www.wiley-vch.de

Uchida

Optical C

omm

unicationw

ith Chaotic Lasers

Starting with an introduction to the fundamental physics in chaotic instabilities in laser systems, this comprehensive and unifi ed reference goes on to present the techniques and technology of synchronization of chaos in coupled lasers, as well as the many applications to lasers and optics, communications, security and information technology. Through-out, it presents the current state of knowledge, including encoding/decoding techniques, performance of chaotic communication systems, random number generation, and novel communication technologies.

From the contents:I. Basic physics of chaos and synchronization in lasers � Introduction � Basics of chaos and laser � Generation of chaos in lasers � Analysis of chaotic laser dynamics: Example of semiconductor laser with optical feedback � Synchronization of chaos in lasers � Analysis of synchronization of chaos: example of unidirectionally coupled semiconductor lasers with optical feedbackII. Application of chaotic lasers to optical communication and information technology � Basic concept of optical communication with chaotic lasers � Implementation of optical communication with chaotic lasers � Secure key distribution based on information-theoretic security with chaotic lasers � Random number generation with chaotic lasers � Controlling chaos in lasers � Other applications with chaotic lasers

Atsushi Uchida received his PhD degree in electrical engineer-ing from Keio University in Yokohama, Japan, in 2000. He is an associate professor at the Department of Information and Computer Sciences at Saitama University in Saitama, Japan, and is currently working on synchronization of chaotic lasers, its applications for optical communication, secure key genera-tion with chaotic lasers, and random number generation with chaotic lasers. Dr. Uchida is a member of the Optical Society of America, IEEE Photonics Society, and American Physical Society.

Atsushi Uchida

Optical Communication with Chaotic Lasers

Applications of Nonlinear Dynamicsand Synchronization

57268File AttachmentCover.jpg

Atsushi Uchida

Optical Communication with

Chaotic Lasers

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Atsushi Uchida

Optical Communication withChaotic Lasers

Applications of Nonlinear Dynamicsand Synchronization

The Author

Dr. Atsushi UchidaSaitama UniversityDepartment of Information and ComputerSciencesSaitama, Japanauchida@mail.saitama-u.ac.jp

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Persistence pays off.

Foreword

In this book, Professor Atsushi Uchida lays the groundwork for many possibleapplications of nonlinear dynamical systems in optical science and engineering –including optical communications with chaotic lasers. The first part of the bookintroduces lasers and the phenomena of chaos and synchronization, while the latterpart explores applications of chaotic lasers to communications, control and randomnumber generation.

That lasers can exhibit irregular dynamics has been known since almost the firstobservations of laser operation. To actually understand and diagnose the origins ofsuch behavior is an effort that still continues today. The realization that deterministicchaos, sources of random noise, and the interactions of multiple frequencies in anonlinear optical medium could generate dynamics that range over many orders ofmagnitude in time scales has gradually grown and blossomed into a new researchfield. We know now that the development of accurate models to understand andpredict the nonlinear dynamical behavior of lasers and optical systems requires newtools for observation, measurement and interpretation that involve both softwareand hardware, numerical models and stochastic simulations. The quantitativedescription of all the variables that describe the spatiotemporal dynamics of electro-magnetic fields interacting with matter require newmathematical concepts, numer-ical simulation techniques, and considerable imagination to think of possible newapplications that may prove useful in the future.

It is truly breathtaking to look back at the period in the 1950s and 1960s inwhich, almost simultaneously, the basic concepts of chaos, computers, informa-tion theory and lasers were discovered and developed. In the 1970s and 1980s,semiconductor lasers, integrated circuits, and fiber optics merged with andtransformed the appearance, properties and applications of lasers. In the lasttwo decades, miniaturized computer technology and digital imaging have begunto transform our world in ubiquitous ways that use lasers and electronics foreveryday tasks and activities.

While almost all the mathematics and engineering needed to understand the vastmajority of devices and instruments are ‘‘linear’’ in concept, we have learned that

VII

almost everything in nature and the laboratory behaves nonlinearly when oneextends the ranges of parameters. As scientists engaged in exploring the unknown,we have glimpsed that in fact, nothing is truly linear in behavior, though linearity is agood and most useful approximation for many purposes. Chaotic and nonlineardynamics appears everywhere we look, from the solar system to the motion ofpendula, from turbulence in fluids to the fluctuations of light emitted by lasers. Newparadigms have been established as the field of nonlinear dynamics of opticalsystems has been developed through the painstaking efforts of engineers, mathe-maticians and physicists.As we begin to enlarge the boundaries of exploration, whether through new

mathematical and computational tools or experimental instruments and measure-ments, we discover unexpected phenomena and begin to piece together parts of theinfinite variety of patterns in space and time that constitute the dynamically evolvingworld around us. One of these new, unexpected phenomena is that of synchroniza-tion of chaotic nonlinear systems. While the concept of synchronization of periodicoscillators may seem natural to most people, the observation that two or morechaotic systems can synchronize is surprising and leads to novel possibilities forcommunication, control and sensing systems.This book is a major step and a systematic effort to bring together those aspects of

mathematics, physics and engineering that are basic in the emerging field ofcommunication with chaotic dynamical systems. It describes many examplesfrom recent literature that illustrate how one can communicate with lasers thatare chaotic, using the phenomenon of synchronization of chaotic systems. Thedynamics of laser and optoelectronic systems that can be used as transmitters andreceivers for chaotic communication are often described by delay-differential equa-tions. The complex and high-dimensional dynamics that can be generated by suchsystems, and their synchronization properties, are a rich and open field. Thetechniques developed to generate chaotic waveforms for communication at highbit rates, and the methods to encode and decode information, are active fields ofresearch, with much work that remains to be done at the device and system levels.Communication networks and the privacy and security of information are importanttopics of exploration.While I have only seen an overview of the text, it is clear that this book contains

many interesting connections of the central topic to related areas in informationscience and communications theory. One of the novel features of this book is thechapter on random number generators using chaotic lasers, a newly emergingfield where the author has made pioneering contributions. Another set of topicsthat one may not expect from the title are presented in the final chapter, on remotesensing, blind-source separation and fractal optics. These represent areas of theauthors active research interests. Another great feature of the book are the manyappendices to the chapters that provide detailed guidance to the student andresearcher through the tricky landscape of computational algorithms. As a colla-borator and colleague who has over a period of time spent many (many!) hoursdiscussing research with Professor Uchida, and as a student of both optics and

VIII Foreword

nonlinear dynamics, it is with much pleasure and anticipation that I lookforward to seeing this book in print, and learning about many of the topicsdeveloped in it.

Rajarshi RoyInstitute for Physical Science and Technologyand Department of PhysicsUniversity of Maryland, College ParkUSA

Foreword IX

Contents

Foreword VIIPreface XXVII

Part One Basic Physics of Chaos and Synchronization in Lasers1 Introduction 3

1.1 Lasers and Chaos 61.1.1 Lasers 61.1.2 Chaos 61.1.3 Connection Between Lasers and Chaos 71.1.4 Chaos and Noise 8

1.2 Synchronization of Chaos and Optical Communication 81.2.1 Synchronization of Chaos 81.2.2 Optical Communication with Synchronized Chaotic Lasers 9

1.3 Random Number Generation with Chaotic Lasers and OtherApplications 111.3.1 Random Number Generation 111.3.2 Controlling Chaos and Other Applications 12

1.4 Research Directions for Engineering Applications with Chaotic Lasers 121.5 Outline of This Book 13

1.5.1 Contents of Chapters: First Parts for the Basics 141.5.2 Contents of Chapters: Second Part for the Applications 16

2 Basics of Chaos and Laser 192.1 History of Instabilities of Laser Output 19

2.1.1 Examples of Laser Instabilities 192.1.1.1 First Observation of Laser Instabilities in a Ruby Laser 192.1.1.2 Green Problem in a Solid-State Laser for Second-

Harmonic Generation 202.1.1.3 Feedback-Induced Instability in a Semiconductor Laser for

Optical Disks and Optical Communication Systems 212.1.2 Inherent Noise-Induced Instabilities 232.1.3 Deterministic Chaos in Lasers and Lorenz-Haken Equations 24

XI

2.2 Basic Chaos Theory 262.2.1 Logistic Map in Discrete-Time System 26

2.2.1.1 Recurrence Formula 262.2.1.2 Chaotic Sequence 272.2.1.3 Bifurcation Diagram 282.2.1.4 Lyapunov Exponent 30

2.2.2 Lorenz Model in a Continuous-Time System 312.2.2.1 Lorenz Equations 312.2.2.2 Temporal Waveform and Attractor 322.2.2.3 Sensitive Dependence on Initial Conditions 322.2.2.4 Bifurcation Diagram 332.2.2.5 Lyapunov Exponent 33

2.2.3 Reconstruction of Attractor in Time-Delayed Phase Space 352.2.4 Types of Bifurcation and Route to Chaos 36

2.2.4.1 Period-Doubling Route to Chaos 362.2.4.2 Quasiperiodicity Route to Chaos 372.2.4.3 Intermittency Route to Chaos 37

2.3 Basic Laser Theory 382.3.1 Light–Matter Interaction for Laser Radiation 38

2.3.1.1 Elements of a Laser 382.3.1.2 Two-Atomic-Level Description and Mechanism of Laser

Oscillation 392.3.2 Radiative Recombination of Electron–Hole Pairs in

Semiconductor Lasers 402.3.3 Rate Equations for Laser Dynamics 412.3.4 Relaxation Oscillation Frequency 442.3.5 Mechanism of Chaotic Instability in Lasers 45

2.4 Connection Between Chaos and Lasers 462.4.1 Single-Mode Laser Model Based on Maxwell–Bloch

Equations 462.4.2 Classification of Laser Models Based on Decay Rates 48

2.4.2.1 Class C Lasers (Three Variables) 482.4.2.2 Class B Lasers (Two Variables) 482.4.2.3 Class A Lasers (One Variable) 49

Appendix2.A.1 General Formula of the Linearized Equations and the

Derivation of the Linearized Equations in the Lorenz Model 512.A.2 General Formula for the Calculation of the Lyapunov Exponent in

Nonlinear Dynamical Systems Without Time Delay 532.A.3 Rössler Model 542.A.4 Derivation of the Relaxation Oscillation Frequency 55

3 Generation of Chaos in Lasers 593.1 Basics of Generation of Chaos in Lasers 59

3.1.1 Classification of Generation Techniques of Chaos in Lasers 59

XII Contents

3.1.1.1 Optical Feedback 603.1.1.2 Optical Coupling and Injection 623.1.1.3 External Modulation 623.1.1.4 Insertion of Nonlinear Element 623.1.1.5 Multimode Lasers and High-Dimensional Laser

Systems 633.1.1.6 Class C Lasers Satisfying the Condition for Lorenz–Haken

Chaos 633.1.1.7 Ikeda-Type Passive Optical Systems 63

3.1.2 Characteristics of Chaos in Lasers 643.1.2.1 High-Frequency Chaos 643.1.2.2 Existence of Two Oscillation Frequencies in Different

Orders of Magnitude 643.1.2.3 Middle Degrees of Freedom 653.1.2.4 Physical Variables 653.1.2.5 Transmission of Chaos 653.1.2.6 Variety of Laser Models 65

3.1.3 How to Distinguish between Chaos and Noise fromExperimental Data? 663.1.3.1 Experimental apparatus for Measurement of Instability

in Laser Intensity 663.1.3.2 Examples of Chaos and Noise from Real Experimental

Data 663.1.3.3 Observation of Bifurcation 673.1.3.4 Comparison with Laser Model 68

3.2 Chaos in Semiconductor Lasers 693.2.1 Semiconductor Laser with Optical Feedback 69

3.2.1.1 Dynamical Regime and L–I Characteristics 693.2.1.2 Coherence Collapse and Chaos in Experiment 713.2.1.3 Numerical Results and Bifurcation Diagram 733.2.1.4 Low-Frequency Fluctuations (LFFs) 753.2.1.5 Short-Cavity Regime 78

3.2.2 Semiconductor Laser with Polarization-Rotated OpticalFeedback 79

3.2.3 Semiconductor Laser with Optoelectronic Feedback 833.2.4 Semiconductor Laser with Optical Injection

and Coupling 883.2.4.1 Temporal Dynamics and Bifurcation Diagram 883.2.4.2 Bandwidth Enhancement of Chaos by Optical

Injection 913.2.5 Semiconductor Laser with Injection Current Modulation 923.2.6 Vertical-Cavity Surface-Emitting Laser (VCSEL) 93

3.2.6.1 Basic Characteristics 933.2.6.2 Optical Feedback 963.2.6.3 Optical Injection 97

Contents XIII

3.3 Chaos in Electro-Optic Systems 983.3.1 Ikeda-Type Nonlinear Delay Dynamics 983.3.2 Wavelength Chaos in Electro-Optic System 1013.3.3 Intensity Chaos in Electro-Optic System 1033.3.4 Phase Chaos in Electro-Optic System 105

3.4 Chaos in Fiber Lasers 1073.4.1 External Modulation 1093.4.2 Multimode (Dual-Wavelength) Dynamics 1123.4.3 Fast Polarization Dynamics 114

3.5 Chaos in Solid-State Lasers 1153.5.1 External Modulation 1163.5.2 Frequency-Shifted Optical Feedback 1173.5.3 Antiphase Dynamics in Multimode Laser 1203.5.4 Insertion of Nonlinear Crystal 122

3.6 Chaos in Gas Lasers 1243.6.1 Lorenz–Haken Chaos 1243.6.2 Insertion of Saturable Absorbers 1283.6.3 External Modulation 1303.6.4 Optical Injection 1303.6.5 Multimode Laser 1323.6.6 Low-Frequency Dynamics with Optical Feedback 134Appendix3.A.1 Numerical Model for Chaotic Dynamics in a

Semiconductor Laser 1353.A.1.1 Coherent Optical Feedback 1353.A.1.2 Polarization-Rotated Optical Feedback 1353.A.1.3 Optoelectronic Feedback (Incoherent Feedback) 136

3.A.2 Mechanism of Low-Frequency Fluctuations (LFFs) inSemiconductor Laser with Optical Feedback 137

3.A.3 Numerical Model for Chaotic Dynamics in a Vertical-CavitySurface-Emitting Laser (VCSEL) 139

3.A.4 Numerical Model for Chaotic Dynamics in an Electro-OpticSystem 139

3.A.5 Numerical Model for Chaotic Dynamics in a Fiber Laser 1403.A.6 Numerical Model for Chaotic Dynamics in a

Solid-State Laser 1403.A.6.1 Single-Mode Laser Model 1403.A.6.2 Multimode Laser Model with Spatial Hole Burning

(Tang–Statz–deMars Equations) 1413.A.7 Numerical Model for Chaotic Dynamics in a Gas Laser 142

4 Analysis of Chaotic Laser Dynamics: Example of SemiconductorLaser with Optical Feedback 1454.1 Experimental Analysis of Semiconductor Lasers with Optical

Feedback 145

XIV Contents

4.1.1 Experimental Setup 1464.1.2 Light Power Versus Injection Current (L–I)

Characteristics 1474.1.3 Temporal Waveforms, RF Spectra, Autocorrelations, and Optical

Spectra 1494.1.4 Two-Dimensional Map 155

4.2 Model for Semiconductor Laser with Optical Feedback 1564.2.1 Lang–Kobayashi Equations 1564.2.2 Derivation of the Electric-Field Amplitude and Phase of the

Lang–Kobayashi Equations from the Complex Electric-FieldEquations 158

4.2.3 Derivation of the Real and Imaginary Electric Fields of theLang–Kobayashi Equations from the Complex Electric-FieldEquations 160

4.3 Analytical Approach of Semiconductor Laser with OpticalFeedback 1614.3.1 Steady-State Solutions without Optical Feedback 1614.3.2 Linear Stability Analysis for Steady-State Solutions without Optical

Feedback 1624.3.2.1 Eigenvalues of Jacobian Matrix 1624.3.2.2 Relaxation Oscillation Frequency 165

4.3.3 Steady-State Solutions with Optical Feedback 1664.3.4 Linear Stability Analysis for Steady State Solutions with Optical

Feedback 1694.4 Numerical Analysis of Semiconductor Laser with Optical

Feedback 1724.4.1 Numerical Results of Chaotic Dynamics 173

4.4.1.1 Temporal Waveforms, FFTs, and Attractors 1734.4.1.2 Bifurcation Diagram and Two-Dimensional Dynamical

Map 1764.4.2 Linear Stability Analysis for Oscillatory Trajectory 1794.4.3 Maximum Lyapunov Exponent 181

4.5 Dimensionless Equations and Further Nonlinear Analysis 1844.5.1 Dimensionless Equations 1844.5.2 Numerical Results of Chaotic Dynamics 1874.5.3 Linear Stability Analysis for Oscillatory Trajectory in

Dimensionless Equations and Maximum LyapunovExponent 188

4.5.4 Lyapunov Spectrum 1904.5.5 Kolmogorov–Sinai Entropy and Kaplan–Yorke Dimension 192

4.6 Lang–Kobayashi Equations with Gain Saturation 1944.6.1 Lang–Kobayashi Equations with Gain Saturation 1944.6.2 Numerical Results and Histogram 1954.6.3 Linear Stability Analysis for Oscillatory Trajectory and

Measurement of Complexity 196

Contents XV

Appendix4.A.1 Derivation of the Rate Equations of Semiconductor Lasers 1994.A.2 Derivation of the Rate Equations of Semiconductor Lasers with

Optical Feedback 2054.A.3 Analytical Approach of the Stability of Steady State Solutions for

Semiconductor Laser with Optical Feedback Under the Limits ofWeak and Short Feedback 206

4.A.4 Runge–Kutta Method for the Integration of Ordinary DifferentialEquations 208

4.A.5 Gram–Schmidt Orthogonalization 210

5 Synchronization of Chaos in Lasers 2115.1 Concept of Synchronization of Chaos in Lasers 211

5.1.1 Introduction 2115.1.2 What is Synchronization? 2125.1.3 Why Should Chaos be Synchronized? 2135.1.4 Characteristics of Synchronization of Chaos in Laser Systems 2155.1.5 Synchronization of Chaos for Communication Applications 216

5.2 History of Synchronization of Chaos in Lasers 2175.2.1 Synchronization of Chaos in Electronic Circuits 217

5.2.1.1 Pecora–Carroll Method 2175.2.1.2 Pyragas Method 218

5.2.2 Synchronization of Chaos in Lasers 2195.3 Coupling Schemes and Synchronization Types 223

5.3.1 Identical Synchronization 2245.3.2 Generalized Synchronization (with High Correlation) 2255.3.3 Synchronization of Chaos in Feedback Systems 225

5.3.3.1 Open-Loop Configuration 2255.3.3.2 Closed-Loop Configuration 227

5.3.4 Mutual Coupling 2285.3.5 Linear Stability Analysis and Conditional Lyapunov Exponent 229

5.4 Examples of Synchronization of Chaos in Semiconductor Lasers 2305.4.1 Semiconductor Lasers with Coherent Optical Feedback 2315.4.2 Semiconductor Lasers with Polarization-Rotated Optical

Feedback 2365.4.3 Semiconductor Lasers with Optoelectronic Feedback 2375.4.4 Semiconductor Lasers with Optical Injection 2395.4.5 Semiconductor Lasers with Mutual Coupling 240

5.4.5.1 Symmetry Breaking and Leader–LaggardRelationship 240

5.4.5.2 Zero-Lag Synchronization 2425.4.6 Vertical-Cavity Surface-Emitting Lasers (VCSELs) 243

5.5 Examples of Synchronization of Chaos in Electro-Optic Systemsand Other Lasers 2455.5.1 Electro-Optic Systems 245

XVI Contents

5.5.2 Fiber Lasers 2485.5.3 Solid-State Lasers 2505.5.4 Gas Lasers 253

5.6 Specific Types of Synchronization 2545.6.1 Phase Synchronization 2545.6.2 Generalized Synchronization (with Low Correlation) 258

5.7 Consistency 2635.7.1 What is Consistency? 2635.7.2 Examples of Consistency in Laser Systems 2655.7.3 Application of Consistency 269

5.7.3.1 Noninvasive Testing 2695.7.3.2 Analysis of Brain Dynamics and Learning Process in

the Brain 2695.7.3.3 Physical One-Way Function 2695.7.3.4 Teaching-Learning Methodologies 2705.7.3.5 Design of Drug Delivery 270

Appendix5.A.1 Pecora–Carroll Method for Synchronization of Chaos 2705.A.2 General Formula of Linearized Equations for Coupled Differential

Equations 2715.A.3 Procedure for the Calculation of the Conditional Lyapunov

Exponent 2735.A.4 Numerical Model for Synchronization of Chaos in

Unidirectionally Coupled Semiconductor Lasers 2745.A.4.1 Coherent Optical Feedback 2745.A.4.2 Polarizaiton-Rotated Optical Feedback 2755.A.4.3 Optoelectronic Feedback (Incoherent Feedback) 276

5.A.5 Numerical Model for Synchronization of Chaos inUnidirectionally Coupled Electro-Optic Systems 277

5.A.6 Numerical Model for Synchronization of Chaos inUnidirectionally Coupled Fiber Lasers 278

5.A.7 Numerical Model for Synchronization of Chaos in UnidirectionallyCoupled Solid-State Lasers 2795.A.7.1 Single-Mode Laser Model 2795.A.7.2 Multimode Laser Model with Spatial Hole Burning

(Tang–Statz–deMars Equations) 2805.A.8 Numerical Model for Synchronization of Chaos in

Unidirectionally Coupled Gas Lasers 2815.A.9 Definition of Phase in Chaotic Temporal Waveform by the

Hilbert Transform 282

6 Analysis of Synchronization of Chaos: Example of UnidirectionallyCoupled Semiconductor Lasers with Optical Feedback 2856.1 Experimental Analysis on Synchronization of Chaos in Two

Semiconductor Lasers with Optical Feedback 285

Contents XVII

6.1.1 Experimental Setup for Synchronization of Chaos 2856.1.2 Experimental Results of Synchronization of Chaos 2876.1.3 Parameter Dependence of Synchronization of Chaos 290

6.2 Model for Synchronization of Chaos in Two Coupled SemiconductorLasers with Optical Feedback 2936.2.1 Lang–Kobayashi Equations for Synchronization of Chaos in

Unidirectionally Coupled Semiconductor Lasers with OpticalFeedback 2936.2.1.1 Coupled Lang–Kobayashi Equations 2936.2.1.2 Identical Synchronous Solution 295

6.2.2 Derivation of the Electric-Field Amplitude and Phase of theCoupled Lang–Kobayashi Equations from the ComplexElectric-Field Equations 296

6.2.3 Derivation of the Real and Imaginary Electric Fields of theCoupled Lang–Kobayashi Equations from the Complex Electric-Field Equations 299

6.3 Numerical Analysis on Synchronization of Chaos in UnidirectionallyCoupled Semiconductor Lasers with Optical Feedback 3006.3.1 Measures for Synchronization of Chaos 300

6.3.1.1 Two Types of Synchronization and Cross-CorrelationValues 300

6.3.1.2 Optical Frequency Detuning 3026.3.2 Numerical Results of Temporal Waveforms and Correlation

Plots 3036.3.3 Parameter Dependence of the Two Types of Synchronization 3056.3.4 Linear Stability Analysis of Synchronous Oscillatory Solutions

for Identical Synchronization 3086.3.4.1 Linearized Equations 3086.3.4.2 Conditional Lyapunov Exponent 3096.3.4.3 Numerical Results of Conditional Lyapunov

Exponent 3106.3.5 Open- Versus Closed-Loop Configurations 312

6.4 Experimental Analysis on Generalized Synchronization with LowCorrelation in Three Semiconductor Lasers in the Auxiliary SystemApproach 3136.4.1 Experimental Setup for Generalized Synchronization with Low

Correlation in the Auxiliary System Approach 3146.4.2 Experimental Results of Generalized Synchronization 3166.4.3 Parameter Dependence of Generalized Synchronization 3206.4.4 Dependence of Generalized Synchronization on Optical Phase of

Feedback Light 3226.5 Model for Generalized Synchronization with Low Correlation in Three

Semiconductor Lasers in the Auxiliary System Approach 3246.5.1 Coupled Lang–Kobayashi Equations for Generalized

Synchronization in the Auxiliary System Approach 325

XVIII Contents

6.5.2 Synchronous Solutions for Generalized Synchronization in theAuxiliary System Approach 327

6.6 Numerical Analysis on Generalized Synchronization of Chaos inThree Semiconductor Lasers in the Auxiliary System Approach 3276.6.1 Temporal Waveforms 3296.6.2 Parameter Dependence of Generalized Synchronization 3296.6.3 Linear Stability Analysis of Synchronous Oscillatory Solutions for

Generalized Synchronization 3316.6.3.1 Linearized Equations 3316.6.3.2 Conditional Lyapunov Exponent 3336.6.3.3 Numerical Results of Conditional Lyapunov

Exponent 3346.6.4 Two-Dimensional Map 3346.6.5 Dependence of Synchronization Quality on Optical Phase of

Feedback Light in the Closed-Loop Configuration 336Appendix6.A.1 Rate Equations for Identical Synchronization in Unidirectionally

Coupled Semiconductor Lasers with Optical Feedback in theClosed-Loop Configuration 337

Part Two Application of Chaotic Lasers to Optical Communication andInformation Technology

7 Basic Concept of Optical Communication with Chaotic Lasers 3437.1 History of Secret Communication 343

7.1.1 Cryptography 3437.1.2 Steganography 3447.1.3 Noise Communication 346

7.2 Concept of Chaos Communication 3467.2.1 Basic Idea of Chaos Communication 3467.2.2 Features of Chaos Communication 3487.2.3 Hardware Keys 3487.2.4 Synchronization for Chaos Communication 349

7.3 Characteristics of Chaos Communication 3517.3.1 Hardware-Based Communication 3517.3.2 Chaos-Synchronization-Based Communication 3517.3.3 Privacy 3527.3.4 Compatibility 3527.3.5 Analog Communication 3537.3.6 Subcarrier Communication 3537.3.7 Coherent Communication 3537.3.8 Multiplexing and Noise Tolerance 353

7.4 Encoding and Decoding Techniques 3547.4.1 Chaos Masking 3547.4.2 Chaos Modulation 3577.4.3 Chaos Shift Keying 360

Contents XIX

7.5 Tools for Quantitative Evaluation of Performance of ChaosCommunication 3627.5.1 Bit Error Rate, Q Factor, and Signal-to-Noise Ratio 3627.5.2 Modulation Format and Eye Diagram 365

8 Implementation of Optical Communication with Chaotic Lasers 3698.1 History of Chaos Communication 369

8.1.1 Chaos Communication in Electronic Circuits 3698.1.2 Chaos Communication in Optical Systems 3708.1.3 European Project for Chaos Communication 375

8.2 Examples of Communication Systems with Various Chaotic Lasers 3778.2.1 Semiconductor Lasers with Optical Feedback 377

8.2.1.1 Field Experiment of Chaos Communication 3778.2.1.2 Transmission of TV Video Signal 381

8.2.2 Semiconductor Lasers with Optoelectronic Feedback 3838.2.3 Electro-Optic Systems 3848.2.4 Fiber Lasers 386

8.3 Performance Evaluation of Optical Communication withChaotic Lasers 3908.3.1 Subcarrier Modulation 3908.3.2 Photonic Integrated Circuit and Forward Error Correction for

High-Performance Chaos Communication 3948.3.2.1 Photonic Integrated Circuit 3948.3.2.2 Chaos Communication Experiment 3968.3.2.3 Forward Error Correction Technique 3978.3.2.4 Bit-Error-Rate (BER) Performance 3978.3.2.5 Analysis for Unauthorized Users 399

8.3.3 Optical Phase Chaos for 10-Gb/s Chaos Communication 4008.3.4 Comparison of the Encoding and Decoding Schemes for Chaos

Communication 4038.4 Privacy Issues in Optical Communication with Chaotic Lasers 405

8.4.1 Introduction 4058.4.2 Reconstruction of Model and Parameter Settings by Time-Series

Analysis 4068.4.3 Parameter Estimation by Using Similar Hardware 4068.4.4 Parameter Estimation by Time-Series Analysis 4088.4.5 Direct Detection of Presence of Message from Time-Series

Analysis 4098.4.6 Summary of Privacy Issues 410

8.5 Photonic Integrated Circuit for Optical Communication withChaotic Lasers 4108.5.1 Photonic Integrated Circuit for a Semiconductor Laser

with all-Optical Feedback 4118.5.2 Photonic Integrated Circuit for two Mutually Coupled

Semiconductor Lasers 413

XX Contents

8.5.3 Photonic Integrated Circuit for Colliding-PulseMode-Locked Lasers 415

8.6 Other Encoding and Decoding Techniques 4178.6.1 Spatiotemporal Encoding 4178.6.2 Polarization Encoding 4198.6.3 Multiplexing Communications 421

8.7 New Perspective of Optical Communication with Chaotic Lasers 4238.7.1 Analogy to Biological Communication Systems 4238.7.2 Towards the World of Scientific Fiction 424

9 Secure Key Distribution Based on Information-TheoreticSecurity with Chaotic Lasers 4279.1 Introduction 427

9.1.1 Secure Key Distribution 4279.1.2 Computational Security and Information-Theoretic Security 428

9.2 Concept of Information-Theoretic Security 4299.2.1 History of Information-Theoretic Security and Maurers Satellite

Scenario 4299.2.2 Bounded Observability 430

9.3 Implementation of Information-Theoretic Security withChaotic Lasers 4319.3.1 Bounded Observability with Chaotic Semiconductor Lasers 4319.3.2 Public-Channel Cryptography with Coupled Chaotic

Semiconductor Lasers 4359.3.3 Bidirectional Message Transmission with Mutually Coupled

Semiconductor Lasers 4399.4 Information-Theoretic Security with Optical Noise 441

9.4.1 Ultralong Fiber-Laser System 441

10 Random Number Generation with Chaotic Lasers 44510.1 Introduction 445

10.1.1 Needs for Random Number Generation 44510.1.2 Extraction of Randomness from Chaotic Lasers 447

10.2 Types of Random Number Generators 44710.2.1 What are Random Numbers? 447

10.2.1.1 Independence 44810.2.1.2 Unpredictability 449

10.2.2 Two Types of Random Number Generators 44910.2.2.1 Physical Random Number Generators (RNG) 44910.2.2.2 Pseudorandom Number Generators (PRNG) 451

10.2.3 Issues of Conventional Random Number Generators 45110.3 Examples of Random Number Generators with Chaotic Lasers 452

10.3.1 Monobit Generation with Two Lasers 45210.3.1.1 Scheme 45210.3.1.2 Parameter Dependence 457

Contents XXI

10.3.2 Monobit Generation with One Laser 46010.3.3 Multibit Generation with One Laser 46110.3.4 Postprocessing for High-Speed Random Number

Generation 46310.3.5 Bandwidth Enhancement of Chaotic Lasers for High-Speed

Random Number Generation 46510.3.6 Photonic Integrated Circuit for Random Bit Generators 469

10.4 Application of Chaotic-Laser-Based Random Number Generatorsto High-Speed Quantum Key Distribution 472

10.5 Numerical Evaluation of Random Number Generator asEntropy Source 47510.5.1 Entropy Generation from Internal Noise by Chaotic

Dynamics 47510.5.2 Estimation of Entropy 47710.5.3 Entropy Rate and Nondeterministic Bit Generation 478

10.6 Conventional Methods for Physical Random Number Generators 48010.6.1 Thermal Noise 480

10.6.1.1 Direct Amplification of Thermal Noise 48010.6.1.2 Metastability 48110.6.1.3 Two Oscillators with Frequency Jitter 482

10.6.2 Quantum Noise 48410.6.3 Optical Noise (Spontaneous Emission Noise) 48510.6.4 Radiation from Radioactive Nuclide 48610.6.5 Chaotic Dynamics in Electronic Circuits 48710.6.6 Traditional Physical Devices 48710.6.7 Other Methods 48810.6.8 Commercial Physical Random Number Generators 488

10.6.8.1 Intel Chip (Intel) 48810.6.8.2 Random Master (Toshiba) 48910.6.8.3 Random Streamer (FDK) 49010.6.8.4 Quantis (ID Quantique) 490

10.7 Postprocessing Techniques for Improvement of Randomness 49010.7.1 von Neumann Method 49110.7.2 Exclusive-OR (XOR) Method 492

10.8 Pseudorandom Number Generators 49310.8.1 Linear Congruential Method 49310.8.2 M sequence and Generalized Feedback Shift Register

(GFSR) 49410.8.3 Combined Tausworthe Method 49610.8.4 Mersenne Twister 496

10.9 Statistical Evaluation of Random Numbers with NIST SpecialPublication 800-22 Test Suite 49710.9.1 Strategies for Statistical Analysis of Random Number

Generators 49710.9.2 Evaluation of p-Values 499

XXII Contents

10.9.3 Interpretation of Empirical Results 50110.9.3.1 Proportion of p-Values 50110.9.3.2 Uniformity of Distribution of p-Values 501

10.9.4 Tendency of Passed/Failed NIST SP 800-22 Tests inLaser-Chaos-Based Random Number Generators 502

10.9.5 Other Statistical Tests of Randomness 503Appendix10.A.1 Recipe for High-Quality Random Number Generators 50310.A.2 Dichtl method for Postprocessing of Random Number

Generators 50410.A.3 Algorithm of Mersenne Twister Pseudorandom Number

Generator 50510.A.4 Detailed Description of NIST Special Publication 800-22 506

10.A.4.1 Frequency (Monobit) Test 50610.A.4.2 Frequency Test within a Block 50610.A.4.3 Cumulative Sums (Cusums) Test 50610.A.4.4 Runs Test 50710.A.4.5 Tests for the Longest-Run-of-Ones in a Block 50710.A.4.6 Binary Matrix Rank Test 50710.A.4.7 Discrete Fourier Transform (Spectral) Test 50710.A.4.8 Non-overlapping Template Matching Test 50710.A.4.9 Overlapping Template Matching Test 50710.A.4.10 Maurers ‘‘Universal Statistical’’ Test 50810.A.4.11 Approximate Entropy Test 50810.A.4.12 Random Excursions Test 50810.A.4.13 Random Excursions Variant Test 50810.A.4.14 Serial Test 50810.A.4.15 Linear Complexity Test 508

11 Controlling Chaos in Lasers 51111.1 Classification of Controlling Chaos 511

11.1.1 Feedback Control Method 51111.1.1.1 OGY Method 51111.1.1.2 Occasional Proportional Feedback (OPF)

Method 51211.1.1.3 Continuous Feedback Control Method 513

11.1.2 Nonfeedback Control Method 51411.2 Examples of Controlling Chaos in Lasers 515

11.2.1 Feedback Control Method for Controlling Chaos in Lasers 51511.2.1.1 Occasional Proportional Feedback (OPF)

Method 51511.2.1.2 Continuous Feedback Control Method 518

11.2.2 Nonfeedback Control Method for Controlling Chaos inLasers 52111.2.2.1 Loss Modulation 521

Contents XXIII

11.2.2.2 High-Frequency Injection (HFI) Method forSemiconductor Lasers 523

11.2.2.3 Stabilization to High-Periodic Oscillations 52511.3 Applications of Controlling Chaos in Lasers 525

11.3.1 Suppression of Relative Intensity Noise (RIN) 52511.3.2 Chaotic Search and Adaptive Mode Selection 52711.3.3 Dynamical Memory 52811.3.4 Communication with Chaos by Controlling Chaos 529Appendix11.A.1 OGY Method for Controlling Chaos 530

12 Other Applications with Chaotic Lasers 53312.1 Remote Sensing with Chaotic Lasers 533

12.1.1 Chaotic Lidar 53312.1.2 Chaotic Radar 53612.1.3 Chaotic Correlation Optical Time-Domain Reflectometer 539

12.2 Blind Source Separation of Chaotic Signals by Using IndependentComponent Analysis 54212.2.1 Motivation for Blind Source Separation 54212.2.2 Principle of Independent Component Analysis 54312.2.3 Examples of Blind-Source Separation with Chaotic

Lasers 54412.3 Fractal Optics 547

12.3.1 Chaos Mirror for Wireless Optical Communications 54812.3.2 Fractal Patterns in Regular Polyhedral Mirror-Ball

Structures 550

References 557

Glossary 575G.1 List of Acronyms 575

G.1.1 Acronyms of Technical Terms 575G.1.2 Acronyms of Units 578

G.2 Source Codes of C Programming Language for NumericalSimulations 579G.2.1 Logistic Map (Chapter 2) 579

G.2.1.1 C Source Code for Sequence of Logistic Map (Figure2.5a) 579

G.2.1.2 C Source Code for Bifurcation Diagram of LogisticMap (Figure 2.8) 580

G.2.1.3 C Source Code for Lyapunov Exponent of Logistic Map(Figure 2.9) 581

G.2.2 Lorenz Euations (Chapter 2) 582G.2.2.1 C Source Code for Time Series of Lorenz Equations

(Figure 2.10) 582

XXIV Contents

G.2.2.2 C Source Code for Bifurcation Diagram of LorenzEquations (Figure 2.12a) 584

G.2.2.3 C Source Code for Lyapnov Spectrum (All theLyapunov Exponents) of Lorenz Equations (Figure2.12b) 586

G.2.2.4 C Source Code for Synchronization of Chaos in LorenzEquations (Diffusive Coupling, Section 5.2.1.2) 590

G.2.3 Lang–Kobayashi Equations for a Semiconductor Laser withTime-Delayed Optical Feedback (Chapter 4) 592G.2.3.1 C Source Code for Time Series of Lang–Kobayashi

Equations (Figure 4.13e) 592G.2.3.2 C Source Code for Bifurcation Diagram of Lang–

Kobayashi Equations (Figure 4.16) 595G.2.3.3 C Source Code for Maximum Lyapunov Exponent of

Lang-Kobayashi Equations (Figure 4.19) 599G.2.4 Synchronization of Chaos in Coupled Lang–Kobayashi

Equations for Unidirectionally Coupled Semiconductor Laserswith Time-Delayed Optical Feedback (Chapter 6) 604G.2.4.1 C Source Code for Time Series of Synchronization

of Chaos in Coupled Lang-Kobayashi Equationsin Open-Loop Configuration (Figure 6.9) 604

G.2.4.2 C Source Code for Time Series of Synchronization ofChaos in Coupled Lang–Kobayashi Equations inClosed-Loop Configuration (Appendix 6.A.1) 609

G.2.4.3 C Source Code for Cross-Correlation Calculation ofSynchronization of Chaos in Coupled Lang–KobayashiEquations in Open-Loop Configuration (Figures 6.10aand c) 614

G.2.4.4 C Source Code for Conditional Lyapunov Exponent ofSynchronization of Chaos in Coupled Lang–KobayashiEquations in Open-Loop Configuration(Figure 6.12) 620

Index 627

Contents XXV

Preface

The aim of this book is to provide a comprehensive overview of research activities onboth chaos (nonlinear dynamics) and lasers (photonics) that can be applied forengineering applications of optical communication and information technology.Many books and review articles have been published in the field of either chaos orlasers, however, the books related to both of these two major interdisciplinary fieldsare limited. This book covers the research fields of both chaos and lasers, and showsthat the combination of chaos and lasers can result in novel applications to opticalcommunications and information technologies.

This book is suitable for graduate students who are interested in learning aboutchaos and lasers and intend to start new research works in the interdisciplinaryresearch fields. The comprehensive overview of experiments and numerical simula-tions for chaotic laser dynamics described in this book would also be very useful forprofessional researchers in the research fields of chaos or lasers and their engineer-ing applications. Complicated mathematical formula are avoided in the main text inorder to understand the essence of the topics treated in this book.

The author strongly acknowledgesmany researchers in the interdisciplinary fieldsof chaos and lasers. The author has been influenced by many enthusiastic discus-sions with these researchers in these research fields. It would be a great pleasure forthe author if this book could contribute a small step by which these research fields ofchaos and lasers would be widely recognized in other research communities. Theauthor strongly feels that many researchers in these fields have contributed toaccumulate a large amount of knowledge for both basic sciences and engineeringapplications.

The author strongly acknowledges Peter Davis, Ingo Fischer, Claudio R. Mirasso,Junji Ohtsubo, Rajarshi Roy, and Kazuyuki Yoshimura for reviewing some chaptersin this book. Their comments and encouragements have been very valuable toimprove the contents in this book.

The author acknowledges Peter Davis and Rajarshi Roy for long-term collabora-tion with the author. The author has been strongly influenced by their philosophieson scientific research.

The author thanks the following researchers for long-term collaboration with theauthor to produce fruitful research results, some of which are included in this book:

XXVII

Tilmann Heil, Tohru Ikeguchi, Fumihiko Kannari, Yun Liu, Ryan McAllister,Toni Pérez, Shigeru Yoshimori, and Kazuyuki Yoshimura.The author thanks the following researchers for helpful discussions and con-

structive comments at different occasions in the interdisciplinary research commu-nities of chaos and lasers: Tahito Aida, Kazuyuki Aihara, F. T. Arecchi, ApostolosArgyris, Stefano Boccaletti, Adonis Bogris, Thomas Carroll, How-Foo Chen,Konstantinos E. Chlouverakis, Muhan Choi, Adam B. Cohen, Pere Colet, Ned J.Corron, Thomas Erneux, Ingo Fischer, Takehiro Fukushima, Jordi García-Ojalvo,Ignace Gatare, Daniel J. Gauthier, Athanasios Gavrielides, Shin-itiro Goto, MichaelHamacher, Takahisa Harayama, Toshimori Honjo, Yoshihiko Horio, GuillaumeHuyet, Sheng-Kwang Hwang, Lucas Illing, Ido Kanter, Chil-Min Kim, Min-YoungKim, Song-Ju Kim, Takeshi Koshiba, Bernd Krauskopf, Jürgen Kurths, FumiyoshiKuwashima, Wing-Shun Lam, Laurent Larger, Tony Lawrance, Min Won Lee, DaanLenstra, Fan-Yi Lin, Jia Ming Liu, Alexander Locquet, Paul Mandel, ChristinaMasoller, Riccardo Meucci, Claudio R. Mirasso, Linda Moniz, Jun Muramatsu,Thomas E. Murphy, Hiroya Nakao, Junji Ohtsubo, Toru Onodera, Kenju Otsuka,Edward Ott, Krassimir Panajotov, Jon Paul, Louis M. Pecora, Michael Peil, LuisPesquera, Arkady Pikovsky, Will Ray, Elizabeth A. Rogers, Fabien Rogister, DamienRontani, Ira B. Schwartz, Marc Sciamanna, K. Alan Shore, S. Sivaprakasam, Paul S.Spencer, Steven H. Strogatz, David W. Sukow, Satoshi Sunada, Dimitris Syvridis,Yoshiyasu Tamura, Shuo Tang, J. R. Tredicce, Ken Umeno, Gregory D. VanWigge-ren, Raúl Vicente, Sebastian Wieczorek, H.-J. Wünsche, and M. Yousefi.The author would strongly like to thank his graduate students, Hiroki Aida,

Kazutaka Kanno, Takuya Mikami, Shinichiro Morikatsu, Haruka Okumura, andTaiki Yamazaki, for providing the original experimental and numerical data inChapters 2, 4, 6, 10, and Glossary G.2 of this book. Without their contributions,this book would not have been completed.The author thanks his graduate students who have enthusiastically worked all day

long with the author: Yasuhiro Akizawa, Kazuya Amano, Kota Aoyama, MasayaArahata, Kenichi Higa, Kunihito Hirano, Hidetoshi Iida, Masaki Inoue, MakitoKawano, Satoshi Kinugawa, Hayato Koizumi, Masahiko Kuraya, Takanori Matsuura,Isao Oowada, Mitsutoshi Ozaki, Satoshi Sano, Nagisa Shibasaki, Hiroyuki Someya,Junichi Takahashi, Toru Yamamoto, and Hoipang Yip.The author would like to thank Valerie Molière and Ulrike Werner at Wiley-VCH

Verlag for their encouragements to complete the book project. It was a long journeyfor this book project as a first experience for the author. The author thanks them fortheir warm patience.Finally, the author thanks his family (Miho, Mikito, Tatsuya, Sadao, and Haruko

Uchida) for their unconditional support for daily life.

Saitama, August 2011 Atsushi Uchida

XXVIII Preface