Dynamics Of Vertical-cavity Surface-emitting Lasers Research Articles

Phase stability diagram, self-similar structures, and multistability in a free-running VCSEL with a small misalignment between the phase and amplitude anisotropies.

We report on the global dynamics of a free-running vertical-cavity surface-emitting laser (VCSEL) with misalignment between the linear phase and amplitude anisotropies due to the fact that this case might occur in practice caused unintentionally by minor manufacturing variations or design, in virtue of high-resolution phase stability diagrams, where two kinds of self-similar structures are revealed. Of interest is that the Arnold tongue cascades covered by multiple distinct periodicities are discovered for the first time in several scenarios specified in the free-running VCSEL, to the best of our knowledge. Additionally, we also uncover the existence of multistability through the basin of the attraction, as well as the eyes of anti-chaos and periodicity characterized by fractal. The findings may shed new light on interesting polarization dynamics of VCSELs, and also open the possibility to detect the above-mentioned structures experimentally and develop some potential applications.

Complex nonlinear dynamics of polarization and transverse modes in a broad-area VCSEL

Lasers can exhibit nonlinear and chaotic dynamics driven by the interaction of multiple lasing modes, and investigating the different scenarios of mode competition and bifurcations of their dynamics is of great interest on a fundamental level as well as in view of applications. We study the dynamics of a broad-area vertical-cavity surface-emitting laser (VCSEL) in solitary continuous-wave operation with a comprehensive investigation of its polarization state, lasing spectra, near-field distributions, and temporal dynamics. Fluctuations at the frequency of birefringence splitting and other frequency components develop in a series of bifurcations. The bifurcations coincide with changes of the transverse lasing modes and/or the polarization state, demonstrating the importance of both the spatial and polarization degrees of freedom for mode competition. As a consequence, the inherent nonlinear dynamics of broad-area VCSELs is significantly more complex than the dynamics of VCSELs with a single spatial mode.

Dynamics of vertical-cavity surface-emitting lasers under AM and FM current modulations

We numerically investigate the dynamics of Vertical-Cavity Surface-Emitting Lasers (VCSELs) described by the current-dependent gain model and subjected to current in the form of amplitude modulation (AM) and frequency modulation (FM). Since VCSELs have two polarization modes (PMs), the competition between them offers more complex dynamical behaviors. It is found that when operating close to the threshold current, the laser converts the AM and FM currents into a series of pulse packages (PPs) that are globally irregular and modulated either in amplitude or frequency. In the particular case of FM current, both modulations occur in a unique signal. Besides, chaotic light is recovered even at low frequencies (few MHz) where it is not usually expected when sinusoidal current modulation is used. But, if the AM and the FM currents evolve over the threshold value, the laser does not alter the form of the current. Therefore, those currents are linearly converted into light signals.

The Winner-Take-All Mechanism for All-Optical Systems of Pattern Recognition and Max-Pooling Operation

The winner-take-all (WTA) mechanism based on the inhibitory dynamics of vertical-cavity surface-emitting laser with an embedded saturable absorber (VCSEL-SA) neurons is proposed for the first time. The WTA mechanism is shown numerically in a photonic spiking neural network (SNN). The effect of bias current on WTA time window is analyzed based on the proposed SNN. Moreover, a pattern recognition approach is presented numerically based on the WTA mechanism in all-optical neural network consisting of VCSEL-SA neurons. For the pattern recognition, the robustness of noisy inputs is examined. The effects of bias current, strength of inhibition and weight matrix on the speed of pattern recognition are analyzed carefully. Furthermore, the max-pooling operation is implemented numerically in the all-optical VCSEL-SA neural network model based on the WTA mechanism for the first time. The results hold great promise for the development of energy-efficient and high-speed photonic spiking neural network.

Real-time optical spike-timing dependent plasticity in a single VCSEL with dual-polarized pulsed optical injection

We propose and numerically realize an optical spike-timing dependent plasticity (STDP) scheme by using a single vertical-cavity surface-emitting laser (VCSEL). In the scheme, the VCSEL is subjected to an orthogonally-polarized continuous-wave optical injection (OPCWOI) and dual-polarized pulsed optical injections (DPPOI). Based on the widely used spin-flip model, the response spiking dynamics of VCSEL is numerically studied, and then the optical STDP in a single VCSEL is explored. The roles of bias current, the strength of OPCWOI and DPPOI, and the frequency detuning on the optical STDP curve are numerically analyzed. It is found that, by simultaneously utilizing the response spiking dynamics in two orthogonal polarization modes, an optical STDP could be achieved by using a single VCSEL. Furthermore, the weight update of STDP curve can be calculated in real-time. Additionally, the STDP curves can also be controlled by adjusting some controllable parameters. The real-time optical STDP based on a single VCSEL is numerically realized for the first time, which paves the way towards fully VCSELs-based photonic neuromorphic systems with low power consumption.

Isochronous cluster synchronization in delay-coupled VCSEL networks subjected to variable-polarization optical injection with time delay signature suppression.

The isochronous cluster synchronization with time delay (TD) signature suppression in delay-coupled vertical-cavity surface-emitting laser (VCSEL) networks subject to variable-polarization optical injection (VPOI) is theoretically and numerically studied. Based on the inherent symmetries of network topology, parameter spaces for stable cluster synchronization are presented, and zero-lag synchronization are achieved for VCSELs in same clusters. Additionally, the TD signature reduction for the dynamics of VCSELs in the stable clusters are systematically discussed. It is shown that both moderate polarizer angle and frequency detuning between different clusters have strengthen the effect of TD signature suppression. Moreover, the isochronous cluster synchronization with TD signature concealment is also verified in another VPOI-VCSEL network with different topology, indicating the generality of proposed results. Our results shed a new light on the research of chaos synchronization and chaos-based secure communications in VCSEL networks.

Chaos-Preserving Reduction of the Spin-Flip Model for VCSELs: Failure of the Adiabatic Elimination of the Spin-Population Difference

When studying the dynamics of Vertical-Cavity Surface-Emitting Lasers, and their polarization properties, the spin-flip model appears to be the simplest model qualitatively reproducing all dynamical features that have been observed experimentally. Nonetheless, because of the fast time-scale of the spin-relaxation processes, the specific role and the importance of the spin-population difference - which is one of the specific feature of the spin-flip model - has been continuously questioned. In fact, the debate regarding the possible adiabatic elimination of the spin-population remains fully open. In this paper, our goal is to bring new light into this issue by demonstrating that this variable is essential to preserve the most complex dynamical features predicted by the spin-flip model, such as polarization chaos, and, therefore, needs to be conserved. To do so, we first perform a detailed analysis, focusing on the chaotic dynamics, to determine the minimal embedding dimension for the spin-flip model. As the latter confirms that a reduction of the model could be envisaged, we then consider the adiabatic elimination of the spin-population difference to highlight and explain its failure to reproduce essential dynamical features obtained in the original model.

Photonic Generation of Neuron-Like Dynamics Using VCSELs Subject to Double Polarized Optical Injection

We propose to generate excitatory and inhibitory neuron-like dynamics in vertical-cavity surface-emitting lasers (VCSELs) by applying simultaneously the orthogonally-polarized CW optical injection (OPCWOI) and parallelly-polarized pulse optical injection stimulus. Based on the spin flip model, excitatory and inhibitory neuron-like dynamics accompanying with reverse polarization switching is numerically investigated. It is found that, due to the injection locking effect or beating effect between two injected fields, the excitatory phasic and tonic spiking dynamics can be obtained in the originally dominated polarization mode. Moreover, the unwanted relaxation oscillation followed by the excitatory spiking dynamics at the end of the stimulus pulse, which is present in previous reported photonic neuron based on the VCSELs subject to a single orthogonally-polarized optical pulse injection, can be completely suppressed. In addition, the inhibition of tonic spiking dynamics can also be achieved, and the transition from tonic spiking dynamics to phasic bursting dynamics can be obtained, when the two injected fields have the same frequency. These results are interesting and valuable for the ultrafast photonic neuromorphic systems and neuron-inspired photonic information processing.

Stable Propagation of Inhibited Spiking Dynamics in Vertical-Cavity Surface-Emitting Lasers for Neuromorphic Photonic Networks

We investigate experimentally and theoretically the communication of inhibited spiking dynamics between two interlinked photonic neurons based upon the vertical-cavity surface-emitting lasers (VCSELs). We show that the sub-nanosecond speed spiking dynamics fired by a Transmitter-VCSEL (T-VCSEL) can be inhibited under the arrival of suitable external stimuli and that the inhibited spiking behaviors are propagated into another Receiver-VCSEL (R-VCSEL). Both VCSELs exhibit analogous inhibited spiking dynamics in response to stimuli with different temporal durations and strength. In addition, a very good agreement is found between theoretical simulations and experiments. These results offer greater prospects for future networks of VCSEL-based photonic neurons for neuromorphic computing platforms.

Controlled Propagation of Spiking Dynamics in Vertical-Cavity Surface-Emitting Lasers: Towards Neuromorphic Photonic Networks

We report experimentally and in theory on the controllable propagation of spiking regimes between two interlinked vertical-cavity surface-emitting lasers (VCSELs). We show that spiking patterns generated in a first transmitter VCSEL (T-VCSEL) are communicated to a second receiver VCSEL (R-VCSEL), which responds by firing the same spiking response. Importantly, the spiking regimes from both devices had analogous temporal and amplitude characteristics, including equal number of spikes fired, same spike and interspike temporal durations, and similar spike intensity properties. These responses are analogous to the spiking communication patterns of biological neurons yet at subnanosecond speeds, this is several (up to 8) orders of magnitude faster than the timescales of biological neurons. We have also carried out numerical simulations reproducing with high degree of agreement the experimental findings. These results obtained with inexpensive, commercially available VCSELs operating at important telecom wavelengths (1300 nm) offer great prospects for the scaling of emerging VCSEL-based photonic neuronal models into network configurations for use in novel neuromorphic photonic systems. This offers high potentials for nontraditional computing paradigms beyond digital systems and able to operate at ultrafast speeds.

Cascadable Neuron-Like Spiking Dynamics in Coupled VCSELs Subject to Orthogonally Polarized Optical Pulse Injection

The cascadability of spiking dynamics in coupled vertical-cavity surface-emitting lasers (VCSELs) subject to orthogonally polarized optical pulse injection (OPOPI) is investigated numerically based on the well-known spin-flip model. For the coupled VCSELs systems, different connection topologies are considered. The effects of stimuli strength, coupling strength, frequency detuning between two coupled VCSELs, and the pump current on the cascadable spiking dynamics are examined by extensive numerical bifurcation analysis. It is found that, the phasic spike generated in one VCSEL with OPOPI can also induce a similar phasic spike in another VCSEL in three considered connection topologies. Besides, such a cascadable spiking behavior can be achieved in wide range of stimuli strength, coupling strength, and frequency detuning. However, failure transfer of the phasic spike between two VCSELs may be caused when the pump currents are relatively large, and hence, low pump currents are desirable for both VCSELs to allow for successful cascadable spiking dynamics. This work is interesting and valuable for constructing reliable optical spiking information processing platform based on polarization dynamics of VCSELs.

Emulation of Spiking Response and Spiking Frequency Property in VCSEL-Based Photonic Neuron

The dynamical response properties of photonic neuron based on vertical-cavity surface emitting lasers (VCSELs) subject to orthogonal polarized optical pulse injection stimuli have been numerically investigated. Based on the well-known spin flip model, we first reproduce some experimental findings of neuron-like dynamics in VCSELs, such as phasic spiking with a single abrupt pulse, and tonic spiking with multiple periodic pulses. Besides, we further go beyond in three directions and obtain several novel results. The operating parameter ranges corresponding to different neuron-like dynamics are identified by extensive bifurcation analysis. In addition, the effect of the time-varying pump current on the neuron-like dynamics for VCSELs under given optical injecting pulse strength is also discussed. For a given pump current, the spiking frequency dependence on the stimuli strength is further revealed in VCSELs with time-varying optical pulse injection. Such controllable neuron-like response dynamics and spiking frequency dependence in VCSELs are interesting and valuable for ultrafast photonic neuromorphic systems and neuron-inspired photonic information processing.

Experimental determination of key parameters in the spin-flip model of 1550 nm vertical-cavity surface-emitting laser

Spin-flip model (SFM) is a mostly used approach to analyzing the nonlinear dynamics of vertical-cavity surface-emitting laser (VCSEL), and therefore the value selections of some key parameters in this model are crucial. In this work, based on experimentally measured dynamical characteristics of a 1550 nm vertical-cavity surface-emitting laser (1550 nm-VCSEL) under free running and parallel optical injection, some key parameters (field decay rate k, total carrier decay rate N, linewidth enhancement factor , active medium birefringence rate p, spin relaxation rate s, and active medium linear dispersion rate a) are estimated. Through experimentally measuring the noise spectrum of the laser, the relaxation oscillation frequency and the damping rate of the relaxation oscillations are calculated, and the photon lifetime can be preliminary estimated. After further amending the photon lifetime by considering the effect of the gain saturation on the damping rate of the relaxation oscillations, the value of k is determined. Based on the function relation between the laser relaxation oscillation frequency and the electrical pumping, the value of N is obtained. By experimentally acquiring the dynamical distribution mapping of the laser under parallel optical injection, the minimum Hopf bifurcation point of the Hopf bifurcation curve can be found, and then the value of is roughly estimated. According to the frequency difference between the two polarization components of the laser in the measured optical spectrum, the value of p can be calculated. The value of s is obtained by using the relationship between s and p. On the basis of the above determined parameter values, the value of a can be specified by numerically simulating the optical spectrum of the laser and comparing with experimentally obtained results. Moreover, by comparing the experimentally measured dynamical mapping of optical injection VCSEL with corresponding dynamical mapping simulated on the basis of the above mentioned parameters, the value of is rectified. Finally, further simulated results agree with relevant experimental observations.

Phase dynamics in vertical-cavity surface-emitting lasers with delayed optical feedback and cross-polarized reinjection

We study theoretically the nonlinear polarization dynamics of vertical-cavity surface-emitting lasers in the presence of an external cavity providing delayed optical feedback and cross-polarized reinjection. We show that, far from the laser threshold, the dynamics remains confined close to the equatorial plane of a Poincar\'e sphere with a fixed radius. It entails that the evolution of the system is described by two phase variables: the orientation phase of the quasilinear polarization and the optical phase of the field. We explore the complex modal structure given by the double reinjection configuration and how it evolves between the cases of single cross-polarized reinjection and single optical feedback, hence disclosing the relationship with the Lang-Kobayashi model. We also reinterpret the square-wave switching observed by J. Mulet et al. [Phys. Rev. A 76, 043801 (2007)] in terms of phase kinks.

Optical Feedback in Vertical-Cavity Surface-Emitting Lasers

Vertical-cavity surface-emitting lasers (VCSELs) are as sensitive to optical feedback (OF) as edge-emitting lasers as their much higher mirror reflectivity is compensated by the much smaller intracavity round-trip time resulting in a similar OF rate. VCSELs may emit light in multiple transverse modes and may exhibit polarization bistability which adds new features to their behavior when subject to OF. We present a survey on experimental and modeling results for three regimes of OF in VCSELs: long, short, and extremely short external cavities (ECs). We show that for long and short ECs, the Lang-Kobayashi model is in good agreement with experiments on OF-induced polarization switching, mode hopping, and antiphase oscillatory dynamics at the delay time. These last dynamics modify polarization residence-time distribution making it oscillatory. We demonstrate numerically and experimentally coherence resonance in such bistable, time-delayed optical system. We present results on low-frequency fluctuation and pulse-package dynamics in VCSELs and discuss the role of light polarization. We discuss polarization rotating OF for generating high-frequency pulses as a result of EC mode beating. Finally, we demonstrate how OF from extremely short EC can strongly modify VCSEL characteristics and can be used to effectively control their polarization properties by subwavelength changes of the EC length.

Polarization dynamics and chaotic synchronization in unidirectionally coupled VCSELs subjected to optoelectronic feedback

Based on the framework of the spin-flip model (SFM), the polarization dynamics of vertical-cavity surface-emitting lasers (VCSELs) subjected to optoelectronic feedback has been investigated numerically. The results show that nonlinear dynamics are similar to what have been observed in VCSELs with negative optoelectronic feedback, such as polarization switching (PS), antiphase and inphase fluctuations, mode competition and quasi-period routes to chaos. The difference between them is that VCSELs subjected to optoelectronic feedback emit only in the yLP mode if the feedback strength is large enough. Moreover, the synchronization performance of unidirectionally coupled VCSELs subjected to optoelectronic feedback has also been investigated. The polarized mode in the single mode region (region A) can achieve complete synchronization. Within the mixed-mode region (region B), the synchronization quality of polarized modes is not ideal.

Square-wave switching in vertical-cavity surface-emitting lasers with polarization-rotated optical feedback: Experiments and simulations

We study experimentally the dynamics of vertical-cavity surface-emitting lasers (VCSELs) with polarization-rotated (PR) optical feedback, such that the natural lasing polarization of a VCSEL is rotated by 90 deg and then is reinjected into the laser. We observe noisy, square-wave-like polarization switchings with periodicity slightly longer than twice the delay time, which degrade to (or alternate with) bursts of irregular oscillations. We present results of simulations that are in good agreement with the observations. The simulations demonstrate that close to threshold the regular switching is very sensitive to noise, while well above threshold is less affected by the noise strength. The frequency splitting between the two polarizations plays a key role in the switching regularity, and we identify wide parameter regions where deterministic and robust switching can be observed.

Nonlinear dynamics in VCSELs driven by a sinusoidally modulated current and Rössler oscillator

We firstly investigate the dynamics of vertical-cavity surface-emitting lasers (VCSELs) subject to sinusoidally modulated current using numerical simulations. For a few-hundred MHz frequency of injection current, the VCSELs emit a single pulse or a package of pulses when the injection current fluctuates around the threshold current. By varying the amplitude of the modulation, we show that the width of the bistability hysteresis loop can be suppressed and replaced by smooth single or multiple switching. For a few GHz frequency of injection current, the polarization modes of VCSELs exhibit a period-doubling transition to chaotic behavior. Then by modulating the VCSELs pump current by one of the outputs of a Rossler oscillator, we generate dynamical behaviors found in VCSELs subject to a sinusoidally modulated current. This new way of modulation of semiconductor lasers would bring a general benefit in the physical equipment, and reduce their cost.

Nonlinear Dynamics of Vertical-Cavity Surface-Emitting Lasers

Nonlinear dynamics of Vertical-Cavity Surface-Emitting Lasers (VCSELs) induced by optical injection, optical feedback, current modulation and mutual coupling is reviewed. Due to the surface emission and cylindrical symmetry VCSELs lack strong polarization anisotropy and may undergo polarization switching. Furthermore, VCSELs may emit light in multiple transverse modes. These VCSEL properties provide new features to the rich nonlinear dynamics induced by an external perturbation. We demonstrate for the case of orthogonal optical injection that new Hopf bifurcation on a two-polarization-mode solution delimits the injection locking region and that polarization switching and injection locking of first-order transverse mode lead to a new resonance tongue for large positive detunings. Similarly, the underlying polarization mode competition leads to chaotic-like behavior in case of gain switching and the presence of two transverse modes additionally reduces the possibility of regular dynamics. The bistable property of VCSEL makes it possible to investigate very fundamental problems of bistable systems with time-delay, such as the coherence resonance phenomenon. We also demonstrate that the synchronization quality between unidirectionally coupled VCSELs can be significantly enhanced when the feedback-induced chaos in the master laser involves both orthogonal LP fundamental transverse modes.

Chaos and pulse packages in current-modulated VCSELs

We numerically study the dynamics of vertical cavity surface-emitting lasers (VCSELs) based on the current-dependent gain model subjected to current modulation. For a few hundred MHz frequency of injection current, the threshold current is reduced and polarization switching is accompanied by an anti-phase dynamics of polarization modes. Pulse packages are found and compared with those found in the literature. The behaviour of pulse packages is studied in terms of the modulation parameters. For a few GHz frequency of injection current, the dynamics of VCSELs presents chaotic behaviour.