臺灣博碩士論文加值系統

同調態在微觀(量子；波)與巨觀(古典；光束)之間的連結上一直以來都扮演著非常重要的角色，而無論在介觀系統的實驗或理論研究上，也都證實了簡併或近似簡併的同調態會產生使波函數落在週期性古典軌跡上的現象。而今許多雷射系統都已經能觀察到類似的量子現象。
本論文主要是利用在離軸激發(off-axis pumping)下的二極體激發摻釹釩酸釔(Nd:YVO4)雷射來探討雷射中的波粒雙重性，並且首次利用 inhomogeneous Helmholtz equation去理論分析當縱向模態頻率和橫向模態頻率的比值為一個分數(fractional)且能量為簡併時，激發光源的分布對於雷射共振模態上的影響。由實驗結果也清楚顯示分數簡併(fractional degeneracy)與波粒雙重性之間的關係，當雷射共振腔長接近簡併腔長時，雷射模態的分布會明顯落在光束軌跡上。而當同調態考慮進非等向性(astigmatism)因素時，也發現微小的對稱破壞會誘發精細的簡併結構，並且使三維同調波落在非平面的週期性軌跡。更進一步地，實驗結果也顯示簡併且分布落在週期軌跡的同調波會產生的穩定的鎖模效應。
此外，藉由隨時間演進的量子波動和隨空間傳遞的同調光在數學上的等價性，光學與力學的相似性也能夠被證實。而本文也藉由物質波的繞射理論來分析二維準晶格圖案，並且利用光學繞射實驗來驗證準晶格圖案的量子理論與循環現象。

Coherent state is of significance for exploring the boundary between the microscopic (quantum; wave) and macroscopic (classical; ray) worlds. In mesoscopic quantum systems, experimental and theoretical studies have verified that the coherent superposition of degenerate or nearly degenerate quantum states can result in wave functions localized on classical periodic orbits. Numerous modern laser systems have been developed as analogous systems to visualize various quantum phenomena.
In this thesis several optical experiments provide fresh insights for the ray-wave duality by use of Nd-doped YVO4 laser with the off-axis pumping scheme. We originally exploit the inhomogeneous Helmholtz equation to perform a theoretical analysis for manifesting the influence of the fractional degeneracy and the pump distribution on the resonant lasing mode. We also explore thorough laser experiments that clearly reveal the relationship between the effect of fractional degeneracy and the emergence of the ray-wave duality. It has been found that the lasing modes have a preference to be localized on the periodic ray trajectories when the cavity lengths are close to the degenerate cavities. When the astigmatism is involved with the coherent superposition, we also observe the tiny symmetry breaking not only induces fine level degeneracy but also leads to nonplanar periodic orbits of three-dimensional coherent waves. Furthermore, experimental results also indicate that the structurally stable mode-locking effects generated from the coherent waves associated with periodic orbits.
In addition to laser resonators, the optical-mechanical analogy has been verified that the free-time evolution of quantum waves exhibits to be mathematically similar to the free-space propagation of the coherent light. We analyze the formation of two-dimensional quasicrystalline patterns from the quantum diffraction of matter waves, and exploit an optical diffraction experiment to analogously demonstrate the time evolution and recurrence of quasicrystalline patterns in the quantum dynamics.

摘要……………………………………………………………………………i
Abstract……………………………………………………………………iii
誌謝……………………………………………………………………………v
Contents……………………………………………………………………vii
List of Figures……………………………………………………………ix

Chapter 1 Introduction……………………………………………………1
1.1 Motivation………………………………………………3
1.2 Overview of Thesis……………………………………4
Reference …………………………………………………………6
Chapter 2 Particle-Wave Duality in Quantum Systems………………7
2.1 One-dimensional Quantum Harmonic Oscillator …9
2.2 Quantum Spectrum and Classical Trajectory……16
2.3 Two-dimensional Quantum Coherence States ……19
2.4 Stationary Coherence States………………………25
Reference…………………………………………………………27
Chapter 3 Ray-Wave Duality in Spherical Cavities ………………29
3.1 Mode-locked States in the Fabry-Pérot Cavity ………………………………………………………………………………31
3.2 Frequency Spectrum and Ray Trajectory…………35
3.3 Two-dimensional Mode-locked Modes………………37
3.4 Stationary Geometric Modes ………………………46
Reference…………………………………………………………49
Chapter 4 Exploration of Laser Modes with Ray-Wave Duality …51
4.1 Inhomogeneous Helmholtz Equation for Selective Pumping………………………………………………………………………54
4.2 Experimental Results for Planar Geometric Modes…………………………………………………………………………73
4.3 Experimental Results for Nonplanar Geometric Modes…………………………………………………………………………84
4.4 Conclusion……………………………………………101
Reference ………………………………………………………103
Chapter 5 Generation of Quasicrystalline Patterns in Optics.107
5.1 Theoretical Analysis………………………………110
5.2 Experimental Results for Generation of Quasicrystal Beams………………………………………………………117
5.3 Analogous Manifestation of the Time Evolution and Recurrence in Quantum Dynamics…………………………………130
5.4 Conclusion……………………………………………141
Reference ………………………………………………………142
Chapter 6 Summary and Future Work …………………………………145
6.1 Summary ………………………………………………147
6.2 Future Work …………………………………………148
Curriculum Vitae…………………………………………………………151
Publication List…………………………………………………………153

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