The response of a nonlinear optical oscillator subject to a delayed broadband bandpass filtering feedback is studied experimentally, numerically, and analytically. The oscillator loop is characterized by a high cutoff frequency with a response time tau approximately 10 ps and by a low cutoff frequency with a response time theta approximately 1 micros. Moreover, the optoelectronic feedback also consists of a significant delay tauD of the order of 100 ns. Depending on two key physical parameters, the loop gain beta and the nonlinearity operating point Phi, a large variety of multiple time scale regimes are reported, including slow or fast periodic oscillations with different waveforms, regular or chaotic breathers, slow time envelope dynamics, complex and irregular self-pulsing, and fully developed chaos. Many of these regimes are exhibiting new features that are absent in the classical first-order scalar nonlinear delay differential equations (DDEs), which differ in the modeling by the low cutoff only. Nearly all kinds of solutions are recovered numerically by a new class of integro-DDE (iDDE) that take into account both the high and low cutoff frequencies of the feedback loop. For moderate feedback gain, asymptotic solutions are determined analytically by taking advantage of the relative values of the time constants tau, theta, and tauD. We confirm the experimental observation of two distinct routes to oscillatory instabilities depending on the value of Phi. One route is reminiscent of the square wave oscillations of the classical first-order DDE, but the other route is quite different and allows richer wave forms. For higher feedback gain, these two distinct regimes merge leading to complex nonperiodic regimes that still need to be explored analytically and numerically. Finally, we investigate the theoretical limits of our iDDE model by experimentally exploring phenomena at extreme physical parameter setting, namely, high-frequency locking at strong feedback gain or pulse packages for very large delays. The large variety of oscillatory regimes of our broadband bandpass delay electro-optic oscillator is attractive for applications requiring rich optical pulse sources with different frequencies and/or wave forms (chaos-based communications, random number generation, chaos computing, and generation of stable multiple GHz frequency oscillations).

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Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators.

We propose a method based on a scalar second-order difference-differential equation to obtain intensity chaos from a laser diode with a nonlinear delayed feedback. The method can be used for encrypting, transmitting, and decrypting a signal in a chaos-based communication system. The core of the chaotic transmitter and receiver is formed by an electrooptic modulator that is used to generate a strong reproducible nonlinearity and chaotic waveforms of extremely high Lyapunov dimensionality. The system opens the way to ultrafast chaotic communications.

Optical communication with synchronized hyperchaos generated electrooptically

Research on the secure communication applications of chaotic synchronization is reviewed. A number of secure communication methods and devices using different types of synchronous behavior are examined. For the purpose of comparing existing methods, quantitative characteristics of operating capacity of various schemes are introduced and estimated. An extremely noise-stable secure information transmission method, based on the phenomenon of generalized chaos synchronization, is proposed. All of the methods considered are systematically checked for efficiency for the first time by numerically simulating unidirectionally coupled chaotic Rossler systems selected for transmitting and receiving oscillators. The key advantages and disadvantages of secure information transmission schemes using synchronized chaotic oscillations are discussed. The experimental data gathered in this field are also reviewed.

On the use of chaotic synchronization for secure communication

Photonic delay systems have revolutionized the hardware implementation of Recurrent Neural Networks and Reservoir Computing in particular. The fundamental principles of Reservoir Computing strongly facilitate a realization in such complex analog systems. Especially delay systems, which potentially provide large numbers of degrees of freedom even in simple architectures, can efficiently be exploited for information processing. The numerous demonstrations of their performance led to a revival of photonic Artificial Neural Network. Today, an astonishing variety of physical substrates, implementation techniques as well as network architectures based on this approach have been successfully employed. Important fundamental aspects of analog hardware Artificial Neural Networks have been investigated, and multiple high-performance applications have been demonstrated. Here, we introduce and explain the most relevant aspects of Artificial Neural Networks and delay systems, the seminal experimental demonstrations of Reservoir Computing in photonic delay systems, plus the most recent and advanced realizations.

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Tutorial: Photonic neural networks in delay systems

https://iopscience.iop.org/article/10.1088/1751-8121/50/10/103001/pdf

Spatio-temporal phenomena in complex systems with time delays

An electronic noise-generator producing high-dimensional narrowband chaotic signals is reported. The generator has the form of a delayed nonlinear feedback circuit. It has potential applications in secure communications.

Bandpass chaotic dynamics of electronic oscillator operating with delayed nonlinear feedback

We demonstrate a chaos generator involving two time delays and two nonlinear functions. Dynamic behaviors are numerically and experimentally observed. The complexity of the dynamics is discussed in terms of Lyapunov exponents and dimensions. The setup can provide a new architecture for enhancing message security in chaos encryption systems.

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Demonstration of a chaos generator with two time delays.

Nonlinear delayed dynamics was first proposed in Optics by Kensuke Ikeda in 1979. Since then, many different setups based on similar dynamical principles were carried out experimentally, first to explore the numerous and various behaviours, and then to use the high complexity chaotic regimes for optical data encryption. After a brief review of the different setups and principles, we will report on 4 different optoelectronics realizations developed in our group, emphasizing on the characteric properties of each setup, and their implementation in chaos-based secure communication systems.

Eric Genin

Papers

Bandpass chaotic dynamics of electronic oscillator operating with delayed nonlinear feedback

Demonstration of a chaos generator with two time delays.

From Ikeda ring cavity to optoelectronic setups dedicated to chaos-based secure communications

From Ikeda ring cavity to optoelectronic set–ups dedicated to chaos based secure communications