非線性光子晶體波導中的光調制不穩定性與光孤子傳播

Abstract

本論文中，我們將對一具有非線性線缺陷的光子晶體波導，研究光調制不穩定性(modulation instability)及光孤子(soliton)的傳播。藉由緊束縛理論，考慮左右三個鄰近點缺陷的耦合影響下，我們導出一不連續的非線性展開方程式，並將其視為一延伸的非線性薛丁格方程(extended NLSE)。對此方程式求解，我們可以完整地以解析解來描述光調制不穩定現象及其增益係數。在負非線性折射介質組成的光子晶體波導中，我們可以發現光調制不穩定性的增益被抑制的現象，此現象不會出現於一般光纖波導的光調制不穩定性的增益頻譜圖。另一方面，我們以四階阮奇庫塔法(4th order Runge-Kutta method)數值解extended NLSE，並與解析解相互印證。首先，我們得到模擬所得到的光調制不穩定性的增益頻譜圖與解析解相符合。利用此方法進行模擬，可在空間及時間軸上作光脈衝傳播的觀察，透過我們的模擬，可歸納出形成光孤子的條件以及光孤子傳播的特性。

We have studied the modulation instability (MI) and solitons propagation in a photonic-crystal waveguide with nonlinear line defect. By tight-binding theory, we consider the coupling effects up to the third order nearest-neighbor defects and obtain the discrete nonlinear evolution equations as a new extended nonlinear Schrödinger equation (NLSE). Solving this equation, MI and MI gain can be analytically determined. We can find the phenomenon of gain suppression when nonlinear coefficient □□< 0 and this is a big different from the gain profile in general. On the other hand, we do the simulation by the 4th order Runge-Kutta method to consistent our analytic solutions and we can observe MI and soliton propagation in the spatial and time domains. The results of analytic solution and simulation are the same. Besides, we also summarize the conditions of forming solitons and some particular behaviors in soliton propagation by our simulation.