The Study Of Disorder And Nonlinearity In One-Dimensional Optical Waveguide Array

As a wave is traveling in a periodical structure, owing to the energy bands in the struc-ture, only a wave with energy inside a band can permeate the structure. When the period-icity is locally broken, some energy levels on the top or bottom of a band will be shifted into forbidden bands and localized states appear. The phenomena was called Anderson localization in solid state physics and subsequently observed in one-dimensional and two-dimensional photonic crystals, becoming a method to control optical beams. When nonlinear effects can be neglected, description of light propagating in a one-dimensional waveguide array is identical to the tight-binding-approximation model in solid state the-ory. Inhomogeneity in geometry or refractive index in the waveguide array behaves as doping in a periodical structure. A doping not only alters the propagation constant of the waveguide but also changes the coupling constant between wave guides, called di-agonal and off-diagonal disorders respectively. Nonlinearity is another effect breaking the periodicity of the waveguide array. The model including the nonlinear effect is the discrete nonlinear Schrodinger (DNLS) equation. In the limit of weak nonlinearity, it approaches the integrable discrete nonlinear Schrodinger (IDNLS) equation which is integrable and supports discrete soliton solution. Now local-izations owing to single diagonal and off-diagonal impurity have been well studied and verified in experiments. Attraction and repulsion to solitons from positive and negative diagonal impurities were verified in experiments. Anderson model concerning diagonal disorder was also experimentally realized. However, there are some problems relevant to off-diagonal disorder remain unsolved.This thesis mainly studies influences of off-diagonal disorder on light propagation in one-dimensional optical waveguide array. By numerical calculations, we find:1. Non-diagonal disorder also exerts attractive or repulsive interaction on optical beams.2. For a negative non-diagonal impurity, though no localized state exists, its inter-action with light beams is still repulsive, which remains repulsive even though a diagonal impurity simultaneously exists.3. In the Anderson model simultaneously concerning diagonal and off-diagonal disor-der, off-diagonal disorder significantly contributes to localization.4. The condition to realize IDNLS soliton and its interaction with various impurities.