本篇論文主要依據M. W. Beijersbergen和L. Allen在1993年所發表的文獻，他們發現使用一對柱狀透鏡組能讓Hermite-Gaussian mode (HG mode)經由相位的改變，將其轉換為Laguerre-Gaussian mode (LG mode)。我們使用的方法則是結合一般常見的球面型共振腔和柱狀透鏡，將球面前鏡更換為柱狀前鏡，稱之為柱面型共振腔。我們觀察此共振腔所產生的基本模態(fundamental mode)、本徵態(eigenmode)，然而柱面型共振腔只在某一軸有曲率半徑，我們得到的模態並非一般的HG modes，而是像散(astigmatic)的HG modes。因其在某一方向有特別大的發散角，我們便在柱面型共振腔外再加一個柱狀透鏡來修正各方向發散角來得到圓對稱的LG modes。

　　在某些特定腔長下，我們透過調整柱面型共振腔的離軸，能夠得到簡併共振腔所產生的高階HG modes疊加模態，經過腔外柱透鏡轉換後呈現出跟基本LG mode有不同對稱性的圖形，此種圖形為LG mode的疊加態。我們再作離焦的動作，讓增益介質上的光斑大小增加，激發出多組模態去作疊加，而得到許多更加複雜的圖形，稱之為flower type modes。

　　在實驗過程中，我們將原本使用的增益介質由a-cut晶體換成c-cut晶體，並使用偏振片來觀察其偏振情形。在理論上我們利用Huygens integral 和ABCD law來進行模擬，表現出模態在經過腔外柱狀透鏡後沿著行進方向上的變化。

In 1993 M. W. Beijersbergen and L. Allen found when Hermite-Gaussian modes pass through a pair of cylindrical lens which can change the phase of light, Laguerre-Gaussian modes would be generated. We combined hemi-spherical cavity and a pair of cylindrical lens to set up a hemi-cylindrical cavity. In our experiment, we verified that astigmatic Hermite-Gaussian modes can be generated from hemi-cylindrical cavity, and can convert to Laguerre-Gaussian modes by using an extra-cavity cylindrical lens.

In degenerate hemi–cylindrical cavity, by controlling the pump offset and the pump size, we observed the unique patterns experimentally. From degenerate cavity theory, we understand that special patterns are superposed by a set of high order Laguerre-Gaussian modes with the same degenerate frequency. Because these unique pattern look like flowers in nature, so we call them flower type modes.

When we use the a-cut Nd:YVO4, the directions of polarization of experimental results are parallel to the c-axis of the crystal. In recent research, the polarized property of c-cut Nd:YVO4 has attracted much attention in laser field. Therefore we change the gain medium from a-cut to c-cut one, and observed the polarization of patterns.

Finally, in simulation analysis we use the Huygens integral and ABCD law to clearly demonstrate the transformation of modes along the propagation directions after they passing through the extra-cavity cylindrical lens.

[1] G. Colin, L. S. Robert, and F.-A. Sonja, "Vacuum Faraday effect for electrons," New J. Phys 14, 103040 (2012).

[2] W. Cheng, X. Hou, and F. Ye, "Use of tapered amplifier diode laser for biological-friendly high-resolution optical trapping," Opt. Lett. 35, 2988-2990 (2010).

[3] Y. Takenaka, J.-i. Nishimae, M. Tanaka, and Y. Motoki, "High-power CO2 laser with a Gauss-core resonator for high-speed cutting of thin metal sheets," Opt. Lett. 22, 37-39 (1997).

[4] B. Dingel, and S. Kawata, "Speckle-free image in a laser-diode microscope by using the optical feedback effect," Opt. Lett. 18, 549-551 (1993).

[5] S. J. van Enk, and G. Nienhuis, "Eigenfunction description of laser beams and orbital angular momentum of light,"Opt Commun 94, 147-158 (1992).

[6] J. Dingjan, M. P. van Exter, and J. P. Woerdman, "Geometric modes in a single-frequency Nd:YVO4 laser," Opt Commun 188, 345-351 (2001).

[7] A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, "A quantum memory for orbital angular momentum photonic qubits," Nat Photon 8, 234-238 (2014).

[8] A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "An optical vortex as a rotating body: mechanical features of a singular light beam," J. Opt. A: Pure Appl. Opt. 6, S170 (2004).

[9] A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams," J. Opt. Soc. Am. A 20, 1635-1643 (2003).

[10] K. T. Gahagan, and G. A. Swartzlander, "Optical vortex trapping of particles," Opt. Lett. 21, 827-829 (1996).

[11] N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52-54 (1997).

[12] K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, "Second-harmonic generation and the orbital angular momentum of light," Phys. Rev. A 54, R3742-R3745 (1996)

[13] W. M. Lee, X. C. Yuan, and W. C. Cheong, "Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation," Opt. Lett. 29, 1796-1798 (2004).

[14] C. N. Alexeyev, B. P. Lapin, A. V. Volyar, and M. A. Yavorsky, "Helical-core fiber analog of a quarter-wave plate for orbital angular momentum," Opt. Lett. 38, 2277-2279 (2013).

[15] A. Y. Bekshaev, and O. V. Orlinska, "Transformation of optical-vortex beams by holograms with embedded phase singularity," Opt Commun 283, 1244-1250 (2010).

[16] J. Courtial, and M. J. Padgett, "Performance of a cylindrical lens mode converter for producing Laguerre–Gaussian laser modes," Opt Commun 159, 13-18 (1999).

[17] R. J. Wiglusz, L. Marciniak, R. Pazik, and W. Strek, "Structural and Spectroscopic Characterization of Nd3+-Doped YVO4 Yttrium Orthovanadate Nanocrystallites," Crystal Growth & Design 14, 5512-5520 (2014).

[18] http://www.chinasupply.net/optical/

[19] H. Kogelnik, and T. Li, "Laser beams and resonators," Proc IEEE 54, 1312-1329 (1966).

[20] Y. F. Chen, K. F. Huang, and Y. P. Lan, "Spontaneous transverse patterns in a microchip laser with a frequency-degenerate resonator," Opt. Lett. 28, 1811-1813 (2003).

[21] Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, "Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity," Phys. Rev. A 69, 053807 (2004).

[22] Y. F. Chen, "Lissajous and trochoidal beam generation from diode pumped solid state lasers," in Lasers and Electro-Optics Pacific Rim (CLEO-PR), 2013 Conference on(2013).

[23] T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, "Generation of multi-axis Laguerre–Gaussian beams from geometric modes of a hemiconfocal cavity," Appl. Phys. B 103, 991-999 (2011).

[24] M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt Commun 96, 123-132 (1993).

[25] T. H. Lu, and Y. C. Wu, "Observation and analysis of single and multiple high-order Laguerre-Gaussian beams generated from a hemi-cylindrical cavity with general astigmatism," Opt. Express 21, 28496-28506 (2013).

[26] T. H. Lu, Y. F. Chen, and K. F. Huang, "Generation of polarization-entangled optical coherent waves and manifestation of vector singularity patterns," Phys. Rev. E 75, 026614 (2007).

[27] H. Kogelnik, "On the Propagation of Gaussian Beams of Light Through Lenslike Media Including those with a Loss or Gain Variation," Appl. Opt. 4, 1562-1569 (1965).

[28] A. Siegman, Lasers ,University Science Boks, Mill Valley.(1986).

[29] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).

[30] A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," PRL 88, 053601 (2002).

[31] P. Galajda, and P. Ormos, "Complex micromachines produced and driven by light," Appl. Phys. Lett. 78, 249-251 (2001).

[32] D. W. Zhang, and X. C. Yuan, "Entangled double-helix phase," Opt. Lett. 28, 1864-1866 (2003).