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

1. Introduction

Localization of light beams in periodic and disordered structures are the foundation of photonics and optics [1–7]. Wavepackets are extended in periodic lattices, however the inclusion of nonlinearity or a structural defect in the lattices leads to the formation of localized states. Wavepackets are exponentially localized in disordered lattices in one- or two-dimensional systems (see [8–11] and references therein for the related theory and experiment).

Quasi-periodic structure featuring long-range order is an intermediate phase between the fully periodic and fully disordered systems [12–20]. In such quasi-periodic structures, localized, critical and extended states have been found to exist or coexist, and the extended modes can be transformed into localized modes upon the change of structural parameters. Such localization-delocalization transition (LDT) in quasi-periodic system has been long explored in the tight-binding model of incommensurable potentials [13–16], the Aubry-André [12] semi-discrete model, and the continuous model with Fibonacci chain [18,19] or bi-chromatic lattice potential [20]. Results from previous studies show that the value of LDT point can be controlled by adjusting the relative strength of two simple lattices forming quasi-periodic structure, the degree of commensurability, and rotation angle of similar structures.

It should be noted that, regardless of the abundant works on LDT in quasi-periodic systems, the existence and properties of the eigenmodes in a quasi-periodic lattice supported by the fractional Schrödinger equation (FSE) has not been addressed yet. The FSE formulated by Laskin is an extension of standard quantum mechanics [21–23], in which the second-order spatial derivative was replaced with a fractional Lévy index. Due to the fractional order derivatives, the wave function inherits a profound change, as has been demonstrated in two important works in optics: fractional quantum harmonic oscillator [24] and the zigzag trajectory propagation [25], supported by the time- and space-dependent model, respectively. Soon after, several intriguing studies based on the linear or nonlinear FSE have been reported, including diffraction-free beams [26], PT-symmetry [27], “accessible solitons” [28], resonant mode conversions and Rabi oscillations [29], propagation management [30], propagation dynamics of Gaussian beam [31], and soliton dynamics [32–39]. These results show that the FSE optical system has a promising application in both the formation or evolution of eigenmodes and creation or stabilization of solitons.

A natural question thus arises: how a fractional Lévy index affects the LDT in a quasi-periodic lattice, and how structural disorders or nonlinearity changes the localization of wavepackets in such lattices?

The aim of this work is to provide answers to the above question. We investigate the eigenmode properties in the 1D FSE in [24, 25], and also consider the influence of the disorder and nonlinearity on the eigenmodes in such fractional-order quasi-crystals.

2. Theoretical model

We start our analysis by considering the propagation of light beams along the z-axis with a transverse quasi-periodic lattice. The evolution of light beam is governed by the fractional Schrödinger equation (FSE),

iΨz=12(2x2)α/2ΨV(x)Ψ.

Here, Ψ is proportional to the electric field envelope and z is a scaled propagation distance. (−2/∂x2)α/2 is the fractional Laplacian with α being the Lévy index (1 < α ≤ 2). The quasi-periodic modulation is described by the function V(x) = p1 cos (Ω1x) + p2 cos(Ω2x), where Ω1 and Ω2 are spatial frequencies of two periodic sublattices, and p1 and p2 are the sublattice depths. In optics, a quasiperiodic lattice can be obtained by superposition of two lattices with incommensurate periods [40]. The factional order can be achieved following the scheme proposed for resonator system [24] or a lensguide system [26], in which the fractional Laplacian is suggested to be realized by one mask with a phase change exp(−i/2|x|αz) at a fixed propagation distance z.

For characteristic transverse scale 8µm and light beams at λ = 1550nm propagating in fused silica, the depth p1 = 1 corresponds to the actual modulation depth δn ~ 6.6 × 10−4, x = 1 and z = 1 correspond to ~ 8µm in transverse direction and ~ 12.7mm in propagation direction, respectively.

Linear eigenmodes of Eq. (1) can be searched numerically in the form of Ψ(x, z) = ϕ(x) exp(iβz), where ϕ(x) is the profile and β is the eigenvalue of linear mode. Substituting this expression into Eq. (1), one can obtain βϕ(x) = [−1/2(−2/∂x2)α/2 + V(x)]ϕ(x) [33]. This linear problem can be solved by the Fourier collocation method [41], and the fractional term in Fourier space satisfies this relation: (Δ)α/2^ϕ(k)=|k|αϕ^(k) [42]. To simplify the discussion, here we focus on the LDT of linear modes with the maximum β and fix the spatial frequencies Ω1=5+1, Ω2 = 2 and the amplitude of the first sublattice p1 = 1. We vary α and p2 in the following analysis. The degree of localization of linear modes can be characterized by both the integral form-factor χ=U1+|ϕ|4dx and the effective width weff=[U1+|ϕ|2(xxc)2dx]1/2. Here, U=+|ϕ|2dx is the power (also called energy flow), xc=U1+x|ϕ|2dx is the central coordinate of the mode. The form-factor of linear mode is inversely proportional to its effective width, hence the higher χ (or smaller weff) means stronger localization.

3. Results and discussions

3.1. LDT in a quasi-periodic lattice

We firstly consider the modes with the largest β (namely, the fundamental mode) in the FSE. The modal profiles with the increasing p2 for three different Lévy index α are shown in Figs. 1(a)-1(c). It is apparent that, with the decreasing of α, the p2 value that requires for the occurrence of LDT increases [Fig. 1(d)]. On the other hand, we also find the degree of localization of the localized modes increases with the decrease of α [for more details see, for instance, the different χ values at p1 = 1 and p2 = 2 in Fig. 1(d)].

 figure: 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, Ω1=5+1, and Ω2 = 2 in all panels.

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We have also made a systematical analysis to reveal generic features of LDT of linear modes in fractional dimensions. Figs. 2(a) and 2(b) show the transition from expansion (blue region) to localization (red region) as a function of the sublattice depth p2. Obviously, with increasing α, the LDT point is gradually decreased, which is in agreement with these results in Figs. 1(a)-1(c). Note that, when the sublattice depth p1 is halved, the LDT points are enhanced in the whole domain [Fig. 2(b)]. It implies that the value of LDT point of linear modes determined by the competition of two sublattice depths, p1 and p2, at a fixed α [Fig. 2(c)]. In another words, the smaller the sublattice depth p1, the greater the LDT point p2, and vice versa. In addition, we can find that the LDT looks smoother as the Lévy index α is decreased [Figs. 1(d), 2(a) and 2(b)]. This phenomenon can be explained by the nonlocal characteristics of the system, i.e., the smaller the Lévy index, the stronger the nonlocal effect [43]. Thus, at a small α, there exists a lager truncated interval ∆p2 for critical states. Although here we only focused on localization and delocalization of fundamental modes, we have also checked the higher-order modes and qualitatively similar characteristics are found too.

 figure: 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.

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These above phenomena are the crucial features of the LDT of linear modes supported by the quasi-periodic lattice in fractional dimensions. However, the localized (or critical) states above (or around) the LDT point may be vulnerable to external effects, such as weak disorder and nonlinearity. Further research is needed to provide greater insight into these external effects for these states.

3.2 Localization and Anderson delocalization in a quasi-periodic lattice with weak disorder

Now we shall include structural disorder into the quasi-crystal, and consider a structure in the form of V(x) = p1 cos(Ω1x) + p2 cos(Ω2x) + ar ρ(x), where ρ(x) is a random function with a zero mean value 〈ρ(x)〉 = 0 and an unit variance 〈ρ2(x)〉 = 1, and ar is the strength of the amplitude stochastic modulation (ASM). As did in part 3.1, we plug the potential V(x) into Eq. (1) and find the stationary solutions of such disordered quasicrystal. In the following results, an ensemble average based on N=103 noise realization is used.

Figure 3(a) presents the dependence of the averaged effective width on the strength of ASM for different α. For all the cases here, we select p2 = 1.2 such that modes in the vanishing disorder cases (ar = 0) are already localized. Interestingly, gradually increasing disorder does not immediately make the modes more localized; rather, the size of the modes increases with the increasing of ar, until the width achieves a maximum at some critical value of ar; with the further increasing ar, the modes becomes more and more localized. The initial increase of the modal width with disorder level is the so-called “Anderson delocalization”, which has been revealed in Penrose-type photonic quasicrystals [17] for the system with the usual dimension α. Interestingly enough, such “Anderson delocalization” becomes even more profound for the fractional-dimension systems, as one can clearly see from Fig. 3(a). This point may be also explained by the fact that the smaller the α value, the stronger the nonlocal effect of the system [43]. Interestingly, the strength of the ASM corresponding to the maximum averaged effective width for different α is basically unchanged (≈0.006).

 figure: 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, Ω1=5+1, Ω2 = 2 in all panels.

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Notice that, at disorder level ar = 0.03, the effective width of the linear mode is still larger than that at ar = 0 for α ≥ 1.3. When the strength ar is large enough, the disorder dominates, and thus the lattice distribution ceases to be the long-range order. Then, we will observe the exponential localized mode in such a system.

Under the weak disorder level, the delocalized state will transform to be localized [Fig. 3(b)]. It is natural to compare the averaged effective width of the localized mode with that of the extended mode at the same disorder level. For ar = 0.03 and α = 1.5, their averaged effective widths are 4.44 and 19.53, as shown in Figs. 3(a) and 3(b), respectively. It indicates that the delocalized mode requires a higher disorder level than the local state if the same averaged effective width is to be achieved. Moreover, in the presence of disorders, the central position and the degree of localization of linear modes are changeable. Typical examples of the field distributions for linear modes are depicted in Figs. 3(c) and 3(d).

3.3 Localization induced by a weak focusing nonlinearity

The self-focusing Kerr nonlinear FSE is written as

iΨz=12(2x2)α/2ΨV(x)Ψ|Ψ|2Ψ.

Substituting this expression Ψ(x,z) = ϕnl(x) exp(iβz) into Eq. (2), one can obtain βϕnl(x)=˜αϕnl(x), in which ˜α=1/2(2/x2)α/2+[V(x)+|ϕnl(x)|2], and ϕnl(x) is the profile of nonlinear eigenmode. For their existence, the nonlinear modes with the maximum β are then found self-consistently as follows [44]: we first assume that the central position of the quasi-periodic lattice is slightly higher than in the other regions (if the weak disorder is introduced, the position of the maximum amplitude for the lattice is fixed), which allows us to find a localized (defect) mode. Then, we weigh the modes, and get the profile |ϕnl(x)|2, finally use |ϕnl(x)|2 to calculate the induced index. For a fixed maximum change of index, the localized mode is computed again. When a convergence is reached, we can obtain the nonlinear mode and eigenvalue.

As shown in Fig. 4, the properties of both localized and delocalized modes under weak nonlinearity are studied. The average eigenvalue 〈β〉 linearly increases with the power U for the localized mode [Fig. 4(a)], while the average effective width 〈weff〉 decrease monotonically as the power increases [Fig. 4(b)]. These suggest that a focusing nonlinearity of the medium further enhances the localization of eigenmodes, even though the weak disorder is present. Whereas for the linear delocalized mode, the impact of the weakly focusing nonlinearity (see details in Figs. 4(c) and 4(d), U < 0.0708, which is larger than that in the above case) on the average eigenvalue and average effective width are very slight. When the power beyond the threshold value U ≈ 0.0708, the focusing nonlinearity becomes dominant, and then, the 〈β〉 increases drastically, and the 〈weff〉 decreases rapidly. Representative examples of the field modulus distribution of nonlinear eigenmodes originating from linear localized and delocalized states are shown in Figs. 4(e) and 4(f), respectively.

 figure: 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, Ω1=5+1, and Ω2 = 2 in all panels.

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3.4 Proof by propagation simulations

To confirm the above discussion for the linear and nonlinear modes in the quasi-periodic lattice, we conduct some direct numerical propagation simulations to reproduce the aforementioned features, using the fourth-order Runge-Kutta method [41]. The linear and nonlinear states can be excited by a Gaussian input field in the form of ϕ(x, z)|z=0 = A exp (−x2/d2), here, d ≡ 0.5. Representative examples of the linear and nonlinear propagations are illustrated in Fig. 5. Firstly, the parameters for a localized eigenmode are considered, such as, p1 = 1, p2 = 1.2, Ω1=5+1, and Ω2 = 2 [Fig. 2(a)]. Under a weak disordered level (ar = 0.006), we monitor the effective width of linear state at z = 1200, and find almost all weff of linear states are larger than that in the purely quasi-periodic lattice [Figs. 5(a), 5(c) and 6(a)]. It indicates that the Anderson delocalization phenomena do occur in the quasi-periodic lattice with small additive random amplitude fluctuations. When the depth of the sublattice p1 = 1 is replaced by p1 = 0.65, the linear eigenmode in such system with ar = 0 is extended [Figs. 2(c), 5(b), and 6(b)]. In the presence of weak disorder (ar = 0.04), almost all weff of linear states at z = 2400 is smaller than that in the vanishing disorder case. And in this case, we find the minimum effective width is 3.044 [Fig. 6(b)]. Therefore, the average result can reveal that the phenomenon of the transformation form delocalization to localization is reproduced by introduced weak disorder [Fig. 5(d)]. Next, to share the same powers with nonlinear modes, which marked by the solid circles in Figs. 4(b) and 4(d), we set the amplitudes A = 0.1642 and A = 0.3519 for the input light beams to excite the nonlinear states [Figs. 5(e) and 5(f)]. We can find that, for the linear localized eigenmode in the purely quasi-periodic lattice, weff = 1.924 (see the red dashed line in Fig. 6(a)), and the corresponding average effective width of the nonlinear states is 〈weff〉 = 1.5539, whereas for the linear delocalized eigenmode in purely quasi-periodic lattice, weff = 25.26 (see the red dashed line in Fig. 6(b)), the corresponding average effective width for the nonlinear states is 〈weff〉 = 2.9703. Therefore, focusing nonlinearity causes the effective width of nonlinear states to be compressed [Figs. 6(c) and 6(d)]. There, we must state that the results at different propagation distances in Fig. 6 could be different, but qualitatively, the above conclusions are unchanged.

 figure: 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).

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 figure: 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.

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4. Summary and outlook

In this work, for the first time to our knowledge, we have reported the LDT and Anderson delocalization of linear modes in fractional dimensions with a quasi-periodic lattice. We have revealed that, the value of LDT point for linear modes is inversely proportional to the Lévy index, and it is also determined by the competition of two sublattices depths. At fixed modulation depths, the degree of localization of the localized mode for a small fractional Lévy index is more significant. When we consider weak disorder in a quasi-periodic lattice, an "Anderson delocalization" phenomenon will be observed. With decreasing Lévy index, it makes that such phenomenon more obvious. In addition, extended (or critical) states can transform into a localized state by introducing a suitable weak disorder. Inclusion of a weak focusing nonlinearity is shown to improve localization. Importantly, the above conclusions have been confirmed by some direct numerical propagation simulations. In view of the above, we believe that our findings reported here motivate the search of other types of linear and nonlinear modes in fractional dimensions with a quasi-periodic lattice, and similar phenomena are expected to be observed in two or higher dimensional systems.

Funding

National Natural Science Foundation of China (NSFC) (Grants No. 11704339, 11374268 and 61475101).

References

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]  

References

  • View by:

  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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