[1] [1] J. F. Nye, M. V. Berry. Dislocations in wave trains[J]. Proc. R. Soc. Ser. A, 1974, 336(1605): 165～190

[2] [2] M. S. Soskin, M. V. Vasnetsov. Singular Optics in Progress in Optics[M]. Amsterdam: Elsevier, 2001. 219～276

[3] [3] H. He, M. E. J. Friese, N. R. Heckenberg et al.. Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity[J]. Phys. Rev. Lett., 1995, 75(5) : 826～829

[4] [4] J. Rogel-Salazar, G. H. C. New, S. Chavez-Cerda. Bessel-Gauss beam optical resonator[J]. Opt. Commun., 2001, 190(1-6): 117～122

[5] [5] K. Sueda, G. Miyaji, N. Miyanagaet et al.. Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses[J]. Opt. Express, 2004, 12(15): 3548～3553

[6] [6] G. D. M. Jeffries, G. Milne, Y. Zhao et al.. Optofluidic generation of Laguerre-Gaussian beams[J]. Opt. Express, 2009, 17(20): 17555～17562

[7] [7] S. Saghafi, C. J. R. Sheppard, J. A. Piper. Characterising elegant and standard Hermite-Gaussian beam modes[J]. Opt. Commun., 2001, 191(5): 173～179

[8] [8] M. Bastiaans, T. Alieva. Propagation law for the generating function of Hermite-Gaussian-type modes in first-order optical systems[J]. Opt. Express, 2005, 13(4): 1107～1112

[9] [9] E. Karimi, G. Zito, B. Piccirillo et al.. Hypergeometric-Gaussian modes[J]. Opt. Lett., 2007, 32(21): 3053～3055

[10] [10] D. Deng, J. Li, Q. Guo. Propagation of truncated modified Laguerre-Gaussian beams[J]. Appl. Phys. B, 2010, 98(1): 211～216

[11] [11] V. Namias. The fractional order Fourier transform and its application to quantum mechanics[J]. J. Inst. Math. Appl., 1980, 25(3): 241～265

[12] [12] A. W. Lohmann. Image rotation, Wigner rotation, and the fractional Fourier transform[J]. J. Opt. Soc. Am. A, 1993, 10(10): 2181～2186

[13] [13] D. Mendlovic, H. M. Ozaktas. Fractional Fourier transforms and their optical implementation: I[J]. J. Opt. Soc. Am. A, 1993, 10(9), 1875～1881

[14] [14] Y. Zhang, B. Z. Dong, B. Y. Gu et al.. Beam shaping in the fractional Fourier transform domain[J]. J. Opt. Soc. Am. A, 1998, 15(5): 1114～1120

[15] [15] Zeng Yangsu, Zhang Yixiao, Gao Fuhua et al.. Multiple fractional Fourier transform hologram recording with holographic lens[J]. Acta Optica Sinica, 2002, 22(8): 947～951

[16] [16] Hongjie Liu, Daomu Zhao, Haidan Mao et al.. Studies of beam propagation characteristics on apertured fractional Fourier transforming systems[J]. Chin. Opt. Lett., 2004, 2(4): 206～209

[17] [17] C. Zheng. Fractional Fourier transform for a hollow Gaussian beam[J]. Phys. Lett. A, 2006, 335(2): 156～161

[18] [18] Chen Tianlu, Wu Ping, Li Haijin et al.. Two-dimensional axial symmetric Fractional Fourier transforms for elliptical gaussian beams employing Wigner distribution function[J]. Chinese J. Lasers, 2010, 37(2): 444～448

[19] [19] D. Zhao, H. Mao, H. Liu et al.. Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems[J]. Optik, 2003, 114(11): 504～508

[20] [20] Chen Senhui, Zhang Yanrong, Feng Xiaofang. Transformation properties of Hermite-Laguerre-Gaussian beam in fractional Fourier transform plane[J]. Chinese J. Lasers, 2009, 36(2): 374～378

[21] [21] C. Zhao, K. Huang, X. Lu. Propagation properties of Bessel and Bessel-Gaussian beams in a fractional Fourier transform optical system[J]. Opt. & Laser Technol., 2010, 42(2): 280～284

[22] [22] Q. Lin, Y. Cai. Fractional Fourier transform for partially coherent Gaussian-Schell model beams[J]. Opt. Lett., 2002, 27(19): 1672～1674

[23] [23] Y. Cai, Q. Lin. The fractional Fourier transform for a partially coherent pulse[J]. J. Opt. A, 2004, 6(4): 307～311

[24] [24] I. S. Gradshteyn, I. M. Ryzhik. Table of Integrals, Series, and Products[M]. New York: Academic, 2007. 706～707