雷射共振腔裝置有別於一般傳統固態雷射不可調，它可根據腔長與離軸變化產生多種各式各樣的雷射本徵態，橫向模態（z軸截面）可呈現出2D圖型 (如Lissajous圖形)或1D圖型(如V-mode、M-mode，或N-mode)。本論文主要研究Nd:YVO_4（摻釹釩酸釔)雷射所產生的特殊簡併模態N-mode，此種本徵態在特定條件下，輸出的三道雷射光束可恰為左旋光、線偏振，以及右旋光。在一個雷射裝置下就可以同時輸出三種不同偏振性的雷射光相當特別，且具有旋光性的雷射光源富有其應用性，可用於發展光學鑷子(optical tweezers)。

雷射共振腔裝置可透過理論推知形成各種穩定簡併態的相位延遲δ（phase retardation)，再依據c-cut Nd:YVO4的雙折射性質，將n_e與n_o帶入雙折射理論計算出等效折射率n_eff，便可計算出該相位延遲δ下的折射角度θ為多少。只要角度符合簡併共振腔的條件，理論結果可有多組。不過實驗上是否可以找到每一組對照的相位延遲δ，則仍需要實驗去做進一步的確認。

透過理論計算後，接下來更關鍵的就是希望透過實驗去驗證理論的推測。多組實驗的模態檢驗線偏振、圓偏振狀況。確認了多組N-modes具有「圓偏振+線偏振」的模態，並得到其張角(折射角)θ，和理論相比較後發現相當吻合，並且是確認了多組不同的張角θ，僅有最小角度的對應關係未在實驗上確認。另外一種N-mode狀況，三道光束皆為「線偏振」，理論上分析主要有兩種狀況：一種是三光束皆為同一線偏振，另一種則是具有兩組線偏振(中間與兩端的光束相異)，這與實驗上所觀察到的結果亦相吻合。

在實驗中我們找尋了多組N-modes，發現有「全線偏」與「圓偏+線偏」兩種簡併態，在此腔長附近或某離軸值附近的模態也多會趨於特定簡併態。又根據共振腔形成共振的條件，亦可推得僅會有此兩種狀況發生，換句話說共振腔會有類似鎖模態的情形發生。最後，還是回到特殊的N-mode，檢測「左、右旋光與線偏振」這種具有三種偏振特性的N-mode功率分佈，觀察三道光束的功率比例各占多少。藉由實驗與理論並行，更清楚理解這種特殊雷射模態的形成之物理機制。

Laser cavity is different with solid state laser of tradition. It can adjust the pattern of beam by off-axis, so it can present many types of entanglement. For example, V-mode, M-mode, W-mode are one dimension or Lissajous is two dimension on the transverse. This thesis focused on N-mode. These modes are on the eigenstates, and the three output laser beams are left circular polarization, right circular polarization, and linear polarization on far field. This device can output three kinds of different polarizations at the same time. Especially the circular polarization has many applications, like optical tweezers.

We use the birefringence theory to know δ of phase retardation at a stable eigenstate, and according to birefringence of Nd:YVO4 put the n_e and n_o into theory to count n_eff. This way would be find the angle θ on every δ. If the angle θ conform to the entanglement condition, we would find many different θ of δ. We investigate the spatial and polarization entanglement of geometric beams generated from a c-cut Nd:YVO4 laser. Experimental results reveal that a geometric mode possesses linearly polarized state and circularly polarized states in opposite directions at the same time and the spatial and polarization entangled geometric beams can be generated systematically by controlling the off-axis magnitude. Another kind of polarization, the experimental results reveal a geometric mode that is a linear polarization. We use the birefringence theory to analyze those data, and the numerical results have a good agreement with the experimental results. In addition, the thesis mentions the power of laser patterns, and analyzes its distribution of power. We understand why the special entanglements of geometric beams have multiple polarizations from theory and experiment.

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