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OPTICAL BISTABILITY IN NONLINEAR KERR DIELECTRIC AND FERROELECTRIC MATERIALS by ABDEL-BASET MOHAMED ELNABAWI ABDEL-HAMID IBRAHIM A Thesis submitted to Universiti Sains Malaysia (USM) in fulfillment of the requirements for the degree of Doctor of Philosophy (Ph.D) in Physics July 009 ACKNOWLEDGMENTS First and foremost I acknowledge my supervisor Professor Junaidah Osman. I thank her for her vision, guidance, involvement, motivation, availability, help and generous support. I am honored and grateful to Assoc. Prof. Lim Siew Choo for being my cosupervisor, for continuous support, for help with the FRGS grant, and thesis submission. I thank Prof David R. Tilley, Prof Y. Ishibashi, and Dr. Ong Lye Hock for friendly, exciting collaboration, as well as for fruitful suggestions and discussions. I thank Dr. Magdy Hussein for continuous support and for exciting conversation about nonlinear optics. I thank all the members of my research group, particularly Tan Teng Yong and Ahmad Musleh for creating a friendly, empathic, helpful and stimulating environment. I am deeply indebted to my wife, my parents, and my father in-law for emotional and continuous motivation. I also acknowledge School of Physics, USM international, and Institute of postgraduate studies (IPS) for friendly environment during various stages of my doctoral studies. Last but not least I thank the Malaysian Ministry of Higher Education and university sains Malaysia (USM) for financial support for two years under the Fundamental Research Grant Scheme (FRGS) project number 03/PFIZIK/ ii TABLE OF CONTENTS ACKNOWLEDGMENTS ii TABLE OF CONTENTS iii LIST OF SYMBOLS viii ABSTRAK xiii ABSTRACT xvi CHAPTER 1 GENERAL INTRODUCTION Motivation of Study Organization of the thesis CHAPTER FUNDAMENTAL ASPECTS IN NONLINEAR OPTICS Introduction Linear Optics of Dielectrics Ionic Crystals Lattice Modes Sources of Polarizability Phenomenological theory for Ionic insulators The Fabry-Pérot resonator Theory of Linear Fabry-Perot resonator Applications Nonlinear Optics of Dielectrics Second-Order Processes Optical Rectification (OR) iii .3.1. Second Harmonic Generation (SHG) Third-Order Processes Optical Kerr Effect Third-Harmonic Generation (THG) The Required Input Optical Intensity Optical Bistability Polarization Optical Bistability and Multistability Phase in Optical Communication Systems CHAPTER 3 DIEELECTRIC OPTICS: OPTICAL BISTAILITY IN KERR NONLNEAR DIELECTRIC MATERIAL BASED ON MAXWELL-DUFFING ANALYSIS Introduction Intrinsic optical bistability (IOB) in nonlinear medium Mechanism for Intrinsic Optical Bistability (IOB) Physical Origin of Intrinsic Optical Bistability (IOB) Optical Bistability in a Fabry-Perot Resonator Mathematical Formulation Fabry-Perot Analysis Numerical Procedure The Input Parameters Results and Discussion Effect of Mirror Effect of Angle of Incidence Effect of Thickness Conclusion iv CHAPTER 4 GENERAL REVIEW ON FERROELECTRICS Introduction Crystal symmetry Types of Ferroelectrics (FE) Displacive FE Order-Disorder FE General Properties Dipole Moment Spontaneous Polarization Ferroelectric Domains and Hysteresis Loop Ferroelectric Phase Transitions Thermodynamic Theory of Ferroelectrics Ferroelectric Soft-Modes Lyddane-Sachs-Teller (LST) relation The Microscopic Model Barium Titanate BaTiO 3 (BT) Optical Applications of Ferroelectrics Applications Related to Intrinsic Optical Properties Electro-optic Applications Nonlinear Optical Applications Applications Related to Extrinsic Optical Properties Waveguides Lasers Photorefractive applications v Ferroelectric Structure-Related Applications CHAPTER 5 FERROELECTRIC OPTICS: OPTICAL BISTABILITY IN KERR FERROELECTRIC MATERIALS Introduction Mathematical Formulation Analysis of the Fabry-Perot Interferometer Intrinsic Optical Bistability in Ferroelectrics Material Aspects Numerical Procedure Results and Discussion Effect of Mirror Reflectivity Effect of Medium Thickness Effect of Temperature Effect of Frequency Conclusion CHAPTER 6 BEHAVIOUR OF DIELECTRIC SUSCEPTIBILITY NEAR THE MORPHOTROPIC PHASE BOUNDARY IN FERROELECTRIC MATERIALS Introduction Background on Morphotropic Phase Boundary (MPB) The Nonlinear Optic Coefficients in Ferroelectric Materials Modeling the MPB Using the Landau-Devonshire Energy Function 6.5 Results and Discussions Linear Dielectric Susceptibility Second-Order Nonlinear Susceptibility vi 6.5.3 Third-Order Nonlinear Susceptibility Conclusion CHAPTER 7 CONCLUDING REMARKS AND PROSPECTIVE STUDEIES REFRENCES APPENDICES APPENDIX A BOUNDARY CONDITIONS BETWEEN TWO MEDIA SEPARATED BY THIN MIRROR APPENDIX B A FLOWCHART DESCRIBING THE NUMERICAL PROCEDURE USED TO OBTAIN THE OPTICAL BISTABILITY CURVES IN CHAPTER APPENDIX C A FLOWCHART DESCRIBING THE NUMERICAL PROCEDURE USED TO OBTAIN THE OPTICAL BISTABILITY CURVES IN CHAPTER LIST OF PUBLICATIONS vii LIST OF SYMBOLS Symbol α α β β 1 β γ Γ Definition Nonlinear coefficient of the second-order term in the free energy Total polarizability Nonlinear coefficient of the fourth-order term in the in isotropic free energy Nonlinear coefficient of the fourth-order pure term in anisotropic free energy Nonlinear coefficient of the fourth-order cross term in the anisotropic free energy Coupling coefficient Damping parameter δ M Mirror thickness Δ Δ s ε ( ω ) The difference between the squared resonance frequency and the squared operating frequency The scaled form of Δ The frequency-dependent linear dielectric function ε s static dielectric constant ε 0 Dielectric permittivity of free space ε The high-frequency limit of the linear dielectric function ε M Dielectric permittivity of the mirror medium. η Complex mirror coefficient η s Scaled complex mirror coefficient η s, r Real part of the scaled mirror coefficient viii η s, i Imaginary part of the scaled mirror coefficient Θ i Angle of incidence Θ r Angle of reflectance Θ Angle of refraction Θ t Exit angle μ 0 Magnetic permeability of free space ν ρ Nonlinear coefficient for the sixth-order term in the isotropic free energy Elementary (Fresnel) reflection coefficient σ (n ω ) Linear response function of ferroelectric material σ M Conductivity of mirror medium τ φ χ ω Complex transmission coefficient Phase function Dielectric susceptibility Operating frequency of the driving field ω 0 Resonance frequency of the material ω T Transverse-optical phonon frequency ω L Longitudinal-optical phonon frequency ω p Plasma frequency of the material a b Inverse of the Curie constant Nonlinear coefficient in Duffing anharmonic equation ix c C d Velocity of light Curie constant Second-order nonlinear coefficient corresponds to second-order susceptibility χ ( ). e 0 Scaled electric field amplitude in free space (Vacuum) E 0 Dimensional Electric field amplitude in free space (Vacuum) E c Coercive field of ferroelectric material e E E f F g Scaled electric field in a medium Dimensional Time-dependent electric Dimensional Time-independent electric field Scaled operating frequency Free energy of ferroelectrics Scaled damping parameter of dielectric medium g F Scaled damping parameter of ferroelectric medium G H k 0 Order parameter in Taylor expansion Time-dependent magnetic field Wavenumber in free space k y-component of the wavenumber y k Extinction coefficient k z-component of the wavenumber z K Wavenumber x K l L m M Wavevector in arbitrary direction Scaled thickness Thickness of the nonlinear medium Scaled reduced mass per unit cell Dimensional Reduced mass per unit cell n Effective refractive index in medium n Transverse refractive index in medium T N p p P P s P q Q r R R M Number density of ions Dipole moment or microscopic polarization Scaled macroscopic polarization amplitude Macroscopic polarization amplitude Spontaneous polarization in ferroelectrics Macroscopic time-dependent polarization Effective charge per ion Polarization of ferroelectric material in tetragonal phase Complex reflection coefficient Reflectance or reflectivity of the nonlinear medium Reflectivity of the mirror medium s( nω ) t t T Linear response function of ferroelectric material corresponds to linear χ nω dielectric susceptibility element ( ) Scaled thermodynamic temperature of ferroelectric material Time Thermodynamic temperature of ferroelectric material zz xi T c Curie temperature T u w x y z Transmittance of a medium Scaled distance in z-direction Scaled plasma frequency Displacement of ions from its equilibrium position Displacement along y-direction in xyz Cartesian coordinate system Displacement along z-direction in xyz Cartesian coordinate system xii DWI-KESTABILAN OPTIK DI DALAM BAHAN-BAHAN DIELEKTRIK TAK LINEAR KERR DAN FEROELEKTRIK. ABSTRAK Kestabilan optiks dwi dan kestabilan optiks pelbagai di dalam hablur Kerr tak linear dikaji. Dua jenis sistem hablur penebat berion dipertimbangkan: bahan dielektrik biasa dan bahan ferroelektrik (FE) dengan struktur Perovskit. Kedua-dua ketakstabilan optiks ekstrinsik dan instrinsik dikaji. Suatu analisis alternatif digunakan untuk memodelkan kestabilan optiks tersebut; berbanding dengan analisis lazim di mana pengkutuban tak linear biasanya dikembangkan sebagai siri Taylor dalam medan elektrik. Formalisma alternatif ini terbukti lebih sesuai untuk bahan tak linear yang tinggi seperti FE dan bagi ketaklinearan resonan. Kestabilan optiks ini terselah pada beberapa pembolehubah-pembolehubah fizikal seperti pengkutuban, pekali-pekali pantulan dan biasan. Kesan-kesan oleh ketebalan sistem, frekuensi operasi, parameter lembapan, dan pekali ketaklinearan ke atas kestabilan optiks bagi setiap sistem hablur dikaji. Persamaan Duffing yang menerangkan pengayun tak harmonik dan persamaan gelombang digunakan untuk memodelkan respons tak linear sistem dielektrik. Applikasi syarat-syarat sempadan lazim memberikan rumus analitik bagi pekalipekali pantulan dan biasan yang diungkap dalam sebutan amplitud medan tuju elektrik, pengkutuban dan parameter-parameter bahan yang lain. Keputusanxiii keputusan simulasi bernombor menunjukkan bahawa kestabilan optiks sistem bergantung kepada ketebalan bahan, pekali ketaklinearan, dan sudut tuju setiap sistem hablur. Bagi kes alat resonan dielektik Fabry-Pérot, kestabilan optiksnya didapati bergantung kepada kepantulan cermin. Bagi sistem FE, respons tak linearnya dimodelkan dengan menggunakan persamaan dinamik Landau-Khalatnikov (LK). Keupayaan tak harmonik tak linearnya diperolehi dari tenaga bebas Landau-Devonshire (LD) yang telah diungkap dalam sebutan pengkutuban sistem. Dengan menggunakan penghampiran frekuensi tunggal, persamaan LK dan persamaan gelombang digunakan untuk menghasilkan persamaan pengkutuban tak linear. Teknik ini membuatkan kestabilan optiks sistem menjadi bergantung kepada suhu. Ungkapan analitik bagi kedua-dua pekali pantulan dan biasan sebagai fungsi suhu dan parameter-parameter bahan yang lain diterbitkan. Parameter-parameter input diperolehi dari data eksperimen yang sedia ada bagi hablur tunggal BaTiO 3 untuk digunakan di dalam simulasi bernombor. Keputusan-keputusan simulasi bersetuju pada prinsipnya dengan pemerhatian eksperimen ke atas kestabilan optiks intrinsik di dalam hablur mono BaTiO 3 dan di dalam bahan-bahan fotorefraktif FE. Didapati bahan-bahan FE ini sentiasa mempamerkan kestabilan jenis threshold. Bahagian terakhir kerja penyelidikan ini tertumpu kepada kajian kerentanan dielektrik bagi bahan pukal FE berhampiran sempadan fasa morfotropik (MPB). Ini termasuk kajian ke atas kerentanan linear dan taklinear (peringkat kedua dan ketiga) bagi kedua-dua had dinamik dan statik. Tabii kerentanan-kerentanan ini dikaji berhampiran MPB dengan menggunakan tenaga bebas LD dan persamaan dinamik LK. Magnitud kerentanan-kerentanan dilakar sebagai fungsi bahan parameter xiv * β β β1 = di mana β 1 and β adalah pekali-pekali di dalam ungkapan tenaga bebas LD. Dalam had statik, MPB diwakilkan oleh nilai * β = 1. Dalam had ini, pemalar dilektrik linear adalah malar, kerentanan-kerentanan tak linear peringkat kedua dan ketiga mencapah di MPB. Kerentanan dielektrik dinamik didapati mempunyai puncak atau puncak-puncak pada sesuatu nilai/nilai-nilai * β. Pertambahan ini boleh difahami dengan menggunakan konsep frekuensi normal dalam mod lembut FE. Pertambahan dalam nilai kerentanan tak linear ini mempunyai potensi di dalam menjuruterakan bahan baru untuk applikasi beberapa peranti-peranti optiks. ω T xv OPTICAL BISTABILITY IN NONLINEAR KERR DIELECTRIC AND FERROELECTRIC MATERIALS ABSTRACT The optical bistability (OB) and multistability in Kerr nonlinear crystals are investigated. Two types of ionic insulating crystals are considered: a typical dielectric and a ferroelectric (FE). Both extrinsic and intrinsic optical instabilities are investigated. An alternative analysis is used to model the OB; rather than the conventional analysis where the nonlinear polarization is usually expanded as the Taylor series in the electric field. The alternative formalism proves to be more suitable for highly nonlinear materials such as FE and for resonant-nonlinearities. The OB has its manifestation in various physical variables such as polarization, reflectance and transmittance coefficients. The effects of system thickness, operating frequency, damping parameter, and nonlinearity coefficient, on the OB of each crystal system are investigated. The Duffing equation describing an anharmonic oscillator and the wave equation are used to model the nonlinear response of the material. Application of standard boundary conditions leads to analytical expressions for both reflectance and transmittance expressed in terms of the electric field incident amplitude, polarization and other material parameters. Numerical simulations show that the OB is dependent on the material thickness, nonlinear coefficient, and the angle of incidence. In the case of a dielectric Fabry-Pérot resonator, the OB is also found to be mirror reflectivity dependent. xvi For FE material, the nonlinear response is modeled using the Landau-Khalatnikov (LK) dynamical equation. The nonlinear anharmonic potential is obtained from the Landau-Devonshire (LD) free energy expressed in terms of polarization. The LK equation and the wave equation are then used to produce a nonlinear polarization equation. This technique makes the OB becomes temperature-dependent. Input parameters from available experimental data of a BaTiO 3 single crystal are used in the numerical simulations. The simulation results agree in principle with the recent experimental observations of intrinsic OB in BaTiO 3 monocrystal and other FE photorefractive materials. It is found that FE always exhibits a threshold-type of bistability. The last part of this work is devoted to studies of dielectric susceptibilities of bulk FE materials in the vicinity of the morphotropic phase boundary (MPB). It includes studies on linear and nonlinear (second-order and third-order) susceptibilities for both the dynamic and static limit. The behaviour of the susceptibilities near the MPB is investigated using the LD free energy and LK dynamical equation. The susceptibility magnitudes are plotted as a function of the material * parameter β = β β where β 1 and β are the coefficients in the LD free energy 1 * expression. In the static limit, the MPB is represented by β = 1. In this limit, the linear dielectric constant, the second third-order nonlinear susceptibilities diverge at the MPB. The dynamic dielectric susceptibility assumes a resonance-like peak(s) or peaks at certain value(s) of * β. The enhancement is explained using the concept of FE soft-mode. The enhancement of the nonlinear susceptibility may have its potential in engineering new materials for various optical device applications. xvii Chapter 1 GENERAL INTRODUCTION 1.1 Motivation of Study Optical technology is employed in CD-ROM drives and their relatives, laser printers, and most photocopiers and scanners. However, none of these devices are fully optical; all rely to some extent on conventional electronic circuits and components. Optical technology has made its most significant inroads in digital communications, where fiber optic data transmission has become commonplace. The ultimate goal is the so-called photonic network, which uses visible and IR energy exclusively between each source and destination. The current communication network operates based on a combination of both electronics and photonics (hybrid). It consists of nodes which are connected by a series of fiber-optic links. These fiber-optic links are used to transmit the information while the nodes decide which path the information should follow from source to destination (routing). While the fiber-optic links carry the information via light, the nodes are still mostly electrical. In the past, this hybrid technology has served the data-communication networks adequately. At present, this technology is predicted to breakdown as the demand for telecommunication services is increasing exponentially. The rapid growth of the Internet, expanding at almost 15% per month, demands faster speeds and larger bandwidths than electronic circuits can provide. Electronic switching limits network speeds to about 50 Gigabits per second (1 Gigabit is 10 9 bits). 1 Fortunately, for long distance communications, a very high capacity dispersion-free transmission can be realized either by implementing solitons and other types of nonlinear pulse transmission in optical fiber. However, it seems that the bottlenecks are the data processing and switching components in the nodes. Therefore, the aim is to replace the nodes based on electrical technology by optical ones. As such, the study of optical phenomena in nonlinear waveguides (optical bistability) has been intensified, in the hope that these can be used for all-optical switching purposes. On the other hand, there has been intensive research for developing an optical computer that might operate 1000 times faster than an electronic computer (Abraham 1983, and, Brenner 1986). An optical computer is a computer that performs its digital computation with photons in visible light or infrared as opposed to the more traditional electron-based computation (silicon technology). The electronic computer have its fundament limit since an electric current flows at only about 10 percent of the speed of light which limits the rate at which data can be exchanged over long distances. The fundamental components of any digital computer are; memory, processor, and terminals for information. The three basic functions of a computer are arithmetic operation, logical operations and the memory. All these functions are done by devices that have bistable states (true and false). In arithmetic operation the two states represent the numbers 0 and 1 of the binary number system. The memory of the computer also stores the results of arithmetic and logical operation in bistable devices. Thus a computer requires an optical transistor that can represent the values 0 and 1 in physical form. Moreover, optical transistors can be assembled into larger scale devices that perform the three logical functions: AND, OR and NOT. Optical switching is a natural candidate to mediate between electronic systems and optical ones. The starting point of the optical transistor is an ingenious and widely used optical apparatus known as the Fabry-Perot interferometer. When the Fabry- Perot apparatus is filled with an intensity-dependent nonlinear medium, and a powerful source of coherent radiation is focused, the transmitted intensity assumes a bistable state when the input laser is varied. Therefore, optical switches can be made in the form of thin nonlinear crystals. Current techniques of crystal growing or thinfilm fabrication make it possible to fabricate very large thin sheets. If an optical image was projected directly through a large sheet in which each small area served as an optical switch, the transmission from the switches would serve as a digital record of the image. Many kinds of p
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