This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

Chaos is good, if you are looking to send encrypted information across a broadband optical network. The idea that the transmission of light-based signals embedded in chaos can provide privacy in data transmission has been demonstrated over short distances in the laboratory. Now it has been shown to work for real, across a commercial fibre-optic channel in the metropolitan area network of Athens, Greece. The results show that the technology is robust to perturbations and channel disturbances unavoidable under real-world conditions. Chaotic signals have been proposed as broadband information carriers with the potential of providing a high level of robustness and privacy in data transmission1,2. Laboratory demonstrations of chaos-based optical communications have already shown the potential of this technology3,4,5, but a field experiment using commercial optical networks has not been undertaken so far. Here we demonstrate high-speed long-distance communication based on chaos synchronization over a commercial fibre-optic channel. An optical carrier wave generated by a chaotic laser is used to encode a message for transmission over 120 km of optical fibre in the metropolitan area network of Athens, Greece. The message is decoded using an appropriate second laser which, by synchronizing with the chaotic carrier, allows for the separation of the carrier and the message. Transmission rates in the gigabit per second range are achieved, with corresponding bit-error rates below 10-7. The system uses matched pairs of semiconductor lasers as chaotic emitters and receivers, and off-the-shelf fibre-optic telecommunication components. Our results show that information can be transmitted at high bit rates using deterministic chaos in a manner that is robust to perturbations and channel disturbances unavoidable under real-world conditions.

/pdf/chaos-based-communications-at-high-bit-rates-using-2z2lxtwks1.pdf

Chaos-based communications at high bit rates using commercial fibre-optic links

Random number generators in digital information systems make use of physical entropy sources such as electronic and photonic noise to add unpredictability to deterministically generated pseudo-random sequences1,2. However, there is a large gap between the generation rates achieved with existing physical sources and the high data rates of many computation and communication systems; this is a fundamental weakness of these systems. Here we show that good quality random bit sequences can be generated at very fast bit rates using physical chaos in semiconductor lasers. Streams of bits that pass standard statistical tests for randomness have been generated at rates of up to 1.7 Gbps by sampling the fluctuating optical output of two chaotic lasers. This rate is an order of magnitude faster than that of previously reported devices for physical random bit generators with verified randomness. This means that the performance of random number generators can be greatly improved by using chaotic laser devices as physical entropy sources. Random-number generators are important in digital information systems. However, the speed at which current sources operate is much slower than the typical data rates used in communication and computing. Chaos in semiconductor lasers might help to bridge the gap.

Fast physical random bit generation with chaotic semiconductor lasers

This Review Article provides an overview of chaos in laser diodes by surveying experimental achievements in the area and explaining the theory behind the phenomenon. The fundamental physics underpinning laser diode chaos and also the opportunities for harnessing it for potential applications are discussed. The availability and ease of operation of laser diodes, in a wide range of configurations, make them a convenient testbed for exploring basic aspects of nonlinear and chaotic dynamics. It also makes them attractive for practical tasks, such as chaos-based secure communications and random number generation. Avenues for future research and development of chaotic laser diodes are also identified.

Physics and applications of laser diode chaos

Complex phenomena in photonics, in particular, dynamical properties of semiconductor lasers due to delayed coupling, are reviewed. Although considered a nuisance for a long time, these phenomena now open interesting perspectives. Semiconductor laser systems represent excellent test beds for the study of nonlinear delay-coupled systems, which are of fundamental relevance in various areas. At the same time delay-coupled lasers provide opportunities for photonic applications. In this review an introduction into the properties of single and two delay-coupled lasers is followed by an extension to network motifs and small networks. A particular emphasis is put on emerging complex behavior, deterministic chaos, synchronization phenomena, and application of these properties that range from encrypted communication and fast random bit sequence generators to bioinspired information processing.

/pdf/complex-photonics-dynamics-and-applications-of-delay-coupled-1llx3j8d4d.pdf

Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers

List of Contributors. Preface. Acknowledgements. 1 Introduction (Deborah M. Kane and K. Alan Shore). 1.1 Semiconductor Laser Basics. 1.2 Nonlinear Dynamical Systems. 1.3 Semiconductor Lasers with Optical Feedback. 1.4 Landmark Results: Theory and Experiment. 1.5 Overview of Feedback Response: Regimes I-V. 1.6 Outline of Applications. References. 2 Theoretical Analysis (Paul Spencer, Paul Rees and Iestyn Pierce). 2.1 Introduction. 2.2 Basic Model: Single Mode Lasers with Weak Optical Feedback. 2.3 Steady State Analysis of the Lang-Kobayashi Equations. 2.4 Multimode Iterative Analysis of the Dynamics of Laser Diodes Subject to Optical Feedback. 2.5 Cavity Length Effects. 2.6 Coupled Cavity Analysis. 2.7 Conclusion. References. 3 Generalized Optical Feedback: Theory (Daan Lenstra, Gautam Vemuri and Mirvais Yousefi). 3.1 Varieties of Optical Feedback. 3.2 Compound-Cavity Analysis: Validity of Lang-Kobayashi Approach. 3.3 Filtered Optical Feedback. 3.4 Phase-Conjugate Feedback. 3.5 Conclusion. Acknowledgements. Note. References. 4 Experimental Observations (A. Tom Gavrielides and David W. Sukow). 4.1 Introduction. 4.2 Experimental Apparatus. 4.3 Extremely Weak Feedback Effects - Regime I. 4.4 Very Weak Feedback Effects - Regime II. 4.5 Weak Feedback Effects - Regime III-IV. 4.6 Moderate Feedback Effects - Low Frequency Fluctuations. 4.7 Short Cavity Regime. 4.8 Double-Cavity Systems. 4.9 Multimode Effects. 4.10 Control. 4.11 Feedback and Modulation. 4.12 Phase Conjugate Feedback. 4.13 Conclusion. References. 5 Bifurcation Analysis of Lasers with Delay (Bernd Krauskopf). 5.1 Introduction. 5.2 Bifurcation Theory of DDEs. 5.3 Numerical Methods. 5.4 Bifurcations in the COF Laser. 5.5 Bifurcations in the PCF Laser. 5.6 Conclusion. Acknowledgements. References. 6 Chaos Synchronization (Siva Sivaprakasam and Cristina Masoller Ottieri). 6.1 Introduction. 6.2 Synchronization of Unidirectionally Coupled Semiconductor Lasers. 6.3 Synchronization of Mutually Coupled Semiconductor Lasers. 6.4 Conclusion. References. 7 Laser Interferometry (Guido Giuliani and Silvano Donati). 7.1 Introduction. 7.2 Laser Diode Feedback Interferometry: Theory and Basic Experiments. 7.3 Application to Measurements. 7.4 Laser Diode Diagnostics Using Self-Mixing Techniques. 7.5 Conclusion. Acknowledgements. References. 8 Single Frequency and Tunable Single Frequency Semiconductor Laser Systems (Esa Jaatinen). 8.1 Introduction. 8.2 Effect of Frequency Filtering the Feedback for Robust Single Frequency Operation. 8.3 Tunable Semiconductor Laser System Designs and Operating Characteristics. 8.4 Frequency Stabilization. 8.5 Tunable Semiconductor Laser System Applications. 8.6 Conclusion. References. 9 Chaotic Optical Communication (Junji Ohtsubo and Peter Davis). 9.1 Introduction. 9.2 Communication Using Synchronized Laser Chaos. 9.3 Methods for Modulation and Recovery of Messages. 9.4 Mechanisms for Synchronization and Signal Recovery. 9.5 Parameter Sensitivity, Robustness and Security for Synchronized Chaos Communication. 9.6 Communication Bandwidth. 9.7 Conclusion. Acknowledgements. References. Index.

Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers

Synchronization of chaos is achieved experimentally in unidirectionally coupled external-cavity vertical-cavity surface-emitting semiconductor lasers operating in an open-loop regime. Synchronization is observed when the polarization of the transmitter is perpendicular to the polarization (x polarization) of the free-running receiver. The ratio of transmitter output to y-polarized receiver output power shows normal (positive-slope) synchronization. However, inverse (negative-slope) synchronization is found to arise between the transmitter output and the x-polarized receiver output power.

Synchronization of chaos in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers

The polarization properties of vertical-cavity surface-emitting lasers (VCSELs) subject to optical feedback are studied experimentally. It is thereby demonstrated that polarization-selective optical feedback can be utilized to entirely eliminate VCSEL polarization switching over the entire device operating range.

Suppression of polarization switching in vertical-cavity surface-emitting lasers by use of optical feedback

White chaos with a colorless spectral bandwidth of 14 GHz contained within a range of 3 dB is experimentally generated by optical heterodyning of two external-cavity laser diodes. Experimental and numerical results show that the signatures of laser relaxation oscillation and external-cavity resonance (i.e., time delay signature), which are intrinsic to the laser dynamics are totally eliminated in the heterodyne-generated chaos. Correlation dimensional analysis shows that the generated chaos has a high dimensionality, being greater than 10. In addition, the effects on the heterodyne spectral bandwidth of variations in the optical feedback strength and frequency detuning of the lasers are studied experimentally and theoretically. Opportunities are identified for the deployment of white chaos.

K. Alan Shore

Papers

Chaos-based communications at high bit rates using commercial fibre-optic links

Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers

Synchronization of chaos in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers

Suppression of polarization switching in vertical-cavity surface-emitting lasers by use of optical feedback

Optical Heterodyne Generation of High-Dimensional and Broadband White Chaos