Abstract

We address the properties of wavepacket localization-delocalization transition (LDT) in fractional dimensions with a quasi-periodic lattice. The LDT point, which is generally determined by the competition between two sub-lattices comprising the quasi-periodic lattice, turns out to be inversely proportional to the Lévy index. Surprisingly, we find that, in the presence of weak structural disorder, anti-Anderson localization occurs, i.e., the introduced disorder results in an increasing of the size of the linear modes. Inclusion of a weak focusing nonlinearity is shown to improve localization. The propagation simulation achieves excellent agreement with the linear and nonlinear eigenmode analysis.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
    [Crossref]
  2. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671 (1997).
    [Crossref]
  3. M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
    [Crossref] [PubMed]
  4. S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
    [Crossref] [PubMed]
  5. A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104, 223903 (2010).
    [Crossref] [PubMed]
  6. C. M. Soukoulis, Photonic crystals and light localization in the 21st century(Springer Science & Business Media, 2012, vol. 563).
  7. 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]
  8. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486 (1987).
    [Crossref] [PubMed]
  9. E. Abrahams, 50 years of Anderson Localization(world scientific, 2010).
  10. M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7, 197 (2013).
    [Crossref]
  11. M. F. Limonov and M. Richard, Optical properties of photonic structures: interplay of order and disorder(Taylor & FrancisGroup, 2016).
  12. S. Aubry and G. André, “Analyticity breaking and anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc 3, 18 (1980).
  13. M. Kohmoto, “Metal-insulator transition and scaling for incommensurate systems,” Phys. Rev. Lett. 51, 1198 (1983).
    [Crossref]
  14. D. Thouless, “Bandwidths for a quasiperiodic tight-binding model,” Phys. Rev. B 28, 4272 (1983).
    [Crossref]
  15. Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
    [Crossref] [PubMed]
  16. M. Modugno, “Exponential localization in one-dimensional quasi-periodic optical lattices,” New J. Phys. 11, 033023 (2009).
    [Crossref]
  17. L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332, 1202977 (2011).
  18. M. Ghulinyan, “One-dimensional photonic quasicrystals,” arXiv preprint arXiv: 1505.02400 (2015).
  19. M. Boguslawski and C. Denz, “Light propagation in optically induced fibonacci lattices,” arXiv preprint arXiv: 1501.04479 (2015).
  20. C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40, 2758–2761 (2015).
    [Crossref] [PubMed]
  21. N. Laskin, “Fractional quantum mechanics and lévy path integrals,” Phys. Lett. A 268, 298–305 (2000).
    [Crossref]
  22. N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E 62, 3135 (2000).
    [Crossref]
  23. N. Laskin, “Fractional schrödinger equation,” Phys. Rev. E 66, 056108 (2002).
    [Crossref]
  24. S. Longhi, “Fractional schrödinger equation in optics,” Opt. Lett. 40, 1117–1120 (2015).
    [Crossref] [PubMed]
  25. 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, 180403 (2015).
    [Crossref]
  26. 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]
  27. Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional schrödinger equation,” Laser Photonics Rev. 10, 526–531 (2016).
    [Crossref]
  28. W.-P. Zhong, M. Belić, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
    [Crossref]
  29. Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Resonant mode conversions and rabi oscillations in a fractional schrödinger equation,” Opt. Express 25, 32401–32410 (2017).
    [Crossref]
  30. C. Huang and L. Dong, “Beam propagation management in a fractional schrödinger equation,” Sci. Rep. 7, 5442 (2017).
    [Crossref]
  31. F. Zang, Y. Wang, and L. Li, “Dynamics of gaussian beam modeled by fractional schrödinger equation with a variable coefficient,” Opt. Express 26, 23740–23750 (2018).
    [Crossref] [PubMed]
  32. 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]
  33. C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional schrödinger equation with a pt symmetric potential,” Europhys. Lett. 122, 24002 (2018).
    [Crossref]
  34. L. Dong and C. Huang, “Double-hump solitons in fractional dimensions with a pt-symmetric potential,” Opt. Express 26, 10509–10518 (2018).
    [Crossref] [PubMed]
  35. J. Xiao, Z. Tian, C. Huang, and L. Dong, “Surface gap solitons in a nonlinear fractional schrödinger equation,” Opt. Express 26, 2650–2658 (2018).
    [Crossref] [PubMed]
  36. L. Dong and C. Huang, “Composition relation between nonlinear Bloch waves and gap solitons in periodic fractional systems,” Mater 11, 1134 (2018).
    [Crossref]
  37. Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-gaussian–like soliton in the nonlocal nonlinear fractional schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
    [Crossref]
  38. X. Yao and X. Liu, “Solitons in the fractional schrödinger equation with parity-time-symmetric lattice potential,” Photon. Res. 6, 875–879 (2018).
    [Crossref]
  39. M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional schrödinger equation with a kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
    [Crossref]
  40. B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M. Lewenstein, “Atomic bose and anderson glasses in optical lattices,” Phys. Rev. Lett. 91, 080403 (2003).
    [Crossref] [PubMed]
  41. J. Yang, Nonlinear waves in integrable and nonintegrable systems (SIAM, 2010).
    [Crossref]
  42. B. Guo, X. Pu, and F. Huang, Fractional partial differential equations and their numerical solutions(World Scientific, 2015).
    [Crossref]
  43. H. Richard, Fractional calculus: an introduction for physicists(World Scientific, 2014).
  44. M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990 (1997).
    [Crossref]

2018 (8)

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional schrödinger equation with a pt symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

L. Dong and C. Huang, “Double-hump solitons in fractional dimensions with a pt-symmetric potential,” Opt. Express 26, 10509–10518 (2018).
[Crossref] [PubMed]

J. Xiao, Z. Tian, C. Huang, and L. Dong, “Surface gap solitons in a nonlinear fractional schrödinger equation,” Opt. Express 26, 2650–2658 (2018).
[Crossref] [PubMed]

L. Dong and C. Huang, “Composition relation between nonlinear Bloch waves and gap solitons in periodic fractional systems,” Mater 11, 1134 (2018).
[Crossref]

Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-gaussian–like soliton in the nonlocal nonlinear fractional schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
[Crossref]

X. Yao and X. Liu, “Solitons in the fractional schrödinger equation with parity-time-symmetric lattice potential,” Photon. Res. 6, 875–879 (2018).
[Crossref]

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional schrödinger equation with a kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[Crossref]

F. Zang, Y. Wang, and L. Li, “Dynamics of gaussian beam modeled by fractional schrödinger equation with a variable coefficient,” Opt. Express 26, 23740–23750 (2018).
[Crossref] [PubMed]

2017 (2)

2016 (4)

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, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional schrödinger equation,” Laser Photonics Rev. 10, 526–531 (2016).
[Crossref]

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

2015 (5)

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

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

M. Ghulinyan, “One-dimensional photonic quasicrystals,” arXiv preprint arXiv: 1505.02400 (2015).

M. Boguslawski and C. Denz, “Light propagation in optically induced fibonacci lattices,” arXiv preprint arXiv: 1501.04479 (2015).

C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40, 2758–2761 (2015).
[Crossref] [PubMed]

2013 (1)

M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7, 197 (2013).
[Crossref]

2012 (1)

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)

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332, 1202977 (2011).

2010 (1)

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104, 223903 (2010).
[Crossref] [PubMed]

2009 (2)

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref] [PubMed]

M. Modugno, “Exponential localization in one-dimensional quasi-periodic optical lattices,” New J. Phys. 11, 033023 (2009).
[Crossref]

2006 (2)

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[Crossref] [PubMed]

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref] [PubMed]

2003 (1)

B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M. Lewenstein, “Atomic bose and anderson glasses in optical lattices,” Phys. Rev. Lett. 91, 080403 (2003).
[Crossref] [PubMed]

2002 (1)

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

2000 (2)

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 (2000).
[Crossref]

1997 (2)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671 (1997).
[Crossref]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990 (1997).
[Crossref]

1987 (1)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486 (1987).
[Crossref] [PubMed]

1983 (2)

M. Kohmoto, “Metal-insulator transition and scaling for incommensurate systems,” Phys. Rev. Lett. 51, 1198 (1983).
[Crossref]

D. Thouless, “Bandwidths for a quasiperiodic tight-binding model,” Phys. Rev. B 28, 4272 (1983).
[Crossref]

1980 (1)

S. Aubry and G. André, “Analyticity breaking and anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc 3, 18 (1980).

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
[Crossref]

Abrahams, E.

E. Abrahams, 50 years of Anderson Localization(world scientific, 2010).

Aegerter, C. M.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[Crossref] [PubMed]

Ahmed, N.

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

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
[Crossref]

André, G.

S. Aubry and G. André, “Analyticity breaking and anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc 3, 18 (1980).

Aubry, S.

S. Aubry and G. André, “Analyticity breaking and anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc 3, 18 (1980).

Bartolini, P.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671 (1997).
[Crossref]

Belic, M.

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

Belic, M. R.

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Resonant mode conversions and rabi oscillations in a fractional schrödinger equation,” Opt. Express 25, 32401–32410 (2017).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional schrödinger equation,” Laser Photonics Rev. 10, 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, 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, 180403 (2015).
[Crossref]

Boguslawski, M.

M. Boguslawski and C. Denz, “Light propagation in optically induced fibonacci lattices,” arXiv preprint arXiv: 1501.04479 (2015).

Chen, M.

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional schrödinger equation with a kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[Crossref]

Christodoulides, D. N.

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

M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7, 197 (2013).
[Crossref]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990 (1997).
[Crossref]

Cianci, E.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref] [PubMed]

Coskun, T. H.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990 (1997).
[Crossref]

Damski, B.

B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M. Lewenstein, “Atomic bose and anderson glasses in optical lattices,” Phys. Rev. Lett. 91, 080403 (2003).
[Crossref] [PubMed]

Davidson, N.

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref] [PubMed]

Deng, H.

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional schrödinger equation with a pt symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

Denz, C.

M. Boguslawski and C. Denz, “Light propagation in optically induced fibonacci lattices,” arXiv preprint arXiv: 1501.04479 (2015).

Dong, L.

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional schrödinger equation with a pt symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

L. Dong and C. Huang, “Double-hump solitons in fractional dimensions with a pt-symmetric potential,” Opt. Express 26, 10509–10518 (2018).
[Crossref] [PubMed]

J. Xiao, Z. Tian, C. Huang, and L. Dong, “Surface gap solitons in a nonlinear fractional schrödinger equation,” Opt. Express 26, 2650–2658 (2018).
[Crossref] [PubMed]

L. Dong and C. Huang, “Composition relation between nonlinear Bloch waves and gap solitons in periodic fractional systems,” Mater 11, 1134 (2018).
[Crossref]

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]

Dreisow, F.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104, 223903 (2010).
[Crossref] [PubMed]

Foglietti, V.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref] [PubMed]

Freedman, B.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332, 1202977 (2011).

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]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104, 223903 (2010).
[Crossref] [PubMed]

Ghulinyan, M.

M. Ghulinyan, “One-dimensional photonic quasicrystals,” arXiv preprint arXiv: 1505.02400 (2015).

Gross, P.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[Crossref] [PubMed]

Guo, B.

B. Guo, X. Pu, and F. Huang, Fractional partial differential equations and their numerical solutions(World Scientific, 2015).
[Crossref]

Guo, Q.

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional schrödinger equation with a kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[Crossref]

Hang, C.

Heinrich, M.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104, 223903 (2010).
[Crossref] [PubMed]

Hu, W.

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional schrödinger equation with a kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[Crossref]

Huang, C.

L. Dong and C. Huang, “Composition relation between nonlinear Bloch waves and gap solitons in periodic fractional systems,” Mater 11, 1134 (2018).
[Crossref]

L. Dong and C. Huang, “Double-hump solitons in fractional dimensions with a pt-symmetric potential,” Opt. Express 26, 10509–10518 (2018).
[Crossref] [PubMed]

J. Xiao, Z. Tian, C. Huang, and L. Dong, “Surface gap solitons in a nonlinear fractional schrödinger equation,” Opt. Express 26, 2650–2658 (2018).
[Crossref] [PubMed]

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional schrödinger equation with a pt symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

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, F.

B. Guo, X. Pu, and F. Huang, Fractional partial differential equations and their numerical solutions(World Scientific, 2015).
[Crossref]

Huang, G.

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486 (1987).
[Crossref] [PubMed]

Kartashov, Y. V.

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]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104, 223903 (2010).
[Crossref] [PubMed]

Kohmoto, M.

M. Kohmoto, “Metal-insulator transition and scaling for incommensurate systems,” Phys. Rev. Lett. 51, 1198 (1983).
[Crossref]

Konotop, V. V.

Lagendijk, A.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671 (1997).
[Crossref]

Lahini, Y.

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref] [PubMed]

Laporta, P.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref] [PubMed]

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 (2000).
[Crossref]

Levi, L.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332, 1202977 (2011).

Lewenstein, M.

B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M. Lewenstein, “Atomic bose and anderson glasses in optical lattices,” Phys. Rev. Lett. 91, 080403 (2003).
[Crossref] [PubMed]

Li, J.

Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-gaussian–like soliton in the nonlocal nonlinear fractional schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
[Crossref]

Li, L.

Limonov, M. F.

M. F. Limonov and M. Richard, Optical properties of photonic structures: interplay of order and disorder(Taylor & FrancisGroup, 2016).

Liu, X.

X. Yao and X. Liu, “Solitons in the fractional schrödinger equation with parity-time-symmetric lattice potential,” Photon. Res. 6, 875–879 (2018).
[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, 180403 (2015).
[Crossref]

Lobino, M.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref] [PubMed]

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]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104, 223903 (2010).
[Crossref] [PubMed]

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref] [PubMed]

Lu, D.

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional schrödinger equation with a kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[Crossref]

Manela, O.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332, 1202977 (2011).

Marangoni, M.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref] [PubMed]

Maret, G.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[Crossref] [PubMed]

Mitchell, M.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990 (1997).
[Crossref]

Modugno, M.

M. Modugno, “Exponential localization in one-dimensional quasi-periodic optical lattices,” New J. Phys. 11, 033023 (2009).
[Crossref]

Morandotti, R.

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref] [PubMed]

Nolte, S.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104, 223903 (2010).
[Crossref] [PubMed]

Pertsch, T.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104, 223903 (2010).
[Crossref] [PubMed]

Pozzi, F.

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref] [PubMed]

Pu, X.

B. Guo, X. Pu, and F. Huang, Fractional partial differential equations and their numerical solutions(World Scientific, 2015).
[Crossref]

Pugatch, R.

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref] [PubMed]

Ramponi, R.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref] [PubMed]

Rechtsman, M.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332, 1202977 (2011).

Richard, H.

H. Richard, Fractional calculus: an introduction for physicists(World Scientific, 2014).

Richard, M.

M. F. Limonov and M. Richard, Optical properties of photonic structures: interplay of order and disorder(Taylor & FrancisGroup, 2016).

Righini, R.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671 (1997).
[Crossref]

Santos, L.

B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M. Lewenstein, “Atomic bose and anderson glasses in optical lattices,” Phys. Rev. Lett. 91, 080403 (2003).
[Crossref] [PubMed]

Schwartz, T.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332, 1202977 (2011).

Segev, M.

M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7, 197 (2013).
[Crossref]

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332, 1202977 (2011).

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990 (1997).
[Crossref]

Silberberg, Y.

M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7, 197 (2013).
[Crossref]

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref] [PubMed]

Sorel, M.

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref] [PubMed]

Soukoulis, C. M.

C. M. Soukoulis, Photonic crystals and light localization in the 21st century(Springer Science & Business Media, 2012, vol. 563).

Störzer, M.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[Crossref] [PubMed]

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]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104, 223903 (2010).
[Crossref] [PubMed]

Szameit, A.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104, 223903 (2010).
[Crossref] [PubMed]

Thouless, D.

D. Thouless, “Bandwidths for a quasiperiodic tight-binding model,” Phys. Rev. B 28, 4272 (1983).
[Crossref]

Tian, Z.

Tünnermann, A.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104, 223903 (2010).
[Crossref] [PubMed]

Wang, Q.

Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-gaussian–like soliton in the nonlocal nonlinear fractional schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
[Crossref]

Wang, R.

Wang, Y.

Wiersma, D. S.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671 (1997).
[Crossref]

Xiao, J.

Xiao, M.

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional schrödinger equation,” Laser Photonics Rev. 10, 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, 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, 180403 (2015).
[Crossref]

Xie, W.

Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-gaussian–like soliton in the nonlocal nonlinear fractional schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
[Crossref]

Yang, J.

J. Yang, Nonlinear waves in integrable and nonintegrable systems (SIAM, 2010).
[Crossref]

Yao, X.

Ye, F.

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional schrödinger equation with a pt symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

Zakrzewski, J.

B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M. Lewenstein, “Atomic bose and anderson glasses in optical lattices,” Phys. Rev. Lett. 91, 080403 (2003).
[Crossref] [PubMed]

Zang, F.

Zeng, S.

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional schrödinger equation with a kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[Crossref]

Zhang, J.

Zhang, L.

Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-gaussian–like soliton in the nonlocal nonlinear fractional schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
[Crossref]

Zhang, W.

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional schrödinger equation with a pt symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

Zhang, Y.

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Resonant mode conversions and rabi oscillations in a fractional schrödinger equation,” Opt. Express 25, 32401–32410 (2017).
[Crossref]

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Resonant mode conversions and rabi oscillations in a fractional schrödinger equation,” Opt. Express 25, 32401–32410 (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, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional schrödinger equation,” Laser Photonics 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]

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

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional schrödinger equation,” Laser Photonics 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, 180403 (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, 180403 (2015).
[Crossref]

Zhong, H.

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Resonant mode conversions and rabi oscillations in a fractional schrödinger equation,” Opt. Express 25, 32401–32410 (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, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional schrödinger equation,” Laser Photonics Rev. 10, 526–531 (2016).
[Crossref]

Zhong, W.

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional schrödinger equation,” Laser Photonics Rev. 10, 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, 180403 (2015).
[Crossref]

Zhong, W.-P.

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

Zhu, Y.

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

Zoller, P.

B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M. Lewenstein, “Atomic bose and anderson glasses in optical lattices,” Phys. Rev. Lett. 91, 080403 (2003).
[Crossref] [PubMed]

Ann. Isr. Phys. Soc (1)

S. Aubry and G. André, “Analyticity breaking and anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc 3, 18 (1980).

Ann. Phys. (1)

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

arXiv preprint arXiv (2)

M. Ghulinyan, “One-dimensional photonic quasicrystals,” arXiv preprint arXiv: 1505.02400 (2015).

M. Boguslawski and C. Denz, “Light propagation in optically induced fibonacci lattices,” arXiv preprint arXiv: 1501.04479 (2015).

Europhys. Lett. (2)

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional schrödinger equation with a pt symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-gaussian–like soliton in the nonlocal nonlinear fractional schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
[Crossref]

Laser Photonics Rev. (1)

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

Mater (1)

L. Dong and C. Huang, “Composition relation between nonlinear Bloch waves and gap solitons in periodic fractional systems,” Mater 11, 1134 (2018).
[Crossref]

Nat. Photonics (1)

M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7, 197 (2013).
[Crossref]

Nature (1)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671 (1997).
[Crossref]

New J. Phys. (1)

M. Modugno, “Exponential localization in one-dimensional quasi-periodic optical lattices,” New J. Phys. 11, 033023 (2009).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Photon. Res. (1)

Phys. Lett. A (1)

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

Phys. Rep. (1)

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]

Phys. Rev. (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
[Crossref]

Phys. Rev. B (1)

D. Thouless, “Bandwidths for a quasiperiodic tight-binding model,” Phys. Rev. B 28, 4272 (1983).
[Crossref]

Phys. Rev. E (3)

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

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

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional schrödinger equation with a kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[Crossref]

Phys. Rev. Lett. (9)

B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M. Lewenstein, “Atomic bose and anderson glasses in optical lattices,” Phys. Rev. Lett. 91, 080403 (2003).
[Crossref] [PubMed]

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

Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, “Observation of a localization transition in quasiperiodic photonic lattices,” Phys. Rev. Lett. 103, 013901 (2009).
[Crossref] [PubMed]

M. Kohmoto, “Metal-insulator transition and scaling for incommensurate systems,” Phys. Rev. Lett. 51, 1198 (1983).
[Crossref]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486 (1987).
[Crossref] [PubMed]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[Crossref] [PubMed]

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104, 223903 (2010).
[Crossref] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990 (1997).
[Crossref]

Sci. Rep. (2)

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]

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

Science (1)

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332, 1202977 (2011).

Other (6)

J. Yang, Nonlinear waves in integrable and nonintegrable systems (SIAM, 2010).
[Crossref]

B. Guo, X. Pu, and F. Huang, Fractional partial differential equations and their numerical solutions(World Scientific, 2015).
[Crossref]

H. Richard, Fractional calculus: an introduction for physicists(World Scientific, 2014).

C. M. Soukoulis, Photonic crystals and light localization in the 21st century(Springer Science & Business Media, 2012, vol. 563).

E. Abrahams, 50 years of Anderson Localization(world scientific, 2010).

M. F. Limonov and M. Richard, Optical properties of photonic structures: interplay of order and disorder(Taylor & FrancisGroup, 2016).

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Figures (6)

Fig. 1
Fig. 1 (a-c) The moduli of the eigenmode with the largest β as a function of x and p2. α = 1.1 in (a), 1.5 in (b), and 1.9 in (c). (d) Integral form-factor of linear eigenmodes with increasing p2 for different α. p1=1.0, Ω 1 = 5 + 1, and Ω2 = 2 in all panels.
Fig. 2
Fig. 2 Integral form-factor of linear eigenmodes as a function of α and p2 for p1 = 1.0 (a) and p1 = 0.5 (b). In (a) and (b), the blue region corresponds to the delocalized state, and the red region corresponds to the localized one. (c) LDT threshold on (p1, p2)-plane defined as the maximum slope of the form factor at α = 1.5.
Fig. 3
Fig. 3 (a) The average effective width of linear eigenmodes with the largest β as a function of ar for 103 disorder realizations. The arrow connotes the increasing α with 1.1, 1.3, 1.5, 1.7, and 1.9. (b) The dependence of the average effective width 〈weff〉 on disorder level ar for a delocalized eigenmode is plotted. (c, d) Any 50 of 103 linear eigenmodes under random perturbation are depicted, which correspond to the red solid circles in (a) and (b). p2 = 1.2 in (a), p2 = 0.65 in (b), and p1 = 1, Ω 1 = 5 + 1, Ω2 = 2 in all panels.
Fig. 4
Fig. 4 The average eigenvalue (a, c) and effective width (b, d) of nonlinear eigenmodes as a function of U for 103 disorder realizations. Two typical examples for nonlinear eigenmodes are plotted in (e) and (f), which correspond to the red solid makers in (b) and (d). p2 = 1.2 in (a, b, e), p2 = 0.4 in (c, d, f), p1 = 1, ar = 0.006, Ω 1 = 5 + 1, and Ω2 = 2 in all panels.
Fig. 5
Fig. 5 Examples of linear and nonlinear Gaussian beams propagation. (a) and (b) corresponding to red dashed lines in Figs. 6(a) and 6(b). (c-f) corresponding to the red solid markers in Figs. 6(a)-6(d), respectively. z = 1200 in (a), (c) and (e), z = 2400 in (b), (d) and (f).
Fig. 6
Fig. 6 The effective width of any 50 linear (a, b) and nonlinear (c, d) excited modes by Gaussian beams with A = 1 in (a, b), A = 0.1642, U = 0.0169 in (c), and A = 0.3519, U = 0.3519 in (d). z = 1200, p1 = 1, p2 = 1.2 in (a, c), and z = 2400, p1 = 1, p2 = 0.65 in (b, d). ar = 0.006 in (a), (c) and (d), ar = 0.04 in (b). The red dashed lines in (a) and (b) correspond to the effective width of linear modes with ar = 0.

Equations (2)

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i Ψ z = 1 2 ( 2 x 2 ) α / 2 Ψ V ( x ) Ψ .
i Ψ z = 1 2 ( 2 x 2 ) α / 2 Ψ V ( x ) Ψ | Ψ | 2 Ψ .

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