Bridges of periodic solutions and tori in semiconductor lasers subject to delay

Subharmonic locking and frequency combs in frequency-swept semiconductor lasers

We analyze a delay differential equation model for a swept semiconductor laser and demonstrate existence of various periodic solutions that are subharmonically locked to the sweep rate. These solutions provide optical frequency combs in spectral domain. We investigate the problem numerically and show that, due to the translational symmetry of the model, there exists a hysteresis loop formed by branches of steady states solutions, bridges of periodic solutions connecting stable and unstable steady state branches, and isolated branches of limit cycles. We discuss the role of bifurcation points and limit cycles embedded into the loop in the formation of the subharmonic dynamics.

Routes to Chaos of a Semiconductor Laser Subjected to External Optical Feedback: A Review

This paper reviews experimental investigations of the route to chaos of a semiconductor laser subjected to optical feedback from a distant reflector. When the laser is biased close to threshold, as the feedback strength is increased, an alternation between stable continuous wave (CW) behavior and irregular, chaotic fluctuations, involving numerous external-cavity modes, is observed. CW operation occurs on an external-cavity mode whose optical frequency is significantly lower than that of the solitary laser. The scenario is significantly different for larger currents as the feedback level is increased. At low feedback, the laser displays periodic or quasiperiodic behavior, mostly around external-cavity modes whose frequency is slightly larger than that of the solitary laser. As the feedback level increases, the RF and optical frequencies involved progressively lock until complete locking is achieved in a mixed external-cavity mode state. In this regime, the optical intensity and voltage oscillate at a frequency that is also equal to the optical frequency spacing between the modes participating in the dynamics. For even higher feedback, the locking cannot be maintained and the laser displays fully developed coherence collapse.

Stability of steady and periodic states through the bifurcation bridge mechanism in semiconductor ring lasers subject to optical feedback

With the development of new applications using semiconductor ring lasers (SRLs) subject to optical feedback, the stability properties of their outputs becomes a crucial issue. We propose a systematic bifurcation analysis in order to properly identify the best parameter ranges for either steady or self-pulsating periodic regimes. Unlike conventional semiconductor lasers, we show that SRLs exhibit both types of outputs for large and well defined ranges of the feedback strength. We determine the stability domains in terms of the pump parameter and the feedback phase. We find that the feedback phase is a key parameter to achieve a stable steady output. We demonstrate that the self-pulsating regime results from a particular Hopf bifurcation mechanism referred to as bifurcation bridges. These bridges connect two distinct external cavity modes and are fully stable, a scenario that was not possible for diode lasers under the same conditions.

Small chimera states without multistability in a globally delay-coupled network of four lasers

We present results obtained for a network of four delay-coupled lasers modeled by Lang-Kobayashi-type equations. We find small chimera states consisting of a pair of synchronized lasers and two unsynchronized lasers. One class of these small chimera states can be understood as intermediate steps on the route from synchronization to desynchronization, and we present the entire chain of bifurcations giving birth to them. This class of small chimeras can exhibit limit-cycle or quasiperiodic dynamics. A second type of small chimera states exists apparently disconnected from any region of synchronization, arising from pair synchronization inside the chaotic desynchronized regime. In contrast to previously reported chimera states in globally coupled networks, we find that the small chimera state is the only stable solution of the system for certain parameter regions; i.e., we do not need to specially prepare initial conditions.

Cycles of self-pulsations in a photonic integrated circuit

We report experimentally on the bifurcation cascade leading to the appearance of self-pulsation in a photonic integrated circuit in which a laser diode is subjected to delayed optical feedback. We study the evolution of the self-pulsing frequency with the increase of both the feedback strength and the injection current. Experimental observations show good qualitative accordance with numerical results carried out with the Lang-Kobayashi rate equation model. We explain the mechanism underlying the self-pulsations by a phenomenon of beating between successive pairs of external cavity modes and antimodes.

Tristability of a semiconductor laser due to time-delayed optical feedback

We present an experimental and theoretical study of multistability of a single-mode laser subject to feedback through phase tuning and amplifier sections integrated on the same chip. Closely above threshold, a regime of tristability of continuous-wave (CW) states is found for multiple ranges of amplifier and phase currents. The separation between the tristable wavelengths agrees with the channel spacing of dense wavelength multiplexing in the C band of optical communication making the device interesting for ternary logic applications. Complementary theoretical investigations in the framework of the paradigmatic Lang-Kobayashi model provide a consistent understanding of the experimental findings and additionally yield an analytic formula expressing the maximum number of coexisting stable CW states by the linewidth-enhancement factor alpha . Tristability belongs to the alpha range from 5 to 8 in good agreement with experiment.

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.

Passive mode locking of lasers by crossed-polarization gain modulation

We report on a novel approach for inducing passive mode locking of lasers without using any saturable absorber but exploiting the polarization degree of freedom of light. In our scheme, passive mode locking is achieved by crossed-polarization gain modulation caused by the reinjection of a polarization-rotated replica of the laser output after a time delay. The reinjection time delay defines resonance tongues that correspond to mode-locking operation. Numerical continuation reveals that the cw solution is destabilized through a Hopf bifurcation that defines the onset of multimode operation which evolves sharply into a mode-locked solution. Our approach can be applied to a large variety of laser systems. For vertical-cavity surface-emitting lasers, we demonstrate stable mode-locked pulses at repetition rates in the GHz range and pulse widths of few tens of picoseconds.

Criterion to identify Hopf bifurcations in maps of arbitrary dimension

The classical Hopf bifurcation criterion is stated in terms of the properties of eigenvalues. In this paper, a criterion without using eigenvalues is proposed for maps of arbitrary dimension. The parameter mechanism of Hopf bifurcation may be explicitly formulated on the basis of the criterion. A numerical example demonstrates that the proposed criterion is preferable to the classical Hopf bifurcation criterion in theoretical analysis and practical applications.

Dynamics of semiconductor lasers with bidirectional optoelectronic coupling: stability, route to chaos, and entrainment

The dynamical behavior of two mutually coupled semiconductor lasers is studied. An optoelectronic coupling including a time delay in the propagation of the signals between the two lasers is considered. Starting from the appropriate rate equations for the photon and carrier densities, we investigate the stability of the fixed points and limit cycles of the system as a function of the coupling strength and the propagation time. From this analysis, a quasiperiodic route to chaos with boundary crisis events is identified as the responsible mechanism leading the system from regular to complex behavior. Several interesting phenomena are predicted for this system. Our analytical and numerical results are supported by experiments which are in good agreement with our predictions.

Bifurcation study of regular pulse packages in laser diodes subject to optical feedback

We study the influence of delayed optical feedback from a short external cavity on the emission dynamics of semiconductor lasers using the Lang and Kobayashi rate equation model. A combination of numerical integration and continuation techniques allows us to bring new light into the bifurcation scenario leading to the regular pulse packages (RPP) regime. We give examples of bistability between RPP and time-periodic or steady state solutions. Our bifurcation study of RPP regime is complemented by an analysis of the dependency of the RPP period on the laser and feedback parameters. We qualitatively study this new dynamical regime by plotting a two-dimensional map in the feedback parameters space. The occurrence of RPP is furthermore associated with a topological change in the bifurcation diagram and accompanied by the creation of a new type of bifurcation bridge between a mode and an antimode.

Hopf bifurcation cascade in small-alpha laser diodes subject to optical feedback

We analyze theoretically the dynamics of a semiconductor laser subject to optical feedback, on the basis of the well-known Lang-Kobayashi equations. Previous investigations on this laser system suggest that a small linewidth enhancement factor (alpha factor) stabilizes the laser dynamics. By contrast, we unveil here optical feedback induced instabilities which are present for a small value of alpha but which disappear when alpha increases above alpha approximately 1. By combining numerical simulations and modern continuation methods for delay-differential equations, we unveil cascades of subcritical and supercritical Hopf bifurcations on the first external-cavity mode (ECM). We unveil for the first time, to our knowledge, the occurrence of subcritical Hopf bifurcation points for intermediate values of the EC length, i.e. close to the boundary between the short and the long EC regimes. They lead to severe laser instabilities such as large intensity and possibly chaotic pulsations. Moreover, these Hopf bifurcation cascades for small values of alpha are shown to be responsible for different bifurcation scenarios leading to restabilization of the first ECM and to ECM bistability.

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.

Self-organization in semiconductor lasers with ultrashort optical feedback

The dynamical behavior of a single-mode laser subject to optical feedback is investigated in the limit, when the delay time is much shorter than the period of the relaxation oscillations. Use of an integrated distributed feedback device allows us to control the feedback phase. We observe two kinds of Hopf bifurcations associated with regular self-pulsations of different frequencies.

External cavity modes of semiconductor lasers with phase-conjugate feedback

External cavity modes (ECMs) of a semiconductor laser with phase-conjugate feedback are defined as time-periodic pulsating intensity solutions exhibiting a frequency close to an integer multiple of the external cavity frequency. As the feedback rate progressively increases from zero, they sequentially appear as stable attractors in the bifurcation diagram. We construct a simple analytical approximation of these pulsating intensity solutions and determine their frequencies. We show that branches of ECMs are isolated. Finally, the validity of our approximation is tested by comparing numerical bifurcation diagrams obtained by simulation and continuation techniques with our analytical results.

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.

Stability and rupture of bifurcation bridges in semiconductor lasers subject to optical feedback

The bifurcation diagram of a single-mode semiconductor laser subject to a delayed optical feedback is examined by using numerical continuation methods. For this, we show how to cope with the special symmetry properties of the equations. As the feedback strength is increased, branches of modes and antimodes appear, and we have found that pairs of modes and antimodes are connected by closed branches of periodic solutions (bifurcation bridges). Such connections seem generically present as new pairs of modes and antimodes appear. We subsequently investigate the behavior of the first connection as a function of the linewidth enhancement factor and the feedback phase. Our results extend and confirm existing results and hypotheses reported in the literature. For large values of the linewidth enhancement factor (alpha=5-6), bridges break through homoclinic orbits. Changing the feedback phase unfolds the bifurcation diagram of the modes and antimodes, allowing different types of connections between modes.

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.