臺灣博碩士論文加值系統

本論文研究介觀系統中具有古典-量子對應性之同調波函數的理論模式，以及雷射系統中同調結構光束的相位奇異點特徵。首先，我們推導出Hermite-Laguerre-Gaissian (HLG) 以及 Zernike 函數的積分形式，使其運算能更為高效且高精準。基於積分的方法，我們不僅將二維簡諧振盪中能的同調態以高斯函數的積分表示，並以此表示式直覺地呈現古典-量子對應現象以及相位結構。在量子彈子球檯的系統中，我們利用同調態得出古典軌跡的方程式。受此方程式啟發，我們進一步從單向激發源的非齊次Helmholtz方程式得出系統的共振模態，此模態成功地描述面射型雷射中橫向模態的結構特徵。
除了量子與古典間的對應性，我們也研究離軸激發下固態雷射系統能產生的結構性光束。利用共振模態與熱透鏡效應，我們成功地找到簡併腔中高功率幾何模態產生的最佳化條件。最佳化的幾何模態能被進一步轉換為帶有大角動量以及特殊相位奇異點的環形幾何模態。我們也提出一套理論與實驗分析的模式，以HLG模態為例，分析模態轉換中相位奇異點的產生與湮滅。最後，我們將大的損耗引入固態雷射共腔腔內，產生了不對稱分布的Laguerre-Gaussian (LG)模態。與傳統的LG模態相比，此模態能呈現重新排列的奇異點分布並帶有新的奇異點。

This thesis demonstrates various methods to construct the coherent wave representation for quantum-classical connection in the mesoscopic systems and phase singularities of coherent structure beams in laser systems. First of all, we originally introduce the integration methodology to effectively and precisely generate the well-known special functions including Hermite-Laguerre-Gaissian (HLG) functions and Zernike polynomials. Based on the integration methodology, we represent the stationary coherent states for two-dimensional (2D) quantum harmonic oscillators as an integral of Gaussian wave packet to intuitively reveal the quantum-classical correspondence and the quantum phase structures. Extending to 2D integrable quantum billiards, the stationary coherent states are exploited to extract the trajectory equations that inspire us to further derive resonant states from the inhomogeneous Helmholtz equation with unidirectional excitation for manifesting the experimental structured transverse patterns of vertical-cavity surface-emitting lasers (VCSELs).
In addition to the quantum-classical correspondence, we systematically explore the structured light generated from the diode-pumped solid state lasers with off-axis pumping scheme. By using the resonant modes derived from inhomogeneous Helmholtz equation and considering the thermal lens effect, we successfully optimize the generation of high-power, stable geometric laser beams under the degenerate cavities. The optimized geometric modes (GMs) can be further transformed to circular-GMs with large orbital angular momentum and unique phase singularities. In non-degenerate cavity, we originally propose a method to investigate the creation and annihilation of phase singularities during the astigmatic transformation of HLG modes and utilize the Mach-Zehnder interferometer to manifest the theoretical simulations. Finally, we introduce large dissipation in to the laser resonator to generate the asymmetrical Laguerre-Gaussian (LG) modes with rearranged and newly formed phase singularities with respect to the common LG modes.

摘要 i
Abstract ii
致謝 iv
Contents vi
List of Figures viii

Chapter 1 Background and General Introduction................................ 1
1.1 Motivation......................................................................................... 2
1.2 Overview of Thesis......................................................................... 6
REFERENCES ........................................................................................................... 8
Chapter 2 Developing the Integral Representation for Special Functions…………………………………………………..... 11
2.1 Generation of Hermite-Laguerre-Gaussian Modes........................... 13
2.2 Generation of Zernike Polynomials.................................................. 24
REFERENCES ........................................................................................................... 44
Chapter 3 Constructing Wave Functions for Quantum-Classical Connections……………………………………………….. 48
3.1 Wave Packet States of 2D Harmonic Oscillator............................... 50
3.2 Stationary Coherent States of 2D Harmonic Oscillator.................... 53
3.3 Stationary Coherent States and Classical Trajectories of Integrable Quantum Billiards............................................................................. 65
REFERENCES ........................................................................................................... 94
Chapter 4 Optimizing the Generation of Geometric Laser Modes...... 97
4.1 Planar and Non-Planar Geometric Modes........................................ 98
4.2 Stability range of the Cavity Length for Geometric Modes…......... 101
4.3 Optimization of the High-Power Geometric Modes......................... 110
REFERENCES ........................................................................................................... 116
Chapter 5 Exploring Phase Singularities of Structured Laser Beams………………..………………………………………. 118
5.1 Creation and Annihilation of Phase Singularities in the Beam Transformation from Hermite-Gaussian Modes to Laguerre-Gaussian Modes……………….………………….…….. 120
5.2 Generation of Asymmetrical Laguerre-Gaussian Modes in the Resonator with Large Dissipation..................................................... 135
REFERENCES ........................................................................................................... 152
Chapter 6 Summary and Future Works................................................. 154
6.1 Summary........................................................................................... 155
6.2 Future Work...................................................................................... 158
Curriculum Viate .................................................................................................. 162
Publication List .................................................................................................. 163

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