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Finite element methods for computational nonlinear optics

Buah, P. A. (1996). Finite element methods for computational nonlinear optics. (Unpublished Doctoral thesis, City, University of London)

Abstract

Unlike linear systems, where knowledge of the eigenvalues and eigenvectors allows one to write a closed-form solution, few nonlinear systems posses closed-form analytical solutions, and therefore numerical simulations play a crucial role in the process of finding and analysing nonlinear phenomena. For the theoretical study of the complex spatial, temporal and spatiotemporal behaviour of nonlinear optical systems, mathematical modelling of the problem under consideration by efficient stepping algorithms is necessary. For the past decade the Finite Element Method has proved to be a very efficient and versatile method in linear and nonlinear modal analysis with the use of variable meshes and infinite elements as some of its greatest strengths, but little work has been done on its application to evolutional analysis in nonlinear optics.

This thesis describes a finite-element-based computer modelling of a wide range of nonlinear optical systems, with a view to developing an understanding of some of the complex but exciting spatial, temporal and spatiotemporal propagation dynamics in such systems. The computer simulation of a wide range of nonlinear optical waveguides and systems in those major areas of nonlinear optics which include nonlinear integrated-optics, nonlinear fiber-optics and nonlinear dynamic systems has been performed. This is carried out through numerical solutions of appropriate wave equations such as the paraxial wave equation, the Maxwell-Debye equations, the infinite-dimensional map of a ring resonator derived from the Maxwell-Bloch equations and coupled nonlinear Schroedinger equations that may include gain terms.

Two well defined problems are addressed in detail. First, the determinations of the modes or characteristic solutions by solving the stationary wave equations through modal analysis of different types of nonlinear optical waveguides. Second, the determination of the paraxial propagation solutions along a nonlinear medium by solving the wave equation as step-by-step initial-boundary value problems through beam propagation analysis.

For this task, current and novel 2D- and 3D- schemes based on the finite element method are presented and described. Particularly, a novel robust time-dependent code which is a combination of the finite-element propagation algorithm coupled to unconditionally stable difference schemes for marching the solutions along the characteristics of the (z,t)-domain is developed as well as accurate propagation schemes for solving generalized coupled nonlinear Schroedinger equations.

Additionally, several novel specific applications involving nonlinear media are thoroughly described. These include the study of nonlinear supermodes of integrated-optics directional couplers, the nonlinear dispersion characteristics of multiple-quantum well waveguides and graded-index fibers with saturable nonlinear cores, controlled spatiotemporal soliton emission, switching and demultiplexing in nonlinear tapered waveguides, temporal optical soliton dynamics in active three-core nonlinear fiber directional couplers and two-dimensional solitary-wave optical memory in fibers and bistable ring cavities. The generation of ultrafast soliton-like pulsetrains from a c.w. dual-frequency input signal with sinusoidal modulation using a proposed novel dual-channel erbium-doped fiber coupler laser is also demonstrated.

Publication Type: Thesis (Doctoral)
Subjects: T Technology > TK Electrical engineering. Electronics Nuclear engineering
Departments: School of Science & Technology > Engineering > Electrical & Electronic Engineering
School of Science & Technology > School of Science & Technology Doctoral Theses
Doctoral Theses
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