Regulation of cluster synchronization in multilayer networks of delay coupled semiconductor lasers with the use of disjoint layer symmetry.

In real-world complex systems, heterogeneous components often interact in complex connection patterns and could be schematized by a formalism of multilayer network. In this work, the synchronization characteristics of multilayer network composed of semiconductor lasers (SLs) are investigated systematically. It is demonstrated that the interplay between different layers plays an important role on the synchronization patterns. We elucidate that the performance of cluster synchronization could be facilitated effectively with the introduction of disjoint layer symmetry into network topology. Intertwined stability of clusters from different layers could be decoupled into independent, and the parameter spaces for stable synchronization are extended significantly. The robustness of our proposed regulation scheme on operation parameters is numerically evaluated. Furthermore, the generality of presented theoretical results is validated in networks with more complex topology and multiple layers.

Wideband chaos synchronization using discrete-mode semiconductor lasers.

Optical chaos communication encounters difficulty in high-speed transmission due to the challenge of realizing wideband chaos synchronization. Here, we experimentally demonstrate a wideband chaos synchronization using discrete-mode semiconductor lasers (DMLs) in a master-slave open-loop configuration. The DML can generate wideband chaos with a 10-dB bandwidth of 30 GHz under simple external mirror feedback. By injecting the wideband chaos into a slave DML, an injection-locking chaos synchronization with synchronization coefficient of 0.888 is realized. A parameter range with frequency detuning of -18.75 GHz to approximately 1.25 GHz under strong injection is identified for yielding the wideband synchronization. In addition, we find it more susceptible to achieve the wideband synchronization using the slave DML with lower bias current and smaller relaxation oscillation frequency.

Numerical and experimental investigation of a dispersive optoelectronic oscillator for chaotic time-delay signature suppression.

Chaos generation from a novel single-loop dispersive optoelectronic oscillator (OEO) with a broadband chirped fiber Bragg grating (CFBG) is numerically and experimentally investigated. The CFBG has much broader bandwidth than the chaotic dynamics such that its dispersion effect rather than filtering effect dominates the reflection. The proposed dispersive OEO exhibits chaotic dynamics when sufficient feedback strength is guaranteed. Suppression of chaotic time-delay signature (TDS) is observed as the feedback strength increases. The TDS can be further suppressed as the amount of grating dispersion increases. Without compromising bandwidth performance, our proposed system extends the parameter space of chaos, enhances the robustness to modulator bias variation, and improves TDS suppression by at least five times comparing to the classical OEO. Experimental results qualitatively agree well with numerical simulations. In addition, the advantage of dispersive OEO is further verified by experimentally demonstrating random bit generation with tunable rate up to 160 Gbps.

Photonic reservoir computing using a self-injection locked semiconductor laser under narrowband optical feedback.

Photonic time-delay reservoir computing (TDRC) using a self-injection locked semiconductor laser under optical feedback from a narrowband apodized fiber Bragg grating (AFBG) is proposed and numerically demonstrated. The narrowband AFBG suppresses the laser's relaxation oscillation and provides self-injection locking in both the weak and strong feedback regimes. By contrast, conventional optical feedback provides locking only in the weak feedback regime. The TDRC based on self-injection locking is first evaluated by the computational ability and memory capacity, then benchmarked by the time series prediction and channel equalization. Good computing performances can be achieved using both the weak and strong feedback regimes. Interestingly, the strong feedback regime broadens the usable feedback strength range and improves robustness to feedback phase variations in the benchmark tests.

Performance improvement of coherent optical chaos communication using probabilistic shaping.

We numerically investigate the effects of probabilistic shaping on the performance improvement of coherent optical chaos communication. Results show that the decryption bit-error ratio (BER) of the 16-ary quadrature amplitude modulation (QAM) signal decreases upon increasing the probabilistic shaping factor. It is predicted that the BER of 10-GBd 16QAM can be decreased by one order of magnitude. On the other hand, for the forward error correction threshold of the BER, the requirement for synchronization quality is no longer strict for successful decryption. This means that probabilistic shaping improves the system's tolerance to residual synchronization error. Thus, the transmission rate can be increased by approximately 30∼60%. The side effect of probabilistic shaping is that the valid masking coefficient range is narrowed.

Modeling of a multi-parameter chaotic optoelectronic oscillator based on the Fourier neural operator.

A model construction scheme of chaotic optoelectronic oscillator (OEO) based on the Fourier neural operator (FNO) is proposed. Different from the conventional methods, we learn the nonlinear dynamics of OEO (actual components) in a data-driven way, expecting to obtain a multi-parameter OEO model for generating chaotic carrier with high-efficiency and low-cost. FNO is a deep learning architecture which utilizes neural network as a parameter structure to learn the trajectory of the family of equations from training data. With the assistance of FNO, the nonlinear dynamics of OEO characterized by differential delay equation can be modeled easily. In this work, the maximal Lyapunov exponent is applied to judge whether these time series have chaotic behavior, and the Pearson correlation coefficient (PCC) is introduced to evaluate the modeling performance. Compare with long and short-term memory (LSTM), FNO is not only superior to LSTM in modeling accuracy, but also requires less training data. Subsequently, we analyze the modeling performance of FNO under different feedback gains and time delays. Both numerical and experimental results show that the PCC can be greater than 0.99 in the case of low feedback gain. Next, we further analyze the influence of different system oscillation frequencies, and the generalization ability of FNO is also analyzed.

Strong cluster synchronization in complex semiconductor laser networks with time delay signature suppression.

Cluster synchronization is a state where clusters of nodes inside the network exhibit isochronous synchronization. Here, we present a mechanism to realize the strong cluster synchronization in semiconductor laser (SL) networks with complex topology, where stable cluster synchronization is achieved with decreased correlation between dynamics of different clusters and time delay signature concealment. We elucidate that, with the removal of intra-coupling within clusters, the stability of cluster synchronization could be enhanced effectively, while the statistical correlation among dynamics of each cluster decreases. Moreover, it is demonstrated that the correlation between clusters can be further reduced with the introduction of dual-path injection and frequency detuning. The robustness of strong cluster synchronization on operation parameters is discussed systematically. Time delay signature in chaotic outputs of SL network is concealed simultaneously with heterogeneous inter-coupling among different clusters. Our results suggest a new approach to control the cluster synchronization in complex SL networks and may potentially lead to new network solutions for communication schemes and encryption key distribution.

Hierarchical-dependent cluster synchronization in directed networks with semiconductor lasers.

Cluster synchronization in complex networks with mutually coupled semiconductor lasers (SLs) has recently been extensively studied. However, most of the previous works on cluster synchronization patterns have concentrated on undirected networks. Here, we numerically study the complete cluster synchronization patterns in directed networks composed of SLs, and demonstrate that the values of the SLs parameter and network parameter play a prominent role on the formation and stability of cluster synchronization patterns. Moreover, it is shown that there is a hierarchical dependency between the synchronization stability of different clusters in directed networks. The stability of one cluster can be affected by another cluster, but not vice versa. Without loss of generality, the results are validated in another SLs network with more complex topology.

Chaotic optical communications at 56 Gbit/s over 100-km fiber transmission based on a chaos generation model driven by long short-term memory networks.

Chaotic optical communication technology is considered as an effective secure communication technology, which can protect information from a physical layer and is compatible with the existing optical networks. At present, to realize long-distance chaos synchronization is still a very difficult problem, mainly because well-matched hardware cannot always be guaranteed between the transmitter and receiver. In this Letter, we introduce long short-term memory (LSTM) networks to learn a nonlinear dynamics model of an opto-electronic feedback loop, and then apply the trained deep learning model to generate a chaotic waveform for encryption and decryption at the transmitter and receiver. Furthermore, to improve the security, we establish a deep learning model pool which consists of different gain trained models and different delay trained models, and use a digital signal to drive chaos synchronization between the receiver and transmitter. The proposed scheme is experimentally verified in chaotic-encrypted 56-Gbit/s PAM-4 systems, and a decrypted performance below 7%FEC threshold (BER = 3.8×10-3) can be achieved over a 100-km fiber transmission.

Instability of optical phase synchronization between chaotic semiconductor lasers.

The instability of optical phase chaos synchronization between semiconductor lasers under master-slave open-loop configuration is investigated. The phase difference between the master and slave lasers is obtained and analyzed in experiment by heterodyne detection and Hilbert transform, and in simulation by solving the rate equations. The results show that the phase difference only maintains in a short duration time and then jumps to another value. A statistical analysis shows that both duration time and jumping values are random, proving that the phase chaos synchronization is unstable. A theoretical analysis shows that the instability of phase synchronization is caused by the jumping of the external cavity mode in the master laser.

High-speed physical key distribution based on dispersion-shift-keying chaos synchronization in commonly driven semiconductor lasers without external feedback.

We propose a scheme of high-speed physical key distribution based on dispersion-shift-keying chaos synchronization in two semiconductor lasers without external feedback (response lasers), which are driven by a common external-cavity semiconductor laser (drive laser). In this scheme, the dispersion introduces a laser field beating-induced nonlinear transformation to the outputs of drive laser and renders the correlation elimination between the drive and response lasers improving the security of key distribution. Moreover, the commonly driven lasers without external feedback constitute an open-loop synchronization configuration and yield a short synchronization recovery time of a subnanosecond supporting the implementation of high-speed key distribution. With these two merits, we numerically demonstrate a 1.2 Gb/s secure key distribution with a bit error ratio below 3.8×10-3.

Scheme of coherent optical chaos communication.

We propose and numerically demonstrate a scheme of coherent optical chaos communication using semiconductor lasers for secure transmission of optical quadrature amplitude modulation (QAM) signals. In this scheme, a laser intensity chaos and its delayed duplicate are used to amplitude-quadrature modulate a continuous-wave light to generate a chaotic carrier. High-quality chaotic carrier synchronization between the transmitter and receiver is guaranteed by laser intensity chaos synchronization, avoiding laser phase fluctuation. Decryption is implemented by a 90 deg optical hybrid using the synchronous chaotic carrier as local light. Secure transmission of an optical 40 Gb/s 16QAM signal is demonstrated by using a laser intensity chaos with a bandwidth of 11.7 GHz. The system performances are evaluated by analyzing a bit error ratio with different masking coefficients, signal rates, synchronization coefficients, parameter mismatches, and dispersion compensation. It is believed that this scheme can pave a way for high-speed optical chaos communication.

Stable period-one oscillations in a semiconductor laser under optical feedback from a narrowband fiber Bragg grating.

Period-one (P1) oscillations in a semiconductor laser under optical feedback from a narrowband fiber Bragg grating (FBG) are numerically investigated. FBG feedback enhances the stability of P1 oscillations compared to the conventional mirror feedback in the form of P1 microwave linewidth and phase noise reduction and residual noise peaks suppression. In the proposed scheme, the FBG has a narrow bandwidth smaller than the laser relaxation oscillation frequency. Then it effectively suppresses the coherence collapse of the laser by filtered feedback. Hence it can keep the laser in P1 operation even under relatively strong feedback. Besides, a uniform FBG has a comb-filtered reflectivity spectrum with a main lobe surrounded by several side lobes. Hence it can limit the external cavity modes by each lobe. As a result, FBG feedback can reduce microwave linewidth and phase noise by sustaining stronger feedback power and improve side-peak suppression ratio (SPSR) by filtering external cavity modes. The effects of stabilization are enhanced by properly increasing grating bandwidth. By fine-tuning the feedback delay time, the microwave linewidth can be reduced to a local minimum which reveals the optimal locking between P1 frequency and one of the external cavity modes. Increasing the feedback delay time, the local minimum linewidth can be further reduced. FBG feedback reduces the microwave linewidth by up to more than an order of magnitude and improves the SPSR by up to more than two orders of magnitude than mirror feedback using the same delay time.

Chaotic time-delay signature suppression with bandwidth broadening by fiber propagation.

Chaotic emission of a semiconductor laser is investigated through propagation over a fiber for achieving broadening of the bandwidth and suppression of the time-delay signature (TDS). Subject to delayed optical feedback, the laser first generates chaos with a limited bandwidth and an undesirable TDS. The laser emission is then delivered over a standard single-mode fiber for experiencing self-phase modulation, together with anomalous group-velocity dispersion, which leads to the broadening of the optical bandwidth and suppression of the TDS in the intensity signal. The effects are enhanced as the input power launched to the fiber increases. By experimentally launching up to 340 mW into a 20 km fiber, the TDS is suppressed by 10 times to below 0.04, while the bandwidth is broadened by six times to above 100 GHz. The improvement of the chaotic signal is potentially useful in random bit generation and range detection applications.

Experimental investigation on feedback insensitivity in semiconductor ring lasers.

The insensitivity to optical feedback is experimentally measured for a semiconductor ring laser (SRL) and compared to that of a Fabry-Perot laser (FPL) fabricated with the same technology and on the same material. An analysis of the optical spectra reveals that the SRL remains nearly unaffected for values of optical feedback as strong as -23 dB. Furthermore, through both optical linewidth and self-mixing measurements, we show that the tolerance to feedback in SRLs is 25-30 dB stronger than in FPLs. This property makes SRLs very interesting candidates for the development of feedback-insensitive optical sources.

Frequency-modulated microwave generation with feedback stabilization using an optically injected semiconductor laser.

Generation of frequency-modulated continuous-wave (FMCW) microwave signals is investigated using the period-one (P1) dynamics of a semiconductor laser. A modulated optical injection drives the laser into P1 oscillation with a modulated microwave frequency, while adding feedback to the injection reduces the microwave phase noise. Using simply a single-mode laser, the tunability of P1 dynamics allows for wide tuning of the central frequency of the FMCW signal. A sweep range reaching 7.7 GHz is demonstrated with a sweep rate of 0.42 GHz/ns. When the external modulation frequency matches the reciprocal of the feedback delay time, feedback stabilization is manifested as an increase of the frequency comb contrast by 30 dB for the FMCW microwave signal.

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.

Square-wave oscillations in a semiconductor ring laser subject to counter-directional delayed mutual feedback.

Square-wave (SW) switching of the lasing direction in a semiconductor ring laser (SRL) is investigated using counter-directional mutual feedback. The SRL is electrically biased to a regime that supports lasing in either counter-clockwise (CCW) or clockwise (CW) direction. The CCW and CW modes are then counter-directionally coupled by optical feedback, where the CCW-to-CW and CW-to-CCW feedback are delayed by τ1 and τ2, respectively. The mutual feedback invokes SW oscillations of the CCW and CW emission intensities with a period of T≈τ1+τ2. When τ1=τ2, symmetric SWs with a duty cycle of 50% are obtained, where the switching time and electrical linewidth of the SWs can be reduced to, respectively, 1.4 ns and 1.1 kHz by strengthening the feedback. When τ1≠τ2, asymmetric SWs are obtained with a tunable duty cycle of τ1/(τ1+τ2). High-order symmetric SWs with a period of T=(τ1+τ2)/n can also be observed for some integer n. Symmetric SWs of order n=13 with a period of T=10.3 ns are observed experimentally.

Random bit generation at tunable rates using a chaotic semiconductor laser under distributed feedback.

A semiconductor laser with distributed feedback from a fiber Bragg grating (FBG) is investigated for random bit generation (RBG). The feedback perturbs the laser to emit chaotically with the intensity being sampled periodically. The samples are then converted into random bits by a simple postprocessing of self-differencing and selecting bits. Unlike a conventional mirror that provides localized feedback, the FBG provides distributed feedback which effectively suppresses the information of the round-trip feedback delay time. Randomness is ensured even when the sampling period is commensurate with the feedback delay between the laser and the grating. Consequently, in RBG, the FBG feedback enables continuous tuning of the output bit rate, reduces the minimum sampling period, and increases the number of bits selected per sample. RBG is experimentally investigated at a sampling period continuously tunable from over 16 ns down to 50 ps, while the feedback delay is fixed at 7.7 ns. By selecting 5 least-significant bits per sample, output bit rates from 0.3 to 100 Gbps are achieved with randomness examined by the National Institute of Standards and Technology test suite.

Polarization-resolved time-delay signatures of chaos induced by FBG-feedback in VCSEL.

Polarization-resolved chaotic emission intensities from a vertical-cavity surface-emitting laser (VCSEL) subject to feedback from a fiber Bragg grating (FBG) are numerically investigated. Time-delay (TD) signatures of the feedback are examined through various means including self-correlations of intensity time-series of individual polarizations, cross-correlation of intensities time-series between both polarizations, and permutation entropies calculated for the individual polarizations. The results show that the TD signatures can be clearly suppressed by selecting suitable operation parameters such as the feedback strength, FBG bandwidth, and Bragg frequency. Also, in the operational parameter space, numerical maps of TD signatures and effective bandwidths are obtained, which show regions of chaotic signals with both wide bandwidths and weak TD signatures. Finally, by comparing with a VCSEL subject to feedback from a mirror, the VCSEL subject to feedback from the FBG generally shows better concealment of the TD signatures with similar, or even wider, bandwidths.