Optical Fibers Research Advances

OPTICAL FIBERS RESEARCH ADVANCES OPTICAL FIBERS RESEARCH ADVANCES JÜRGEN C. SCHLESINGER EDITOR Nova Science Publish...

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OPTICAL FIBERS RESEARCH ADVANCES

OPTICAL FIBERS RESEARCH ADVANCES

JÜRGEN C. SCHLESINGER EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2007 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Schlesinger, Jürgen C. Optical fibers research advances / Jürgen C. Schlesinger, Editor. p. cm. Includes index. ISBN-13: 978-1-60692-607-9 1. Optical communications. 2. Fiber optics. 3. Optical fibers. I. Title. TK5103.59.S35 2008 621.36'92--dc22 2007031168

Published by Nova Science Publishers, Inc.

New York

CONTENTS Preface

vii

Short Communication Ignition with Optical Fiber Coupled Laser Diode Shi-biao Xiang, Xu Xiang , Wei-huan Ji and Chang-gen Feng Research and Review Studies

1 3 13

Chapter 1

Evanescent Field Tapered Fiber Optic Biosensors (TFOBS): Fabrication, Antibody Immobilization and Detection Angela Leung, P. Mohana Shankar and Raj Mutharasan

15

Chapter 2

New Challenges in Raman Amplification for Fiber Communication Systems P.S. André, A.N. Pinto, A.L.J. Teixeira, B. Neto, S. Stevan Jr., Donato Sperti, F. da Rocha, Micaela Bernardo, J.L. Pinto, Meire Fugihara, Ana Rocha and M. Facão

51

Chapter 3

Fiber Bragg Gratings in High Birefringence Optical Fibers Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski

83

Chapter 4

Applications of Hollow Optical Fibers in Atom Optics Heung-Ryoul Noh and Wonho Jhe

119

Chapter 5

Advances in Physical Modeling of Ring Lasers Vittorio M.N. Passaro and Francesco De Leonardis

161

Chapter 6

Investigation of Optical Power Budget of Erbium-Doped Fiber Hideaki Hayashi, Setsuhisa Tanabe and Naoki Sugimoto

187

Chapter 7

Recent Developments in All-Fibre Devices for Optical Networks Nawfel Azami and Suzanne Lacroix

205

vi

Contents

Chapter 8

Advances in Optical Differential Phase Shift Keying and Proposal for an Alternative Receiving Scheme for Optical Differential Octal Phase Shift Keying M. Sathish Kumar, Hosung Yoon and Namkyoo Park

231

Chapter 9

A New Generation of Polymer Optical Fibers Rong-Jin Yu and Xiang-Jun Chen

257

Chapter 10

Dissipative Solitons in Optical Fiber Systems Mário F.S. Ferreira and Sofia C.V. Latas

279

Chapter 11

Bright - Dark and Double - Humped Pulses in Averaged, Dispersion Managed Optical Fiber Systems K.W. Chow and K. Nakkeeran

301

Chapter 12

Dynamics and Interactions of Gap Solitons in Hollow Core Photonic Crystal Fibers Javid Atai and D. Royston Neill

315

Chapter 13

Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier Ring Lasers Byoungho Lee and Ilyong Yoon

335

Chapter 14

Aging and Reliability of Single-Mode Silica Optical Fibers M. Poulain, R. El Abdi and I. Severin

355

Index

369

PREFACE An optical fiber is a glass or plastic fiber designed to guide light along its length by confining as much light as possible in a propagating form. In fibers with large core diameter, the confinement is based on total internal reflection. In smaller diameter core fibers, (widely used for most communication links longer than 200 meters) the confinement relies on establishing a waveguide. Fiber optics is the overlap of applied science and engineering concerned with such optical fibers. Optical fibers are widely used in fiber-optic communication, which permits transmission over longer distances and at higher data rates than other forms of wired and wireless communications. They are also used to form sensors, and in a variety of other applications. The term optical fiber covers a range of different designs including graded-index optical fibers, step-index optical fibers, birefringent polarization-maintaining fibers and more recently photonic crystal fibers, with the design and the wavelength of the light propagating in the fiber dictating whether or not it will be multi-mode optical fiber or single-mode optical fiber. Because of the mechanical properties of the more common glass optical fibers, special methods of splicing fibers and of connecting them to other equipment are needed. Manufacture of optical fibers is based on partially melting a chemically doped preform and pulling the flowing material on a draw tower. Fibers are built into different kinds of cables depending on how they will be used. This new book presents the latest research in the field. Optical fibers, an important and promising material, have attracted more and more attention and extended their applications to various scientific and practical aspects. In the short communication, the key role of fibers, as the carriers of information and energy in our times, was briefly summarized. Afterwards, the configuration of fiber coupled laser diode ignition system was elucidated as well as the advantages, developments and applications of this technology. Furthermore, the energy-transmitting characteristics of single-mode fibers and multi-mode ones and the key points of fiber-coupled technology were analyzed. In a practical case, the effect of the diameters of core on laser ignition, from both theory and experiments, was studied. The findings suggest that the smaller the diameters of core, the lower the ignition threshold under the same laser power. That is to say, the ignition becomes easier while using fibers with smaller core. Finally, the issue on selection of core was clarified based on the consideration of both laser power density and the endurance of fibers. Tapered Fiber Optic Biosensors (TFOBS) are sensors that operate based on fluctuations in the evanescent field in the tapered region. In the laboratory, TFOBS are made by heat

viii

Jürgen C. Schlesinger

pulling commercially-available single mode optical fibers. They have been investigated for various applications, including measurement of physical characteristics (refractive index, temperature, pressure, etc.), chemical concentrations, and biomolecule detection. In this chapter, an up-to-date review of TFOBS research is provided, with emphasis on applications in biosensing such as pathogen, proteins, and DNA detection. The physics of sensing and optical behavior based on taper geometry is discussed. Methods of fabrication, antibody immobilization, sample preparation, and detection from our laboratory are described. This chapter presents results on the non-specific response, simulation, and detection of E.coli O157:H7 and BSA. Chapter 1 will conclude with an analysis of the future direction of the Tapered Fiber Optic Biosensors. Raman fiber amplifiers (RFA) are among the most promising technologies in lightwave systems. In recent years, Raman optical fiber amplifiers have been widely investigated for their advantageous features, namely the transmission fiber can be itself used as the gain media reducing the overall noise figure and creating a lossless transmission media. The introduction of RFA based on low cost technology will allow the consolidation of this amplification technique and its use in future optical networks. Chapter 2 reviews the challenges, achievements, and perspectives of Raman amplification in optical communication systems. In Raman amplified systems, the signal amplification is based on stimulated Raman scattering, thus the peak of the gain is shifted by approximately 13.2 THz with respect to the pump signal frequency. The possibility of combining many pumps centered on different wavelengths brings a flat gain in an ultra wide bandwidth. An initial physical description of the phenomenon is presented as well as the mathematical formalism used to simulate the effect on optical fibers. The review follows with one section describing the challenging developments in this topic, such as using low cost pump lasers, in-fiber lasing, recurring to fiber Bragg grating cavities or broadband incoherent pump sources and Raman amplification applied to coarse wavelength multiplexed networks. Also, one of the major issues on Raman amplifier design, which is the determination of pump powers in order to realize a specific gain will be discussed. In terms of optimization, several solutions have been published recently, however, some of them request extremely large computation time for every interaction, what precludes it from finding an optimum solution or solve the semi-analytical rate equation under strong simplifying assumptions, which results in substantial errors. An exhaustive study of the optimization techniques will be presented. This paper allows the reader to travel from the description of the phenomenon to the results (experimental and numerical) that emphasize the potential applications of this technology. Fiber Bragg gratings (FBG) are a key element in optical communication devices and in fiber sensors. This is mainly due to its intrinsic characteristics, which include low insertion loss, passive operation and immunity to electromagnetic interferences. Basically a FBG is a periodic modulation of the core refractive index formed by exposure of a photosensitive fiber to a spatial pattern of ultraviolet light in the region of 244–248 nm. The lengths of FBGs are normally within the region of 1–20 mm. Usually a FBG operates as a narrow reflection filter, where the central wavelength is directly proportional to the periodicity of the spatial modulation and to the effective refractive index of the fiber. The production technology of these devices is now in a mature state, which enables the design of gratings with custom-

Preface

ix

made transfer functions, crucial for all-optical processing. Recently, some work has been done in the application of FBG written in highly birefringent fibers (HiBi). Due to the birefringence, the effective refractive index of the fiber will be different for the two transversal modes of propagation. Therefore, the reflection spectrum of a FBG will be different for each polarization. This unique property can be used for advanced optical processing or advanced fiber sensing. Chapter 3 will describe in detail this unique device. The chapter will also analyze the device and demonstrate different applications that take advantage of its properties, like multiparameter sensors, devices for optical communications or in the optimization of certain architectures in optics communications systems. A hollow optical fiber (HOF) has a lot of interesting applications in atom optics experiments such as atom guiding and the generation of hollow laser beam (HLB). In this article the authors present theoretical and experimental works on the use of hollow optical fibers in atom optics. Chapter 4 is divided into two parts: One is devoted to the atom guide using HOFs and the other describes the atom optics researches that utilizes laser lights emanated from the HOF. First, the authors describe the electromagnetic fields inside the HOF and characterize the electromagnetic modes diffracted from the HOF. Then they describe two guiding schemes using red and blue detuned laser lights. Finally, they describe the various relevant experiments using LP01 or LP11 modes such as the generation of HLB from the HOF, funneling atoms using the diffracted fields, diffraction-limited dark laser spot, and a dipole trap using LP01 mode of the diffracted field from the HOF. In Chapter 5, an overview on fiber ring lasers and III/V semiconductor integrated ring lasers is presented. In particular, some aspects of mathematical modelling of these devices are reviewed. In the first part of the chapter, the authors have focused our attention on the more recent theoretical and experimental studies concerning fiber ring laser architectures. Then, a complete quantum-mechanical model for integrated ring lasers is presented, including the evaluation of all the involved physical parameters, such as self and cross saturation and backscattering. Finally, the influence of sidewall roughness on either unidirectional or bidirectional regime in multi-quantum-well III/V semiconductor ring lasers is demonstrated. In Chapter 6, the authors investigated optical power budget of an erbium-doped fiber (EDF). In addition to the output signal and amplified spontaneous emission (ASE) powers from the fiber end, lateral spontaneous emissions and scattering laser powers in the EDF were measured quantitatively by using an integrating sphere. Compared with the signal and ASE powers, it was found that considerable powers were consumed by the laterally emitting lights. As an optically undetected loss which limits power conversion efficiency (PCE) of the fiber amplifier, the effect of nonradiative decay from the termination level of pump excited state absorption (pump ESA) was estimated from decay rate analyses of the relevant levels. The nonradiative loss was comparable to amplified signal power in the EDF when pumped with a 980 nm LD. Nonradiative decay following cooperative upconversion (CUP) process is also discussed using rate equations analysis. All-fibre components are essential components of optical networks systems. Development of such devices is of great importance to allow network functions to be performed in the glass of the optical fibre itself. Among of all fabrication techniques, the Fused Fibre Biconical Taper (FBT) technique allows optical devices with high performances. Although fibre devices are mainly based on the passive directional coupler basic structure, research is made to design components that perform complex functionalities in today optical

x


networks systems. Recent developments on all-fibre devices in network systems are presented. Research is mainly focused on enhanced fabrication and stability of FBT fabrication technique, passive thermal compensation for stable interferometer optical structure, broadband spectral operation for multi-wavelength operations and new interferometer designs. An overview of recent fused fibre devices for optical telecommunications is presented to understand the main functionalities of these fibre devices. The limiting factors are explained in Chapter 7, to understand challenges on fibre devices development. Optical Differential Phase Shift Keying (oDPSK) with delay interferometer based direct detection receiver was proposed as an alternative for the conventional On-Off Keying (OOK) modulation schemes. Compared to OOK, oDPSK was predicted to have a 3dB improvement in performance due to its balanced detection receiver structure. It was also predicted that due to the optical signal occupying all the symbol slots, unlike in OOK, symbol pattern dependent fiber nonlinear effects will make less of an impact on long haul optical transmission schemes based on oDPSK. Subsequent successful demonstrations of these positive attributes of oDPSK resulted in active investigations into multilevel formats of oDPSK namely, optical Differential Quadrature Phase Shift Keying (oDQPSK) and optical Differential Octal Phase Shift Keying (oDOPSK). Significant developments in theoretical models of optically amplified lightwave communication systems based on the Karhunen-Loeve Series Expansion (KLSE) method assisted such investigations. In Chapter 8, the authors discuss some of the recent advances in oDPSK and its multilevel formats that have been achieved such as proposals for receiver schematics, theoretical analysis of receiver schematics, electronic techniques to counter polarization mode dispersion induced penalties, and application of coded modulation techniques. The chapter also proposes an alternative receiver schematic for oDOPSK which can separately detect the three constituent bits from an oDOPSK symbol. Chapter 9 describes the background to the development of Polymer Optical fibers (POFs), discusses the optical and temperature resistant properties of polymers while emphasizing the intrinsic high attenuation of them. The first generation of POFs which consists of a solid-core surrounded by cladding and transmits light by total internal reflection, is puzzled by the difficulty of high attenuation. Then, the method of using a specific structure (i.e. hollow-core Bragg fiber) to solve the problem is presented. A new generation of POFs based on the hollow-core Bragg fibers with cobweb-structured cladding can guide light with low transmission loss and high bandwidth in the wavelength range of visible to terahertz ( THz ) radiation. Efficient hollow-core guiding for delivery of power laser radiation and solar radiation can be achieved by replacing the traditional polymethylmethacrylate (PMMA) with heat-resistant polymers. Lastly, this chapter concludes with a discussion of applications in diverse areas. Chapter 10 introduces the concept of dissipative solitons, which emerge as a result of a double balance: between nonlinearity and dispersion and also between gain and loss. Such dissipative solitons have many unique properties which differ from those of their conservative counterparts and which make them similar to living things. The authors focus our discussion on dissipative solitons in optical fiber systems, which can be described by the cubic-quintic complex Ginzburg-Landau equation (CGLE). The conditions to have stable solutions of the CGLE are discussed using the perturbation theory. Several exact analytical solutions, namely in the form of fixed-amplitude and arbitrary-amplitude solitons, are presented. The numerical solutions of the quintic CGLE include plain pulses, flat-top pulses, and composite pulses,

Preface

xi

among others. The interaction between plain and composite pulses is analyzed using a twodimensional phase space. Stable bound states of both plain and composite pulses are found when the phase difference between them is ± π / 2 . The possibility of constructing multisoliton solutions is also demonstrated. As explained in Chapter 11, the envelope of the axial electric field in a dispersion managed (DM) fiber system is governed by a nonlinear Schrödinger model. The group velocity dispersion (GVD) varies periodically and thus realizes both the anomalous and normal dispersion regimes. Kerr nonlinearity is assumed and a loss / gain mechanism will be incorporated. Due to the big changes in the GVD parameter, the correspondingly large variation in the quadratic phase chirp of the DM soliton will be identified. An averaging procedure is applied. In many DM systems, an amplifier at the end of the dispersion map will compensate for the energy dissipated in that map. Here the case of gain not exactly compensating the loss is considered, in other words, a small residual amplification / attenuation is permitted. The present model differs from other similar ones on variable coefficient NLS, as the inhomogeneous features involve both time and the spatial coordinate. The goal here is to extend the model further, by permitting coupled modes or additional degree of freedom. More precisely, the coupling of fiber loss and initial chirping is considered for a birefringent fiber. The corresponding dynamics is governed by variable coefficient, coupled NLS equations for the components of the orthogonal polarization of the pulse envelopes. When the self phase and cross phase modulation coefficients are identical for special angles, several new classes of wave patterns can be found. New stationary wave patterns which possess multiple peaks within each period are found, similar to those found for the classical Manakov model. For situations where the self- and cross-phase modulation coefficients are different, symbiotic solitary pulses are studied. A pair of bright-dark pulses exists, where either or both pulse(s) cannot propagate in that waveguide if coupling is absent. The existence and stability of gap solitons in a model of hollow core fiber in the zero dispersion regime are analyzed in Chapter 12. The model is based on a recently introduced model where the coupling between the dispersionless core mode and nonlinear surface mode (in the presence of the third order dispersion) results in a bandgap. It is found that similar to the anomalous and normal dispersion regimes, the family of solitons fills up the entire bandgap. The family of gap solitons is found to be formally unstable but in a part of family the instability is very weak. Consequently, gap solitons belonging to that part of the family are virtually stable objects. The interactions and collisions of in-phase and the π -out-ofphase quiescent solitons and moving solitons in different dispersion regimes are investigated and compared. Chapter 13 reviews various schemes for multiwavelength fiber lasers and semiconductor optical amplifier (SOA) ring lasers. Multiwavelength fiber lasers have applications in wavelength division multiplexing (WDM) optical communication systems, optical fiber sensors and optical spectroscopy. Erbium-doped fiber amplifiers (EDFAs), Raman amplifiers and SOAs are mainly used as gain media for multiwavelength fiber lasers. Because EDFAs are homogeneously broadened gain media, various methods have been researched to enable the multiwavelength generation. Due to the introduction of liquid nitrogen cooling, four-wave mixing, frequency shifted feedback, and so on, multiwavelength erbium-doped fiber lasers could become realized.

xii


On the other hand, because SOA and Raman amplifiers are gain media with inhomogeneous broadening, multiwavelength generation is relatively easy. The useful features of the multiwavelength lasers are mainly dependent on a comb filter. One of the most important features of multiwavelength lasers is tunability. The tunability of wavelengths and channel spacing is required for WDM optical communication systems. Much research has been conducted to enable implementation of tunable multiwavelength fiber lasers. Various comb filters such as Fabry-Perot filters, fiber Bragg gratings, and polarization-maintaining fiber loop mirrors can be used for multiwavelength fiber lasers. The authors review several schemes for multiwavelength SOA-fiber and Raman fiber lasers in this chapter. The optical fiber reliability in telecommunication networks has been still an issue, that’s why the question of how long an optical fibers might been used without a significant probability of failure isn’t out of interest. Much work was developed around this issue, but the optical fiber fatigue and aging process has not been yet fully understood. The reliability of the optical fibers depends on various parameters that have been identified: time, temperature, applied stress, initial fiber strength and environmental corrosion. The major and usually unique corrosion reagent is water, either in the liquid state or as atmospheric moisture. Glass surface contains numerous defects, either intrinsic, the socalled “Griffith’s flaws and extrinsic, in relation to fabrication process. Under permanent or transient stress, microcracks grow from these defects, and growth kinetics depend on temperature and humidity. Although polymeric coating efficiently protects glass surface from scratches, it does not prevent water to reach glass fiber. The work carried out during the last years made possible to apprehend in a more coherent way the problems of failure and rupture of fibers subjected to severe aging conditions. In Chapter 14, some informations on the used characterization methodology for the silica optical fibers are given. In addition, Optical fibers analysis advantages, expected percussions and theoretical background are given to enlighten the potential concerned persons. The principal optical fiber test benches are described and some results are commented. Finally, final remarks are noted.

SHORT COMMUNICATION

In: Optical Fibers Research Advances Editor: Jurgen C. Schlesinger, pp. 3-11

ISBN: 1-60021-866-0 © 2007 Nova Science Publishers, Inc.

IGNITION WITH OPTICAL FIBER COUPLED LASER DIODE Shi-biao Xiang1,2*, Xu Xiang3 , Wei-huan Ji2 and Chang-gen Feng4 1

Department of Technical Physics, Zhengzhou Institute of Light Industry, No.5 Dongfeng Road, Zhengzhou 450002, P.R. China 2 Key Laboratory of Informationalized Electric Apparatus of Henan Province, Zhengzhou 450002, P.R. China 3 State Key Laboratory of Chemical Resource Engineering, Beijing University of Chemical Technology, P.O. BOX 98, Beijing 100029, P.R. China 4 School of Mechanics and Engineering, Beijing Institute of Technology, Beijing 100081, P.R. China

Abstract Optical fibers, an important and promising material, have attracted more and more attention and extended their applications to various scientific and practical aspects. In this article, the key role of fibers, as the carriers of information and energy in our times, was briefly summarized. Afterwards, the configuration of fiber coupled laser diode ignition system was elucidated as well as the advantages, developments and applications of this technology. Furthermore, the energy-transmitting characteristics of single-mode fibers and multi-mode ones and the key points of fiber-coupled technology were analyzed. In a practical case, the effect of the diameters of core on laser ignition, from both theory and experiments, was studied. The findings suggest that the smaller the diameters of core, the lower the ignition threshold under the same laser power. That is to say, the ignition becomes easier while using fibers with smaller core. Finally, the issue on selection of core was clarified based on the consideration of both laser power density and the endurance of fibers.

*

E-mail address: [email protected]. Tel: 86-371-63557226 (Corresponding author: S. B. Xiang)

4

Shi-biao Xiang, Xu Xiang, Wei-huan Ji et al.

1. Introduction Optical fibers as carrier of information and energy have intrigued intensive interest worldwide due to its scientific and technological significance in various practical fields. For instance, in optical communications, fibers have received tremendous attention from both experimental and theoretical aspects not only on the type of fiber materials but also on various communicating techniques [1-4], in which the most primary function of fibers is to transmit information like voice, images and videos from one place to another. A wide variety of optical fiber devices have been designed and exploited in the field of fiber-based communications, such as fiber optical amplifiers, frequency or phase modulators, planar waveguides and fiber polarizers. Furthermore, the developments of microstructured optical fibers (MOFs) and photonic crystal fibers [5-9] enable a number of potential functionalities including tunability and enhanced nonlinearity, and extend novel fiber device applications to fiber Bragg gratings, tunable resonant filters, variable optical attenuators and nonlinear optics devices owing to their unique characteristics [10-15]. More interestingly, chemical sensors based on optical fibers have been widely explored in the past few years [16-18]. For example, sensors for gases or vapors [19-20], humidity [2122], metallic ions, specific chemical compounds [23], viscosity [24], intensity [25] and miniature pressure [26] have been delicately designed and rapidly developed. Also, biosensors [27, 28] for enzymes, antibodies or antigens, DNA [29] and bacteria are becoming a prevailing research topic on the basis of fiber materials. They have been exhibiting promising applications in a variety of fields such as chemical analysis, biological monitoring and environmental detection. In this article, the emphasis has been highlighted on the fundamental principles and the important practice of fiber-coupled laser diode ignition.

2. Fiber Coupled Laser Diode Ignition 2.1. Brief Review on Laser Diode (LD) Ignition Laser ignition is a kind of ignition technique, which refers to detonation or ignition of energetic materials such as solids or fluids [30-33] by laser beam. At early stage of laser ignition technique, the types of laser used for the experimental and application research are mostly Nd:YAG, Nd: GSGG, Nd: glass laser and CO2 laser [34-40]. These lasers possess the characteristics of high output power or energy, small radiation angle of light, long life-span and low price. However, the obvious disadvantages of this kind of laser are low energy conversion efficiency, in which the ratio of output light energy and input electric energy is usually lower than 3%, as well as large volume and heavy weight. With the born of LD and the naissance of LD ignition, the research and evolution of laser ignition technique come into a new era. The experimental studies for laser diode ignition began in the middle of 1980s. Ewick, Kunz, Kramer, Jungst, Merson, Glass and Roman et al have made great devotion to the field of LD ignition, of which Ewick [41] and Kunz [42] published their literatures firstly. LD belongs to a kind of semiconductor laser stimulated by current. In LD ignition, LD is utilized as energy source, and the energy is transmitted to powders by using optical fiber, which detonates or ignites the energetic materials. This ignition configuration has the

Ignition with Optical Fiber Coupled Laser Diode

5

characteristics of safety, reliability, and strong capability of anti-interference of electromagnetism. In addition, the following advantages are also realized. (1) It is easy for LD ignition system to realize miniaturization of apparatus due to its small volume and light weight. (2) LD ignition system has excellent adaptability to the ambient environment because of the input of low voltage and electric energy. (3) LD ignition system can output multi-channel laser signals by using LD arrays and consequently control multi-point ignition through the selection to time and order of signals. As a result, LD ignition has received extensive attention, and exhibits promising application especially in the field of aviation and aerospace. Fig. 1 illustrates the schematic diagram of ignition system induced by laser diode. Laser diode is employed as light source, and energy is transmitted to powders by optical fibers. The powders are ignited and subsequently exploded while enough energy is provided.

powder fiber

aperture

fiber coupler

lock device

connecting laser

Figure 1. Schematic illustration of ignition system induced by laser diode.

2.2. Optical Fiber and Fiber Coupled Technology As a carrier to transmit laser, optical fiber plays a crucial role in LD ignition. The materials of optical fiber should possess the favorable characteristics of optical and mechanical properties as well as the characteristic of temperature. The widely used fibers are made of silica glass or plastic. The fibers can be classified into two types, one is step-index fiber and the other is grade-index one according to the distribution of refractive-index of fiber core. The refractive index of core is a constant for step-index fiber, schematically shown in Fig. 2. However, for the grade-index fiber, the refractive index of core gradually decreases outwards along the radial direction. Due to the self-focusing characteristic of the grade-index fiber, the output beam has higher energy density close to the axis of fiber. As a consequence, the laser power density can be enhanced by using the grade-index fiber.

6

Shi-biao Xiang, Xu Xiang, Wei-huan Ji et al.

y

y Claddin g

φ Core

n2 n1

z Fiber axis

r

n

Figure 2. The schematic diagram of step-index fibers.

Both theoretical analysis and experimental results indicate that the increase of power density is considerably favorable to LD ignition. That is to say, the combination of thin diameter, low attenuation, small numerical aperture and grade-index fiber is advantageous to LD ignition. Ewick and coworkers found that the threshold of ignition using grade-index fiber was decreased by around 30% than that using step-index fiber in the ignition experiments of Ti/KClO4 and CP/carbon black. Generally, optical fibers can be classified into single-mode fibers (SMFs) and multi-mode fibers (MMFs) according to the transmission modes. SMFs exhibit excellent capability in optical communications. And the light energy transmitted by SMFs presents to be Gauss distributions, which means the more centralized energy can be obtained, and is thus favorable to LD ignition. Nevertheless, the diameter of core in SMFs is confined to a large extent. The fiber waveguide parameters can be expressed as V = kr ( n1 − n2 ) 2

2 1/ 2

, where n1 and

n2 are the refractive indices of the core and the cladding, respectively, and r is the core radius. And k = 2π / λ0 is the wave number, where λ0 represents the wavelength in vacuum. Single mode operation is obtained for V 0 IIIrd quadrant

1 < Γ1 = -, Γ2 = +, if s = 3π/4 and k = π/2; therefore, y1 − y 2 0 and y1 + y 2 = > 0 IVth quadrant

0 < Γ1 = +, Γ2 = +, if s = 0 and k = π/4; therefore, y1 + y 2 0 and y1 − y 2 = + > 1

252

M. Sathish Kumar, Hosung Yoon and Namkyoo Park

From the above we note that either (y1-y2) or (y1+y2) is the deciding factor and when one is the deciding factor the other has a constant numerical sign. From this observation, the four different decision rules given above can be combined into a single decision rule for estimating the third bit as given below.

0 < ( y1 − y 2 )( y1 + y 2 ) 0 > 1

(17)

The above equation suggests that a binary decision on the sum and difference of y1 and y2 followed by an XNOR operation on those decisions can readily provide an estimate of the third bit. In fact, this is the receiver schematic suggested in [25] with a minor variation in that the XNOR is replaced by the XOR apparently due to the swap in positions of 0s and 1s in the third bit of the triplet as compared to what it is herein. The receiver schematic depicted in figure (4.b) also works as per the same principle as discussed above. The two inputs to the XOR are effectively (y1-y2) and (y1+y2) [25]. Also, with an appropriate precoding of the binary data as given in [10], it is possible to directly obtain the three constituent data bits from the detected binary levels of y1, y2 and the product (y1-y2)(y1+y2). Further, if equation (17) is rewritten as

0 < ( y12 − y 22 ) 0 > 1

(18)

it becomes obvious that the decisions can be taken depending solely on the difference of the absolute values of y1 and y2. The conversion of the detected samples to their absolute values can be achieved in effect by considering the fact that the detected analog samples y1 and y2 are in fact dealt with in the receiver electronics in the digital domain through an analog to digital converter. More the resolution of the analog to digital converter better will be the resultant digital representation of the detected analog voltage. This is the methodology used in almost all the electrical soft decision decoding receivers [3]. In an analog to digital converter, it is possible in principle to identify the numerical sign of the digitally converted sample and as such it is possible to alter that numerical sign. Thus, if the detected sample y1 or y2 is negative, its numerical sign can be altered and the following decision rule can be applied to detect the third bit.

0 < ( y1 − y 2 ) 0 > 1

(19)

Advances in Optical Differential Phase Shift Keying…

253

The advantage of this receiving scheme is that the dependence of the decision making variable on the sum as well as difference of y1 and y2 is removed and is now dependent only on the difference between y1 and y2. This is of course at the cost of an additional electronic operation of changing the numerical sign of the detected samples. It may also be noted that a mere change in numerical sign does not alter the pdf of the detected samples. The complete schematic representation of this receiver is as given in figure (13).

Figure 13. Schematic representation of an oDOPSK receiver which employs only two delay interferometers.

The BER or probability of error for this receiver schematic can be readily arrived at as BER = ( P(y1>0/ b1= 0)+P(y10/b2=0)+P(y20/b3=0)+ P((y1-y2) 200MHz at 50m ), which is currently commercialized. It meets the requirement of standardization of 156 Mb/s , transmission 50m approved by the ATM forum in May 1997. It is a common knowledge that the main limitation on the bandwidth of multimode optical fibers is modal dispersion, which means that different optical modes propagate at different velocities and the dispersion grows linearly with length. One way to overcome the modal dispersion is to use single mode (SM) POF. The first SMPOF was reported in 1991, which was successfully prepared by the interfacial-gel polymerization technique [9]. In the fiber, the core diameter was 3 − 15 μ m and the attenuation of the transmission was about 200 dB/Km at 652nm wavelength. Another way to solve the problem for POFs with large cores is to use multi-layer step-index (ML-SI) POF [10], multi-core step-index (MC-SI) POF [11], or GI-POF [12]. In ML-SI POFs, the core region is composed of several layers with different refractive index. This concentric multilayer structure decreases modal dispersion compared to conventional SI type POF and a data rate as high as 500 Mb/s for 50m transmission is achieved experimentally. MC-SI POF has a low numerical aperture (0.25) and a core region composed of 19 cores of small-core. By reducing the core diameter, not only modal dispersion but also bending loss is decreased. A data transmission at 500 Mb/s for 50m is also achieved by the MC-SI POF. For GI POF, the refractive index of the fiber core is graded parabola-like from a high index at the fiber core center to a low index in the outer core region. For the GI POF produced by the interfacial-gel polymerization method, its bandwidth measured is 3GHz for a fiber length of 100m . A lowloss PF polymer based GI POF has been developed and PF polymer based GI POF is able to transmit a data rate of 10Gb/s or higher because of its material dispersion property [13]. For the temperature resistance of POFs, the high-temperature performance of a polymer is limited by its glass transition temperature ( Tg ). For PMMA, Tg is about 105℃. Maximum operating temperature for PMMA-core SI-POFs is 80℃. Ziemann et al. [14] had carried out a

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Rong-Jin Yu and Xiang-Jun Chen

accelerated aging test for the fibers from the three leading POF manufacturers. The results of the test show that for all fibers and wavelengths ( 650nm , 590nm , 525nm and 520nm ) the estimated possible operating temperature for 20 years use is over 70℃. Some applications, such as in automobiles, aerospace environments and transmitted power demand performance at temperature in excess of 80℃. The Tg of polycarbonate (PC) is around 170℃. The use of PC and partially fluorinated PC as core material enables temperatures of up to 115℃ and 145℃, respectively. The Tg of polyethersulfone (PES) is about 225℃, maximum operating temperature of PES is 197℃. Polyimide material has even more high operating temperature (316℃). The attenuation of these high temperature resistant polymers is generally larger than that of PMMA, therefore making the polymers useless in fabricating the fibers, but using the polymers (such as polyimide) as the coating of high temperature resistant silica fiber. In a word, in the forty years development of POFs, there is no better position in both performance (especially attenuation) and cost comparing with silica glass fibers. Thus, first generation POFs have limited their penetration in important market-segments, and are only suited to ornament, illumination, sensors and short-distance data transmission applications.

3. Hollow-Core Fibers Hollow-core fibers reported to date in the literature can generally be classified into four types: (1) those in which the refractive index of the cladding is greater than that of the core, (2) those in which inner wall coating has high reflectivity, (3) hollow-core photonic bandgap fiber, and (4) hollow-core Bragg fiber.

3.1. Those in Which the Refractive Index of the Cladding Is Greater Than That of the Core As is known to all, waveguiding is achieved in conventional solid-core fibers due to the total internal reflection from the interface between the core with the refractive index ncore and the cladding with the refractive index nclad ( ncore > nclad ) . For the hollow-core fiber in which the refractive index of the core is lower than that of the cladding, the propagation of light is achieved by the regime of grazing incidence and is accompanied by radiation losses (leaky guide). In fact, this hollow-core fiber is a capillary tube, as shown in Fig.1. The coefficient of optical losses in the hollow fibers scales as λ 2 /a 3 , where λ is the radiation wavelength and a is the core radius of the fiber. Thus, most of applications are performed by using the hollow-core fibers with large inner radii and short length. For example, a 10cm -long and 150μ m -diameter hollow-core fiber filled with argon gas is used on extreme ultraviolet (EUV) light generated through the process of high-harmonic up-conversion of femtosecond laser [15].

A New Generation of Polymer Optical Fibers

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cladding (glass)

hollow-core

Figure 1. Cross section of the hollow-core fiber with nclad > ncore (leaky guide).

3.2. Those in Which Inner Wall Coating Has High Reflectivity

structural tube

structural tube

n 1120nm ), specifically at 1390nm wavelength, at which the transmission loss is only 40 dB/m compared with the 420 dB/m material loss [30]. We proposed a modified cladding structure, i.e. a hollow-core Bragg fiber with cobwebstructured cladding [26]. The structure uses a single dielectric material and may solve the problem of structural support by using a certain number of supporting strips. The supporting strips are always symmetric in the cross-section and use the same dielectric material as alternating layers. Our research shows that the field profiles are slightly deformed due to the introduction of supporting structure. Although a small fraction of power is leaked out as a

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result of the introduction of supporting structure, properly selected parameters of supporting structure will keep the loss at a low level, neglecting the presence of the supporting strips. The number and width of supporting strips should be as small as possible, generally, m (number of supporting strips) = 6~12 and ws (width of supporting strips) = λ /3 ~ λ / 30 , where λ is the operating wavelength of the fibers. In comparison to “OmniGuide” fibers, the feasibility of cobweb-structured fibers is greatly improved. For ring-structured fibers, the refractive index of low-index layers in the cladding is between high-index (host material) and 1 (air). As a result, the cladding indices contrast of ring-structured fibers is smaller than that of cobweb-structured fibers. As far as the ability to confine the transverse leakage of guided wave is concerned, the ring-structured fibers are smaller compared to the cobweb-structured fibers. In order to compare the confinement losses of hollow-core ring-structured Bragg fiber with hollow-core cobwebstructured Bragg fiber, we make the design of analogous structure. Argyros et al. [25] have presented the design that supports a single-polarization, circularly symmetric nondegenerate mode in an air-core ring-structured Bragg fiber. The design presented has Λ i = 0.403μ m , Λ e = 0.578 μ m , d = 0.355 μ m and core radius( ro ) = 2.89 μ m , giving d / Λ i = 0.83 . The host

material was assumed to be lossless with a refractive index of 1.49 (corresponding to PMMA material). When N (number of rings in cladding) = 9, the confinement losses of the TE01 mode (lowest-loss mode) and TE02 mode (second-lowest-loss mode) are about 0.83dB/m and 57.14dB/m , respectively. The ratio of the loss of the TE02 mode to the loss of the TE01 mode reaches approximately 70. In our design, the same parameters: n2 (PMMA) = 1.49,

rco (core radius) = 2.89 μ m , d 2 (thickness of high-index layers) = 0.243 μm , d1 (thickness of low-index layers) = 0.335 μ m and N (number of alternating layers in cladding) = 9, as well as n1 = 1 are used. The host material was also assumed to be lossless. The calculated results show that the least-loss wavelength of the TE01 mode is located at 0.72 μ m . The confinement losses of the TE01 mode and TE02 mode at 0.72 μ m wavelength are 5.32 × 10 −5 dB/m and 2.97 × 10 −3 dB/m , respectively. The ratio of the loss of the TE02 mode to the loss of the TE01

mode reaches approximately 56. Thus it can be seen that the confinement loss of the TE01 mode in the hollow-core cobweb-structured Bragg fiber is reduced by 15600 times in comparison to that of the air-core ring-structured Bragg fiber. These hollow-core Bragg fibers not only can reduce unwanted material properties, such as absorption, scattering, dispersion and nonlinearity to a large extent, but also can act as a modal filter [3]. Sterke et al. [31] found that such Bragg fibers can be guaranteed to be effectively single-moded. Johnson et al. [23] presented their work of “how the lowest-loss TE01 mode can propagate in a single-mode fashion through even large-core fibers, with other modes eliminated asymptotically by their higher losses and poor coupling, analogous to hollow metallic microwave waveguides.” The single-mode operation of the Bragg fibers is achieved through asymptotic way during the transmission of guided waves, i.e. the number of modes in large-core Bragg fibers causes the change as follows, at the beginning, the transmission with multimode is followed by a few modes, and then the transmission becomes


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single moded at last. Thus the single mode is achieved in a certain length range of the fiber. Moreover, Bassett and Argyros [32] presented a method for calculating the single-mode length range: “The individual modes are characterized by two lengths, L1% at which the transmitted power in that mode is reduced to 1%, and L0.01% = 2 L1% , at which the power is reduced to 0.01%. We characterize each fiber as a whole by two lengths, Lmax = L1% for the best guided mode, and Lsm = L0.01% for the second best guided mode. We consider the usefully single moded for lengths between Lsm and Lmax .”

4. Hollow-Core Bragg Fiber with Cobweb-Structured Claadding The refractive index profiles of hollow-core Bragg fiber with cobweb-structured cladding, together with those of ring-structured and “OmniGuide” hollow-core Bragg fibers are shown in Fig.5 for comparison. The parameters of the fiber with cobweb-structured cladding are rco (hollow-core radius), n1 (=1, air), n2 (high-index), d1 (thickness of air layers),

d 2 (thickness of high-index layers), η ( = d 2 / d1 ) , Λ(= d1 + d 2 ) , N , m and ws , where N is the number of alternating layers in cladding, m and ws are the number and the width of the supporting strips, respectively.

(a) cobweb-structured fiber

(b) ring-structured fiber

(c) “OmniGuide” fiber

Figure 5. Profiles of refractive index for hollow-core Bragg fibers.

In cylindrical waveguides, modes can be labeled by their ‘angular momentum’ integer m ; the ( z , t , ϕ ) dependence of the modes is given by e j ( β z −ωt + mf ) . In the hollow-core fiber with cobweb-structured cladding the modes will be affected by the supporting strip. Because the supporting strip is periodic in ϕ , the modes can be written as e j ( β z −ωt + mϕ ) ∑ n e j 2π nϕ / φ , where

n is integer, φ is the periodicity of supporting strip in ϕ direction. The effective wavevector kϕ = m / r in the ϕ direction goes to zero for r → ∞ . So the bandgap of this structure is the same as “OmniGuide” Bragg fiber in Ref. [23] and purely depends on k r and β as long as

ws is small enough. For designing hollow-core Bragg fiber with cobweb-structured cladding, some important structural parameters related to the permitted normalized frequency range of the TE01 mode, and their varying rule were analyzed by using a plane wave expansion method [27]. The

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lowest-loss mode in Bragg fiber is TE01 mode. The simulated results for hollow-core Bragg fiber with cobweb-structured cladding show that leakage losses of TE02 mode for the fibers with η = 0.05 , d 2 = 0.25μ m , n2 = 1.49 , N = 4 and different core radii ( rco = 10 μ m and 6 3 50 μ m ) at λ = 0.65μ m are 5.4 ×10 and 8.7 ×10 times larger than those of TE01 mode,

respectively. Thus, the permitted frequency range of TE01 mode is of especially interest. The most commonly used material in POF is PMMA, its refractive index is 1.49. Using PMMA as the high-index material of the Bragg reflection layers, the first two TE modes in the Bragg reflection layers are calculated with the plane wave expansion method [33]. Figure 6 shows the mode index of the first two TE modes in the Bragg reflection layers. TE01 mode is the fundamental mode in the hollow core. Its mode index must be below 1 and approach to 1. The frequency range formed by two intersecting points ( P and Q ) of the two TE mode curves and the air line ( neff = 1 ) is approximately the permitted frequency range of TE01 mode in hollow core. For η = 0.01 and 0.05, n1 (air), n2 = 1.49 (PMMA), we can see from Fig.6 that this kind of structure can guide light in the hollow core over a wide frequency range. Different η have a strong effect on the permitted normalized frequency range of the TE01 mode. For η = 0.01 , normalized frequency can achieve the range from 2.91 to 45.76, while for η = 0.05 , normalized frequency is within the range 1.34 to 9.5. The permitted normalized frequency range of TE01 mode shrinks more than 5 times as η changes from 0.01 to 0.05. In order to figure out the influence of the structural parameter η of Bragg reflection layers on the permitted normalized frequency range of TE01 mode, the permitted normalized frequency range of TE01 mode with different η at a fixed n2 (1.49) was calculated. The results are listed in Table 1.

Figure 6. Permitted normalized frequency range of TE01 mode for Bragg fiber with

η =0.01, 0.05. The

two curves indicate the first two TE modes in the Bragg reflection layers, and the solid line is the air line ( n =1) [27].


267

Table 1. The permitted normalized frequency range of TE01 mode vs different η at a fixed n2 (1.49) [27]

η Q point value P point value Λ / λ range (Q ~ P)

0.5

0.4

0.3

0.2

0.1

0.08

0.06

0.05

0.04

0.02

0.01

1.36

1.59

1.96

2.72

4.98

6.11

8.00

9.5

11.76

23.07

45.76

0.57 0.79

0.60 0.99

0.65 1.31

0.75 1.97

0.99 3.99

1.09 5.02

1.24 6.76

1.34 8.16

1.49 10.27

2.07 21.00

2.91 42.85

0.5 0.4 ′

0.3

upper limit of d2

′

lower limit of d2

0.2 n2=1.49

0.1 0 0

0.1

0.2

η

0.3

0.4

0.5

Figure 7. Range of normalized high index layer thickness ( d 2′ = d 2 λ ) vs. η [27]

Figure 8. Permitted normalized frequency range of TE01 mode vs. d1 [27]

In regard to the range of allowed values of d 2 , we define d 2′ = d 2 / λ as the normalized high-index-layer thickness, where λ is the operating wavelength of the fiber. The upper and lower limit of d 2′ can be obtained by means of the Q and P point values for each η in Table 1. Take η = 0.05 as an example, the upper limit ( Q point value) of the permitted normalized frequency range of TE01 mode is 9.5, which means ( d1 + d 2 ) / λ = 9.5 . Substituting d1 = d 2 / 0.05 into it, we can obtain d 2′ = d 2 / λ = 0.4524 . The relationship between d 2′ and η is shown in Fig.7. From Fig.7, we can see that the values of d 2′ for the

268


upper line are approximately 0.45, indicating that the maximum thickness of d 2 cannot go beyond 0.45λ . In general, d 2 takes 0.4λ ~ 0.3λ . The minimum thickness of d 2′ decreases when decreasing η .

Figure 9. Permitted normalized frequency range of TE01 mode as a function of n2 for η = 0.05 [27].

In regard to the relationship between the d1 and permitted normalized frequency range of

TE01 mode, the permitted normalized frequency range of TE01 mode with d 2 = 0.25μ m and

n2 = 1.49 at different d1 is illustrated in Fig.8. One obvious feature of Fig.8 is that the permitted normalized frequency ranges of TE01 mode and the corresponding thickness d1 of air layer are approximately a linear relationship. Thus, so long as the thickness d1 of air layer increases at a fixed d 2 , the normalized frequency range broadens. In regard to the relationship between n2 and permitted normalized frequency range of

TE01 mode, a series of n2 ranging from 1.45 to 5.8 at a fixed η (0.05) are calculated, as shown in Fig.9. The permitted normalized frequency range of TE01 mode increases when n2 decreases. Most of polymers have the refractive index smaller than 1.8. Therefore, they are advantageous as the materials of the fiber with a large transmission frequency range. In regard to the tolerance of the parameters, we take a dielectric material PMMA as an example. The design objective is a hollow-core fiber to use as optical fiber communication in the wavelength range from 0.65μ m to 1.65 μ m . The design parameters are η = 0.05 ,

d 2 = 0.25μ m and n2 = 1.49 . Its normalized frequency Λ / λ is in the range from 5.25/0.65=8.1 to 5.25/1.65=3.2, all within the permitted normalized frequency range of TE01 mode (9.5-1.34) as shown in Fig.6(b). If d 2 has a error of d 2 ± 20% in the production process, this corresponds to d 2 = 0.2 μ m and 0.3μ m . For d 2 = 0.2 μ m , the normalized frequency range is from 5.2/0.65=8 to 5.2/1.65=3.15. This is within the permitted normalized frequency range of TE01 mode (11.76-1.49) as shown in Table 1 for η = 0.04 . For d 2 = 0.3μ m , the normalized frequency range is from 5.3/0.65=8.15 to 5.3/1.65=3.21. This is


269

almost within the permitted normalized frequency range of TE01 mode (8.00-1.24) as shown in Table 1 for η = 0.06 . If d1 has a error of d1 ± 25% , this corresponds to d1 = 3.75 μ m and 6.25 μ m . For d1 = 3.75μ m , the normalized frequency range is from 4/0.65=6.15 to

4/1.65=2.42. This is within the permitted normalized frequency range of TE01 mode (7.211.18) for η = 0.067 . For d1 = 6.25μ m , the normalized frequency range is from 6.5/0.65=10

to 6.5/1.65=3.94. This is within the permitted normalized frequency range of TE01 mode (11.76-1.49) for η = 0.04 . Finally, polymers are considered to have different refractive indices for the same material, due to different molecular weight or polymerization condition. If the index of PMMA has a variation of n2 ± 0.02 , which corresponds to n2 = 1.47 and 1.51, then the permitted normalized frequency range of TE01 mode are (9.74-1.38) and (9.27-1.31), respectively. They are essentially consistent with the normalized frequency range (9.5-1.34) as originally designed for n2 = 1.49 and η = 0.05 . The confinement loss and transmission loss for hollow-core Bragg fiber with cobwebstructured cladding were modelled by using an asymptotic formalism [34]. Many results show that the fibers with only 3-4 alternating layers in cladding can achieve the low confinement loss and transmission loss, and the confinement and transmission losses decrease with increasing the hollow-core radius ( rco ). In order to achieve both low loss and wide wavelength range, fiber design should adopt smaller d 2 value and lager d1 value, besides increasing rco and N .

5. Functional Exploiting of Hollow-Core Bragg Fiber with Cobweb - Structured Cladding With the appealing properties described above, the possibility of using hollow-core Bragg fiber with cobweb-structured cladding for transmitting the information and delivering the laser energy was analyzed.

5.1. Fibers for Use in Optical Communications from Visible to near Infrared Region Today, the capacity of optical fiber communications has expanded gigabits per second into terabits per second, enough to meet the current traffic demand due to the explosive growth of data transfer and internet services. Large-capacity and long-distance optical fiber communication trunk line has been installed in many countries. The next big step will be extending the network from fiber-to-the-curb into every building and home. In the area of fiber to the home (FTTH) or fiber to the premises (FTTP) application, passive optical networks (PON), especially ethernet passive optical networks (EPON) and gigabit ethernet passive optical networks (GEPON), are generally preferred for home fiber connections. Usually, the transmission bandwidth and transmission distance required for the

270


networks are 100MHz - 10GHz and 100m-10km , respectively. Therefore, the fibers with lower loss, higher bandwidth and cheaper cost are in demand. People have been trying to find materials and methods to meet those requirements and POF is one of the major approaches being explored in addition to silica glass single-mode fiber and multi-mode hard plastic clad fiber (HPCF). The simulated results for hollow-core Bragg fibers with cobweb-structured cladding had proved that depending on the modal-filtering effect, they may realize the transmission of TE01 single-mode or a few modes, thus achieving the transmission of higher bandwidth ( GHz ) [35].

Figure 10. Absorption loss spectrum of PMMA [36].

A fiber design for use in optical communication from visible to near infrared region is presented. The fiber parameters are n2 = 1.49 (PMMA), n1 = 1 , d 2 = 0.25μ m , d1 = 5μ m ,

rco = 75 μ m and N = 3 . According to absorption loss spectrum of PMMA [36] as shown in Fig.10, we calculate the transmission losses of the fiber. The absorption losses of PMMA at the wavelengths of 0.65 , 0.85 , 1.3 and 1.55μ m are about 100dB/km , 2.5 ×103 dB/km ,

2.5 × 10 4 dB/km and 7.8 × 10 4 dB/km , respectively. The transmission losses of TE01 mode at these wavelengths are 3.9 × 10 −4 dB/km , 4.3 ×10−3 dB/km , 0.13dB/km and 0.80dB/km , respectively. The results show that after inevitable factors (material purity, imperfection and nonuniformity of fiber structure and existence of supporting strips) being considered, the transmission losses of the fiber are still very low. Thus, by using an inexpensive material (PMMA), it allows the fibers to meet the needs of the transmission distance and bandwidth for EPON and GEPON, and to realize the wavelength division multiplexing (WDM).


271

5.2. Fibers for Use in THz Waveguiding The THz radiation, whose frequency range is about 0.1 − 10THz , has important applications in spectroscopy, imaging, space science and information transmission. To date, progress in THz wave generation and detection techniques has been enormous. However, most of the present THz systems rely on free space propagation due to the absence of low loss waveguides and transparent materials in the THz region. The waveguides constructed with some metals suitable for microwave guides or some dielectrics (such as silica) suitable to optical waveguiding have very high losses for THz wave. Even if for high-resistivity silicon, the most common material for the passive devices in the THz technology, its absorption coefficient is of the order of 0.04cm −1 . In recent years, THz waveguides have been fabricated from some dielectrics (such as sapphire, plastics) except from metals such as Cu, brass, and stainless steel. The loss coefficients of high-index core (solid-core) photonic crystal fibers using high-density polyethylene (HDPE) [37] and polytetrafluoroethylene (Teflon) [38] are less than 0.5cm −1 ( 0.1 − 3THz ) and approximately 0.12cm −1 , respectively. Hollow polycarbonate waveguides with inner Cu coatings for broadband THz transmission have been reported [39]. The lowest loss 3.9dB/m ( 0.00898cm −1 ) was obtained from a 3mm core diameter fiber at 158.51μ m wavelength. Recently, a simple subwavelength-diameter ( 200μ m ) plastic (polyethylene) wire, similar to an optical fiber for guiding a THz wave has been reported as well [40]. Its attenuation constant is reduced to less than 0.01cm −1 in the frequency range near 0.3THz . A fiber design for use in THz waveguiding is presented. The structural parameters of fibers (A, B, C) are as follows: rco = 9mm , n2 = 1.52 , n1 = 1 , d 2 = 25μ m , d1 = 500μ m and

N = 3 (fiber A); rco = 12mm , n2 = 1.52 , n1 = 1 , d 2 = 70 μ m , d1 = 1050 μ m and N = 3 (fiber B); rco = 16mm , n2 = 1.52 , n1 = 1 , d 2 = 150 μ m , d1 = 2250 μ m and N = 3 (fiber C). The host material was assumed to be lossless with a refractive index of 1.52 (corresponding to HDPE material). The confinement loss as a function of wavelength for TE01 , TE02 , TM 01 and

TM 02 modes is shown in Fig.11. The lowest-loss mode is TE01 mode. The confinement loss of the TE01 mode at the least-loss wavelength is 1.13 × 10 −8 dB/km at 83.5μ m (fiber A),

2.15 × 10 −7 dB/km at 233μ m (fiber B), and 5.05 × 10−7 dB/km at 500 μ m (fiber C).

Figure 11. Confinement loss as a function of wavelength for TE01 , TE02 , TM 01 , and TM 02 modes.

272


Figure 12. Transmission loss as a function of wavelength for TE01 , TE02 , TM 01 , and TM 02 modes.

Then, we attempt to take the calculation of losses further by including the material absorption. Based on the absorption spectra of HDPE in wavelength range 50 μ m − 1200 μ m [41], the transmission losses of three hollow-core fibers (A, B, C) with cobweb cladding are calculated. The calculated results are shown in Fig.12. The data in Fig.12 show that the transmission losses of TE01 mode for fiber A in the wavelength range of

65μ m − 200 μ m are below 5.5dB/km . The lowest loss is 0.63dB/km (corresponding to loss coefficient 1.45 × 10 −6 cm −1 ) at 90 μ m . The transmission losses of TE01 mode for fiber B in the wavelength range of 200 μ m − 450 μ m are below 5.0dB/km . The lowest loss is

2.0dB/km at 270 μ m . The transmission losses of TE01 mode for fiber C in the wavelength range of 420 μ m − 1000 μ m are below 5.6dB/km . The lowest loss is 2.09dB/km at 560 μ m . The above transmission losses were taken into account only the absorption spectra of the material (HDPE). In fact, certain spectral region in the THz waves may not be available for signal transmission due to the strong absorption of water present in the constituent materials and air-core for the polymer fibers [42]. Therefore, while using hollow-core polymer Bragg fiber with cobweb-structured cladding in transmitting light through air-core, it is very important to eliminate the water from the constituent material and avoid moist air in the environment during fabrication and storage.

5.3. Fibers for Infrared (IR) Applications IR optical fibers may be defined as fiber optics transmitting wavelengths greater than approximately 2 μ m . IR fibers can be useful for the medical, industrial, civil, and military arenas. For example, they are used in surgical applications by transmitting CO2 laser radiation (10.6 μ m) and Er : YAG laser radiation (2.94 μ m) . When used as fiber sensors, IR fibers are generally used either to transmit blackbody radiation for temperature measurements or as an active or passive link for chemical sensing, achieving non-contact temperature monitoring and remote spectroscopic chemical sensing. The application in the industrial arena includes welding and cutting. Scanning near-field microscopy by using high-quality singlemode and multimode IR fiber-tapered tips can obtain 20-nm topographic resolution and about 200-nm optical resolution for a variety of samples. IR fibers are also used for military applications including anti-aircraft missile defense. The development of infrared fiber optics


273

began in the year 1960. The first IR fibers were fabricated in the mid-1960’s using arsenicsulphur glasses [43]. So far, there are four classes of infrared fibers: (i) fluoride, germanate, tellurite or chalcogenide glass based solid-core fibers; (ii) crystalline silver halide solid-core fibers; (iii) hollow-core fibers in which inner wall coatings have high reflectivity; and (iv) solid-core photonic crystal fibers and hollow-core photonic bandgap fibers. The optical-loss values of the sulfide based chalcogenide glass fibers at the Naval Research Laboratory have been reduced to only 0.1 to 0.2 dB/m in fiber lengths of about 500m by using improved chemical purification and better fiber fabrication techniques [44]. The optical losses of crystalline silver halide solid-core fibers by an extrusion process have been reduced to lower than 50dB/km in a broad IR region from 9 to 14 μ m and lower than

1dB/m in the region from 3 to 20 μ m [45]. The losses of rectangular hollow waveguides with 1- m -long and 1mm × 1mm cross-section by first depositing thin-film coatings of PbF2 on phosphor bronze strips and then soldering four of these phosphor bronze metal strips together are as low as 0.1dB/m at 10.6 μ m [46]. Photonic crystal fibers for the middle infrared were fabricated by multiple extrusions of silver halide crystalline materials [47]. These fibers are composed of two solid materials: the core consists of pure AgBr (n=2.16) and the cladding includes AgCl (n=1.98) fiberoptic elements arranged in two concentric hexagonal rings around the core. IR transmissive As-S glass and As-Se glass triangular photonic band gap fiber structures were theoretically modeled [48]. From numerical simulations, Pottage et al. [49] discovered a new type of air-line bandgap that is of considerable importance in the design of practical hollow-core photonic bandgap fibers made from high-index glass (n≥2.0) for guidance in the mid/far-IR. A silica based hollow-core photonic bandgap fiber in which fiber-core diameter is 40 μ m (nineteen capillaries were omitted from the centre of the stack to form the core), the overall outside diameter is 150 μ m and the nearest-neighbor hole spacing is around 7 μ m , has been fabricated [50]. The peak of the bandgap is at 3.14 μ m with a typical attenuation of 2.6dB/m . By further optimization of the structure, modeling suggests that a loss below 1dB/m should be achievable. The design is a hollow-core Bragg fiber with cobweb-structured cladding for the mid-IR region. In the wavelength region between 100 μ m and 1μ m , many longitudinal and rotational resonances of molecules are present in almost all substances, especially the long-chain polymers [2]. Polymers such as teflon and polyethylene show relatively strong absorption at 1000cm −1 ( 10 μ m ). The absorption coefficient α at 10 μ m wavelength is about 100cm −1 for teflon and about 50cm −1 for polyethylene [16]. A fiber design for use in infrared is presented. The structural parameters of fibers (A, B) are as follows: rco = 1500 μ m , d 2 = 1.4μ m ,

d1 = 30 μ m , n1 = 1 , n2 = 1.37 (teflon) and N = 3 (fiber A); rco = 1200 μ m , d 2 = 2.8μ m , d1 = 28μ m , n1 = 1 , n2 = 1.55 (PES) and N = 3 (fiber B). The absorption coefficient of the host material (teflon) is 100cm −1 (corresponding to absorption loss 4.343 × 107 dB/km ). The calculated results are shown in Fig.13(a). The data in Fig.13(a) show that the transmission

274


loss of TE01 mode for fiber A in the wavelength range of 2.8μ m to 10.6 μ m are below

39.5dB/km . The lowest loss is 1.47dB/km at 3.9 μ m . The absorption loss of the host material (PES) is 3 × 107 dB/km . The calculated results are shown in Fig.13(b). The data in Fig. 13(b) show that the transmission loss of TE01 mode for fiber B in the wavelength range of 8 μ m to 13μ m are below 30.9dB/km . The loss for the 10.6 μ m wavelength of CO2 laser is 18.9dB/km .

Figure 13. Transmission losses of the mid-IR region for fibers (A, B).

The numerical results show that despite the strong absorption of the polymers in the midIR region, the transmission losses of the fibers are lower by comparison with those of other IR fibers reported in the literature. And the polymer fibers have an advantage over other fibers in flexibility.

5.4. Circular-Polarization-Maintaining Single-Mode Fibers Standard single-mode fibers support two degenerate, orthogonally polarized modes ( HE11 mode). Random imperfections in the fiber structure and external forces on the fibers can create asymmetries that break the polarization degeneracy, resulting in polarization mode dispersion and polarization fading in interferometers. Conventional polarization-maintaining fibers (highly birefringent fibers) and some single-polarization single-mode photonic crystal fibers supported a linear polarization mode. The fibers require accurate alignment of the birefringence axes of the two fibers when coupling, splicing and some sensing applications are considered. Therefore, in the year 1980, Jeunhomme and Monerie [51] have suggested the design of a circular-polarization-maintaining single-mode fiber cable . Recently, Argyros et al. [25] have presented the design that supports a single-polarization, circularly symmetric nondegenerate mode in an air-core ring-structured Bragg fiber. We presented the design that supports a circular-polarization-maintaining single mode in a hollow-core and cobweb-structured cladding Bragg fiber. The structural parameters of the fiber are rco = 10 μ m , n1 = 1 , n2 = 1.585 (PC), d 2 = 0.21μ m , d1 = 2.1μ m and N = 3 . The intrinsic losses of the host material (PC) are 166dB/km at 650 − 656nm and 224dB/km at 764nm [52]. The calculated results show that the transmission losses of TE01 mode (lowest


275

loss mode) are 0.226dB/km at 650 − 656nm and 0.170dB/km at 764nm , those of TE02 mode (second-lowest loss mode) are 3.513dB/km at 650 − 656nm and 1.848dB/km at 764nm . The ratio of the loss of the TE02 mode to the loss of the TE01 mode is 15.54 ( 650 − 656nm ) and 10.87 ( 764nm ). In accordance with the research reported in Ref.32, the fiber is single moded for lengths between 11.4km and 88.5km ( 650 − 656nm ), and 21.7km and 117.6km ( 764nm ). We expect that this type of hollow-core Bragg fibers with circular-polarizationmaintaining single-mode and low-losses will find many applications, such as gyroscopes, current sensors and coherent communication systems.

6. Applications of Hollow-Core Bragg Fiber with CobwebStructured Cladding A new generation of POFs has the advantages of both low-cost and high-performance in terms of attenuation, bandwidth and flexibility. It will find many applications in diverse areas and increases market acceptance. As respects information transmissions, the new generation of POFs can guide the light of visible to terahertz radiation, and can be applied to optical fiber communications and optical fiber sensing, such as LANs, specially FTTH, THz wave fiber communications. It can also be used as an active or passive links for chemical sensing and remote spectroscopic chemical sensing, a variety of physical quantity sensing as well as medical diagnostics including noninvasive blood glucose monitoring and detection of tumors. As respects delivery of power laser radiation and solar radiation, hollow-core Bragg fibers with cobweb-structured cladding can deliver solar radiation into darkroom, be used for indoor illumination, replacing former guided light tube or solid-core polymer fiber. Efficient hollow-core guiding for delivery of power laser radiation ( 10.6 μ m CO2 laser, 2.94 μ m Er : YAG laser, etc) can be achieved by replacing the traditional PMMA with heat-resistant polymers, and can be used for medical therapy and processing including micro-processing and material processing. By using gas-filled hollow-core Bragg fibers with cobweb-structured cladding, it is possible to obtain the EUV light generated through the process of high-harmonic upconversion of femtosecond laser and ultrahigh efficiency laser wavelength conversion by pure stimulated rotational Raman scattering, as well as to use laser light to levitate and guide particles through the hollow-core fiber, etc. Circular-polarization-maintaining single-mode low-loss fibers and high-strength, flexibility and resistance to shock fibers will provide the possibilities for some new applications. These fibers will stimulate further progress, both in fiber and allied systems technologies. The new generation of POFs based on hollow-core Bragg fiber with cobweb-structured cladding will find many applications and is irreplaceable for some applications such as THz wave low-loss transmission.

276


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Chapter 10

DISSIPATIVE SOLITONS IN OPTICAL FIBER SYSTEMS Mário F.S. Ferreira and Sofia C.V. Latas Department of Physics, University of Aveiro, 3800-193 Aveiro, Portugal

Abstract We introduce the concept of dissipative solitons, which emerge as a result of a double balance: between nonlinearity and dispersion and also between gain and loss. Such dissipative solitons have many unique properties which differ from those of their conservative counterparts and which make them similar to living things. We focus our discussion on dissipative solitons in optical fiber systems, which can be described by the cubic-quintic complex Ginzburg-Landau equation (CGLE). The conditions to have stable solutions of the CGLE are discussed using the perturbation theory. Several exact analytical solutions, namely in the form of fixed-amplitude and arbitrary-amplitude solitons, are presented. The numerical solutions of the quintic CGLE include plain pulses, flat-top pulses, and composite pulses, among others. The interaction between plain and composite pulses is analyzed using a twodimensional phase space. Stable bound states of both plain and composite pulses are found when the phase difference between them is ± π / 2 . The possibility of constructing multisoliton solutions is also demonstrated.

1. Introduction Solitary waves have been the subject of intense theoretical and experimental studies in many different fields, including hydrodynamics, nonlinear optics, plasma physics, and biology [1][5]. In fact, the history of solitons dates back to 1834, the year in which James Scott Russell observed that a heap of water in a canal propagated undistorted over several kilometres [6]. However, the term “soliton” was coined only in 1965, to reflect the particle-like nature of solitary waves that remain intact even after mutual collisions [7]. Such waves correspond to localized solutions of integrable equations such as the Korteveg de Vries and nonlinear Schrödinger equations. In these circumstances, solitons were usually attributed only to integrable systems. However, the concept of soliton was subsequently broaden to include also the localized solutions of non-integrable systems.

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Mário F.S. Ferreira and Sofia C.V. Latas

Concerning the field of nonlinear optics, one can distinguish between temporal and spatial solitons [8]. Spatial optical solitons are beams of light in which nonlinearity counteracts diffraction, leading to a robust structure which propagates without change of form. Such structures will play a major role in the future in the field of all-optical processing and logic. Temporal solitons, on the other hand, represent shape invariant (or breathing) pulses, formed by a balance between nonlinearity and dispersion. It is believed that temporal solitons will play a major role in future all-optical high-capacity transmission systems [9] [10]. Until now, the main emphasis has been given to the well-known conservative soliton systems, where only the diffraction or dispersion needs to be balanced by the nonlinearity. However, a new field has emerged in the last few years concerning the formation of solitons in systems far from equilibrium [11]. These solitons are termed dissipative solitons or autosolitons and they emerge as a result of a double balance: between nonlinearity and dispersion and also between gain and loss. Such dissipative solitons have many unique properties which differ from those of their conservative counterparts. For example, except for very few cases [5], they form zero-parameter families and their properties are completely determined by the external parameters of the optical system. They can exist indefinitely in time, as long as these parameters stay constant. However, they cease to exist when the source of energy or matter is switched off, or if the parameters of the system move outside the range which provides their existence. Even if it is a stationary object, a dissipative soliton shows non-trivial energy flows with the environment and between different parts of the pulse. Hence the dissipative soliton is an object which is far from equilibrium and which presents characteristics similar to a living thing. In fact, we can consider animal species in nature as elaborate forms of dissipative solitons. An animal is a localized and persistent “structure” which has material and energy inputs and outputs and complicated internal dynamics. Moreover, it exists only for a certain range of parameters (pressure, temperature, humidity, etc.) and dies if the supply of energy is switched off. The same analogy can be applied to individual organs within an animal, since each maintains its shape and function over time. Many non-equilibrium phenomena, such as convection instabilities, binary fluid convection and phase transitions, can be described by the complex Ginzburg-Landau equation (CGLE) [12]-[14]. In the field of nonlinear optics, the CGLE can describe various systems, namely optical parametric oscillators, free-electron laser oscillators, spatial and temporal soliton lasers, and all-optical transmission lines [9][15]-[27]. In these systems there are dispersive elements, linear and nonlinear gain, as well as losses. In some cases, the CGLE admits a multiplicity of solutions for the same range of system parameters. This reality again resembles the world of biology, where the number of species existing in the same environment is trully impressive. In this chapter we will discuss the cubic-quintic CGLE and the characteristics of some of its solutions. In Section 2 we present the CGLE and in Section 3 the perturbation approach to solve this equation is discussed. Some analytical and numerical solutions of the CGLE are presented in Sections 4 and 5, respectively. Finally, Section 6 summarizes the main conclusions.

Dissipative Solitons in Optical Fiber Systems

281

2. The Complex Ginzburg-Landau Equation In one of the forms used in nonlinear optics, the cubic-quintic complex Ginzburg-Landau equation (CGLE) can be written as [5][19]-[27]:

i

∂q D ∂ 2 q ∂ 2q 2 2 4 4 + + q q = i δ q + i β + iεq q q + iμ q q − ν q q 2 2 ∂Z 2 ∂T ∂T

(1)

where Z is the propagation distance or the normalized number of round trips, T is the retarded time, q is the normalized envelope of the electric field, β stands for spectral filtering ( β >0),

δ is the linear gain or loss coefficient, ε accounts for nonlinear gain-absorption processes (for example, two-photon absorption), μ represents a higher order correction to the nonlinear gain-absorption, and ν is a higher order correction term to the nonlinear refractive index. The parameter D is the group velocity dispersion coefficient, with D = ±1 , depending on whether the group velocity dispersion (GVD) is anomalous or normal, respectively. The CGLE is rather general, as it includes dispersive and nonlinear effects, in both conservative and dissipative forms. It is known in many branches of physics, including fluid dynamics, nonlinear optics and laser physics. Equation (1) becomes the standard nonlinear Schrödinger equation (NLSE) when the right-hand side is set to zero. When this does not happen, Eq. (1) is non-integrable, and only particular exact solutions can be obtained. In the case of the cubic CGLE ( μ = ν = 0 ), exact solutions can be obtained using a special ansatz [28], Horota bilinear method [29] or reduction to systems of linear PDEs [30]. Concerning the quintic CGLE, the existence of soliton-like solutions in the case ε > 0 has been demonstrated both analytically and numerically [5][20][26][31]. Exact solutions of the quintic CGLE, including solitons, sinks, fronts and sources, were obtained in [32], using Painlevé analysis and symbolic computations. It must be noted that Eq. (1) can not be used as it stands to describe the behaviour of femtosecond optical pulses. For such ultrashort pulses, some higher-order nonlinear and dispersive effects must be taken into account, which results in additional terms to be added to the right-hand side of Eq. (1) [33]-[38].

3. Results from the Soliton Perturbation Theory Assuming that D=+1 and that all the other coefficients in the right-hand side of Eq. (1) are small, we can use the adiabatic soliton perturbation theory [9][34][39][40] to evaluate the dynamical evolution of the soliton parameters the amplitude η and the frequency κ , with which the one soliton solution is given by:

[

]

i ⎧ ⎫ q (T , Z ) = η ( Z ) sec h{η ( Z )[T + κ ( Z ) − θ ]}exp⎨− iκ ( Z )T + η ( Z ) 2 − κ ( Z ) 2 Z − iσ ⎬ (2) 2 ⎩ ⎭

282


Applying the perturbation procedure, we get the following set of ordinary-differential equations:

16 dη ⎛1 ⎞ 4 = 2δη − 2βη ⎜ η 2 + κ 2 ⎟ + εη 3 + μη 5 15 dZ ⎝3 ⎠ 3

(3)

dκ 4 = − βη 2κ dZ 3

(4)

As can be seen from Eq. (4), the soliton frequency approaches asymptotically to κ = 0 (stable fixed point) if η ≠ 0 . The stable fixed points for the soliton amplitude, on the other hand, are given by minimums of the potential function φ defined by:

dη dφ =− dZ dη

(5)

Considering the Eq. (3), we have the following expression for the potential function:

φ (η ) = −δη 2 +

η

1 (β − 2ε )η 4 − 8 μη 6 6 45

(6)

For the zero-amplitude state to be stable, the potential function must have a minimum at = 0 , in addition to a minimum at η = η s ≠ 0 . These objectives can be achieved if the

following conditions are verified [20]:

δ < 0 , μ < 0 , ε > β / 2 , 15δ > 8μη s4

(7)

We can verify from the above conditions that the inclusion of the quintic term in Eq. (1) is necessary to have the double minimum potential. The stationary value for the soliton amplitude can be obtained from Eq. (6) and is given by:

η s2 where

=

− 5(ε − ε s ) − 5 (ε − ε s ) 2 − 24δμ / 5 8μ

(8)

ε s = β / 2 for small values of β . However, the result given by Eq. (8) can be

generalized for arbitrary values of

β using ε s given by [20][26][41]:


283

β 3 1 + 4β 2 − 1 εs = 2 2 + 9β 2

(9)

From Eq. (8) it can be seen that a stationary amplitude

η s = 1 occurs when the

coefficients satisfy the relation:

15δ + 5(2ε − β ) + 8μ = 0

(10)

The discriminant in Eq. (8) must be greater than or equal to zero for the solution to exist. For given values of β , μ , and ε , the allowed values of δ to guarantee a stable pulse propagation must satisfy the condition

δ min ≤ δ ≤ 0 , where

δ min When

5(ε − ε s ) = 24μ

2

(11)

δ = 0 , the peak amplitude is found to achieve a maximum value:

η max = −

5 (ε − ε s ) 4 μ

(12)

μ = 0 and ε = ε s the peak amplitude becomes arbitrary. On the other hand, for given values of β , μ , and δ , the minimum value of allowed ε For

becomes

ε min = ε s + 24δμ / 5

(13)

Considering the last condition in Eq. (7) or, alternatively, from Eq.s (8) and (13) we find that there is a minimum value for the peak amplitude, given by:

η min = 4

15δ 8μ

(14)

Fig. 1 shows the potential function given by Eq. (6) when the relation (10) is satisfied for

β = 0.3 , ε = 0.5 , μ = −0.25 (curve a), μ = −0.34375 (curve b) and μ = −0.5 (curve c). Curves a and b present a minimum at η = 1 and η = 0 since they satisfy the conditions (7), corresponding to negative values of the linear gain ( δ = −0.05 and δ = −0.1 , respectively). However, curve c has no minimum at η = 0 , since the corresponding value of δ is positive ( δ = 0.033 ).

284


φ versus soliton amplitude when the relation (10) is satisfied for β = 0.3 , ε = 0.5 , ν = 0 , μ = −0.5 (curve a), μ = −0.34375 (curve b) and μ = −0.25 (curve c).

Figure 1. Potential

Figure 2. Phase portrait of Eq.s (3) and (4) corresponding (A) to curve c and (B) to curve b of Figure 1.


285

Fig. 2 illustrates the stability characteristics of the stationary solutions using the phaseplane formalism. Fig. 2a corresponds to curve c in Fig. 1, and we observe that, in this case, soliton propagation can be affected by background instability due to the amplification of small-amplitude waves. The steady-state solution shows a limited basin of attraction. For example, initial conditions with ηi = 0.7 and κ i = ±1 evolve toward the trivial solution

η s = 0 of Eq.s (3) and (4). For these initial conditions, the nonlinearity is not sufficiently strong to balance dispersion, and the pulse disperses away. The dashed curves in Fig. 2a give approximate limits between different basins of attraction. From a perturbation analysis of Eq.s (3) and (4) around η = 0 , one can show that these curves cross the η = 0 axis at

κ c = ±0.33 . Thus, waves weak initial amplitudes grow up to η s = 1 if κ i < 0.33 . In this case, soliton propagation can be severely affected by the background instability. Fig. 2b corresponds to curve b in Fig. 1, and we can see that, in this case, the background instability is avoided, since the small-amplitude waves are attenuated, irrespective of their frequency κ . Besides the stable stationary point at η s = 1 , we note, in this case, the existence of another stationary point at η s ≈ 0.5 , which is unstable. This simple approach shows that, in general, the CGLE has stationary soliton-like solutions, and that, for the same set of equation parameters, there may be two of them simultaneously (one stable and one unstable). Moreover, this approach shows that soliton parameters are fixed.

4. Exact Analytical Solutions Several types of exact analytical solutions of the CGLE have been obtained considering a particular ansatz [5][26]. However, due to restrictions imposed by the ansatz, these solutions do not cover the whole range of parameters. In the following, we will assume a stationary solution of Eq. (1) in the form:

q(T , Z ) = a(T ) exp{id ln[a (T )] − iωZ } where a(T) is a real function and d,

(15)

ω are real constants.

4.1. Solutions of the Cubic CGLE The cubic CGLE is given by Eq. (1) with

μ = ν = 0 . Inserting Eq. (15) in this equation we

obtain the following solution for a(T):

a (T ) = A sec h( BT ) where

(16)

286


A=

B 2 (2 − d 2 ) + 3dβ B 2 2 B=

(17)

δ

(18)

βd + d − β 2

and d is given by

d=

3(1 + 2εβ ) ± 9(1 + 2εβ ) 2 + 8(ε − 2β ) 2 2(ε − 2 β )

(19)

On the other hand, we have

ω=−

δ (1 − d 2 + 4 βd ) 2(d − β + β d 2 )

(20)

The solution (16)-(18) is known as the solution of Pereira and Stenflo [28]. Although the amplitude profile of the solution (16)-(18) is an hyperbolic secant as in the case of the NLSE solitons, two important differences exist between the CGLE and the NLSE solitons. First, for CGLE pulses the amplitude and width are independently fixed by the parameters of (1), whereas for NLSE solitons A=B. The second difference is that the CGLE solitons are chirped. The solution given by Eq.s (16)-(18) has a singularity at d − β + βd 2 = 0 , which takes place on the line

ε s ( β ) in the plane ( β , ε ) defined by Eq. (9). For a given value of β , the

denominator in the expression for B in Eq. (18) is positive for

ε < ε s and negative for

ε > ε s . Hence, for solution (16)-(18) to exist, the excess linear gain δ must be positive for ε < ε s and negative for ε > ε s . In the last case, both numerical simulations and the soliton perturbation theory show that the soliton is unstable relatively to any small amplitude fluctuations [20][26]. On the other hand, for δ > 0 and ε < ε s the solution (16)-(18) is stable, since after any small perturbation it approaches the stationary state. However, the background state is unstable in this case, since the positive excess gain also amplifies the linear waves coexistent with the soliton trains. The general conclusion is that either the soliton itself or the background state is unstable at any point in the plane ( β , ε ) , which means that the total solution is always unstable. The stationary value of the pulse width 1/B can be significantly reduced by a convenient choice of the system parameters [42]. In fact, it can be verified from Eq.s (11) and (12) that, for a given value of the filter strength β , as the nonlinear gain coefficient approaches the value

ε s given by Eq. (9), the amplitude A increases to infinity and its width 1/B tends to


287

zero. This singularity can be used in soliton lasers to vary the pulse parameters by a small variation of the material parameters. If β and ε satisfy the Eq. (9) and δ = 0, a solution of the cubic CGLE with arbitrary amplitude exists, given by [5][26]:

a(T ) = C sec h( DT )

(21)

where C is an arbitrary positive parameter and C/D is given by:

C = D

( 1 + 4β − 1) 2 β (3 1 + 4 β − 1)

(2 + 9β ) 1 + 4β 2

2

2

2

2

(22)

We have also

d=

1 + 4β 2 − 1 2β (23)

ω = −d

1 + 4β D2 2β 2

Figure 3. Simultaneous propagation of four arbitrary-amplitude solitons with with amplitudes 2, 1.5, 1, and 0.5, for

δ = 0 , β = 0.2 ν = μ = 0

and

ε = εs .

288

Mário F.S. Ferreira and Sofia C.V. Latas It can be verified that the cubic CGLE becomes invariant under the scale transformation

q → Dq , T → DT , Z → D 2 Z when δ = 0 . This is the reason for the existence of the arbitrary-amplitude solitons. On the ther hand, we can see that the limiting value of the amplitude-width product A/B for the fixed-amplitude solitons coincide with the value C/D on the line (22) [20]. This shows that arbitrary amplitude solitons can be considered as a limiting case of fixed amplitude solitons when δ → 0 . However, the arbitrary amplitude solitons have stability properties different from those for fixed amplitude solitons. In fact, arbitrary amplitude solitons are stable pulses, which propagate in a stable background because δ = 0 . This feature is illustrated in Fig. 3, which shows the simultaneous propagation of four stable solitons with amplitudes 2, 1.5, 1, and 0.5, for δ = 0 , β = 0.2 and ε = ε s .

4.2. Solutions of the Quintic CGLE Considering the quintic CGLE and inserting Eq. (15) in Eq. (1), the following general solution can be obtained for f = a [5][26]: 2

f (T ) =

2 f1 f 2

(

( f1 + f 2 ) − ( f1 − f 2 ) cosh 2α

f1 f 2 T

)

(24)

where

α=

μ 3β − 2d − βd 2

(25)

and d is given by Eq. (19). The parameters f1 and f1 are the roots of the equation:

δ 2ν 2(2 β − ε ) f2+ f − =0 2 2 8β d − d + 3 3d (1 + 4β ) d − β + βd 2

(26)

and the coefficients are connected by the relation:

⎡12εβ 2 + 4ε − 2 β ⎤ ⎡ 2εβ − 16 β 2 − 3 ⎤ d − 2β ⎥ + μ ⎢ d + 1⎥ = 0 ε − 2β ε − 2β ⎣ ⎦ ⎣ ⎦

ν⎢

(27)

One of the roots of Eq. (26) must be positive for the solution (24) to exist, while the other can have either sign. When the two roots are both positive, the general solution given by Eq. (24) becomes wider and flatter as they approach each other. These flat-top solitons correspond to stable pulses, whereas the solution (24) is generally unstable for arbitray choice of parameters. If f1 = f 2 , the width of the flat-top soliton tends to infinity and the soliton splits into two


289

fronts. The formation and stable propagation of a flat-top soliton will be demonstrated numerically in Section 5. If β and ε satisfy the Eq. (9) and δ = 0, a solution of the quintic CGLE with arbitrary amplitude exists, given by:

(

)

(

)

3d 1 + 4 β 2 P f (T ) = [a(T )] = (2β − ε ) + S cosh 2 PT 2

(

)

(28)

where P is an arbitrary positive parameter and

S=

(2β − ε )

2

d=

9d 2 μ 1 + 4 β 2 + P 3β − 2d − βd 2 2

(29)

1 + 4β 2 − 1 2β (30)

ω = −d When

1 + 4β P 2β 2

μ → 0 , the solution (28) transforms to the arbitrary-amplitude solution of the

cubic CGLE, given by Eq. (21)-(22).

5. Numerical Solutions Due to restrictions imposed by the ansatz, the analytic solutions of the quintic CGLE presented above do not cover the whole range of parameters and almost all of them are unstable. To find stable solutions in other regions of the parameters, different approximate methods [41], a variational approach [43]-[45], or numerical techniques must be used. As shown by the perturbative analysis presented in Section 3, the parameter space where stable solitons exist has certain limitations. We must have β > 0 in order to stabilize the soliton in frequency domain. The linear gain coefficient δ must be zero or negative in order to avoid the background instability. The parameter μ must be negative in order to stabilize the soliton against collapse. Concerning the parameter ν , it can be positive or negative. Stable solitons can be found numerically from the propagation equation (1) taking as the initial condition a pulse of somewhat arbitrary profile. In fact, such profile appears to be of little importance. For example, Fig. 4 illustrates the formation of a fixed amplitude soliton of the cubic CGLE starting from an initial pulse with a rectangular profile. It must be noted that, in this case, the linear gain is positive but relatively small ( δ = 0.003 ) and the soliton propagation remains stable within the displayed distance.

290


In general, if the result of the numerical calculation converges to a stationary solution, it can be considered as a stable one, and the chosen set of parameters can be deemed to belong to the class of those which permit the existence of solitons. In the following we show some examples of stable soliton solutions found with this method.

Figure 4. Formation of a fixed-amplitude soliton solution of the cubic CGLE starting from an initial pulse with a rectangular profile of amplitude Ao = 0.7 (a) and Ao = 1.0 (b), when δ = −0.003 , β = 0.2 , and ε = 0.09 .

δ = −0.01 , ε = 0.4 , μ = −0.3875 (full

Figure 5. (a) Evolution of the peak amplitude and (b) the final pulse profile when

β = 0.15 , ε = 0.2 , ν = 0 , μ = −0.1375 curves), considering an input pulse

(dashed curves) or

q( 0,T ) = sec h( T ) .

Fig. 5 shows (a) the evolution of the peak amplitude and (b) the final pulse profile obtained numerically from Eq. (1), assuming an input pulse with a sech profile and considering the following parameter values: δ = −0.01 , β = 0.15 , ν = 0 , ε = 0.2 ,

μ = −0.1375 (dashed curves) or ε = 0.4 , μ = −0.3875 (full curves). When inserted in Eq. (8), these values provide a stationary amplitude η s = 1 . This prediction of the


291

perturbation theory, as well as the stability of the stationary solution are confirmed by the numerical results of Fig. 5. For small values of the parameters in the right-hand side of Eq. (1) the stable soliton solutions of the CGLE have a sech profile, similar to the soliton solutions of the NLSE, and correspond to the so-called plain pulses (PPs). However, rather different pulse profiles can be obtained for non small values of those parameters. As an example, Fig. 6 illustrates the formation and stable propagation of a flat-top soliton, starting from an initial pulse with a sech profile. The following parameter values were considered: δ = −0.1 , β = 0.5 ,

ε = 0.66 , μ = ν = −0.01 .

Figure

6.

Formation

and

evolution

of

a

flat-top

soliton,

considering

q( 0,T ) = sec h( T ) , for δ = −0.1 , β = 0.5 , ε = 0.66 , μ = ν = −0.01 .

an

input

pulse

Fig. 7 shows (a) the amplitude profiles and (b) the spectra of a plain pulse, as well as of two composite pulses (CPs). The following parameter values were considered: δ = −0.01 , β = 0.5 , μ = −0.03 , ν = 0 , ε = 1.5 (plain pulse), ε = 2.0 (narrow composite pulse) and ε = 2.5 (wide composite pulse). Fig. 7c illustrates the formation and propagation of the wide composite pulse starting from the plain pulse solution represented in a) and b). A composite pulse exhibits a dual-frequency but symmetric spectrum (Fig. 6b) and can be considered as a bound state of a plain pulse and two fronts attached to it from both sides [5]. The “hill” between the two fronts should be counted as a source, because it follows from the phase profile that energy flows from the centre to the CP wings. If one of the fronts of a CP is missing one has a moving soliton (MS) [5]. The MS always moves with a velocity smaller than the velocity of the front for the same set of parameters. In fact, the front tends to move with its own velocity but the soliton tends to be stationary, due to the spectral filtering. The resulting velocity of the MS is determined by competition between these two processes. Increasing slightly the nonlinear gain coefficient and keeping the values of the other parameters equal to those used in Fig. 7 the stationary wide composite pulse shown in Fig. 7c is lost and a non stationary expanding structure appear, as illustrated in Fig. 8.

292


Figure 7. (a) Amplitude profiles and (b) spectra of a plain pulse and of two composite pulses when

δ = −0.01 , β = 0.5 , μ = −0.03 , ν = 0 , ε = 1.5 (plain pulse), ε = 2.0 (narrow composite pulse) and ε = 2.5 (wide composite pulse). Figure 7c illustrates the formation and propagation of the wide composite pulse, starting from the plain pulse solution.

Figure 8. Nonstationary expanding structure obtained from an initial plain pulse when

β = 0.5 , μ = −0.03 , ν = 0

and ε

= 2.183 .

δ = −0.01 ,


293

Pulsating and exploding soliton solutions of the CGLE were also observed recently [46]. Pulsating solitons correspond to fixed solutions in the same way as the stationary pulses and can be found when the parameters of the CGLE are far enough from the NLSE limit. On the other hand, exploding solitons appear for a wide range of parameters of the CGLE and originate from soliton solutions which remain stationary only for a limited period of time. Following the explosion, there is a “cooling” period, after which the solution becomes “stationary” again. This is a periodic phenomenon, like other phenomena occurring in the nature. It can be verified that different stable stationary solutions of the quintic CGLE can exist simultaneously for the same set of parameters [5][19]. This can be understood considering that solitons, fronts and sources are elementary building units which can be combined to form more complicated structures. In more complex systems, the number of solutions may be very high. This reality again resembles the world of biology, where the number of species is trully impressive.

6. Soliton Bound States After finding the conditions for the existence of stable solitary-pulse solutions of the CGLE equation, the next natural step is to consider their interactions and, in particular, the possibility of the existence of bound states of these pulses [19][25][47]-[52]. In fact, the problem of soliton interaction is crucial for the transmission of information. In the case of Hamiltonian systems, the interaction between the pulses is inelastic. Energy exchange between the pulses is one of the mechanisms that makes the two-soliton solutions of these systems unstable, even when such stationary solutions do exist. The situation is rather different for dissipative systems. In this case, all solutions are a result of a double balance: between nonlinearity and dispersion and also between gain and loss. Moreover, the properties of dissipative solitons are completely determined by the external parameters of the optical system. For given values of the CGLE parameters, the amplitude and width of its soliton solutions are fixed. As a consequence, during the interaction of two solitons, basically only two parameters may change: their separation r and the phase difference, φ , between them. These two parameters provide a two-dimensional plane in which we may analyze of pulse interaction, namely the formation of bound states, their stability and their global dynamics [19][25][51][52]. This reduction in the number of degrees of freedom is a unique feature of systems with gain and loss. In the case of Hamiltonian systems, the amplitudes of the solitons can also change, which can affect the stability of the possible bound states. In order to analyze numerically the soliton interaction in the 2-D space provided by the separation, r, and phase difference, φ , between the two solitons, Eq. (1) can be solved with an initial condition

q(T ) = q0 (T − r / 2) + q0 (T + r / 2) exp(iφ )

(31)

294


where q0 is the stationary solution obtained numerically from Eq. (1) when the values of its parameters are specified. Initial condition (31) with arbitrary values for r and φ will result in a trajectory on the interaction plane. Bound states will be singular points of this plane.

Figure 9. Trajectories on the interaction plane showing the evolution of two plain pulses for

δ = −0.01 , β = 0.5 , ε = 1.5 ,ν = 0 , Y = r sin(φ ) .

and

μ = −0.03 .

We have X =

r cos(φ )

and

Fig. 9 shows an example of a numerical simulation of an interaction between the two solitons on the interaction plane, considering the following parameter values: δ = −0.01 , β = 0.5 , ε = 1.5 , ν = 0 , μ = −0.03 . This figure indicates that, for the given set of parameters, there are at least four singular points. The points P3 and P4 are saddles and correspond to unstable bound states. In these states, the phase difference between the solitons is zero or π . In addition, there are two symmetrically located stable foci (points P1 and P2 ), which correspond to stable bound states of two solitons with a phase difference

φ = ±π / 2

between them. The stationary pulse separation in these bound states is r ≈ 1.62 . As a consequence of its asymmetric phase profile, the two-soliton solution corresponding to the stable bound states P1 and P2 in Fig. 9 moves with a constant velocity. The direction

φ . An example of stable propagation of a two-soliton bound state with a phase difference of π / 2 between the pulses is given in Fig. 10.

of motion depends on the sign of

Dissipative Solitons in Optical Fiber Systems Stable bound states of two CPs, with a phase difference

295

φ = ±π / 2 between them, can

also be observed. This is illustrated in Fig. 11, which shows the stable propagation of a bound state of two composites pulses with a phase difference π / 2 . The following parameter values were assumed: δ = −0.01 , β = 0.5 , ε = 2.0 , ν = 0 , μ = −0.3 . In contrast with the behaviour of the plain pulse bound state shown in Fig. 10, the CP bound state moves at the group velocity.

Figure 10. Propagation of a bound state of two plain pulses with a phase difference of them.

π / 2 between

Figure 11. Propagation of a bound state of two composite pulses with a phase difference of between them.

π /2

296


The two-soliton solution can be assumed as the building block to construct various multisoliton solutions. An example is given in Fig. 12, corresponding to a four-plain pulse solution, with a phase difference of π / 2 between adjacent pulses. As observed in the case of the two-PP solution, multisoliton solutions formed by plain pulses move with a constant velocity along the T axis.

Figure 12. Four-plain pulse solution with a phase difference of π / 2 between adjacent pulses. The dash-dotted (full) lines in (b) correspond to the initial (final) phase profiles.

Figure 13. Five-plain pulse solution and the correspondent phase profiles. The dash-dotted (full) lines in (b) correspond to the initial (final) phase profiles of the pulses in (a).

Multisoliton solutions formed by plain pulses with zero velocity can be obtained by choosing appropriately its phase profile. Fig. 13a illustrates the evolution of a five-soliton solution whose initial phase profile is given by the dash-dotted line in Fig. 13b. This phase profile evolves during the propagation, and achieves a final profile given by the full curve in Fig. 13b. In spite of some oscillations, this multisoliton bound state remains relatively stable


297

and propagates with zero velocity. The final phase profile shown in Fig. 13b corresponds indeed to a stationary stable solution. From Fig. 13 we can infer that a zero velocity multisoliton solution formed by plain pulses must present a symmetric and concave phase profile, such that the temporal displacement of half of the structure is balanced by the opposite displacement of the other half. These solutions can be the basic building blocks for more complicated structures.

7. Conclusion The concept of dissipative solitons was explained in this chapter. In fact, this concept is wideranging and provides a new paradigm for the investigation of phenomena involving stable structures in nonlinear systems far from equilibrium. Here, we have considered the particular case of nonlinear optical fiber systems with gain and loss, which can be described by the cubic-quintic complex Ginzburg-Landau equation (CGLE). These include spatial and temporal soliton lasers, parametric amplifiers and optical transmission lines. However, the model can also be applied in other fields of physics. The conditions to have stable solutions of the CGLE were discussed using the perturbation theory. Several exact analytical solutions, namely in the form of fixed-amplitude and arbitrary-amplitude solitons, were presented. The numerical solutions of the quintic CGLE include plain pulses, flat-top pulses, and composite pulses, among others. We used the two-dimensional phase space (distance-phase difference) to analyze the dynamics of the two soliton system. We have found stable bound states of both plain pulses and composite pulses when the phase difference between them is ± π / 2 . Two-composite pulses bound states have zero velocity, which is in contrast with the behaviour of the bound states formed by plane pulses. As a consequence of the existence of two-soliton bound states, three-soliton and other multisoliton bound states also exist. In particular, we have shown the possibility of constructing stable bound states of multiple plain pulses with zero velocity by choosing appropriately the phase profile of the whole solution.

References [1] Ablowitz, M. J.; Clarkson, P. A. (1991). Solitons, Nonlinear Evolution Equations, and Inverse Scattering, Cambridge University Press, New York. [2] Taylor, J. T., Editor (1992). Optical Solitons – Theory and Experiment, Cambridge University Press, New York. [3] Drazin, P. G. (1993). Solitons: An Introduction, Cambridge University Press, New York. [4] Gu, C. H. (1995). Soliton Theory and its Applications, Springer, New York. [5] Akhmediev, N. N., and Ankiewicz, A. A. (1997). Solitons: Nonlinear Pulses and Beams, Chapman and Hall, London. [6] Russell, J. S. (1844). Report of 14th Meeting of the British Association for Advancement of Science, York, September, 311-390. [7] Zabusky, N. J., and Kruskal, M. D. (1965). Interactions of “solitons” in a collisionless plasma and the recurrence of initial statea, Phys. Rev. Lett., 15, 240-243.

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[8] Kivshar, Y. A., and Agrawal, G. P. (2003). Optical Solitons. From Fibers to Photonic Crystals, Chapman and Hall, London. [9] Hasegawa, A., and Kodama, Y. (1995). Solitons in Optical Communications, Clarendon Press, Oxford. [10] Mollenauer, L. F., and Gordon, J. P. (2006). Solitons in Optical Fibers. Fundamentals and Applications. Elsevier Academic Press, San Diego. [11] Akhmediev, N. N., and Ankiewicz, A. A. (2005). Dissipative Solitons, Springer, Berlin. [12] Normand, C., and Pomeau, Y. (1977). Convective instability: a physicist’s approach. Rev. Mod. Phys., 49, 581-623. [13] Kolodner, P. (1991). Collisions between pulses of travelling-wave convection, Phys. Rev. A, 44, 6466-6479. [14] Graham, R. (1975). Fluctuations, Instabilities and Phase Transictions, Springer, Berlin. [15] Staliunas, K. (1993). Laser Ginzburg-Landau equation and laser hydrodynamics, Phys. Rev. A, 48, 1573-1581. [16] Jian, P.-S., Torruellas, W. E., Haelterman, M., Trillo, S., Peschel, U., and Lederer, F. (1999). Solitons of singly resonant optical parametric oscillators, Optics Letters, 24, 400-402. [17] Dunlop, A. M., Wright, E. M., and Firth, W. J. (1998). Spatial soliton laser, Optics Commun., 147, 393-401 [18] Ng, C. S., and Bhattacharjee, A. (1999). Ginzburg-Landau model and single-mode operation of a free-electron laser oscillator, Phys. Rev. Lett., 82, 2665-2668. [19] Akhmediev, N. N., Rodrigues, A., and Townes, G. (2001). Interaction of dual-frequency pulses in passively mode-locked lasers, Opt. Commun. 187, 419-426. [20] Ferreira, M. F., Facão, M. V., and Latas, S. V. (2000). Stable soliton propagation in a system with spectral filtering and nonlinear gain, Fiber Integrated Opt. 19, 31-41. [21] Ferreira, M. F., Facão, M. V., Latas, S. V., and Sousa, M. H. (2005). Optical solitons in fibers for Communication systems, Fiber Integrated Opt., 24, 287-314. [22] Ferreira, M. F., Facão, M. V., and Latas, S. V. (1999). Solitonlike pulses in a system with nonlinear gain, Photonics and Optoelect. 5, 147-153. [23] Matsumoto, M., Ikeda, H., Uda, T., and Hasegawa, A. (1995). Stable soliton transmission in the system with nonlinear gain, J. Lightwave Technol., 13, 658-665. [24] Latas, S. V., and Ferreira, M. F. (1999). Soliton stability and compression in a system with nonlinear gain, SPIE Proc., 3899, 396-405. [25] Akhmediev, N. N., Ankiewicz, A. A., and Soto-Crespo, J. M. (1998). Stable soliton pairs in optical transmission lines and fiber lasers, J. Opt. Soc. Am. B 15, 515-523. [26] Akhmediev, N. N., Afanasjev, V. V., and Soto-Crespo, J. M. (1996). Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation, Phys. Rev. E 53, 1190-1201. [27] Soto-Crespo, J. M., Akhmediev, N. N., and Afanasjev, V. V. (1996). Stability of the pulselike solutions of the quintic complex Ginzburg-Landau equation, J. Opt. Soc. Am. B 13, 1439-1448. [28] Pereira, N. R., and Stenflo, L. (1977). Nonlinear Schrödinger equation including growth and damping,” Phys. Fluids, 20, 1733-1734. [29] Nozaki, K., and Bekki, N. (1984). Exact solutions of the generalized Ginzburg-Landau equation, Phys. Soc. Japan, 53, 1581-1582.


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[30] Conte, R., and Musette, M. (1993). Linearity inside nonlinearity: exact solutions to the complex Ginzburg-Landau equation, Physica, D, 69, 1-17. [31] Thual, O. and Fauvre, S. (1988). Localized structures generated by subcritical instabilities, J. Phys, 49, 1829-1833. [32] Marcq, P., Chaté, H., and Conte, R. (1994). Exact solutions of the one-dimensional quintic complex Ginzburg-Landau equation, Physica D, 73, 305-317. [33] Agrawal, G. P. (1989). Nonlinear Fiber Optics, Academic Press, San Diego. [34] Ferreira, M. F. (1994). Analysis of femtosecond optical soliton amplification in fiber amplifiers, Opt. Commun., 107, 365-368. [35] Latas, S. V., and Ferreira, M. F. (2005). Soliton propagation in the presence of intrapulse Raman scattering and nonlinear gain, Opt. Commun., 251, 415-422. [36] Facão, M. V. and Ferreira, M. F. (2001). Analysis of the timing jitter for ultrashort soliton communication systems using perturbation methods, J. Nonlinear Math. Phys., 8, 112-117 [37] Latas, S. V., and Ferreira, M. F. (2007). Stable soliton propagation with self-frequency shift, J. Mathematics and Computers in Simulation, 74, 370-387 [38] Ferreira, M. F. (1997). Ultrashort soliton stability in distributed fiber amplifiers with different pumping configurations. In Applications of Photonic Technology. Ed.s G. Lampropoulos and R. Lessard, Plenum Press, New York, 2, 249-254. [39] Karpman, V. I., and Maslov, E. M. (1977). Perturbation theory for solitons. Zh. Eksp. Teor. Fiz. 73, 537-559 (1977. Sov. Phys. JETP. 46, 281-291). [40] Essiambre, R. J., and Agrawal, G. P. (1997). Soliton communication systems. In Progress in Optics XXXVII. Ed. E. Wolf. Amsterdam: North Holland Physics and Elsevier Science. [41] Soto-Crespo, J. M., and Pesquera, L. (1977). Analytical approximation of the soliton solutions of the quintic complex Ginzburg-Landau equation, Phys. Rev. E 56 , 72887295. [42] Ferreira, M. F., and Latas, S. V. (2002). Soliton stability and compression in a system with nonlinear gain, Optical Eng. 41, 1696-1703. [43] Anderson, D, (1983). Variational approach to nonlinear pulse propagation in optical fibers, Phys. Rev. A, 27, 3135-3145. [44] Kaup, D. J., and Malomed, B. A. (1995). The variational principle for nonlinear waves in dissipative systems, Physica D, 87, 155-159. [45] Ankiewicz, A. A., Akhmediev, N. N., and Devine, N. (2007). Dissipative solitons with a Lagrangian approach, Optical Fiber Technol., 13, 91-97. [46] Akhmediev, N., Soto-Crespo, J. M., and Town, G. (2001). Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation, Phys. Rev. E 63, 056602. [47] Malomed, B. A. (1991). Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation, Phys. Rev. A 44, 6954-6957. [48] Malomed, B. A. (1993). Bound states of envelope solitons, Phys. Rev. E 47, 2874-2880. [49] Afanasjev, V. V., and Akhmediev, N. N. (1996). Soliton interaction in nonequilibrium dynamical systems, Phys. Rev. E 53, 6471-6475. [50] Ferreira, M. F., and Latas, S. V. (2001). Interaction and bound states of pulses in the Gnizburg-Landau equation, SPIE Proc. 4271, 268-279.

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[51] Ferreira, M. F., and S. V. Latas. 2002. Bound states of plain and composite pulses in optical transmission lines and fiber lasers. In Applications of Photonic Technology. Ed.s R. Lessard, G. Lampropoulos, and G. Schinn. SPIE. 4833:845-854. [52] Latas, S. V. , Ferreira, M. F., and Rodrigues, A. (2005). Bound states of plain and composite pulses: multi-soliton solutions, Optical Fiber Technol., 11, 292-305


ISBN 1-60021-866-0 c 2007 Nova Science Publishers, Inc. °

Chapter 11

B RIGHT - DARK AND D OUBLE - H UMPED P ULSES IN AVERAGED , D ISPERSION M ANAGED O PTICAL F IBER S YSTEMS K.W. Chow† and K. Nakkeeran‡ †

Department of Mechanical Engineering University of Hong Kong, Pokfulam, Hong Kong ‡

School of Engineering, Fraser Noble Building, King’s college University of Aberdeen, Aberdeen AB24 3UE, UK Abstract

The envelope of the axial electric field in a dispersion managed (DM) fiber system is governed by a nonlinear Schrödinger model. The group velocity dispersion (GVD) varies periodically and thus realizes both the anomalous and normal dispersion regimes. Kerr nonlinearity is assumed and a loss / gain mechanism will be incorporated. Due to the big changes in the GVD parameter, the correspondingly large variation in the quadratic phase chirp of the DM soliton will be identified. An averaging procedure is applied. In many DM systems, an amplifier at the end of the dispersion map will compensate for the energy dissipated in that map. Here the case of gain not exactly compensating the loss is considered, in other words, a small residual amplification / attenuation is permitted. The present model differs from other similar ones on variable coefficient NLS, as the inhomogeneous features involve both time and the spatial coordinate. The goal here is to extend the model further, by permitting coupled modes or additional degree of freedom. More precisely, the coupling of fiber loss and initial chirping is considered for a birefringent fiber. The corresponding dynamics is governed by variable coefficient, coupled NLS equations for the components of the orthogonal polarization of the pulse envelopes. When the self phase and cross phase modulation coefficients are identical for special angles, several new classes of wave patterns can be found. New stationary wave patterns which possess multiple peaks within each period are found, similar to those found for the classical Manakov model. For situations where the self- and cross-phase modulation coefficients are different, symbiotic solitary pulses are studied. A pair of bright-dark pulses exists, where either or both pulse(s) cannot propagate in that waveguide if coupling is absent.

K.W. Chow and K. Nakkeeran

302

1.

Introduction

Transmission of information (voice, video, and data) over distances (short, moderate, long, and ultra-long) is a common requirement in the past, present and future. Carrier communication of information using the electromagnetic waves is the best technology for high-speed transmission. Out of different frequency bands in the electro-magnetic wave spectrum, optical regime has various advantages. Optical fibers are commonly used in optical communication for channelling the light pulses for digital transmission. Both linear and nonlinear optical effects in fibers play vital roles in determining the dynamics of pulse propagation. The field of nonlinear optics has blossomed and is undergoing a new revolution in recent years. The nonlinear optical response is now a key element for new emerging technologies. This is particularly true for soliton and other types of nonlinear pulse transmission in optical fibers/nonlinear materials, since this form of light propagation can be used to realize the long-held dream of very high capacity dispersion-free communications. In the recent past, it has been proved beyond doubt that solitons do exist not only in optics but also in many other areas of science. Solitons that exist in optics called “optical solitons” have been drawing a greater attention among the scientific community, as they seem to be right candidates for transferring information across the world through optical fibers. Nonlinear pulse propagation in a long-distance, high speed optical fiber transmission system can be described by the (perturbed) nonlinear Schrödinger equation (NLSE). NLSE includes linear dynamics due to group velocity dispersion of the pulse, and nonlinear mechanism due to the Kerr effect [1]. Much research efforts on the development of such a system have been made with the intention to overcome or control these effects [2, 3]. In this direction, recent numerical studies [4–6] and experiments [7] have shown that a periodic dispersion compensation seems to be the most effective way for improving the optical transmission system. The main purpose of dispersion management is to reduce several effects, such as radiation due to lumped amplifiers compensating the fiber loss [8, 9], resonant fourwave mixing [10,11], modulational instability [12], jitters caused by the collisions between signals [13], and the Gordon-Haus effect resulting from the interaction with noise [14], also to decide a desired average value for the dispersion [12]. Basically, dispersion-management technique utilizes a transmission line with a periodic dispersion map, such that each period consists of two types of fiber, generally with different lengths and opposite group-velocity dispersion (GVD) [4]. Lakoba has proved the nonintegrability of the system equation governing the pulse propagation in dispersion-managed (DM) fibers [15]. As analytical solution for DM solitons is not available, researchers have so far utilized the Lagrangian method to study the dynamics of DM solitons [4]. Very recently we have developed a complete collective variable theory for DM solitons which effectively includes the residual field due to soliton dressing and radiation [16]. Many works have reported on fitting a Hermite-Gaussian ansatz function for the oscillating tails of the numerical stationary solution (fixed point) of the DM solitons [4, 17–19]. From numerical studies [5, 6] of DM fiber line, the pulse is deformed from the ideal soliton, has a chirp and requires an enhanced power for the average dispersion. Meanwhile Kumar and Hasegawa [20] have obtained a new nonlinear pulse (quasi-soliton) by programming the dispersion profile such that the wave equation has a combination of the usual quadratic potential and the linear potential.

Bright - Dark and Double - Humped Pulses...

303

The envelope of the axial electric field in a DM fiber system is governed by a NLS model. The GVD varies periodically and thus realizes both the anomalous and normal dispersion regimes. Kerr nonlinearity is assumed and a loss / gain mechanism will be incorporated. Due to the big changes in the GVD parameter, the correspondingly large variation in the quadratic phase chirp of the DM soliton will be identified. An averaging procedure is applied [21]. In many DM systems, an amplifier at the end of the dispersion map will compensate for the energy dissipated in that map. Here the case of gain not exactly compensating the loss is considered, in other words, a small residual amplification / attenuation is permitted. The present model differs from other similar ones on variable coefficient NLS [22], as the inhomogeneous features involve both time and the spatial coordinate. The goal here is to extend the model further, by permitting coupled modes or additional degree of freedom. More precisely, the coupling of fiber loss and initial chirping is considered for a birefringent fiber [23]. The corresponding dynamics is governed by variable coefficient, coupled NLS equations for the components of the orthogonal polarization of the pulse envelopes. When the self phase and cross phase modulation coefficients are identical for special angles, several new classes of wave patterns can be found. The first result will be a stationary wave pattern which possesses multiple peaks within each period, similar to those found for the classical Manakov model [24]. Another new result is the family of symbiotic solitary pulses, and this novel finding is applicable to configuration where the self phase and cross phase modulation coefficients are different. Indeed the constraints imposed on these coefficients extend or generalize results obtained earlier in the literature. As a simple example, a pair of bright - dark pulses exists where each individual wave guide separately will only admit bright solitons. This coupling nonlinearity is truly remarkable. As a second example, bright or dark solitons are allowed to propagate in waveguides which would otherwise prohibit their existence.

2.

Double-Hump Bright - Dark Periodic and Solitary Pulses

We consider the averaged, dispersion management system for coupled waveguides: ∂A ∂ 2 A + 2 + (AA∗ + σBB ∗ )A + iβA + β 2 t2 A = 0, ∂z ∂t ∂B ∂ 2 B + + (σAA∗ + BB ∗ )B + iβB + β 2 t2 B = 0. i ∂z ∂t2 i

(1) (2)

A, B are the complex envelopes of the axial electric fields, z is the distance and t is the retarded time. The quantity β measures the quadratic phase chirp, and the residual gain or loss is specifically selected to match this parameter. The parameter σ represents cross phase modulation coefficient arising from the coupling. The derivation of system (1, 2) from the first principle of averaging over a dispersion map can be found in our earlier work [21]. System (1, 2) can be solved exactly by several techniques, but we shall focus on the Hirota bilinear method. As the description has been given in our earlier work, the presentation here will be brief. The quadratic phase factor or chirp is first isolated as


304 Ã

iβt2 A = exp 2

!

Ã

iβt2 B = exp 2

ϕ,

!

ψ.

(3)

The reduced governing equations for the auxiliary variables ϕ and ψ will be free of quadratic terms in time. To search for the special modes of optical pulses, we further constrain the wave pattern to be expressed as ϕ=

g exp(−iΩ1 ) , f

ψ=

G exp(−iΩ2 ) . f

(4)

G, g and f are dependent variables for the bilinear operation with the restriction that f is real. Typically they are combinations of exponential functions for solitary pulses but elliptic functions for periodic patterns. The phase factors Ω1 , Ω2 are functions of the distance (z) only. They will have their derivatives determined in the bilinear equations, and hence they themselves are readily recovered by quadrature. The resulting bilinear equations are then ·µ

¶

¸

∂Ω1 + 2iβ g · f + g(−Dt2 f · f + gg ∗ + σGG∗ ) = 0, f + 2iβtDt + ∂z ·µ ¶ ¸ ∂Ω2 2 + 2iβ G · f + G(−Dt2 f · f + σgg ∗ + GG∗ ) = 0. f Dt + 2iβtDt + ∂z Dt2

(5) (6)

They are solved by using rather straightforward differentiation formulas developed from first principles. D is the bilinear operator, with its definition and properties described more fully in Appendix A.. For periodic wave patterns, Hirota derivatives of theta functions can be simplified by identities involving products of theta functions (Appendix B.). As illustrative examples, the simplest periodic wave pattern will be given by the choice,

g = A0 θ1 (t[h1 (z)]),

G = B0 θ3 (t[h1 (z)]),

f = θ4 (t[h1 (z)]).

(7)

The theta functions are Fourier series with exponentially decaying coefficients and the classical Jacobi functions can be expressed as ratios of theta functions. The amplitude parameters, A0 , B0 , isolated here for convenience will also be functions of z. The distance dependent wave number function h1 (z) will render the period of the pattern to change with location, and the precise form is determined by forcing the odd Hirota derivatives to vanish. The loss / gain factor is not arbitrary as it has to match the precise forms of the functions A0 , B0 . Theta functions will be convenient in the intermediate calculations. However, the Jacobi elliptic functions are preferred in the final expressions, as they can be easily handled by most established routines in computer algebra. A summary on existing results will be instructive: 1. When the cross phase modulation coefficient, σ, is arbitrary, the wave system will permit periodic patterns in terms of a single elliptic function. The long wave limit will, not surprisingly, yield solitary bright or dark pulses.


305

2. When σ is constrained to be unity, there are other varieties of solutions. In particular, one class of wave patterns can be expressed in terms of products of elliptic functions. The physical implication is that the intensity will display two, or perhaps more, peaks per period. The goal of this section is to derive still further new wave patterns by choosing products of elliptic functions as the starting point of these calculations, while still assuming the cross phase modulation coefficient, σ, to be one. The motivation comes from the choice of wave patterns for the case of coupled nonlinear Schrödinger models with constant coefficients. Proceeding along the lines of reasoning just described will yield " # " # √ √ dn2 (rte−2βz ) iβt2 6r 1 − k 2 c− √ − 2βz − iΩ1 , exp A= q √ 2 1 − k2 1 − 2c 1 − k 2 " # √ iβt2 6 rk sn(rte−2βz )dn(rte−2βz ) q exp − 2βz − iΩ2 . B= √ 2 1 − 2c 1 − k 2 " # √ 2 1 − k2 6c2 (1 − k 2 ) r2 exp(−4βz) √ + , Ω1 = 4β c 1 − 2c 1 − k 2

(8) (9) (10)

#

"

6c2 (1 − k 2 ) r2 exp(−4βz) √ − 2(5 − 4k 2 ) , Ω2 = 4β 1 − 2c 1 − k 2

(11)

r is a free parameter and represents the wave number at the initial location (z = 0). The quantity c will be one of the roots of the quadratic equation ·p

¸

1 + 1 = 0, 1− + √ (12) 3c − 2c 1 − k2 k is the modulus of the elliptic function. Waveguide B will generally exhibit two peaks per period. Waveguide A will degenerate to a dark solitary pulse with multiple peaks in the long wave period. Figures 1a, 1b illustrate this pattern. 2

3.

k2

A Generalized Model with Different Dispersion Coefficients

In many applications involving wave propagation along different channels or waveguides, the optical pulses will experience different measures of group velocity dispersion. Hence the coefficients of the second derivative terms of the coupled NLS equations will generally be different. Remarkably a special model system will still permit analytical progress, and we shall consider pairs of bright - dark solitons in this model. Generally bright (dark) solitons occur for the conventional NLS model in the anomalous (normal) dispersion regimes respectively. However, due to the special nonlinear effects in coupled NLS systems, these bright / dark solitons can occur in the appropriate waveguide which are otherwise prohibited in the single mode NLS. They have sometimes been termed ‘symbiotic solitons’ in the literature. In optical physics, such waves have indeed been studied for configurations associated with conventional NLS with Kerr nonlinearity [25], intra-pulse stimulated Raman scattering


306

|A|2 [Norm. Unit]

2

1.5

1

0.5

0 0.0

z [N orm

10 5

0.5

.U nit]

0

-5 1.0 -10

t [No

rm. U

nit]

|B|2 [Norm. Unit]

(a)

0.1 0.08 0.06 0.04 0.02 0 0.0

z [N orm 0.5 .U nit]

5 0 -5 1.0

t [No

rm

t] . Uni

(b) Figure 1. Evolution of the periodic solution (8) and (9) for the parameters β = 0.05, r = 1, k = 0.9 and c = 1.61. [26], quasi-phase matched parametric oscillator [27], second harmonic generation [28], and three-wave solitons [29]. In other systems, symbiotic solitons also occur in phenomena connected with Bose - Einstein condensates [30], discrete systems [31], multi-dimensional NLS by separation of variables [32], and quadratic, nonlinear media with loss and gain [33]. More precisely, we shall consider

i

β 2 t2 A ∂A ∂2A = 0, + δ 2 + (AA∗ + σBB ∗ )A + iβA + ∂z ∂t δ ∂B ∂ 2 B + + (σAA∗ + BB ∗ )B + iβB + β 2 t2 B = 0. i ∂z ∂t2

(13) (14)


307

Here A and B are again complex envelopes but the first waveguide is permitted to have a dispersion coefficient δ (positive or negative) relative to waveguide B. The chirp factors, however, must be modified to Ã

iβt2 A = exp 2δ

!

Ã

iβt2 B = exp 2

ϕ,

!

ψ.

A periodic pattern is obtained earlier in the literature as #

"

A = rkQ1 sn(rte

−2βz

iβt2 ir2 e−4βz 2 − (Q1 + δ(1 − k 2 )) , (15) ) exp −2βz + 2δ 4β #

"

−2βz

B = rQ2 dn(rte

iβt2 ir2 e−4βz 2 − (Q2 − k 2 ) , ) exp −2βz + 2 4β

s

s

2(δ − σ) , σ2 − 1

Q1 =

(16)

Q2 =

2(σδ − 1) . σ2 − 1

(17)

The restrictions are either δ>σ

if

σ > 1,

(18)

δ

1 σ

if

σ < 1,

(25)

or 1 if σ > 1, (26) σ (25) and (26) are different from (18) and (19). Ω1 , Ω2 are angular frequency parameters given by δ
1). For σ > 1, δ can be either positive or negative. In particular, negative values of δ here will imply that waveguide A is in normal dispersion regime. Remarkably, a bright (dark) soliton now propagates in the normal (anomalous) dispersion regime respectively. These phenomena are quite contrary to the well known results.

4.

Conclusions

A class of periodic and solitary waves has been studied for a system of coupled envelope equations. This system can model averaged, dispersion managed systems where the residual gain / loss in each cycle of the dispersion map has been carefully chosen. Waves with multiple peaks per period or symbiotic pairs of solitary pulses are obtained analytically. They will enhance our capability in modeling such systems and strengthen our understanding in this and similar optical systems.


309

Several aspects still allow rooms for future work and expansions. In particular, configurations where both waveguides are in the normal dispersion regime have not been worked out in details yet, although the same physics is expected to hold true qualitatively. One issue which has not been addressed is the stability of these wave patterns. Numerical simulations of perturbed wave trains must be pursued. Recent works and experience have indicated that stability will probably still prevail in some parameter regimes. The precise elucidation will await further efforts.

Acknowledgement Partial financial support has been provided by the Research Grants Council through the contract HKU 7123/05E. KWC and KN wish to thank The Royal Society for their support in the form of an International Joint Project Grant. KWC and KN are very grateful to Prof. John Watson for his valuable support for this research collaboration. KN also wishes to thank the Nuffield Foundation for financial support through the Newly Appointed Lecturer Award.

A. Hirota Bilinear Operator The Hirota operator for any two functions f and g is defined as [34, 35] µ

Dxm Dtn g · f =

∂ ∂ − 0 ∂x ∂x

¶m µ

∂ ∂ − 0 ∂t ∂t

¶n

¯ ¯

g(x, t)f (x0 , t0 )¯¯

,

(31)

x=x0 ,t=t0

and the properties in association with differentiation of exponential functions are especially striking (m, n are constants):

Dx [exp(imx)g · exp(inx)f ] = [Dx g · f + i(m − n)gf ] exp[i(m + n)x], 2 Dx [exp(imx)g · exp(inx)f ] = [Dx2 g · f + 2i(m − n)Dx g.f − (m − n)2 gf ]

(32)

× exp[i(m + n)x].

(33)

Most existing works on the Hirota operator focus on the case of constant wave number or frequencies. The important point in this work is to extend Hirota derivatives to the case of time or space dependent wavenumbers. The bilinear identities for Hirota derivatives, even for the case of variable wave number, can be obtained from simple, straightforward differentiation. Examples are:

=

{t[ξ10 (z)

−

Dz exp[tξ1 (z) + ξ2 (z)] · exp[tη1 (z) + η2 (z)] 0 η1 (z)] + ξ20 (z) − η20 (z)} · exp{t[ξ1 (z) + η1 (z)] + ξ2 (z) ·

Dz exp(a) · m(z) exp(b) = m Dz exp(a) · exp(b) −

+ η20 (z)}, (34) ¸

1 ∂m exp(a + b) . m ∂z

(35)


310

B. Theta Functions The theta functions θn (x), n = 1, 2, 3, 4 in terms of the parameter q (the nome) are defined by [36–38]: θ1 (x) = 2

∞ X

2

(−1)n q (n+1/2) sin [(2n + 1)x] ,

(36)

n=0

θ2 (x) = 2

∞ X

2

q (n+1/2) cos [(2n + 1)x] ,

(37)

n=0

θ3 (x) = 1 + 2

∞ X

2

q n cos (2nx) ,

(38)

n=1

θ4 (x) = 1 + 2

∞ X

2

(−1)n q n cos (2nx) ,

0 < q < 1.

(39)

n=1

Basically they are Fourier series with exponentially decaying coefficients. Relationships between the theta and elliptic functions are: θ3 (0)θ1 (z) θ4 (0)θ2 (z) , cn(u) = , θ2 (0)θ4 (z) θ2 (0)θ4 (z) u θ2 (0) z= 2 , k = 22 . θ3 (0) θ3 (0) sn(u) =

dn(u) =

θ4 (0)θ3 (z) , (40) θ3 (0)θ4 (z) (41)

Arguments of the theta and elliptic functions are related by a scale factor. The modulus of the elliptic functions, k, is related to the theta constants by (41). Theta functions possess a huge number of identities involving addition and subtraction of arguments: θ3 (x + y)θ3 (x − y)θ22 (0) = θ42 (x)θ12 (y) + θ32 (x)θ22 (y), θ4 (x + y)θ4 (x −

y)θ22 (0)

=

θ42 (x)θ22 (y)

+

θ32 (x)θ12 (y),

(42) (43)

Such identities can be proven by re-arranging terms of the multiple sums [37]. By considering the leading and quadratic terms in the Taylor series of y in identities of the form (42-43), one obtains θ400 (0) θ300 (0) − = θ24 (0), θ4 (0) θ3 (0)

θ400 (0) θ200 (0) − = θ34 (0), θ4 (0) θ2 (0)

θ300 (0) θ200 (0) − = θ44 (0). (44) θ3 (0) θ2 (0)

2θ200 (0)θ32 (x) + 2θ32 (0)θ42 (0)θ42 (x), θ2 (0) 2θ00 (0)θ42 (x) . Dx2 θ4 (x) · θ4 (x) = 2θ32 (0)θ42 (0)θ32 (x) + 2 θ2 (0)

Dx2 θ3 (x) · θ3 (x) =

(45) (46)

Hence formulas for Dx θm · θn , Dx2 θm · θn can be developed for m, n integers using this line of reasoning.


311

References [1] A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett., 23, 142 (1973). [2] A. Hasegawa and Y. Kodama, Solitons in Optical Communication, (Oxford University Press, New York, 1995). [3] G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, San Diego, 1989). [4] V. E. Zakharov and S. Wabnitz, Optical Solitons: Theoretical Challenges and Industrial Perspectives, (Springer-Verlag, Heidelberg, 1998). [5] N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett., 32, 54 (1996). [6] T. Georges and B. Charbonnier, “Reduction of the dispersive wave in periodically amplified links with initially chirped solitons,” IEEE Photon. Technol. Lett., 9, 127 (1997). [7] M. Suzuki, I. Morita, N. Edagawa, S. Yamamoto, H. Toga and S. Akiba, “Reduction of Gordon-Haus timing jitter by periodic dispersion compensation in soliton transmission,” Electron. Lett., 31, 2027 (1995). [8] W. Forysiak, F. M. Knox and N. J. Doran, “Stepwise dispersion profiling of periodically amplified soliton systems,” J. Lightwave Technol., 12, 1330 (1994). [9] S. Kumar, A. Hasegawa and Y. Kodama, “Adiabatic soliton transmission in fibers with lumped amplifier: Analysis,” J. Opt. Soc. Am. B, 14, 888 (1997). [10] C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” Photon. Technol. Lett., 5, 1250 (1993). [11] P. V. Mamyshev and L. F. Mollenauer, “Pseudo-phase-matched four-wave mixing in soliton wavelength-division multiplexing transmission,” Opt. Lett., 21, 396 (1996). [12] N. J. Doran, N. J. Smith, W. Forysiak and F. M. Knox, in Physics and Applications of Optical Solitons in Fibers ’95, (Kluwer Academic Press, 1996). [13] A. Hasegawa, S. Kumar and Y. Kodama, “Reduction of collision-induced time jitters in dispersion-managed soliton transmission systems,” Opt. Lett., 21, 39 (1996). [14] W. Forysiak, K. J. Blow and N. J. Doran, “Reduction of Gordon-Haus jitter by posttransmission dispersion compensation,” Electron. Lett., 29, 1225 (1993). [15] T. I. Lakoba, “Non-integrability of equations governing pulse propagation in dispersion-managed optical fibers,” Phys. Lett. A, 260, 68 (1999).

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[16] P. Tchofo Dinda, A. B. Moubissi and K. Nakkeeran, “A collective variable approach for dispersion-managed solitons,” J. Phys. A, 34, L103 (2001). [17] T. I. Lakoba and D. J. Kaup, “Hermite-Gaussian expansion for pulse propagation in strongly dispersion managed fibers,” Phys. Rev. E, 58, 1998 (1998). [18] S. K. Turitsyn, T. Schäfer, K. H. Spatschek and V. K. Mezentsev, “Path-averaged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun., 163, 122 (1999). [19] P. Tchofo Dinda, K. Nakkeeran and A. B. Moubissi, “Optimized Hermite-gaussian ansatz functions for dispersion-managed solitons,” Opt. Commun., 187, 427 (2001). [20] S. Kumar and A. Hasegawa, “Quasi-soliton propagation in dispersion-managed optical fibers,” Opt. Lett., 22, 372 (1997). [21] C. C. Mak, K. W. Chow and K. Nakkeeran, “Soliton Pulse Propagation in Averaged Dispersion-managed Optical Fiber System,” J. Phys. Soc. Japan, 74, 1449 (2005). [22] V. N. Serkin and A. Hasegawa, “Novel Soliton Solutions of the Nonlinear Schrödinger Equation Model,” Phys. Rev. Lett., 85, 4502 (2000). [23] R. Ganapathy, V. C. Kuriakose and K. Porsezian, “Soliton propagation in a birefringent optical fiber with fiber loss and frequency chirping,” Opt. Commun., 194, 299 (2001). [24] K. W. Chow and D. W. C. Lai, “Periodic solutions for systems of coupled nonlinear Schrdinger equations with five and six components,” Phys. Rev. E, 65, 026613 (2002). [25] M. Lisak, A. Höo¨ k and D. Anderson, “Symbiotic solitary-wave pairs sustained by cross-phase modulation in optical fibers,” J. Opt. Soc. Am. B, 7, 810 (1990). [26] K. Hayata and M. Koshiba, “Bright-kink symbions resulting from the combined effect of self-trapping and intra-pulse stimulated Raman-scattering,” J. Opt. Soc. Am. B, 11, 61 (1994). [27] A. Picozzi and M. Haelterman, “Spontaneous formation of symbiotic solitary waves in a backward quasi-phase-matched parametric oscillator,” Opt. Lett., 23, 1808 (1998). [28] S. Trillo, “Bright and dark simultons in second-harmonic generation,” Opt. Lett., 21, 1111 (1996). [29] C. Durniak, C. Montes and M. Taki, “Temporal walk-off for self-structuration of three-wave solitons in CW-pumped backward optical parametric oscillators,” J. Opt. B: Quantum and Semi-Classical Optics, 6, S241 (2004). [30] V. M. Perez-Garcia and J. B. Beitia, “Symbiotic solitons in heteronuclear multicomponent Bose-Einstein condensates,” Phys. Rev. A, 72, 033620 (2005).


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[31] E. P. Fitrakis, P. G. Kevrekidis, B. A. Malomed and D. J. Frantzeskakis, “Discrete vector solitons in one-dimensional lattices in photorefractive media,” Phys. Rev. E, 74, 026605 (2006). [32] K. Hayata and M. Koshiba, “Bright-dark solitary-wave solutions of a multidimensional nonlinear Schrdinger equation,” Phys. Rev. E, 48, 2312 (1993). [33] S. Darmanyan, L. Crasovan and F. Lederer, “Double-hump solitary waves in quadratically nonlinear media with loss and gain,” Phys. Rev. E, 61, 3267 (2000). [34] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, (SIAM, Philadelphia, 1981). [35] Y. Matsuno, The Bilinear Transformation Method, (Academic Press, New York, 1984). [36] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1964). [37] D. F. Lawden, Elliptic Functions and Applications, (Springer, New York, 1989). [38] K. W. Chow, “A class of doubly periodic waves for nonlinear evolution equations,” Wave Motion, 35, 71–90.


ISBN 1-60021-866-0 c 2007 Nova Science Publishers, Inc.

Chapter 12

DYNAMICS AND I NTERACTIONS OF G AP S OLITONS IN H OLLOW C ORE P HOTONIC C RYSTAL F IBERS Javid Atai and D. Royston Neill School of Electrical and Information Engineering The University of Sydney, NSW 2006 Australia Abstract The existence and stability of gap solitons in a model of hollow core fiber in the zero dispersion regime are analyzed. The model is based on a recently introduced model where the coupling between the dispersionless core mode and nonlinear surface mode (in the presence of the third order dispersion) results in a bandgap. It is found that similar to the anomalous and normal dispersion regimes, the family of solitons fills up the entire bandgap. The family of gap solitons is found to be formally unstable but in a part of family the instability is very weak. Consequently, gap solitons belonging to that part of the family are virtually stable objects. The interactions and collisions of in-phase and theπ-out-of-phase quiescent solitons and moving solitons in different dispersion regimes are investigated and compared.

1.

Introduction

Gap solitons (GSs) were originally introduced in Ref. [1]. Recent years have witnessed an upsurge of research activity on gap solitons in various areas of physics such as nonlinear optics and Bose-Einstein condensation (BEC). In optics, a nonlinear dispersive medium whose spectrum contains one or more forbidden bands can support gap solitons. An example of such a system is a fiber Bragg grating (FBG). The periodic variation of linear dielectric constant in an FBG leads to a photonic band structure. The linear cross coupling between the counter-propagating waves results in a large effective dispersion (5 to 6 orders of magnitude larger than the dispersion of standard optical fiber) [2, 3]. For sufficiently high light intensities, the large Bragg grating induced dispersion may be counterbalanced by Kerr nonlinearity resulting in a gap soliton. Significant theoretical [3–6] and experimental [7–10] efforts have been directed toward understanding and characterizing GSs in periodic media. In particular, it has been

316

Javid Atai and D. Royston Neill

shown that GSs in an FBG form a two-parameter family of solutions [4]. It has also been shown that approximately half of the soliton family is stable against oscillatory perturbations [11–13]. Experimental activities in this area have focused on generating zerovelocity (quiescent) GSs due to their potential applications in optical buffers and storage elements. To date, GSs with a velocity of 0.23 of the speed of light in the fiber have been observed [14]. GSs have been studied in more sophisticated systems such as in the presence of higher order dispersion [15], quadratic nonlinearity [16], cubic-quintic nonlinearity [18], dual core fibers with FBGs [17] and in waveguide arrays [19] and photonic crystal fibers [20]. Since their demonstration in 1996 [21], photonic crystal fibers (PCFs) have been the subject of extensive research due their interesting and peculiar properties. PCFs are specially designed optical fibers with many microstructured air holes running along the fiber’s length. They can be divided into two main categories depending on the mechanism of light guidance, namely the solid core and hollow core PCFs. Solid core PCFs are similar to conventional optical fibers in that they guide light through total internal reflection. On the other hand, in hollow core PCFs (HC-PCFs) the microstructured cladding surrounding the hollow core creates a photonic bandgap that guides the light [22–24]. Introduction of atomic or molecular gases into the core of HC-PCF results in efficient nonlinear optical interactions due to strong confinement of light in the core region. Some recent results include demonstrations of generation of stimulated Raman scattering (SRS) in hydrogen [25], and electromagnetically-induced transparency (EIT) [26, 27]. They have also been utilized in delivery of high energy pulses [28–30] and in soliton lasers [31]. In Ref. [34] a model for pulse propagation in HC-PCF based on experimental [32] and numerical [33] results was proposed. The model took into account the coupling of a linear dispersionless mode propagating in a gas-filled core with a nonlinear dispersive surface mode propagating in silica. In Ref. [35] a simpler model was considered where the second and third order group velocity dispersion terms were absent. The model contained a linear loss term which accounted for the power leakage from the core to the cladding. In both models a bandgap opens in the system’s spectrum. The models in Ref. [34, 35] belong to a general class of models that give rise to wavenumber bandgap [36, 37]. A wavenumber bandgap arises as a result of the coupling between the co-propagating waves (in this case the core and surface modes). On the other hand, a frequency bandgap (e.g. the abovementioned bandgap structure in a FBG) is due to the coupling of counterpropgating waves. The stability of GSs in the model of Ref. [34] has been investigated in both anomalous [38] and normal [39] dispersion regimes. It is shown that, strictly speaking, GSs in both anomalous and normal dispersion regimes are unstable. However, due to the fact that instability is weak in a part of the soliton family, the GSs belonging to that part of the family are “virtually” stable objects. In addition, an important finding reported in Ref. [39] is that GSs in the normal dispersion are far more stable than their counterparts in the anomalous dispersion. In this article, we will first investigate the existence and stability of gap solitons in a HC-PCF in the special case when the second order group velocity dispersion is negligible. The model, which is based on the model of Ref. [34], and the characteristics of the bandgap and soliton solutions will be discussed in Section 2.. In Section 3. stability of quiescent and moving GSs will be presented and their stability will be compared with that of GSs in the

Dynamics and Interactions of Gap Solitons...

317

anomalous and normal dispersion regimes. In Section 4. we will investigate and compare the interactions of quiescent GSs and collision dynamics of moving solitons in the normal, anomalous and zero dispersions. In particular we will analyze the effect of initial phase difference and separation on the outcome of collisions and interactions. The results are summarized in Section 5..

2.

The Model and Gap Soliton Solutions

The system of equations governing the propagation of the above-mentioned surface and core modes in the zero GVD are based on the model introduced in Ref. [34]. In the normalized form it reads: iuz − icuτ + iγuτ τ τ + |u|2 u + v = 0,

(1)

ivz + icvτ + u = 0,

(2)

where u and v are the amplitudes of the surface and core waves, respectively, z and τ are the propagation distance and reduced time and c represents the group velocity mismatch between the modes. The coefficients of Kerr nonlinearity and the linear coupling between the modes have been scaled to unity. Therefore, there are two free parameters in the model namely γ and c. It should be noted that when γ = 0 Eqs. (1) and (2) reduce to the model of Ref. [35]. However, as was pointed out in Refs. [34,38,39], due to the small temporal width of solitons and that the carrier wavelength may be close or exactly equal to zero GVD point, the third order dispersion needs to be present. Undoing the rescalings and using a typical value of |β3 | = 0.2 ps3 /km the soliton’s width is found to be in the range of 100-300 fs. This value of β3 corresponds to a normalized value of γ ≈ 0.3. Also, ∆z = 1 and ∆τ = 1 correspond to ranges 1-10 cm and 30-100 fs in physical units. In order to determine the linear spectrum of the system, we substitute (u, v) ∼ exp(ikz − iωτ ) into the linearized Eqs. (1) and (2). This results in the following dispersion relation: 2k± = −ω 3 γ ±

q (ω 3 γ + 2ωc)2 + 4.

(3)

By definition we set c > 0 in which case the wavenumber bandgap exists for γ > 0. Straightforward analysis of Eq. (3) shows that the bandgap is −1 < k < +1 provided that 1 c2 ≥ . The solid curves in Fig. 1 represent the branches of the dispersion relation (3) for 4 c = 1 and γ = 0.3 with the bandgap being −1 < k < 1. In the gap, soliton solutions to (1) and (2) were sought in the form of {u(z, τ ), v(z, τ )} = {U (τ ), V (τ )} exp(ikz). Substituting this ansatz into (1) and (2) results in a set of equations for complex functions U (τ ) and V (τ ). These equations can be solved numerically using the relaxation method. It is found that, similar to the anomalous and normal dispersion regimes, the family of gap solitons completely fill the bandgap, and

318

Javid Atai and D. Royston Neill 10

k

5

0

−5

−10 −5

−2.5

0

2.5

5

ω Figure 1. Dispersion diagram corresponding to c = 1 and γ = 0.3 for quiescent gap solitons (solid lines) and moving ones with δ = 0.7 (dashed lines). The bandgap for quiescent solitons is −1 < k < 1. The bandgap for moving solitons is −0.77 < k < 0.78. |U (τ )| and |V (τ )| are always single-humped. As is shown in Fig. 2, the real and imaginary parts of U (τ ) and V (τ ) are even and odd functions of τ , respectively. The GSs in the model of Eqs. (1) and (2), similar to their counterparts in the anomalous and normal dispersion regimes [34, 39], satisfy Vakhitov-Kolokolov criterion [40]. This criterion states that a necessary condition for the stability of solitons against nonoscillatory dE perturbations with purely real growth rates is > 0 where E is the energy of the soliton dk family and is given by:

E(k) =

Z

+∞

−∞

|U (τ ; k|2 + |V (τ ; k|2 dτ.

(4)

However, the soliton family or part thereof may be unstable against oscillatory perturbations. A finding of Ref. [39] was that the energy of the soliton family in the normal dispersion regime is considerably lower than that of the anomalous dispersion regime. As is shown in Fig. 3, the energy of solitons in (1) and (2) is less than that of the anomalous case and greater than the normal dispersion case. Based on the results of Ref. [39] one may conjecture that the solitons in the zero dispersion regime are more stable than their counterparts in the anomalous dispersion and less so compared with the ones in the normal dispersion regime. This issue will be considered in the next section. Moving solitons can be obtained by rewriting Eqs. (1) and (2) in the boosted reference frame through the coordinate transform (z, τ ) −→ (z, τ − δz) where δ is the velocity shift. The dispersion relation of the transformed system of equations is given by:


319

2 Im(U) Re(U) 1

0

−1

−2 −20

−10

0

10

20

τ (a) 4

3

Im(V) Re(V)

2

1

0

−1

−2 −20

−10

0

10

20

τ (b)

Figure 2. The real and imaginary parts of the U (τ ) and V (τ ) for a quiescent gap soliton with c = 1 and γ = 0.3 and k = 0.

q 2k± = −(ω γ + 2ωδ) ± (ω 3 γ + 2ωc)2 + 4. 3

(5)

The bandgap defined by Eq. (5) varies with δ. The dashed curves in Fig. 1 display the branches of Eq. (5) for δ = 0.7. The bandgap for moving solitons exists in the range

320

Javid Atai and D. Royston Neill 320 Anomalous Normal Zero

Energy

240

160

80

0 − 0.9

− 0.6

− 0.3

0

0.3

0.6

0.9

k Figure 3. The total energy of gap solitons with c = 1 and γ = 0.3 in the anomalous dispersion (see Ref. [34, 38]), the normal dispersion (see Ref. [39]) and the zero dispersion regime (Eqs. (1) and (2)). δmin < δ < c where δmin is negative and can be obtained numerically. It is also found that the bandgap defined by (5) is completely filled with soliton solutions all of which satisfy VK criterion. In addition, similar to the case of quiescent solitons, the energy of moving solitons in this model is found to be greater than that in the normal dispersion and less than that of moving GSs in the anomalous dispersion regime.

3.

Stability of Solitons

In this section we investigate the stability of GSs in this model by means of direct numerical simulations and linear stability analysis. Evolution of GSs were simulated by numerically solving Eqs. (1) and (2) using the symmetrized split-step Fourier method. Absorbing boundary conditions were implemented in order to attenuate any radiation that reaches the boundaries of computational window. To seed any inherent instability in the system, the GSs found by the above-mentioned relaxation algorithm were initially perturbed asymmetrically and then propagated. It is found that, the GSs in the model of (1) and (2), like their counterparts in the anomalous and normal dispersion regimes, are unstable against oscillatory perturbations. But, in a part of the GS family the instability is weak and as a result solitons may propagate for long distances before the instability is manifested. As a consequence, the GSs belonging to this part of the family can be considered as being “practically” stable A key result of Ref. [39] was that GSs in the normal dispersion regime are significantly more stable than their counterparts in the anomalous dispersion. Moreover, it was con-


321

jectured that the higher degree of stability of GSs in the normal dispersion regime was, at least in part, due to the fact that their total energy was considerably smaller than GSs in the anomalous dispersion. Based on this conjecture and since the total energy of GSs in Eqs. (1) and (2) is greater (smaller) than those in the normal (anomalous) dispersion (see Fig. 3), one expects the GSs in the zero dispersion to be more stable than those in the anomalous and less stable compared to the GSs in the normal dispersion regime. Our simulations corroborate this prediction. A comparison between the propagation of GSs in different dispersion regime is provided in Fig. 4. To quantify the degree of instability of GSs in this model, we have utilized a linear stability analysis to calculate the instability growth rates for small perturbations. Substituting the following perturbed soliton solution {u (z, τ ) , v (z, τ )} = {Uδ (τ ) + f (τ ) eσz , Vδ (τ ) + g (τ ) eσz } ekz

(6)

into the boosted equations (see Section 2.) and linearizing, we arrive at the following eigenvalue problem: Ay = σy

(7)

where Uδ (τ ) and Vδ (τ ) are the soliton solutions corresponding to velocity δ and f (τ ) and g (τ ) are the eigenmodes of the small perturbations and σ is the corresponding complex eigenvalue. y = [f, f ? , g, g ? ]T , and −ik + 2i |Uδ |2 + Df  iUδ?2 A=  i 0

 iUδ2 i 0  ik − 2i |Uδ |2 + Df ? 0 −i .  0 −ik + Dg 0 −i 0 ik + Dg?



3

d d d with Df = Df ? = (c + δ) dτ − γ dτ 3 , and Dg = Dg? = − (c − δ) dτ . In the above expressions, asterisk represents complex conjugate.

The eigenvalue problem posed by Eq. (7) can be solved using standard numerical techniques. The results of the stability analysis are summarized in Fig. 5 as graphs of Re(σ) vs. k for c = 1, γ = 0.3 and δ = 0, 0.25 and 0.5. Since the instability growth rate for all the cases is positive the GSs are formally unstable. However, one observes that the growth rates for a part of the family, particularly toward the lower edge of the bandgap, are very small. In this case, the instability will only be observable after a long propagation distance. Solitons exhibiting this character are therefore “virtually stable” objects. We have adopted the definition of Ref. [39] for virtual stability and quasi-stability. That is, for a soliton to be virtually 1 stable it must remain stable for at least 300Znonlin (where Znonlin ∼ ) and for it to be |Uδ |2 quasi-stable it must remain stable for propagation distances 50Znonlin < z < 300Znonlin . There are a number of noteworthy features in Fig. 5. Firstly, we note that the growth rates in the zero dispersion regime are greater than those in the normal dispersion region (c.f. Fig. 5 in Ref. [39]) and smaller than those in the anomalous dispersion regime (c.f. Figs. 3 and 4 in Ref. [38]). This is consistent with the results of the direct numerical simulations shown in Fig. 4. Secondly, increasing δ gives rise to larger growth rates, particularly for

322


solitons near the upper edge of the bandgap. Nevertheless, varying δ does not have an appreciable effect on the border of stable and quasi-stable regions. The weak dependence of boundary of stable and unstable regions on the velocity of solitons has also been reported for GSs in a FBG [11, 13].

z

z 250

48

0

− 60

− 40

− 20

0

20

40

60

0

− 60

− 40

− 20

τ (a)

0

20

40

60

τ (b)

z 1600

0

− 60

− 40

− 20

0

20

40

60

τ (c)

Figure 4. Examples of propagation of asymmetrically perturbed quiescent gap soliton corresponding to k = −0.4, c = 1, and γ = 0.3 in (a) anomalous dispersion, (b) zero dispersion and (c) normal dispersion. In (c) the initial perturbation causes the soliton to acquire a small velocity. In this figure and all others below, only the u-component is shown.


323

0.4

Stable

Quasi-Stable

Re(σ)

0.3

0.2

0.1

0 − 0.8

− 0.4

0

0.4

0.8

k (a) 0.5

Stable

Quasi-Stable

0.4

Re(σ)

0.3

0.2

0.1

0 − 0.8

− 0.4

0

0.4

0.8

k (b) 1

Stable

Quasi-Stable

0.8

Unstable

Re(σ)

0.6

0.4

0.2

0 − 0.8

− 0.4

0

0.4

0.8

k (c)

Figure 5. Instability growth rate of GSs in the model of Eqs. (1) and (2) with c = 1 and γ = 0.3 versus k for (a) quiescent gap solitons, (b) moving gap solitons with δ = 0.25 and (c) moving gap solitons with δ = 0.5. In the “Stable” region, the solitons propagate for long distances i.e. z & 300Znonlin without any conspicuous instability development. In the “Quasi-Stable” region, instability occurs in the range 50Znonlin < z < 300Znonlin .

324

4.


Interactions and Collisions of Solitons

In view of nonintegrability of the model, the collision dynamics and interactions between the solitons may be quite complex. In Refs. [38, 39], the collisions between in-phase GSs in the anomalous and normal dispersion regimes were considered. Moreover, in [39] the interaction of in-phase and π-out-of-phase quiescent GSs in the normal dispersion regime was investigated and it was shown that in the case of π-out-of-phase solitons the outcome of interaction depends on k and the initial separation of solitons. In this section we will investigate the interaction of quiescent solitons in the anomalous and zero dispersion regimes. In addition, the collisions of in-phase and π-out-of-phase moving GSs in anomalous, normal and zero dispersion regimes will also be studied.

4.1.

Interactions of Quiescent Solitons in Anomalous and Zero Dispersions

The interaction of GSs in zero and anomalous dispersion regimes was simulated by propagating two identical quiescent solitons belonging to the “Stable” regions with a time separation of ∆τ and a phase difference of ∆φ. It is found that, irrespective of dispersion regime, when ∆φ = 0 (i.e. GSs are in-phase), the solitons always repel each other. This behavior was reported for the GSs in the normal dispersion regime (see [39]) . In the case of ∆φ = π, the interaction of solitons becomes dependent on ∆τ and k.These interactions can be divided into three types, denoted here as Types A, B, and C. In the Type A interactions the pulses initially attract each other and collide without merging and then bounce back. An example of this type of interaction in the zero dispersion regime is shown in Fig. 6(a). In the Type B interactions the pulses attract and temporarily merge and form a “lump” which subsequently disintegrates into two separating solitons with different amplitudes and velocities. Fig. 6(b) displays an example this type of interaction. In the Type C interactions the pulses repel each other resulting in two separating pulses with different amplitudes and velocities (Fig. 6(c)). It should be noted that the velocity and amplitude of resulting solitons in the Type C interaction as well as the interaction between in-phase solitons depend on the degree of initial overlap of solitons. If the solitons are initially weakly overlapping then the difference between the velocities and amplitudes of the eventual moving solitons will be small (see for example Fig. 7 and Fig 8(b) in Ref. [39]). On the other hand, increasing the initial overlap between solitons (i.e. reducing ∆τ ) leads to generation of solitons whose velocities and amplitudes differ considerably. Fig. 7 displays the regions in the plane of (∆τ ,k) where the types A, B and C interactions occur in zero and anomalous dispersion regimes. A noteworthy feature in Fig. 7(a) is that in the zero dispersion the boundary between the types C and A is very weakly dependent on the initial separation of solitons.

4.2.

Collisions of Moving Solitons

In Refs. [38, 39] it was shown that in the anomalous and normal dispersion regimes the collisions between in-phase counterpropagating solitons belonging to the “Stable” region

Dynamics and Interactions of Gap Solitons... z

z

50

50

0

− 60

− 40

− 20

0

20

40

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− 60

− 40

− 20

325

0

20

40

60

τ (b)

τ (a) z 50

0

0

20

40

60

τ (c)

Figure 6. Examples of interaction of quiescent gap solitons in the zero dispersion regime with c = 1, γ = 0.3, and ∆φ = π. (a) k = −0.64, ∆τ = 10; (b) k = −0.64, ∆τ = 8; (c) k = −0.9, ∆τ = 8.

with δ = ±0.5 are always elastic. In particular, It was also found the relative collision induced loss of energy is ≈ 0.1%. In this section the effect of phase and velocity shift on the collisions will be considered. First, we consider the collisions between GSs with initial velocities ±0.25. As shown in Fig. 8, in-phase GSs in different dispersion regimes with δ = ±0.25 bounce off each other elastically. On the other hand, as is displayed in Fig. 9, the π-out-of-phase solitons with δ = ±0.25 collide and merge temporarily and form a “lump” which then quickly breaks

326


− 0.5 A

k

− 0.6

− 0.7

B

− 0.8

A

C − 0.9

8

10 ∆τ (a)

11

12

10 ∆τ (b)

11

12

9

− 0.6 B

− 0.7 k

A

− 0.8 C − 0.9

8

9

Figure 7. Regions of different types of interaction in the plane of (∆τ , k) for (a) the zero dispersion and (b) the anomalous dispersion.

up into two separating solitons. In addition, the collisions do not generate any noticeable radiation. Figs. 10 and 11 show that the collisions of in-phase and π-out-of-phase solitons with δ = ±0.5 in different dispersion regimes . In this case, regardless of the initial phase

Dynamics and Interactions of Gap Solitons... z

z

50

50

0

0 − 60

− 40

− 20

0

20

40

60

− 60

− 40

− 20

− 60

− 40

− 20

327

0

20

40

60

τ (b)

τ (a) z 50

0

0

20

40

60

τ (c)

Figure 8. Examples of collisions of moving gap solitons with k = −0.6, c = 1, γ = 0.3, δ = ±0.25 and ∆φ = 0. (a) Anomalous dispersion; (b) zero dispersion; (c) normal dispersion.

difference, the solitons collide and form a lump which breaks up into two solitons which travel at almost the same velocity as the initial solitons. The effect of the initial phase difference is that in the case of ∆φ = π the emerging solitons have different velocity shifts compared to those with ∆φ = 0.

328

5.


Conclusion

In this article, we have characterized the gap soliton solutions in a recently introduced model in the absence of the second order dispersion. Similar to the anomalous and normal dispersion regimes, the family of GSs in this case is found to be formally unstable but in a part of the family the instability is very weak and the solitons belonging to that part of the family are therefore virtually stable. Interactions of quiescent solitons and collisions of moving

z z

50

0

50

− 60

− 40

− 20

0

20

40

60

0

τ (a) z

− 60

− 40

− 20

0

20

40

60

τ (b)

50

0

− 60

− 40

− 20

0

20

40

60

τ (c)

Figure 9. Examples of collisions of moving gap solitons with k = −0.6, c = 1, γ = 0.3, δ = ±0.25 and ∆φ = π. (a) Anomalous dispersion; (b) zero dispersion; (c) normal dispersion.


329

z z

50

50

0

0 − 60

− 40

− 20

0

20

40

60

− 60

− 40

− 20

− 60

− 40

− 20

0

20

40

60

τ (b)

τ (a) z 50

0

0

20

40

60

τ (c)

Figure 10. Examples of collisions of moving gap solitons with k = −0.6, c = 1, γ = 0.3, δ = ±0.5 and ∆φ = 0. (a) Anomalous dispersion; (b) zero dispersion; (c) normal dispersion.

solitons in zero, anomalous and normal dispersion regimes are analyzed. Depending on the initial separation and the wavenumber, the solitons may either attract and bounce, attract and merge temporarily and break up into separating solitons, or repel each other. We also find that the outcome of the collisions of moving solitons depends on the initial phase and the velocity shift. In all dispersion regimes, when δ = 0.25, the in-phase solitons collide and bounce off each other elastically whereas the π-out-of-phase solitons collide and form a lump which subsequently disintegrates into two separating solitons. On the other hand,

330

Javid Atai and D. Royston Neill z

z

50

50

0

− 60

− 40

− 20

0

20

40

60

0

τ (a) z

− 60

− 40

− 20

0

20

40

60

τ (b)

50

0

− 60

− 40

− 20

0

20

40

60

τ (c)

Figure 11. Examples of collisions of moving gap solitons with k = −0.6, c = 1, γ = 0.3, δ = ±0.5 and ∆φ = π. (a) Anomalous dispersion; (b) zero dispersion; (c) normal dispersion. when δ = 0.5, the solitons always collide elastically and two separating solitons emerge. In this case, the velocity shifts of the emerging solitons depend on the initial phase difference.


331

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Chapter 13

MULTIWAVELENGTH OPTICAL FIBER LASERS AND SEMICONDUCTOR OPTICAL AMPLIFIER RING LASERS Byoungho Lee* and Ilyong Yoon School of Electrical Engineering, Seoul National University Gwanak-Gu Sinlim-Dong, Seoul 151-744, Korea

Abstract We review various schemes for multiwavelength fiber lasers and semiconductor optical amplifier (SOA) ring lasers. Multiwavelength fiber lasers have applications in wavelength division multiplexing (WDM) optical communication systems, optical fiber sensors and optical spectroscopy. Erbium-doped fiber amplifiers (EDFAs), Raman amplifiers and SOAs are mainly used as gain media for multiwavelength fiber lasers. Because EDFAs are homogeneously broadened gain media, various methods have been researched to enable the multiwavelength generation. Due to the introduction of liquid nitrogen cooling, four-wave mixing, frequency shifted feedback, and so on, multiwavelength erbium-doped fiber lasers could become realized. On the other hand, because SOA and Raman amplifiers are gain media with inhomogeneous broadening, multiwavelength generation is relatively easy. The useful features of the multiwavelength lasers are mainly dependent on a comb filter. One of the most important features of multiwavelength lasers is tunability. The tunability of wavelengths and channel spacing is required for WDM optical communication systems. Much research has been conducted to enable implementation of tunable multiwavelength fiber lasers. Various comb filters such as Fabry-Perot filters, fiber Bragg gratings, and polarization-maintaining fiber loop mirrors can be used for multiwavelength fiber lasers. We review several schemes for multiwavelength SOA-fiber and Raman fiber lasers in this chapter.

*

E-mail address: [email protected]. Tel: +82-2-880-7245, Fax: +82-2-873-9953

336

Byoungho Lee and Ilyong Yoon

1. Introduction The realization of the laser has made many applications possible. Among those applications, a light source for optical communication systems is one of the most important applications. As wavelength-division-multiplexing (WDM) optical communication systems have become more developed, multiwavelength light sources have also been widely researched. In the first stage, multiwavelength lasers could be made as a simple structure consisting of the array of lasers and a multiplexer [1, 2]. However, there have been difficulties with these lasers such as large insertion loss and bulky size. Therefore, multiwavelength fiber lasers using a single gain medium are desired. There are many possible gain media for optical communication such as erbium-doped fiber (EDF), semiconductor optical amplifier (SOA), and stimulated Raman scattering (SRS). In this chapter we review a wide variety of multiwavelength fiber lasers employing a single gain medium. Because EDF is a homogeneously broadened gain medium, a laser using EDF normally lases at a single wavelength. Various methods have been researched to enable the multiple wavelength generation, such as the introduction of liquid nitrogen cooling, four-wave mixing, frequency shifted feedback, and so on. For the multiwavelength EDF laser (EDFL), schemes to suppress mode competition are a main subject. On the other hand, Raman amplifiers and SOAs are inhomogeneously broadened gain media. Therefore, multiwavelength generation is relatively simple compared with an EDFL. Many methods have also been proposed to implement multiwavelength lasers using these technologies. One of the useful characteristics of multiwavelength fiber lasers is tuning capability. The tunability of lasing wavelengths and channel spacing is required for WDM optical communication systems. Therefore, much research has been conducted for the implementation of tunable multiwavelength fiber lasers. For multiwavelength SOA-fiber and Raman fiber lasers, the schemes for tuning and these lasers’ characteristics are main subjects. We classify and review many schemes for the design of tunable fiber lasers in this chapter.

2. Multiwavelength Fiber Lasers Using EDFA The most challenging difficulty of an EDF amplifier (EDFA) for a multiwavelength laser is that the EDFA is a homogeneously broadened medium. In a homogeneously broadened medium, all atoms in the excited state have the same gain spectrum. Therefore, when a laser employs a homogeneously broadened gain medium, only the wavelength which has the largest net gain (gain minus cavity loss) can survive. The other wavelengths decay due to loss. When a number of wavelengths are in a cavity, each channel experiences mode competition. However, because Er3+ ions are surrounded by a glass host, the interaction with the silica and other dopants leads to some degree of inhomogeneous broadening contribution. Therefore, the schemes for the multiwavelength EDFL involve increasing of the competitiveness of weak wavelengths or decreasing of the homogeneously broadened linewidth of the EDFA.

Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier…

337

2.1. Cavity Loss Balancing The first multiwavelength operation of an EDFL was demonstrated in 1992 [3]. The cavity losses of the lasing wavelengths are carefully controlled to suppress single channel lasing as shown in Fig. 1. The cavities of lasing wavelengths are separated and the losses of cavities are controlled independently so that many wavelengths can lase. This is equivalent to the net gain flattening. There is no dominant wavelength due to the flattened net gain. However, this method requires a careful control of cavity losses. Thus it can be easily expected that lasers employing this scheme are relatively unstable and sensitive to environmental conditions.

FLM λ1

Gain medium

λ2

W

PC D FLM

M λ8

λ1, λ2, …, λ8 (a)

λ1 λ1, λ2, …, λ8

Polarizer

W

W

D

D λ8

M

M

Isolator Gain medium (b) Figure 1. (a) An eight-channel laser configuration based on a linear cavity. (b) An eight-channel laser configuration based on a ring cavity (PC: polarization controller, FLM: fiber loop mirror, WDM: wavelength division multiplexer) [3].

338


2.2. Liquid Nitrogen Cooling When an EDFA is cooled, the homogeneous linewidth of the EDFA is narrowed. Spectral hole burning and homogeneous linewidth were measured as a function of temperature in Ref. [4]. The homogeneous linewidth was measured as 1.3 nm at 61 K. It was shown that the homogeneous linewidth exceeded 11.5 nm at room temperature. For a multiwavelength application of an EDFA, there was other research to make an inhomogeneously broadened EDFA by liquid nitrogen cooling [5]. In Ref. [5], the main idea was the suppression of dynamic crosstalk between adjacent channels. The liquid nitrogen cooling made an 11 dB suppression of crosstalk. The channel spacing was 4 nm and the homogeneous linewidth was measured as ~1 nm.

Pump WDM coupler 77K Comb filter

Isolator

Doped fiber

75:25 coupler Output

Figure 2. A multiwavelength EDF ring laser configuration using a comb filter in the cavity [6].

The first multiwavelength EDFL by liquid nitrogen cooling (77 K) was presented in 1996 [6]. Figure 2 shows the configuration. 11 stable laser lines were demonstrated with 0.65 nm channel spacing around 1535 nm. Two types of comb filters were used in the experiment. Those were a chirped fiber Bragg grating (CFBG) Fabry-Perot filter and a sampled grating.

2.3. Four-Wave Mixing The self-stabilizing effect of four-wave mixing (FWM) can be used for a multiwavelength EDFL. The powers of lasing wavelengths are automatically balanced by several degenerated FWMs, 2ω1 = ω2 + ω3 , or nondegenerated FWMs, ω1 + ω2 = ω3 + ω4 . For a phase matching condition, dispersion-shifted fiber (DSF) or photonic crystal fiber (PCF) are required. The self-stabilizing effect may be described as a “photonic Robin Hood.” This means that FWM takes the energy of a rich wavelength and gives it to a poor wavelength. Because power is transferred between wavelengths by the nonlinear process, this scheme can be thought of as automatic net gain equalization.


HN-PCF

WDM coupler EDF

980nm Pump laser diode

3-dB coupler

Output

PC1

Output

VOA1

VOA3 FBG1

Fused coupler1

Fused coupler2

PC3

339

FBG3

3-dB coupler

PC2

VOA4

VOA2

FBG2

FBG4

Figure 3. An experimental setup for four-wavelength EDFL (HN-PCF: highly nonlinear photonic crystal fiber, VOA: variable optical attenuator, FBG: fiber Bragg grating, PC: polarization controller) [7].

Liu and Lu demonstrated a four-wavelength EDFL using a highly nonlinear PCF to suppress the mode competition at room temperature as shown in Fig. 3 [7]. Experimental results showed lasing wavelengths of 1540.28, 1543.58, 1546.79 and 1550.08 nm.

EDFA Isolator

PC

Output 10 90

FBG1

λ1

λ2

FBGn

λn DSF

PC AWG(1xn)

Figure 4. A schematic of the multiwavelength EDFL based on degenerate four-wave mixing in the DSF (EDFA: erbium-doped fiber amplifier, FBG: fiber Bragg grating, AWG: arrayed waveguide grating, PC: polarization controller) [8].

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Han et al. also presented similar multiwavelength EDFLs by using a DSF [8, 9]. Figure 4 shows a schematic diagram of the multiwavelength EDFL employing multiple fiber Bragg gratings (FBGs) and a 1 km DSF for 10 channels’ lasing with 0.8 nm channel spacing. In addition, they showed channel spacing tunability through the elimination of the effects of several FBGs.

980nm pump laser diode

EDF

Isolator PC1 Output 10 90 PMF2 L2

PC2

PMF1 L1

DSF

PC3 PC4

Lyot-Sagnac filter Figure 5. Schematics of the multiwavelength EDFL using DSF and Lyot-Sagnac filter (PMF: polarization-maintaining fiber, EDF: erbium-doped fiber, PC: polarization controller) [9].

A tunable multiwavelength EDFL was demonstrated in 2005 [9]. The tunability originated from a tunable Lyot-Sagnac filter as shown in Fig. 5. The wavelength spacing of the two-segment Lyot-Sagnac filter was Δλ = λ / [ Δn ⋅ ( L1 ± L2 ) ] , where Δn was the 2

effective birefringence between two orthogonal polarization modes and L1 , L2 were the lengths of the two polarization-maintaining fibers (PMFs) shown in Fig. 5. Thus, the channel spacing was switchable by polarization control. In the Lyot-Sagnac filter, clockwise and counterclockwise lights experienced optical path difference due to PMF segments. Therefore, the optical path difference between two lights led to comb-like filter characteristics. The birefringence of the two PMF segments may be summed or subtracted depending on the state of the polarization controllers (PCs). Experimental results showed 11 laser lines with 1 nm spacing and 17 laser lines with 0.8 nm spacing. Stable lasing characteristics and tuning capability were obtained due to FWM.


341

2.4. Frequency Shifting Technique Another scheme for a multiwavelength EDFL is a frequency shifted feedback scheme [10]. Figure 6 is a schematic diagram of the multiwavelength EDFL. In the scheme, an acoustooptic modulator (frequency shifter) shifts the frequency of light by 100 MHz for each round trip. This prevents single frequency lasing. Experimental results showed stable ~13 laser lines with 0.8 nm spacing. A Fabry-Perot etalon with a CFBG or sampled grating was used for the periodic filter. The experimental results showed good agreement with the simulation results.

Isolator Output EDFA1 3-dB coupler

Periodic filter

EDFA2 Frequency shifter

Figure 6. A schematic diagram of the multiwavelength EDFL employing a frequency shifted feedback scheme (EDFA: erbium-doped fiber amplifier) [10].

Table 1. Multiwavelength EDFLs Comments

Channel number

Channel spacing (nm)

The first multiwavelength EDFL

6

4.8

Cavity loss balancing

1996 J. Chow [6]

11

0.65

Liquid nitrogen cooling

2000 A. Bellemare [10]

~13

0.8

Frequency shifting

18

0.8

Frequency shifting

4

3.3

Four-wave mixing

0.8, 1

Four-wave mixing

Year

First author [Reference]

1992 N. Park [3]

2002 R. Slavik [11]

High uniformity

2005 X. Liu [7] 2005 Y.-G. Han [9]

Channel spacing switching

17, 11

2006 Y.-G. Han [8]

Channel spacing switching

10

Scheme

0.8, 1.6, 2.4 Four-wave mixing

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For the frequency shifting technique, a more in-depth study was published in 2002 [11]. These researchers improved the uniformity of the lasing wavelengths in the EDFL. Uniform 18 laser lines with 0.8 nm channel spacing were obtained in the experiments. Similar to the frequency shifted feedback scheme, the phase-modulation feedback scheme was also presented [12, 13]. A LiNbO3 phase modulator was used for phase modulation. In Ref. [12], the sawtoothed and sinusoidal phase modulation of a few tens of kHz generated a multiwavelength operation. In Ref. [13], the authors reported that sinusoidal, sawtoothed, triangular and square waveforms are all suitable for multiwavelength lasing. They also indicated that the phase modulation of 500 Hz to a few tens of kHz is good. Important features of the above multiwavelength EDFLs are shown in Table 1 as a summary.

3. Multiwavelength Fiber-SOA and Fiber-Raman Lasers In a SOA, the gain medium is a semiconductor and not a single atom or ion. The recombination of electron-hole pairs makes spontaneous or stimulated emission. The SOA is electrically pumped. More electrons in the conduction band and more holes in the valance band lead to higher gain. The gain spectrum of the SOA depends on materials and structure. The intrinsic inhomogeneous broadening is an advantage of the SOA in its application to multiwavelength lasers. High gain per unit length and compact size are other advantages. On the other hand, the rectangular structure leads to a coupling loss for optical fiber and polarization-dependent gain. The fast carrier lifetime (~200 ps) leads to cross saturation and stronger nonlinear processes. SRS is an interaction between photon energy and molecular vibrational energy (optical phonon). The amplification is performed by the energy transfer from a pump beam to the signal beam (or light to lase). Unlike that of the EDFA and the SOA, Raman scattering does not require a population inversion for amplification. Very broad gain bandwidth is the main characteristic of the Raman amplification process. In the SRS, specific resonant frequency does not exist in contrast to the EDFA and the SOA. The wavelength of the pump beam determines the location of the gain spectrum which has a peak at 13.2 THz off the pump wavelength. It is a main advantage of a Raman amplifier that a specific gain medium is not required, i.e., amplification occurs in a common optical fiber. Therefore, lumped or distributed schemes are all possible. If several pump wavelengths are used properly, a flat gain over a wide bandwidth can be obtained [14]. Because Raman scattering is a weak effect, the SRS requires very high pump power (typically a few Watts) and a long length of fiber. Intrinsic inhomogeneous linewidth broadening is very attractive for a multiwavelength laser. In the SOA and Raman fiber lasers, multiwavelength generation is relatively easy because of their inhomogeneous broadening. Therefore, tunable capability has been a main subject of research involving these multiwavelength SOA and Raman fiber lasers. Tunability was even considered in the demonstration of the first multiwavelength Raman fiber laser [15, 16]. Thus, in this section, we focus on the tunable capability of SOAs and Raman fiber lasers. To avoid confusion, we use two different terminologies: switchability and tunability. Switchability and tunability denote discrete tuning and continuous tuning, respectively. Thus a wavelength switchable laser means a laser which can shift the spectral position of lasing wavelengths by some discrete steps.


343

3.1. Wavelength or Channel Spacing Switchability By using a sampled high-birefringence (Hi-Bi) fiber grating as a switchable comb filter, the wavelength switchable laser was demonstrated by Yu et al. as shown in Fig. 7 [17]. We can think of this as if two different sampled FBGs (SFBGs) are used due to the difference of the refractive indexes in the fast and slow axes of the Hi-Bi fiber. The control of a rotatable polarizer is equivalent to the selection of one of two SFBGs. In a SFBG, the center Bragg wavelength is

λB = 2neff Λ and the wavelength separation is Δλ = λB2 / 2neff p , where neff

is the effective refractive index of the fiber core, Λ the individual grating pitch, and p is the sampling period. Therefore, while Δλ is maintained at nearly the same value, it is possible to move only the center Bragg wavelength. Because the birefringence Δn is of the order of 10-4, the channel spacing is hardly influenced by the choice of polarization axis. However, if we control the polarization of light incident on the sampled grating by using the rotatable polarizer, transmission peaks can be shifted by 2ΔnΛ . The experimental result showed an interleaving characteristic. The laser output was shifted by 0.4 nm with the 0.8 nm channel spacing fixed. It had a disadvantage in that the number of switchable wavelength set was intrinsically limited to two. The amount of switchable wavelengths was determined by the choice of PMF. Thus, the maximum switchable range was limited by the birefringence of the PMF.

Isolator

SOA

Isolator Polarizer

PC Variable coupler Output

SMF

Sampled Hi-Bi fiber grating

Figure 7. A schematic diagram of a wavelength switchable SOA-fiber ring laser employing sampled HiBi FBG (SMF: single mode fiber) [17]

Lee et al. presented wavelength switchable SOA fiber lasers employing two SFBGs [18] and a reflection type interleaver [19]. The former used two SFBGs connected to a polarization beam splitter (PBS) for waveband switching as shown in Fig. 8. A rotatable linear polarizer selected one of the two SFBGs. Contrary to the work by Yu et al. [17], the amount of waveband switching depended on the design of the SFBGs. In other words, the amount of wavelength shift was not limited by the choice of a PMF. The experimental result in Fig. 9 showed that 5 laser lines with 0.8 nm spacing could be switched by a spectral displacement of 10 nm.

344


Isolator

SOA

75: 25 coupler

PC

Rotatable liner polarizer

Output

Sampled fiber Bragg gratting1

PBS

Light absorber Sampled fiber Bragg gratting2 Figure 8. A schematic diagram of the waveband-switchable SOA-fiber laser using two SFBGs (PBS: polarization beam splitter, PC: polarization controller) [18].

-10 -20 -30 -40

1535

1540

1545

1550

1555

1560

1565

Wavelength [nm] Figure 9. Output optical spectra showing the waveband switching operation.

Optical power [dBm]

0


Isolator

SOA

345

Isolator

75: 25 coupler Output PC1 2

50:50 coupler

1

PC2 PMF

Figure 10. A schematic diagram of a SOA-fiber laser employing a reflective type interleaver (PMF: polarization maintaining fiber, PC: polarization controller) [19].

Tunable filter λ/2

L1

a

λ/2 λ/2 L2

Pump laser

c

b λ/4

WDM1

λ/2

Raman fiber

λ/4 50:50 coupler

WDM2 Output

Figure 11. An experimental setup for a tunable Raman fiber ring laser (WDM: wavelength division multiplexer) [21].

Another scheme using a reflective interleaver is shown in Fig. 10 [19]. The interleaver is composed of a PBS and a PMF loop mirror. A PC in the PMF loop mirror consists of two quarter-wave plates. The control of waveplates leads to an interleaving characteristic. The feature of this filter is that the transmission and reflection characteristics show interleaved sets of multiple wavelength peaks. Theoretically, for transmission, the filter has infinite

346


channel isolation and 3 dB insertion loss. On the other hand, for reflection, the filter shows 3 dB channel isolation and 0 insertion loss. 17 wavelengths were generated with 0.8 nm channel spacing. The laser lines could be shifted by 0.4 nm with channel spacing fixed. The PMF Lyot-Sagnac filter was also used for a multiwavelength SOA-fiber laser [20]. With the PMF Lyot-Sagnac filter, the SOA-fiber laser could have channel spacing switchability. In addition, the rotation of a quarter-wave plate made the lasing wavelength shift. Channel spacing switchability from 0.8 nm to 4.1 nm was demonstrated. Laser lines from 5 to 20 were observed. Continuous wavelength tuning was also shown. There have also been intensive research efforts for tunable multiwavelength Raman fiber lasers. Kim et al. demonstrated a multiwavelength Raman fiber ring laser with switchable channel spacing and a tunable lasing wavelength [21]. The multiwavelength source was composed of a Raman fiber and a Lyot-Sagnac filter as shown in Fig. 11. The experimental results showed a multiwavelength generation of up to 20 laser lines with 0.43 nm spacing.

Raman gain fiber WDM coupler Fiber grating 97%

PC1 (λ/2) PMF1

PC2 (λ/2) PMF2

Lyot-Sagnac filter

Fiber grating 90%

Output

Pump Pump combiner

Pump laser

Figure 12. An experimental setup for a tunable multiwavelength Raman laser based on an FBG cavity incorporating PMF Lyot-Sagnac filter (PC: polarization controller, WDM: wavelength division multiplexer) [22].

Han et al. demonstrated a multiwavelength Raman laser with a similar filter as shown in Fig. 12 [22]. Although a similar PMF Lyot-Sagnac filter was used, there were two differences: a linear cavity structure employing FBGs and the use of PMFs with different birefringences. In the experimental result, the multiwavelength laser generated 7 channels with 0.6 nm spacing and 5 channels with 0.8 nm spacing. A phase modulator loop mirror filter (PM-LMF) could be used for wavelength switchability [23]. The PM-LMF is a sort of PMF loop mirror where a phase modulator is inserted. Because DC bias, RF power, or modulation frequency changes the birefringence of the phase modulator slightly, the spectral comb position can be shifted while the channel spacing is fixed. Experimentally, 21 laser lines with a 0.8 nm channel spacing were obtained.


Optical amplifier Programmable Hi-Bi FLM

Pump laser

DCF 6.9 km Input

Output WSC

10% Output

Isolator

Residual pump power

Coupler

Hi-Bi, L1 PC1 In Out 3 dB coupler

347

Hi-Bi, L2 PC2

Hi-Bi, Ln PCn

Combiner

2× 2 switch

Figure 13. A schematic of a multiwavelength laser and a programmable Hi-Bi fiber loop mirror (FLM) (DCF: dispersion compensation fiber, WSC: wavelength selection coupler) [24].

Chen demonstrated channel spacing switchable fiber lasers by using a programmable HiBi fiber loop mirror as shown in Fig. 13 [24]. He employed a SOA or a Raman amplifier as an optical amplifier. Although many switchable sections are theoretically possible in Fig. 13, the two sections of the PMF were demonstrated. The use of 2×2 switches changes the combination of the PMF section more flexibly. The channel spacing expression is essentially equivalent to that of the PMF Lyot-Sagnac filter except that it has a more flexible birefringence combination. The experimental result showed 3.2 nm and 1.6 nm channel spacing switching. A tunable Raman fiber laser employing an electro-optical tuning scheme was presented in 2004 [25]. The comb filter uses an electro-optic polarization controller (EOPC) inserted in the PMF Sagnac loop filter. The PMF Sagnac loop filter is sometimes called a Lyot-Sagnac filter. Due to the PMF Sagnac loop filter, the channel spacing was switchable. When a driving voltage was applied to the EOPC, the additional birefringence was induced. This effect is equivalent to changing the length of the PMF slightly. Therefore, lasing wavelengths can be shifted as channel spacing is nearly fixed. This Raman laser had both channel spacing switchability and wavelength tunability. Experimental results indicated channel spacing switchability between 0.95 and 2.95 nm. Interleaved switching operation of 11 laser lines was also demonstrated with 0.88 nm channel spacing.

3.2. Wavelength Tunability In a SOA fiber laser, wavelength tuning possibility was presented in 2001 [26]. In fact, the work in Ref. [26] was about a multiplexed sensor. The sensor was a SOA ring laser using a transmission-type filter consisting of a circulator and multiple FBGs. Eight laser lines were

348


observed when 8 FBGs were used. The center wavelengths of the FBGs were 1534.44, 1543.68, 1546.32, 1549.38, 1552.5, 1554.06, 1556.28 and 1558.92 nm. The strain on each FBG changed each lasing wavelength. Therefore, the sensor was indeed a sort of a tunable laser although it was not clarified in the reference. Output

Isolator

HWP1

QWP

75:25 coupler

CW Port 2

GCSOA

PMF Port 1

PBS

CCW

PC

Isolator

HWP2

Figure 14. The schematic diagram of a multiwavelength SOA-fiber ring laser employing a PDLC (GCSOA: gain-clamped semiconductor optical amplifier, HWP: half-wave plate, QWP: quarter-wave plate, PMF: polarization-maintaining fiber, PBS: polarization beam splitter, PC: polarization controller, CW: clockwise, CCW: counterclockwise) [27].

80°

-10 -30

-30 -50 -10 -30 -50 -10

0°

-40

-50 1545

1550

1555

1560

1565

Wavelength [nm]

(a)

1570

1575

-30

-50 1557 1558 1559 1560 1561 1562 1563

Wavelength [nm]

(b)

Figure 15. (a) The output spectrum of a multiwavelength laser (b) The tuning characteristic.

Optical power [dBm]

-30

-50 -10

40°

-20

-30

20°

The angle of HWP 1

Optical power [dBm]

-10

60°

-50 -10


349

Yoon et al. published their research on the wavelength tunable SOA fiber ring laser [27]. They employed a polarization-diversity loop configuration (PDLC) comb filter as shown in Fig. 14. The PDLC comb filter consists of two half-wave plates, a quarter-wave plate, a polarization beam splitter, and a PMF. The PDLC filter can tune lasing wavelengths with the control of a half-wave plate alone. A single polarization which comes out of the PBS enters the PMF after passing through wave plates. The light experiences polarization change according to its wavelength by the PMF. When the light meets the PBS again, the transmittivity is determined by its polarization. The role of the wave plates is to make the polarization-change by the rotation of the first half-wave plate reproduce the polarizationchange by the PMF. When we represent polarization change in the Poincare sphere, the trajectory of polarization change becomes a circle. Because the rotation of the first half-wave plate makes the trajectory rotate, the rotation of the first half-wave plate can shift the position of the filter comb. In the PDLC filter, clockwise and counterclockwise lights exist. Two counter propagating lights experience the same transmittivity and reflectivity. A 90° rotation of the angle of the first HWP corresponds to the sweep of the entire channel spacing. The channel spacing is determined by the length and birefringence of the PMF. In the experiment, 18 laser lines were observed with 0.8 nm channel spacing. Figs. 15 (a) and (b) show the output spectrum and tuning characteristics. The rotation of a half-wave plate can shift the position of lasing wavelengths linearly.

PC

Raman fiber (SMF 50 km)

Tunable chirped FBG Output

Few-mode Bragg grating

Pump Pump combiner

Pump laser 1425nm 1435 1455 1465

(a) Compression (Negative bending) d

Tension (Positive bending)

Flexible metal plate

(b) Figure 16. (a) An experimental setup for a multiwavelength Raman fiber laser based on few-mode FBGs (b) A tuning method based on the symmetrical bending of a flexible metal plate (FBG: fiber Bragg grating, PC: polarization controller) [28].

350


For a wavelength-tunable Raman fiber laser, Han et al. demonstrated the few-mode FBG scheme as shown in Fig. 16 [28]. Because the few-mode FBG has multiple resonant wavelengths, a multiwavelength Raman fiber laser could be obtained without additional multichannel filters. The CFBG is used to form a linear cavity because of the broad reflection spectrum. Tuning of the CFBG is needed to match the reflection spectrum to that of a few-mode FBG. The lasing wavelength shift ( Δλ ) can be defined as Δλ = (1 − ρ )ελ p , where

λ p is the lasing wavelength of the Raman fiber laser, ρ is the photo-elastic coefficient, and ε is the strain induced by the bending of the fiber. The experimental results showed 3 laser lines with 3.5 nm spacing and the wavelength tuning characteristics. The number of laser lines was limited by the few-mode FBG. Han et al. also demonstrated temperature tuning of the lasing wavelength of a multiwavelength Raman laser using the few-mode FBG [29]. Three laser lines were also obtained and the temperature sensitivity was measured as 10.5 pm/°C. They also applied a similar structure to a temperature and strain sensor using a multiwavelength Raman laser with a phase-shifted FBG [30]. The experimental result showed that two lasing wavelengths could be shifted by strain and temperature with a fixed spacing.

3.3. Channel Spacing Tunability Dong et al. presented a multiwavelength SOA-fiber laser and Raman fiber laser employing a fiber Fabry-Perot filter based on a superimposed CFBG [31, 32]. Two super imposed CFBGs form the Fabry-Perot filter when the writing positions of the two CFBGs are slightly different. The tuning of the chirp rate changes the channel spacing. The SOA-fiber laser and Raman fiber laser could be implemented by using the same filter. The SOA-fiber laser generated 10~13 laser lines with 0.3~0.6 nm channel spacing and the Raman fiber laser generated 2~10 laser lines with 0.3~0.6 nm channel spacing.

3.4. Both Wavelength and Channel Spacing Tunability Roh et al. demonstrated a SOA-fiber laser with both wavelength and channel spacing tunability [33]. They employed a PDLC comb filter with a differential delay line (DDL) as shown in Fig. 17. The use of the DDL instead of a PMF could lead to spacing tunability as well as wavelength tunability. Both the channel spacing and lasing wavelength are continuously tunable. Channel spacing is tuned electrically and wavelength is tuned by the rotation of a half-wave plate. Because this laser adopted a PDLC comb filter, the wavelength tuning characteristics are totally equivalent to Ref. [27]. Experimental results showed channel spacing tunability of 0.4 ~ 1.6 nm with up to 23 laser lines.


SOA

Isolator

S

S

F

θh1

Isolator

PC

F

θq

351

2

PBS

75:25 coupler

CCW 1

Output QWP

CW

HWP1 HWP2 DDL S

F

S

θD

F θh2

Figure 17. The schematic diagram of the channel spacing and wavelength tunable SOA-fiber ring laser (DDL: differential delay line, QWP: quarter-wave plate, HWP: half-wave plate, PBS: polarization beam splitter, PC: polarization controller, CW: clockwise, CCW: counterclockwise) [33]

The features of the demonstrated SOA-fiber lasers and Raman fiber lasers are summarized in Tables 2 and 3. Table 2. Multiwavelength SOA-fiber lasers Year


Advantage

2001 S. Kim [26] 2003 B.-A. Yu [17]

Channel number

Channel spacing (nm)

8 Wavelength switching

4

0.8

2004 L. R. Chen [24] Spacing tuning

11, 6

1.6, 3.2

2004 Y. W. Lee [19] Waveband tuning 2005 M. P. Fok [23] Wavelength tuning

17 21

0.8 0.8

2005 Y.-G. Han [20] Spacing and waveband switching 5~20

0.8~4.1

Scheme Fiber Bragg grating Sampled fiber Bragg grating Programmable HiBi FLM Hi-Bi FLM PM-LMF PMF Lyot-Sagnac filter

2005 X. Dong [31]

Spacing tuning

13

0.4

Fabry-Perot

2006 I. Yoon [27]

Wavelength tuning Wavelength and channel spacing tuning

18

0.8

PDLC

23

0.8

PDLC with DDL

2006 S. Roh [33]

352

Byoungho Lee and Ilyong Yoon Table 3. Multiwavelength Raman fiber lasers

Year


Advantage

Channel Channel spacing number (nm)

2001 F. Koch [15] Potential angle tuning 24 C. J. S. de Matos 2001 Potential individual tuning 4 [16] Channel spacing switching and 2003 C.-S. Kim [21] 20 wavelength tuning 2004 C.-S. Kim [25]

Channel spacing switching and 11 wavelength tuning

Scheme

0.8

Fabry-Perot

~4.5

FBG

0.4~3

Sagnac

PMF Sagnac loop filter 0.95, 0.88, with electro-optic 2.95 polarization controller

2004 Y.-G. Han [22] Channel spacing switching

7, 5

0.6, 0.8

PMF Lyot-Sagnac

2005 Y.-G. Han [28] Wavelength tuning

3

3.5

Few-mode fiber

2005 Y.-G. Han [30] Sensing (wavelength tuning)

2

1.4

Phase-shifted fiber

2006 X. Dong [32]

2~10

0.3~0.6

Sample fiber Bragg grating

Channel spacing tuning

4. Conclusion As multiwavelength light sources become more important in WDM optical communication, there is an increasing amount of research on the multiwavelength fiber lasers. In this chapter we reviewed various schemes for a multiwavelength fiber laser to date. Feasible gain media are the EDFA, the SOA and the SRS. Each of these schemes has different challenging difficulties for multiwavelength generation. For the EDFA, the most difficult problem is its homogeneous broadening. The unstable lasing characteristic of the EDFA results from homogeneous broadening. As possible schemes to overcome homogeneous broadening, we reviewed the techniques of cavity loss balancing among wavelengths, self stabilization from FWM, liquid nitrogen cooling and frequency shifted feedback. On the other hand, inhomogeneous broadening of a SOA and Raman amplifier makes multiwavelength generation easy. The tunability in lasing wavelength and channel spacing becomes more important as optical communication systems become more flexible and efficient. We focused on the challenging issues of tuning multiwavelength SOAs and Raman fiber lasers. Practically, tunable channel spacing and tunable lasing wavelengths are required features in WDM optical communication systems. Tuning capabilities can be classified into switchability and tunability. Today’s tunable SOA-fiber and Raman fiber lasers were classified and reviewed. Because the tunable characteristic originates from characteristics of a filter, various filters for tunable lasers were introduced.


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References [1] Hamakawa. A.; Kato, T.; Sasaki, G.; Shigehara, M. Conf. Opt. Fiber Comm. 1997, 297-298. [2] Takahashi, H.; Toba, H.; Inoue, Y. Electron. Lett. 1994, 30, 44-45. [3] Park, N.; Dawson, J. W.; Vahala, K. J. IEEE Photon. Technol. Lett. 1992, 4, 540-541. [4] Desurvire, E.; Zyskind, J. L.; Simpson, J. R. IEEE Photon. Technol. Lett. 1990, 2, 246 – 248. [5] Goldstein, E. L.; Eskildsen, L.; da Silva, K.; Andrejco, M.; Silberberg, Y. IEEE Photon. Technol. Lett. 1993, 5, 937 – 939. [6] Chow, J.; Town, G.; Eggleton, B.; Ibsen, M.; Sugden, K.; Bennion, I. IEEE Photon. Technol. Lett. 1996, 8, 60-62. [7] Liu, X.; Lu, C. IEEE Photon. Technol. Lett. 2005, 17, 2541-2543. [8] Han, Y.-G.; Tran, T. V. A.; Lee, S. B. Opt. Lett. 2006, 31, 697-699. [9] Han, Y.-G.; Lee, S. B. Opt. Express 2005, 13, 10134-10139. [10] Bellemare, A.; Karasek, M.; Rochette, M.; LaRochelle, S.; Têtu, M. J. Lightwave Technol. 2000, 18, 825-831. [11] Slavik, R.; LaRochelle, S.; Karasek, M. Opt. Commun. 2002, 206, 365-371. [12] Zhou, K.; Zhou, D.; Fengzhong, F.; Ngo, Q. N. Opt. Lett. 2003, 28, 893-895. [13] Ahmed, F.; Kishi, N.; Miki, T. IEEE Photon. Technol. Lett. 2005, 17, 753-755. [14] Liu, X.; Lee, B. IEEE Photon. Technol. Lett. 2004, 16, 428-430. [15] Koch, F.; Reeves-Hall, P. C.; Chernikov, S. V.; Taylor, J. R. Conf. Opt. Fiber Comm. 2001, 54, WDD7/1-WDD7/3. [16] De Matos, C. J. S.; Chestnut, D. A.; Reeves-Hall, P. C.; Koch, F.; Taylor, J. R. Electron. Lett. 2001, 37, 825-826. [17] Yu, B.-A.; Kwon, J.; Chung, S.; Seo, S.-W.; Lee, B. Electron. Lett. 2003, 39, 649-650. [18] Lee, Y. W.; Yu, B.-A.; Lee, B.; Jung, J. Opt. Eng., 2003, 42, 2786-2787. [19] Lee, Y. W.; Jung, J.; Lee, B. IEEE Photon. Technol. Lett. 2004, 16, 54-56. [20] Han, Y.-G.; Kim, G.; Lee, J. H.; Kim, S. H.; Lee, S. B. IEEE Photon. Technol. Lett. 2005, 17, 989-991. [21] Kim, C.-S.; Sova, R. M.; Kang, J. U. Opt. Commun. 2003, 218, 291-295. [22] Han, Y.-G.; Lee, J. H.; Kim, S. H.; Lee, S. B. Electron. Lett. 2004, 40, 1475-1476. [23] Fok, M. P.; Lee, K. L.; Shu, C. IEEE Photon. Technol. Lett. 2005, 17, 1393-1395. [24] Chen, L. R. IEEE Photon. Technol. Lett. 2004, 16, 410-412. [25] Kim, C.-S.; Kang, J. U. Appl. Opt. 2004, 43, 3151-3157. [26] Kim, S.; Kwon, J.; Kim, S.; Lee, B. IEEE Photon. Technol. Lett. 2001, 13, 350-351. [27] Yoon, I.; Lee, Y. W.; Jung, J.; Lee, B. J. Lightwave Technl. 2006, 24, 1805-1811. [28] Han, Y.-G.; Moon, D. S.; Chung, Y.; Lee, S. B. Opt. Express 2005, 13, 6330-6335. [29] Han, Y.-G.; Lee, S. B.; Moon, D. S.; Chung, Y. Opt. Lett. 2005, 30, 2200-2202. [30] Han, Y.-G.; Tran, T. V. A.; Kim, S.-H.; Lee, S. B. Opt. Lett. 2005, 30, 1114-1116. [31] Dong, X.; Shum, P.; Xu, Z.; Lu, C. IEEE LEOS Ann. Meeting 2005, 814 – 815. [32] Dong, X.; Shum, P.; Ngo, N. Q.; Chan, C. C. Opt. Express 2006, 14, 3288-3293. [33] Roh, S.; Chung, S.; Lee, Y. W.; Yoon, I.; Lee, B. IEEE Photon. Technol. Lett. 2006, 18, 2302-2304.



Chapter 14

AGING AND RELIABILITY OF SINGLE-MODE SILICA OPTICAL FIBERS M. Poulain1, R. El Abdi2 and I. Severin3 1

UMR 6226, Université de Rennes1, F-35042 Rennes, France LARMAUR, Fre-Cnrs 2717, Université de Rennes1, F-35042 Rennes, France 3 Universita Politechnica, Splaiul Independentei, IMST, 06042 Bucarest, Romania 2

Abstract The optical fiber reliability in telecommunication networks has been still an issue, that’s why the question of how long an optical fibers might been used without a significant probability of failure isn’t out of interest. Much work was developed around this issue, but the optical fiber fatigue and aging process has not been yet fully understood. The reliability of the optical fibers depends on various parameters that have been identified: time, temperature, applied stress, initial fiber strength and environmental corrosion. The major and usually unique corrosion reagent is water, either in the liquid state or as atmospheric moisture. Glass surface contains numerous defects, either intrinsic, the socalled “Griffith’s flaws and extrinsic, in relation to fabrication process. Under permanent or transient stress, microcracks grow from these defects, and growth kinetics depend on temperature and humidity. Although polymeric coating efficiently protects glass surface from scratches, it does not prevent water to reach glass fiber. The work carried out during the last years made possible to apprehend in a more coherent way the problems of failure and rupture of fibers subjected to severe aging conditions. In the proposed chapter, some informations on the used characterization methodology for the silica optical fibers are given. In addition, Optical fibers analysis advantages, expected percussions and theoretical background are given to enlighten the potential concerned persons. The principal optical fiber test benches are described and some results are commented. Finally, final remarks are noted.

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M. Poulain, R. El Abdi and I. Severin

1. Introduction Terrestrial and submarine telecommunication networks depend critically on optical fibers. While main emphasis is put on transmission and signal characteristics [1], more basic features such as reliability and expected lifetime has appeared also as major concerns [2, 3]. However, these concerns become less popular with the deep crisis that occurred in the telecom market and the emergence of more advanced fibers: one may think that new fibers should replace the existing ones earlier than expected. In addition, until now, operators did not face serious problems in relation to fiber failure. Nevertheless the reliability issue remains more than ever a topical question for several reasons. Firstly, the impressive increase of the bit rate is accompanied by a power increase which is supported by the fiber core and can generate catastrophic failure phenomena and generate damage of the fiber ends or losses in the connectors. Secondly, current models include humidity, applied stress and temperature as major aging factors, but their accuracy for lifetime prediction is questionable. Aging of silica fibers is now rather well understood as numerous studies have been implemented in this area [2-17]. One must separate the case of the fatigue static behavior where fibers are subjected to a permanent strain, e.g. bended fibers, and the dynamic fatigue corresponding to an unexpected tensile stress arising from environmental changes. Failure mechanism involves surface phenomena, which raise fundamental questions. Surface defects, initiator for cracks grow have not yet been identified neither by Scanning Electron Microscopy (SEM) nor by Atomic Force Microscopy (AFM). The current random network model used actually to describe glass structure gives no explanation for the so-called Griffith’s flaws [18] and does not account for density fluctuations and inhomogeneities in glass. While other models, such as the vacancy model [14, 19], may provide a physical picture of these defects, they are still in an emerging state that limits their application. Water is also critical in fiber failure: fiber strength may increase by 100 % if water is missing, for example under vacuum, in a dry box or at liquid nitrogen temperature [3, 4, 16]. It is assumed that water molecules break the Si-O-Si chemical bonds of the vitreous network. This simple and logical model may be incomplete and ignore some aspects of the whole phenomenon. Polymeric coatings are largely used to inhibit surface flaws and proved to be efficient. However the reinforcement mechanism is not well understood. The general use of Weibull’s statistics in data processing may be inappropriate in some cases: it is widely observed that fiber strength value calculated from Weibull plots decreases as sample length increases, but Weibull formula is precisely expressed to be independent on fiber length. These questions, and others, have not only a fundamental interest, but still could have notable economic implications. The technology evolution and the research for low cost solutions lead to use new fibers and new components. Thus, polymeric fibers are being considered for the local distribution, while Bragg grating fiber components are now largely used in optical amplifiers. However, the reliability of these new components has still to be evaluated. A traditional stake is referred to the future fibers for the local distribution networks (Fiber To The Home, FTTH). These fibers will be submitted to notable permanent stresses, for

Aging and Reliability of Single-Mode Silica Optical Fibers

357

example at door corners, thus exposure to temperature, humidity and sudden stress may be larger than in classical cables. The better understanding of the factors ruling aging and reliability of optical fibers should lead not only to scientific advances, but also to economical spin-offs. The telecom market will require light cables for local area networks, and the design of mini cables can be optimized on this basis. In addition various markets are likely to open in other fields where optical fiber components are key elements. This concerns optical fiber sensors, laser power delivery, fiber lasers, monitoring and control, remote spectroscopy, in line imaging, etc… Of particular interest is automotive industry in those case reliability and cost make essential points. On the fundamental level, the principal goal is to collect new information elements that could help answering recurrent questions.

2. Background Failure of fibers is rather well understood as it might be considered as a particular case of fragile material fracture [2-18]. While such materials exhibit a large resistance to compressive stress, they are much more sensitive to traction. Failure originates from surface flaws that may be described as microcracks. Their cracks growth under tensile stress as effective stress is amplified at the bottom of the crack. This growth is enhanced by water activity in most materials, including oxide glasses. It is generally assumed that water acts by breaking the chemical bonds between oxygen and silicium or other cations. For this reason the intrinsic strength K1C of the material can be observed only in extremely dry conditions, e.g. vacuum or liquid nitrogen. In practice, optical fiber aging depends on various factors that may decrease effective fiber strength: residual applied stress, temperature and water. It is assumed that surface flaws are enlarged, consequently crack growth promotes. Maximum water activity is in aqueous solutions and it is expressed by the relative humidity (RH) in current atmosphere.

Figure 1. Fracture morphology of silica optical fibre (see silica core – typical fragile surface fracture surrounded by the two layer epoxi-acrylate polymer coating).

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As fiber surface has determined fracture to a large extent, external coating appears critical. This coating is polymeric in most cases, and modern optical fibers are coated by two different layers, a soft coating at glass surface and a hard coating at external surface (Fig. 1). The coating first makes a protection against scratches that occur in normal handling; it also fills the surface flaws gluing in some a way the two sides of the micro cracks and finally, it reduces water activity at glass surface. Ideally, coating should prevent any water molecule to reach glass surface. Unfortunately, polymeric coatings, including hydrophobic coatings, are very permeable to water. Only inorganic hermetic coating could make an efficient barrier against water. Polymeric coatings (e.g. epoxyacrylates) are preferred in practice because they are more efficient to inhibit surface defects [13, 20-24]. Various theoretical models are applied for mechanical characterization of optical fibers [25- 26], but the most common one is based on Weibull's statistics. The Weibull law expresses the failure probability F of a fiber with a length L subjected to an applied stress σ :

⎡1 ⎧ 1 ⎫⎤ Ln ⎢ ⎨Ln[ ]⎬⎥ = m [Ln(σ ) − Ln(σ o )] ⎣ L ⎩ 1 − F ⎭⎦

(1)

where m is a size parameter and σο is a scale parameter.

⎡1 ⎧ 1 ⎫⎤ ]⎬⎥ in function of Ln (σ) is known as the Weibull ⎨Ln[ ⎣ L ⎩ 1 − F ⎭⎦

The evolution of Ln ⎢

plot. The values of m and σο are calculated from the slope of the curve and the intersection with the stress axis. The m parameter characterizes the defect size dispersion [26]. A high m value indicates that the distribution of the defect size is homogeneous while a low m value means that surface defects are varying in size. When the curve appears as a broken line with two distinct slopes – one small for low stress and the second one large, respectively – one has assumed two different families of defects, the first one corresponding to large extrinsic defects, and the second one relating to intrinsic flaws. Other plots encompass several straight lines relating to different groups of defects. The failure probability F is calculated from the relation:

Fi =

i − 0.5 N

(2)

where i represents the rank of the measurement and N the total number of values. The σο parameter represents the stress corresponding to the fiber cumulative fracture probability F of is 50%. In the static fatigue measurements, the fiber is subject to a constant stress and one measures the time to failure. This time tf is ruled by the following relation:


tf = B

S in − 2

359

(3)

σ an

where B is a constant that depends on environment – typically water, S the initial inert strength of the fiber and n the stress corrosion parameter, and σa the failure stress. The failure probability of F can be written as: m ⎛ ⎞ ⎛ t f .σ an ⎞ n − 2 ⎟ ⎜ ⎟ F (t f , L) = 1 − exp⎜ − L ⎜⎜ ⎟ n−2 ⎟ B . S ⎜ ⎟ ⎝ ⎠ o ⎝ ⎠

(4)

Or, in the logarithmic form:

( )

(

n−2

Ln t f = − nLnσ a + LnB + Ln Si

)

(5)

The plot Ln (t f ) as a function of Ln (σ a ) describes the static fatigue behavior and gives n− 2 access to the fatigue parameters n and BS i .

3. Experimental: Mechanical Measurements The mechanical strength of the fibers may be measured in different ways corresponding to static fatigue and dynamic fatigue [27, 28]. In the static fatigue tests, fibers are subject to a permanent stress, and the time to failure is recorded for a set a fibers. Then a statistical analysis gives values for the mean failure strength and the mean lifetime in the testing conditions: type of fiber, temperature, applied stress and water activity. The dynamic fatigue test consists in applying an increasing tensile strength until fiber breaks. From a convenient data processing one finds the mean fiber strength. Special equipments are used for these measurements, presented as follows.

3.1. Vertical Bench The static fatigue under axial tensile loading consists in subjecting a fiber sample to a uniform load as a suspending weight of known value. The two fiber ends are rolled up on a pulley provided with a system allowing to block the fiber sample ends and to avoid any slip (Fig. 2a). The higher pulley (noted 1) is fixed on a support, while the lower pulley (noted 2) is mobile and interdependent of a plate on which a chosen mass is applied. This set up leads to carry out static tensile tests on high length fiber samples (usually 4 m in length) for applied loads ranging between 5 and 50 N. A number of forty samples can simultaneously be tested (Fig. 2b). The measurement of the fiber fracture time for different

360


loads allows determining the static stress corrosion parameter. Thus, the fibers are submitted to different aging conditions and subsequently to mechanical tensile testing.

Pulley1 Silica fiber

Pulley2 Mass Plate Optical Beam of light sensor

(a)

(b)

Figure 2. Vertical static tensile-test bench: (a) diagram of sample fiber under mass loading, (b) general view.

3.2. Static Fatigue under Permanent Curvature Another testing bench can be used for static testing [29]. Optical fibers, one meter in length, are subjected to bending stresses by winding around alumina mandrel with calibrated diameter sizes (Fig. 3a). The constant level of applied stress is adjusted by the proper choice of the mandrel size. The time to failure is measured, and this corresponds to the time required for the fiber strength to degrade until it equals the stress applied through winding round the mandrel. The time to failure is measured by optical detection when the ceramic mandrel moves out of the special holder. When fiber breaks, the mandrel rocks from its vertical static position and the time to failure is directly recorded with an accuracy of ±1 s. The testing setup consists of a large number of vats containing 16 holders each (Fig. 3b). The applied stress on the fiber depends on the mandrel diameter according to the Mallinder and Proctor relation [30] as follows: 'ε ⎞ ⎛ σ = E 0 ε ⎜⎜1 + α ⎟⎟ ;

⎝

2 ⎠

3 4

α'= α ;

ε=

d glass φ + d fiber

(6)


361

where σ is the applied stress (in GPa), E0 is Young modulus (equal to 72 GPa for the silica), ε is the relative fiber deformation, α is the constant of the elastic nonlinearity (equal to 6), φ is the mandrel diameter (in μm), dglass is the glass fiber diameter and dfiber is the fiber diameter including the polymer coating. For example, for standard silica optical fibers used for telecommunication networks, dglass is equal to 125 μm and dfiber is equal to 250 μm; this leads to the corresponding stress of 3.92, 3.76, 3.34 and 3.22 GPa for the calibrated diameter mandrel of 2.3, 2.4, 2.7 and 2.8 mm respectively. The testing environmental conditions during static fatigue measurements (temperature and relative humidity) should be also taken into account.

E

R

Clamping rings

Light beam

Wound fibre on calibrated mandrel

(a)

(b) Figure 3. Static bending test.

3.3. Vertical Dynamic Tensile Test

Dynamometric cell Device speed control

500 mm

Higher plate

Engine

Movable pulley

Fiber Fixed pulley

(a)

(b)

Figure 4. Schematic description of the dynamic tensile-test bench.

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During a dynamic tensile test, the fiber is subjected to a deformation under a constant speed until the rupture. The two fiber ends are rolled up on pulleys, having 65 mm in diameter and covered with a powerful adhesive so as to prevent any fiber slip during the test (Fig. 4). The lower pulley is fixed while the higher pulley is mobile and its displacement velocity (v, mm/min) corresponds to the chosen deformation speed to carry out the test. Typical fiber length is 500 mm. During the test, the deformation and the tensile load are measured using a dynamometric cell while the fiber deformation is deduced from the displacement between the fixed lower pulley and the mobile higher plate. The test velocity has an important influence for the failure stresses as this might be seen in Fig. 5. High speeds lead to failure cracks with the same geometry (not curve slope variation for v=500 mm/min), while the low speeds lead to various crack forms. 2

v- 50mm/min

Ln (- ln(1-F))

1

v- 150mm/min v-300mm/min

0

v - 500mm/min

-1 -2 -3 -4 0,60

0,80

1,00

1,20

1,40

Failure stress (GPa)

(F represents the cumulative fracture probability)

Figure 5. Evolution of failure stresses for different tensile test velocities v (mm/min).

3.4. Long Length Dynamic Tensile Bench This mechanical bench (Fig. 6) allows to carry out tensile tests on fibers with high lengths (from 0.5 m to 18 m) with broad speeds (ranging between 30 mm/min to 30 m/min with an accuracy of less than 2 per 1000) and under very diverse environmental conditions (temperature, aqueous solution...). Using the set up, one can obtain information on the defect size dispersion onto the fiber surface and can determine the dynamic stress corrosion parameter n. Indeed, this parameter is related to the velocity by the following relation:

V = A K nI

(7)

where A is a parameter environment dependent, KI is the stress intensity factor and n is a parameter characterizing the material capacity to resist to a stress.


363

3.5. Two Point Bending Bench Fibers can also be characterized by using a two point bending testing device (Fig. 7). Samples of 10 cm in length are bent and placed between the grooved faceplates of the testing apparatus, in order to avoid the fiber slipping during the faceplate displacements and to maintain the fiber ends in the same vertical plane.

Figure 6. Thirty meters long dynamic tensile bench.

Generally, a series of 30 samples are tested for different faceplate velocities (for example, 100, 200, 400 and 800 μm/s, respectively). The failure stress is calculated from the distance separating the faceplates, using the Proctor and Mallinder relation, improved by Griffioen [8]. Subsequently failure stress is obtained for each tested sample and tracing the classical Weibull plots one might calculate the statistical parameters. Due to a very short fiber sample part subjected under stress, this testing method is preferentially used to study the intrinsic defects or selected flaws. Fiber

Computer Fiber Piezo Electric Sensor

Faceplates Stepper motor Control Block

Fiber Faceplates

Figure 7. Dynamic two point bending bench.

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4. Results The work carried out during the last years made possible to apprehend in a more coherent way the failure of the fibers subjected to severe aging conditions. An empirical relation defining the fiber lifetime tf according to the temperature, residual stress and water content was established. For this purpose, silica optical fibers were either immersed in hot water or heated in wet atmosphere with a controlled relative humidity (RH). The two relations derived from this set of measurements are the followings [14]:

⎛ E + (φ ⋅ T − β ) ⋅ σ a ⎞ t f = A0 exp⎜ 0 ⎟ RT ⎝ ⎠

(8)

⎛ E + (φ ⋅ T − β ) ⋅ σ a ⎞ t f = A0 Z −δ exp⎜ 0 ⎟ RT ⎝ ⎠

(9)

The first relation (8) applies to fibers aged in liquid water, while the second one (9) concerns fibers exposed to humid atmosphere with variable RH (RH = Z), temperature T and applied stress σ. It is worth noting that the factor Φ corresponds to some kind of relaxation that decreases the effect of the applied stress. Its magnitude grows with temperature and applied stress. In a second set of measurements, fibers were immersed in hot water at 65°C or at 85°C for a long time (up to 24 and 27 months). Then they were characterized in static and dynamic fatigue. As one could expect, fiber strength in dynamic fatigue decreases as aging time increases. More surprisingly, the time to failure of aged fibers subjected to static fatigue increased enormously by comparison to non-aged fibers [31]. However this unexpected effect did not follow a regular evolution versus time, but a rather cyclic one (Fig. 8).

(In legend calibrated mandrel diameter, in mm, in the case of the static fatigue testing set-up – Fig.3)

Figure 8. Evolution of the fiber failure time in function of different aging conditions.


365

The explanation for this lifetime increase can be found in the structural relaxation identified in the previous relations. The failure mechanism of the aged fibres involves surface phenomena, in relation to water activity. A layer of hydrated silica is likely to be formed at fibre surface [15]. This vitreous hydrated phase may relax under stress at room temperature, which partly compensates the external applied stress in static fatigue. The change of the glass surface was exemplified by the indentation behaviour that is different from that of normal silica [32]. New experiments are carried out as well on standard silica fibers as on new fibers [33]. Fibers with a hermetic coating, fibers before and after photo-printing, fibers of polymer, on average have diameters between 85 µm and 125 µm. The influence of temperature, water and various corrosive agents on the mechanical fiber strength is determined. The coating aging is also taken into account. Characterizations are also carried out on fibers belonging to different vitreous systems (fluorides, oxides, sulphides) to detect and analyze less visible phenomena when silica fibers are studied. For several silica fibers subjected to vertical static tensile testing (see Fig. 2) under various loadings, one can notice that more the suspended mass value is high; more the time of rupture is large (Fig. 9). For weak loads (15 N), two families of cracks exist (a slope break indicates the dispersion of the microcrack shapes). 2 1 ln(-ln(1-F))

0 -1

20N

-2

15N 25N

-3

30N -4 -5 0

2

4

6 8 ln (time to rupture) (h)

Figure 9. Time to rupture evolution for different loadings (F represents the cumulative fracture probability).

5. Final Remarks The huge development of the telecommunication networks has been made possible by the availability of low cost and high quality silica optical fibers. As industrial production reaches millions of km, research rather focuses on networks and advanced components. Fiber reliability is not a critical issue at this time because few problems were encountered, most of them being accidental. However future fiber local loops will put fibers under large and

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permanent stress. In addition, lighter and less expansive cables could be manufactured if transient or permanent stresses have no significant influence on fiber lifetime. Fiber aging has been the subject of numerous studies leading to theoretical models for lifetime assessment. While ground observations do not contradict these predictions, the accuracy of the models is questionable due to the complexity of the aging mechanism. In this respect, experiments implemented on a long time scale are likely to bring new information. There are some questions underlying the reliability studies. Aging parameters encompass time, temperature, applied stress and water activity. While the critical part of water in failure mechanism is well known, its real impact varies according to the physical state - liquid or vapor - and partial vapor pressure. The stress applied to the fiber may be temporary, for example during the proof test or network installation, or permanent when the fiber is bent in cable and connecting areas. The aging mechanism is assumed to enlarge or to extend the "Griffith flaws" which are spread at the fiber surface. These defects may be described as micro-cracks which grow under applied stress in wet environment. Although this mechanism is believed to be irreversible, water may also induce some curing effect which could correspond to the geometrical smoothing of the crack tip [34, 35]. There is a practical interest in collecting quantitative information on the aging of the commercial optical fibers over a long period of time. Such observations should allow a more accurate comparison between experimental and calculated strengths and make lifetime assessments more realistic as testing periods (> 2 years) become closer to the lifetime required by network users that is at least 20 years.

Acknowledgments Authors express their gratitude to France Telecom for technical assistance and equipment supply and to Region Bretagne for financial support.

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INDEX A absorption spectra, 272 access, 53, 75, 112, 216, 359 accounting, 254 accuracy, 120, 149, 201, 215, 356, 360, 362, 366 acetone, 33 acetylene, 332 achievement, 206 acid, 31, 33, 36, 44, 104, 105 acrylate, 357 adaptability, 5 adjustment, 59, 215 adriamycin, 31, 48 adsorption, 40 aerospace, 5, 260 AFM, 356 agent, 31 aging, xii, 260, 355, 356, 357, 360, 364, 365, 366, 367, 368 aging process, xii, 355 albumin, 45 algorithm, 59, 60, 65, 68, 74, 77, 243, 246, 247, 320 alternative(s), x, 54, 65, 146, 231, 233, 238, 249, 255, 261 aluminum, 144 amplitude, x, 19, 78, 84, 85, 128, 131, 166, 171, 172, 177, 209, 279, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 293, 297, 304, 324 AN, 175, 231 annealing, 65 antibody, viii, 15, 28, 29, 30, 31, 36, 37, 38, 39, 40, 42, 43, 45, 46 antigen, 31, 38, 39, 40, 43 antimony, 188 APC, 220 apoptosis, 31 argon, 31, 260

arsenic, 273 assessment, 366 assignment, 251 assumptions, viii, 43, 52 asymmetry, 105, 152 atoms, ix, 119, 120, 122, 124, 128, 130, 131, 132, 133, 134, 136, 137, 138, 139, 140, 142, 143, 146, 147, 148, 149, 150, 151, 152, 153, 154, 336 attachment, 21, 38, 39, 40, 41, 42, 43, 44 attention, vii, ix, 3, 4, 5, 161, 163, 164, 176, 217, 258, 302 attenuated total reflectance, 261 Australia, 315 automobiles, 260 availability, 365 averaging, xi, 301, 303 avoidance, 108

B Bacillus, 22, 28, 29, 47 Bacillus subtilis, 22 backscattering, ix, 54, 70, 71, 72, 73, 161, 162, 164, 170, 171, 172, 176, 177, 179, 180, 181, 182, 185 bacteria, 4, 28, 41, 48 bandgap, xi, 258, 260, 262, 265, 273, 315, 316, 317, 318, 319, 320, 321, 322, 332, 333 bandwidth, viii, x, 51, 54, 62, 63, 65, 68, 69, 72, 74, 76, 77, 84, 114, 115, 168, 170, 188, 206, 207, 216, 218, 221, 222, 232, 233, 241, 247, 257, 258, 259, 269, 270, 275, 342 beams, 120, 132, 140, 146, 150, 151, 154, 178, 179, 180, 181, 217, 280 beef, 22, 28, 46 behavior, viii, 15, 18, 35, 40, 72, 78, 97, 102, 115, 324, 356, 359, 367 Beijing, 3 bending, 177, 259, 262, 349, 350, 360, 361, 363

370

Index

bias, 346 binary decision, 252 binding, 29, 31, 39, 40, 41, 42, 47 biomarkers, 30 biomechanics, 84 biomolecule(s), viii, 15, 46 biosensors, 4, 15, 35 biotin, 29 birefringence, ix, 83, 84, 87, 88, 89, 90, 91, 92, 94, 98, 101, 102, 104, 106, 107, 109, 110, 170, 184, 225, 274, 340, 343, 346, 347, 349 bismuth, 184 blackbody radiation, 272 blocks, 297 blood, 26, 31, 275 BN, 175 BNP, 25, 26 Boltzmann constant, 58 bonding, 36, 44, 259 bonds, 356, 357 boundary value problem, 59 Bragg grating, viii, xii, 4, 51, 54, 83, 84, 85, 92, 97, 98, 101, 104, 106, 115, 161, 169, 170, 182, 184, 213, 227, 228, 315, 331, 332, 335, 338, 339, 349, 351, 356 branching, 210 brass, 271 breast carcinoma, 31 breathing, 280 broadband, viii, x, 51, 53, 63, 64, 79, 85, 113, 182, 187, 205, 208, 227, 261, 271 buffer, 22, 23, 24, 25, 26, 27, 29, 30, 38, 39, 177 building blocks, 297 burning, 169, 338

C cabinets, 53 cables, vii, 53, 357, 366 calibration, 33, 94, 96, 98, 107, 192 Canada, 205 cancer screening, 31 candidates, 182, 244, 302 capillary, 120, 121, 127, 128, 131, 132, 133, 134, 135, 260, 262 carbohydrate, 37 carbon, 6, 8, 259 carboxylic groups, 36 carcinogenicity, 31 carcinogens, 31 carcinoma, 31, 48 cardiovascular disease, 30, 49

carrier, 4, 5, 114, 115, 165, 167, 206, 233, 234, 236, 317, 342 cDNA, 32 cell, 21, 26, 27, 28, 31, 35, 40, 44, 48, 148, 262, 361, 362 cell culture, 27, 31 cell growth, 28 ceramic, 360 CFBG, 338, 341, 350 CGLE, x, 279, 280, 281, 285, 286, 287, 288, 289, 290, 291, 293, 297 channels, 53, 54, 57, 63, 75, 76, 77, 84, 115, 169, 206, 213, 215, 216, 218, 219, 227, 236, 305, 338, 346 chemical bonds, 356, 357 chemical etching, 33, 98, 105, 106 chemical properties, 15 China, 3, 257 Chinese, 24 cladding, x, 6, 16, 17, 18, 19, 31, 88, 105, 106, 121, 122, 143, 144, 146, 167, 168, 177, 190, 209, 210, 215, 218, 232, 257, 258, 260, 261, 262, 263, 264, 265, 266, 269, 270, 272, 273, 274, 275, 316, 332 cladding layer, 177 classes, xi, 263, 273, 301, 303 cleaning, 36, 42 clustering, 201 CO2, 4, 207, 208, 261, 272, 274, 275 coagulation, 30, 48 coatings, 190, 258, 261, 271, 273, 356, 358, 367 codes, 112, 113, 114, 247 coding, 114, 244, 246, 247, 249 coherence, 89, 109, 151, 224, 225, 226 collaboration, 134, 309 collisions, xi, 149, 151, 279, 302, 315, 317, 324, 325, 326, 327, 328, 329, 330, 333 combined effect, 39, 312 communication, vii, viii, x, xi, xii, 51, 52, 55, 56, 79, 83, 84, 108, 169, 187, 206, 215, 231, 232, 234, 236, 238, 239, 242, 244, 246, 268, 269, 270, 275, 299, 302, 312, 335, 336, 352 communication systems, viii, x, xi, xii, 51, 52, 55, 56, 79, 108, 231, 232, 234, 238, 239, 244, 246, 275, 299, 335, 336, 352 community, 302 compatibility, 84, 205 compensation, x, 205, 206, 213, 218, 220, 221, 302, 311, 347 competition, 111, 169, 174, 179, 180, 291, 336, 339 competitiveness, 336 complementary DNA, 32 complexity, 165, 366 complications, 30

Index components, ix, xi, 7, 18, 54, 57, 58, 68, 84, 87, 89, 90, 91, 93, 98, 104, 111, 122, 164, 166, 173, 176, 188, 205, 206, 207, 208, 212, 218, 220, 224, 225, 226, 233, 234, 236, 238, 250, 301, 303, 312, 356, 357, 365 composites, 295 composition, 190, 215 compounds, 4 computation, viii, 52, 61, 67 computing, 206, 253 concentration, 28, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 44, 104, 105, 106, 107, 164, 168, 200, 215 condensation, 315 conduction, 173, 342 conductivity, 8, 172, 221 configuration, vii, 3, 4, 49, 57, 65, 72, 75, 77, 78, 135, 142, 164, 165, 166, 169, 178, 189, 207, 216, 218, 220, 222, 226, 227, 303, 337, 338, 349 confinement, vii, 16, 52, 177, 264, 269, 271, 316 confusion, 342 Congress, 202 conjecture, 318, 321 consolidation, viii, 51 constituent materials, 258, 263, 272 constraints, 303, 307 contaminant, 47 contamination, 7, 32, 48, 143 continuity, 122, 173 control, 5, 68, 71, 72, 81, 154, 166, 170, 202, 212, 213, 215, 219, 220, 232, 244, 247, 302, 337, 340, 343, 345, 349, 357, 361 convergence, 8, 59, 65, 73 conversion, ix, 4, 108, 164, 187, 188, 202, 220, 233, 252, 260, 275, 332 cooling, xi, 68, 97, 111, 148, 154, 293, 335, 336, 338, 341, 352 Copenhagen, 255 corn, 26 coronary heart disease, 30 correlation(s), 176, 179, 180, 182, 193, 194, 201 correlation function, 176, 180 corrosion, xii, 84, 355, 359, 360, 362 costs, 68, 84 couples, 55 coupling, xi, 7, 16, 18, 58, 85, 87, 114, 128, 134, 136, 146, 147, 177, 178, 179, 182, 185, 187, 209, 216, 218, 264, 274, 301, 303, 315, 316, 317, 333, 342 covalent bonding, 36, 44 coverage, 40, 41, 43 crack, 357, 362, 366, 367, 368 C-reactive protein, 25, 30

371

critical value, 170, 179 cross-phase modulation, xi, 166, 301, 312 CRP, 25 crystal growth, 154 crystalline, 59, 273 culture, 27, 28, 31 curing, 366 CVD, 30 cytochrome, 30, 31, 48 cytokines, 30, 31, 47 cytoplasm, 31

D damping, 298 data processing, 356, 359 data transfer, 269 decay, ix, 56, 120, 187, 188, 189, 196, 197, 199, 201, 202, 336 decibel, 62 decision making, 253 decisions, 246, 252 decoding, 114, 238, 246, 247, 252 decoupling, 108 defects, xii, 7, 164, 355, 356, 358, 363, 366 defense, 15, 272 deficiency, 30 definition, 78, 146, 240, 304, 317, 321 deformation, 84, 96, 99, 103, 207, 361, 362 degenerate, 71, 141, 274, 305, 339 degradation, 52, 54, 62, 220, 367 delivery, x, 6, 257, 258, 261, 275, 316, 333, 357 demand, 7, 52, 53, 260, 269, 270 Denmark, 255 density, vii, 3, 5, 6, 7, 9, 53, 56, 73, 154, 162, 163, 164, 172, 173, 175, 176, 181, 198, 199, 200, 202, 240, 271, 356 density fluctuations, 356 dependent variable, 59 depolarization, 222, 225 deposition, 45, 139, 154 derivatives, 199, 304, 309 destruction, 53 detection, viii, x, 4, 15, 21, 28, 29, 30, 37, 38, 40, 42, 44, 45, 46, 47, 48, 49, 131, 134, 217, 225, 231, 232, 233, 234, 235, 236, 237, 238, 239, 241, 249, 250, 255, 271, 275, 360 detection techniques, 271 detonation, 4 deviation, 68, 76, 77, 176, 179, 180, 182, 221 dielectric constant, 315 dielectric permittivity, 175 dielectrics, 271

372

Index

differential equations, 162, 165, 171, 175, 198 differentiation, 304, 309 diffraction, ix, 119, 120, 124, 125, 126, 143, 144, 152, 153, 154, 280 diffusion, 367 digital communication, 242 diode laser, 29, 133, 134, 165 diodes, 64, 65, 162, 165, 170 dipole, ix, 119, 120, 121, 128, 130, 134, 136, 146, 150, 151, 152, 154, 155, 171, 173 dipole moment, 173 dispersion, x, xi, 53, 58, 68, 71, 72, 110, 122, 123, 163, 165, 183, 206, 216, 218, 220, 226, 227, 228, 231, 232, 238, 241, 259, 261, 264, 274, 279, 280, 281, 285, 293, 301, 302, 303, 305, 307, 308, 309, 311, 312, 315, 316, 317, 318, 320, 321, 322, 324, 325, 326, 327, 328, 329, 330, 331, 333, 338, 347, 358, 362, 365 displacement, 297, 343, 362 distortions, 244 distribution, 5, 59, 69, 70, 120, 124, 126, 139, 140, 141, 142, 143, 144, 146, 150, 152, 153, 154, 173, 176, 192, 193, 198, 206, 246, 250, 356, 358 distribution function, 173 divergence, 206 diversity, 349 division, xi, 53, 112, 169, 182, 183, 187, 217, 227, 228, 270, 311, 335, 336, 337, 345, 346 DNA, viii, 4, 15, 21, 22, 27, 31, 32, 44, 48, 49 DOP, 221, 226 dopants, 215, 336 Doppler, 148 dream, 302 drinking water, 31 DRS, 72 DSC, 221 duration, 207, 244 dyes, 30 dynamic control, 220 dynamical systems, 299

E E. coli, viii, 15,23, 28, 29, 32, 35, 37, 38, 40, 41, 44 earth, 183, 202 EEA, 79 eigenvalue, 240, 321 Einstein, Albert, 154, 306, 312, 315 elaboration, 208 elasticity, 257 electric current, 34 electric energy, 4, 5

electric field, xi, 19, 121, 124, 127, 142, 171, 176, 177, 281, 301, 303 electrical power, 221 electrodes, 34 electromagnetic, viii, ix, 83, 84, 119, 120, 121, 124, 154, 171, 302 electromagnetic fields, ix, 120, 121, 154 electromagnetic waves, 302 electromagnetism, 5 electron(s), 55, 128, 131, 134, 144, 162, 173, 175, 280, 298, 342 ELISA, 31 elongation, 208, 209 emission, ix, 28, 30, 56, 57, 72, 94, 111, 165, 187, 189, 191, 192, 193, 194, 195, 196, 197, 198, 199, 201, 202, 219, 342 encoding, 114, 251 endotoxins, 29 endurance, vii, 3, 9 energetic materials, 4 energy, vii, xi, 3, 4, 5, 6, 7, 8, 55, 62, 77, 87, 134, 139, 151, 162, 164, 166, 173, 176, 178, 189, 190, 197, 198, 199, 200, 216, 257, 269, 280, 291, 301, 303, 316, 318, 320, 321, 325, 338, 342 energy density, 5 energy transfer, 342 enlargement, 54 environment, 5, 16, 30, 44, 272, 280, 359, 362, 366, 367 environmental change, 356 environmental conditions, 212, 337, 361, 362 enzyme(s), 4, 48 epoxy, 37 equilibrium, 39, 56, 280, 297 equipment, vii, 99, 100, 101, 104, 108, 366 erbium, ix, xi, 94, 111, 162, 164, 165, 166, 167, 168, 169, 182, 183, 184, 187, 188, 201, 219, 335, 336, 339, 340, 341 Escherichia coli, 22, 28, 45, 46 estimating, 251, 252 etching, 28, 33, 44, 98, 104, 105, 106, 107, 181 ethylene glycol, 38, 39 Euro, 276 European Union, 79 evanescent waves, 132, 133, 134, 151 evaporation, 143, 144 evidence, 31, 182 evolution, 4, 52, 54, 57, 59, 61, 62, 67, 68, 73, 74, 84, 89, 94, 96, 97, 105, 106, 110, 151, 162, 165, 171, 183, 281, 290, 291, 294, 296, 313, 356, 358, 364, 365 excitation, 29, 30, 32, 46, 127, 130, 131, 132, 134, 144, 196, 210

Index exercise, 254 exploitation, 258 exponential functions, 197, 309 exposure, viii, 83, 85, 104, 106, 213, 214, 357 extinction, 169, 222, 223, 224 extrusion, 273

F fabrication, viii, ix, x, xii, 15, 33, 44, 181, 196, 205, 206, 213, 214, 216, 219, 226, 227, 229, 257, 272, 273, 355 Fabry-Perot filters, xii, 335 failure, xii, 84, 222, 355, 356, 358, 359, 360, 362, 363, 364, 365, 366 family, xi, 303, 315, 316, 317, 318, 320, 321, 328 fatigue, xii, 355, 356, 358, 359, 361, 364, 365, 366, 367, 368 feedback, xi, 54, 56, 161, 162, 163, 170, 182, 331, 335, 336, 341, 342, 352 FFT, 241 fiber aging, 357 fiber optics, 101, 104, 272, 366 fibers, vii, viii, ix, x, xii, 3, 4, 5, 6, 7, 8, 9, 15, 16, 17, 18, 21, 28, 29, 32, 34, 35, 37, 41, 42, 45, 51, 53, 55, 58, 59, 68, 72, 83, 84, 86, 87, 88, 89, 91, 92, 96, 97, 98, 104, 105, 106, 107, 108, 111, 119, 120, 121, 123, 124, 132, 133, 139, 154, 163, 167, 187, 188, 190, 191, 227, 228, 232, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 269, 270, 271, 272, 273, 274, 275, 298, 299, 302, 311, 312, 316, 332, 333, 340, 355, 356, 357, 358, 359, 360, 361, 362, 364, 365, 366, 367, 368 fibre laser, 206 fibrinogen, 27, 30 film(s), 144, 215, 261, 273 filters, xii, 4, 109, 165, 166, 206, 209, 218, 219, 227, 228, 239, 335, 338, 350, 352 financial support, 309, 366 first generation, x, 257, 258 flame, 33, 34, 44, 208 flatness, 54, 69, 219, 220 flexibility, 15, 84, 110, 212, 257, 258, 274, 275 flight, 148 fluctuations, vii, 15, 55, 166, 215, 227, 286, 356 fluid, 280, 281 fluorescence, 26, 28, 29, 30, 31, 44, 48, 49, 56, 148, 150, 197 fluorine, 259 fluorophores, 48 focusing, 5, 119 food, 29, 44, 47 food poisoning, 29

373

Fourier, 125, 157, 219, 227, 239, 304, 310, 320 Fourier transformation, 125 four-wave mixing, xi, 169, 170, 311, 335, 336, 338, 339 France, 157, 161, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 276, 355, 366 freedom, xi, 240, 293, 301, 303 FTTH, 269, 275, 356 function values, 65 functionalization, 37 fusion, 19, 33, 34, 35, 38, 44, 190, 206, 207

G gases, 4, 316 Gaussian, 127, 143, 144, 146, 147, 149, 153, 176, 180, 181, 239, 240, 241, 246, 250, 302, 312 gel, 45, 259 generalization, 165, 170 generation, ix, x, xi, xii, 54, 57, 108, 111, 113, 119, 120, 140, 141, 143, 152, 154, 257, 258, 260, 271, 275, 306, 312, 316, 324, 335, 336, 342, 346, 352 geometrical parameters, 219 germanium, 93, 122, 261 Germany, 81 Ginzburg-Landau equation, x, 279, 280, 297, 298, 299 Ginzburg-Landau equation (CGLE), x, 279, 297 glass(es), vii, ix, xii, 4, 5, 7, 8, 30, 88, 91, 120, 121, 122, 127, 131, 132, 133, 142, 188, 190, 192, 196, 197, 198, 200, 205, 227, 257, 258, 259, 260, 261, 262, 270, 273, 336, 355, 356, 357, 358, 360, 361, 365, 366, 367, 368 glass transition temperature, 259 global communications, 83 glucose, 35, 44, 275 glycine, 38 glycol, 38, 39 gold, 26, 37, 44, 48 graph, 41, 77, 96, 127, 181 gratings, viii, xii, 4, 54, 72, 83, 84, 91, 92, 98, 101, 102, 104, 108, 109, 114, 115, 161, 170, 184, 213, 218, 227, 232, 331, 332, 335, 340 gravity, 147, 151 grazing, 120, 132, 260 Green’s function, 176 Griffith’s flaws, xii, 355, 356 groups, 30, 36, 37, 55, 132, 163, 166, 196, 358 growth, xii, 28, 44, 46, 154, 232, 258, 269, 298, 318, 321, 323, 355, 357, 367 growth rate, 318, 321, 323 guidance, 17, 120, 127, 132, 133, 134, 136, 139, 146, 154, 155, 273, 316, 332

374

Index

H halogen, 192 Hamiltonian, 293 HD, 35 HDPE, 271, 272 HE, 122 healing, 31, 47 heart disease, 30 heat(ing), vii, x, 15, 17, 19, 33, 35, 38, 44, 97, 130, 136, 150, 151, 207, 208, 209, 213, 214, 257, 258, 275 height, 152, 177 helium, 133 hemoglobin, 30 high power density, 164 hip, 24, 26, 113 homeland security, 44 homogeneity, 111, 112 Hong Kong, 301 host, 264, 271, 273, 274, 336 house(ing), 33, 115, 202 humidity, xii, 4, 45, 280, 355, 356, 357, 361, 364, 366, 367 hybrid, 53, 65, 66, 68, 77 hybridization, 32, 44, 48 hydrocarbons, 45 hydrochloric acid, 36 hydrofluoric acid, 33, 44, 104 hydrogen, 45, 213, 259, 316, 332 hydrogenation, 213 hydrolysis, 36 hydroxide, 36 hydroxyl groups, 36, 196 hypothesis test, 249, 250

I identification, 233, 238 ignition energy, 8 IL-6, 27, 31 illumination, 260, 275 images, 4, 124, 142 imaging, 31, 120, 126, 143, 144, 146, 147, 150, 155, 271, 357 immobilization, viii, 15, 28, 29, 36, 37, 40, 43, 47 immunity, viii, 83, 84 impairments, 238 implementation, xii, 55, 57, 62, 66, 68, 112, 113, 115, 228, 335, 336 impurities, 55 in situ, 164

in vivo, 31, 48 incidence, 16, 120, 132, 260 inclusion, 254, 282 India, 55 indication, 31 indices, 6, 121, 122, 212, 213, 214, 263, 264, 269 indirect measure, 84 industrial production, 365 industry, 357 inelastic, 55, 293 infarction, 30 infinite, 121, 127, 207, 212, 345 information processing, 206 information technology, 83 inhibition, 48 initial state, 246, 297 inoculation, 23 insertion, viii, 83, 84, 108, 114, 165, 188, 190, 200, 201, 206, 215, 216, 218, 220, 221, 223, 336, 346 insight, 18 instability, xi, 111, 285, 289, 298, 302, 315, 316, 320, 321, 323, 328 instruments, 191 integrated optics, 178 integration, 59, 60, 218, 225, 226, 238, 240, 254 intensity, 4, 28, 30, 44, 53, 56, 102, 120, 122, 123, 124, 126, 127, 128, 130, 131, 132, 133, 134, 138, 139, 141, 142, 144, 145, 146, 147, 150, 152, 153, 154, 165, 180, 181, 188, 211, 212, 217, 232, 305, 331, 362 interaction(s), viii, xi, 17, 29, 43, 52, 53, 54, 55, 57, 64, 76, 77, 133, 136, 152, 166, 171, 279, 293, 294, 298, 299, 302, 315, 316, 317, 324, 325, 326, 332, 336, 342 interface, 16, 53, 124, 132, 142, 209, 210, 258, 260 interference, 5, 84, 108, 113, 114, 115, 152, 174, 176, 185, 217, 232, 241 internet, 80, 187, 255, 269 interpretation, 21 interval, 232, 233, 236, 237, 240, 243, 246, 247, 249 inversion, 63, 183, 233, 342 investment, 52 ionization, 134, 139 ions, 4, 128, 131, 189, 190, 198, 200, 201, 219, 261, 336 IR, 35, 42, 80, 272, 273, 274 Islam, 81 isolation, 212, 216, 218, 220, 223, 224, 346 isotope(s), 136, 137 Italy, 51, 81, 161, 227 iterative solution, 165

Index

J Japan, 33, 134, 187, 276, 298, 312

K Karhunen-Loeve Series Expansion (KLSE), x, 231, 238, 239, 240, 241, 243, 253, 254 kernel, 240, 241, 253 kinetics, xii, 32, 40, 48, 355 Korea, 119, 134, 231, 276, 335

L laser(s), vii, viii, ix, x, xi, xii, 3, 4, 5, 6, 7, 8, 9, 29, 31, 42, 51, 53, 54, 56, 64, 65, 68, 69, 70, 71, 72, 73, 74, 75, 81, 108, 111, 112, 115, 119, 120, 121, 123, 124, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 187, 189, 191, 196, 201, 202, 206, 207, 208, 215, 221, 222, 226, 233, 234, 257, 258, 260, 261, 269, 272, 274, 275, 280, 281, 287, 297, 298, 299, 300, 316, 332, 333, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 357 laser radiation, x, 257, 258, 272, 275 lasing effect, 72, 73 lattices, 313 leakage, 53, 143, 172, 175, 177, 208, 258, 262, 264, 266, 316 lectin, 29 LED, 206 lens, 7, 120, 126, 147, 150, 155 leprosy, 31 lifetime, 84, 152, 196, 197, 198, 342, 356, 359, 364, 365, 366 light beam, 136, 146 light scattering, 136 light transmission, 18, 28, 39 likelihood, 238, 246, 249 limitation, 164, 227, 259 linear function, 109 linkage, 29, 38 links, vii, 53, 275, 311 liquid nitrogen, 336, 338, 352, 356, 357 liquid phase, 45 Listeria monocytogenes, 23, 28, 46, 47 literature, 16, 98, 164, 165, 166, 170, 200, 260, 274, 303, 305, 307

375

local area networks, 357 location, 18, 228, 304, 305, 342 long distance, 120, 221, 232, 320, 323 LPS, 29 luminescence, 189, 196, 197, 201 lysine, 28

M magnetic field, 150 malaria, 31 management, 52, 302, 303, 311 Manakov model, xi, 301, 303 manipulation, 119, 154, 206 manufacturer, 37 manufacturing, 207, 257, 258 mapping, 176, 236 market(s), 84, 258, 260, 275, 356, 357 Marx, 48 masking, 144 Massachusetts, 81 matrix, 59, 73, 81, 99, 100, 103, 108, 162, 173, 210, 211, 223 Maxwell's equations, 162 measurement, viii, 15, 32, 46, 47, 84, 98, 99, 100, 104, 108, 126, 148, 162, 169, 201, 202, 358, 359 measures, 60, 303, 305, 358 mechanical properties, vii, 5, 263 mechanical stress, 94 mechanical testing, 367 media, viii, xi, xii, 35, 47, 51, 169, 306, 313, 315, 331, 335, 336, 352 medical diagnostics, 44, 275 medicine, 84 melting, vii, 190 memory, 184 metals, 271 methanol, 26 microarray, 48 microcavity, 149 micrometer, 96 microorganisms, 28 microscope, 7, 31, 33, 105, 144, 146, 208 microscopy, 272 microspheres, 31 microwave, 108, 264, 271 military, 272 miniaturization, 5 mitochondria, 31 mixing, xi, 54, 169, 170, 183, 232, 302, 311, 335, 336, 338, 339, 341 model system, 305 modeling, 46, 59, 63, 68, 166, 183, 273, 308

376

Index

models, x, 162, 164, 165, 170, 231, 305, 316, 356, 358, 366, 367 modules, 52, 206, 212, 218, 222, 227 modulus, 261, 305, 310, 361 moisture, xii, 355, 367 molecular weight, 269 molecules, 15, 32, 43, 44, 49, 55, 259, 273, 332, 356 moon, 353 Morocco, 205 morphology, 357 motion, 131, 294 motivation, 305 multidimensional, 313 multiples, 54, 237 multiplicity, 280 multiplier, 128, 131, 134 muscles, 30 mutation, 65, 68 myocardial infarction, 30 myoglobin, 30, 47

N NADH, 24 nanometers, 222 National Science Foundation, 45 national security, 29 Nd, 4, 6, 164, 196 necrosis, 27, 31 neodymium, 164 Netherlands, 255, 367 network(ing), ix, x, 52, 114, 188, 205, 207, 216, 218, 226, 227, 269, 356, 366 neural networks, 65 New York, 79, 115, 155, 156, 157, 158, 255, 297, 299, 311, 313, 366 next generation, 108 NIR, 192 nitrogen, xi, 133, 335, 336, 338, 341, 352, 356, 357 nodes, 124, 146, 154 noise, viii, 51, 52, 54, 56, 57, 62, 63, 64, 75, 162, 165, 166, 167, 168, 169, 182, 184, 189, 191, 206, 218, 220, 221, 228, 238, 239, 240, 241, 246, 247, 250, 302 nonequilibrium, 299 nonlinear dynamics, 154 nonlinear optical response, 302, 331 nonlinear optics, 4, 279, 280, 281, 302 nonlinear Schrödinger model, xi nonlinear systems, 297 normalization constant, 127 numerical analysis, 123, 262 numerical aperture, 6, 7, 168, 190, 198, 259

numerical computations, 65

O observations, 244, 366 OFS, 188 oligomers, 48 one dimension, 233, 234 On-Off Keying (OOK), x, 231, 232, 233, 238 operator, 65, 171, 173, 304, 309 optical communications, ix, 4, 6, 76, 83, 91, 115, 120, 164 Optical Differential Phase Shift Keying (oDPSK), x, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 245, 246, 247, 248, 249, 251, 253, 254, 255 optical fiber, vii, viii, ix, x, xi, xii, 4, 5, 6, 7, 15, 17, 30, 31, 33, 34, 45, 46, 48, 51, 53, 54, 55, 57, 72, 76, 84, 85, 87, 102, 119, 120, 121, 123, 124, 127, 132, 136, 137, 138, 139, 146, 151, 154, 163, 164, 165, 183, 187, 188, 227, 232, 234, 257, 258, 259, 268, 269, 271, 272, 275, 279, 297, 302, 311, 312, 315, 316, 333, 335, 342, 355, 356, 357, 358, 361, 364, 365, 366, 367, 368 optical gain, 56, 221 optical polarization, 222 optical properties, 55, 56, 84 optical pulses, 163, 233, 281, 305, 311 optical solitons, 280, 302, 332 optical systems, 215, 308 optics, vii, ix, 4, 83, 101, 104, 109, 115, 119, 120, 126, 178, 207, 272, 279, 280, 281, 302, 315, 366 optimization, viii, ix, 52, 60, 64, 65, 68, 69, 76, 77, 83, 115, 225, 244, 273 optimization method, 65 ordinary differential equations, 198 orientation, 98, 236 orthogonality, 18, 19, 233 OSA, 69, 94, 167, 192, 193, 201, 208, 228 oscillation, 153, 162, 165, 166, 170, 209 oscillograph, 8 oxides, 365 oxygen, 357

P palladium, 45, 46 PAN, 192 parameter, xi, 6, 17, 41, 71, 177, 199, 219, 266, 280, 281, 287, 289, 290, 291, 294, 295, 301, 303, 305, 307, 309, 310, 316, 358, 359, 360, 362 Paris, 276

Index particles, 154, 275 particulate matter, 21 partition, 246 passive, viii, ix, x, 54, 83, 84, 165, 166, 205, 224, 269, 271, 272, 275 pathogens, 28, 44, 48 PBC, 113, 222, 223, 224 PCF, 169, 170, 262, 316, 332, 338, 339 PDEs, 281 penalties, x, 70, 231, 233 performance, x, 7, 8, 62, 63, 65, 113, 115, 162, 165, 166, 169, 171, 182, 189, 218, 219, 220, 228, 231, 232, 233, 236, 238, 239, 240, 241, 242, 243, 244, 247, 248, 254, 255, 258, 259, 260, 275 periodicity, viii, 83, 89, 166, 265 permit, 290, 304, 305 permittivity, 172, 175, 177 pH, 38, 39, 42 phase shifts, 234, 236 phase transitions, 280 phonons, 55, 56, 63 phosphate, 28 photoelastic effect, 87, 110 photographs, 91, 105 photonic crystal fiber, vii, 258, 262, 273, 316, 332, 333, 338 photonic crystal fiber (PCF), 338 photons, 55, 56, 151, 189, 194, 196, 197, 198 photosensitivity, 84, 213, 227 physics, viii, 15, 44, 183, 279, 281, 297, 305, 309, 315 pitch, 343 plane of polarization, 141 plasma, 23, 26, 30, 279, 297 plasminogen, 30 plastics, 271 PM, 69, 92, 223, 226, 346, 351 PMMA, x, 257, 258, 259, 260, 262, 263, 264, 266, 268, 269, 270, 275 POFs, x, 257, 258, 259, 260, 275 polarity, 238 polarization, vii, ix, x, xi, xii, 57, 58, 64, 83, 84, 87, 88, 89, 90, 94, 96, 97, 98, 99, 100, 102, 103, 106, 108, 109, 110, 111, 112, 113, 114, 115, 140, 141, 148, 165, 166, 167, 169, 170, 171, 172, 173, 175, 176, 183, 206, 216, 218, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 231, 264, 274, 275, 301, 303, 335, 337, 339, 340, 342, 343, 344, 345, 346, 347, 348, 349, 351, 352 polarized light, 87, 88 polycarbonate, 260, 261, 271 polyethylene, 271, 273 polyimide, 260

377

polymer(s), x, 257, 258, 259, 260, 261, 262, 263, 268, 269, 272, 273, 274, 275, 357, 361, 365, 367 polymer molecule, 259 polymeric materials, 257 polymerization, 259, 269 polymethylmethacrylate, x, 257 polystyrene, 22, 23, 29 poor, 35, 264, 338 population, 63, 65, 66, 68, 165, 198, 342 population size, 66 ports, 207, 223, 224 Portugal, 48, 51, 81, 83, 279 positive correlation, 194, 201 positive feedback, 56 power, vii, ix, x, 3, 4, 5, 6, 7, 8, 9, 19, 28, 30, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 88, 111, 114, 115, 132, 134, 135, 138, 139, 142, 146, 147, 149, 150, 152, 153, 162, 163, 164, 165, 167, 169, 170, 176, 177, 179, 182, 187, 188, 189, 191, 192, 193, 194, 195, 196, 197, 199, 200, 201, 202, 206, 207, 208, 209, 210, 211, 216, 218, 219, 220, 221, 222, 225, 226, 227, 234, 254, 257, 258, 260, 261, 263, 265, 275, 302, 311, 316, 338, 342, 344, 346, 347, 348, 356, 357 prediction, 290, 321, 356 pressure, viii, 4, 15, 45, 68, 131, 149, 213, 227, 280, 366 prices, 68 probability, xii, 7, 173, 197, 198, 238, 240, 247, 250, 253, 355, 358, 359, 362, 365 probability density function, 240 probe, 27, 29, 32, 57, 61, 63, 64, 67, 68, 69, 70, 72, 75, 77, 148, 150 production, viii, 83, 84, 91, 108, 268, 365 production costs, 84 production technology, viii, 83, 108 progesterone, 31, 47 prognosis, 49 program(ming), 19, 35, 65, 79, 302 propagation, ix, 18, 19, 52, 61, 65, 74, 75, 83, 84, 85, 87, 89, 108, 109, 122, 127, 144, 146, 147, 165, 171, 172, 177, 189, 191, 206, 208, 210, 211, 225, 226, 257, 258, 260, 262, 271, 281, 283, 285, 287, 288, 289, 291, 292, 294, 295, 296, 298, 299, 302, 305, 311, 312, 316, 317, 321, 322, 331 propane, 33 protein(s), viii, 15, 24, 25, 26, 28, 29, 30, 31, 42, 44, 46, 47 protocol, 32, 37, 39 prototype, 54 PTT, 367

378

Index

pulse(s), x, xi, 6, 108, 128, 131, 163, 164, 165, 166, 169, 183, 196, 232, 233, 238, 241, 243, 244, 279, 280, 281, 283, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 308, 311, 312, 316, 324, 331, 333 pumps, viii, 51, 53, 54, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 70, 74, 76, 77, 78, 79, 108, 112, 221, 222, 226 purification, 273 pyrene, 31

Q QED, 136, 139, 154 quantum phenomena, 171 quantum well, 161 quartz, 26

R radial distribution, 153 radiation, x, 4, 7, 80, 149, 166, 171, 172, 176, 179, 198, 213, 214, 257, 258, 260, 271, 272, 275, 302, 320, 326 radio, 114 radius, 6, 17, 18, 19, 33, 121, 127, 144, 145, 146, 147, 162, 171, 172, 176, 177, 181, 182, 260, 264, 265, 269 Raman, viii, xi, xii, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 163, 164, 165, 182, 206, 207, 218, 219, 221, 222, 227, 228, 229, 232, 275, 299, 305, 312, 316, 332, 335, 336, 342, 345, 346, 347, 349, 350, 351, 352 Raman and Brillouin scattering, 55 Raman spectroscopy, 55 Raman-scattering, 312 range, vii, x, 7, 28, 29, 30, 35, 41, 56, 58, 64, 65, 76, 84, 102, 108, 110, 132, 133, 162, 163, 166, 167, 168, 180, 182, 184, 191, 193, 207, 208, 212, 214, 215, 221, 222, 224, 226, 228, 234, 249, 257, 258, 261, 265, 266, 267, 268, 269, 271, 272, 274, 280, 285, 289, 293, 317, 319, 323, 343 reactant, 41 reactive sites, 41 reality, 56, 280, 293 reasoning, 305, 307, 310 reception, 76, 77 receptors, 48 recognition, 29, 32, 44 recombination, 175, 217, 342

recovery, 167 recurrence, 297 redistribution, 176 reduction, 40, 53, 54, 61, 131, 232, 244, 259, 262, 281, 293 redundancy, 238 reflection, vii, viii, ix, x, 7, 16, 70, 72, 83, 85, 90, 91, 93, 94, 96, 97, 98, 101, 102, 103, 105, 106, 108, 113, 136, 142, 168, 182, 189, 191, 216, 227, 257, 258, 260, 266, 316, 343, 345, 346, 350 reflectivity, 71, 85, 86, 90, 114, 172, 177, 258, 260, 261, 273, 349 refraction index, 85, 88, 89 refractive index(ices), viii, ix, 5, 6, 8, 15, 16, 17, 21, 35, 40, 42, 43, 44, 83, 84, 85, 87, 89, 121, 122, 132, 146, 168, 175, 190, 212, 213, 214, 215, 218, 220, 232, 258, 259, 260, 261, 264, 265, 266, 268, 271, 281, 343 regenerate, 36, 218 regeneration, 52 rehydration, 39 reinforcement, 356 relationship, 8, 9, 19, 146, 147, 150, 172, 173, 177, 178, 180, 199, 267, 268 relaxation, 59, 63, 175, 196, 317, 320, 364, 365 reliability, xii, 5, 355, 356, 357, 365, 366, 367 remote sensing, 44 reparation, viii, 15 resistance, 84, 258, 259, 275, 357 resolution, 61, 84, 119, 144, 162, 167, 192, 252, 272 resonator, 162, 163, 166, 172, 176, 177, 178, 179, 183 response time, 221 RF, 346 rings, viii, 51, 166, 262, 263, 264, 273, 361 RNA, 27, 32 robustness, 44, 249 Romania, 355 room temperature, 8, 36, 111, 338, 339, 365 root-mean-square, 139 roughness, ix, 161, 162, 171, 172, 176, 177, 178, 179, 180, 181, 182, 262 routines, 304 routing, 108, 205 Royal Society, 309 rubidium, 122, 134 Russia, 55

S SA, 43, 44, 51 safety, 5, 53 Salmonella, 22, 28, 46

Index sample(ing), viii, 15, 16, 17, 23, 36, 37, 39, 40, 42, 43, 44, 94, 102, 105, 154, 196, 197, 233, 237, 239, 252, 343, 356, 359, 360, 363 sapphire, 8, 131, 147, 261, 271, 333 saturation, ix, 43, 44, 54, 120, 161, 162, 168, 174, 176, 179, 221, 342 scaling, 164 scattered light, 55, 70, 132, 143 scattering, viii, ix, 28, 39, 51, 53, 54, 55, 56, 57, 62, 79, 136, 143, 151, 153, 162, 164, 174, 176, 177, 187, 189, 191, 192, 193, 194, 197, 200, 201, 219, 221, 232, 264, 275, 299, 305, 312, 316, 332, 336, 342 schema, x, 231, 233, 234, 235, 236, 237, 241, 247, 249, 254 scholarship, 79 Schrödinger equation, 279, 281, 298 science, vii, 271, 302 scientific community, 302 search, 65, 166 security, 29, 44 seed(ing), 54, 74, 165, 320 selecting, 7, 63, 235 selectivity, 32, 40, 44 self-phase modulation, 183 semiconductor, ix, xi, 4, 54, 161, 162, 168, 169, 170, 171, 172, 173, 178, 180, 182, 184, 185, 335, 336, 342, 348 semiconductor lasers, 173 sensing, viii, ix, 15, 16, 21, 30, 31, 44, 45, 46, 83, 84, 98, 104, 115, 169, 182, 205, 272, 274, 275 sensitivity, 28, 29, 35, 41, 44, 94, 96, 97, 98, 114, 162, 170, 178, 182, 217, 227, 244, 249, 350 sensors, vii, viii, ix, xi, 4, 15, 16, 21, 41, 44, 45, 46, 47, 83, 84, 93, 98, 108, 111, 115, 260, 272, 275, 335, 357 separation, 93, 98, 136, 137, 213, 293, 294, 306, 317, 324, 329, 343 sepsis, 29 series, 134, 219, 240, 268, 304, 310, 363 serum, 23, 26, 31, 45 shape(ing), 17, 57, 88, 108, 127, 146, 151, 180, 208, 219, 280 sharing, 167 shock, 275 sign(s), 57, 120, 122, 237, 238, 250, 251, 252, 253, 288, 294 signaling, 49, 232, 233, 236, 237, 243, 246, 247, 249 signals, 5, 28, 53, 54, 57, 59, 61, 62, 63, 64, 67, 68, 69, 70, 73, 77, 98, 114, 130, 134, 148, 150, 164, 187, 188, 189, 206, 207, 216, 218, 221, 226, 236, 250, 302 signal-to-noise ratio, 54, 165

379

silane, 32 silica, xii, 5, 6, 16, 23, 24, 25, 26, 31, 32, 48, 87, 93, 122, 182, 183, 188, 191, 199, 202, 206, 213, 215, 221, 226, 227, 257, 258, 260, 261, 262, 270, 271, 273, 316, 332, 333, 336, 355, 356, 357, 361, 364, 365, 366, 367, 368 silicon, 154, 271 silver, 6, 183, 273 simulation, viii, 15, 18, 19, 20, 21, 64, 66, 67, 68, 69, 73, 74, 77, 81, 86, 90, 92, 110, 145, 149, 152, 162, 171, 178, 294, 341 SiO2, 190 sites, 7, 29, 41, 52, 71, 216 smoothing, 366 sodium, 36 sodium hydroxide, 36 software, 33, 34 solitons, x, xi, 183, 279, 280, 281, 286, 287, 288, 289, 290, 293, 294, 297, 298, 299, 302, 303, 305, 306, 307, 311, 312, 313, 315, 316, 317, 318, 319, 320, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333 solvent, 37 species, 28, 29, 196, 201, 280, 293 specificity, 40 spectral component, 58 spectroscopy, xi, 30, 55, 162, 271, 335, 357 spectrum, ix, 35, 56, 58, 65, 69, 70, 71, 72, 74, 76, 77, 83, 91, 92, 93, 94, 98, 101, 102, 103, 106, 112, 131, 132, 136, 137, 140, 163, 166, 167, 188, 192, 201, 216, 218, 219, 226, 270, 291, 302, 315, 316, 317, 336, 342, 348, 349, 350 speed, 18, 65, 162, 187, 188, 205, 208, 217, 227, 255, 302, 316, 361, 362 speed of light, 18, 316 spin, 357 sprouting, 22, 46 stability, x, xi, 44, 59, 68, 73, 149, 165, 167, 205, 285, 288, 291, 293, 298, 299, 309, 315, 316, 318, 320, 321 stabilization, 72, 73, 74, 75, 178, 352 stages, 247 standard deviation, 179, 180, 182 standardization, 259 standards, 68, 149 staphylococcal, 47 Staphylococcus, 22, 29, 47 statistics, 356, 358 steel, 271 sterile, 37 stimulant, 29 stock, 39 storage, 272, 316

380

Index

strain, 40, 45, 84, 92, 93, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104, 107, 108, 115, 348, 350, 356 strength, xii, 7, 17, 32, 34, 58, 132, 275, 286, 355, 356, 357, 359, 360, 364, 365, 366, 367 stress, xii, 7, 87, 88, 91, 93, 94, 104, 106, 107, 109, 110, 115, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366 stress intensity factor, 362 stretching, 259 strong interaction, 29 structural relaxation, 365 subtraction, 310 sulphur, 273 sun, 116, 183, 202, 277 supply, 44, 192, 280, 366 suppression, 162, 167, 170, 185, 338 surface layer, 43 susceptibility, 56 suspensions, 35 switching, 108, 170, 331, 341, 343, 344, 347, 351, 352 symbols, 217, 236, 237, 238, 244, 246, 247, 249, 250, 251, 253 symmetry, 211 synchronization, 163 synthesis, 32 systems, viii, ix, x, xi, xii, 51, 52, 53, 54, 55, 56, 63, 69, 70, 75, 76, 77, 78, 79, 83, 84, 108, 114, 162, 164, 165, 169, 171, 182, 187, 188, 205, 206, 207, 215, 221, 222, 231, 232, 234, 236, 238, 239, 241, 243, 244, 246, 247, 254, 255, 271, 275, 279, 280, 281, 293, 297, 298, 299, 301, 303, 305, 306, 308, 311, 312, 316, 332, 335, 336, 352, 365, 366

T technical assistance, 366 technology, vii, viii, 3, 7, 51, 52, 55, 68, 83, 84, 108, 187, 213, 222, 232, 259, 271, 302, 356 teflon, 273 telecommunication networks, xii, 206, 218, 355, 356, 361, 365 telecommunications, x, 53, 56, 111, 162, 166, 205 temperature, viii, x, xii, 5, 7, 8, 15, 33, 36, 45, 57, 58, 84, 92, 93, 97, 98, 99, 100, 101, 102, 103, 104, 107, 108, 110, 111, 115, 140, 148, 150, 152, 166, 169, 184, 191, 206, 208, 212, 213, 214, 215, 219, 220, 226, 257, 258, 259, 260, 261, 272, 280, 338, 339, 350, 355, 356, 357, 359, 361, 362, 364, 365, 366 temperature dependence, 57, 97, 102 temperature gradient, 110 tensile strength, 359

tensile stress, 356, 357 tension, 33, 34 theory, vii, x, 3, 18, 85, 86, 121, 124, 125, 127, 144, 154, 162, 171, 249, 279, 281, 286, 291, 297, 299, 302 therapy, 31, 275 thermal expansion, 213 threat, 29, 44 threshold(s), vii, 3, 6, 8, 9, 53, 56, 72, 74, 132, 138, 139, 163, 164, 168, 170, 178, 179, 180, 181, 233 threshold level, 163 thrombin, 30, 31, 48 tics, x, 231, 233, 234, 235, 236, 237, 241, 247, 249, 254 time(ing), viii, xi, xii, 5, 8, 30, 32, 33, 37, 41, 42, 43, 45, 46, 47, 52, 57, 59, 61, 64, 65, 66, 67, 72, 84, 104, 105, 106, 108, 112, 113, 114, 115, 148, 150, 151, 152, 162, 163, 165, 170, 171, 175, 187, 188, 192, 197, 198, 199, 213, 214, 217, 221, 228, 233, 239, 244, 254, 258, 262, 280, 281, 293, 299, 301, 303, 309, 311, 317, 324, 355, 358, 359, 360, 364, 365, 366 tin, 175 TIR, 16 TNF-alpha, 27, 31 Tokyo, 158, 203 topology, 68 total energy, 320, 321 total internal reflection, vii, x, 16, 257, 258, 316 toxin, 23, 29 tracking, 183 trade, 258 traffic, 53, 187, 269 trajectory, 294, 349 transference, 81 transformation, 125, 288 transition(s), 130, 134, 138, 148, 150, 151, 164, 173, 189, 190, 197, 198, 199, 208, 245, 246, 259, 280 transition rate, 198 transition temperature, 259 translation, 93, 96 transmission, vii, viii, x, 6, 7, 16, 18, 19, 20, 21, 28, 33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 51, 52, 53, 54, 55, 57, 58, 63, 70, 71, 75, 76, 77, 78, 84, 114, 115, 136, 137, 138, 139, 182, 187, 188, 205, 206, 207, 208, 211, 212, 213, 215, 217, 220, 221, 223, 226, 228, 231, 232, 244, 246, 247, 255, 257, 258, 259, 260, 261, 262, 263, 264, 268, 269, 270, 271, 272, 273, 274, 275, 280, 293, 297, 298, 300, 302, 311, 343, 345, 347, 356 transmission path, 78 transmits, x, 8, 21, 232, 257 transparency, 316, 331, 332

Index transport, 30, 48 transverse section, 91, 94, 105 trend, 43, 106, 130 triggers, 56 tuberculosis, 31 tumor necrosis factor, 27, 31 tumors, 275 tunneling, 176 typhoid, 31

381

vector, 127, 172, 173, 175, 176, 183, 210, 211, 250, 313 vehicles, 84 velocity, xi, 57, 87, 105, 133, 134, 139, 150, 163, 172, 212, 281, 291, 294, 295, 296, 297, 301, 302, 305, 316, 317, 318, 321, 322, 324, 325, 327, 329, 330, 362 vibration, 259 viscosity, 4 voice, 4, 302

U W UK, 81, 301 ultraviolet light, viii, 83, 85 uniform, 16, 17, 18, 21, 22, 23, 68, 85, 86, 87, 110, 152, 218, 331, 359 urine, 23 users, 112, 113, 114, 366 UV, 213, 214, 227 UV radiation, 213, 214

V vacuum, 6, 36, 124, 128, 131, 132, 134, 142, 175, 356, 357 Vakhitov-Kolokolov criterion, 318 valence, 173 values, 19, 20, 21, 59, 60, 65, 67, 68, 70, 76, 87, 92, 96, 97, 99, 100, 101, 104, 108, 126, 139, 147, 162, 176, 178, 179, 180, 181, 200, 237, 244, 247, 248, 249, 252, 267, 273, 282, 283, 290, 291, 293, 294, 295, 308, 358, 359 vapor, 45, 131, 148, 366 variability, 214 variable(s), xi, 4, 59, 174, 220, 240, 253, 301, 302, 303, 306, 309, 312, 339, 364 variance, 240, 254 variation, xi, 92, 93, 94, 98, 142, 167, 200, 219, 226, 229, 252, 269, 287, 301, 303, 315, 362

water absorption, 224 wave number, 6, 122, 132, 304, 305, 309 wave propagation, 257, 305 wavelengths, viii, xii, 42, 51, 53, 54, 63, 64, 68, 71, 72, 75, 76, 77, 87, 89, 97, 98, 106, 107, 109, 112, 113, 114, 163, 170, 188, 193, 208, 213, 216, 220, 221, 222, 260, 261, 262, 270, 272, 335, 336, 337, 338, 339, 342, 343, 346, 347, 348, 349, 350, 352 welding, 272 wells, 177 windows, 7, 8, 53 workers, 164, 259 wound healing, 31 writing, 350

X X-axis, 95, 97

Y Y-axis, 95, 97, 102 yield, 304, 305 ytterbium, 164, 333

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