OPTICAL SOLITON GENERATION IN FIBER OPTICS: FREE & FORCED .
OPTICAL SOLITON GENERATION IN FIBER OPTICS:FREE & FORCED NONLINEAR SCHRÖDINGER EQUATIONKEE BOON LEEA dissertation submitted in partial fulfillment of therequirements for the award of the degree ofMaster of Science (Engineering Mathematics)Faculty of ScienceUniversiti Teknologi MalaysiaJANUARY 2013
iiiMy utmost dedication to mum and dad.Thank you always being there for me.I love you.
ivACKNOWLEDGEMENTFirst at all, words cannot express my gratitude towards my supervisor, Assoc.Prof. Dr. Ong Chee Tiong. Through his enthusiasm, inspiration and great efforts toguide me throughout my dissertation journey, he helped to make Mathematics fun.He has also provided unlimited encouragement, motivation, sound advice andvaluable suggestions in guiding me during the process of completing this researchsmoothly.I would also like to thank my loving family members especially my parentswho have given me their unflagging love and moral support, which has provided meabsolute confidence and courage to confront problems while conducting this research.In addition, I would like to thank all the librarians of Perpustakaan SultanahZanariah for their assistance in retrieving the relevant literature guidance.I would like to extend my sincere appreciation to all my friends especially Mr.Darrel Wee Chia Kee who have lent me a hand in providing valuable and helpfulcomments in completing this research.Their supports have proven to be verybeneficial.Last but not least, my gratitude goes to those who are involved directly orindirectly in helping me throughout the tough hurdle of writing this dissertation.
vABSTRACTFuture telecommunication will depend on effective data transmission, highquality video encoding, and efficient networking using optical solitons in fiber optic.At this moment, we encounter slow data transmission, fiber loss and chromaticdispersion that can be considered as forcing terms. Fiber optics is a man-made toolthat has changing refractive index and this cause fiber loss. Nonlinear Schrödinger(NLS) equation which is a nonlinear partial differential equation can models the nonforced system effectively that combines the effect of nonlinearity and dispersion. Inthis research, a numerical method that consists of semi-implicit Pseudo-Spectralmethod scheme will be implemented to solve NLS equation. Comparing the resultsfrom analytical solution between numerical solution of NLS equation to determine anaccurate and stable code so that it can be used to solve forced nonlinear Schrödinger(fNLS) equation that models forcing system. MATLAB computer programmingwhich is user friendly will be used to implement the numerical scheme that producesvarious graphical outputs to simulate the propagation of solitons.
viABSTRAKTelekomunikasi masa depan akan bergantung kepada keberkesananpenghantaran, pengekodan video yang berkualiti tinggi dan rangkaian cekap yangmenggunakan soliton optik dalam gentian optik.Pada masa ini, kita dapatipenghantaran data yang lambat, kehilangan gentian dan serakan kromatik yang bolehdianggap sebagai rintangan.Gentian optik adalah alat buatan manusia yangmempunyai indeks biasan yang berubah-ubah dan menyebabkan kehilangan gentian.Persamaan tak linear Schrödinger (NLS) merupakan satu persamaan pembezaansepara tak linear yang boleh memodelkan sistem bebas rintangan dengan berkesanyang memggabungkan kesan ketaklinearan dan penyebaran.Dalam kajian ini,kaedah berangka (pseudo-Spectral) dikemukaan untuk menyelesaikan persamaanNLS. Hasil daripada penyelesaian analitikal antara berangka untuk persamaan NLSyang bebas dibandingkan untuk menentukan kod yang tepat dan stabil supaya isboleh digunakan bagi menyelesaikan sistem paksaan (fNLS). MATLAB adalah satupengaturacaraan yang mesra pengguna dan dapat menjana pelbagai grafik yangmenunjukkan interaksi solitons.
viiTABLE OF iACKNOWLEDGEMENTSivABSTRACTvABSTRAKviTABLE OF CONTENTSviiLIST OF TABLESxiLIST OF FIGURESxiiLIST OF SYMBOLSxviiLIST OF kground of Study21.3Statement of the Problem31.4Objectives of the Study31.5Scope of the Study31.6Significance of the Study41.7Outline of the Study51.8Conclusion6
viii23LITERATURE REVIEW72.1Introduction72.2Discovery of Solitons72.2.1 Properties of Solitons92.2.2 Spatial Solitons and Temporal Solitons102.3Fiber optics102.4Reviews of Fiber Optics122.5Advantages of Fiber Optics152.6Optical Communication System162.7Fiber Losses172.7.1 Absorption Loss182.7.2 Rayleigh Scatter192.7.3 Bending Loss192.7.4 Insertion Loss (IL)202.7.5 Return Loss (RL)20NONLINEAR SCHRÖDINGER EQUATION223.1Mathematical Modeling of Fiber Optics223.2Analytical Solution of Nonlinear Schrödinger33(NLS) Equation3.33.4Parameter Testing443.3.1 Parameter443.3.2 Parameter463.3.3 Parameter48Conclusion50
ix4DEVELOPMENT OF NUMERICAL51COMPUTATION4.1Numerical Method514.2Flow of the Computational Process534.3Defining Input Data574.4MATLAB SimBiology584.5Numerical Solution for Nonlinear Schrödinger62(NLS) Equation4.64.5.1 Initialization Block654.5.2 Forward Scheme Block67Comparison between Analytical and Numerical73Solution4.75Von Neumann Stability Analysis78FORCED NONLINEAR SCHRÖDINGER82EQUATION5.1Numerical Solution for Forced Nonlinear82Schrödinger (fNLS) Equation5.1.1 Dirac-Delta Forcing5.2825.1.1.1Initialization Block845.1.1.2Forward Scheme Block85Other types of forcing term935.2.1 Initialization Block935.2.2 Forward Scheme Block945.2.2.1Gaussian Forcing955.2.2.2Secant Hyperbolic Forcing99
x5.36Experiment for Input Dataand103SUMMARY AND nclusion1076.4Suggestions and Recommendations108REFERENCES111APPENDIX114
xiLIST OF TABLESTABLE NO.TITLEPAGE4.1Percentage error for Pseudo-Spectral method775.1The observation for ratio103data ,andto investigate input
xiiLIST OF FIGURESFIGURE NO.2.1TITLEPAGEConcept of solitary waves was first introduced7by Scott Russell2.2John Scott Russell82.3The two solitary waves before, during and9after a collision2.4The fiber optic is made up of 5 distinct layers112.5Principles of reflection and refraction122.6Total internal reflection122.7Spectral Distribution of Losses for a Typcal18Multimode Silica 9Light Scattered during Transmission193.1Optical Fiber22
xiiiFIGURE NO.3.2TITLEPAGE41Travelling wave solution of NLS at time,and3.3Envelope Solitons423.4Graphical Output for at time,43and3.5Graphical Output for at,46and3.6GraphicalOutputfor at,48and3.7Graphical Output for at,50and4.1Flow of computational process534.2Initialization block554.3Forward scheme564.4Computational structures for forward scheme57of NLS and fNLS4.5Import Wizard59
xivFIGURE NO.TITLEPAGE4.6Import data from MATLAB to SimBiology604.7Main frame of Simbiology614.8Zero boundary condition at624.9Coordinate transformation624.10Graphical output of numerical results for NLS70equation4.11Main frame of numerical results for NLS71equation4.12Graphical Output for NLS Solution at4.13Transformationsolution75Graphical results for NLS equation for nd(b)NumericalSolution5.1Graphical output for Dirac-delta forcing at87
xvFIGURE NO.5.25.3TITLEPAGE88Main frame of Dirac-delta forcing forGraphical Output of Dirac-delta forcing for89at5.4Graphical output for Dirac-delta forcing at905.5Main Frame of Dirac-delta forcing for915.6Graphical output of Dirac-delta forcing for92at,,,,,5.7Graphical output for Gaussian forcing965.8Main Frame for Gaussian forcing975.9Graphical output for Gaussian forcing at985.10Graphical output for secant hyperbolic forcing1005.11Main frame for secant hyperbolic forcing101
xviFIGURE NO.5.12TITLEPAGEGraphical output for secant hyperbolic forcing102at5.13The relationship between5.14Graphical Output for fNLS solution with (a), (b)5.15and104, and (c)Graphical output for fNLS solution with(a)103105and (b)6.1Compensation of pulse dispersion via DCF1096.2Waveform after 50km of input109
xviiLIST OF SYMBOLS2Dt2-Dimensional-Envelope function-Time propagation-Distance-Nonlinearity coefficient-Dispersion coefficient-Fiber attenuation coefficient-Imaginary unit-Power in the fiberNumber of loopsNumber of discrete points
xviiiLIST OF APPENDICESAPPENDIXATITLEPAGEMATLAB Code: Analytical Solution for NLS114EquationBMATLAB Code : Numerical Solution for NLS115EquationCMATLAB Code: Numerical Solution for fNLSEquation118
CHAPTER 1INTRODUCTION1.1IntroductionWe now live in a world where computers, television, cell phones and othertools are a necessity. This advancement has been made possible via the progressionof information technology.Fiber optics technology has played a major role incontributing to this rapid advancement of technology. Without us realizing it, fiberoptics is an essential part of our everyday lives and is becoming an increasinglycommon replacement for traditional standard copper wire. Both materials are used totransmit signals from one location to another via fiber optic cables. Transmission ofinformation includes data transmission, image transmission, and energy transmissionfrom one source to another. However, a fiber-optic line has significant advantagescompared to copper wire, including the ability to carry a larger amount of bandwidthover a greater distance at faster speeds, all for a lower maintenance cost and withincreased resistance to electromagnetic interference from objects, such as radios andother cables. Optical fiber is also the safest way in transmitting data as it does notleak data or information due to the fact that the transmissions signals are guidedthrough the optic fiber and not through copper wires like it's done in a cable.
21.2Background of StudyThere are two different materials used to fabricate optics which operate as aguide for the light waves to travel within the cables which carry information sentfrom one end to another. Polymeric fibers are commonly used for short distancetransfer and installations in rough surroundings, whereas glass fibers are used forhigh quality and long distance data transfer. The light transmitted through the fiberis confined due to total internal reflection within the material.Opticalcommunication use “wavelength division multiplexing with different wavelengths tocarry different signals in the same fiber”. In other words, information is encodedinto different wavelengths of light to allow information to travel in differentdirections without interference, therefore, making it possible for high speeds of datatransmission within one small strand of fiber optic cable. The information containedby fiber these cables travel with the light which reflects off the inner walls of thecable and is guided throughout a fiber optic circuit.Fiber optics can transmit information in the same way that copper wire cantransmit electricity. However, copper transmits only a few million electrical pulsesper second, compared to an optical fiber that carries up to 20 billion light pulses persecond. This means telephone, cable and computer companies can handle hugeamounts of data transfers at once, compared to the limited capabilities ofconventional wires. Fiber optic cable was developed due to the massive surge in thequantity of data over the past 20 years. Without fiber optics cable, the modernInternet and World Wide Web would not be possible. If you were to make a phonecall to Europe, traditionally the signal would go up to a satellite and then back downto Europe. With fiber optics, if the call is transmitted through a transatlantic fiberoptic cable, there is a direct connection.
31.3Statement of the ProblemThe nonlinear Schrödinger (NLS) equation describes the phenomena that isvery important nowadays; which is the propagation of waves in relation to the designof optical long-distance communications lines and all optical signal processingdevices for reliable and high-bit-rate transmission of information. We will determinehow the propagation of optical pulses in optical fiber can be modeled into NLSequation and also what are the main obstacles that affect the propagation of stablesoliton pulses. Besides that, we will also solve the forced nonlinear Schrödinger(fNLS) equation to minimize fiber loss.1.4Objectives of the StudyThe objectives of this research are:a) obtaining mathematical modeling of optical soliton transmission in fiberoptics.b) lly andnumerically.c) solving forced nonlinear Schrödinger (fNLS) equation numerically.1.5Scope of the StudyThe main focus of this study is nonlinear Schrödinger (NLS) equation, apartial differential equation which has the nonlinearity and dispersion effect given by: (1.1)Based on the physical model of fiber optics, the following assumptions are made:a) There is no fiber loss along this fiber.
4b) The shape will be maintained the whole time during propagation.If the fiber optics is facing the fiber loss, the NLS equation with an addedterm representing the optical loss is observed. Forced nonlinear Schrödinger (fNLS)equation (forced system) is defined in the equation below: where , and(1.2)are respectively the normalized parameters of the envelope function,the time propagation and the distance, meanwhile,andmagnitude of forcing,andis a function in terms ofare real constant, is ais an imaginaryunit.These equations focus on the physical problem of fiber optics.1.6Significance of the StudyThe results of this research play an important role because sophisticated datatransfer systems nowadays rely more and more on fiber optics for digital signaltransmission.Gbits/seconds.Central information superhighways can transmit up to 10High speed data transfer, highly reduced data loss, noelectromagnetic problems, small dimensions and a low weight are the main featuresthat characterize the use of fiber optic technology. There is a link between thephysical phenomenon and mathematic
6.1 Compensation of pulse dispersion via DCF 109 6.2 Waveform after 50km of input 109 . xvii LIST OF SYMBOLS 2D 2-Dimensional Envelope function- t - Time propagation Distance - . B MATLAB Code : Numerical Solution for NLS Equation 115 C MATLAB Code: Numerical Solution for fNLS Equation 118 . CHAPTER 1 INTRODUCTION
Fiber optic collimators are components designed to collimate/focus light exiting a fiber to a desired optical beam. G&High powered fiber optic collimators offer 's SM h high reliability with low optical loss. Ideal for use in fiber sensors and fiber lasers. The fiber collimators are available in several single mode fiber types with operating .
2.7 Fabrication of Optical Fiber 53 2.8 Drawing of Optical Fibers 54 2.9 Classification of Fiber Cables 57 2.10 The Selection of Fiber Cable 62 2.11 Installation Procedures 63 2.12 Performance of Optical Fiber Cables 64 2.12.1 Mini Bundle Cable 64 2.13 High Performance Optical Communications Cable 66 Problems 68 References 69 3.
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and 245 5 Pm with the protective coating and photosensitive optical fiber PS1250/1500 co-doped fiber, from FiberCore, with 125 1 P m diameter without coating. . values can be used in the design and modeling of optical fiber sensors were the fiber can be under some kind of stress, like FBG based sensors. 3. Optical fiber degradation behavior
and 245 5 m with the protective coating and photosensitive optical fiber PS1250/1500 co-doped fiber, from FiberCore, with 125 1 m diameter without coating. . values can be used in the design and modeling of optical fiber sensors were the fiber can be under some kind of stress, like FBG based sensors. 3. Optical fiber degradation behavior
From the above-mentioned features of using optical fiber link as a transmitted media, it is proven that it is the best communication link. The following sections review the different components that constitute the optical fiber communication system, namely the optical transmitter,the optical fiber links and the optical receiver. II.
lasers and optical fiber sensors, which are the subject of this chapter. An optical fiber is an optical waveguide in the shape of a filament and is generally made of glass (although it can also be made of plastic materials). An optical fiber is composed of three parts: the core, the cladding, and the coating or buffer. Fibers can be produced in a
Fiber damage, changes in the fiber wall structure, reduced single softwood kraft fiber strength and fiber deformations (curl, kinks and dislocations) all affected the fiber network properties. Mechanical treatment at the end of kraft cooking conditions resulted in fiber damage such that single fiber strength was reduced.
