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Dynamics of a Passively Mode-Locked Fiber Laser Containing a Long-Period Fiber Grating

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Dynamics of a Passively Mode-Locked Fiber Laser Containing a Long-Period Fiber Grating Dynamics of a Passively Mode-Locked Fiber Laser containing a Long-Period Fiber Grating by Abdullah S. Karar A Thesis submitted to the Faculty of Graduate Studies and Research in partial fulfilment of the requirements for the degree of Master of Applied Science Ottawa-Carleton Institute for Electrical and Computer Engineering Department of. Electronics Carleton University Ottawa, Ontario, Canada September 2007 Copyright © 2007 - Abdullah S. Karar Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Library and Bibliotheque et Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-33654-0 Our file Notre reference ISBN: 978-0-494-33654-0 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non­ sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission. In compliance with the Canadian Conformement a la loi canadienne Privacy Act some supporting sur la protection de la vie privee, forms may have been removed quelques formulaires secondaires from this thesis. ont ete enleves de cette these. While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. i* i Canada Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract A numerical investigation of the nonlinear dynamics of a passively mode-locked fiber laser containing a long period fiber grating was conducted. The model was based on the complex Ginzburg-Landau equation and the nonlinear coupled mode equations of the grating. The numerical results indicated the existence of passive mode-locking and autosoliton generation in the cavity of the laser. Both single and bound soliton pulse trains were found to exhibit period doubling bifurcations and a route to chaos as the saturated gain was increased. Furthermore, the presence of long period pulsation, soliton sidebands and possible coexisting attractors excited by multisoliton formation and soliton energy quantization was observed. The laser dynamics were presented through plotting the pulse energy against the linearly increasing gain, so obtaining bifurcation diagrams. Higher order nonlinear and dispersive effects, such as Raman self-frequency shift, self-steepening and third order dispersion, were found to change these bifurcation diagrams. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dedicated to Dr. Sayed Karar iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments I would like to acknowledge the academic and moral support of my supervisor pro­ fessor A lan Steele and co-supervisor professor Tom Smy. I would like to extend my thanks to Carleton university’s department of electronics, the Canadian Foundation for Innovation and Canada’s National Science and Engineering Research Council for financial support and availability of high computing facilities. Also, I would like to thank S. Lynch (Manchester Metropolitan University, United Kingdom) and J. Albert (Carleton University, Canada) for their valued input and useful discussions. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Abstract ii Acknowledgments iv Table of Contents v List of Figures viii 1 Introduction 1 1.1 Fiber Lasers........................................................................................................ 1 1.2 Thesis Motivation .............................................................................................. 6 1.3 C o n trib u tio n s ..................................................................................................... 7 1.4 Organization ..................................................................................................... 7 2 Laser M odel 9 2.1 Laser S y s te m ..................................................................................................... 9 2.2 Fiber E q u a tio n s ................................................................................................. 11 2.2.1 M axw ell’s E q u a tio n s ......................................................................... 11 2.2.2 Nonlinear Pulse Propagation ...................................................................... 13 2.2.3 Erbium-Doped Fiber Amplifier ..................................................... 18 2.2.4 Fiber Parameters ................................................................................ 23 2.3 LPFG E q u a tio n s .............................................................................................. 26 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3.1 Nonlinear Mode C o u p lin g ................................................................ 28 2.3.2 Pulse Shaping ........................................................................................ 32 3 Numerical Methods and Implementation 38 3.1 Split-Step Fourier M e th o d ............................................................................ 39 3.1.1 Fiber E q u a tio n .................................................................................... 39 3.1.2 LPFG E q u a tio n ..................................................................................... 42 3.2 Object Oriented Programming ...................................................................... 43 4 Laser Dynamics 48 4.1 Passive Mode-Locking .................................................................................. 49 4.2 Autosoliton G eneration .................................................................................. 52 4.3 Coexisting A ttra c to rs ...................................................................................... 56 4.3.1 Bifurcation D ia g ra m s ........................................................................... 56 4.3.2 Period Doubling ..................................................................................... 60 4.3.3 Long Period Pulsation ....................................................................... 61 4.3.4 Chaos Dynamics ................................................................................. 64 4.4 Dynamics of the LPFG ................................................................................... 67 4.5 Soliton Sidebands ............................................................................................. 70 5 High Order Effects 74 5.1 Higher Order Terms ...................................................................................... 74 5.2 Raman Self-Frequency S h i f t ......................................................................... 76 5.3 S e lf-S tee pening ................................................................................................ 78 5.4 T h ird Order D ispersion ................................................................................... 80 5.5 Ultrashort Pulse Generation ......................................................................... 82 6 Conclusion 84 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of References 86 Appendix A C++ Code 95 v ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures 1.1 Plot of the number of journal publications on fiber lasers between 1975- 2007. Source: Engineering Village - title search ......................................... 2 1.2 Schematic of a Fabry-Perot cavity fiber laser ............................................... 4 1.3 Schematic of a unidirectional ring cavity fiber laser .................................. 4 1.4 Schematic of a figure of eight ring cavity fiber laser................................... 4 2.1 Passive mode-locking laser model containing a LPFG ................................ 10 2.2 ED FA two energy level m odel .......................................................................... 19 2.3 Coupling of the core mode into copropagating cladding mode in a 25 mm LPFG written in a single mode fiber (a) schematic, (b) transmis­ sion spectrum ......................................................................................................
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