Complex dynamics in photonic delay systems: a story of consistency and unpredictability

Abstract

- Introducció Photonic systems are powerful test-beds for the investigation of complex dynamics emerging from delays in feedback or coupling. The aim of this thesis was to study two relevant properties of photonic delay systems with direct applications in current information processing and encryption systems: consistency and unpredictability. We characterized the ability of laser systems with delay to exhibit, on the one hand, reliable complex dynamics when an external stimulus is applied, and on the other hand, unpredictable complex behavior, depending on the operating conditions. Consistency properties have been studied as a necessary condition for the implementation of Reservoir Computing schemes. The property of unpredictability has been exploited in the application of random bit generation. - Contingut de la investigació For the characterization of the consistency properties, we used three different experiments based on a semiconductor laser systems with delay that followed the drive-response scheme. Through the analysis of the responses to a repeated drive, a consistent or inconsistent behavior can be identified. In the first setup, we investigated consistency of a semiconductor laser to its own time-delayed feedback, so that the drive was the self-generated complex signal, and the response system was the semiconductor laser itself. A high accuracy in the repetition of the drive was achieved with the design of a fiber-optic setup with two feedback loops. This allowed us the extraction of measures like the sub-Lyapunov exponent. We extended the study to the use of a semiconductor laser system with excitability properties subject to electrical input pulse trains. Here, two different pulse trains modulating the pump current of the laser were used as drives, while the semiconductor laser with delayed feedback operating in the chaotic regime of Low Frequency Fluctuations acted as response system. The purpose of this experiment was to study the possibility and requirements to induce a consistent response, particularly the power drop-out, with the injection of a short pulse. To complement the investigations on consistency, the last drive-response scheme used was an electro-optic intensity oscillator driven by 3 scalar signals: an harmonic waveform, a sequence of pulses, and recorded time-traces from the autonomous dynamics. Under certain conditions, the response system can show hysteresis and coexistence of multistable states. We introduced new tools to quantify consistency and identified common features of the setups investigated. Our study showed that when the autonomous dynamics were periodic or period doubled, a consistent response was obtained independent of the drive. In cases of bistable dynamics, new sorts of consistency were discovered, like reproducible time-position transitions between the two states. The last Chapter of this thesis was devoted to the use of the unpredictable dynamics for the generation of random bits. A semiconductor laser with polarization rotated feedback was utilized to provide chaos characterized by randomness-like features, including a flat broad spectrum, suppression of the delay echoes of the autocorrelation function and no recurrences in the temporal oscillations. Nevertheless, we found that other factors like the data acquisition and the postprocessing of the signal also affected the randomness of the finally generated bits. The validity of our guidelines was proven with our random bit generator, enhancing its generation rate up to 160Gbit/s. - Conclusió With this work, we explored the emerging complex behavior in laser systems with delay. In particular, we characterized their ability to display a consistent behavior, and deterministic chaos that can lead to unpredictable dynamics. We introduced new tools and measures to quantify consistency, and guidelines for an optimum performance of a random bit generator. Altogether, our results represent a significant contribution to the areas where these two properties play a role, such as information processing and secure optical communications. Contents: Resumen vii Summary ix List of publications xiii 1 Introduction 1 1.1 Delay systems 2 1.2 Photonic delay systems 3 1.2.1 Photonic delay systems in the drive-response scheme 3 1.3 About Consistency 4 1.3.1 Applications of consistency 5 1.4 About Unpredictability 6 1.4.1 Applications of unpredictable dynamics 7 1.5 Overview of this Thesis 8 2 Concepts and tools 11 2.1 Consistency and Generalized Synchronization 11 2.2 Tools to measure consistency 13 2.2.1 The sub-Lyapunov exponent 13 2.2.2 Inter- and intra-correlations 15 2.2.3 Review of the consistency measures 19 2.3 Tools to measure unpredictability 21 3 Consistency of a laser to time delayed feedback 23 3.1 Introduction 23 3.2 Experimental implementation 24 3.2.1 Dynamical performance 26 3.3 Quantifying consistency from experimental data 29 3.3.1 Consistency correlation 29 3.3.2 Sub-Lyapunov exponents from transverse distribution functions 33 3.4 Sub-Lyapunov exponent in numerical simulations 39 3.5 Summary and conclusions 41 4 Consistency of a laser system to input pulse trains 43 4.1 Introduction 43 4.2 Experimental realization 44 4.2.1 A study at slow timescales 48 4.3 Influence of the history of pulses 48 4.3.1 Consistency for a bimodal distribution of drive pulses 51 4.4 Influence of the inter-pulse intervals 54 4.4.1 Consistency for a uniform distribution of drive pulses 55 4.5 Filtering the responses 58 4.6 Summary and conclusions 59 5 Consistency of an electro-optic intensity oscillator 61 5.1 Introduction 61 5.2 Experimental realization 62 5.2.1 Methodology 64 5.3 Dynamics without modulation 65 5.3.1 Changing : from fixed point to chaos 65 5.3.2 Changing and 0: the bifurcation diagrams 66 5.4 Consistency with an external drive 71 5.4.1 Harmonic drive 72 5.4.2 Pseudo-random pulse distribution 74 5.4.3 Recorded time traces 77 5.5 Summary and conclusions 83 6 Random bit generation with a chaotic laser 87 6.1 What are random numbers? 88 6.1.1 Types of random bit generators 89 6.1.2 Why a random bit generator based on a semiconductor laser? 89 6.2 Experimental implementation 90 6.3 Dynamical properties for a good Random Bit Generator 91 6.3.1 RF power spectrum 92 6.3.2 Autocorrelation conditions 93 6.3.3 Systematic study of the AC properties 96 6.3.4 Role of noise 97 6.4 Acquisition conditions 97 6.4.1 Sampling rate and data acquisition . 99 6.5 Postprocessing 100 6.6 Assessing the randomness 102 6.7 Generation of random bit sequences 105 6.8 Interplay of postprocessing and sampling rate 106 6.9 Optimizing the bit generation rate 108 6.9.1 Extension to 16-bit digitization 110 6.9.2 Information Theoretic limits 110 6.10 Discussion 112 6.11 Summary 115 7 Conclusions and future work 117 7.1 Conclusions 117 7.2 Future work 119 Bibliography 121