Manipulation of Gaussian Beam Based on Fractional Schrdinger Equation

WANG Dong-dong; ZANG Feng; LI Lu

doi:10.3788/gzxb20194810.1048005

[1] LAUGHLIN R B. Anomalous quantum hall effect: an incompressible quantum fluid with fractionally chared excitations［J］. Physical Review Letters, 1983, 50: 1395-1398.

[2] WEN Jian-ming, ZHANG Yong, XIAO Min. The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics［J］. Advances in Optics and Photonics, 2016, 5: 83-130.

[3] LEONID P R,LIU Xin-yu, JACEK K F. The fractional a.c. Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles［J］. Nature Physics, 2012, 8: 795-799.

[4] OLIVAR-ROMERO F, ROSAS-ORTIZ O. Factorization of the quantum fractional oscillator［J］. Journal of Physics: Conference Series, 2016, 698: 012025.

[5] LASKIN N. Fractional quantum mechanics andLévy path integrals［J］. Physics Letters A, 2000, 268: 298-305.

[6] LASKIN N. Fractional quantum mechanics［J］. Physical Review E, 2000, 62: 3135-3145.

[7] LASKINN. Fractional Schrdinger equation［J］. Physical Review E, 2002, 66: 056108.

[8] DONG Jian-ping, XU Ming-yu. Some solutions to the space fractional Schrdinger equation using momentum representation method［J］. Journal of Mathematical Physics, 2007, 48: 072105.

[9] OLIVEIRA E C DE, VAZ JR J. Tunneling in fractional quantum mechanics［J］. Journal of Physics A: Mathematical and Theoretical, 2011, 44: 185303.

[10] LUCHKO Y. Fractional Schrdinger equation for a particle moving in a potential well［J］. Journal of Mathematical Physics, 2013, 54: 012111.

[11] LI Zhi-feng, LIU Jin-jin, LUCHA W,et al. SCHBERL F F. Relativistic harmonic oscillator［J］. Journal of Mathematical Physics, 2005, 46: 103514.

[12] KOWALSKI K, REMBIELINSKI J. Relativistic massless harmonic oscillator［J］. Physical Review A, 2010, 81: 012118.

[13] LRINCZI J, MALECKI J. Spectral properties of the massless relativistic harmonic oscillator［J］. Journal of Differential Equations, 2012, 253: 2846.

[14] LONGHI S. Frational Schrdinger equation in optics［J］. Optics Letters, 2015, 40: 1117-1120.

[15] ZHANG Yi-qi, LIU Xin, BELIC M R, et al. Propagation dynamics of a light beam in a fractional Schrdinger equation［J］. Physical Review Letters, 2015, 115: 180403.

[16] ZHANG Yi-qi, ZHONG Hua, BELIC M R, et al. Diffraction-free beams in fractional Schrdinger equation［J］. Scientific Reports, 2016, 6: 23645.

[17] ZHANG Yi-qi, ZHONG Hua, BELIC M R,et al. PT symmetry in a fractional Schrdinger equation［J］. Laser Photonics Reviews, 2016, 10: 526-531.

[18] ZHANG Li-fu, LI Chu-xin, ZHONG Hai-zhe,et al. Propagation dynamics of super-Gaussian beanms in fractional Schrdinger equation: from linear to nonlinear regimes［J］. Optics Express, 2016, 24: 14406-14418.

[19] LIEMERT A, KIENLE A. Fractional Schrdinger equation in the presence of the linear potential［J］. Mathematics, 2016, 4: 31.

[20] HUANG Xian-wei, DENG Zhi-xiang, FU Xi-quan. Dynamics of finite Airy beams modeled by the fractional Schrdinger equation with a linear potential［J］. Journal of the Optical Society of America B, 2017, 34: 976-982.

[21] HUANG Chang-ming, DONG Liang-wei. Beam propagation management in fractional Schrdinger equation［J］. Scientific Reports, 2017, 7: 5442.

[22] ZHANG Yi-qi, WANG Rong, ZHONG Hua, et al. Optical Bloch oscillation and Zener tunneling in the fractional Schringer equation［J］. Scientific Reports, 2017, 7: 17872.

[23] ZANG Feng, WANG Yan, LI Lu.Dynamics of Gaussian beam modeled by fractional Schrdinger equation with a variable coefficient［J］. Optics Express, 2018, 26: 23740-23750.

[24] GUO Bo-ling, HUANG Dai-wen. Existence and stability of standing waves for nonlinear fractional Schrdinger equations［J］. Journal of Mathematical Physics, 2012, 53(8): 083702.

[25] WANG Dong-ling, XIAO Ai-guo, YANG Wei. Crank-Nicolson difference scheme for the coupled nonlinear Schrdinger equations with the Riesz space fractional derivative［J］. Journal of Computational Physics, 2013, 242: 670-681.

[26] KLEIN C, SPARBER C, MARKOWICH P. Numerical study of fractional nonlinear Schrdinger equations［J］. Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences, 2014, 470(2172): 20140364.

[27] WANG Peng-de, HUANG Cheng-ming, ZHAO Long-bin. Point-wise error estimate of a conservative difference scheme for the fractional Schrdinger equation［J］. Journal Computational and Applied Mathematics, 2016, 306: 231-247.

[28] WANG Peng-de, HUANG Cheng-ming. An energy conservative difference scheme for the nonlinear fractional Schrdinger equations［J］. Journal Computational Physics, 2015, 293: 238-251.

[29] ZHANG Li-fu, He Zeng-hui, CONTI C, et al. Modulational instability in fractional nonlinear Schrdinger equation［J］. Commun Nonlinear Sci Numer Simulat, 2017, 48: 531-540.

[30] DONG Liang-wei, HUANG Chang-ming. Double-hump solitons in fractional dimensions with a pt-symmetric potential［J］. Optics Express, 2018, 26: 10509-10518.

[31] YAO Xian-kun, LIU Xue-ming. Solitons in the fractional Schrdinger equation with parity-time-symmetric lattice potential［J］. Photonics Research, 2018, 6: 875-879.

[32] ZHONG Wei-ping, BELIC M R, ZHANG Yi-qi. Accessible solitons of fractionaldimension［J］. Annals of Physics, 2016, 368: 110-116.

[33] XIAO Jing, TIAN Zhao-xia, HUANG Chang-ming, et al. Surface gap solitons in a nonlinear fractional Schrdinger equation［J］. Optics Express, 2018, 26: 2650-2658.

[34] CHEN Man-na, ZENG Shi-hao, LU Da-quan, et al. Optical solitons, self-focusing, and wave collapse in a space-fractional Schrdinger equation with a kerr-type nonlinearity［J］. Physical Review E, 2018, 98: 022211.

[35] HUANG Chang-ming, SHANG Ce, LI Jing, et al. Localization and Anderson delocalization of light in fractional dimensions with a quasi-periodic lattice［J］. Optics Express, 2019, 27: 6259-6269.

[36] ZHANG Li-fu, ZHANG Xiang, WU Hao-zhe,et al. Anomalous interaction of Airy beams in the fractional nonlinear Schrdinger equation［J］. Optics Express, 2019, 27: 27936-27945.

[37] ZANG Feng, WANG Yan, LI Lu.Self-induced periodic interfering behavior of dual Airy beam in strongly nonlocal medium［J］. Optics Express, 2019, 27: 15079-15090.