Abstract

In the paper, we investigate the propagation dynamics of the Gaussian beam modeled by the fractional Schrödinger equation (FSE) with a variable coefficient. In the absence of the beam’s chirp, for smaller Lévy index, the Gaussian beam firstly splits into two beams, however under the action of the longitudinal periodic modulation, they exhibit a periodically oscillating behaviour. And with the increasing of the Lévy index, the splitting behaviour gradually diminishes. Until the Lévy index equals to 2, the splitting behaviour is completely replaced by a periodic diffraction behaviour. In the presence of the beam’s chirp, one of the splitting beams is gradually suppressed with the increasing of the chirp, while another beam on the opposite direction becomes stronger and exhibits a periodically oscillating behaviour. Also, the oscillating amplitude and period are investigated and the results show that the former is dependent on the modulation frequency, the Lévy index and the beam’s chirp, the latter depends only on the modulation frequency. Thus, the evolution of the Gaussian beam can be well manipulated to achieve the beam management in the framework of the FSE by controlling the system parameters and the chirp parameter.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2018 (1)

2017 (5)

2016 (7)

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

C. M. Huang and L. W. Dong, “Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice,” Opt. Lett. 41, 5636–5639 (2016).
[Crossref] [PubMed]

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

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

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Resonant mode conversions and Rabi oscillations in a fractional Schrödinger equation,” Opt. Express 25, 032401 (2016).
[Crossref]

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

F. Olivar-Romero and O. Rosas-Ortiz, “Factorization of the quantum fractional oscillator,” J. Phys.: Conf. Ser. 698, 012025 (2016).

2015 (5)

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

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

C. Y. Li, R. Cui, F. W. Ye, Y. V. Kartashov, L. Torner, and X. F. Chen, “Self-deflecting plasmonic lattice solitons and sur-face modes in chirped plasmonic arrays,” Opt. Lett. 40, 898–901 (2015).
[Crossref] [PubMed]

K. Y. Zhou, T. T. Wei, H. P. Sun, Y. J. He, and S. T. Liu, “Soliton dynamics in a PT-symmetric optical lattice with a longitudinal potential barrier,” Opt. Express 23, 16903–16911 (2015).
[Crossref] [PubMed]

Y. Q. Zhang, X. Liu, M. R. Belić, W. P. Zhong, M. S. Petrović, and Y. P. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Annals of Physics 363, 305–315 (2015).
[Crossref]

2014 (2)

H. F. Zhang, F. Xu, D. S. Zhu, L. Zhang, D. H. Xu, and Y. H. Tian, “Soliton controlling and steering in asymmetric nonlocal media with optical lattices,” Opt. Express 35, 995–1007 (2014).
[Crossref]

Z. Mai, S. Fu, J. Wu, and Y. Li, “Discrete solitons in waveguide arrays with long-range linearly coupled effect,” J. Phys. Soc. Japan 83, 034404 (2014).
[Crossref]

2013 (2)

2012 (2)

H. G. Li, X. J. Jiang, X. Zhu, and Z. W. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[Crossref]

I. D. Chremmos and N. K. Efremidis, “Surface optical Bloch oscillations in semi-infinite waveguide arrays,” Opt. Lett. 37, 1892–1894 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (2)

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength Plasmonic Lattice Solitons in Arrays of Metallic Nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
[Crossref] [PubMed]

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Twin-vortex solitons in nonlocal nonlinear media,” Opt. Lett. 35, 628–630 (2010).
[Crossref] [PubMed]

2009 (2)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

K. Shandarova, C. E. Rüer, and D. Kip, “Experimental observation of rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

2008 (2)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. E. Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

2007 (1)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Resonant mode oscillations in modulated waveguiding structures,” Phys. Rev. Lett. 99, 233903 (2007).
[Crossref]

2004 (2)

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
[Crossref] [PubMed]

2003 (2)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[Crossref] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref] [PubMed]

1999 (1)

T. Pertsch, P. Dannberg, W. Elflein, and A. Brauer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

1997 (1)

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[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]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic, San Diego, 2003.

Ahmed, N.

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

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Aitchison, J. S.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
[Crossref] [PubMed]

Bai, Y. F.

Belic, M. R.

Y. Q. Zhang, D. Zhang, Z. Y. Zhang, C. B. Li, Y. P. Zhang, F. L. Li, M. R. Belić, and M. Xiao, “Optical Bloch oscillation and Zener tunneling in an atomic system,” Optica 4, 571–575 (2017).
[Crossref]

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (2017).
[Crossref]

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Resonant mode conversions and Rabi oscillations in a fractional Schrödinger equation,” Opt. Express 25, 032401 (2016).
[Crossref]

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

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

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

Y. Q. Zhang, X. Liu, M. R. Belić, W. P. Zhong, M. S. Petrović, and Y. P. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Annals of Physics 363, 305–315 (2015).
[Crossref]

Brauer, A.

T. Pertsch, P. Dannberg, W. Elflein, and A. Brauer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Chen, L.

Chen, X. F.

Chremmos, I. D.

Christodoulides, D. N.

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

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. E. Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
[Crossref] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[Crossref] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref] [PubMed]

Cui, R.

Dannberg, P.

T. Pertsch, P. Dannberg, W. Elflein, and A. Brauer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Deng, Z. X.

Dong, L. W.

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Efremidis, N. K.

I. D. Chremmos and N. K. Efremidis, “Surface optical Bloch oscillations in semi-infinite waveguide arrays,” Opt. Lett. 37, 1892–1894 (2012).
[Crossref] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[Crossref] [PubMed]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, and A. Brauer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

El-Ganainy, R.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Fan, D.Y.

Fleischer, J. W.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[Crossref] [PubMed]

Fu, S.

Z. Mai, S. Fu, J. Wu, and Y. Li, “Discrete solitons in waveguide arrays with long-range linearly coupled effect,” J. Phys. Soc. Japan 83, 034404 (2014).
[Crossref]

Fu, X. Q.

Ganainy, R. E.

Z. H. Musslimani, K. G. Makris, R. E. Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

He, Y. J.

Hu, B.

Huang, C. M.

Huang, X. W.

Iwanow, R.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

Jeng, C. C.

Jiang, X. J.

H. G. Li, X. J. Jiang, X. Zhu, and Z. W. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[Crossref]

Kartashov, Y. V.

Kienle, A.

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

Kip, D.

K. Shandarova, C. E. Rüer, and D. Kip, “Experimental observation of rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic, San Diego, 2003.

Krolikowski, W.

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).
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Lederer, F.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref] [PubMed]

Lee, R. K.

Lei, D. J.

Li, C. B.

Li, C. X.

Li, C. Y.

Li, F. L.

Li, H. G.

H. G. Li, X. J. Jiang, X. Zhu, and Z. W. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[Crossref]

Li, Y.

Liemert, A.

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

Lin, Y. Y.

Liu, S. T.

Liu, X.

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

Y. Q. Zhang, X. Liu, M. R. Belić, W. P. Zhong, M. S. Petrović, and Y. P. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Annals of Physics 363, 305–315 (2015).
[Crossref]

Longhi, S.

Mai, Z.

Z. Mai, S. Fu, J. Wu, and Y. Li, “Discrete solitons in waveguide arrays with long-range linearly coupled effect,” J. Phys. Soc. Japan 83, 034404 (2014).
[Crossref]

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. E. Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

Meier, J.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
[Crossref] [PubMed]

Mihalache, D.

F Ye, D. Mihalache, B. Hu, and N. C Panoiu, “Subwavelength vortical plasmonic lattice solitons,” Opt. Lett. 36, 1179–1181 (2011).
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F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength Plasmonic Lattice Solitons in Arrays of Metallic Nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
[Crossref] [PubMed]

Min, Y.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
[Crossref] [PubMed]

Musslimani, Z. H.

Z. H. Musslimani, K. G. Makris, R. E. Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Olivar-Romero, F.

F. Olivar-Romero and O. Rosas-Ortiz, “Factorization of the quantum fractional oscillator,” J. Phys.: Conf. Ser. 698, 012025 (2016).

Panoiu, N. C

Panoiu, N. C.

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength Plasmonic Lattice Solitons in Arrays of Metallic Nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
[Crossref] [PubMed]

Pertsch, T.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, and A. Brauer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Petrovic, M. S.

Y. Q. Zhang, X. Liu, M. R. Belić, W. P. Zhong, M. S. Petrović, and Y. P. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Annals of Physics 363, 305–315 (2015).
[Crossref]

Rosas-Ortiz, O.

F. Olivar-Romero and O. Rosas-Ortiz, “Factorization of the quantum fractional oscillator,” J. Phys.: Conf. Ser. 698, 012025 (2016).

Rüer, C. E.

K. Shandarova, C. E. Rüer, and D. Kip, “Experimental observation of rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

Salamo, G.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
[Crossref] [PubMed]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Schiek, R.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

Segev, M.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[Crossref] [PubMed]

Shandarova, K.

K. Shandarova, C. E. Rüer, and D. Kip, “Experimental observation of rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

Shen, M.

Shi, X. H.

Shi, Z. W.

H. G. Li, X. J. Jiang, X. Zhu, and Z. W. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[Crossref]

Silberberg, Y.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
[Crossref] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref] [PubMed]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

Sohler, W.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

Sorel, M.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
[Crossref] [PubMed]

Stegeman, G. I.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
[Crossref] [PubMed]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

Sun, H. P.

Tian, Y. H.

H. F. Zhang, F. Xu, D. S. Zhu, L. Zhang, D. H. Xu, and Y. H. Tian, “Soliton controlling and steering in asymmetric nonlocal media with optical lattices,” Opt. Express 35, 995–1007 (2014).
[Crossref]

Torner, L.

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Resonant mode oscillations in modulated waveguiding structures,” Phys. Rev. Lett. 99, 233903 (2007).
[Crossref]

Wang, Q.

Wang, R.

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (2017).
[Crossref]

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Resonant mode conversions and Rabi oscillations in a fractional Schrödinger equation,” Opt. Express 25, 032401 (2016).
[Crossref]

Wei, T. T.

Wen, J.

Wu, J.

Z. Mai, S. Fu, J. Wu, and Y. Li, “Discrete solitons in waveguide arrays with long-range linearly coupled effect,” J. Phys. Soc. Japan 83, 034404 (2014).
[Crossref]

Xiao, M.

Y. Q. Zhang, D. Zhang, Z. Y. Zhang, C. B. Li, Y. P. Zhang, F. L. Li, M. R. Belić, and M. Xiao, “Optical Bloch oscillation and Zener tunneling in an atomic system,” Optica 4, 571–575 (2017).
[Crossref]

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

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

Y. Q. Zhang, X. Liu, M. R. Belić, W. P. Zhong, Y. P. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115, 180403 (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. W.

Xu, D. H.

H. F. Zhang, F. Xu, D. S. Zhu, L. Zhang, D. H. Xu, and Y. H. Tian, “Soliton controlling and steering in asymmetric nonlocal media with optical lattices,” Opt. Express 35, 995–1007 (2014).
[Crossref]

Xu, F.

H. F. Zhang, F. Xu, D. S. Zhu, L. Zhang, D. H. Xu, and Y. H. Tian, “Soliton controlling and steering in asymmetric nonlocal media with optical lattices,” Opt. Express 35, 995–1007 (2014).
[Crossref]

Yang, H.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
[Crossref] [PubMed]

Ye, F

Ye, F.

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Twin-vortex solitons in nonlocal nonlinear media,” Opt. Lett. 35, 628–630 (2010).
[Crossref] [PubMed]

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength Plasmonic Lattice Solitons in Arrays of Metallic Nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
[Crossref] [PubMed]

Ye, F. W.

Zhang, D.

Zhang, H. F.

H. F. Zhang, F. Xu, D. S. Zhu, L. Zhang, D. H. Xu, and Y. H. Tian, “Soliton controlling and steering in asymmetric nonlocal media with optical lattices,” Opt. Express 35, 995–1007 (2014).
[Crossref]

Zhang, J. W.

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (2017).
[Crossref]

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Resonant mode conversions and Rabi oscillations in a fractional Schrödinger equation,” Opt. Express 25, 032401 (2016).
[Crossref]

Zhang, L.

H. F. Zhang, F. Xu, D. S. Zhu, L. Zhang, D. H. Xu, and Y. H. Tian, “Soliton controlling and steering in asymmetric nonlocal media with optical lattices,” Opt. Express 35, 995–1007 (2014).
[Crossref]

Zhang, L. F.

Zhang, Y.

Zhang, Y. P.

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (2017).
[Crossref]

Y. Q. Zhang, D. Zhang, Z. Y. Zhang, C. B. Li, Y. P. Zhang, F. L. Li, M. R. Belić, and M. Xiao, “Optical Bloch oscillation and Zener tunneling in an atomic system,” Optica 4, 571–575 (2017).
[Crossref]

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Resonant mode conversions and Rabi oscillations in a fractional Schrödinger equation,” Opt. Express 25, 032401 (2016).
[Crossref]

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

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

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

Y. Q. Zhang, X. Liu, M. R. Belić, W. P. Zhong, M. S. Petrović, and Y. P. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Annals of Physics 363, 305–315 (2015).
[Crossref]

Zhang, Y. Q.

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (2017).
[Crossref]

Y. Q. Zhang, D. Zhang, Z. Y. Zhang, C. B. Li, Y. P. Zhang, F. L. Li, M. R. Belić, and M. Xiao, “Optical Bloch oscillation and Zener tunneling in an atomic system,” Optica 4, 571–575 (2017).
[Crossref]

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

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Resonant mode conversions and Rabi oscillations in a fractional Schrödinger equation,” Opt. Express 25, 032401 (2016).
[Crossref]

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

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

Y. Q. Zhang, X. Liu, M. R. Belić, W. P. Zhong, M. S. Petrović, and Y. P. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Annals of Physics 363, 305–315 (2015).
[Crossref]

Zhang, Z. Y.

Zhao, H.

Zhong, H.

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (2017).
[Crossref]

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

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

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Resonant mode conversions and Rabi oscillations in a fractional Schrödinger equation,” Opt. Express 25, 032401 (2016).
[Crossref]

Zhong, H. Z.

Zhong, W. P.

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

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

Y. Q. Zhang, X. Liu, M. R. Belić, W. P. Zhong, M. S. Petrović, and Y. P. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Annals of Physics 363, 305–315 (2015).
[Crossref]

Zhou, K. Y.

Zhu, D. S.

H. F. Zhang, F. Xu, D. S. Zhu, L. Zhang, D. H. Xu, and Y. H. Tian, “Soliton controlling and steering in asymmetric nonlocal media with optical lattices,” Opt. Express 35, 995–1007 (2014).
[Crossref]

Zhu, X.

H. G. Li, X. J. Jiang, X. Zhu, and Z. W. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[Crossref]

Zhu, Y.

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

Adv. Opt. Photon. (1)

Annals of Physics (1)

Y. Q. Zhang, X. Liu, M. R. Belić, W. P. Zhong, M. S. Petrović, and Y. P. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Annals of Physics 363, 305–315 (2015).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. Soc. Japan (1)

Z. Mai, S. Fu, J. Wu, and Y. Li, “Discrete solitons in waveguide arrays with long-range linearly coupled effect,” J. Phys. Soc. Japan 83, 034404 (2014).
[Crossref]

J. Phys.: Conf. Ser. (1)

F. Olivar-Romero and O. Rosas-Ortiz, “Factorization of the quantum fractional oscillator,” J. Phys.: Conf. Ser. 698, 012025 (2016).

Laser Photonics Rev. (1)

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

Mathematics (1)

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

Nature (2)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[Crossref] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (7)

Optica (1)

Phys. Rev. A (1)

H. G. Li, X. J. Jiang, X. Zhu, and Z. W. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[Crossref]

Phys. Rev. Lett. (11)

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

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Resonant mode oscillations in modulated waveguiding structures,” Phys. Rev. Lett. 99, 233903 (2007).
[Crossref]

K. Shandarova, C. E. Rüer, and D. Kip, “Experimental observation of rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, and A. Brauer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. E. Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
[Crossref] [PubMed]

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength Plasmonic Lattice Solitons in Arrays of Metallic Nanowires,” Phys. Rev. Lett. 104, 106802 (2010).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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

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[Crossref]

Y. Q. Zhang, R. Wang, H. Zhong, J. W. Zhang, M. R. Belić, and Y. P. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (2017).
[Crossref]

Science (1)

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[Crossref]

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Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic, San Diego, 2003.

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Figures (8)

Fig. 1
Fig. 1 (a) The numerical evolution and (b) analytical result of the Gaussian beam without the chirp in the system with the longitudinal periodic modulation D (z) = cos(Ωz), where the modulation frequency is set as Ω = 0.1. (c) The oscillating amplitude and (d) period versus the modulation frequency Ω. Here, the other parameters are σ = 0.25 and x0 = 0.
Fig. 2
Fig. 2 The evolution of the beam power, where the black and red curves represent the numerical and analytical results, respectively. The parameters are the same as in Fig. 1.
Fig. 3
Fig. 3 The evolutions of the Gaussian beam in the system with the longitudinal periodic modulation D (z) = cos(Ωz) for different Lévy index. (a–d) The results of the numerical simulation for Eq. (1). (a1–d1) The numerical integration for Eq. (4). The Lévy index is respectively α = 1.25, 1.5, 1.75, 2 from left to right. Here, the other parameters are the same as in Fig. 1.
Fig. 4
Fig. 4 The dependence of the chirp parameter |C| on the beam width σ for different ε.
Fig. 5
Fig. 5 The numerical evolution of the initial Gaussian beam with the chirp for different ε in the system with the longitudinal periodic modulation D (z) = cos(Ωz). (a) ε = 0.2, C = 1.44; (b) ε = 0.2, C = −1.44; (c) ε = 0.1, C = 1.84; (d))ε = 0.(1, C) = 1.84; (e) ε = 0.01, C = 2.88; (f) ε = 0.01, C = −2.88. The other parameters are σ = 0.62−, Ω = 0.1 and x0 = 0.
Fig. 6
Fig. 6 (a) The numerical evolution and (b) analytical result of the Gaussian beam with the chirp in the system with the longitudinal periodic modulation D(z) = cos(Ωz), where Ω = 0.3. (c) The oscillating amplitude and (d) period versus the modulation frequency Ω. Here, the other parameters are σ = 0.25, C = 3 and x0 = 0.
Fig. 7
Fig. 7 The evolution plots of the Gaussian beam with the chirp in the system with the longitudinal periodic modulation D(z) = cos(Ωz), where the modulation frequency is Ω = 0.3. Here, α = 1.25, 1.5, 1.75, 2 from top row to bottom row, and C = 3, 6, 9 from left column to right column, respectively, and the other parameters are σ = 0.25 and x0 = 0.
Fig. 8
Fig. 8 The oscillating amplitude versus (a) the Lévy index α for different chirp parameter C, and (b) the chirp for different Lévy index α. Here, the other parameters are σ = 0.25, Ω = 0.3 and x0 = 0.

Equations (16)

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i ψ z 1 2 D ( z ) ( 2 x 2 ) α / 2 ψ ( x , z ) = 0 ,
i ψ ^ ( k , z ) z 1 2 | k | α D ( z ) ψ ^ ( k , z ) = 0 ,
ψ ^ ( k , z ) = ψ ^ ( k , 0 ) e i 2 | k | α 0 z D ( ς ) d ς ,
ψ ( x , z ) = 1 2 π + ψ ^ ( k , 0 ) e i 2 | k | α 0 z D ( ς ) d ς e i k x d k .
ψ ( x , 0 ) = e σ ( x x 0 ) 2 + i C ( x x 0 )
ψ ^ ( k , 0 ) = π σ e ( k C ) 2 4 σ i k x 0 .
ψ ( x , z ) = 1 2 π σ 0 + e k 2 4 σ + i k [ x X + ( z ) ] d k + 1 2 π σ 0 e k 2 4 σ + i k [ x X ( z ) ] d k ,
ψ ( x , z ) 1 4 π σ + e k 2 4 σ i k | x X + ( z ) | d k + 1 4 π σ e k 2 4 σ + i k | x X ( z ) | d k = 1 2 e σ [ x X + ( z ) ] 2 + 1 2 e σ [ x X ( z ) ] 2 .
P = + | ψ ( x , z ) | 2 d x = π 8 σ ( 1 + e σ sin 2 ( Ω z ) 2 Ω 2 ) ,  
ψ ( x , z ) = 1 B ( z ) e σ ( x x 0 ) 2 B ( z ) ,
ψ ( x , z ) = 1 B ( z ) + e ( k C ) 2 4 σ + i k ( x x 0 ) i 2 | k | 0 z D ( ς ) d ς d k = e i C [ x X + ( z ) ] 2 π σ C + e k 2 4 σ + i k [ x X + ( z ) ] d k + e i C [ x X ( z ) ] 2 π σ C e k 2 4 σ + i k [ x X ( z ) ] d k = { e σ [ x X + ( z ) ] 2 + i C [ x X + ( z ) ] + g + ( x , z ) , C > 0 , e σ [ x X ( z ) ] 2 + i C [ x X ( z ) ] + g ( x , z ) , C < 0 ,
g ± ( x , z ) = e i C [ x X + ( z ) ] 2 π σ | C | + e k 2 4 σ i k [ x X + ( z ) ] d k ± e i C [ x X ( z ) ] 2 π σ | C | + e k 2 4 σ i k [ x X ( z ) ] d k
| g ± ( x , z ) | 1 π σ | C | + e k 2 4 σ d k
ψ ( x , z ) e σ [ x X ± ( z ) ] 2 + i C [ x X ± ( z ) ] .
ε = 1 π σ | C | + e k 2 4 σ d k
ψ ( x , z ) = 1 B ( z ) e σ X ( x , z ) 2 B ( z ) e i C [ X ( x , z ) + C 2 Ω sin ( Ω z ) ] ,

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