光學漩渦(Optical vortex)在中心具有相位基點，有著環形結構，光束內部光子還帶有軌道角動量(Orbital angular momentum)。在近年來雷射應用與物理研究有備受矚目，在生醫、顯微鏡、顆粒處理、天文學與光通訊上有應用的方面。有許多方式建構光學漩渦，可簡單分為腔內或腔外激發出光束，利用π ⁄ 2 轉換器、全相片、q-plate、螺旋相位板等等來激發出光學漩渦，但至今仍沒有一個有效轉換光束的方法。
本研究將提出 2-fold 簡併型渦流雷射，架構為接近半球型共振腔(near hemi-spherical resonator)直接由腔內產生高階OV。首先在沒有腔內SPP的情況下，可以利用泵浦光束故意偏離腔軸的方式激發雷射晶體，產生拉蓋爾高斯模式的OV，其拓撲電荷（TC）為1且符號（正或負）為隨機出現，並且容易跳動。此外，我們利用錐狀透鏡對泵浦光束進行整形，則可以產生簡併態的高階LG光束。然而其相位結構為不穩定。當引入腔內SPP時，單次馳返的拉蓋爾高斯模式無法存在於這樣的共振腔中。共振腔支持的模態為經兩次馳返並通過SPP調製的V形軌道的光束，此模態為一多次馳返橫向模式（MPT模式）。OV光束在本渦流諧振器中產生的是鎖相MPT模式的疊加而不是LG模式，且OV的拓撲電荷與SPP的拓撲電荷（TC）相同。產生鎖相的原因為這些MPT模態在雷射晶體中具有高度重疊並且發生強烈的耦合，經過受激發射過程，可以使共振的光束之間的相位關系固定，其相位關係相位結構遵循Kuramoto模型。在實驗中當腔內SPP拓撲電荷是正1時，產生的渦旋光束前方正形成之拓撲電荷為正1，而在向後方的拓撲電荷是0。改置入拓撲電荷為正2的SPP，則激光器渦旋光束前方正形成之拓撲電荷為正1，而在向後方的拓撲電荷是負1，雖然未如預期產生拓撲電荷為正2的光學漩渦，這些OV具有穩定的相位結構。推測是由於拓撲電荷為正2的光學漩渦不符合最低能量條件。

Optical vortex (OV) is more than a beam of donut-shaped intensity profile. With phase singularities in the center, OVs thus carry orbital angular momentum (OAM) in the photons within which distinguishes OV beams from vector beams of radail or azimuthal polarizations. The unique property of OV beam have attracted growing attentions due to the wide range of promising applications including biomedical Science, microscopy, particle manipulation, astronomy and cosmology and optical communications. Although a number of techniques are devised to generate OV with helical wavefront by beam conversion, such as π ⁄ 2 conversion, holograms, anisotropic media (q-plate), spiral phase plate, but not one of these approaches achieve a satisfactory conversion efficiency and beam quality. This work proposes to use a near hemi-spherical resonator of 2-fold degeneracy for the direct generation of high order OV without and with intracavity spiral phase element, for example SPP(charges), for direct high azimuthal order OV excitation.
Without intra-cavity SPP, OV in the form of Laguerre Gaussian mode of topological charge (TC) with undetermined sign can be generated as the pumping beam is deliberately off-set from the cavity axis. Moreover, by shaping the pump beam with an axicon lens, degenerate high order LG beams are experimentally demonstrated with unstable phase structure.
When an intra-cavity SPP is introduced, single pass transverse verse mode such as LG modes are forbiden. In such a cavity any one of the beamlets inside retraces itself twice to complete a closed orbital following a V-shaped trajectories, and beamlet is modulated in by the SPP. This cavity therefore support the oscillation of multi-pass transverse mode (MPT mode). The OV beam created in the proposed vortex resonator is the superposition of phase-locked MPT modes instead of LG modes and will thus mimic the topological charge (TC) of the SPP. For the MPT modes following the closed v-shaped orbitals, they overlap and are strongly coupled in the laser crystal as a result of stimulated emission process. When the intra-cavity SPP is of a positive unit TC, the generated vortex beam carries the positive unit TC in the forward direction while zero TC in the backward direction. Replacing the SPP of TC number of 2, the laser emits vortex beam of a positive unit TC in the forward direction and a negative unit TC in the backward direction. It is believed the phase structure of the OV beam follows the Kuramoto model with the minimum energy criterion.

中文論文審定書 i
英文論文審定書 ii
摘要 iii
Abstract v
目錄 vii
圖目錄 x
表目錄 xiv
第一章 緒論 1
1-1 光學漩渦簡介 1
1-2 文獻回顧 4
1-2-1 Hermite-Gauss轉換至Laguerre-Gaussian 4
1-2-2 q-plate 產生高階Poincaré sphere 光束 6
1-2-3空間光調變器置入共振腔調變光場 8
1-2-4內部環形共振腔產生渦流光束 10
1-2-5晶體為增益介質輸出環形光場 12
1-2-6螺旋相位元件產生渦流光束 15
1-3 研究動機 19
1-4論文架構 22
第二章、簡易簡併行雷射共振腔 23
2-1光線追跡法分析共振腔與Laguerre-Gaussian分析 23
2-2產生渦流雷射材料介紹 28
2-2-1泵浦光源 28
2-2-2準直聚焦透鏡組與錐狀透鏡 30
2-2-3增益介質之晶體特性 32
2-2-4螺旋相位調變元件 37
2-3簡併型共振腔介紹與激發出高斯光束 39
2-3-1螺旋相位板放置腔外調變之干涉分析 40
2-4結果與討論 42
2-4-1簡併型共振腔實驗架構 42
2-4-1-1、光場相位干涉特性分析 43
2-4-2環形泵浦之簡併型共振腔實驗架構 45
2-4-2-1光場分佈之分析 46
2-4-2-2相位干涉特性分析 47
2-2-4-3光束發散特性分析 49
第三章、2-fold高拓樸電荷數簡併型渦流雷射 50
3-1簡併型共振腔內放置螺旋相位板之光束傳波模型 50
3-2簡併型共振腔在腔內放置調變元件架構圖 51
3-2-1驗證渦流共振腔光束傳播軌跡 52
3-2-1-1光束間的同調性驗證之分析 54
3-2-2相位干涉與光譜的分析 56
3-2-3光束發散特性 58
3-2-4高拓樸電荷數之渦流雷射輸出結果 59
3-2-5光場穩定度測試 60
3-3L型共振腔設計圖並驗證高拓樸電荷數之光學漩渦 64
3-3-1前後端輸出光場進行相位干涉分析 66
3-3-3遮蔽軌跡光束觀察相位分析與相位穩定度 68
3-3-3光束發散特性分析 70
3-3-4不同拓樸電荷數產生渦流光束輸出結果 70
3-4結果分析與討論 72
3-4-1不同腔長在腔內調變光學漩渦 72
3-4-2相位干涉之分析 73
3-4-3光束發散特性 74
第四章、結論與未來展望 75
4-1結論 75
4-2未來展望 76
參考文獻 77

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