Short delay limit of the delayed Duffing oscillator.

The delayed Duffing equation, x^{″}+ɛx^{'}+x+x^{3}+cx(t-τ)=0, admits a Hopf bifurcation which becomes singular in the limit ɛ→0 and τ=O(ɛ)→0. To resolve this singularity, we develop an asymptotic theory where x(t-τ) is Taylor expanded in powers of τ. We derive a minimal system of ordinary differential equations that captures the Hopf bifurcation branch of the original delay differential equation. An unexpected result of our analysis is the necessity of expanding x(t-τ) up to third order rather than first order. Our work is motivated by laser stability problems exhibiting the same bifurcation problem as the delayed Duffing oscillator [Kovalev et al., Phys. Rev. E 103, 042206 (2021)2470-004510.1103/PhysRevE.103.042206]. Here we substantiate our theory based on the short delay limit by showing the overlap (matching) between our solution and two different asymptotic solutions derived for arbitrary fixed delays.

Optical information processing using dual state quantum dot lasers: complexity through simplicity.

We review results on the optical injection of dual state InAs quantum dot-based semiconductor lasers. The two states in question are the so-called ground state and first excited state of the laser. This ability to lase from two different energy states is unique amongst semiconductor lasers and in combination with the high, intrinsic relaxation oscillation damping of the material and the novel, inherent cascade like carrier relaxation process, endows optically injected dual state quantum dot lasers with many unique dynamical properties. Particular attention is paid to fast state switching, antiphase excitability, novel information processing techniques and optothermally induced neuronal phenomena. We compare and contrast some of the physical properties of the system with other optically injected two state devices such as vertical cavity surface emitting lasers and ring lasers. Finally, we offer an outlook on the use of quantum dot material in photonic integrated circuits.

Short-delayed-feedback semiconductor lasers.

We consider the laser rate equations describing the evolution of a semiconductor laser subject to an optoelectronic feedback. We concentrate on the first Hopf bifurcation induced by a short delay and develop an asymptotic theory where the delayed variable is Taylor expanded. We determine a nearly vertical branch of strongly nonlinear oscillations and derive ordinary differential equations that capture the bifurcation properties of the original delay differential equations. An unexpected result is the need for Taylor expanding the delayed variable up to third order rather than first order. We discuss recent laser experiments where sustained oscillations have been clearly observed with a short-delayed feedback.

Turbulent coherent structures in a long cavity semiconductor laser near the lasing threshold: publisher's note.

This publisher's note contains corrections to Opt. Lett.45, 4903 (2020)OPLEDP0146-959210.1364/OL.397840.

Turbulent coherent structures in a long cavity semiconductor laser near the lasing threshold.

We report on the formation of novel turbulent coherent structures in a long cavity semiconductor laser near the lasing threshold. Experimentally, the laser emits a series of power dropouts within a roundtrip, and the number of dropouts per series depends on a set of parameters including the bias current. At fixed parameters, the drops remain dynamically stable, repeating over many roundtrips. By reconstructing the laser electric field in the case where the laser emits one dropout per roundtrip and simulating its dynamics using a time-delayed model, we discuss the reasons for long-term sustainability of these solutions. We suggest that the observed dropouts are closely related to the coherent structures of the cubic complex Ginzburg-Landau equation.

Pulse dynamics in SESAM-free electrically pumped VECSEL.

Self-starting pulsed operation in an electrically pumped (EP) vertical-external-cavity surface-emitting-laser (VECSEL) without intracavity saturable absorber is demonstrated. A linear hemispherical cavity design, consisting of the EP-VECSEL chip and a 10% output-coupler, is used to obtain picosecond output pulses with energies of 2.8 pJ and pulse widths of 130 ps at a repetition rate of 1.97 GHz. A complete experimental analysis of the generated output pulse train and of the transition from continuous-wave to pulsed operation is presented. Numerical simulations based on a delay-differential-equation (DDE) model of mode-locked semiconductor lasers are used to reproduce the pulse dynamics and identify different laser operation regimes. From this, the measured single pulse operation is attributed to FM-type mode-locking. The pulse formation is explained by strong amplitude-phase coupling and spectral filtering inside the EP-VECSEL.

Highly reconfigurable hybrid laser based on an integrated nonlinear waveguide.

The ability of laser systems to emit different adjustable temporal pulse profiles and patterns is desirable for a broad range of applications. While passive mode-locking techniques have been widely employed for the realization of ultrafast laser pulses with mainly Gaussian or hyperbolic secant temporal profiles, the generation of versatile pulse shapes in a controllable way and from a single laser system remains a challenge. Here we show that a nonlinear amplifying loop mirror (NALM) laser with a bandwidth-limiting filter (in a nearly dispersion-free arrangement) and a short integrated nonlinear waveguide enables the realization and distinct control of multiple mode-locked pulsing regimes (e.g., Gaussian pulses, square waves, fast sinusoidal-like oscillations) with repetition rates that are variable from the fundamental (7.63 MHz) through its 205th harmonic (1.56 GHz). These dynamics are described by a newly developed and compact theoretical model, which well agrees with our experimental results. It attributes the control of emission regimes to the change of the NALM response function that is achieved by the adjustable interplay between the NALM amplification and the nonlinearity. In contrast to previous square wave emissions, we experimentally observed that an Ikeda instability was responsible for square wave generation. The presented approach enables laser systems that can be universally applied to various applications, e.g., spectroscopy, ultrafast signal processing and generation of non-classical light states.

Resonances between fundamental frequencies for lasers with large delayed feedbacks.

High-order frequency locking phenomena were recently observed using semiconductor lasers subject to large delayed feedbacks. Specifically, the relaxation oscillation (RO) frequency and a harmonic of the feedback-loop round-trip frequency coincided with the ratios 1:5 to 1:11. By analyzing the rate equations for the dynamical degrees of freedom in a laser subject to a delayed optoelectronic feedback, we show that the onset of a two-frequency train of pulses occurs through two successive bifurcations. While the first bifurcation is a primary Hopf bifurcation to the ROs, a secondary Hopf bifurcation leads to a two-frequency regime where a low frequency, proportional to the inverse of the delay, is resonant with the RO frequency. We derive an amplitude equation, valid near the first Hopf bifurcation point, and numerically observe the frequency locking. We mathematically explain this phenomenon by formulating a closed system of ordinary differential equations from our amplitude equation. Our findings motivate experiments with particular attention to the first two bifurcations. We observe experimentally (1) the frequency locking phenomenon as we pass the secondary bifurcation point and (2) the nearly constant slow period as the two-frequency oscillations grow in amplitude. Our results analytically confirm previous observations of frequency locking phenomena for lasers subject to a delayed optical feedback.

Convective Nozaki-Bekki holes in a long cavity OCT laser.

We show, both experimentally and theoretically, that the loss of coherence of a long cavity optical coherence tomography (OCT) laser can be described as a transition from laminar to turbulent flows. We demonstrate that in this strongly dissipative system, the transition happens either via an absolute or a convective instability depending on the laser parameters. In the latter case, the transition occurs via formation of localised structures in the laminar regime, which trigger the formation of growing and drifting puffs of turbulence. Experimentally, we demonstrate that these turbulent bursts are seeded by appearance of Nozaki-Bekki holes, characterised by the zero field amplitude and π phase jumps. Our experimental results are supported with numerical simulations based on the delay differential equations model.

Customizing supercontinuum generation via on-chip adaptive temporal pulse-splitting.

Modern optical systems increasingly rely on complex physical processes that require accessible control to meet target performance characteristics. In particular, advanced light sources, sought for, for example, imaging and metrology, are based on nonlinear optical dynamics whose output properties must often finely match application requirements. However, in these systems, the availability of control parameters (e.g., the optical field shape, as well as propagation medium properties) and the means to adjust them in a versatile manner are usually limited. Moreover, numerically finding the optimal parameter set for such complex dynamics is typically computationally intractable. Here, we use an actively controlled photonic chip to prepare and manipulate patterns of femtosecond optical pulses that give access to an enhanced parameter space in the framework of supercontinuum generation. Taking advantage of machine learning concepts, we exploit this tunable access and experimentally demonstrate the customization of nonlinear interactions for tailoring supercontinuum properties.

Wavelength stability in a hybrid photonic crystal laser through controlled nonlinear absorptive heating in the reflector.

The need for miniaturized, fully integrated semiconductor lasers has stimulated significant research efforts into realizing unconventional configurations that can meet the performance requirements of a large spectrum of applications, ranging from communication systems to sensing. We demonstrate a hybrid, silicon  photonics-compatible photonic crystal (PhC) laser architecture that can be used to implement cost-effective, high-capacity light sources, with high side-mode suppression ratio and milliwatt output output powers. The emitted wavelength is set and controlled by a silicon PhC cavity-based reflective filter with the gain provided by a III-V-based reflective semiconductor optical amplifier (RSOA). The high power density in the laser cavity results in a significant enhancement of the nonlinear absorption in silicon in the high Q-factor PhC resonator. The heat generated in this manner creates a tuning effect in the wavelength-selective element, which can be used to offset external temperature fluctuations without the use of active cooling. Our approach is fully compatible with existing fabrication and integration technologies, providing a practical route to integrated lasing in wavelength-sensitive schemes.

Class-A mode-locked lasers: Fundamental solutions.

We consider a delay differential equation (DDE) model for mode-locked operation in class-A semiconductor lasers containing both gain and absorber sections. The material processes are adiabatically eliminated as these are considered fast in comparison to the delay time for a long cavity device. We determine the steady states and analyze their bifurcations using DDE-BIFTOOL [Engelborghs et al., ACM Trans. Math. Software 28, 1 (2002)]. Multiple forms of coexistence, transformation, and hysteretic behavior of stable steady states and fundamental periodic regimes are discussed in bifurcation diagrams.

Two-color bursting oscillations.

Neurons communicate by brief bursts of spikes separated by silent phases and information may be encoded into the burst duration or through the structure of the interspike intervals. Inspired by the importance of bursting activities in neuronal computation, we have investigated the bursting oscillations of an optically injected quantum dot laser. We find experimentally that the laser periodically switches between two distinct operating states with distinct optical frequencies exhibiting either fast oscillatory or nearly steady state evolutions (two-color bursting oscillations). The conditions for their emergence and their control are analyzed by systematic simulations of the laser rate equations. By projecting the bursting solution onto the bifurcation diagram of a fast subsystem, we show how a specific hysteresis phenomenon explains the transitions between active and silent phases. Since size-controlled bursts can contain more information content than single spikes our results open the way to new forms of neuron inspired optical communication.

Coherence and incoherence in an optical comb.

We demonstrate a coexistence of coherent and incoherent modes in the optical comb generated by a passively mode-locked quantum dot laser. This is experimentally achieved by means of optical linewidth, radio frequency spectrum, and optical spectrum measurements and confirmed numerically by a delay-differential equation model showing excellent agreement with the experiment. We interpret the state as a chimera state.

Bistable regimes in an optically injected mode-locked laser.

We study experimentally the dynamics of quantum-dot (QD) passively mode-locked semiconductor lasers under external optical injection. The lasers demonstrated multiple dynamical states, with bifurcation boundaries that depended upon the sign of detuning variation. The area of the hysteresis loops grew monotonically at small powers of optical injection and saturated at moderate powers. At high injection levels the hysteresis decreased and eventually disappeared.

Low-frequency fluctuations in two-state quantum dot lasers.

We study the feedback-induced instabilities in a quantum dot semiconductor laser emitting in both ground and excited states. Without optical feedback the device exhibits dynamics corresponding to antiphase fluctuations between ground and excited states, while the total output power remains constant. The introduction of feedback leads to power dropouts in the ground state and intensity bursts in the excited state, resulting in a practically constant total output power.

Synchronization and clustering in a multimode quantum dot laser.

We analyze experimentally the intensity oscillations of the longitudinal modes of quantum dot semiconductor lasers. We show that the modal intensities can oscillate chaotically with different average frequencies, but obey a highly organized antiphase dynamics leading to a constant total output power. The fluctuations are in the MHz range. We report the first experimental observation of frequency clustering associated with synchronization. We also observe the propagation of perturbations across the optical spectrum from blue to red.

Feedback induced instabilities in a quantum dot semiconductor laser.

We analyse the properties of GaAs based quantum dot semiconductor lasers emitting near 1310 nm. The line-width enhancement factor is shown to depend strongly on device temperature, ranging from 1.5 at 20 degrees C to 5 at 50 degrees C. With optical feedback from a distant reflector, devices remained stable at 20 degrees C but displayed a range of instabilities at 50 degrees C, including irregular power drop--outs and periodic pulsations, before entering a chaotic regime. Such dynamical features are unique to quantum dot lasers -- quantum well lasers are significantly more unstable under optical feedback making such a clear route to chaos difficult to observe.