本實驗利用二極體泵源及Nd：YVO4晶體來架設雷射系統並產生幾何環形的離軸模態，利用ABCD Law的計算來完成四面鏡雷射系統的共振腔設計。隨著調整共振腔長，會使共振腔結構操作從穩定區到穩定區邊緣，因此會依序得到基本模態、高階模態、環形模態及幾何模態，然而我們發現光形會沿著光軸逐漸擴展並到最後變成兩個光點；在加入一個腔內遮蔽物，我們發現離軸模態特性會隨著腔長接近幾何模態的操作腔長而產生。藉由將光形從架構上做劃分及加入調制訊號，並量測相關係數、分岔圖動態行為、Lyapunov exponents及極端事件比例去探討穩定區到穩定區邊緣模態的空間分布與動態行為上的關係。透過分析不同模態的相對位置與相關係數及分岔圖趨勢，我們發現兩正交方向上模態的非線性動態行為會隨著腔長調整離開基本模態而逐漸產生差異，然而，水平方向位置的兩個訊號會因為幾何封閉路徑的產生而導致動態行為也會開始產生不同。接著模態轉換的過程會觀察到Lyapunov exponents及極端事件比例突然上升的趨勢，最後也有發現超過6倍標準差的極端事件

In this thesis, a diode-pumped Nd:YVO4 laser is utilized that can obtain an off-axis geometric-ring-mode. By applying the ABCD Law, the four-elements laser system can be designed. By adjusting the cavity length, the cavity configuration can be operated from the stable region to the edge of the stable region and therefore the fundamental mode, high order mode, ring mode and geometric mode can be obtained in sequence. However, we find the pattern extend along the optical axis and finally turned into two spots. By inserting a cavity barrier, we observe that the off-axis mode characteristics can be gradually generated with the cavity length adjusted in a region close to the geometric mode cavity configuration. In a simple way of division of the pattern structure and adding the modulated signal, we analyze the correlation coefficient, bifurcation, Lyapunov exponents and the ratio of extreme events to discuss the relation between the spatially distribution and the dynamic behaviors of various modes from the stable region to the edge of the stable region. By analyzing the trends of the correlation coefficient bifurcation of the various modes, we explore that gradually different nonlinear dynamics evolution in the two orthogonal modes when the cavity length is tuned away from the fundamental mode configuration. However, the two signals in the horizontal relative position will start to have different evolutions of dynamic behaviors due to the generation of geometric closed trajectory. Next, we find that the sudden increase of the positive Lyapunov exponents and the ratio of extreme events when crossing a transition two different modes. Finally, extreme events are found with peak intensities exceeding six standard deviations over the average peak intensities of the events.

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