Abstract

In a theoretical and numerical analysis, we report resonant mode conversions and Rabi oscillations in the fractional Schrödinger equation through the longitudinal modulation of the transverse potential. As specific systems of interest, we select eigenmodes of the transverse Gaussian and periodic potentials. In the Gaussian potential, we find that an increasing number of eigenmodes can be supported as the Lévy index α is reduced from 2 to 1, and that the conversion distance between the first and third eigenmodes first decreases and then increases. In the periodic potential, we obtain a cascade conversion between the neighboring eigenmodes because the parity of eigenmodes remains the same. We also find that the conversion distances between the first and second eigenmodes, as well as between the second and third eigenmodes, decrease monotonously, while that between the first and third eigenmodes first decreases and then increases with increasing α. In addition, we find that for a certain α, these conversion distances can be equal to each other.

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

Full Article  |  PDF Article
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    [Crossref]
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  3. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
    [Crossref]
  4. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
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  5. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
    [Crossref]
  6. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
    [Crossref]
  7. Y. V. Kartashov, V. V. Konotop, D. A. Zezyulin, and L. Torner, “Bloch oscillations in optical and Zeeman lattices in the presence of spin-orbit coupling,” Phys. Rev. Lett. 117, 215301 (2016).
    [Crossref] [PubMed]
  8. 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]
  9. K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electronics Letters 26, 1270–1272 (1990).
    [Crossref]
  10. K. S. Lee and T. Erdogan, “Fiber mode coupling in transmissive and reflective tilted fiber gratings,” Appl. Opt. 39, 1394–1404 (2000).
    [Crossref]
  11. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Resonant mode oscillations in modulated waveguiding structures,” Phys. Rev. Lett. 99, 233903 (2007).
    [Crossref]
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    [Crossref] [PubMed]
  13. X. Zhang, F. Ye, Y. V. Kartashov, and X. Chen, “Rabi oscillations and stimulated mode conversion on the subwavelength scale,” Opt. Express 23, 6731–6737 (2015).
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  14. M. Ornigotti, G. D. Valle, T. T. Fernandez, A. Coppa, V. Foglietti, P. Laporta, and S. Longhi, “Visualization of two-photon Rabi oscillations in evanescently coupled optical waveguides,” J. Phys. B: At., Mol. Opt. Phys. 41, 085402 (2008).
    [Crossref]
  15. K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express 16, 10309–10314 (2008).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  18. G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
    [Crossref] [PubMed]
  19. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Dynamics of topological light states in spiraling structures,” Opt. Lett. 38, 3414–3417 (2013).
    [Crossref] [PubMed]
  20. Y. V. Kartashov, A. Szameit, V. A. Vysloukh, and L. Torner, “Light tunneling inhibition and anisotropic diffraction engineering in two-dimensional waveguide arrays,” Opt. Lett. 34, 2906–2908 (2009).
    [Crossref] [PubMed]
  21. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Enhancement and inhibition of light tunneling mediated by resonant mode conversion,” Opt. Lett. 39, 933–936 (2014).
    [Crossref] [PubMed]
  22. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3, 91–94 (2009).
    [Crossref]
  23. N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Phys. Lett. A 268, 298–305 (2000).
    [Crossref]
  24. N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E 62, 3135–3145 (2000).
    [Crossref]
  25. N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E 66, 056108 (2002).
    [Crossref]
  26. 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]
  27. 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]
  28. 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 Photon. Rev. 10, 526–531 (2016).
    [Crossref]
  29. A. Liemert and A. Kienle, “Fractional Schrödinger equation in the presence of the linear potential,” Mathematics 4, 31 (2016).
    [Crossref]
  30. C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation,” Sci. Rep. 7, 5442 (2017).
    [Crossref]
  31. B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53, 083702 (2012).
    [Crossref]
  32. C. Klein, C. Sparber, and P. Markowich, “Numerical study of fractional nonlinear Schrödinger equations,” Proc. R. Soc. A 470, 20140364 (2014).
    [Crossref]
  33. 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]
  34. 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]
  35. S. Longhi, “Fractional Schrödinger equation in optics,” Opt. Lett. 40, 1117–1120 (2015).
    [Crossref] [PubMed]
  36. D. Zhang, Y. Q. Zhang, Z. Y. Zhang, N. Ahmed, Y. P. Zhang, F. L. Li, M. R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529, 1700149 (2017).
    [Crossref]

2017 (3)

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]

C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation,” Sci. Rep. 7, 5442 (2017).
[Crossref]

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

2016 (6)

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]

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]

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 Photon. Rev. 10, 526–531 (2016).
[Crossref]

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

Y. V. Kartashov, V. V. Konotop, D. A. Zezyulin, and L. Torner, “Bloch oscillations in optical and Zeeman lattices in the presence of spin-orbit coupling,” Phys. Rev. Lett. 117, 215301 (2016).
[Crossref] [PubMed]

2015 (4)

2014 (3)

2013 (1)

2012 (3)

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref] [PubMed]

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

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
[Crossref]

2011 (1)

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[Crossref]

2009 (4)

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[Crossref]

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

Y. V. Kartashov, A. Szameit, V. A. Vysloukh, and L. Torner, “Light tunneling inhibition and anisotropic diffraction engineering in two-dimensional waveguide arrays,” Opt. Lett. 34, 2906–2908 (2009).
[Crossref] [PubMed]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3, 91–94 (2009).
[Crossref]

2008 (3)

M. Ornigotti, G. D. Valle, T. T. Fernandez, A. Coppa, V. Foglietti, P. Laporta, and S. Longhi, “Visualization of two-photon Rabi oscillations in evanescently coupled optical waveguides,” J. Phys. B: At., Mol. Opt. Phys. 41, 085402 (2008).
[Crossref]

K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express 16, 10309–10314 (2008).
[Crossref] [PubMed]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

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]

2002 (1)

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

2000 (3)

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

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

K. S. Lee and T. Erdogan, “Fiber mode coupling in transmissive and reflective tilted fiber gratings,” Appl. Opt. 39, 1394–1404 (2000).
[Crossref]

1990 (1)

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electronics Letters 26, 1270–1272 (1990).
[Crossref]

1936 (1)

I. I. Rabi, “On the process of space quantization,” Phys. Rev. 49, 324–328 (1936).
[Crossref]

1929 (1)

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
[Crossref]

Ahmed, N.

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

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

Belic, M. R.

D. Zhang, Y. Q. Zhang, Z. Y. Zhang, N. Ahmed, Y. P. Zhang, F. L. Li, M. R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529, 1700149 (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, Y. Zhu, W. P. Zhong, Y. P. Zhang, D. N. Christodoulides, and M. Xiao, “Pt symmetry in a fractional Schrödinger equation,” Laser Photon. 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]

Biancalana, F.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref] [PubMed]

Bilodeau, F.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electronics Letters 26, 1270–1272 (1990).
[Crossref]

Bloch, F.

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
[Crossref]

Chen, X.

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 Photon. Rev. 10, 526–531 (2016).
[Crossref]

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express 16, 10309–10314 (2008).
[Crossref] [PubMed]

Conti, C.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref] [PubMed]

Coppa, A.

M. Ornigotti, G. D. Valle, T. T. Fernandez, A. Coppa, V. Foglietti, P. Laporta, and S. Longhi, “Visualization of two-photon Rabi oscillations in evanescently coupled optical waveguides,” J. Phys. B: At., Mol. Opt. Phys. 41, 085402 (2008).
[Crossref]

Dong, L.

C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation,” Sci. Rep. 7, 5442 (2017).
[Crossref]

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]

Erdogan, T.

Fan, D.

Fan, S.

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3, 91–94 (2009).
[Crossref]

Fernandez, T. T.

M. Ornigotti, G. D. Valle, T. T. Fernandez, A. Coppa, V. Foglietti, P. Laporta, and S. Longhi, “Visualization of two-photon Rabi oscillations in evanescently coupled optical waveguides,” J. Phys. B: At., Mol. Opt. Phys. 41, 085402 (2008).
[Crossref]

Foglietti, V.

M. Ornigotti, G. D. Valle, T. T. Fernandez, A. Coppa, V. Foglietti, P. Laporta, and S. Longhi, “Visualization of two-photon Rabi oscillations in evanescently coupled optical waveguides,” J. Phys. B: At., Mol. Opt. Phys. 41, 085402 (2008).
[Crossref]

Garanovich, I. L.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
[Crossref]

Guo, B.

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

Hill, K. O.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electronics Letters 26, 1270–1272 (1990).
[Crossref]

Huang, C.

C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation,” Sci. Rep. 7, 5442 (2017).
[Crossref]

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]

Huang, D.

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

Johnson, D. C.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electronics Letters 26, 1270–1272 (1990).
[Crossref]

Kang, M. S.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref] [PubMed]

Kartashov, Y. V.

Y. V. Kartashov, V. V. Konotop, D. A. Zezyulin, and L. Torner, “Bloch oscillations in optical and Zeeman lattices in the presence of spin-orbit coupling,” Phys. Rev. Lett. 117, 215301 (2016).
[Crossref] [PubMed]

X. Zhang, F. Ye, Y. V. Kartashov, and X. Chen, “Rabi oscillations and stimulated mode conversion on the subwavelength scale,” Opt. Express 23, 6731–6737 (2015).
[Crossref] [PubMed]

V. A. Vysloukh, Y. V. Kartashov, and K. Staliunas, “Efficient mode conversion in guiding structures with longitudinal modulation of nonlinearity,” Opt. Lett. 40, 4631–4634 (2015).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Enhancement and inhibition of light tunneling mediated by resonant mode conversion,” Opt. Lett. 39, 933–936 (2014).
[Crossref] [PubMed]

V. A. Vysloukh and Y. V. Kartashov, “Resonant mode conversion in the waveguides with unbroken and broken PT symmetry,” Opt. Lett. 39, 5933–5936 (2014).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Dynamics of topological light states in spiraling structures,” Opt. Lett. 38, 3414–3417 (2013).
[Crossref] [PubMed]

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[Crossref]

Y. V. Kartashov, A. Szameit, V. A. Vysloukh, and L. Torner, “Light tunneling inhibition and anisotropic diffraction engineering in two-dimensional waveguide arrays,” Opt. Lett. 34, 2906–2908 (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]

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. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express 16, 10309–10314 (2008).
[Crossref] [PubMed]

Kivshar, Y. S.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
[Crossref]

Klein, C.

C. Klein, C. Sparber, and P. Markowich, “Numerical study of fractional nonlinear Schrödinger equations,” Proc. R. Soc. A 470, 20140364 (2014).
[Crossref]

Konotop, V. V.

Y. V. Kartashov, V. V. Konotop, D. A. Zezyulin, and L. Torner, “Bloch oscillations in optical and Zeeman lattices in the presence of spin-orbit coupling,” Phys. Rev. Lett. 117, 215301 (2016).
[Crossref] [PubMed]

Laporta, P.

M. Ornigotti, G. D. Valle, T. T. Fernandez, A. Coppa, V. Foglietti, P. Laporta, and S. Longhi, “Visualization of two-photon Rabi oscillations in evanescently coupled optical waveguides,” J. Phys. B: At., Mol. Opt. Phys. 41, 085402 (2008).
[Crossref]

Laskin, N.

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

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

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

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

Lee, H. W.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref] [PubMed]

Lee, K. S.

Lei, D.

Li, C.

Li, C. B.

Li, F. L.

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]

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

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]

Longhi, S.

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

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
[Crossref]

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[Crossref]

M. Ornigotti, G. D. Valle, T. T. Fernandez, A. Coppa, V. Foglietti, P. Laporta, and S. Longhi, “Visualization of two-photon Rabi oscillations in evanescently coupled optical waveguides,” J. Phys. B: At., Mol. Opt. Phys. 41, 085402 (2008).
[Crossref]

Makris, K. G.

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express 16, 10309–10314 (2008).
[Crossref] [PubMed]

Malo, B.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electronics Letters 26, 1270–1272 (1990).
[Crossref]

Malomed, B. A.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[Crossref]

Markowich, P.

C. Klein, C. Sparber, and P. Markowich, “Numerical study of fractional nonlinear Schrödinger equations,” Proc. R. Soc. A 470, 20140364 (2014).
[Crossref]

Ornigotti, M.

M. Ornigotti, G. D. Valle, T. T. Fernandez, A. Coppa, V. Foglietti, P. Laporta, and S. Longhi, “Visualization of two-photon Rabi oscillations in evanescently coupled optical waveguides,” J. Phys. B: At., Mol. Opt. Phys. 41, 085402 (2008).
[Crossref]

Peleg, O.

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express 16, 10309–10314 (2008).
[Crossref] [PubMed]

Rabi, I. I.

I. I. Rabi, “On the process of space quantization,” Phys. Rev. 49, 324–328 (1936).
[Crossref]

Russell, P. S. J.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref] [PubMed]

Rüter, C. E.

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

Segev, M.

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express 16, 10309–10314 (2008).
[Crossref] [PubMed]

Shandarova, K.

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

Skinner, I.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electronics Letters 26, 1270–1272 (1990).
[Crossref]

Sparber, C.

C. Klein, C. Sparber, and P. Markowich, “Numerical study of fractional nonlinear Schrödinger equations,” Proc. R. Soc. A 470, 20140364 (2014).
[Crossref]

Staliunas, K.

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

Sukhorukov, A. A.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
[Crossref]

Szameit, A.

Torner, L.

Y. V. Kartashov, V. V. Konotop, D. A. Zezyulin, and L. Torner, “Bloch oscillations in optical and Zeeman lattices in the presence of spin-orbit coupling,” Phys. Rev. Lett. 117, 215301 (2016).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Enhancement and inhibition of light tunneling mediated by resonant mode conversion,” Opt. Lett. 39, 933–936 (2014).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Dynamics of topological light states in spiraling structures,” Opt. Lett. 38, 3414–3417 (2013).
[Crossref] [PubMed]

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[Crossref]

Y. V. Kartashov, A. Szameit, V. A. Vysloukh, and L. Torner, “Light tunneling inhibition and anisotropic diffraction engineering in two-dimensional waveguide arrays,” Opt. Lett. 34, 2906–2908 (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]

Valle, G. D.

M. Ornigotti, G. D. Valle, T. T. Fernandez, A. Coppa, V. Foglietti, P. Laporta, and S. Longhi, “Visualization of two-photon Rabi oscillations in evanescently coupled optical waveguides,” J. Phys. B: At., Mol. Opt. Phys. 41, 085402 (2008).
[Crossref]

Vineberg, K. A.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electronics Letters 26, 1270–1272 (1990).
[Crossref]

Vysloukh, V. A.

Weiss, T.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref] [PubMed]

Wong, G. K. L.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref] [PubMed]

Xiao, M.

D. Zhang, Y. Q. Zhang, Z. Y. Zhang, N. Ahmed, Y. P. Zhang, F. L. Li, M. R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. (Berlin) 529, 1700149 (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, Y. Zhu, W. P. Zhong, Y. P. Zhang, D. N. Christodoulides, and M. Xiao, “Pt symmetry in a fractional Schrödinger equation,” Laser Photon. 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]

Xu, C.

Ye, F.

Yu, Z.

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3, 91–94 (2009).
[Crossref]

Zezyulin, D. A.

Y. V. Kartashov, V. V. Konotop, D. A. Zezyulin, and L. Torner, “Bloch oscillations in optical and Zeeman lattices in the presence of spin-orbit coupling,” Phys. Rev. Lett. 117, 215301 (2016).
[Crossref] [PubMed]

Zhang, D.

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]

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

Zhang, L.

Zhang, X.

Zhang, Y. P.

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

Zhang, Y. Q.

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]

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

Zhang, Z. Y.

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

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. 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 Photon. 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]

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 Photon. 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]

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 Photon. Rev. 10, 526–531 (2016).
[Crossref]

Ann. Phys. (Berlin) (1)

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

Appl. Opt. (1)

Electronics Letters (1)

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electronics Letters 26, 1270–1272 (1990).
[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, 083702 (2012).
[Crossref]

J. Phys. B: At., Mol. Opt. Phys. (1)

M. Ornigotti, G. D. Valle, T. T. Fernandez, A. Coppa, V. Foglietti, P. Laporta, and S. Longhi, “Visualization of two-photon Rabi oscillations in evanescently coupled optical waveguides,” J. Phys. B: At., Mol. Opt. Phys. 41, 085402 (2008).
[Crossref]

Laser Photon. Rev. (2)

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 Photon. Rev. 10, 526–531 (2016).
[Crossref]

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[Crossref]

Mathematics (1)

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

Nat. Photon. (1)

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3, 91–94 (2009).
[Crossref]

Opt. Express (3)

Opt. Lett. (7)

Optica (1)

Phys. Lett. A (1)

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

Phys. Rep. (2)

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
[Crossref]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

Phys. Rev. (1)

I. I. Rabi, “On the process of space quantization,” Phys. Rev. 49, 324–328 (1936).
[Crossref]

Phys. Rev. E (2)

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

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

Phys. Rev. Lett. (4)

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. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[Crossref] [PubMed]

Y. V. Kartashov, V. V. Konotop, D. A. Zezyulin, and L. Torner, “Bloch oscillations in optical and Zeeman lattices in the presence of spin-orbit coupling,” Phys. Rev. Lett. 117, 215301 (2016).
[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]

Proc. R. Soc. A (1)

C. Klein, C. Sparber, and P. Markowich, “Numerical study of fractional nonlinear Schrödinger equations,” Proc. R. Soc. A 470, 20140364 (2014).
[Crossref]

Rev. Mod. Phys. (1)

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[Crossref]

Sci. Rep. (2)

C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation,” Sci. Rep. 7, 5442 (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]

Science (1)

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref] [PubMed]

Z. Phys. (1)

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 (a)–(c) Eigenmodes of the Gaussian potential, corresponding to α = 2, 1.5 and 1, respectively. The blue, green, red, magenta, and cyan curves indicate the first, second, third, fourth, and fifth modes, respectively. All the modes are shifted vertically, to show the corresponding energy levels in the potential. (d)–(f) Resonant mode conversions between the first and third modes, corresponding to α = 2, 1.5 and 1, respectively. (g) Resonant mode conversion among the first, third and fifth modes, corresponding to α = 1.
Fig. 2
Fig. 2 (a)–(c) Relation between the conversion period zcr13 and the detuning δ, corresponding to α = 2, 1.5 and 1, respectively. (d) Relation between the conversion period zcr13/2 and the Lévy index α at different longitudinal modulation depths µ. (e) Relation between the conversion period zcr13 and the longitudinal modulation depth µ, for different Lévy indices α.
Fig. 3
Fig. 3 (a)–(c) Band structure in the first Brillouin zone, corresponding to α = 2, 1.5 and 1, respectively. The black, blue and red curves represent the first, second and third bands. (d)–(f) Eigenmodes corresponding to α = 2, 1.5 and 1, respectively. The black, blue and red modes are corresponding to the black, blue and red dots (kx = 0.55) in the band structure.
Fig. 4
Fig. 4 (a) Cascading mode conversion at α = 2. (b) Weight of the eigenmode during propagation. (c)&(d) and (e)&(f) are same as (a)&(b), but for α = 1.5 and α = 1, respectively. Black, blue and red curves in (b), (d) and (f) represent the weights of the first, second and third eigenmodes, respectively. The other parameter: µ = 0.04.
Fig. 5
Fig. 5 (a) Conversion distance between different modes. (b) Weight of the eigenmode during propagation. The blue curve refers to the left y axis, while the red curves (solid and dashed curves are zcr23 and zcr13, respectively) refer to the right y axis. (b)–(d) Mode conversion between the first and third eigenmodes with α = 2, 1.5 and 1, respectively. (e) Eigenmodes at α ≈ 1.242. The setup is as in Fig. 3(b). The other parameter: µ = 0.04.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

i ψ z = 1 2 ( 2 x 2 ) α / 2 ψ + p R ( x ) [ 1 + μ cos ( Ω z z ) ] ψ ,
ψ ( x , z ) = ϕ ( x ) exp ( i β z ) .
β ϕ = 1 2 ( 2 x 2 ) α / 2 ϕ + p R ( x ) ϕ .
P m = 1 d 0 d 0 V ( x ) exp ( i K m x ) d x .
n [ β + 1 2 | k + K n | α ] c n exp [ i ( k + K n ) x ] + m , n P m c n exp [ i ( k + K n + K m ) x ] = 0 .
1 2 | k + K q | α c q m P m c q m = β c q ,
ψ ( x , z ) = C m ( z ) ϕ m ( x ) exp ( i β m z ) + C n ( z ) ϕ n ( x ) exp ( i β n z ) ,
i C m z = 1 2 p μ ϕ m R ϕ n ϕ m ϕ m C n , i C n z = 1 2 p μ ϕ n R ϕ m ϕ n ϕ n C m ,
z crmn = π | Ω x | ,
Ω x = μ p 2 ϕ m R ϕ n ϕ m ϕ m ϕ n ϕ n .

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