光學漩渦為一環型光束，除了具有環型結構外，這個光束光子內部同時具有定義完善的軌道角動量。由於光學漩渦的獨特特性以及應用，在現今的研究中引起越來越多的關注，其中包括顯微鏡、顆粒處理、天文學、宇宙學或者是Sub-Peta赫茲的光通訊上，儘管在現今有許多建構光學漩渦的方式，例如螺旋相位板、全相片或者是Q板來產生光學漩渦，但到目前為止還沒有一個有效轉換光束的方法。
光傳輸矩陣目前可以廣泛被運用在分析近軸光束共振腔特性，但將之用在方相角對稱破壞(Azimuthal symmetry breaking, ASB)的非均勻共振腔中則是個難題，因此，在這項研究中，我們建構出一個迭代法與ASB傳輸矩陣來解決雷射共振腔的非線性特徵值問題，利用這個方法可以有效的分辨ASB共振腔不同週期(或模態)N的軌跡，或調整共振腔參數如腔鏡曲率半徑、腔長或者是ASB元件之後的軌跡。
我們這裡還提出一特定相對角動量的概念，主要是在粒子模型軌跡來獲得，而這裡則是用來研究光子經過螺旋相位板後在軌道上獲得之角動量，主要是藉由探討螺旋相位板兩邊特定相對角動量的間距來直接表示螺旋相位板攜帶的拓樸電荷。

Optical vortex (OV) is more than a beam of donor-shaped intensity pro- file. It carries well-defined orbital angular momentum (OAM) in the photons within. The unique property of OV beam have attracted growing attentions due to the wide range of promising applications including microscopy, particle manipulation, astronomy, cosmology and sub-Peta hertz bit-rate optical communications. Although a number of techniques are devised to generate OV with helical wavefront from gaussian beams by exploiting external resonator conversion element such as spiral phase plate (SPP), holograms, anisotropic media (Q-plate), not one of these approaches achieve a satisfactory conversion efficiency and beam quality.
Ray transfer matrix (RTM) are widely adopted technique to analyses the property of a laser resonator of paraxial bundles of lights. Nonetheless, the inhomogeneous feature of azimuthal symmetry breaking (ASB) element make the resonator design a difficult problem. In this study we develop an iterative approach to solve the nonlinear eigenvalue problem in laser
resonator with intra-cavity ASB elements within. This approach efficiently identify the trajectories of stable periodic orbitals (modes) of ASB laser resonators of arbitrary order N. Manipulation of the trajectories is accomplished by tuning the cavity parameters such as radius of curvature of the cavity mirror, cavity length and the position of the ASB element inside the cavity.
Notably the specific relative angular momentum (SRAM, angular mom- entum over mass) in the orbital is obtained with a proposed specious particle model for the trajectories. The SRAM introduced by the SPP is investigated to be linearly to the pitch height of the SPP. It is in good agreement in wave-optics picture, in which the pitch directly indicates the topological charge carried by the SPP.
Further phase space characterization of the cavity orbitals is considered for possible chaotic orbitals and scar modes to mimic the quantum billiard problem in ASB laser resonator in classical way.

中文審定書 i
摘要 ii
目錄 v
圖次 vii

第一章 緒論 1
1-1 光學漩渦簡介 1
1-4 文獻回顧 3
1-4-1 不同模態疊加 3
1-4-2 利用全相片(Hologram)產生光學漩渦 4
1-4-3 利用Q板(Q-plate)產生光學漩渦 5
1-3 研究動機 6
1-4 論文架構 8
第二章 光線追跡法研究具有螺旋相位板之雷射共振腔 9
2-1 光線追跡法理論基礎 9
2-2 螺旋相位版之光轉化矩陣 11
2-3 非均勻光傳輸矩陣的非線性特徵值問題 14
2-4 光線追跡法特性研究 17
2-4-1 不同模態之光線追跡 17
2-4-2 光線追跡之共振腔設計 18
第三章 光角動量 24
3-1 光角動量概述 24
3-2 光追跡法之光角動量 25
3-3 模擬結果與分析 28
3-4 結果分析與討論 32
第四章 光場分析 33
4-1 光線追跡轉換光場 33
4-2 一般高斯光束在共振腔之光場 35
4-3 拉蓋爾高斯光束模態分析 37
4-4 結果分析與討論 40
第五章 結論與未來展望 48
5-1 結論 48
5-2 未來展望 49
參考文獻 50

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