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
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  semi-discrete model, and the continuous model with Fibonacci chain [18,19] or bi-chromatic lattice potential . 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  and the zigzag trajectory propagation , 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 , -symmetry , “accessible solitons” , resonant mode conversions and Rabi oscillations , propagation management , propagation dynamics of Gaussian beam , 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),
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 . The factional order can be achieved following the scheme proposed for resonator system  or a lensguide system , 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) . This linear problem can be solved by the Fourier collocation method , and the fractional term in Fourier space satisfies this relation: . To simplify the discussion, here we focus on the LDT of linear modes with the maximum β and fix the spatial frequencies , 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 and the effective width . Here, is the power (also called energy flow), 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)].
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 . 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.
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 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  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 . Interestingly, the strength of the ASM corresponding to the maximum averaged effective width for different α is basically unchanged (≈0.006).
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
Substituting this expression Ψ(x,z) = ϕnl(x) exp(iβz) into Eq. (2), one can obtain , in which , 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 : 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 , finally use 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.
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 . 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, , 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.
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.
National Natural Science Foundation of China (NSFC) (Grants No. 11704339, 11374268 and 61475101).
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