雷射共振腔系統提供了比傳統固態雷射更多元的變化性，藉由腔長與激發源(pumping source)離軸變化可產生各式各樣的雷射本徵態。本論文主要研究的方向為c-cut Nd：YVO4(摻釹釩酸釔)雷射所產生的三維六點環形模態。此種雷射模態為Laguerre-Gaussian modes疊加而成，其本徵態在特定的條件下，雷射模態在遠場時會產生左旋光(left-handed light)與右旋光(right-handed light)。而在近場時因為幾何光束會有重疊的情況，此時會產生單一光點同時擁有左旋與右旋光的現象。在單一的固態雷射系統能同時產生左旋光與右旋光是一件非常特別的情形，應用上能發展於冷原子系統(cold atom system)相關的研究中。
雷射共振腔裝置可以從理論推知各種穩定簡併態的相位延遲δ(phase retardation)。藉由c-cut Nd：YVO4的雙折射特性，可以將ne與no帶入雙折射理論公式計算出等效折射率neff，如此一來即可了解相位延遲δ下的折射角θ值。理論上能符合簡併共振腔條件的折射角存在很多組，本實驗也確實完整找到每一組對照的相位延遲δ之圖像。藉由理論計算能先訂出實驗的目標，接著透過實驗去驗證理論上的計算。藉由四分之一波片與線偏振片的配合檢驗與多次實驗觀察，確定了三維六點環形模態同時具有「左旋圓偏振和右旋圓偏振」的模態。由計算得到實驗與理論的相位延遲δ值相互比較的結果發現實驗與理論計算相當吻合。並且確認了多組不同階數的情形。

本實驗一併對不同三維模態進行觀察，在本文後段中將展現橢圓形或是複合形式的模態，並觀察其相關光學特性。藉由實驗與理論的相互印證，能更加的了解此種特殊的雷射模態形成的物理機制，在未來基礎科學的研究應用上能給予相當的貢獻與啟發。

The laser cavity system mentioned in the thesis is different from the traditional solid state laser. We can manipulate the pump offset to generate a variety of the laser beams with complex spatial structures. We investigate the geometric beams generated from a c-cut Nd:YVO4 laser. The Nd:YVO4 acts as a birefringence crystal. In this work, we focused on six spots of the circular geometric mode. In the far field, the structured laser beam is left-handed circularly polarized and right-handed circularly polarized at the same time. This particular property has many applications, like cold atom system.
The phase retardation (δ) of a stable geometric mode can be derived by the birefringence theory. According to the birefringence property of Nd:YVO4, the effective refractive index neff can be calculated by the coefficient ne and no. It leads to get the refraction angle θ of the geometric mode. In the numerical simulation, we fit twelve orders of six spots of circular geometric modes and get the patterns.
In the experiment we use the quarter-wave plate (QWP) and the linear plate(LP) to detect the laser polarization. Experimental results reveal that a geometric mode possesses circularly polarized states in opposite directions at the same time and the superposition of orthogonally polarized geometric beams can be generated systematically by controlling the off-axis magnitude. The numerical results have a good agreement with the experimental results. The research may make some contributions for the application of structured beams.

[1] Andrews, David L, Structured light and its applications: An introduction to phase-structured beams and nanoscale optical forces, Academic Press (2011)
[2] M. Nisoli, S. De Silvestri, O. Svelto, R. Szipöcs, K. Ferencz, Ch. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Optics Letter 22, 522 (1997)
[3] AE Siegman, Lasers, University Science Books, (1986).
[4] S.J. van Enk, G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Optics Communications 94, 147 (1992)
[5] A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Physical Review Letters 24, 156 (1970)
[6] A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and anipulation of single cells using infrared laser beams,” Nature 330, 6150 (1987)
[7] 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,” Physical Review A 45, 8185 (1992)
[8] B. L. Johnson and G. Kirczenow, “Enhanced dynamical symmetries and quantum degeneracies in mesoscopic quantum dots: Role of the symmetries of closed classical orbits,” Euro physics Letters 51, 367 (2000)

[9] Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, “Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces,” Physical review letters 96, 213902 (2006)
[10] B. L. Johnson and G. Kirczenow, “Spatial transformation of Laguerre–Gaussian laser modes,” Journal of Modern Optics 48, 783 (2000)
[11] A. Normatov, P. Ginzburg, N. Berkovitch, G. M. Lerman, A. Yanai, U. Levy and M. Orenstein, “Efficient coupling and field enhancement for the nano-scale: plasmonic needle,” Optics express 18, 14079 (2010)
[12] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics : Wiley Series in Pure and Applied Optics, NY, NY (1991).
[13] Kuhn, J. Kelin, Laser engineering (1998)
[14] 盧廷昌、王興宗著。半導體雷射導論。五南圖書出版股份有限公司台北(2008）。
[15] 國立中央大學晶體生長與分析實驗室。2014,05,13 <http://lhpg138.me.ncu.edu.tw>
[16] 銘金科技股份有限公司-產品頁面。2014,05,16
<http://tw.acesuppliers.com/company/information_6664.html>
[17] H. Zhanga, X Menga, L Zhua, C. Wangb, Y. T. Chowb and M. Lua, “Growth, spectra and influence of annealing effect on laser properties of Nd:YVO4 crystal Optical Materials,” Optical Materials 1, Issues 1, pp. 25-30 (2000)

[18] 福建福晶科技股份有限公司-產品頁面。2014,05,16 <http://gb.castech.com/products_detail/&productId=d822e599-ba7b-4ab2-ab6b-4a4560e78bad&comp_stats=comp-FrontProducts_list01-1316410981381.html>
[19] F. L. Pedrotti, L. S. Pedrotti, L. M. Pedrotti, Introduction to optics, Englewood Cliffs: Prentice-Hall (1993)..
[20] Smith, F. Graham, T. A. King, and D. Wilkins, Optics and photonics: an introduction. John Wiley & Sons (2007).
[21] R. K. Bhaduri, S. Li, K Tanaka and J. C. Waddington, “Quantum gaps and classical orbits in a rotating two-dimensional harmonic oscillator,” Journal of Physics A 27, 553 (1994)
[22] 馬軍山, 侯琳琳, 付東翔, 陳家璧與莊松林, “雙熒光標記生物芯片激光共聚焦檢測系統,” 光學精密工程 13, 727 (2005)
[23] Milonni, W. Peter and J. H. Eberly, Laser Physics: Laser Resonators and Gaussian Beams, New York (1988)
[24] H. Kogelnik and L. Tingye, Applied Optics: Laser beams and resonators, optics infobase (1966)
[25] Hodgson, Norman and H. Weber, Laser resonators and beam propagation. Springer (2005)
[26] 楊寶賡。雷射工程。新文京開發出版股份有限公司台北(2010)。
[27] Sasada Lab研究室頁面。2014,05,18
<http://www.phys.keio.ac.jp/guidance/labs/sasada/research/orbangmom-en.html>

[28] Y. F. Chen, C. H. Jiang, Y. P. Lan and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Physical Review A 69, 053807 (2004)
[39] V. Bužek and T. Quang, “Generalized coherent state for bosonic realization of SU (2) Lie algebra,” Journal of the Optical Society of America B 6, 2447 (1989)
[30] J. Banerji and G. S. Agarwal, “Non-linear wave packet dynamics of coherent states of various symmetry groups,” Optics express 5, 220 (1999)
[31] M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J.P. Woerdman. “Astigmatic laser mode converters and transfer of orbital angular momentum,” Optics Communications 96, 123 (1993)
[32] I. V. Zozoulenko and K.-F. Berggren, “Quantum scattering, resonant states, and conductance fluctuations in an open square electron billiard,” Physical Review B 56, 6931 (1997)
[33] R. Narevich, R. E. Prange and Oleg Zaitsev, “Square billiard with a magnetic flux,” Physical Review E 62, 2046 (2000)