Abstract

We have investigated the propagation dynamics of super-Gaussian optical beams in fractional Schrödinger equation. We have identified the difference between the propagation dynamics of super-Gaussian beams and that of Gaussian beams. We show that, the linear propagation dynamics of the super-Gaussian beams with order m > 1 undergo an initial compression phase before they split into two sub-beams. The sub-beams with saddle shape separate each other and their interval increases linearly with propagation distance. In the nonlinear regime, the super-Gaussian beams evolve to become a single soliton, breathing soliton or soliton pair depending on the order of super-Gaussian beams, nonlinearity, as well as the Lévy index. In two dimensions, the linear evolution of super-Gaussian beams is similar to that for one dimension case, but the initial compression of the input super-Gaussian beams and the diffraction of the splitting beams are much stronger than that for one dimension case. While the nonlinear propagation of the super-Gaussian beams becomes much more unstable compared with that for the case of one dimension. Our results show the nonlinear effects can be tuned by varying the Lévy index in the fractional Schrödinger equation for a fixed input power.

© 2016 Optical Society of America

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References

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2016 (5)

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 23645 (2016).
[Crossref] [PubMed]

A. Liemert and A. Kienle, “Fractional Schrödinger equation in the presence of the linear potential,” Mathematics 4(2), 31 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT-symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

W. Zhong, M. Belić, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
[Crossref]

P. Wang, C. Huang, and L. Zhao, “Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation,” J. Comput. Appl. Math. 306, 231–247 (2016).
[Crossref]

2015 (3)

P. Wang and C. Huang, “An energy conservative difference scheme for the nonlinear fractional Schrödinger equations,” J. Comput. Phys. 293, 238–251 (2015).
[Crossref]

S. Longhi, “Fractional Schrödinger equation in optics,” Opt. Lett. 40(6), 1117–1120 (2015).
[Crossref] [PubMed]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

2014 (1)

C. Klein, C. Sparber, and P. Markowich, “Numerical study of fractional nonlinear Schrödinger equations,” Proc. Math. Phys. Eng. Sci. 470(2172), 20140364 (2014).
[Crossref] [PubMed]

2013 (1)

D. Wang, A. Xiao, and W. Yang, “Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative,” J. Comput. Phys. 242, 670–681 (2013).
[Crossref]

2012 (2)

B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53(8), 083702 (2012).
[Crossref]

C. Maher-McWilliams, P. Douglas, and P. F. Barker, “Laser-driven acceleration of neutral particles,” Nat. Photonics 6(6), 386–390 (2012).
[Crossref]

2011 (1)

2010 (1)

S. Gras, D. G. Blair, and L. Ju, “Opto-acoustic interactions in gravitational wave detectors: comparing flat-top beams with Gaussian beams,” Phys. Rev. D Part. Fields Gravit. Cosmol. 81(4), 042001 (2010).
[Crossref]

2009 (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

2008 (1)

2007 (1)

W. Wang, P. X. Wang, Y. K. Ho, Q. Kong, Y. Gu, and S. J. Wang, “Vacuum electron acceleration and bunch compression by a flat-top laser beam,” Rev. Sci. Instrum. 78(9), 093103 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (1)

G. Fibich, N. Gavish, and X. Wang, “New singular solutions of the nonlinear Schrödinger equation,” Physica D 211(3–4), 193–220 (2005).
[Crossref]

2004 (2)

G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93(3), 035003 (2004).
[Crossref] [PubMed]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

2002 (1)

N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056108 (2002).
[Crossref] [PubMed]

2000 (4)

H. Kröger, “Fractal geometry in quantum mechanics, field theory and spin systems,” Phys. Rep. 323(2), 81–181 (2000).
[Crossref]

R. Metzler and J. Klafte, “The random walk’s guide to anomalous diffusion: a fractional dynamics approach,” Phys. Rep. 339(2), 1–77 (2000).
[Crossref]

N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(3), 3135–3145 (2000).
[Crossref] [PubMed]

N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Phys. Lett. A 268(4–6), 298–305 (2000).
[Crossref]

1999 (1)

A. W. Albrecht, J. D. Hybl, S. M. Gallagher, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111(24), 10934–10956 (1999).
[Crossref]

1980 (1)

R. E. Grojean, D. Feldman, and J. F. Roach, “Production of flat top beam profiles for high energy lasers,” Rev. Sci. Instrum. 51(3), 375–376 (1980).
[Crossref] [PubMed]

Ahmed, N.

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 23645 (2016).
[Crossref] [PubMed]

Albrecht, A. W.

A. W. Albrecht, J. D. Hybl, S. M. Gallagher, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111(24), 10934–10956 (1999).
[Crossref]

Barker, P. F.

C. Maher-McWilliams, P. Douglas, and P. F. Barker, “Laser-driven acceleration of neutral particles,” Nat. Photonics 6(6), 386–390 (2012).
[Crossref]

Belic, M.

W. Zhong, M. Belić, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
[Crossref]

Belic, M. R.

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT-symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 23645 (2016).
[Crossref] [PubMed]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

Bergé, L.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

Blair, D. G.

S. Gras, D. G. Blair, and L. Ju, “Opto-acoustic interactions in gravitational wave detectors: comparing flat-top beams with Gaussian beams,” Phys. Rev. D Part. Fields Gravit. Cosmol. 81(4), 042001 (2010).
[Crossref]

Bourayou, R.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

Christodoulides, D. N.

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT-symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

Couairon, A.

G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93(3), 035003 (2004).
[Crossref] [PubMed]

Douglas, P.

C. Maher-McWilliams, P. Douglas, and P. F. Barker, “Laser-driven acceleration of neutral particles,” Nat. Photonics 6(6), 386–390 (2012).
[Crossref]

Feldman, D.

R. E. Grojean, D. Feldman, and J. F. Roach, “Production of flat top beam profiles for high energy lasers,” Rev. Sci. Instrum. 51(3), 375–376 (1980).
[Crossref] [PubMed]

Fibich, G.

Franco, M.

G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93(3), 035003 (2004).
[Crossref] [PubMed]

Gaeta, A. L.

Gallagher, S. M.

A. W. Albrecht, J. D. Hybl, S. M. Gallagher, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111(24), 10934–10956 (1999).
[Crossref]

Gao, W.

Gavish, N.

Gras, S.

S. Gras, D. G. Blair, and L. Ju, “Opto-acoustic interactions in gravitational wave detectors: comparing flat-top beams with Gaussian beams,” Phys. Rev. D Part. Fields Gravit. Cosmol. 81(4), 042001 (2010).
[Crossref]

Grojean, R. E.

R. E. Grojean, D. Feldman, and J. F. Roach, “Production of flat top beam profiles for high energy lasers,” Rev. Sci. Instrum. 51(3), 375–376 (1980).
[Crossref] [PubMed]

Grow, T. D.

Gu, Y.

W. Wang, P. X. Wang, Y. K. Ho, Q. Kong, Y. Gu, and S. J. Wang, “Vacuum electron acceleration and bunch compression by a flat-top laser beam,” Rev. Sci. Instrum. 78(9), 093103 (2007).
[Crossref] [PubMed]

Guo, B.

B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53(8), 083702 (2012).
[Crossref]

Ho, Y. K.

W. Wang, P. X. Wang, Y. K. Ho, Q. Kong, Y. Gu, and S. J. Wang, “Vacuum electron acceleration and bunch compression by a flat-top laser beam,” Rev. Sci. Instrum. 78(9), 093103 (2007).
[Crossref] [PubMed]

Huang, C.

P. Wang, C. Huang, and L. Zhao, “Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation,” J. Comput. Appl. Math. 306, 231–247 (2016).
[Crossref]

P. Wang and C. Huang, “An energy conservative difference scheme for the nonlinear fractional Schrödinger equations,” J. Comput. Phys. 293, 238–251 (2015).
[Crossref]

Huang, D.

B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53(8), 083702 (2012).
[Crossref]

Hybl, J. D.

A. W. Albrecht, J. D. Hybl, S. M. Gallagher, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111(24), 10934–10956 (1999).
[Crossref]

Ishaaya, A. A.

Jonas, D. M.

A. W. Albrecht, J. D. Hybl, S. M. Gallagher, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111(24), 10934–10956 (1999).
[Crossref]

Ju, L.

S. Gras, D. G. Blair, and L. Ju, “Opto-acoustic interactions in gravitational wave detectors: comparing flat-top beams with Gaussian beams,” Phys. Rev. D Part. Fields Gravit. Cosmol. 81(4), 042001 (2010).
[Crossref]

Kasparian, J.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

Kemao, Q.

Kienle, A.

A. Liemert and A. Kienle, “Fractional Schrödinger equation in the presence of the linear potential,” Mathematics 4(2), 31 (2016).
[Crossref]

Klafte, J.

R. Metzler and J. Klafte, “The random walk’s guide to anomalous diffusion: a fractional dynamics approach,” Phys. Rep. 339(2), 1–77 (2000).
[Crossref]

Klein, C.

C. Klein, C. Sparber, and P. Markowich, “Numerical study of fractional nonlinear Schrödinger equations,” Proc. Math. Phys. Eng. Sci. 470(2172), 20140364 (2014).
[Crossref] [PubMed]

Kong, Q.

W. Wang, P. X. Wang, Y. K. Ho, Q. Kong, Y. Gu, and S. J. Wang, “Vacuum electron acceleration and bunch compression by a flat-top laser beam,” Rev. Sci. Instrum. 78(9), 093103 (2007).
[Crossref] [PubMed]

Kröger, H.

H. Kröger, “Fractal geometry in quantum mechanics, field theory and spin systems,” Phys. Rep. 323(2), 81–181 (2000).
[Crossref]

Laskin, N.

N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056108 (2002).
[Crossref] [PubMed]

N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(3), 3135–3145 (2000).
[Crossref] [PubMed]

N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Phys. Lett. A 268(4–6), 298–305 (2000).
[Crossref]

Lederer, F.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

Liemert, A.

A. Liemert and A. Kienle, “Fractional Schrödinger equation in the presence of the linear potential,” Mathematics 4(2), 31 (2016).
[Crossref]

Liu, X.

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

Longhi, S.

Maher-McWilliams, C.

C. Maher-McWilliams, P. Douglas, and P. F. Barker, “Laser-driven acceleration of neutral particles,” Nat. Photonics 6(6), 386–390 (2012).
[Crossref]

Markowich, P.

C. Klein, C. Sparber, and P. Markowich, “Numerical study of fractional nonlinear Schrödinger equations,” Proc. Math. Phys. Eng. Sci. 470(2172), 20140364 (2014).
[Crossref] [PubMed]

Méchain, G.

G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93(3), 035003 (2004).
[Crossref] [PubMed]

Méjean, G.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

Metzler, R.

R. Metzler and J. Klafte, “The random walk’s guide to anomalous diffusion: a fractional dynamics approach,” Phys. Rep. 339(2), 1–77 (2000).
[Crossref]

Mysyrowicz, A.

G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93(3), 035003 (2004).
[Crossref] [PubMed]

Peschel, U.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

Prade, B.

G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93(3), 035003 (2004).
[Crossref] [PubMed]

Roach, J. F.

R. E. Grojean, D. Feldman, and J. F. Roach, “Production of flat top beam profiles for high energy lasers,” Rev. Sci. Instrum. 51(3), 375–376 (1980).
[Crossref] [PubMed]

Rodriguez, M.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

Salmon, E.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

Sauerbrey, R.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

Schrauth, S. E.

Shim, B.

Skupin, S.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

Slepkov, A. D.

Sparber, C.

C. Klein, C. Sparber, and P. Markowich, “Numerical study of fractional nonlinear Schrödinger equations,” Proc. Math. Phys. Eng. Sci. 470(2172), 20140364 (2014).
[Crossref] [PubMed]

Vuong, L. T.

Wang, D.

D. Wang, A. Xiao, and W. Yang, “Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative,” J. Comput. Phys. 242, 670–681 (2013).
[Crossref]

Wang, H.

Wang, P.

P. Wang, C. Huang, and L. Zhao, “Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation,” J. Comput. Appl. Math. 306, 231–247 (2016).
[Crossref]

P. Wang and C. Huang, “An energy conservative difference scheme for the nonlinear fractional Schrödinger equations,” J. Comput. Phys. 293, 238–251 (2015).
[Crossref]

Wang, P. X.

W. Wang, P. X. Wang, Y. K. Ho, Q. Kong, Y. Gu, and S. J. Wang, “Vacuum electron acceleration and bunch compression by a flat-top laser beam,” Rev. Sci. Instrum. 78(9), 093103 (2007).
[Crossref] [PubMed]

Wang, S. J.

W. Wang, P. X. Wang, Y. K. Ho, Q. Kong, Y. Gu, and S. J. Wang, “Vacuum electron acceleration and bunch compression by a flat-top laser beam,” Rev. Sci. Instrum. 78(9), 093103 (2007).
[Crossref] [PubMed]

Wang, W.

W. Wang, P. X. Wang, Y. K. Ho, Q. Kong, Y. Gu, and S. J. Wang, “Vacuum electron acceleration and bunch compression by a flat-top laser beam,” Rev. Sci. Instrum. 78(9), 093103 (2007).
[Crossref] [PubMed]

Wang, X.

G. Fibich, N. Gavish, and X. Wang, “New singular solutions of the nonlinear Schrödinger equation,” Physica D 211(3–4), 193–220 (2005).
[Crossref]

Wolf, J. P.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

Wöste, L.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

Xiao, A.

D. Wang, A. Xiao, and W. Yang, “Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative,” J. Comput. Phys. 242, 670–681 (2013).
[Crossref]

Xiao, M.

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT-symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 23645 (2016).
[Crossref] [PubMed]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

Yang, W.

D. Wang, A. Xiao, and W. Yang, “Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative,” J. Comput. Phys. 242, 670–681 (2013).
[Crossref]

Yu, J.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
[Crossref] [PubMed]

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Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
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Zhang, Y.

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT-symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

W. Zhong, M. Belić, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 23645 (2016).
[Crossref] [PubMed]

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 23645 (2016).
[Crossref] [PubMed]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT-symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

Zhao, L.

P. Wang, C. Huang, and L. Zhao, “Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation,” J. Comput. Appl. Math. 306, 231–247 (2016).
[Crossref]

Zhong, H.

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 23645 (2016).
[Crossref] [PubMed]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT-symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

Zhong, W.

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT-symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

W. Zhong, M. Belić, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

Zhu, Y.

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT-symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

Adv. Opt. Photonics (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Ann. Phys. (1)

W. Zhong, M. Belić, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
[Crossref]

Appl. Opt. (1)

J. Chem. Phys. (1)

A. W. Albrecht, J. D. Hybl, S. M. Gallagher, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111(24), 10934–10956 (1999).
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J. Comput. Appl. Math. (1)

P. Wang, C. Huang, and L. Zhao, “Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation,” J. Comput. Appl. Math. 306, 231–247 (2016).
[Crossref]

J. Comput. Phys. (2)

P. Wang and C. Huang, “An energy conservative difference scheme for the nonlinear fractional Schrödinger equations,” J. Comput. Phys. 293, 238–251 (2015).
[Crossref]

D. Wang, A. Xiao, and W. Yang, “Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative,” J. Comput. Phys. 242, 670–681 (2013).
[Crossref]

J. Math. Phys. (1)

B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53(8), 083702 (2012).
[Crossref]

Laser Photonics Rev. (1)

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT-symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
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Mathematics (1)

A. Liemert and A. Kienle, “Fractional Schrödinger equation in the presence of the linear potential,” Mathematics 4(2), 31 (2016).
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Nat. Photonics (1)

C. Maher-McWilliams, P. Douglas, and P. F. Barker, “Laser-driven acceleration of neutral particles,” Nat. Photonics 6(6), 386–390 (2012).
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Opt. Express (2)

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R. Metzler and J. Klafte, “The random walk’s guide to anomalous diffusion: a fractional dynamics approach,” Phys. Rep. 339(2), 1–77 (2000).
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S. Gras, D. G. Blair, and L. Ju, “Opto-acoustic interactions in gravitational wave detectors: comparing flat-top beams with Gaussian beams,” Phys. Rev. D Part. Fields Gravit. Cosmol. 81(4), 042001 (2010).
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Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046602 (2004).
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N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056108 (2002).
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G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93(3), 035003 (2004).
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Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref] [PubMed]

Physica D (1)

G. Fibich, N. Gavish, and X. Wang, “New singular solutions of the nonlinear Schrödinger equation,” Physica D 211(3–4), 193–220 (2005).
[Crossref]

Proc. Math. Phys. Eng. Sci. (1)

C. Klein, C. Sparber, and P. Markowich, “Numerical study of fractional nonlinear Schrödinger equations,” Proc. Math. Phys. Eng. Sci. 470(2172), 20140364 (2014).
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[Crossref] [PubMed]

Sci. Rep. (1)

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 23645 (2016).
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A. Forbes, P. Mthunzi, M. McLaren, and T. Khanyile, “Optical trapping with Super-Gaussian beams,” in Optics in the Life Sciences, OSA Technical Digest (online) (Optical Society of America, 2013), paper JT2A.34.

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Supplementary Material (4)

NameDescription
» Visualization 1: AVI (3556 KB)      Movie for the linear propagation dynamics of the SG beam in the FSE with two dimensions
» Visualization 2: AVI (3117 KB)      Movie for the nonlinear propagation dynamics of the SG beam in the FSE with two dimensions
» Visualization 3: AVI (2268 KB)      Movie for the spatial-frequency distributions of the SG beams in the fractional linear SE with a=1.
» Visualization 4: AVI (1907 KB)      Movie for the spatial-frequency distributions of the SG beams in the fractional linear SE with a=2

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Figures (10)

Fig. 1
Fig. 1 The spatial evolution of SG beams of different order m during propagation in fractional linear SE with α=1 . Top panel: the results obtained by solving of Eq. (1) with γ=0 directly; bottom panel: numerical integration of Eq. (6).
Fig. 2
Fig. 2 The beam shapes at four propagation distances ζ= (a) 5, (b) 10, (c) 20, and (d) 30 for the incident SG beams with different order of m.
Fig. 3
Fig. 3 (a) Peak intensity is plotted as a function of propagation distance ζ for several values of m. (b) Peak intensity and the corresponding position as a function of parameter m. α=1 in (a) and (b).
Fig. 4
Fig. 4 The spatial evolution of a SG beam with m=4 during propagation in fractional linear SE with different values of α.
Fig. 5
Fig. 5 (a) Peak intensity is plotted as a function of propagation distance ζ for several values of α. (b) Peak intensity and its corresponding position are shown as a function of parameter α. Both (a) and (b) for the case m=4 .
Fig. 6
Fig. 6 The spatial evolution of the SG beams with different m during propagation in fractional nonlinear SE with α=1 and γ=1 .
Fig. 7
Fig. 7 The spatial evolutions of the SG beams with different m during propagation in fractional nonlinear SE with α=1 and (top panel) γ=2 , (bottom panel) γ=3 .
Fig. 8
Fig. 8 The spatial evolution of the SG beams with m=4 during propagation in (a) linear (Visualization 1) and (b) nonlinear (Visualization 2) two dimensional fractional SE with α=1 and γ=2 .
Fig. 9
Fig. 9 The spectral phase and group delay evolutions are plotted as a function of the spatial frequency κ and propagation distance ζ.
Fig. 10
Fig. 10 The spatial-frequency distributions of the SG beams with m=4 in the fractional linear SE with α=1 (Visualization 3) and α=2 (Visualization 4) at six different propagation distances. The insets are the spectral profile.

Equations (12)

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i ψ ζ = 1 2 ( ) α/2 ψγ | ψ | 2 ψ,
i ψ ^ ζ = 1 2 | κ | α ψ ^ ,
ψ( μ,ζ )= 1 2π ψ ^ ( κ,0 )exp( i 2 | κ | α ζ )exp( iκζ )dκ .
ψ( μ,0 )=exp( μ 2m ),
ψ ^ ( κ,0 )= 2 κ n=0 ( 1 ) n κ 2n+1 ( 2n+1 )! Γ( 2n+1 2m +1 ) ,
ψ( μ,ζ )= 1 π n=0 ( 1 ) n κ 2n ( 2n+1 )! Γ( 2n+1 2m +1 ) exp( i ζ 2 | κ | α )exp( iμκ )dκ .
i ψ ζ =[ 1 2 ( 2 μ 2 2 v 2 ) α/2 γ | ψ | 2 ]ψ.
ψ ^ ( κ,ζ )= ψ ^ + ψ ^ + ,
Φ( κ,ζ )=0.5ζ | κ | α .
T g ( κ )= Φ κ .
{ Φ α=1 ( κ,ζ )=0.5ζκθ( κ )0.5ζκθ( κ ) Φ α=2 ( κ,ζ )=0.5ζ κ 2 T g α=1 ( κ,ζ )=0.5ζθ( κ )+0.5ζθ( κ ) T g α=2 ( κ,ζ )=ζκ ,
Sf( κ,μ )= | f( x )g( xμ ) e iκx dx | 2 ,

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