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

Instability in optical injection locking semiconductors lasers using multiparametric bifurcation analysis

In this paper, we investigate bifurcations of equilibria and transients by using modified rate equations of semiconductor lasers (SCLs) subjected to optical injection. An analytical study is performed to demonstrate some two-parameter bifurcations, inter alia, Bogdanov-Takens and Gavrilov-Guckenheimer bifurcations. A detailed numerical study based on the multiparametric bifurcation method and using 3D-plots and projections reveal a rich locking dynamics of SCLs. In this way, a so-called zero frequency detuning well is highlighted in the vicinity of a Hopf bifurcation confining minimal states of the larger Lyapunov exponent in injection locking curves. Three-parameter bifurcation curves mainly underscore cusp bifurcation and resizing of its multi-equilibrium region by the specific control parameter defined in this model. The bursting phenomenon observed in the transient regime is discussed by using various numerical approaches wherefrom another quantifying method tapping into two-parameter bifurcation analysis is proposed. Thereafter, metastable chaos dynamics supported by spiraling relaxation oscillations is also investigated as well as planar saddle-node bifurcations with three homoclinic orbits for high positive and negative detunings. At last, zero α-factor effects contribute to drastically shrink the unlocking region of SCLs, twofold increase in Hopf bifurcation along with evidencing of complex chaotic sine-shaped and folded torus-shaped attractors.

Tunable, coherent optical comb source via on-chip bidirectional coupling

A tunable comb source is demonstrated through gain switching on a three-sectioned photonic integrated circuit (PIC). The PIC consists of two mutually coupled lasers connected by a passive waveguide. One of these is a tunable, two-section, single mode laser. The second laser is a simple Fabry-Perot cavity laser which can be phase-locked with the single mode laser via bidirectional coupling. Frequency combs are produced by gain switching the Fabry-Perot laser by applying a high-power radio frequency signal. Combs are generated with line spacings ranging from 3.5 to 8 GHz. The on-chip bidirectional coupling causes the comb to also be generated in the two-section device. Despite the lack of on-chip optical isolation between the lasers, the resulting combs are stable. Numerical simulations using a delay-differential model reproduce the results and reveal the important role played by the short delay times inherent to on-chip integration in this stability.

How to entrain a selected neuronal rhythm but not others: open-loop dithered brain stimulation for selective entrainment

Objective.While brain stimulation therapies such as deep brain stimulation for Parkinson's disease (PD) can be effective, they have yet to reach their full potential across neurological disorders. Entraining neuronal rhythms using rhythmic brain stimulation has been suggested as a new therapeutic mechanism to restore neurotypical behaviour in conditions such as chronic pain, depression, and Alzheimer's disease. However, theoretical and experimental evidence indicate that brain stimulation can also entrain neuronal rhythms at sub- and super-harmonics, far from the stimulation frequency. Crucially, these counterintuitive effects could be harmful to patients, for example by triggering debilitating involuntary movements in PD. We therefore seek a principled approach to selectively promote rhythms close to the stimulation frequency, while avoiding potential harmful effects by preventing entrainment at sub- and super-harmonics.Approach.Our open-loop approach to selective entrainment, dithered stimulation, consists in adding white noise to the stimulation period.Main results.We theoretically establish the ability of dithered stimulation to selectively entrain a given brain rhythm, and verify its efficacy in simulations of uncoupled neural oscillators, and networks of coupled neural oscillators. Furthermore, we show that dithered stimulation can be implemented in neurostimulators with limited capabilities by toggling within a finite set of stimulation frequencies.Significance.Likely implementable across a variety of existing brain stimulation devices, dithering-based selective entrainment has potential to enable new brain stimulation therapies, as well as new neuroscientific research exploiting its ability to modulate higher-order entrainment.

Nonlinear dynamics of a semiconductor microcavity laser subject to frequency comb injection

The nonlinear dynamical behaviors of a semiconductor microcavity laser with frequency comb injection have been experimentally and numerically investigated. The microcavity laser is harmonically locked to a unit fraction of the comb spacing due to the undamped relaxation oscillation at certain conditions, creating additional comb lines with reduced frequency spacing. The stability maps indicating various locking states are obtained based on rate equations, which demonstrates that the locking regions are closely related to the relaxation oscillation. Moreover, the microcavity laser with comb injection leads to spectral broadening of the original comb and the number of comb lines raises from 3 to 13. Owing to the large modulation bandwidth of the microcavity laser, the comb lines and the frequency spacing can be tailored over a wide range by varying the injection parameters.

Noise-induced stabilization of the FitzHugh-Nagumo neuron dynamics: Multistability and transient chaos

The nonlinear dynamics of a FitzHugh-Nagumo (FHN) neuron driven by an oscillating current and perturbed by a Gaussian noise signal with different intensities D is investigated. In the noiseless case, stable periodic structures [Arnold tongues (ATS), cuspidal and shrimp-shaped] are identified in the parameter space. The periods of the ATSs obey specific generating and recurrence rules and are organized according to linear Diophantine equations responsible for bifurcation cascades. While for small values of D, noise starts to destroy elongations ("antennas") of the cuspidals, for larger values of D, the periodic motion expands into chaotic regimes in the parameter space, stabilizing the chaotic motion, and a transient chaotic motion is observed at the periodic-chaotic borderline. Besides giving a detailed description of the neuronal dynamics, the intriguing novel effect observed for larger D values is the generation of a regular dynamics for the driven FHN neuron. This result has a fundamental importance if the complex local dynamics is considered to study the global behavior of the neural networks when parameters are simultaneously varied, and there is the necessity to deal the intrinsic stochastic signal merged into the time series obtained from real experiments. As the FHN model has crucial properties presented by usual neuron models, our results should be helpful in large-scale simulations using complex neuron networks and for applications.

Mode selection and amplification from optical frequency combs by optical injection locking: a numerical study of noise

Noise, except for residual side modes, in mode selection and amplification from optical frequency combs by optical injection locking has not been studied in detail, to our knowledge. We report a numerical study of noise behaviors, including both residual side modes and the noise between them. It reveals that comb laser injection can introduce excessive noise, compared to optical injection with a single mode master laser through the interference between residual side modes and resonances within Arnol'd tongues created by periodic forcing in the optical phase, which can be a severe problem for the case of sub-GHz comb mode spacing. The dependences of residual side mode rejection and phase noise variance on comb mode spacing, seeding power, and detuning are discussed to help in low-noise system design and operation.

Two Polarization Comb Dynamics in VCSELs Subject to Optical Injection

Optical frequency comb technologies have received intense attention due to their numerous promising applications ranging from optical communications to optical comb spectroscopy. In this study, we experimentally demonstrate a new approach of broadband comb generation based on the polarization mode competition in single-mode VCSELs. More specifically, we analyze nonlinear dynamics and polarization properties in VCSELs when subject of optical injection from a frequency comb. When varying injection parameters (injection strength and detuning frequency) and comb properties (comb spacing), we unveil several bifurcation sequences enabling the excitation of free-running depressed polarization mode. Interestingly, for some injection parameters, the polarization mode competition induces a single or a two polarization comb with controllable properties (repetition rate and power per line). We also show that the performance of the two polarization combs depends crucially on the injection current and on the injected comb spacing. We explain our experimental findings by utilizing the spin-flip VCSEL model (SFM) supplemented with terms for parallel optical injection of frequency comb. We provide a comparison between parallel and orthogonal optical injection in the VCSEL when varying injection parameters and SFM parameters. We show that orthogonal comb dynamics can be observed in a wide range of parameters, as for example dichroism linear dichroism (γa=−0.1 ns−1 to γa=−0.8 ns−1), injection current (μ=2.29 to μ=5.29) and spin-flip relaxation rate (γs=50 ns−1 to γs=2300 ns−1).

Tailoring frequency combs through VCSEL polarization dynamics

We investigate experimentally the nonlinear polarization dynamics of a VCSEL subject to optical injection of a frequency comb. By tuning the polarization of the injected comb to be orthogonal to that of the VCSEL, we demonstrate the generation of either a single polarization or a dual polarization frequency comb. The injection parameters (injected power and detuning frequency) are then used either to generate harmonics of the initial comb spacing or to increase the number of total output frequency lines up to 15 times the number of injected comb lines. Optimisation of the injection parameters yields a comb extending over 60 GHz for a comb spacing of 2 GHz with a carrier to noise ratio (CNR) of up to 60 dB. Our technique allows us to separately control the comb spacing, comb bandwidth, CNR and polarization. Our finding can be used for spectroscopy measurement and also for polarization division multiplexing in optical data communications.