光學渦旋(optical vortex)是一種具有軌道角動量的特殊形態光束，它具有中心沒有能量的環形結構，在近年的雷射發展上有不少的應用，特殊的光學顯微鏡、具有不同角動量的光通訊、利用光學渦旋限制粒子的光鉗等等都是現今發展出來的應用。而產生光學渦旋的方法也有許多不同的方法，利用特殊形態光纖產生、利用給予極化光纖壓力使不同模態光束疊加產生、將螺旋相位板置入共振腔產生等方法。而本實驗使用的方法是利用接近半球型(near hemi-spherical)的簡併型共振腔，再將螺旋相位板置放至其中，使光束在共振腔中走一個V型的路線，並透過增益介質內的光束相互耦合後再經相位鎖模產生多模態(muti-modes)的光學渦旋。
本實驗利用Tang-Statz-deMars(TSD)方程式來模擬多模態光束的鬆弛震盪(relaxation oscillation)，發現在模擬上因為不同模態光束得到增益的能力不同，鬆弛震盪會有競爭的現象，最後會由特定幾個模態的光束來主導整個鬆弛震盪的結果。
然而在實驗上我們產生的光學渦旋所量測的鬆弛震盪與模擬有很大的不同，因為本實驗產生的光學渦旋是經由多模態同調合併產生，彼此的耦合能力很強，所以看到的鬆弛震盪現象幾乎同步，於是再由此結果在模擬的模型上加上耦合項，當耦合項越強時會發現不同模態的光束鬆弛震盪開始同步，以此結果修正模擬模型。此外，再雷射尖峰的區間，多模合併組成的渦漩光束呈現出高低相間的結果，恰與一般的高斯光束呈現的狀況不同，這是由於不同模態之間在增益中的拮抗與雷射的非線性系統多個模態耦合與競爭的結果。

Optical vortex (OV) is a special morphological beam with orbital angular momentum. It has a ring shape with no energy at the center. In recent years, there have been many applications associated with optical vortex beam, such as STED microscopy, OAM multiplexing optical communication, optical tweezers and the like. Upon the demand of the applications, high quality optical vortex is necessary and there are many different methods to achieve this goal, such as the pump beam shaping, fiber-based amplifier with mode conversion, and degenerate cavity with intra-cavity spiral phase plate (SPP). In particular, a nearly hemi-spherical cavity with SPP produces OV with ultrahigh vortex purity of more than 99.99%. In such a laser configuration, the OV is formed as a coherent superposition of off-axis multiple pass transverse modes circulating in the resonator. They retrace a periodic orbital when travels twice roundtrips and form a V-shaped optical path. The phases of the beamlets traveling a MPT modes are then correlated by the strong coupling during the stimulated emission process in the laser gain medium and can be understood by the Kuromoto model under the minimum energy criterion. In this new laser configuration, built up dynamics has never been explored.
Multi-mode relaxation oscillations in lasers can be modeled by Tang-Statz-deMars (TSD) equation. It is found that modes with unbalanced gain volume results in irregular relaxation oscillation for each mode while the intensity sum of all modes remains regular. However, the relaxation dynamics in our optical vortex produces irregular but synchronized spiking and relaxation oscillation across the spatial measurement. By introducing coupling terms between modes in to the photon equations in TSD equations, the spiking and relaxation synchronizations among modes can be explained. Experiments with a varying gain volume and coupling strength showed degradation in the correlation of signals taken from different oscillating modes as can be expected by TSD　equations with photon mode couplings. Moreover it is also found that during the laser spiking phase, the optical vortex output shows a regular high and low changes, which we believe is due to the interplay between competition and couplings among the resonating modes.

中文審定書 i
英文審定書 ii
摘要 iii
Abstract v
目錄 vii
圖目錄 ix
表目錄 xiii
第一章、緒論 1
1-1光學渦漩簡介 1
1-2雷射鬆弛震盪 2
1-3 文獻回顧 3
1-4 研究動機 13
1-5 論文架構 15
第二章、簡併型共振腔產生渦流雷射 16
2-1 平凹簡併型共振腔 16
2-2 two fold簡併型共振腔 17
2-3 實驗元件介紹 18
2-4 雷射輸出結果 23
2-5 渦流雷射光束軌跡及多模態驗證 27
第三章、雷射鬆弛震盪模擬 31
3-1 Tang-Statz-deMars (TSD) equations模擬 31
第四章、渦流雷射鬆弛震盪量測與分析 40
4-1 量測實驗架構 40
4-2 量測結果與分析 41
4-3 渦流雷射不同模態間相關性量測 54
4-4 模擬模型的修正 60
第五章、結論與未來展望 65
5-1 結論 65
5-2 未來展望 65
參考文獻 66

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