Interacting pairs of periodic solutions lead to tori in lasers subject to delayed feedback

Electro-optic delay oscillator with nonlocal nonlinearity: Optical phase dynamics, chaos, and synchronization

We demonstrate experimentally how nonlinear optical phase dynamics can be generated with an electro-optic delay oscillator. The presented architecture consists of a linear phase modulator, followed by a delay line, and a differential phase-shift keying demodulator (DPSK-d). The latter represents the nonlinear element of the oscillator effecting a nonlinear transformation. This nonlinearity is considered as nonlocal in time since it is ruled by an intrinsic differential delay, which is significantly greater than the typical phase variations. To study the effect of this specific nonlinearity, we characterize the dynamics in terms of the dependence of the relevant feedback gain parameter. Our results reveal the occurrence of regular GHz oscillations (approximately half of the DPSK-d free spectral range), as well as a pronounced broadband phase-chaotic dynamics. Beyond this, the observed dynamical phenomena offer potential for applications in the field of microwave photonics and, in particular, for the realization of novel chaos communication systems. High quality and broadband phase-chaos synchronization is also reported with an emitter-receiver pair of the setup.

Stability near threshold in a semiconductor laser subject to optical feedback: a bifurcation analysis of the Lang-Kobayashi equations

Through the use of analytical and numerical techniques, we investigate the interaction between the trivial off-state and the continuous-wave (CW) operation of a semiconductor laser subject to conventional optical feedback. More specifically, using numerical continuation tools, the stability and bifurcations of the CW states, or external-cavity modes (ECMs), are analyzed in dependence on the parameters of feedback phase, feedback strength, pump current, and the linewidth enhancement factor. In this way, curves of codimension-one Hopf bifurcations are shown to destabilize the off-state and lead to stable ECM operation. Moreover, self-intersections of these Hopf curves in codimension-two Hopf-Hopf bifurcation points are seen to give rise to curves of codimension-one torus bifurcations (Hopf bifurcations of the ECMs), and degenerate-Hopf points to the birth of saddle-node bifurcations of the ECMs, as parameters are varied. These codimension-two points are shown to come together at a codimension-three degenerate Hopf-Hopf point (a Bogdanov-Takens bifurcation of the ECMs): a limiting point for which a stable off-state can exist.

Numerical stability analysis of a large-scale delay system modeling a lateral semiconductor laser subject to optical feedback

This paper highlights the use of advanced numerical tools to study the stability of large-scale systems of delay differential equations (DDEs). Specifically, we consider a model describing a semiconductor laser subject to conventional optical feedback and lateral carrier diffusion. The symmetry of the governing rate equations allows external cavity mode solutions (ECMs) to be computed as steady state solutions. Using the software package DDE-BIFTOOL, branches of ECMs are computed as a function of varying feedback strength. The stability along these branches is computed by solving eigenvalue problems, the size of which is governed by a step-length heuristic. In this paper, we employ an improved heuristic which substantially reduces the size of these eigenvalue problems. This approach makes the stability analysis of large-scale systems of DDEs computationally feasible.

Delay dynamics of semiconductor lasers with short external cavities: bifurcation scenarios and mechanisms

We present a comprehensive study of the emission dynamics of semiconductor lasers induced by delayed optical feedback from a short external cavity. Our analysis includes experiments, numerical modeling, and bifurcation analysis by means of computing unstable manifolds. This provides a unique overview and a detailed insight into the dynamics of this technologically important system and into the mechanisms leading to delayed feedback instabilities. By varying the external cavity phase, we find a cyclic scenario leading from stable intensity emission via periodic behavior to regular and irregular pulse packages, and finally back to stable emission. We reveal the underlying interplay of localized dynamics and global bifurcations.

Bifurcation diagram of a complex delay-differential equation with cubic nonlinearity

We reduce the Lang-Kobayashi equations for a semiconductor laser with external optical feedback to a single complex delay-differential equation in the long delay-time limit. The reduced equation has a time-delayed linear term and a cubic instantaneous nonlinearity. There are only two parameters, the real linewidth enhancement factor and the complex feedback strength. The equation displays a very rich dynamics and can sustain steady, periodic, quasiperiodic, and chaotic regimes. We study the steady solutions analytically and analyze the periodic solutions by using a numerical continuation method. This leads to a bifurcation diagram of the steady and periodic solutions, stable and unstable. We illustrate the chaotic regimes by a direct numerical integration and show that low frequency fluctuations still occur.

Dynamical properties of lasers coupled face to face

We derive a reduced model to describe two identical lasers coupled face to face. Two limits are introduced in the Maxwell-Bloch equations: adiabatic elimination of the material polarization and large distance between the two lasers. The resulting model describes coupled homogeneously broadened lasers, including semiconductor lasers. It consists of two coupled delay differential equations with delayed linear cross-coupling and an instantaneous self-coupling nonlinearity. The study is analytical and numerical. We focus on the properties of steady and periodic amplitudes of the electric fields. In steady state, there are symmetric, antisymmetric, and asymmetric solutions with respect to a permutation of the two fields. A similar classification holds for the periodic states. The stability of these solutions is determined partly analytically and partly numerically. A homoclinic point is associated with the asymmetric periodic solutions.

Low-frequency pulsations in class-B solid-state lasers with delayed feedback

We report on the numerical observation of short-duration low-frequency pulsations in class-B solid-state lasers subject to delayed feedback. The period of these pulsations is much larger than either the delay induced by the feedback or the relaxation oscillation period of the laser without feedback. A link is established between the low-frequency pulsations and the low-frequency fluctuations observed in semiconductor lasers subject to coherent optical feedback.

Global bifurcations and bistability at the locking boundaries of a semiconductor laser with phase-conjugate feedback

We investigate dynamics and bifurcations of a single-mode semiconductor laser subject to phase-conjugate feedback near the locking region. The system is described by rate equations which are a three-dimensional system with a delay. With tools that go much beyond mere simulation, we find and follow steady states regardless of their stability and compute unstable manifolds of saddle points. Furthermore, we identify heteroclinic bifurcations, which turn out to be responsible for bistability and excitability at the locking boundaries.