角動量雷射為具有光學軌道角動量、螺旋型的波前與相位分布及光旋渦之雷射光，其擁有與傳統雷射截然不同的空間分布及光學特性。將不同參數之角動量雷射做疊加，能製造單一光場擁有多個光旋渦之光源。因角動量雷射的參數可以不斷增加，與傳統控制光學偏振及其基本參數的實驗相比，前者多了更高的自由度，也不易受到實驗上的侷限。角動量雷射具有的特殊性質，在不同領域中，它能提供不同的應用價值。舉例來說，討論光學操控(optical manipulation)的領域時，角動量雷射由於螺旋型的波前以及其特有的波印亭向量(Poynting vector)，使其能產生光力矩(optical torque)並抓住粒子，且帶動粒子隨著其強度分布做逆時針或順時針的旋轉，而其方向取決於角動量雷射的參數。討論光通訊的領域時，角動量雷射能攜帶比傳統雷射更高的資訊量。傳統光纖通訊，使用時間範疇(time domain)的方式(1或0)，1為光傳播訊號，0為光不傳播訊號。藉由此方式傳播數據，需有很大的重複率來提高傳播速度，而角動量雷射能在採取1的動作時，給予更高的資訊量，進而提高整體傳播速度。另外，角動量雷射也能應用在大氣傳播中(free propagation)，因其具有特殊的螺旋型波前，使其不易受到空氣擾動(air turbulence)的影響，進而能進行遠距離的傳播，再者，若將角動量雷射與空間非齊性(space-inhomogeneous)的偏振分佈做結合(向量場之角動量雷射)，更能提高其傳播效率。角動量雷射也能應用於量子解密或量子糾纏中，此應用在未來通訊協定或是宇宙科學的方面都提供很高的應用潛力。因此，利用不同光學系統製造空間齊性偏振分佈(Scalar beam)及空間非齊性(Vector beam)的角動量雷射為本論文的重要工作。在本論文中，空間齊性偏振分佈的角動量雷射能藉由半柱面型共振腔搭配腔外柱透鏡、空間光調制器搭配電腦產生之全像片(computer-generated hologram)來產生。另一方面，空間非齊性的角動量雷射可以藉由共振腔及空間光調制器與其他光學元件的結合來產生。如: 干涉儀、四分之一波片等…。另外，我們也將產生之角動量雷射光源與層狀二維材料做結合，探究其特有光學特性與材料交互作用後，產生的光致發光及聲子震動(拉曼光譜)，進而研究產生的物理甚至與元件的製程做連接。

Structured light possesses properties such as orbital angular momentum (OAM), spiral wavefront, spiral phase, and optical vortex. The crucial properties of the generated beams extend its potential applications to various research fields, such as optical manipulation, optical communication, and optical cryptography.
In general, a scalar structured light, which is an optical field with space-homogenous polarized distribution, is directly generated either from a spatial light modulator (SLM) with computer-generated holograms (CGHs) or a hemi-cylindrical resonator with an extra-resonator cylindrical lens. The order of the structured light is determined by the different grating phase of the structured light, as well as the pump offset and defocus magnitude of the laser resonator, respectively. On the other hand, a vector structured light, which is an optical field with space-inhomogeneous polarized distribution, is passively generated by combining the SLM with other optical systems such as Mach-Zehnder interferometer, dove prism, and optical retarder. It could also be directly generated from the vortex phase plate (VPP). Structured light with space-inhomogeneous polarized distribution provides an extra degree of freedom to further extend its potential applications. All states of polarization of the scalar and vector structured light are described by the fundamental Poincaré sphere and high-order Poincaré sphere (HOPs).
Meanwhile, the scalar and vector structured light beam have been utilized to interact with layered Molybdenum Disulfide (MoS2). The resulting phonon behavior in the material is analyzed using polarized Raman spectroscopy, which shows that the symmetry of the phonon can be broken by using elliptically polarized light. The exciton behavior in the layered material is also observed via the optical spectra induced by the light-matter interaction, which demonstrates that the photon energy is increased when the order of structured light is incremented. These results introduce a probable increase of the degree of freedom in various material science applications.

1. Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C. &Woerdman, J. P. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys. Rev. A 45, 8185–8189 (1992).
2. Djordjevic, I. B. Deep-space and near-Earth optical communications by coded orbital angular momentum (OAM) modulation. Opt. Express 19, 14277 (2011).
3. Gibson, G. et al. Free-space information transfer using light beams carrying orbital angular momentum. Opt. Express 12, 5448 (2004).
4. Krenn, M. et al. Communication with spatially modulated light through turbulent air across Vienna. New J. Phys. 16, 113028 (2014).
5. Fickler, R. et al. Quantum Entanglement of High Angular Momenta. Science (80-. ). 338, 640–643 (2012).
6. Leonhard, N., Sorelli, G., Shatokhin, V. N., Reinlein, C. &Buchleitner, A. Protecting the entanglement of twisted photons by adaptive optics. Phys. Rev. A 97, 012321 (2018).
7. Mirhosseini, M. et al. High-dimensional quantum cryptography with twisted light. New J. Phys. 17, 033033 (2015).
8. Padgett, M. &Bowman, R. Tweezers with a twist. Nat. Photonics 5, 343–348 (2011).
9. Tao, S. H., Yuan, X.-C., Lin, J., Peng, X. &Niu, H. B. Fractional optical vortex beam induced rotation of particles. Opt. Express 13, 7726 (2005).
10. Simpson, N. B., Allen, L. &Padgett, M. J. Optical tweezers and optical spanners with Laguerre–Gaussian modes. J. Mod. Opt. 43, 2485–2491 (1996).
11. Milione, G., Nguyen, T. A., Leach, J., Nolan, D. A. &Alfano, R. R. Using the nonseparability of vector beams to encode information for optical communication. Opt. Lett. 40, 4887 (2015).
12. Zhao, Y. &Wang, J. High-base vector beam encoding/decoding for visible-light communications. Opt. Lett. 40, 4843 (2015).
13. Milione, G. et al. 4 × 20 Gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de)multiplexer. Opt. Lett. 40, 1980 (2015).
14. Cheng, W., Haus, J. W. &Zhan, Q. Propagation of vector vortex beams through a turbulent atmosphere. Opt. Express 17, 17829 (2009).
15. Wang, Q., Sun, X. W. &Shum, P. Generating doughnut-shaped beams with large charge numbers by use of liquid-crystal spiral phase plates. Appl. Opt. 43, 2292 (2004).
16. Hui, X. et al. Ultralow Reflectivity Spiral Phase Plate for Generation of Millimeter-wave OAM Beam. IEEE Antennas Wirel. Propag. Lett. 14, 966–969 (2015).
17. Huang, T. D. &Lu, T. H. Large astigmatic laser cavity modes and astigmatic compensation. Appl. Phys. B 124, 72 (2018).
18. Lu, T. H., Huang, T. D. &Chiou, G. Y. Kaleidoscope vortex lasers generated from astigmatic cavities with longitudinal-transverse coupling. Opt. Express 26, 31464 (2018).
19. Mirhosseini, M. et al. Rapid generation of light beams carrying orbital angular momentum. Opt. Express 21, 30196 (2013).
20. Gruneisen, M. T., Miller, W. A., Dymale, R. C. &Sweiti, A. M. Holographic generation of complex fields with spatial light modulators: application to quantum key distribution. Appl. Opt. 47, A32 (2008).
21. Barreiro, J. T., Wei, T.-C. &Kwiat, P. G. Remote Preparation of Single-Photon “Hybrid” Entangled and Vector-Polarization States. Phys. Rev. Lett. 105, 030407 (2010).
22. Fickler, R., Lapkiewicz, R., Ramelow, S. &Zeilinger, A. Quantum entanglement of complex photon polarization patterns in vector beams. Phys. Rev. A 89, 060301 (2014).
23. Zhan, Q. Cylindrical vector beams: from mathematical concepts to applications. Adv. Opt. Photonics 1, 1 (2009).
24. Lu, T. H., Huang, T. D., Wang, J. G., Wang, L. W. &Alfano, R. R. Generation of flower high-order Poincaré sphere laser beams from a spatial light modulator. Sci. Rep. 6, 39657 (2016).
25. Naik, D. N., Saad, N. A., Rao, D. N. &Viswanathan, N. K. Ultrashort vortex from a Gaussian pulse – An achromatic-interferometric approach. Sci. Rep. 7, 2395 (2017).
26. Liu, R. et al. Compact, robust, and high-efficiency generator of vector optical fields. Opt. Lett. 44, 2382 (2019).
27. Marrucci, L. et al. Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications. J. Opt. 13, 064001 (2011).
28. Marrucci, L. The q-plate and its future. J. Nanophotonics 7, 078598 (2013).
29. Heckenberg, N. R., McDuff, R., Smith, C. P. &White, A. G. Generation of optical phase singularities by computer-generated holograms. Opt. Lett. 17, 221 (1992).
30. Ford, J. E., Xu, F., Urquhart, K. &Fainman, Y. Polarization-selective computer-generated holograms. Opt. Lett. 18, 456 (1993).
31. Dudley, D., Duncan, W. M. &Slaughter, J. Emerging digital micromirror device (DMD) applications. in MOEMS Display and Imaging Systems (ed. Urey, H.) 14 (2003).
32. Chen, Y., Fang, Z.-X., Ren, Y.-X., Gong, L. &Lu, R.-D. Generation and characterization of a perfect vortex beam with a large topological charge through a digital micromirror device. Appl. Opt. 54, 8030 (2015).
33. Ross, W. E., Psaltis, D. &Anderson, R. H. Two-Dimensional Magneto-Optic Spatial Light Modulator For Signal Processing. Opt. Eng. 22, (1983).
34. Nolte, D. D. Resolution of electro-optic spatial light modulators: the role of lateral transport. Opt. Commun. 92, 199–204 (1992).
35. Allen, L., Courtial, J. &Padgett, M. J. Matrix formulation for the propagation of light beams with orbital and spin angular momenta. Phys. Rev. E 60, 7497–7503 (1999).
36. Beth, R. A. Mechanical Detection and Measurement of the Angular Momentum of Light. Phys. Rev. 50, 115–125 (1936).
37. Allen, P. J. A Radiation Torque Experiment. Am. J. Phys. 34, 1185–1192 (1966).
38. He, H., Friese, M. E. J., Heckenberg, N. R. &Rubinsztein-Dunlop, H. Direct Observation of Transfer of Angular Momentum to Absorptive Particles from a Laser Beam with a Phase Singularity. Phys. Rev. Lett. 75, 826–829 (1995).
39. Friese, M. E. J., Enger, J., Rubinsztein-Dunlop, H. &Heckenberg, N. R. Optical angular-momentum transfer to trapped absorbing particles. Phys. Rev. A 54, 1593–1596 (1996).
40. Garcés-Chávez, V. et al. Observation of the Transfer of the Local Angular Momentum Density of a Multiringed Light Beam to an Optically Trapped Particle. Phys. Rev. Lett. 91, 093602 (2003).
41. Dienerowitz, M., Mazilu, M., Reece, P. J., Krauss, T. F. &Dholakia, K. Optical vortex trap for resonant confinement of metal nanoparticles. Opt. Express 16, 4991 (2008).
42. Barnett, S. M. et al. On the natures of the spin and orbital parts of optical angular momentum. J. Opt. 18, 064004 (2016).
43. Fang, N., Wang, L., Guo, S. &Huang, Z. Security of polarization-shift keying chaos optical communication system. Front. Optoelectron. China 1, 64–69 (2008).
44. Benedetto, S. &Poggiolini, P. Theory of polarization shift keying modulation. IEEE Trans. Commun. 40, 708–721 (1992).
45. Gong, L. et al. Optical orbital-angular-momentum-multiplexed data transmission under high scattering. Light Sci. Appl. 8, 27 (2019).
46. Wang, J. et al. Ultra-High 230-bit/s/Hz Spectral Efficiency using OFDM/OQAM 64-QAM Signals over Pol-Muxed 22 Orbital Angular Momentum (OAM) Modes. in Optical Fiber Communication Conference W1H.4 (OSA, 2014).
47. Beijersbergen, M. W., Allen, L., van derVeen, H. E. L. O. &Woerdman, J. P. Astigmatic laser mode converters and transfer of orbital angular momentum. Opt. Commun. 96, 123–132 (1993).
48. Huang, T. D. &Lu, T. H. Large astigmatic laser cavity modes and astigmatic compensation. Appl. Phys. B 124, 72 (2018).
49. Lu, T. H. &Wu, Y. C. Observation and analysis of single and multiple high-order Laguerre-Gaussian beams generated from a hemi-cylindrical cavity with general astigmatism. Opt. Express 21, 28496 (2013).
50. Arnaud, J. A. &Kogelnik, H. Gaussian Light Beams with General Astigmatism. Appl. Opt. 8, 1687 (1969).
51. Chen, Y. F. et al. Characterizing the propagation evolution of wave patterns and vortex structures in astigmatic transformations of Hermite–Gaussian beams. Laser Phys. 28, 015002 (2018).
52. Lu, T. H., Huang, T. D. &Chiou, G. Y. Kaleidoscope vortex lasers generated from astigmatic cavities with longitudinal-transverse coupling. Opt. Express 26, 31464 (2018).
53. Chen, Y. F., Lu, T. H., Su, K. W. &Huang, K. F. Devil’s Staircase in Three-Dimensional Coherent Waves Localized on Lissajous Parametric Surfaces. Phys. Rev. Lett. 96, 213902 (2006).
54. Lu, T. H., Lin, Y. C., Chen, Y. F. &Huang, K. F. Three-Dimensional Coherent Optical Waves Localized on Trochoidal Parametric Surfaces. Phys. Rev. Lett. 101, 233901 (2008).
55. Huang, T. D. &Lu, T. H. Partial Poincaré beams generated from wavelength-mismatched vortex plates. Opt. Express 25, 33184 (2017).
56. Lu, T. H., Huang, T. D., Wang, J. G., Wang, L. W. &Alfano, R. R. Generation of flower high-order Poincaré sphere laser beams from a spatial light modulator. Sci. Rep. 6, 39657 (2016).
57. Huang, T. D. &Lu, T. H. Controlling an optical vortex array from a vortex phase plate, mode converter, and spatial light modulator. Opt. Lett. 44, 3917 (2019).
58. Allen, L., Courtial, J. &Padgett, M. J. Matrix formulation for the propagation of light beams with orbital and spin angular momenta. Phys. Rev. E 60, 7497–7503 (1999).
59. Lin, Y. C., Lu, T. H., Huang, K. F. &Chen, Y. F. Generation of optical vortex array with transformation of standing-wave Laguerre-Gaussian mode. Opt. Express 19, 10293 (2011).
60. Chu, S.-C., Chen, Y.-T., Tsai, K.-F. &Otsuka, K. Generation of high-order Hermite-Gaussian modes in end-pumped solid-state lasers for square vortex array laser beam generation. Opt. Express 20, 7128 (2012).
61. Bolotin, K. I. et al. Ultrahigh electron mobility in suspended graphene. Solid State Commun. 146, 351–355 (2008).
62. Papageorgiou, D. G., Kinloch, I. A. &Young, R. J. Mechanical properties of graphene and graphene-based nanocomposites. Prog. Mater. Sci. 90, 75–127 (2017).
63. Singh, V. et al. Graphene based materials: Past, present and future. Prog. Mater. Sci. 56, 1178–1271 (2011).
64. Balog, R. et al. Bandgap opening in graphene induced by patterned hydrogen adsorption. Nat. Mater. 9, 315–319 (2010).
65. Dvorak, M., Oswald, W. &Wu, Z. Bandgap Opening by Patterning Graphene. Sci. Rep. 3, 2289 (2013).
66. Nourbakhsh, A. et al. Bandgap opening in oxygen plasma-treated graphene. Nanotechnology 21, 435203 (2010).
67. Zhou, S. Y. et al. Substrate-induced bandgap opening in epitaxial graphene. Nat. Mater. 6, 770–775 (2007).
68. Ouyang, F., Peng, S., Liu, Z. &Liu, Z. Bandgap Opening in Graphene Antidot Lattices: The Missing Half. ACS Nano 5, 4023–4030 (2011).
69. Manzeli, S., Ovchinnikov, D., Pasquier, D., Yazyev, O.V. &Kis, A. 2D transition metal dichalcogenides. Nat. Rev. Mater. 2, 17033 (2017).
70. Wang, Q. H., Kalantar-Zadeh, K., Kis, A., Coleman, J. N. &Strano, M. S. Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nat. Nanotechnol. 7, 699–712 (2012).
71. Kang, J., Zhang, L. &Wei, S.-H. A Unified Understanding of the Thickness-Dependent Bandgap Transition in Hexagonal Two-Dimensional Semiconductors. J. Phys. Chem. Lett. 7, 597–602 (2016).
72. Zhu, Z. Y., Cheng, Y. C. &Schwingenschlögl, U. Giant spin-orbit-induced spin splitting in two-dimensional transition-metal dichalcogenide semiconductors. Phys. Rev. B 84, 153402 (2011).
73. Kormányos, A., Zólyomi, V., Drummond, N. D. &Burkard, G. Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides. Phys. Rev. X 4, 011034 (2014).
74. Xia, F., Wang, H. &Jia, Y. Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics. Nat. Commun. 5, 4458 (2014).
75. Kim, J. et al. Anomalous polarization dependence of Raman scattering and crystallographic orientation of black phosphorus. Nanoscale 7, 18708–18715 (2015).
76. Ling, X. et al. Low-Frequency Interlayer Breathing Modes in Few-Layer Black Phosphorus. Nano Lett. 15, 4080–4088 (2015).
77. Huang, S. et al. In-Plane Optical Anisotropy of Layered Gallium Telluride. ACS Nano 10, 8964–8972 (2016).
78. Huang, T.-D., Simbulan, K. B., Chiang, Y.-F., Lan, Y.-W. &Lu, T.-H. Symmetry breaking of in-plane Raman scattering by elliptically polarized light in MoS2. Phys. Rev. B 100, 195414 (2019).
79. Zhang, X. et al. Phonon and Raman scattering of two-dimensional transition metal dichalcogenides from monolayer, multilayer to bulk material. Chem. Soc. Rev. 44, 2757–2785 (2015).
80. Yu, H., Liu, G.-B., Gong, P., Xu, X. &Yao, W. Dirac cones and Dirac saddle points of bright excitons in monolayer transition metal dichalcogenides. Nat. Commun. 5, 3876 (2014).
81. Qiu, D. Y., Cao, T. &Louie, S. G. Nonanalyticity, Valley Quantum Phases, and Lightlike Exciton Dispersion in Monolayer Transition Metal Dichalcogenides: Theory and First-Principles Calculations. Phys. Rev. Lett. 115, 176801 (2015).
82. Bowman, D. et al. High-fidelity phase and amplitude control of phase-only computer generated holograms using conjugate gradient minimisation. Opt. Express 25, 11692 (2017).
83. Moreno, I., Davis, J. A., Hernandez, T. M., Cottrell, D. M. &Sand, D. Complete polarization control of light from a liquid crystal spatial light modulator. Opt. Express 20, 364 (2012).
84. Schaibley, J. R. et al. Valleytronics in 2D materials. Nat. Rev. Mater. 1, 16055 (2016).
85. Zeng, H., Dai, J., Yao, W., Xiao, D. &Cui, X. Valley polarization in MoS2 monolayers by optical pumping. Nat. Nanotechnol. 7, 490–493 (2012).
86. Mak, K. F., He, K., Shan, J. &Heinz, T. F. Control of valley polarization in monolayer MoS2 by optical helicity. Nat. Nanotechnol. 7, 494–498 (2012).