Abstract

The dynamics of Airy beams modeled by the fractional Schrödinger equation (FSE) with the potential barrier are numerically investigated. Adjusting the Lévy index provides a convenient way to control the diffraction of Airy beams which presents the non-diffraction and splitting property. It has been found that the total reflection of beams occurs when the depth of the single potential barrier exceeds the threshold, and that the number of reflected waves is influenced by the Lévy index and the location of potential barrier. However, the periodic self-imaging phenomenon of Airy beams is shown under a symmetric potential barrier when the Lévy index is equal to one, and the self-imaging period of the asymmetric Airy beams is analytically demonstrated and is as twice as that of symmetric Gaussian beams, moreover, the chaoticon of light field is formed during propagation as the Lévy index increases. All the properties of Airy beams modeled by FSE confirm the potential application in optical manipulation.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. R. B. Laughlin, “Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations,” Phys. Rev. Lett. 50, 1395–1398 (1983).
    [Crossref]
  2. J. Wen, Y. Zhang, and M. Xiao, “The talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photon. 5, 83–130 (2013).
    [Crossref]
  3. L. P. Rokhinson, X. Liu, and J. K. Furdyna, “Josephson effect in a semiconductor-superconductor nanowire as a signature of majorana,” Nat. Phys. 6, 795–799 (2012).
    [Crossref]
  4. H. Kröger, “Fractal geometry in quantum mechanics, field theory and spin systems,” Phys. Rep. 323(2), 81–181 (2000).
    [Crossref]
  5. 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]
  6. N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E 62, 3135–3145 (2000).
    [Crossref]
  7. N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Phys. Lett. A 268, 298–305 (2000).
    [Crossref]
  8. N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E 66, 056108 (2002).
    [Crossref]
  9. J. Dong and M. Xu, “Some solutions to the space fractional Schrödinger equation using momentum representation method,” J. Math. Phys. 51, 678–682 (2011).
  10. K. Kowalski and J. Rembielinski, “The relativistic massless harmonic oscillator,” Phys. Rev. A 81, 15780–15787 (2010).
    [Crossref]
  11. Y. Luchko, “Fractional Schrödinger equation for a particle moving in a potential well,” J. Math. Phys. 54, 012111 (2013).
    [Crossref]
  12. S. Longhi, “Fractional Schrödinger equation in optics,” Opt. Lett. 40, 13749–13754 (2015).
    [Crossref]
  13. 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]
  14. X. Huang, Z. Deng, and X. Fu, “Dynamics of finite energy Airy beams modeled by the fractional Schrödinger equation with a linear potential,” J. Opt. Soc. Am. B 34(5), 976–982 (2017).
    [Crossref]
  15. L. Zhang, C. Li, H. Zhong, C. Xu, D. Lei, Y. Li, and D. Fan, “Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: from linear to nonlinear regimes,” Opt. Express 24, 14406–14418 (2016).
    [Crossref] [PubMed]
  16. 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, 1–5 (2015).
    [Crossref]
  17. Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, and M. Xiao, “Symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 526–531 (2016).
    [Crossref]
  18. C. Huang and L. Dong, “Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice,” Opt. Lett. 41, 5636–5639 (2016).
    [Crossref] [PubMed]
  19. L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
    [Crossref]
  20. D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M.R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529(9), 1700149 (2017).
    [Crossref]
  21. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
    [Crossref] [PubMed]
  22. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008).
    [Crossref] [PubMed]
  23. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
    [Crossref] [PubMed]
  24. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
    [Crossref]
  25. Y. Liang, Y. Hu, D. Song, C. Lou, X. Zhang, Z. Chen, and J. Xu, “Image signal transmission with Airy beams,” Opt. Lett. 40(23), 5686–5689 (2015).
    [Crossref] [PubMed]
  26. F. Bleckmann, A. Minovich, J. Frohnhaus, D. N. Neshev, and S. Linden, “Manipulation of Airy surface plasmon beams,” Opt. Lett. 38(9), 1443–1445 (2013).
    [Crossref] [PubMed]
  27. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
    [Crossref] [PubMed]
  28. I. Oreshnikov, R. Driben, and A. V. Yulin, “Interaction of high-order solitons with external dispersive waves,” Opt. Lett. 40(23), 5554–5557 (2015).
    [Crossref] [PubMed]
  29. S. F. Wang, A. Mussot, M. Conforti, X. L. Zeng, and A. Kudlinski, “Bouncing of a dispersive wave in a solitonic cage,” Opt. Lett. 40(14), 3320–3323 (2015).
    [Crossref] [PubMed]
  30. Z. Deng, X. Fu, J. Liu, C. Zhao, and S. Wen, “Trapping and controlling the dispersive wave within a solitonic well,” Opt. Express 24(10), 10302–10312 (2016).
    [Crossref] [PubMed]
  31. X. Guo and M. Xu, “Some physical applications of fractional Schrödinger equation,” J. Math. Phys. 47(8), 082104 (2006).
    [Crossref]
  32. R. Herrmann, Fractional Calculus: An Introduction for Physicists (World Scientific, 2011).
    [Crossref]
  33. L. Zhang, K. Liu, H. Zhong, J. Zhang, Y. Li, and D. Fan, “Effect of initial frequency chirp on Airy pulse propagation in an optical fiber,” Opt. Express 23(3), 2566–2576 (2015).
    [Crossref] [PubMed]
  34. X. Huang, Z. Deng, X. Shi, and X. Fu, “Propagation charateristics of ring Airy beams modeled by fractional Schödinger equation,” J. Opt. Soc. Am. B 34(10), 2190–2197 (2017).
    [Crossref]
  35. X. Chu, Z. Liu, and P. Zhou, “Generation of a high-power Airy beam by coherent combining technology,” Laser Phys. Lett. 10(12), 5102 (2013).
    [Crossref]

2017 (4)

X. Huang, Z. Deng, and X. Fu, “Dynamics of finite energy Airy beams modeled by the fractional Schrödinger equation with a linear potential,” J. Opt. Soc. Am. B 34(5), 976–982 (2017).
[Crossref]

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M.R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529(9), 1700149 (2017).
[Crossref]

X. Huang, Z. Deng, X. Shi, and X. Fu, “Propagation charateristics of ring Airy beams modeled by fractional Schödinger equation,” J. Opt. Soc. Am. B 34(10), 2190–2197 (2017).
[Crossref]

2016 (5)

2015 (6)

2013 (4)

X. Chu, Z. Liu, and P. Zhou, “Generation of a high-power Airy beam by coherent combining technology,” Laser Phys. Lett. 10(12), 5102 (2013).
[Crossref]

F. Bleckmann, A. Minovich, J. Frohnhaus, D. N. Neshev, and S. Linden, “Manipulation of Airy surface plasmon beams,” Opt. Lett. 38(9), 1443–1445 (2013).
[Crossref] [PubMed]

Y. Luchko, “Fractional Schrödinger equation for a particle moving in a potential well,” J. Math. Phys. 54, 012111 (2013).
[Crossref]

J. Wen, Y. Zhang, and M. Xiao, “The talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photon. 5, 83–130 (2013).
[Crossref]

2012 (1)

L. P. Rokhinson, X. Liu, and J. K. Furdyna, “Josephson effect in a semiconductor-superconductor nanowire as a signature of majorana,” Nat. Phys. 6, 795–799 (2012).
[Crossref]

2011 (1)

J. Dong and M. Xu, “Some solutions to the space fractional Schrödinger equation using momentum representation method,” J. Math. Phys. 51, 678–682 (2011).

2010 (1)

K. Kowalski and J. Rembielinski, “The relativistic massless harmonic oscillator,” Phys. Rev. A 81, 15780–15787 (2010).
[Crossref]

2009 (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

2008 (3)

2007 (1)

2006 (1)

X. Guo and M. Xu, “Some physical applications of fractional Schrödinger equation,” J. Math. Phys. 47(8), 082104 (2006).
[Crossref]

2002 (1)

N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E 66, 056108 (2002).
[Crossref]

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 62, 3135–3145 (2000).
[Crossref]

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

1983 (1)

R. B. Laughlin, “Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations,” Phys. Rev. Lett. 50, 1395–1398 (1983).
[Crossref]

Ahmed, N.

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M.R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529(9), 1700149 (2017).
[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]

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Belic, M. R.

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]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, and M. Xiao, “Symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 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, 1–5 (2015).
[Crossref]

Belic, M.R.

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M.R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529(9), 1700149 (2017).
[Crossref]

Bleckmann, F.

Broky, J.

Chen, Z.

Christodoulides, D. N.

Chu, X.

X. Chu, Z. Liu, and P. Zhou, “Generation of a high-power Airy beam by coherent combining technology,” Laser Phys. Lett. 10(12), 5102 (2013).
[Crossref]

Conforti, M.

Conti, C.

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

Deng, Z.

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Dogariu, A.

Dong, J.

J. Dong and M. Xu, “Some solutions to the space fractional Schrödinger equation using momentum representation method,” J. Math. Phys. 51, 678–682 (2011).

Dong, L.

Driben, R.

Fan, D.

Frohnhaus, J.

Fu, X.

Furdyna, J. K.

L. P. Rokhinson, X. Liu, and J. K. Furdyna, “Josephson effect in a semiconductor-superconductor nanowire as a signature of majorana,” Nat. Phys. 6, 795–799 (2012).
[Crossref]

Guo, X.

X. Guo and M. Xu, “Some physical applications of fractional Schrödinger equation,” J. Math. Phys. 47(8), 082104 (2006).
[Crossref]

He, Z.

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

Herrmann, R.

R. Herrmann, Fractional Calculus: An Introduction for Physicists (World Scientific, 2011).
[Crossref]

Hu, Y.

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

Y. Liang, Y. Hu, D. Song, C. Lou, X. Zhang, Z. Chen, and J. Xu, “Image signal transmission with Airy beams,” Opt. Lett. 40(23), 5686–5689 (2015).
[Crossref] [PubMed]

Huang, C.

Huang, X.

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]

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Kowalski, K.

K. Kowalski and J. Rembielinski, “The relativistic massless harmonic oscillator,” Phys. Rev. A 81, 15780–15787 (2010).
[Crossref]

Kröger, H.

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

Kudlinski, A.

Laskin, N.

N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E 66, 056108 (2002).
[Crossref]

N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E 62, 3135–3145 (2000).
[Crossref]

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

Laughlin, R. B.

R. B. Laughlin, “Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations,” Phys. Rev. Lett. 50, 1395–1398 (1983).
[Crossref]

Lei, D.

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

L. Zhang, C. Li, H. Zhong, C. Xu, D. Lei, Y. Li, and D. Fan, “Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: from linear to nonlinear regimes,” Opt. Express 24, 14406–14418 (2016).
[Crossref] [PubMed]

Li, C.

Li, F.

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M.R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529(9), 1700149 (2017).
[Crossref]

Li, Y.

Liang, Y.

Linden, S.

Liu, J.

Liu, K.

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, 1–5 (2015).
[Crossref]

L. P. Rokhinson, X. Liu, and J. K. Furdyna, “Josephson effect in a semiconductor-superconductor nanowire as a signature of majorana,” Nat. Phys. 6, 795–799 (2012).
[Crossref]

Liu, Z.

X. Chu, Z. Liu, and P. Zhou, “Generation of a high-power Airy beam by coherent combining technology,” Laser Phys. Lett. 10(12), 5102 (2013).
[Crossref]

Longhi, S.

S. Longhi, “Fractional Schrödinger equation in optics,” Opt. Lett. 40, 13749–13754 (2015).
[Crossref]

Lou, C.

Luchko, Y.

Y. Luchko, “Fractional Schrödinger equation for a particle moving in a potential well,” J. Math. Phys. 54, 012111 (2013).
[Crossref]

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

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]

Minovich, A.

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Mussot, A.

Neshev, D. N.

Oreshnikov, I.

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Rembielinski, J.

K. Kowalski and J. Rembielinski, “The relativistic massless harmonic oscillator,” Phys. Rev. A 81, 15780–15787 (2010).
[Crossref]

Rokhinson, L. P.

L. P. Rokhinson, X. Liu, and J. K. Furdyna, “Josephson effect in a semiconductor-superconductor nanowire as a signature of majorana,” Nat. Phys. 6, 795–799 (2012).
[Crossref]

Shi, X.

Siviloglou, G. A.

Song, D.

Wang, S. F.

Wang, Z.

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

Wen, J.

Wen, S.

Xiao, M.

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M.R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529(9), 1700149 (2017).
[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]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, and M. Xiao, “Symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 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, 1–5 (2015).
[Crossref]

J. Wen, Y. Zhang, and M. Xiao, “The talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photon. 5, 83–130 (2013).
[Crossref]

Xu, C.

Xu, J.

Xu, M.

J. Dong and M. Xu, “Some solutions to the space fractional Schrödinger equation using momentum representation method,” J. Math. Phys. 51, 678–682 (2011).

X. Guo and M. Xu, “Some physical applications of fractional Schrödinger equation,” J. Math. Phys. 47(8), 082104 (2006).
[Crossref]

Yulin, A. V.

Zeng, X. L.

Zhang, D.

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M.R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529(9), 1700149 (2017).
[Crossref]

Zhang, J.

Zhang, L.

Zhang, X.

Zhang, Y.

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M.R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529(9), 1700149 (2017).
[Crossref]

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M.R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529(9), 1700149 (2017).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, and M. Xiao, “Symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 526–531 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, and M. Xiao, “Symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 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]

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]

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, 1–5 (2015).
[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, 1–5 (2015).
[Crossref]

J. Wen, Y. Zhang, and M. Xiao, “The talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photon. 5, 83–130 (2013).
[Crossref]

Zhang, Z.

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M.R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529(9), 1700149 (2017).
[Crossref]

Zhao, C.

Zhong, H.

L. Zhang, C. Li, H. Zhong, C. Xu, D. Lei, Y. Li, and D. Fan, “Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: from linear to nonlinear regimes,” Opt. Express 24, 14406–14418 (2016).
[Crossref] [PubMed]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, and M. Xiao, “Symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 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]

L. Zhang, K. Liu, H. Zhong, J. Zhang, Y. Li, and D. Fan, “Effect of initial frequency chirp on Airy pulse propagation in an optical fiber,” Opt. Express 23(3), 2566–2576 (2015).
[Crossref] [PubMed]

Zhong, W.

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, and M. Xiao, “Symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 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, 1–5 (2015).
[Crossref]

Zhou, P.

X. Chu, Z. Liu, and P. Zhou, “Generation of a high-power Airy beam by coherent combining technology,” Laser Phys. Lett. 10(12), 5102 (2013).
[Crossref]

Zhu, Y.

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, and M. Xiao, “Symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 526–531 (2016).
[Crossref]

Adv. Opt. Photon. (1)

Ann. Phys. (Berlin) (1)

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M.R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529(9), 1700149 (2017).
[Crossref]

Commun. Nonlinear Sci. Numer. Simul. (1)

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

J. Math. Phys. (3)

Y. Luchko, “Fractional Schrödinger equation for a particle moving in a potential well,” J. Math. Phys. 54, 012111 (2013).
[Crossref]

J. Dong and M. Xu, “Some solutions to the space fractional Schrödinger equation using momentum representation method,” J. Math. Phys. 51, 678–682 (2011).

X. Guo and M. Xu, “Some physical applications of fractional Schrödinger equation,” J. Math. Phys. 47(8), 082104 (2006).
[Crossref]

J. Opt. Soc. Am. B (2)

Laser Photon. Rev. (1)

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, and M. Xiao, “Symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 526–531 (2016).
[Crossref]

Laser Phys. Lett. (1)

X. Chu, Z. Liu, and P. Zhou, “Generation of a high-power Airy beam by coherent combining technology,” Laser Phys. Lett. 10(12), 5102 (2013).
[Crossref]

Nat. Photonics (1)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Nat. Phys. (1)

L. P. Rokhinson, X. Liu, and J. K. Furdyna, “Josephson effect in a semiconductor-superconductor nanowire as a signature of majorana,” Nat. Phys. 6, 795–799 (2012).
[Crossref]

Opt. Express (4)

Opt. Lett. (8)

Phys. Lett. A (1)

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

Phys. Rep. (2)

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]

Phys. Rev. A (1)

K. Kowalski and J. Rembielinski, “The relativistic massless harmonic oscillator,” Phys. Rev. A 81, 15780–15787 (2010).
[Crossref]

Phys. Rev. E (2)

N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E 66, 056108 (2002).
[Crossref]

N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E 62, 3135–3145 (2000).
[Crossref]

Phys. Rev. Lett. (2)

R. B. Laughlin, “Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations,” Phys. Rev. Lett. 50, 1395–1398 (1983).
[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, 1–5 (2015).
[Crossref]

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).
[Crossref]

Science (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Other (1)

R. Herrmann, Fractional Calculus: An Introduction for Physicists (World Scientific, 2011).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 The intensity distribution of Airy beams in FSE without potential under different Lévy index, (a0)–(d0) are the numerical results by solving Eq. (1) with V(x) = 0, and (a1)–(d1) are the results of the numerical integration of Eq. (4); (e) and (f) are the corresponding center of gravity and the peak intensity during propagation.
Fig. 2
Fig. 2 The intensity distribution of Airy beams with the potential barrier under α = 1, L = 20 for (a) V0 = 0.5 and (b) V0 = 2; (c) is the relation between the threshold of potential depth and the alpha under the complete reflection of Airy beams; (d) is the reflectivity (R) and transmissivity (T) versus the potential depth under α = 1.
Fig. 3
Fig. 3 The reflection of Airy beams under different Lévy index when the depth potential is over the threshold with L = 20.
Fig. 4
Fig. 4 The reflection of Airy beams under different L with V0 = 3.5, for (a0)–(a3) α = 1 and (b0)–(b3) α = 1.6.
Fig. 5
Fig. 5 The intensity distribution under α = 1 and V0 = 5 with the symmetric potential barrier for (a)–(c) Airy beams and (d) Gaussian beams; (e) and (f) are the one-dimensional intensity distribution at different propagation distances.
Fig. 6
Fig. 6 The schematics of the Airy beams in FSE with the symmetric potential barrier for (a), (b) is the period of self-imaging for Airy beams and Gaussian beams, the blue and purple curve are the analytical result, and the red and the black diamond are the simulation result.
Fig. 7
Fig. 7 The intensity distribution of Airy beams during propagation for different Lévy index under the symmetric potential barrier with L = 20, V0 = 5.

Equations (11)

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i z φ ( x , z ) = 1 2 ( 2 x 2 ) α / 2 φ ( x , z ) + V ( x ) φ ( x , z ) .
φ ( x , 0 ) = F ( a ) Ai ( x ) exp ( a x ) ,
Ψ 0 ( k x , 0 ) = F ( a ) 2 exp ( a k x 2 ) exp [ 1 3 ( k x 3 3 a 2 k x i a 3 ) ] ,
φ ( x , z ) = F ( a ) 2 π exp ( a k x 2 ) exp [ 1 3 ( k x 3 3 a 2 k x i a 3 ) ] exp ( i k x x i z | k x | α 2 ) d k x .
V ( x ) = { V 0 x > L 0 x < L .
V ( x ) = { V 0 | x | > L 0 | x | < L .
φ ( x , z ) = Ψ 0 ( k x , 0 ) exp ( i k x x i z | k x | 2 ) d k x = 0 Ψ 0 ( k x , 0 ) exp [ i k x ( x z 2 ) ] d k x + 0 Ψ 0 ( k x ) exp [ i k x ( x z 2 ) ] d k x .
φ ( x , 0 ) = m = 1 N ( 1 ) m A m exp [ ( x b m ) 2 w 0 m 2 ] .
φ ( x , z ) m = 1 N ( 1 ) m A m 2 { exp [ w 0 m ( b m + x + z 2 ) 2 ] + exp [ w 0 m ( b m + x z 2 ) 2 ] } .
φ ( x , z ) 1 2 { exp [ ( x + z 2 ) 2 ] + exp [ ( x z 2 ) 2 ] } .
φ ( x , z ) = φ 1 ( x z 2 , z ) + φ 2 ( x + z 2 , z ) .

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