近年來量子糾纏以及量子通訊越來越重要，去年的諾貝爾物理獎也頒給了相關的主題，為了實現光學量子糾纏，一次產生多光源且具有偏振依賴相關性 (correlation) 的現象是很重要的。因此，本工作以共振腔產生多重雷射源，除了探討多個雷射源的偏振關聯性之外，這些雷射源的模態及偏振目前都能有效且穩定的被操縱。
我們以柱面鏡組成凹平共振腔，並以雙折射晶體c-cut Nd:YVO_4 為增益介質，產生多重偏振雷射源，如Hermite-Gaussian (HG) 模態以及簡併態。由於使用雙折射晶體，共振腔可以產生較為豐富的偏振組態，調控離軸可使HG 模態的階數增加，其偏振皆保持線性偏振，且會隨著階數增加而有規律的變化。給予相同量值的正負離軸會得到一樣的HG 模態，但是線性偏振方向會相差90度。當離軸程度更大時，所產生的高階模態中央的部分維持線性偏振，外圍則觀察到橢圓偏振狀態，這是來自於發散角變大，雙折射現象引發的相位差所致。
在特殊腔長下雷射模態會形成簡併態，為多個能量相同的HG 模態疊加而成，其腔內軌跡可以古典幾何光學解釋。隨著離軸增加，模態會向外伸展，垂直輸出的光源保持線性偏振。而具有相同發散角但朝著不同方向輸出的光源，偏振態從線性偏振變為橢圓偏振，當發散角夠大時，又會變回線性偏振。並且稍微增加腔長時會使所有偏振態具有幾乎90度的偏轉。
若把簡併態的偏振結果繪製於龐加萊球 (Poincaré sphere) 上可發現，具有相同發散角但不同輸出方向的光源的偏振，會位於南北半球的相對位置上，這個多重雷射的偏振在龐加萊球上的現象非常相似於在布洛赫球 (Bloch sphere) 的兩個糾纏量子位元 (qubits)。因此，使用共振腔產生具有特殊偏振關聯的多重偏振雷射在光學量子糾纏的工作裡具有一定的潛力與應用。

Quantum entanglement and quantum communication are becoming increasingly important. Even last year, the Nobel Prize was awarded for research in this field. To achieve optical quantum entanglement, it is crucial to have multiple laser sources with correlated polarization states.
In this work, a cylindrical-plano laser cavity with a c-cut Nd:YVO_4 crystal is used to generate multi-laser sources, like Hermite-Gaussian modes and degenerate modes. Attribute to the birefringence gain medium, the modes possess versatile polarization configurations. The order of Hermite-Gaussian modes can be tuning by increasing the offsets, the polarization states is all linear and the angle increase when the order increase, it still observes same mode while giving the offsets with opposite direction, but the polarization states possess a π phase shift. As the order increase, the polarization state of the outer part of the mode becomes elliptical polarized.
Degenerate states can be found at special cavity length, the modes extend as the offsets increase, the trajectories can be explained by the geometrical optics. As the offsets increase, the polarization states of the normal incident ray consistently remain linear polarized; the polarization states of rays with an incident angle change from linear polarized to elliptical polarized, and back to linear polarized again. Slightly increase the offsets will tune the polarization states for a π phase shift.
The polarization results are mapped onto the Poincaré sphere, the polar state of mode with same incident angle but different direction is on the opposite side of the hemisphere, the behavior of the polarization states on Poincaré sphere is similar to the entangled qubits on the Bloch sphere, hence the multi-beams generated by cylindrical cavity have a potential candidate for quantum optics.

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