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Light polarization is an inherent property of the coherent laser output that finds applications, for example, in vision, imaging, spectroscopy, cosmology, and communications. We report here on light polarization dynamics that repeatedly switches between a stationary state of polarization and an irregularly pulsating polarization. The reported dynamics is found to result from the onset of chimeras. Chimeras in nonlinear science refer to the counterintuitive coexistence of coherent and incoherent dynamics in an initially homogeneous network of coupled nonlinear oscillators. The existence of chimera states has been evidenced only recently in carefully designed experiments using either mechanical, optomechanical, electrical, or optical oscillators. Interestingly, the chimeras reported here originate from the inherent coherent properties of a commercial laser diode. The spatial and temporal properties of the chimeras found in light polarization are controlled by the laser diode and feedback parameters, leading, e.g., to multistability between chimeras with multiple heads and to turbulent chimeras.

International audience; We report on a bifurcation scenario from which emerges oscillations at a frequency higher than the relaxationoscillation frequency in an edge-emitting laser diode subjected to polarized optical feedback; these oscillations appear in the square-wave regime. The study has been performed experimentally and numerically based on Lang-Kobayashi model. We unveil bifurcation diagrams showing a clear bifurcation point that marks the transition between sustained and damped oscillations on the plateaus of square-waves. We extend this first bifurcation study by providing additional analytical insight to characterize the main parametric dependence of the frequency of these undamped oscillations. © 2018 SPIE.

International audience; We investigate the stability boundaries of a quantum cascade laser subject to phase-conjugate optical feedback. From a three-level model, we reduce our set of equations to the usual modified Lang-Kobayashi equations describing a semiconductor laser subject to phase-conjugate feedback. We then determine the Hopf bifurcation conditions, which we explore by using asymptotic methods. In the limit of large delays, we find approximations of the first Hopf bifurcation that is responsible for the destabilization of the system. We obtain an expression that depends only on three parameters: the feedback strength, the line-width enhancement factor, and the pump current. From this expression, we study the stability boundaries of our system. We compare our results with the initial three-level model using a continuation method. We find qualitative and quantitative agreements of the stability boundaries with the two methods. Finally, we compare our findings with the ones obtained for a quantum cascade laser subject to conventional optical feedback.

International audience; We report experimentally and theoretically the emergence of sustained oscillations over a slow and periodic polarization switching in a laser subjected to polarization rotated optical feedback. This phenomenon originates from a clear bifurcation point that marks the transition between sustained and damped oscillations on the plateaus. Analytical study reveals also that the frequency of this new oscillatory dynamics is independent of the time delay.

We numerically investigate the dynamics of a semiconductor laser subject to phase-conjugate optical feedback. We explore the effects of the laser model and feedback parameters for the generation of time-periodic oscillations of the output power at harmonics of the external cavity frequency, i.e., dynamical solutions that have been named external cavity modes. We point out that both the experimentally tunable and other parameters have an influence on the frequency of such dynamics. Since the delay has to exist, it is not the relevant parameter as we show that the feedback rate fixes the frequency of the periodic self-pulsations. The interaction length of the crystal and the ratio between carrier and photon lifetimes tend to filter out high frequencies as they increase. Finally, the linewidth enhancement factor unlocks high frequencies as it increases. We conclude by providing a situation which leads to periodic solutions with higher frequencies using a set of realistic values of parameters.

We investigate the coexistence of low- and high-frequency oscillations in a delayed optoelectronic oscillator. We identify two nearby Hopf bifurcation points exhibiting low and high frequencies and demonstrate analytically how they lead to stable solutions. We then show numerically that these two branches of solutions undergo higher order instabilities as the feedback rate is increased but remain separated in the bifurcation diagram. The two bifurcation routes can be followed independently by either progressively increasing or decreasing the bifurcation parameter., SCOPUS: ar.j, info:eu-repo/semantics/published

We study numerically and experimentally the transition from convective to absolute dynamical instabilities in an optical system composed of a bulk photorefractive crystal subjected to a single optical feedback. We demonstrate that the convective regime is directly related to the bistability area in which the homogeneous steady state coexists with a Turing pattern solution. Outside this domain, the system exhibits either a homogeneous steady state or an absolute dynamical regime. Moreover, an external background illumination applied onto the nonlinear medium is used as an external parameter for controlling the size of the bistability area. We question the role of this parameter and show how the background illumination makes the bistability area even larger.

International audience; We experimentally report the sequence of bifurcations destabilizing and restabilizing a laser diode with phase-conjugate feedback when the feedback rate is increased. Specifically, we successively observe the initial steady state, undamped relaxation oscillations, quasi-periodicity, chaos, and oscillating solutions at harmonics up to 13 times the external cavity frequency but also the restabilization to a steady state. The experimental results are qualitatively well reproduced by a model that accounts for the time the light takes to penetrate the phase-conjugate mirror. The theory points out that the system restabilizes through a Hopf bifurcation whose frequency is a harmonic of the external cavity frequency.

We theoretically investigate a weakly birefringent all-fiber cavity subject to linearly polarized optical injection. We describe the propagation of light inside the cavity using, for each linear polarization component of the electric field, the Lugiato-Lefever model. These two components are coupled by cross-phase modulation. We show that, for a wide range of parameters, there is a coexistence between a homogeneous steady state and two different types of temporal vector cavity solitons, which can be hosted in the same system. They differ by their polarization state and peak intensity. We construct their bifurcation diagram and show that they are connected through a saddle-node bifurcation. Finally, we show that vector cavity solitons exhibit multistability involving different polarization states with different energies.

We report on a bifurcation mechanism following which an external-cavity laser diode emits regular oscillating output power at a high frequency. This frequency does not vary with the external-cavity length and it can be adjusted by varying the feedback strength. We observe this phenomenon numerically by investigating the external-cavity modes generated by a semiconductor laser subject to a phase-conjugate optical feedback. Particularly, we explore the effects of both the feedback rate and the time delay induced by the feedback on the frequency of the external-cavity modes. Counterintuitively, we evidence that having a short cavity does not necessarily yield oscillations at higher frequencies. We show that the key parameter in order to generate high-frequency solutions is the feedback rate. This parameter fixes the frequency of the solutions obtained independently of the time delay. We finally relate our observations to Hopf bifurcation phenomena.

We analyze the transition from convective to absolute dynamical instabilities in a nonlinear optical system forming patterns, i.e., a photorefractive crystal in a single feedback configuration. We demonstrate that the convective regime is directly related to the bistability area in which the homogeneous steady state coexists with a pattern solution. Outside this domain, the system exhibits either a homogeneous steady state or an absolute dynamical regime. We evidence that the bistability area can be greatly increased by adjusting the mirror tilt angle and/or by applying an external background illumination on the photorefractive crystal.

International audience; We analyze numerically and experimentally the pattern formation process in an optical system composed of a bulk photorefractive crystal subjected to a single optical feedback. In this configuration, the system admits an homogeneous solution for low coupling strength. Increasing the coupling strength leads to a sub-critical bifurcation which leads to a pattern state. Such a bifurcation gives access to a well-defined hysteresis. In this paper we demonstrate that the size of the bistable area can be adjusted by different system parameters such as the intensity of the input beam, the power of an external background illumination and more interestingly by the feedback mirror tilt angle.

International audience; We study both analytically and numerically the rate equations of a laser diode subject to a filtered phase-conjugate optical feedback (FPCF). We formulate dimensionless equations for the FPCF and determine the Hopf bifurcation conditions. The coalescence of different Hopf bifurcations as the filter width decreases suggests the disappearance of the external cavity modes for a narrow width. We confirm our analytical predictions with direct numerical simulations of the FPCF equations.

We analyze the FitzHugh-Nagumo equations subject to time-delayed self-feedback in the activator variable. Parameters are chosen such that the steady state is stable independent of the feedback gain and delay τ. We demonstrate that stable large-amplitude τ-periodic oscillations can, however, coexist with a stable steady state even for small delays, which is mathematically counterintuitive. In order to explore how these solutions appear in the bifurcation diagram, we propose three different strategies. We first analyze the emergence of periodic solutions from Hopf bifurcation points for τ small and show that a subcritical Hopf bifurcation allows the coexistence of stable τ-periodic and stable steady-state solutions. Second, we construct a τ-periodic solution by using singular perturbation techniques appropriate for slow-fast systems. The theory assumes τ=O(1) and its validity as τ→0 is investigated numerically by integrating the original equations. Third, we develop an asymptotic theory where the delay is scaled with respect to the fast timescale of the activator variable. The theory is applied to the FitzHugh-Nagumo equations with threshold nonlinearity, and we show that the branch of τ-periodic solutions emerges from a limit point of limit cycles., SCOPUS: ar.j, info:eu-repo/semantics/published

A time-delayed FitzHugh-Nagumo (FHN) system exhibiting a threshold nonlinearity is studied both experimentally and theoretically. The basic steady state is stable but distinct stable oscillatory regimes may coexist for the same values of parameters (multirhythmicity). They are characterized by periods close to an integer fraction of the delay. From an asymptotic analysis of the FHN equations, we show that the mechanism leading to those oscillations corresponds to a limit-point of limit-cycles. In order to investigate their robustness with respect to noise, we study experimentally an electrical circuit that is modeled mathematically by the same delay differential equations. We obtain quantitative agreements between numerical and experimental bifurcation diagrams for the different coexisting time-periodic regimes.

The rate equations for a laser diode subject to a filtered phase-conjugate optical feedback are studied both analytically and numerically. We determine the Hopf bifurcation conditions, which we explore by using asymptotic methods. Numerical simulations of the laser rate equations indicate that different pulsating intensity regimes observed for a wide filter progressively disappear as the filter width increases. We explain this phenomenon by studying the coalescence of Hopf bifurcation points as the filter width increases. Specifically, we observe a restabilization of the steady-state solution for moderate width of the filter. Above a critical width, an isolated bubble of time-periodic intensity solutions bounded by two successive Hopf bifurcation points appears in the bifurcation diagram. In the limit of a narrow filter, we then demonstrate that only two Hopf bifurcations from a stable steady state are possible. These two Hopf bifurcations are the Hopf bifurcations of a laser subject to an injected signal and for zero detuning., SCOPUS: ar.j, info:eu-repo/semantics/published

The rate equations for a laser with a polarization rotated optical feedback are investigated both numerically and analytically. The frequency detuning between the polarization modes is now taken into account and we review all earlier studies in order to motivate the range of values of the fixed parameters. We find that two basic Hopf bifurcations leading to either stable sustained relaxation or square-wave oscillations appear in the detuning versus feedback rate diagram. We also identify two key parameters describing the differences between the total gains of the two polarization modes and discuss their effects on the periodic square-waves.

We investigate the onset of time-periodic oscillations for a system of two identical delay-coupled excitable (nonoscillatory) units. We first analyze these solutions by using asymptotic methods. The oscillations are described as relaxation oscillations exhibiting successive slow and fast changes. The analysis highlights the determinant role of the delay during the fast transition layers. We then study experimentally a system of two coupled electronic circuits that is modeled mathematically by the same delay differential equations. We obtain quantitative agreements between analytical and experimental bifurcation diagrams.

Square-wave oscillations exhibiting different plateau lengths have been observed experimentally by investigating an electro-optic oscillator. In a previous study, we analysed the model delay differential equations and determined an asymptotic approximation of the two plateaus. In this paper, we concentrate on the fast transition layers between plateaus and show how they contribute to the total period. We also investigate the bifurcation diagram of all possible stable solutions. We show that the square waves emerge from the first Hopf bifurcation of the basic steady state and that theymay coexist with stable lowfrequency periodic oscillations for the same value of the control parameter. © 2013 The Author(s) Published by the Royal Society. All rights reserved., SCOPUS: ar.j, info:eu-repo/semantics/published

Time-delayed systems are known to exhibit symmetric square waves oscillating with a period close to twice the delay. Here, we show that strongly asymmetric square waves of a period close to one delay are possible. The plateau lengths can be tuned by changing a control parameter. The problem is investigated experimentally and numerically using a simple bandpass optoelectronic delay oscillator modeled by nonlinear delay integrodifferential equations. An asymptotic approximation of the square-wave periodic solution valid in the large delay limit allows an analytical description of its main properties (extrema and square pulse durations). A detailed numerical study of the bifurcation diagram indicates that the asymmetric square waves emerge from a Hopf bifurcation. © 2012 American Physical Society., SCOPUS: ar.j, info:eu-repo/semantics/published

An electro-optic oscillator subject to two distinct delayed feedbacks has been designed to develop pronounced broadband chaotic output. Its route to chaos starts with regular pulsating gigahertz oscillations that we investigate both experimentally and theoretically. Of particular physical interest are the transitions to various crenelated fast time-periodic oscillations, prior to the onset of chaotic regimes. The two-delay problem is described mathematically by two coupled delay-differential equations, which we analyze by using multiple-time-scale methods. We show that the interplay of a large delay and a relatively small delay is responsible for the onset of fast oscillations modulated by a slowly varying square-wave envelope. As the bifurcation parameter progressively increases, this envelope undergoes a sequence of bifurcations that corresponds to successive fixed points of a sine map. © 2012 American Physical Society., SCOPUS: ar.j, info:eu-repo/semantics/published

An unusual bifurcation sequence exhibited in a recently proposed nonlinear delay electro-optic phase dynamics setup, is explored both experimentally and analytically. Strong agreements indicate that well-controlled dynamical complexity can be usefull in many emerging applications.

A fundamental property of nonlinear dynamical systems controlled by a delayed feedback is their tendency to exhibit 2τ -periodic square-wave oscillations of equal plateau lengths (τ is the delay) [1]. The question was recently raised whether an optical system could exhibit stable square-wave oscillations with different plateau lengths [2]. We have experimentally found these regimes using a single optoelectronic oscillator (OEO) with a bandpass feedback [3]. See Fig. 1. The period is close to τ (and not 2τ). The response of the OEO is accurately described by the following delay differential equations (time s = t/τ ) [4]