具有週期性結構的系統在自然界可說是隨處可見。帶有週期性結構的系統與連續性系統在許多物理行為表現上有著根本上的不同。在所有週期性結構中,由於光的可調變性(adaptability)以及可控制性(controllability),讓波導陣列正好為研究週期系統開啟了一條康莊大道。在波導陣列中,當光束之擴散效應與非線性效應達到平衡時,光束將形成自定域態(self-localized modes)或稱做離散光孤子(discrete solitons)。在此論文中,將一一介紹光束在波導陣列中的行為,包括非尋常繞射(anomalous diffraction)以及無繞射傳播(diffraction-free propagation)等。 本論文主要針對「離散孤子之交互作用」以及「對稱孤子之對稱性破壞」進行研究。由於在非線性光學領域中,孤子之交互作用一直是國際上研究的焦點,因此,我們利用數值運算,針對離散孤子之交互作用作了一系列的探討。研究中發現,當離散光孤子的同調性降低時,將弱化其交互作用。此結果與先前研究在均勻介質中,利用光孤同調性控制其交互作用有相似的結果。最後,本論文亦針對對稱孤子之對稱性破壞進行研究。研究發現,當孤子之同調性下降時,將可抑制孤子之對稱性破壞。
Periodic systems are ubiquitous in nature and are known to exhibit behaviors that differ fundamentally from those of homogenous systems. Among all periodic systems, optical waveguide array opens a wide door for investigating the dynamics in nonlinear periodic systems for its adaptability and controllability of light. Self-localized modes in periodically modulated structures, or discrete solitons, form when the broadening effects (discrete diffraction) and the nonlinear effects are balanced. Many properties about the wave propagation in the (nonlinear) periodic systems are demonstrated in this thesis, such as anomalous diffraction, diffraction-free propagation, and staggered and unstaggered modes. This thesis mainly focus on two topics: discrete soliton interactions and symmetry breaking of even discrete solitons. Since one of the most intriguing phenomena in the nonlinear optics is the soliton interaction, we perform numerical simulations of the discrete soliton interactions, and study the threefold interplay between statistical properties (coherence), the periodic refractive index, and the nonlinear effects. We show that when the two beams are made partially incoherent, the interaction force will become much weaker, and this result is similar to the previous study, done by T.S. Ku, that focuses on the coherence-controlled soliton interactions in the homogenous nonlinear media. In the final section, we discuss the symmetry breaking instability of discrete solitons with even parity and show how incoherence can suppress the instability.
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