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

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.

Numerical Analysis of Nonlinear Dynamics Based on Spin-VCSELs with Optical Feedback

In this paper, the nonlinear dynamics of a novel model based on optically pumped spin-polarized vertical-cavity surface-emitting lasers (spin-VCSELs) with optical feedback is investigated numerically. Due to optical feedback being the external disturbance component, the complex nonlinear dynamical behaviors can be enhanced and the regions of different nonlinear dynamics in size can be extended with appropriate parameters of spin-VCSELs. According to the equations of the modified spin-flip model (SFM), the comparison of bifurcation diagrams is first presented for the clear presentation of different routes to chaos. Meanwhile, numerous bifurcation diagrams in color are illustrated to demonstrate the rich dynamical regimes intuitively, and the crucial effects of optical feedback strength, feedback delay, linewidth enhancement factor, and spin-flip relaxation rate on the region evolvement of complex dynamics of the proposed model are revealed to investigate the dependence of dynamical behaviors on external and internal parameters when the optical feedback scheme is introduced. These parameters play a remarkable role in enhancing the mechanism of complex dynamic oscillations. Furthermore, utilizing combination with time series, power spectra, and phase portraits, the various dynamical behaviors observed in the bifurcation diagram are simulated numerically. Correspondingly, the powerful measure 0–1 test is employed to distinguish between chaos and non-chaos.

Optical feedback induced oscillation bursts in two-state quantum-dot lasers

We investigate the impact of short optical feedback on a two-state quantum dot laser. A region in the feedback parameter space is identified, where the laser emission periodically alternates between oscillation bursts from the quantum dot ground and excited state, i.e. two-color anti-phase oscillation bursts. We compare these results to the low-frequency fluctuations and regular pulse packages of single-color semiconductor lasers and show via an in-depth bifurcation analysis, that the two-color oscillation bursts originate from a torus-bifurcation of a two-state periodic orbit. A cascade of further period-doubling bifurcations produces chaotic dynamics of the burst envelope. Our findings showcase the rich dynamics and complexity, which can be generated via the interaction of electronic and photonic time scales in quantum dot lasers with optical feedback.

Manipulating coherence resonance in a quantum dot semiconductor laser via electrical pumping

Excitability and coherence resonance are studied in a semiconductor quantum dot laser under short optical self-feedback. For low pump levels, these are observed close to a homoclinic bifurcation, which is in correspondence with earlier observations in quantum well lasers. However, for high pump levels, we find excitability close to a boundary crisis of a chaotic attractor. We demonstrate that in contrast to the homoclinic bifurcation the crisis and thus the excitable regime is highly sensitive to the pump current. The excitability threshold increases with the pump current, which permits to adjust the sensitivity of the excitable unit to noise as well as to shift the optimal noise strength, at which maximum coherence is observed. The shift adds up to more than one order of magnitude, which strongly facilitates experimental realizations.

Influence of carrier lifetimes on the dynamical behavior of quantum-dot lasers subject to optical feedback

We examine changes in the dynamics of a semiconductor quantum-dot (QD) laser subject to optical feedback that correlate to changes in the QD laser band structure. By employing a microscopic model for the carrier-carrier scattering processes between the QDs and the carrier reservoir we are able to tune the carrier lifetimes in the QDs, e.g., by modifying the QD confinement energies or the pump current. By using numerical continuation methods as well as asymptotic theory we demonstrate that the feedback sensitivity crucially depends on these lifetimes through the damping of the turn-on oscillations, and small lifetimes on the order of this relaxation time scale lead to an increased feedback resistivity. Thus intelligent band structure engineering can lead to stable continuous wave operation of the laser over a large parameter range.

Delay and periodicity

Systems with time delay play an important role in modeling of many physical and biological processes. In this paper we describe generic properties of systems with time delay, which are related to the appearance and stability of periodic solutions. In particular, we show that delay systems generically have families of periodic solutions, which are reappearing for infinitely many delay times. As delay increases, the solution families overlap leading to increasing coexistence of multiple stable as well as unstable solutions. We also consider stability issue of periodic solutions with large delay by explaining asymptotic properties of the spectrum of characteristic multipliers. We show that the spectrum of multipliers can be split into two parts: pseudocontinuous and strongly unstable. The pseudocontinuous part of the spectrum mediates destabilization of periodic solutions.

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.

Frequency versus relaxation oscillations in a semiconductor laser with coherent filtered optical feedback

We investigate the dynamics of a semiconductor laser subject to coherent delayed filtered optical feedback. A systematic bifurcation analysis reveals that this system supports two fundamentally different types of oscillations, namely relaxation oscillations and external roundtrip oscillations. Both occur stably in large domains under variation of the feedback conditions, where the feedback phase is identified as a key quantity for controlling this dynamical complexity. We identify two separate parameter regions of stable roundtrip oscillations, which occur throughout in the form of pure frequency oscillations.

Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation

We study the continuous-wave (cw) operation of a semiconductor laser subject to optical feedback from a Fabry-Perot resonator in a case where the emission is resonant to a reflection minimum of the resonator. This configuration is treated in the framework of Lang-Kobayashi equations. The nature of bifurcations and the stability of steady state solutions is analyzed in terms of the dependence on magnitude and phase of the feedback. In contrast to conventional optical feedback from a single mirror, the locus of external cavity modes is not elliptic but represents a tilted eight with possible satellite bubbles. Below a critical feedback strength, which is analytically given, only one single mode exists representing the completely unchanged cw emission of the laser. In this weak-feedback regime, the feedback phase allows noninvasive control of the cw emission and a tailoring of its small-signal response within wide limits. The results obtained are a prototype for all-optical realizations of delayed feedback control.

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.

Frequency-shifted optical feedback in a pumping laser diode dynamically amplified by a microchip laser

Compared with conventional optical heterodyne detection, laser optical feedback imaging (LOFI) allows for a several orders of magnitude higher intensity modulation contrast. The maximum contrast amplification is typically 10(3) for a diode laser in the gigahertz range and 10(6) for a microchip laser in the megahertz range. To take advantage of the wavelength tunability of a laser diode and of the lower resonant detection frequency of a microchip laser, we used LOFI modulation induced by the frequency-shifted optical feedback in a laser diode as a modulated pumping power for a microchip laser for resonant dynamic amplification. In this way, we were able to transfer the optical feedback sensitivity of the laser diode to the megahertz range. Application to telemetry is also reported.

Global bifurcation investigation of an optimal velocity traffic model with driver reaction time

We investigate an optimal velocity model which includes the reflex time of drivers. After an analytical study of the stability and local bifurcations of the steady-state solution, we apply numerical continuation techniques to investigate the global behavior of the system. Specifically, we find branches of oscillating solutions connecting Hopf bifurcation points, which may be super- or subcritical, depending on parameters. This analysis reveals several regions of multistability.

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.

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.