Lendert Gelens

Experimental and numerical study of square wave oscillations due to asymmetric optical feedback in semiconductor ring lasers

We study experimentally and numerically a new dynamical regime in the operation of semiconductor ring lasers (SRLs) subject to delayed optical feedback. When employing an asymmetric feedback scheme, we find experimentally that the SRL can show square-wave intensity oscillations with a 50 % duty cycle. In this scheme, where the output in one direction is delay-coupled to the other direction but not vice versa, the laser switches regularly between the clockwise (CW) and counter-clockwise (CCW) propagating modes. The measured period of the square-waves is slightly longer than twice the roundtrip time in the external cavity. We analyze the regularity and the shape of the square-waves as a function of the pumping current and the feedback strength. For higher pump currents on the SRL,the output displays stochastic mode hopping between the square waves attractor and stable unidirectional operation in the CW mode. To understand the origin of this dynamical regime, we rely on numerical simulations based on the Lang-Kobayashi equations. We demonstrate a novel mechanism leading to square wave oscillations based on the cross-feedback overcoming backscattering asymmetries present in the devices structure. Our numerical results are in close agreement with the experimental ones.

Semiconductor ring lasers as optical neurons

Semiconductor Ring Lasers (SRLs) are a modern class of semiconductor lasers whose active cavity is characterized by a circular geometry. This enables the laser to support two counterpropagating modes, referred to as the clockwise (CW) and the counterclockwise (CCW) mode. Semiconductor ring lasers have been shown to have a regime of operation in which they are excitable, when the linear coupling between the counterpropagating modes is asymmetric. This can be achieved by increasing the reflection of, for example, the CW mode into the CCW mode. This will stabilize lasing in the CCW mode. In the excitable regime, the SRL will fire optical pulses (spikes) in the CW mode as a response to noise perturbations. In this contribution we experimentally and theoretically characterize these spikes. Our experiments reveal a statistical distribution of the characteristics of the optical pulses that is not observed in regular excitable systems. In particular, an inverse correlation exists between the pulse amplitude and duration. Numerical simulations and an interpretation in an asymptotic phase space confirm and explain these experimentally observed pulse characteristics [L. Gelens et al., Phys. Rev. A 82 063841, 2010]. We will also theoretically consider asymmetric SRLs coupled through a single bus waveguide. This is a first step towards an integrated optical neural network using semiconductor ring lasers as building blocks. We will show that for weak coupling, excitatory excursions still persist due to the similar phase space structure. Moreover, the coupled SRLs can excite pulses in each other and can thus function as communicating neurons [W. Coomans et al., Phys. Rev. E 84 036209, 2011]. This type of neural network can be fully integrated on chip and does not suffer from the drawback of needing extra-cavity measures, such as optical injection or saturable absorbers.

Two semiconductor ring lasers coupled by a single-waveguide for optical memory operation

Semiconductor ring lasers are semiconductor lasers where the laser cavity consists of a ring-shaped waveguide. SRLs are highly integrable and scalable, making them ideal candidates for key components in photonic integrated circuits. SRLs can generate light in two counterpropagating directions between which bistability has been demonstrated. Hence, information can be coded into the emission direction. This bistable operation allows SRLs to be used in systems for all-optical switching and as all-optical memories. For the demonstration of fast optical flip-flop operation, Hill et al. [Nature 432, 206 (2004)] fabricated two SRLs coupled by a single waveguide, rather than a solitary SRL. Nevertheless, the literature shows that a single SRL can also function perfectly as an all-optical memory. In our recent paper [W. Coomans et al., Phys. Rev. A 88, 033813, (2013)], we have raised the question whether coupling two SRLs to realize a single optical memory has any advantage over using a solitary SRL, taking into account the obvious disadvantage of a doubled footprint and power consumption. To provide the answer, we have presented in that paper a numerical study of the dynamical behavior of semiconductor ring lasers coupled by a single bus waveguide, both when weakly coupled and when strongly coupled. We have provided a detailed analysis of the multistable landscape in the coupled system, analyzed the stability of all solutions and related the internal dynamics in the individual lasers to the field effectively measured at the output of the waveguide. We have shown which coupling phases generally promote instabilities and therefore need to be avoided in the design. Regarding all-optical memory operation, we have demonstrated that there is no real advantage for bistable memory operation compared to using a solitary SRL. An increased power suppression ratio has been found to be mainly due to the destructive interference of the SRL fields at the low power port. Also, multistability between several modal configurations has been shown to remain unavoidable.

Nonlinear dynamics in directly modulated semiconductor ring lasers

In this paper, we have theoretically studied the dynamical behavior of current modulated semiconductor ring lasers (SRLs). As we vary the amplitude and frequency of the modulation around a fixed bias current, difference dynamical states including periodic, quasi-periodic and chaotic states are found. As in other single mode semiconductor lasers, the modal intensities in an SRL present chaotic behavior for driving frequencies comparable to the relaxation oscillation frequency. In this regime the two counter-propagating modes vary in phase. However, for modulation frequencies significantly lower than the relaxation oscillation frequency, we reveal the existence of chaotic oscillations where the two counter-propagating modes are in anti-phase.

Dynamical regimes in an optically injected semiconductor ring laser

We theoretically investigate optical injection in semiconductor ring lasers. Starting from a rate-equation model for semiconductor ring lasers, we use numerical simulations and a bifurcation analysis to reveal all the relevant dynamical regimes that will unfold for different parameter values. Our numerical simulations reproduced the saddle-node and Hopf bifurcation observed in other optically injected laser systems, which typically yield the boundaries of the parameter region in which stable locking can occur. Nevertheless, the bifurcation diagram of the optically injected semiconductor ring laser shows differences with the ones of other semiconductor lasers. For low injection power, we not only observe the regular saddle-node locking bifurcation, we also reveal the presence of an additional family of saddle-node bifurcations and a new Hopf bifurcation. These new bifurcations lead to the coexistence of two injection-locked states in two separate parameter regions and a parameter region is revealed in which a frequency-locked limit cycle coexists with an injection-locked solution, providing an additional route to stable locking. Finally, a chaotic regime that extends to low values of the detuning and injection power is revealed.

Analysis of multistability in semiconductor ring lasers

We present both an experimental and theoretical investigation of multistable states in a single-longitudinal mode and single transverse mode semiconductor ring laser (SRL). Our experiments have been performed on an InP-based multiquantum-well SRL with a racetrack geometry and a free-spectral-range of 53.6 Ghz. The power emitted from the chip is collected with a multimode fiber and detected with a 2.4 GHz photodiode connected to an oscilloscope. We show how the operation of the device can be steered to either monostable, bistable or multistable dynamical regimes in a controlled way. The diverse multistable dynamical regimes are shown to be organized in well reproducible sequences [Gelens et al., Phys. Rev. Lett. 102, 193904 (2009)]. These sequences are demonstrated to match the bifurcation diagrams of an asymptotic two-dimensional Z2-symmetric model for SRLs. Apart from predicting the different measured multistable time series, we demonstrate how the stochastic transitions between multistable states take place by analyzing the phase space in this model.

Theoretical and experimental investigation of mode-hopping in semiconductor ring lasers

We investigate both theoretically and experimentally the noise-induced transitions between the counter-rotating lasing modes of a semiconductor ring laser (SRL). Our experiments reveal that the residence time distribution (RTD) cannot be described by a simple one-parameter Arrhenius exponential law, due to the presence of two well-separated time scales in the process. Time-series of the mode-resolved power reveal an intricate mode-hopping dynamics and the connection between the time scales in the RTD and different mode-hopping scenarios. A theoretical approach is proposed in order to elucidate the origin of the two time scales, as well as the features of the mode-hopping events. We argue that the presence of two time-scales in the system is due to the finiteness of the noise intensity in the system which allows for diffusion between different folds of the invariant manifolds. Our approach is based on a double asymptotic reduction which is valid in the limit of slow dynamics and low noise-intensity. The theoretical predictions agree well with the results of numerical simulations and the experiments.

Study of excitability in semiconductor ring lasers: theory and experiment

Semiconductor Ring Lasers (SRLs) are a novel class of semiconductor lasers whose active cavity is characterized by a circular geometry. SRLs have attracted attention due to the possibility of monolithical integration of thousands of them on the same chip in a cheap and reliable way. SRLs are interesting for applications that rely on the presence of two counter-propagating modes inside the optical cavity. For instance, fully symmetric coupled SRLs have been proposed as candidates for the realisation of small and fast all-optical memories. At the same time, a wealth of nonlinear and stochastic dynamics have been predicted and observed in symmetric SRLs which is a consequence of the underlying Z2-symmetry of the device. However, unavoidable fabrication defects, material roughness and chip-cleaving break the device symmetry in an uncontrolled and unpredictable way, which may result in a deterioration of the devices performance in applications such as all-optical signal-processing. Despite their importance, the effects of symmetry breaking in SRLs remain unaddressed. In this contribution we investigate theoretically and experimentally the stochastic dynamics of SRLs with weakly broken Z2-symmetry . We show how the symmetry of an SRL can be experimentally manipulated using the reflection from a cleaved facet of a multi-mode optical fibre and a control electrode on the bus waveguide. The experiments are performed on an InP-based multi-quantum well SRL operating in single-longitudinal mode regime. The power at the CCW output is collected using a fast photodiode connected to an oscilloscope with a sampling rate of 4.0 ns. For a not-too-weak symmetry breaking, we reveal that SRLs become excitable and therefore can emit large, deterministic power bursts as a response to stochastic fluctuations. The origin of excitability is explained by investigating the topology of the invariant manifolds of an asymptotic two-dimensional phase space model with broken Z2-invariance. The results of the experiments confirm the prediction of the theory.

The dynamic behavior of a semiconductor ring laser

We review theoretical results on the dynamics of solitary single longitudinal mode and single transversal mode semiconductor ring lasers. These analyses are based on a rate equation model for the slowly varying envelopes of the counter-propagating fields in the ring cavity which has been proposed by Sorel et al. [Opt. Lett. 27, 1992 (2002); IEEE J. Quantum Electron. 39, 1187 (2003)]. The model shows several operating regimes. The lasers are found to operate bidirectionally up to twice the threshold, where unidirectional operation starts. Just above threshold, the lasers operate in a regime where the two counterpropagating modes are continuous wave, while as the injected current is increased, a regime appears where the intensities of the two counterpropagating modes undergo alternate sinusoidal oscillations. To understand these dynamical features, we discuss a reduction of this basic rate equation model derived by Van der Sande et al. [accepted for publication in J. Phys. B (2008)]. The reduction has been achieved using asymptotic methods based on the typical relative scaling of the dynamical time scales of the system. Physical conditions for the emergence of the operating regimes are assessed quantitatively in terms of nonlinear (saturation processes) and linear coupling (backscattering) between the counter-propagating modes.

Sub-diffraction-limited localized structures: influence of linear non-local interactions

Cavity solitons are controllable two-dimensional transverse Localized Structures (LS) in dissipative optical cavities. Such LS have been suggested for use in optical data storage and information processing. Typically, diffraction constrains the size of these light spots to be of the order of the square root of the diffraction coefficient of the system. Due to recent advances in the development of metamaterials, the diffraction strength in a cavity could be controlled by adding a left-handed material layer in a Fabry-Perot resonator together with a traditional nonlinear material. This system thus potentially allows for LS beyond the size limit imposed by natural diffraction. However, when the diffraction strength becomes smaller, the non-local response of the left-handed metamaterial starts to dominate the nonlinear spatiotemporal dynamics. Considering a typical linear non-local response, we develop a mean-field model describing the spatiotemporal evolution of LS. First, the influence of this non-local response on the minimal attainable width of the LS is studied [Gelens et al., Phys. Rev. A 75, 063812 (2007)]. Secondly, we elaborate on the different possible mechanisms that can destabilize the LS, leading to stable oscillations, expanding patterns, or making the LS disappear. Furthermore, we also show multiple routes towards excitability present in the system. We demonstrate that these different regions admitting stationary, oscillating or excitable LS unfold from two Takens-Bogdanov codimension-2 points [Gelens et al., Phys. Rev. A 77 (2008)].