||A study on the characteristics of cylindrical vector polarization and nonlinear dynamics behavior in Nd-doped laser systems
||Department of Photonics
cylindrical vector beams
本博士論文主旨為研究利用半導體雷射激發摻雜銣的雷射晶體所產生的雷射輸出特性—在不同的共振腔架構中產生圓柱向量光束以及動力學之行為探討。產生圓柱向量光束的機制皆以雷射晶體所提供本質的雙折射效應作為出發點，並且操作在穩定區邊緣，可使得非尋常光束(extraordinary ray)穩定，尋常光束(ordinary ray)不穩定；或是尋常光束穩定，非尋常光束不穩定。此時將一相位元件-軸稜錐透鏡置入共振腔中，可使得離軸模態之生成以及放大了c軸雷射晶體所貢獻的雙折射效應，當共振腔長移動時，可以觀察到偏振轉變的現象，此特性利用簡單的光束追跡即可解釋。再者，我們提出在三面鏡系統下，不需要外加任何元件，只需要共振腔設計就可以產生圓柱向量光束，此設計則根據幾何光學的ABCD矩陣以及雷射光來回一趟後達成自恰的特性，可以在不同的架構下描述其穩定區的邊界，並且隨著共振腔長的移動，從不穩定區移至穩定區可以觀察到偏振態的轉變，並計算非尋常光束(extraordinary ray)和尋常光束(ordinary ray)各自的模態疊加積分(overlap integrals)之貢獻，進而關注能量擷取效益之問題，解釋了實驗觀察到的偏振轉換行為；更進一步地，將飽和吸收體放入共振腔中即可產生以脈衝形式輸出的圓柱向量光束。另一方面，我們利用軸稜錐透鏡加上透鏡的組合將泵浦形狀修飾為瓶型光束來激發微晶片雷射，可以直接產生圓柱向量光束，此時熱透鏡的效應便不可忽視。以主動式的機制來說，在此架構只需單一元件即可達成。在動力學的研究上，我們將時序上的泵浦調制訊號進行整型，研究其抑制混沌的能力，首先使用單一弦波調制讓雷射系統進入混沌狀態，第二調制訊號為弦波、方波以及三角波，我們發現抑制混沌的區域會有不同，對於控制動態行為這是一個可用的參數，並且在抑制混沌的路徑中伴隨著兩次鎖相的現象。在理論分析部分，則以一般化的惠更斯繞射積分搭配來回一趟的ABCD矩陣來描述光場在共振腔內的傳播及速率方程式的疊代計算光在增益介值中的放大，根據不同的參數我們進行模擬與實驗互相驗證，而實驗結果與數值模擬互相吻合，此部分則為純非線性動態行為之探討。最後，我們在可產生圓柱向量光束的三面鏡雷射系統中加入調制訊號控制其動態行為，在週期二以及混沌的輸出訊號中探討其空間偏振特性與動態行為的關連性。再者，我們利用線性疊加的概念進行定性分析，成功地解釋在實驗上所觀察到的結果，我在此部份的工作是以模擬為主。已初步結果來看，發現其動態行為是屬於與空間和偏振特性有關之時序上的混沌現象。
In this dissertation, we have investigated the two characteristics of laser output in laser-diode-pumped Nd-doped laser systems that one is the generation of cylindrical vector beams under different cavity configuration, and the other one is dynamical behaviors. For our work, the mechanism of producing cylindrical vector beams is based on the intrinsic characteristic of birefringence, which is provided from laser crystal. When the laser cavity was operated nearby the boundary of stable region, the range in which the extraordinary ray (e-ray) is stable but the ordinary ray (o-ray) is unstable will be achieved, and vice versa. An intra-cavity axicon can generate the off-axis modes with ring or arc oscillation and enlarge the contribution of birefringence effect provided from the laser crystal. When the laser cavity was increased, the transformation of polarization state was observed. This feature can be explained by using ray-tracing method for both rays. Furthermore, we offered the method that without inserted any optical elements to directly realize cylindrical vector beams in three-element cavity under the specific configuration. Base on the round-trip ABCD transfer matrices and self-consistent q-parameter, theboundary of stable region can be computed and analyzed under different cavity configurations. The experimental results reveal that the transformations of polarization are observed by properly tuning cavity length from unstable to stable region. The overlap integrals for the e-ray and the o-ray were calculated with the dependence of cavity length, further concerning the extraction efficiency from pump energy. The behavior of transforming polarization can be interpreted. Further inserting the saturable absorber in the laser cavity, the pulsed type of cylindrical vector beams can be demonstrated. On the other hand, we used the bottle shape which was formed with a combination of the axicon and the lens to excite the microchip laser. It directly produces the cylindrical vector beams; however, the thermal effect cannot be ignored. For active type to generate cylindrical vector beams, just a single element can be implemented in this setup. In the study of the laser dynamics behavior, we reshaped the pump modulation with dual waveforms in time sequences to investigate the suppression of chaos. The first sinusoidal modulation with the specific conditions allows laser system to change the chaotic output. The waveform of the second modulation was varied to be sinusoidal, square, and triangle. We found that the suppression region was changed; therefore, the modulated profile can be used as a parameter to control the dynamics. In the chaos-suppressed route, double phase locking was occurred when the modulation depth of the second modulation increased. By numerical simulations, the generalized Huygens diffraction integral with the round-trip ABCD matrix was employed to describe the propagation of the optical field. The amplification of optical field through the gain medium can be calculated by using iterative method with the rate equation. According to the different parameters, the results of simulation and experiment were verified mutually. The experimental results are consistent with the numerical simulations. This part was the pure investigation of nonlinear dynamic behavior. Finally, we generated cylindrical vector beam from three-element cavity laser system, and then adding single modulation to control its dynamic behavior. The relationship between the spatial polarization characteristics and dynamic behavior in period-2 and chaos signal were further explored. Furthermore, we use the concept of linear superposition for qualitative analysis to successfully explain the observed experimental results. I am responsible for constructing the simple simulation for this part. We found that the dynamics behavior belongs to temporal chaos with the correlation between spatial and polarization. The characteristics of spatial-polarization temporal dynamics should be further researched in detail.
誌 謝 vi
List of Figures xii
List of Table xx
Chapter 1. Introduction 1
1.1 Progress in diode-laser pumped solid-state lasers 1
1.1.1 Laser crystal 2
1.2 The literature review of generation of Cylindrical Vector beams 6
1.2.1 Mathematical expressions for CVBs 7
1.2.2 Passive generation methods 11
1.2.3 Active generation methods 15
1.2.4 The principle of birefringence 20
1.3 The background of controlling chaos 22
1.4 Aim of this research 26
Chapter 2. Production of azimuthally and radially polarized off-axis laser modes with intra-cavity axicon and birefringent laser crystal 29
2.1 Experimental setup and the results of slope efficiency 29
2.2 Results and discussion 31
2.2.1 The ring-pattern beams with radial and azimuthal polarization 31
2.2.2 Investigating the off-axis characteristics. 37
2.2.3 The arc-pattern beams with radial and azimuthal polarization 38
2.3 Mode analysis of off-axis modes 39
2.3.1 Comparison of laser patterns without adding the intra-cavity axicon 44
2.4 Conclusion 45
Chapter 3. Generation and transformation of azimuthally and radially polarized beams in a diode end-pumped Nd:GdVO4 laser by use of three element cavity configuration 46
3.1 Experimental setup 47
3.2 Results of transformation between azimuthal and radial polarization 48
3.3 Analysis and discussion 53
3.4 Conclusion 56
Chapter 4. Azimuthal polarization for intrinsic lasing mode generated in a passively Q-switched Nd:GdVO4 laser with a Cr4+:YAG saturable absorber 57
4.1 Experimental setup 58
4.2 Theory 60
4.3 Results and Discussion 64
4.4 Conclusion 70
Chapter 5. Generation of radially polarized beam in a microchip Nd：YVO4 laser by shaping the pump profile 71
5.1 Experimental construction for the shape of bottle pump profile 71
5.2 Thermal lens effect 73
5.3 Results of radial polarization 77
5.4 Conclusion 81
Chapter 6. Chaos suppression and phase locking generated by reshaping pump biwaveform modulation from an Nd:YVO4 laser 82
6.1 Numerical Simulations 83
6.1.1 Model 83
6.1.2 Numerical results 85
6.2 Experimental setup with biharmonic modulation 92
6.3 Experimental results 94
6.3.1 Chaotic intensity output for first modulation with sinusoidal waveform 94
6.3.2 Chaos suppress regions for second modulation with the various modulation shapes 95
6.3.3 Phase locking to subharmonic oscillation and chaos 96
6.4 Conclusion 98
Chapter 7. Generation of cylindrical vector beams and related dynamic behaviors in a three-element cavity laser 99
7.1 Analysis method and results 99
Chapter 8. Conclusions and future works 104
8.1 Conclusions 104
8.2 Future works 108
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