Universal generation of devil's staircases near Hopf bifurcations via modulated forcing of nonlinear systems.

The discrete circle map is the archetypical example of a driven periodic system, showing a complex resonance structure under a change of the forcing frequency known as the devil's staircase. Adler's equation can be seen as the direct continuous equivalent of the circle map, describing locking effects in periodic systems with continuous forcing. This type of locking produces a single fundamental resonance tongue without higher-order resonances, and a devil's staircase is not observed. We show that, with harmonically modulated forcing, nonlinear oscillations close to a Hopf bifurcation generically reproduce the devil's staircase even in the continuous case. Experimental results on a semiconductor laser driven by a modulated optical signal show excellent agreement with our theoretical predictions. The locking appears as a modulation of the oscillation amplitude as well as the angular oscillation frequency. Our results show that by proper implementation of an external drive, additional regions of stable frequency locking can be introduced in systems which originally show only a single Adler-type resonance tongue. The induced resonances can be precisely controlled via the modulation parameters.

Dynamics of on-chip asymmetrically coupled semiconductor lasers.

We investigate the dynamics of asymmetrically coupled semiconductor lasers on photonic integrated circuits in experiment and theory. The experimental observations are explained using a rate-equation model for coupled lasers incorporating a saturable coupling waveguide. We perform a bifurcation analysis of the coupled laser dynamics, focusing on the effects of the coupling phase and the dynamical difference between passive and saturable coupling waveguides. For a passive waveguide, we find a bifurcation scenario closely resembling the well-known optical injection setup, which is largely insensitive to the coupling phase. When the coupling waveguide is saturable, the dynamics become increasingly complex and unpredictable, with a strong phase-dependence. Our results show the possibility of a simple layout for reproducible laser dynamics on a chip.

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.

Harmonic frequency locking and tuning of comb frequency spacing through optical injection.

We show, both experimentally and theoretically, that a slave laser injected with an optical frequency comb can undergo two distinct locking mechanisms, both of which decrease the output optical comb's frequency spacing. We report that, for certain detuning and relative injection strengths, slave laser relaxation oscillations can become undamped and lock to rational frequencies of the optical comb spacing, creating extra comb tones by nonlinear dynamics of the injected laser. We also study the frequency locking of the slave laser at detunings in between the injected comb lines, which add the slave laser's frequency to the comb. Our results demonstrate the effect of the α parameter and stability of the locked states and indicate how the frequency of the relaxation oscillations affect both of these locking mechanisms. These optical locking mechanisms can be applied to regenerate or multiply optical combs.

Stochastic polarization switching induced by optical injection in bimodal quantum-dot micropillar lasers.

Mutual coupling and injection locking of semiconductor lasers is of great interest in non-linear dynamics and its applications for instance in secure data communication and photonic reservoir computing. Despite its importance, it has hardly been studied in microlasers operating at µW light levels. In this context, vertically emitting quantum dot micropillar lasers are of high interest. Usually, their light emission is bimodal, and the gain competition of the associated linearly polarized fundamental emission modes results in complex switching dynamics. We report on selective optical injection into either one of the two fundamental mode components of a bimodal micropillar laser. Both modes can lock to the master laser and influence the non-injected mode by reducing the available gain. We demonstrate that the switching dynamics can be tailored externally via optical injection in very good agreement with our theory based on semi-classical rate equations.

Class-C semiconductor lasers with time-delayed optical feedback.

We perform a linear stability analysis and numerical bifurcation diagrams of a class-C laser with time-delayed optical feedback. We employ a rate equation system based on the Maxwell-Bloch equations, and study the influence of the dephasing time on the laser dynamics. We find a stabilizing effect of an intermediate dephasing time, i.e. when moving from a class-B to a class-C laser. At long dephasing times, a destabilization of the laser solution occurs by a feedback-induced unlocking of Rabi oscillations at the second laser threshold. We predict an optimum resistance to time-delayed optical feedback for dephasing times close to the photon cavity lifetime. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Mutual coupling and synchronization of optically coupled quantum-dot micropillar lasers at ultra-low light levels.

Synchronization of coupled oscillators at the transition between classical physics and quantum physics has become an emerging research topic at the crossroads of nonlinear dynamics and nanophotonics. We study this unexplored field by using quantum dot microlasers as optical oscillators. Operating in the regime of cavity quantum electrodynamics (cQED) with an intracavity photon number on the order of 10 and output powers in the 100 nW range, these devices have high ß-factors associated with enhanced spontaneous emission noise. We identify synchronization of mutually coupled microlasers via frequency locking associated with a sub-gigahertz locking range. A theoretical analysis of the coupling behavior reveals striking differences from optical synchronization in the classical domain with negligible spontaneous emission noise. Beyond that, additional self-feedback leads to zero-lag synchronization of coupled microlasers at ultra-low light levels. Our work has high potential to pave the way for future experiments in the quantum regime of synchronization.

Ultra-Short Pulse Generation in a Three Section Tapered Passively Mode-Locked Quantum-Dot Semiconductor Laser.

We experimentally and theoretically investigate the pulsed emission dynamics of a three section tapered semiconductor quantum dot laser. The laser output is characterized in terms of peak power, pulse width, timing jitter and amplitude stability and a range of outstanding pulse performance is found. A cascade of dynamic operating regimes is identified and comprehensively investigated. We propose a microscopically motivated traveling-wave model, which optimizes the computation time and naturally allows insights into the internal carrier dynamics. The model excellently reproduces the measured results and is further used to study the pulse-generation mechanism as well as the influence of the geometric design on the pulsed emission. We identify a pulse shortening mechanism responsible for the device performance, that is unique to the device geometry and configuration. The results may serve as future guidelines for the design of monolithic high-power passively mode-locked quantum dot semiconductor lasers.

Tailoring the mode-switching dynamics in quantum-dot micropillar lasers via time-delayed optical feedback.

Microlasers are ideal candidates to bring the fascinating variety of nonlinear complex dynamics found in delay-coupled systems to the realm of quantum optics. Particularly attractive is the possibility of tailoring the devices' emission properties via non-invasive delayed optical coupling. However, until now scarce research has been done in this direction. Here, we experimentally and theoretically investigate the effects of delayed optical feedback on the mode-switching dynamics of an electrically driven bimodal quantum-dot micropillar laser, characterizing its impact on the micropillar's output power, optical spectrum and photon statistics. Feedback is found to influence the switching dynamics and its characteristics time scales. In addition, stochastic switching is reduced with the subsequent impact on the microlaser photon statistics. Our results contribute to the comprehension of feedback-induced phenomena in micropillar lasers and pave the way towards the external control and tailoring of the properties of these key systems for the nanophotonics community.

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.

Amplitude-phase coupling and chirp in quantum-dot lasers: influence of charge carrier scattering dynamics.

We investigate the dependence of the amplitude-phase coupling in quantum-dot (QD) lasers on the charge-carrier scattering timescales. The carrier scattering processes influence the relaxation oscillation parameters, as well as the frequency chirp, which are both important parameters when determining the modulation performance of the laser device and its reaction to optical perturbations. We find that the FM/AM response exhibits a strong dependence on the modulation frequency, which leads to a modified optical response of QD lasers when compared to conventional laser devices. Furthermore, the frequency response curve changes with the scattering time scales, which can allow for an optimization of the laser stability towards optical perturbations.

Quantum coherence induces pulse shape modification in a semiconductor optical amplifier at room temperature.

Coherence in light-matter interaction is a necessary ingredient if light is used to control the quantum state of a material system. Coherent effects are firmly associated with isolated systems kept at low temperature. The exceedingly fast dephasing in condensed matter environments, in particular at elevated temperatures, may well erase all coherent information in the material at timescales shorter than a laser excitation pulse. Here we show for an ensemble of semiconductor quantum dots that even in the presence of ultrafast dephasing, for suitably designed condensed matter systems quantum-coherent effects are robust enough to be observable at room temperature. Our conclusions are based on an analysis of the reshaping an ultrafast laser pulse undergoes on propagation through a semiconductor quantum dot amplifier. We show that this pulse modification contains the signature of coherent light-matter interaction and can be controlled by adjusting the population of the quantum dots via electrical injection.

Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers.

We show that the long-established concept of a linewidth-enhancement factor α to describe carrier-induced refractive index changes in semiconductor lasers breaks down in quantum-dot (QD) lasers when describing complex dynamic scenarios, found, for example, under high-excitation or optical injection. By comparing laser simulations using a constant α factor with results from a more complex nonequilibrium model that separately treats gain and refractive index dynamics, we examine the conditions under which an approximation of the amplitude-phase coupling by an α factor becomes invalid. The investigations show that while a quasiequilibrium approach for conventional quantum well lasers is valid over a reasonable parameter range, allowing one to introduce an α factor as a constant parameter, the concept is in general not applicable to predict QD laser dynamics due to the different time scales of the involved scattering processes.