSEL. In agreement with previous experiments [Z. G. Pan et al., Appl. Phys. Lett. 63, 2999 (1993)], an increase of the injection strength may lead to a polarization switching accompanied by an injection locking. We find that this route to polarization switching is typically accompanied by a cascade of bifurcations to wave-mixing dynamics and time-periodic and possibly chaotic regimes. A detailed mapping of the polarization dynamics in the plane of the injection parameters (detuning, injection strength) unveils a large richness of dynamical scenarios. Of particular interest is the existence of another injection-locked solution for which the two LP modes both lock to the master laser frequency, i.e., an elliptically polarized injection-locked (EPIL) steady state. Modern continuation techniques allow us to unveil an unfolding mechanism of the EPIL solution as the detuning varies and also to link the existence of the EPIL solution to a resonance condition between the master laser frequency and the free-running frequency of the normally depressed LP mode in the slave laser. We furthermore report an additional case of bistability, in which the EPIL solution may coexist with the second injection-locked solution (the one being locked to the master polarization). This case of bistability is a result of the interaction between optical injection and the two-polarization-mode characteristics of VCSEL devices

International audienceA critical issue in optical chaos-based communications is the possibility to identify the parameters of the chaotic emitter and, hence, to break the security. In this paper, we study theoretically the identification of a chaotic emitter that consists of a semiconductor laser with an optical feedback. The identification of a critical security parameter, the external-cavity round-trip time (the time delay in the laser dynamics), is performed using both the auto-correlation function and delayed mutual information methods applied to the chaotic time-series. The influence on the time-delay identification of the experimentally tunable parameters, i.e., the feedback rate, the pumping current, and the time-delay value, is carefully studied. We show that difficult time-delay-identification scenarios strongly depend on the time-scales of the system dynamics as it undergoes a route to chaos, in particular on how close the relaxation oscillation period is from the external-cavity round-trip time

The onset of nonlinear dynamics and chaos is evidenced in a mid-infrared distributed feedback quantum cascade laser both in the temporal and frequency domains. As opposed to the commonly observed route to chaos in semiconductor lasers, which involves undamping of the laser relaxation oscillations, quantum cascade lasers first exhibit regular self-pulsation at the external cavity frequency before entering into a chaotic low-frequency fluctuation regime. The bifurcation sequence, similar to that already observed in class A gas lasers under optical feedback, results from the fast carrier relaxation dynamics occurring in quantum cascade lasers, as confirmed by numerical simulations. Such chaotic behavior can impact various practical applications including spectroscopy, which requires stable single-mode operation. It also allows the development of novel mid-infrared high-power chaotic light sources, thus enabling secure free-space high bit-rate optical communications based on chaos synchronization.

In this paper, we report on theoretical and experimental investigation on polarization and transverse mode behavior of vertical-cavity surface-emitting lasers (VCSELs) under orthogonal optical injection as a function of the injection strength and of the detuning between the injection frequency and the free-running frequency of the solitary laser. As the injection strength increases the VCSEL switches to the master laser polarization. We find that the injection power necessary to obtain such polarization switching is minimum at two different values of the frequency detuning: the first one corresponds to the frequency splitting between the two linearly polarized fundamental transverse modes, and the second one appears at a larger positive frequency detuning, close to the frequency difference between the first-order and the fundamental transverse modes of the solitary VCSEL. We show theoretically that both the depth and the frequency corresponding to the second minimum increase when the relative losses between the two transverse modes decrease. Bistability of the polarization switching is obtained for the whole frequency detuning range. Such a bistability is found for the fundamental mode only or for both transverse modes, depending on the value of the detuning. The theoretical and experimental optical spectra are in good agreement showing that the first-order transverse mode appears locked to the external injection.

International audienceIn this contribution we provide an in depth theoretical analysis of the bifurcations leading to nonlinear polarization dynamics in a free-running vertical-cavity surface-emitting laser (VCSEL). We detail the sequence of bifurcations that occurs when increasing the injection current, and which brings the laser from linear to elliptical polarization emission and then self-pulsating or even more complex chaotic dynamics of the light intensity. Continuation techniques allow us to follow the stable and unstable limit cycle solutions emerging from Hopf bifurcations, and therefore to interpret the frequency of the self-pulsating polarization dynamics. The fundamental frequency of the pulsating dynamics is either close to the laser relaxation oscillation frequency or close to the linear-birefringence-induced polarization mode frequency splitting, depending on the laser parameters. A systematic analysis of the parameter space allows us to identify two scenarios that are in excellent qualitative agreement with those reported in recent experiments. Our results provide, moreover, evidence for an interesting polarization mode hopping mechanism, i.e., a so-called deterministic mode hopping where the laser exhibits a chaotic and therefore random-like hopping between two states that are elliptically polarized and nonorthogonal