Stability analysis of a quantum cascade laser subject to phase-conjugate feedback.

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.

Sustained oscillations accompanying polarization switching in laser dynamics.

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.

Mapping of external cavity modes for a laser diode subject to phase-conjugate feedback.

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.

Coexistence of cavity solitons with different polarization states and different power peaks in all-fiber resonators.

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.

Bistability Controlled by Convection in a Pattern-Forming System.

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.

High-order external cavity modes and restabilization of a laser diode subject to a phase-conjugate feedback.

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.

Two distinct bifurcation routes for delayed optoelectronic oscillators.

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.

Short-time-delay limit of the self-coupled FitzHugh-Nagumo system.

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.

Laser diode nonlinear dynamics from a filtered phase-conjugate optical feedback.

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.

Multirhythmicity in an optoelectronic oscillator with large delay.

An optoelectronic oscillator exhibiting a large delay in its feedback loop is studied both experimentally and theoretically. We show that multiple square-wave oscillations may coexist for the same values of the parameters (multirhythmicity). Depending on the sign of the phase shift, these regimes admit either periods close to an integer fraction of the delay or periods close to an odd integer fraction of twice the delay. These periodic solutions emerge from successive Hopf bifurcation points and stabilize at a finite amplitude following a scenario similar to Eckhaus instability in spatially extended systems. We find quantitative agreements between experiments and numerical simulations. The linear stability of the square waves is substantiated analytically by determining the stable fixed points of a map.

Analytical and experimental study of two delay-coupled excitable units.

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.

Relaxation and square-wave oscillations in a semiconductor laser with polarization rotated optical feedback.

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.

Slow-fast dynamics of a time-delayed electro-optic oscillator.

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 they may coexist with stable low-frequency periodic oscillations for the same value of the control parameter.

Strongly asymmetric square waves in a time-delayed system.

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.

Crenelated fast oscillatory outputs of a two-delay electro-optic oscillator.

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.