Strong and weak chaos in networks of semiconductor lasers with time-delayed couplings

Wideband and high-dimensional chaos generation using optically pumped spin-VCSELs

We propose and numerically demonstrate wideband and high-dimensional chaos signal generation based on optically pumped spin-polarized vertical-cavity surface-emitting lasers (spin-VCSELs). Here, we focus on the chaotic characteristics of spin-VCSELs under two scenarios: one is a spin-VCSEL with optical feedback and the other is optical heterodyning the outputs of two free-running spin-VCSELs. Specifically, we systematically investigate the influence of some key parameters on the chaotic properties, i.e., bandwidth, spectral flatness (SF), time delay signature (TDS), correlation dimension (CD), and permutation entropy (PE), and reveal the route to enhance these properties simultaneously. Our simulation results demonstrate for the first time that spin-VCSELs with simple auxiliary configurations allow for chaos generation with desired properties, including effective bandwidth up to 30 GHz and above, no TDS of greater than 0.2, the flatness of 0.75 and above, and the high complexity/dimensionality over a wide range of parameters under both schemes. Therefore, our study may pave the way for potential applications requiring wideband and high-dimensional chaos.

Nonlinear Dynamics of a Single-Mode Semiconductor Laser with Long Delayed Optical Feedback: A Modern Experimental Characterization Approach

Semiconductor lasers can exhibit complex dynamical behavior in the presence of external perturbations. Delayed optical feedback, re-injecting part of the emitted light back into the laser cavity, in particular, can destabilize the laser’s emission. We focus on the emission properties of a semiconductor laser subject to such optical feedback, where the delay of the light re-injection is large compared to the relaxation oscillations period. We present an overview of the main dynamical features that emerge in semiconductor lasers subject to delayed optical feedback, emphasizing how to experimentally characterize these features using intensity and high-resolution optical spectra measurements. The characterization of the system requires the experimentalist to be able to simultaneously measure multiple time scales that can be up to six orders of magnitude apart, from the picosecond to the microsecond range. We highlight some experimental observations that are particularly interesting from the fundamental point of view and, moreover, provide opportunities for future photonic applications.

Reconstruction of parameters and unobserved variables of a semiconductor laser with optical feedback from intensity time series

We propose a method for the reconstruction of time-delayed feedback systems having unobserved variables from scalar time series. The method is based on the modified initial condition approach, which allows one to significantly reduce the number of starting guesses for an unobserved variable with a time delay. The proposed method is applied to the reconstruction of the Lang-Kobayashi equations, which describe the dynamics of a single-mode semiconductor laser with external optical feedback. We consider the case where only the time series of laser intensity is observable and the other two variables of the model are hidden. The dependence of the quality of the system reconstruction on the accuracy of assignment of starting guesses for unobserved variables and unknown laser parameters is studied. The method could be used for testing the security of information transmission in laser-based chaotic communication systems.

The reservoir's perspective on generalized synchronization

We employ reservoir computing for a reconstruction task in coupled chaotic systems, across a range of dynamical relationships including generalized synchronization. For a drive-response setup, a temporal representation of the synchronized state is discussed as an alternative to the known instantaneous form. The reservoir has access to both representations through its fading memory property, each with advantages in different dynamical regimes. We also extract signatures of the maximal conditional Lyapunov exponent in the performance of variations of the reservoir topology. Moreover, the reservoir model reproduces different levels of consistency where there is no synchronization. In a bidirectional coupling setup, high reconstruction accuracy is achieved despite poor observability and independent of generalized synchronization.

Consistency properties of chaotic systems driven by time-delayed feedback

Consistency refers to the property of an externally driven dynamical system to respond in similar ways to similar inputs. In a delay system, the delayed feedback can be considered as an external drive to the undelayed subsystem. We analyze the degree of consistency in a generic chaotic system with delayed feedback by means of the auxiliary system approach. In this scheme an identical copy of the nonlinear node is driven by exactly the same signal as the original, allowing us to verify complete consistency via complete synchronization. In the past, the phenomenon of synchronization in delay-coupled chaotic systems has been widely studied using correlation functions. Here, we analytically derive relationships between characteristic signatures of the correlation functions in such systems and unequivocally relate them to the degree of consistency. The analytical framework is illustrated and supported by numerical calculations of the logistic map with delayed feedback for different replica configurations. We further apply the formalism to time series from an experiment based on a semiconductor laser with a double fiber-optical feedback loop. The experiment constitutes a high-quality replica scheme for studying consistency of the delay-driven laser and confirms the general theoretical results.

Chaotic bursting in semiconductor lasers

We investigate the dynamic mechanisms for low frequency fluctuations in semiconductor lasers subjected to delayed optical feedback, using the Lang-Kobayashi model. This system of delay differential equations displays pronounced envelope dynamics, ranging from erratic, so called low frequency fluctuations to regular pulse packages, if the time scales of fast oscillations and envelope dynamics are well separated. We investigate the parameter regions where low frequency fluctuations occur and compute their Lyapunov spectra. Using the geometric singular perturbation theory, we study this intermittent chaotic behavior and characterize these solutions as bursting slow-fast oscillations.

Recovery of couplings and parameters of elements in networks of time-delay systems from time series

We propose a method for the recovery of coupling architecture and the parameters of elements in networks consisting of coupled oscillators described by delay-differential equations. For each oscillator in the network, we introduce an objective function characterizing the distance between the points of the reconstructed nonlinear function. The proposed method is based on the minimization of this objective function and the separation of the recovered coupling coefficients into significant and insignificant coefficients. The efficiency of the method is shown for chaotic time series generated by model equations of diffusively coupled time-delay systems and for experimental chaotic time series gained from coupled electronic oscillators with time-delayed feedback.

Randomness evaluation for an optically injected chaotic semiconductor laser by attractor reconstruction

State-space reconstruction is investigated for evaluating the randomness generated by an optically injected semiconductor laser in chaos. The reconstruction of the attractor requires only the emission intensity time series, allowing both experimental and numerical evaluations with good qualitative agreement. The randomness generation is evaluated by the divergence of neighboring states, which is quantified by the time-dependent exponents (TDEs) as well as the associated entropies. Averaged over the entire attractor, the mean TDE is observed to be positive as it increases with the evolution time through chaotic mixing. At a constant laser noise strength, the mean TDE for chaos is observed to be greater than that for periodic dynamics, as attributed to the effect of noise amplification by chaos. After discretization, the Shannon entropies continually generated by the laser for the output bits are estimated in providing a fundamental basis for random bit generation, where a combined output bit rate reaching 200 Gb/s is illustrated using practical tests. Overall, based on the reconstructed states, the TDEs and entropies offer a direct experimental verification of the randomness generated in the chaotic laser.

Consistency in experiments on multistable driven delay systems

We investigate the consistency properties in the responses of a nonlinear delay optoelectronic intensity oscillator subject to different drives, in particular, harmonic and self-generated waveforms. This system, an implementation of the Ikeda oscillator, is operating in a closed-loop configuration, exhibiting its autonomous dynamics while the drive signals are additionally introduced. Applying the same drive multiple times, we compare the dynamical responses of the optoelectronic oscillator and quantify the degree of consistency among them via their correlation. Our results show that consistency is not restricted to conditions close to the first Hopf bifurcation but can be found in a broad range of dynamical regimes, even in the presence of multistability. Finally, we discuss the dependence of consistency on the nature of the drive signal.

Complexity and bandwidth enhancement in unidirectionally coupled semiconductor lasers with time-delayed optical feedback

We numerically investigate the frequency bandwidth and the autocorrelation characteristics of chaotic temporal wave forms in unidirectionally coupled semiconductor lasers with time-delayed optical feedback. We evaluate the complexity of the chaotic temporal wave forms by using Lyapunov exponents. We found that larger maximum Lyapunov exponents can be obtained for smaller peak values of the autocorrelation function at the delay time of the optical feedback. On the contrary, the maximum Lyapunov exponent is independent from the frequency bandwidth of the chaotic temporal wave forms.

Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection

The concealment of the time-delay signature (TDS) of chaotic external-cavity lasers is necessary to ensure the security of optical chaos-based cryptosystems. We show that this signature can be removed simply by optically injecting an external-cavity laser with a large linewidth-enhancement factor into a second, noninjection-locked, semiconductor laser. Concealment is ensured both in the amplitude and in the phase of the optical field, satisfying a sought-after property of optical chaos-based communications. Meanwhile, enhancement of the dynamical complexity, characterized by permutation entropy, coincides with strong TDS suppression over a wide range of parameters, the area for which depends sensitively on the linewidth-enhancement factor.

Reconfigurable semiconductor laser networks based on diffractive coupling

Networks of optical emitters are highly sought-after, both for fundamental investigations as well as for various technological applications. We introduce and implement a novel scheme, based on diffractive optical coupling, allowing for the coupling of large numbers of optical emitters with adjustable weights. We demonstrate its potential by coupling emitters of a 2D array of semiconductor lasers with significant efficiency.

Determining the sub-Lyapunov exponent of delay systems from time series

For delay systems the sign of the sub-Lyapunov exponent (sub-LE) determines key dynamical properties. This includes the properties of strong and weak chaos and of consistency. Here we present a robust algorithm based on reconstruction of the local linearized equations of motion, which allows for calculating the sub-LE from time series. The algorithm is inspired by a method introduced by Pyragas for a nondelayed drive-response scheme [K. Pyragas, Phys. Rev. E 56, 5183 (1997)]. In the presented extension to delay systems, the delayed feedback takes over the role of the drive, whereas the response of the low-dimensional node leads to the sub-Lyapunov exponent. Our method is based on a low-dimensional representation of the delay system. We introduce the basic algorithm for a discrete scalar map, extend the concept to scalar continuous delay systems, and give an outlook to the case of a full vector-state system, from which only a scalar observable is recorded.

The transition between strong and weak chaos in delay systems: Stochastic modeling approach

We investigate the scaling behavior of the maximal Lyapunov exponent in chaotic systems with time delay. In the large-delay limit, it is known that one can distinguish between strong and weak chaos depending on the delay scaling, analogously to strong and weak instabilities for steady states and periodic orbits. Here we show that the Lyapunov exponent of chaotic systems shows significant differences in its scaling behavior compared to constant or periodic dynamics due to fluctuations in the linearized equations of motion. We reproduce the chaotic scaling properties with a linear delay system with multiplicative noise. We further derive analytic limit cases for the stochastic model illustrating the mechanisms of the emerging scaling laws.

Consistency properties of a chaotic semiconductor laser driven by optical feedback

We experimentally study consistency properties of a semiconductor laser in response to a coherent optical drive originating from delayed feedback. The laser is connected to a short and a long optical fiber loop, switched such that only one is providing input to the laser at a time. This way, repeating the exact same optical drive twice, we find consistent or inconsistent responses depending on the pump parameter and we relate the kind of response to strong and weak chaos. Moreover, we are able to experimentally determine the sub-Lyapunov exponent, underlying the consistency properties.

Autocorrelation properties of chaotic delay dynamical systems: A study on semiconductor lasers

We present a detailed experimental characterization of the autocorrelation properties of a delayed feedback semiconductor laser for different dynamical regimes. We show that in many cases the autocorrelation function of laser intensity dynamics can be approximated by the analytically derived autocorrelation function obtained from a linear stochastic model with delay. We extract a set of dynamic parameters from the fit with the analytic solutions and discuss the limits of validity of our approximation. The linear model captures multiple fundamental properties of delay systems, such as the shift and asymmetric broadening of the different delay echoes. Thus, our analysis provides significant additional insight into the relevant physical and dynamical properties of delayed feedback lasers.

Statistics of the optical intensity of a chaotic external-cavity DFB laser

We study experimentally and theoretically the first- and second-order statistics of the optical intensity of a chaotic external-cavity semiconductor DFB laser in fully developed coherence-collapse. The second-order statistic is characterized by the autocorrelation, where we achieve consistent experimental and theoretical results over the entire parameter range considered. For the first-order statistic, we find that the experimental probability-density function is significantly more concentrated around the mean optical power and robust to parameter changes than theory predicts.

Reconstruction of ensembles of coupled time-delay systems from time series

We propose a method to recover from time series the parameters of coupled time-delay systems and the architecture of couplings between them. The method is based on a reconstruction of model delay-differential equations and estimation of statistical significance of couplings. It can be applied to networks composed of nonidentical nodes with an arbitrary number of unidirectional and bidirectional couplings. We test our method on chaotic and periodic time series produced by model equations of ensembles of diffusively coupled time-delay systems in the presence of noise, and apply it to experimental time series obtained from electronic oscillators with delayed feedback coupled by resistors.

Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy

A semiconductor laser with delayed optical feedback is an experimental implementation of a nominally infinite dimensional dynamical system. As such, time series analysis of the output power from this laser system is an excellent test of complexity analysis tools, as applied to experimental data. Additionally, the systematic characterization of the range and variation in complexity that can be obtained in the output power from the system, which is available to be used in applications like secure communication, is of interest. Output power time series from a semiconductor laser system, as a function of the optical feedback level and the laser injection current, have been analyzed for complexity using permutation entropy. High resolution maps of permutation entropy as a function of optical feedback level and injection current have been achieved for the first time. This confirms prior research that identifies a coherence collapse region which is found to be uninterrupted with respect to any embedded islands with different dynamics. The results also show new observations of low optical feedback dynamics which occur in a region below that for coherence collapse. The map of the complexity shows a strong dependence on the delay time used in the permutation entropy calculation. Short delay times, which sample information at the complete measurement bandwidth, produce maps with drastically different systematic variation in complexity throughout the coherence collapse region, compared to maps generated with a delay time that matches the optical feedback delay. Evaluating the complexity with a permutation entropy delay equal to the external cavity delay produces results consistent with the notion of weak/strong chaos, as well as categorizing the dynamics as being of high complexity where the external cavity delay time is harder to identify. These are both desirable features for secure communication applications. The results also show permutation entropy as a function of delay time can be used to detect key frequencies driving the dynamics, including any that may exist due to, or arise from, technicalities of device fabrication and/or noise. A more complete insight into complexity as measured by permutation entropy is gained by considering multiple delay times.

Relation between delayed feedback and delay-coupled systems and its application to chaotic lasers

We present a systematic approach to identify the similarities and differences between a chaotic system with delayed feedback and two mutually delay-coupled systems. We consider the general case in which the coupled systems are either unsynchronized or in a generally synchronized state, in contrast to the mostly studied case of identical synchronization. We construct a new time-series for each of the two coupling schemes, respectively, and present analytic evidence and numerical confirmation that these two constructed time-series are statistically equivalent. From the construction, it then follows that the distribution of time-series segments that are small compared to the overall delay in the system is independent of the value of the delay and of the coupling scheme. By focusing on numerical simulations of delay-coupled chaotic lasers, we present a practical example of our findings.

Characterizing the deterministic nature of individual power dropouts in semiconductor lasers subject to delayed feedback

We implement a method to identify the deterministic nature of specific events in the dynamics of a semiconductor laser subject to time-delayed optical feedback. Specifically, we study the power dropouts in the low-frequency fluctuations regime on an individual event basis and identify whether the underlying dominant mechanism is deterministic. Our approach is based on sychronization with a twin system in a symmetric relay configuration. We investigate the dependence of the fraction of deterministically driven (i.e., synchronized) dropouts on the laser's pump current as a key parameter. Our experimental results are corroborated by numerical modeling based on rate equations. Our numerical findings also provide insights into the influence of spontaneous emission noise.