本論文探討內部圓錐折射現象 (internal conical refraction)。首先利用Hamilton原理分析圓錐折射的光線成像，並解釋Poggendorff暗環及Raman光斑，再透過Belsky和Khapalyuk的精確近軸理論計算高斯光束通過雙軸晶體 (biaxial crystal) 的內部圓錐折射現象。利用近軸近似的條件，計算光所走的光程長度，並考慮入射光束腰寬的影響，透過疊加各角度平面波計算出光強度。
對於非磁性、非手徵 (non-chiral) 的雙軸晶體，在線性偏振情況下，於焦平面 (focal image plane) 位置出射光為新月形 (crescent-shaped) 的圓環，並證明偏振方向會隨著方位角改變以一種迷人的方式旋轉。當雙軸晶體加入手徵性 (chirality) 後，光束會在某一特定位置聚焦，且對應於線偏振光的成像會有類似咖啡漩渦 (coffee swirl) 的圖形。本研究所發現的特殊偏振旋轉現象可供實驗學家在未來進行實驗上的驗證。

In this thesis, we discuss internal conical refraction. First, we review the theory based on Hamilton’s principle for analyzing the image of rays due to conical refraction, and explain the mechanisms of Poggendorff’s dark ring and Raman spot. Next, we calculate the image intensity using Belsky and Khapalyuk’s exact paraxial theory for internal conical refraction of a Gaussian beam passing through a biaxial crystal along an optical axis. In this theory, the image intensity is obtained from absolute-squaring the electric-displacement field (D-field), which is a superposition of many plane waves of D-field, each carries a phase of the optical path length of the corresponding ray, modified by the Gaussian phase function of the incident beam of a given beam width.
For non-magnetic and non-chiral biaxial crystal, if the input beam is linearly polarized, the output beam is a crescent-shaped ring on the focal image plane, and the polarization direction rotates in a fascinating way as the azimuth angle changes. When chirality is included, the imaging beam will focus at some specific location, and for linearly polarized incident beam the image pattern resembles a coffee swirl. The behavior of the polarization rotation found in this study might be an interesting phenomenon to be verified experimentally in the future.

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