本論文旨在運用數值模擬的方法，將模擬退火法應用在數位雷射的架構中，我們透過分析輸出光束與目標光束的差異，即時調整共振腔中的相位邊界，將其最佳化以實現不同形狀的結構光束輸出。於此方法中，透過評價函數的設定可以調整輸出光束的場型和能量之間最佳化的權重，在保持輸出光束的光強度分布盡可能接近目標光束的同時，最小化能量損失。本研究展示了三種光束的模擬結果：平頂光束、圓錐形光束和球形光束。在僅最佳化輸出光束場型的情況下，儘管這三種光束在共振腔中產生了一些繞射損失，但平頂光束和圓錐形光束的輸出場型與目標場型的相關係數可達0.9999，球形光束則可達0.9990，顯示本方法所選擇的相位邊界能夠使共振腔的輸出場型非常接近目標場型。這項研究為雷射光束整形提供了一種有效的方法，並展示了其在製作各種結構光束的潛力。

In this study, we presented a numerical method to generate various shapes of structured beams in a digital laser system using the simulated annealing (SA) method. This method enables real-time adjustments to the phase boundaries within the resonant cavity by analyzing deviations between the output beam and the desired target beam. The relative weight between the output beam's field pattern and energy can be tuned by configuring the cost function to minimize the energy losses while maintaining the output beam's light intensity distribution as closely as possible to the target beam. This research introduces an efficient strategy for laser beam shaping and demonstrates its potential in fabricating various structured beams.

1 M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and Manipulation of Three-Dimensional Optically Trapped Structures," Science 296 (5570), 1101-1103 (2002).
2 R. L. Eriksen, V. R. Daria, P. J. Rodrigo, and J. Glückstad, "Computer-controlled orientation of multiple optically-trapped microscopic particles," Microelectronic Engineering 67-68, 872-878 (2003).
3 V. R. Daria, R. L. Eriksen, and J. Glückstad, "Dynamic optical manipulation of colloidal systems using a spatial light modulator," Journal of Modern Optics 50 (10), 1601-1614 (2003).
4 F. Iwata, M. Kaji, A. Suzuki, S. Ito, and H. Nakao, "Local electrophoresis deposition of nanomaterials assisted by a laser trapping technique," Nanotechnology 20 (23), 235303 (2009).
5 K. Obata, J. Koch, U. Hinze, and B. N. Chichkov, "Multi-focus two-photon polymerization technique based on individually controlled phase modulation," Optics Express 18 (16), 17193-17200 (2010).
6 L. Yang, A. El-Tamer, U. Hinze, J. Li, Y. Hu, W. Huang, J. Chu, and B. N. Chichkov, "Parallel direct laser writing of micro-optical and photonic structures using spatial light modulator," Optics and Lasers in Engineering 70, 26-32 (2015).
7 N. J. Jenness, K. D. Wulff, M. S. Johannes, M. J. Padgett, D. G. Cole, and R. L. Clark, "Three-dimensional parallel holographic micropatterning using a spatial light modulator," Optics Express 16 (20), 15942-15948 (2008).
8 S. D. Gittard, A. Nguyen, K. Obata, A. Koroleva, R. J. Narayan, and B. N. Chichkov, "Fabrication of microscale medical devices by two-photon polymerization with multiple foci via a spatial light modulator," Biomed. Opt. Express 2 (11), 3167-3178 (2011).
9 T. Takai, H. Nakao, and F. Iwata, "Three-dimensional microfabrication using local electrophoresis deposition and a laser trapping technique," Optics Express 22 (23), 28109-28117 (2014).
10 T. Matsuura, T. Takai, and F. Iwata, "Local electrophoresis deposition assisted by laser trapping coupled with a spatial light modulator for three-dimensional microfabrication," Japanese Journal of Applied Physics 56 (10), 105502 (2017).
11 C. David, J. Wei, T. Lippert, and A. Wokaun, "Diffractive grey-tone phase masks for laser ablation lithography," Microelectronic Engineering 57-58, 453-460 (2001).
12 G. Valentin, B. Gérard, V. Colette, C. Anthony, V. Francis, J. Yves, and L. Michel, "Fast dynamic interferometric lithography for large submicrometric period diffraction gratings production," Optical Engineering 52 (9), 091712 (2013).
13 P. Visconti, C. Turco, R. Rinaldi, and R. Cingolani, "Nanopatterning of organic and inorganic materials by holographic lithography and plasma etching," Microelectronic Engineering 53 (1), 391-394 (2000).
14 T. D. Huang, and T. H. Lu, "Controlling an optical vortex array from a vortex phase plate, mode converter, and spatial light modulator," Opt. Lett. 44 (16), 3917-3920 (2019).
15 X. Chen, S. Liu, Z. Lin, Z. Chen, and J. Pu, "Dual-cavity digital laser for intra-cavity mode shaping and polarization control," Optics Express 26 (14), 18182-18189 (2018).
16 A. S. Rao, T. Morohashi, W. R. Kerridge-Johns, and T. Omatsu, "Generation of higher-order Laguerre–Gaussian modes from a diode-pumped Pr3+:LiYF4 laser with an intra-cavity spherical aberration," J. Opt. Soc. Am. B 40 (2), 406-411 (2023).
17 C. Hu, Y. Xiao, Y. He, Y. Liu, Y. Song, and X. Tang, "Optimization of brightness in a Nd:YAG laser by maximizing the single-mode power factor with an intra-cavity spatial light modulator," Applied Optics 61 (6), 1482-1491 (2022).
18 D. Wang, B. Jin, Y. Wang, P. Jia, D. Cai, and Y. Gao, "Adaptive Flattop Beam Shaping With a Spatial Light Modulator Controlled by the Holographic Tandem Method," IEEE Photonics Journal 8 (1), 1-7 (2016).
19 S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, "A digital laser for on-demand laser modes," Nature Communications 4 (1), 2289 (2013).
20 C.-Y. Huang, K.-C. Chang, and S.-C. Chu, "Experimental Investigation of Generating Laser Beams of on-Demand Lateral Field Distribution from Digital Lasers," Materials 12 (14), 2226 (2019).
21 S.-C. Chu, Y.-X. Fu, K.-C. Chang, and C.-Y. Huang, "Generating a geometric structure light field from a digital laser by specifying a laser cavity phase boundary with a Gaussian-convoluted target field," Optics Express 29 (22), 35980-35992 (2021).
22 Z. Fayyaz, N. Mohammadian, F. Salimi, A. Fatima, M. R. R. Tabar, and M. R. N. Avanaki, "Simulated annealing optimization in wavefront shaping controlled transmission," Applied Optics 57 (21), 6233-6242 (2018).
23 A. Einstein, "Zur Quantentheorie der Strahlung (On the Quantum Theory of Radiation)." Physika Zeitschrift 18, 121-128 (1917).
24 T. H. Maiman, "Stimulated Optical Radiation in Ruby," Nature 187, 493-494 (1960).
25 S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by Simulated Annealing," Science 220 (4598), 671-680 (1983).
26 J. W. Goodman, Introduction to Fourier Optics, Chap. 4, pp. 67 (W. H. Freeman, 2005).
27 M. Endo, S. Yamaguchi, T. Uchiyama, and T. Fujioka, "Numerical simulation of the w-axicon type optical resonator for coaxial slab CO2 lasers," Journal of Physics D: Applied Physics 34, 68 (2001).
28 M. Endo, M. Kawakami, K. Nanri, S. Takeda, and T. Fujioka, "Two-dimensional simulation of an unstable resonator with a stable core," Applied Optics 38 (15), 3298-3307 (1999).
29 M. Endo, "Numerical simulation of an optical resonator for generation of a doughnut-like laser beam," Optics Express 12 (9), 1959-1965 (2004).
30 Matlab, "corrcoef" https://www.mathworks.com/help/matlab/ref/corrcoef.html.
31 N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, "Equation of State Calculations by Fast Computing Machines," The Journal of Chemical Physics 21 (6), 1087-1092 (1953).
32 Matlab, "Evaluating Goodness of Fit" https://www.mathworks.com/help/curvefit/evaluating-goodness-of-fit.html.
33 Matlab, "rmse" https://www.mathworks.com/help/matlab/ref/rmse.html.