前人於雷射共振腔內系統中放入空間調製器，成功地在半導體雷射激發固態雷射產生多種典型模態的光束輸出，如: 厄米-高斯光束（Hermite-Gaussian beams）、 拉蓋爾-高斯光束（Laguerre-Gaussian beams）、平頂光束 (flat-top beam) 和艾瑞光束 (Airy beam) 等模態，並稱之為數位雷射 (Digital Laser)。該發明是共振腔內雷射模態動態控制的極大的進步。但對於在數位雷射中產生贗無繞射光束，以及產生具有拓普電荷的近軸雷射光束之相位調製機制如何無法放入數位雷射架構中等問題尚未被解決。
本論文的主要工作是利用數值模擬的方法探討數位雷射之模態選擇性激發的機制。主要分為三個部分，第一部分探討贗無繞射光束在數位雷射中的產生。在先前數位雷射的架設中，在雷射共振腔系統中無法加入增益範圍的控制。橫向均勻的增益分佈會使的高階贗無繞射光束在雷射共振腔內無法順利收斂到一個穩定的場型，無法順利產生高模態純度的贗無繞射光束。在本研究中，改進先前數位雷射的架構為末端光激發固態L型數位雷射。末端光激發的激發形式，經由控制聚焦光束大小和離軸位置，可以達到控制增益範圍的目的。並成功在此數位雷射架構中產生四種類形的贗無繞射光束，且模態純度高達99%以上。利用數值的方法驗證，q=5三階偶數馬修-高斯光束增益位置選擇在指定光束的最亮點可以有最好的模態匹配和能量提取效率。相較於增益位置於最亮點處，增益位置在第三亮點處，相對輸出能量下降75%。增益範圍直徑從100µm增加到1200µm，模態純度從99%以上下降到85%以下。證實增益範圍控制的必要性。a=5偶數拋物線-高斯光束在雷射共振腔有效腔長誤差±20%，模態純度還維持在98%以上。
第二部分探討結構光束在數位雷射中的產生。使用相同的雷射架構，把高斯光束視為結構光束的最小像素。增益範圍在光軸上固定直徑200µm，利用光腰大小100µm的高斯光束，成功產生九點陣列、倒三角形、矩形和tw字母等結構光束。並利用高斯光束的瑞立範圍z_R，探討在傳播範圍±0.5zR內，光束場型維持設之結構光束場型。
第三部分進一步改進數位雷射架構，從原本共振腔內的光束往返一次只通過一次空間光調制器，改為光束往返一次通過兩次空間光調制器。可在相位調制加入螺旋相位的調制，可以順利利用空間光調制器來控制產生之雷射光束的拓撲電荷的階數或螺旋相位的左右旋性。恩司-高斯光束或馬修-高斯光束與本身旋轉90度正交光束疊加而成的光渦流陣列雷射光束成功在此數位雷射產生。並成功產生不同模態階數且高斯光腰或橫向波數皆不同之疊加態光束，且產生之模態純度皆為99%以上。

This study proposes a complete method for selectively exciting any specified beam in a digital laser. The proposed method is verified by using the Endo method to simulate the convergent oscillating light field in a digital laser cavity. We propose two new digital lasers. The first configuration is an end-pumped digital laser with a planar spatial light modulator (SLM) L-shaped resonator. This setup provides control of the laser gain range and can be easily controlled by off-axis pumping. The numerical simulation results show that the nearly nondiffracting beams can be generated directly from the digital laser only by controlling the phase boundary conditions of the laser cavity provided by the spatial light modulator when the lateral range of the laser gain is controlled at the same time. Second configuration is the digital laser of dual SLM n-shaped resonators which provides the feasibility of spiral phase modulation. In this setup, the vortex array beams and arbitrarily complex laser mode can be generated. Gain range control or dual SLM modulation significantly improves the controllability of digital lasers and the mode purity of the output laser mode.

1. M. Woerdemann, C. Alpmann, and C. Denz, “Optical assembly of microparticles into highly ordered structures using Ince–Gaussian beams,” Appl. Phys. Lett. 98, 111101 (2011).
2. M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21–28 (2002).
3. N. Eckerskorn, L. Li, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “Hollow Bessel-like beam as an optical guide for a stream of microscopic particles,” Opt. Express 21, 30492–30499 (2013).
4. C. Alpmann, R. Bowman, M. Woerdemann, M. Padgett, and C. Denz, “Mathieu beams as versatile light moulds for 3D micro particle assemblies,” Opt. Express 18, 26084–26091 (2010).
5. P. Visconti, C. Turco, R. Rinaldi, R. Cingolani, “Nanopatterning of organic and inorganic materials by holographic lithography and plasma etching,” Microelectron. Eng. 53, 391-394 (2000).
6. C. David, J. Wei, T. Lippert, and A. Wokaun, “Diffractive grey-tone phase masks for laser ablation lithography,” Microelectron. Eng. 57–8, 453-460 (2001).
7. V. Gate, G. Bernaud, C. Veillas, “Fast dynamic interferometric lithography for large submicrometric period diffraction gratings production,” Opt. Eng. 52, 091712 (2013).
8. J. Ni, C. Wang, C. Zhang, Y. Hu, L. Yang, Z. Lao, B. Xu, J. Li, D. Wu, and J. Chu, “Three-dimensional chiral microstructures fabricated by structured optical vortices in isotropic material,” Light Sci. Appl. 6, e17011 (2017).
9. S. J. Zhang, Y. Li, Z. P. Liu, J. L. Ren, Y. F. Xiao, H. Yang, and Q. H. Gong, “Two-photon polymerization of a three dimensional structure using beams with orbital angular momentum,” Appl. Phys. Lett. 105, 061101 (2014).
10. 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,” Opt. Express 16, 15942-15948 (2008).
11. I. Iglesias, “Using axicons for depth discrimination in excitation-emission laser scanning imaging systems,” Opt. Laser Technol. 95, 8-13 (2017).
12. X. L. Chen, C. Zhang, P. Lin, K. C. Huang, J. M. Liang, J. Tian and J. X. Cheng, “Volumetric chemical imaging by stimulated Raman projection microscopy and tomography,” Nat. Commun. 8, 15117 (2017).
13. N. Shiokawa, E. Tokunaga, “Quasi first-order Hermite Gaussian beam for enhanced sensitivity in Sagnac interferometer photothermal deflection spectroscopy,” Opt. Express 24, 1961-1974 (2016).
14. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987).
15. F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
16. J. C. Gutiérrez-Vega, and M. A. Bandres, “Helmholtz–Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005).
17. S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4, 2289 (2013).
18. L. Burger, I. Litvin, S. Ngcobo, A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17, 015604 (2015)
19. Z. Bouchal, J. Wagner, and M. Chlup, ‘‘Self-reconstruction of a distorted nondiffracting beam,’’ Opt. Commun. 151, 207–211 (1998).
20. G. Milione, A. Dudley, T. A. Nguyen, O. Chakraborty, E. Karimi, A. Forbes, and R. R. Alfano, “Measuring the self-healing of the spatially inhomogeneous states of polarization of vector Bessel beams,” J. Opt. 17(3), 035617 (2015).
21. S. Li and J. Wang, “Adaptive free-space optical communications through turbulence using self-healing Bessel beams,” Sci. Rep. 7, 43233 (2017).
22. C. López-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 45(6), 068001 (2006).
23. R. J. Hernández-Hernández, R. A. Terborg, I. Ricardez-Vargas, and K. Volke-Sepúlveda, “Experimental generation of Mathieu-Gauss beams with a phase-only spatial light modulator,” Appl. Opt. 49, 6903–6909 (2010).
24. A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based Bessel resonator: Analytical description and experiment,” J. Opt. Soc. Am. A 18(8), 1986–1992 (2001).
25. J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001).
26. M. B. Alvarez-Elizondo, R. Rodr´ıguez-Masegosa, and J. C. Guti´errez-Vega, “Generation of Mathieu-Gauss modes with an axicon-based laser resonator,” Opt. Express 16, 18770-18775 (2008).
27. K. Tokunaga, S.-C. Chu, H.-Y. Hsiao, T. Ohtomo, and K. Otsuka, “Spontaneous Mathieu-Gauss mode oscillation in micro-grained Nd:YAG ceramic lasers with azimuth laser-diode pumping,” Laser Phys. Lett. 6, 635–638 (2009).
28. S. Li and J. Wang, “Adaptive free-space optical communications through turbulence using self-healing Bessel beams,” Sci. Rep. 7, 43233 (2017).
29. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
30. J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003).
31. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
32. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000).
33. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004).
34. I. Litvin and A. Forbes, “Intra–cavity flat–top beam generation,” Opt. Express 17, 15891-15903 (2009).
35. I. A. Litvin and A. Forbes, “Gaussian mode selection with intracavity diffractive optics,” Opt. Lett. 34, 2991-2993 (2009).
36. P. A. Be´langer, R. L. Lachance, and C. Pare, “Super-Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett. 17, 739-741 (1992).
37. J. R. Leger, D. Chen, and Z. Wang, “Diffractive optical element for mode shaping of a Nd:YAG laser,” Opt. Lett. 19, 108-110 (1994).
38. S. Ngcobo, K. Alt-Ameur, N. Passilly, A. Hasnaoui, and A. Forbes, “Exciting higher-order radial Laguerre–Gaussian modes in a diode-pumped solid-state laser resonator,” Appl. Opt. 52, 2093-2101 (2013).
39. H. Yang, J. Meng, W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015).
40. J. Bourderionnet, A. Brignon, J. –P. Huignard, A. Delboulbe, and B. Loiseaux, “Spatial mode control of a diode-pumped Nd:YAG laser by an intracavity liquid-crystal light valve,” Opt. Lett. 26, 1958-1960 (2001).
41. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453-488 (1961).
42. A. Fox and T. Li, “Modes in a maser interferometer with curved mirrors,” Proc. IEEE 51, 80-89 (1963).
43. A. G. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron. 4, 460-465 (1968).
44. A. Bhowmik, “Closed-cavity solutions with partially coherent fields in the space-frequency domain,” Appl. Opt. 22, 3338–3346 (1983).
45. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2004), Chap. 4.
46. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2004) pp.97–101.
47. G. Stephan and M. Trumper, “Inhomogeneity effects in a gas laser,” Phys. Rev. A 28, 2344–2362 (1983).
48. J. Seurin and S. L. Chuang, “Discrete Bessel transform and beam propagation method for modeling of verticalcavity surface-emitting lasers,” J. Appl. Phys. 82, 2007–2016 (1997).
49. T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, “Electromagnetic field computation in semiconductor laser resonators,” J. Opt. Soc. Am. A 23, 906–911 (2006).
50. J. Chao, B. Li, Y. Cheng, and Y. Wang, “Simulation of optical field in laser resonators cavity by eigenvector method,” Opt. Laser Technol. 39, 490–499 (2007).
51. N. Elkin, A. Napartovich, D. Vysotsky, and V. Troshchieva, “Round-trip operator technique applied for optical resonators with dispersion elements,” in Numerical Methods and Applications, Vol. 4310 of Lecture Notes in Computer Science (Springer Berlin, 2007), pp. 542–549.
52. W. Kong, A. Sugita, and T. Taira, “Generation of Hermite–Gaussian modes and vortex arrays based on two-dimensional gain distribution controlled microchip laser,” Opt. Lett. 37, 2661 (2012).
53. T. Sato, Y. Kozawa, and S. Sato, “Transverse-mode selective laser operation by unicursal fast-scanning pumping,” Opt. Lett. 40, 3245 (2015).
54. J. Dong, J. Ma, Y. Y. Ren, G. Z. Xu and A. A. Kaminskii, “Generation of Ince-Gaussian beams in highly efficient, nanosecond Cr, Nd:YAG microchip lasers,” Laser Phys. Lett. 10, 085803 (2013).
55. S. C. Chu, Y. T. Chen, K. F. Tsai, and K. Otsuka, “Generation of high-order Hermite-Gaussian modes in end-pumped solid-state lasers for square vortex array laser beam generation,” Opt. Express 20(7), 7128–7141 (2012).
56. S.-C. Chu and K. Otsuka, “Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers,” Opt. Express 15(25), 16506–16519 (2007).
57. T. Ohtomo, S. Chu, and K. Otsuka, “Generation of vortex beams from lasers with controlled Hermite- and Ince- Gaussian modes,” Opt. Express 16, 5082–5094 (2008).
58. M. Endo, “Numerical simulation of an optical resonator for generation of a doughnut-like laser beam,” Opt. Express 12, 1959-1965 (2004).
59. M. Endo, M. Kawakami, K. Nanri, S. Takeda, and T. Fujioka, “Two-dimensional simulation of an unstable resonator with a stable core,” Appl. Opt. 38, 3298-3307 (1999).
60. M. Endo, S. Yamaguchi, T. Uchiyama, and T. Fujioka, “Numerical simulation of the w-axicon type optical resonator for coaxial slab CO2 lasers,” J. Phys. D 34, 68 (2001).
61. A. E. Siegman, Lasers (University Science Books, 1986), pp.295.
62. J. Hoffnagle and C. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39, 5488-5499 (2000).
63. V. Gandhi, J. Orava, H. Tuovinen, T. Saastamoinen, J. Laukkanen, S. Honkanen, and M. Hauta-Kasari, “Diffractive optical elements for optical identification,” Appl. Opt. 54(7), 1606-1611 (2015).
64. T. Gissibl, M. Schmid, and H. Giessen, “Spatial beam intensity shaping using phase masks on single-mode optical fibers fabricated by femtosecond direct laser writing,” Optica 3(4), 448-451 (2016).
65. Lee, W.M., Yuan, X.-C., Cheong, W.C., “Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,” Opt. Lett. 29, 1796-1798 (2004)
66. R. S. Rodrigues Ribeiro, P. Dahal, A. Guerreiro, P. Jorge, and J. Viegas, “Optical fibers as beam shapers: from Gaussian beams to optical vortices,” Opt. Lett. 41, 2137-2140 (2016).
67. C. Béchet, A. Guesalaga, B. Neichel, V. Fesquet, H. González-Núñez, S. Zúñiga, P. Escarate, and D. Guzman, “Beam shaping for laser-based adaptive optics in astronomy,” Opt. Express 22, 12994-13013 (2014).
68. O. Boyko, Th. A. Planchon, P. Mercere, C. Valentin, and Ph. Balcou, “Adaptive shaping of a focused intense laser beam into a doughnut mode,” Opt. Commun. 246, 131-140 (2005).
69. D. Brousseau, J. Drapeau, M. Piché, and E. F. Borra, “Generation of Bessel beams using a magnetic liquid deformable mirror,” Appl. Opt. 50, 4005-4010 (2011).
70. Y.-X. Ren, Z.-X. Fang, L. Gong, K. Huang, Y. Chen, and R.-D. Lu, “Dynamic generation of Ince-Gaussian modes with a digital micromirror device,” J. Appl. Phys. 117, 133106 (2015).
71. Y. Chen, Z.-X. Fang, Y.-X. Ren, L. Gong, and R.-D. Lu, “Generation and characterization of a perfect vortex beam with a large topological charge through a digital micromirror device,” Appl. Opt. 54, 8030-8035 (2015).
72. V. Arrizón, U. Ruiz, D. Sánchez-de-la-Llave, G. Mellado-Villaseñor, and A. S. Ostrovsky, “Optimum generation of annular vortices using phase diffractive optical elements,” Opt. Lett. 40, 1173-1176 (2015).
73. J. A. Davis, E. Carcole, and D. M. Cottrell, “Nondiffracting interference patterns generated with programmable spatial light modulators,” Appl. Opt. 35, 599-602 (1996).
74. S. C. Chu, C. S. Yang, and K. Otsuka, “Vortex array laser beam generation from a Dove prism-embedded unbalanced Mach-Zehnder interferometer,” Opt. Express 16(24), 19934–19949 (2008).
75. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
76. Alexander Cerjan and Charles Cerjan, "Orbital angular momentum of Laguerre–Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 28, 2253-2260 (2011)
77. K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclass. Opt. 4, S82 (2002).
78. D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
79. R. Oron, Y. Danziger, N. Davidson, A. A. Friesem, and E. Hasman, “Laser mode discrimination with intra-cavity spiral phase elements,” Opt. Commun. 169(1–6), 115–121 (1999).
80. R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182, 205–208 (2000).
81. Y. Lin, C. Yeh, and H. Lee, "Vortex laser generation in a degenerate optical resonator with an intra-cavity spiral phase plate," in Laser Congress 2017 (ASSL, LAC), OSA Technical Digest (online) (Optical Society of America, 2017), paper JTu2A.52.