在這個研究中，我們用數值方法spectral method計算了馬克士威爾方程式和薛丁格方程式，討論了電磁波和電子波在不同種類的一維無序晶格的散射與局域，我們也計算Lyapunov exponent和層數不同的無序晶格的平均穿透率來分析波長在能帶和能帶邊緣的電磁波在不同無序程度的晶格的局域程度，我們也分析了因為無序而產生的能隙模態的局域特性，然後，我們也用牛頓法在位能是無序晶格的非線性薛丁格方程式找到一些無序晶格能隙孤立子。

There have been numerical investigations for electron and electromagnetic waves in the one-dimensional disordered lattice. The electron eigenmodes of one-dimensional disordered lattice are investigated for three different models. The electromagnetic eigenmodes of one-dimensional disordered lattice are also investigated. The degree of localization of the wavelength at the band, gap edge and gap are studied. Finally, the disordered lattice solitons are presented.

Table of Contents

List of Figures vi
List of Symbols viii
Chapter 1: Introduction
1.1 Overview 1
1.2 Description of the work 2

Chapter 2: Schrödinger equation in one-dimensional disordered lattices
2.1 Electron wave dynamic in 1D random potential 3
2.2 Randomness in the width and position of the barriers 4
2.3 Randomness in the width and position of the wells 6
2.4 Randomness in the width and position of barriers and wells 8

Chapter 3: Maxwell’s equations in one-dimensional disordered lattices
3.1 Electromagnetic wave dynamic in 1D dielectric lattices 9
3.2 Randomness in the layer thickness 11

Chapter 4: Propagation of electromagnetic wave through a 1D disordered lattice: Transmittance, Lyapunov exponent, and its Variance
4.1 The transfer matrix method for a multilayered structure 14
4.2 Anderson localization in disordered lattice 19

Chapter 5: Nonlinear Schrödinger equation in one-dimensional disordered lattices
5.1 Disordered lattice solitons 26

Chapter 6: Conclusion 30

Bibliography 31

[1] Mordechai Segev, Yaron Silberberg and Demetrios N. Christodoulides. Anderson localization of light. Nature photon 7, 197-204 (2013).
[2] Schwartz, T., Bartal, G., Fishman, S. & Segev, M. Transport and Anderson
localization in disordered two-dimensional photonic lattices. Nature 446,
52–55 (2007).
[3] Ad Lagendijk, Bart van Tiggelen, and Diederik S. Wiersma. Fifty years of Anderson localization. Physics Today 62, 24-29 (2009).
[4] Alain Aspect and Massimo Inguscio. Anderson localization of ultracold atoms. Physics Today 62, 30-35 (2009).
[5] Lev I. Deych,D. Zaslavsky, and A. A. Lisyansky. Statistics of the Lyapunov Exponent in 1D Random Periodic-on-Average Systems. Phys.Rev.Lett.81, 5390-5394 (1999).
[6] Pi-Gang Luan and Zhen Ye. Statistics of Lyapunov exponent in one-dimensional layered systems. Phys. Rev. E. 64, 066609 (2001).
[7] Yoav Lahini, Assaf Avidan, Francesca Pozzi, Marc Sorel, Roberto Morandotti, Demetrios N. Christodoulides, and Yaron Silberberg. Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices. Phys.Rev.Lett. 100, 013906 (2008).
[8] Nikolaos K. Efremidis and Kyriakos Hizanidis. Disordered Lattice Solitons. Phys.Rev.Lett. 101, 143903 (2008).
[9] D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg. Gap Solitons inWaveguide Arrays. Phys.Rev.Lett. 92, 1093904 (2004).
[10] 黃孟傑，波在一維系統中的傳播與局域化，中央大學碩士論文(民國94年)