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系統識別號 U0026-2506201813023000
論文名稱(中文) 探討孤子在非均勻光纖中的理論和數值模擬
論文名稱(英文) Theoretical and computational studies of solitons in inhomogeneous optical fibers
校院名稱 成功大學
系所名稱(中) 太空與電漿科學研究所
系所名稱(英) Institute of Space and Plasma Sciences
學年度 106
學期 2
出版年 107
研究生(中文) 陳秋芸
研究生(英文) Ciou-Yun Chen
學號 LA6041129
學位類別 碩士
語文別 英文
論文頁數 109頁
口試委員 指導教授-西村泰太郎
口試委員-黃勝廣
口試委員-曾碩彥
口試委員-張世慧
中文關鍵字 非線性薛丁格  非均勻介質  光孤子  逆散射法 
英文關鍵字 nonlinear Schrödinger  inhomogeneity  optical soliton  inverse scattering method 
學科別分類
中文摘要 在這篇論文中,我們分別於數值解及解析解兩方面探討光孤子在非均勻光纖下的現象。在這裡我們利用隨位置變化之折射率當作我們非均勻的來源。
我們利用非線性薛丁格方程來描述光孤子。光孤子通訊依賴著孤子在傳遞過程中不改其形狀之特性,而此性質是由於非線性薛丁格系統之運動積分具固有性。在光纖的非線性效應(克爾效應)與色散效應的平衡之下,孤子的包絡將得以維持。在數值模擬方面,我們利用狄利克雷邊界條件以及有限差分法之中的蛙跳格式作為數值模擬的手法來探討光孤子的行為。在非均勻介質下,光孤子之訊號寬度可被控制,以防止訊號尾端的交疊(孤子的交疊導致孤子間的交互作用)。當折射率隨光纖的位置下降,色散效應亦隨之下降,這使光孤子被局部化。
另一方面,非線性薛丁格方程可由逆散射法求得解析解。由於逆散射法始於求解KdV方程(其解為實數解),因此在了解逆散射法的使用上,我們著重在求解KdV方程的過程,隨後應用於求解非線性薛丁格方程(其解為複數解,需要使用2×2矩陣方程)。最後運用了變數變換法來求解非均勻介質下非線性薛丁格方程(正如對於伯格斯方程作霍普夫-科爾變換求解一樣,對於非線性薛丁格方程,我們可利用巴克隆德變換或者達布變換求解)。我們亦比較了解析解及數值解的結果。
英文摘要 In this work, solitons in inhomogeneous optical fibers is studied both numerically and analytically. Spatial variation of index of refraction is considered as our source of inhomogeneity.
Optical solitons can be described by nonlinear Schrödinger (NLS) equation. Due to inherent existence of integrals of motion in the NLS system, the solitons can propagate without changing their shapes. Soliton communication heavily relies on this latter nature of NLS system. The envelope soliton is preserved due to balance between the nonlinear drive (Kerr effect) and the dispersion effect of optical fiber. For our numerical simulation, finite difference methods (the leapfrog scheme) is employed with Dirichlet boundary conditions. For the inhomogeneous optical fibers, it is demonstrated that, the widths of optical solitons can be controlled so as to prevent unwanted overlapping of the soliton tails (the overlapping of solitons leads to the interaction of solitons). When the index of refraction “n” is reduced spatially along the optical fiber (and thus the permeability “ε”) dispersion effect is reduced which makes the soliton to be localized.
On top of the numerical analysis, the homogeneous NLS equation can be solved by inverse scattering method (ISM) which was initially applied for Korteweg–de Vries (KdV) equation. To recapitulate ISM, the process of solving KdV equation is presented (which for real values), then for NLS (which is for the complex values. A 2×2 matrix equation is required). Finally, a variable transformation (Bäcklund transformation or Darboux transformation as in Cole-Hopf transformation for Burgers equation) is applied to incorporate inhomogeneity coefficients multiplied to the dispersion term of the NLS equation. We compare the analytical solutions with the simulation results.
論文目次 摘要……………………………………………………………………………………………………………………………………………I
Abstract………………………………………………………………………………………………………………………………II
Contents………………………………………………………………………………………………………………………………IV
Chapter1 Introduction and Motivation………………………………………………………1
Chapter 2 Theoretical and Computational Model………………………………5
2-1 Introduction to Solitons……………………………………………………………………………5
2-2 Wave Equations……………………………………………………………………………………………………10
2-3 Nonlinear Polarization and Kerr Effect……………………………………12
2.4 Nonlinear Schrodinger Equation in Optical Fibers…………16
Chapter 3 Analytical Soliton Solutions………………………………………………24
3.1 Solutions of KdV Equation by Inverse Scattering Method……………………………………………………………………………………………………………………………………25
3.2 Solutions of Nonlinear Schrödinger Equation………………………37
Chapter 4 Numerical Simulation Results of Nonlinear Schrödinger Equation………………………………………………………………………………………………45
4.1 Numerical Solutions of Nonlinear Schrödinger Equation with Homogeneous Background……………………………………………………………………………45
4.2 Numerical Solutions of Nonlinear Schrödinger Equation with Inhomogeneous Background………………………………………………………………………62
4.3 Analytical Solutions of Nonlinear Schrödinger Equation with Inhomogeneous Background………………………………………………………………………72
Chapter 5 Summary and Future Work……………………………………………………………95
Appendix………………………………………………………………………………………………………………………………98
Reference…………………………………………………………………………………………………………………………107
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