Abstract

We investigated the existence and stability of fundamental and multipole solitons supported by amplitude-modulated Fibonacci lattices with self-focusing nonlinearity. Owing to the quasi-periodicity of Fibonacci lattices, families of solitons localized in different waveguides have different properties. We found that the existence domain of fundamental solitons localized in the central lattice is larger than that of solitons localized in the adjacent central waveguide. The former counterparts are completely stable in their existence region, while the latter have a narrow unstable region near the lower cut-off. Two families of dipole solitons were also comprehensively studied. We found the outer lattice distribution can significantly change the existence region of solitons. In addition, we specifically analyzed the properties of four complicated multipole solitons with pole numbers 3, 5, 7, and 9. In the Fibonacci lattice, their field moduli of multipole solitons are all asymmetrically distributed. The linear-stability analysis and direct simulations reveal that as the number of poles of the multipole soliton increases, its stable domain is compressed. Our results provide helpful insight for understanding the dynamics of nonlinear localized multipole modes in Fibonacci lattices with an optical nonlinearity.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Since the $1980$s, the properties of localized states in one- and two-dimensional quasi-periodic lattices have been extensively investigated [16], owing to the spatial ordering of the quasi-periodic lattices between periodicity and disorder. Kohmoto et al. [7] described the first example of a quasi-periodic system with long-range order in optics, in which the photonic quasicrytal can form the Fibonacci sequence by dielectric multilayers. Many interesting phenomena have been revealed in quasi-periodic optical structures, such as the diffraction spectrum of a general family of linear quasiperidic arrays [8], localization and optical transmission of light waves [9], bandgap structure of optical Fibonacci lattices after light diffraction [10], disorder-enhanced transport [11], diffraction suppression [12], localized quantum walks [13], and localization-delocalization transition of light [1418] in photonic quasicrystals.

In the presence of the nonlinear Kerr effect, focusing nonlinear interactions in quasiperiodic photonic lattices can increase the width of localized wave packets [14]. The existence of gap solitons was investigated in a $12$-fold quasicrystal [19] and in bichromatic lattice potentials [20,21]. Optical soliton formation controlled by angle twisting in photonics Moiré lattices was also reported [16,22]. Despite notable recent interest in the optical properties of nonlinear localized states in quasi-periodic optical lattices, the underlying existence and propagation of these states are still poorly understood. More attention should be paid to them in optics, given the diversity of optical signals and noise immunity.

Multipole solitons belong to a family of the nonlinear localized states that can be formed by a combination of several out-of-phase peaks (or spots). In different optical structures, such as nonlinear saturated bulk media [23,24], axisymmetric Bessel lattices in a medium with defocusing cubic nonlinearity [25], the surface between two distinct periodic lattices imprinted in Kerr-type nonlinear media [26], the interface between uniform or layered thermal media and a linear dielectric [27], bulk nonlocal nonlinear media [28,29], and two-dimensional metallic nanowire arrays with Kerr-type nonlinearities [30], the unique and extraordinary optical properties of multipole spatial solitons have been investigated. However, to the best of our knowledge, multipole solitons have not been investigated in quasi-periodic optical lattices.

In this study, we mainly analyzed the optical properties of multipole solitons in amplitude-modulated Fibonacci lattices with a Kerr nonlinearity, including their existence and stability. Given that solitons can be localized in lattice channels of different amplitudes, fundamental and dipole solitons would see different existence domains. Moreover, four more complicated multipole solitons with peak numbers $3$, $5$, $7$, and $9$ were also studied. Compared with the multipole-mode solitons in periodic lattices [31], the field moduli of multipole solitons supported by the Fibonacci lattices are not axisymmetric. Equally important, we found that the solitons mentioned above can propagate stably under appropriate parameters.

2. Model

We consider beam propagation which can be described by the nonlinear Schrödinger equation for dimensionless amplitude of the light field $q$:

$$i\frac{\partial q}{\partial z}={-}\frac{1}{2}\frac{\partial^2 q}{\partial x^2}-g|q|^2q-pR(x)q.$$

Here, the transverse coordinate $x$ is normalized to the characteristic transverse scale $x_0$; the longitudinal coordinate $z$ is normalized to $2\pi nx^2_0/\lambda _0$, where $n$ is the refractive index and $\lambda _0$ is the wavelength of light beam. $g\equiv 1$ represents the focusing nonlinearity. The parameter $p=4\pi ^2\delta n x^2_0n/\lambda ^2_0$ represents the modulation depth of optical lattice, where $\delta n$ is the real variation of the refractive index; and the function $R(x)$ is the profile of the optical Fibonacci lattice.

To obtain an aperiodic Fibonacci pattern $R(x)$, we can vary the amplitudes of optical lattices or the distance between two adjacent lattice channels. In this study, the Fibonacci lattice distribution corresponding to the former was considered. The Fibonacci lattice under study is a sequence of depths of two types: a low one with amplitude $A$, and a high one with an amplitude $B$. According to the Fibonacci series, the $n$-th sequence $F_n$ can be constructed using a deterministic generation scheme expressed as $F_{n}=\{F_{n-1}F_{n-2}\}$, where the initial elements are given by $F_{0}=A$ and $F_{1}=AB$. In the case $F_{8}$, there exist $55$ elements in the Fibonacci series. Thus, the profile of Fibonacci optical lattices can be described by the expression $R(x)=\sum _{m=-27}^{m=+27}F(m+28)Q(x-mw_d)$, where $m \in [-27, +27]$ is an integer, $w_d$ is the distance between two adjacent lattice channels, and $Q(x)=\exp (-x^6/d^6)$ describes the supper-Gaussian profiles of individual waveguides of width $d$. The quasi-periodic Fibonacci lattices can be created by the femtosecond laser direct-writing technique in fused silica samples [32,33]. Without loss of generality, we selected an intermediate segment $\{AABAABABAABAABA\}$ and set $B=A/\kappa$, $\kappa =(\sqrt {5}-1)/2$, $w_d=2.5$ and $d=0.4$ to study the properties of nonlinear localized states.

For parameter $x_0=12\mu m$, the diffraction length at $\lambda _0=633 nm$ and $n=1.45$ amounts to about $2.07mm$, the lattice depeth $p=1$ corresponds to $\delta n\sim 4.9\times 10^{-5}$. $x=1$ and $z=1$ correspond to $\sim 12\mu m$ in transverse direction and $\sim 2.07mm$ in propagation direction, respectively.

The stationary soliton solutions in Eq. (1) are searched in the form of $q(x,z)=w(x)\exp (ibz)$, where $w(x)$ is the real profile of a soliton and $b$ is the real propagation constant. By substituting this expression into Eq. (1), an ordinary differential equation for the function $w(x)$ can be obtained that can be solved numerically by the squared operator iteration method or by the Newton iterative method.

To elucidate the stability of soliton families, we used both the beam-propagation method and linear stability analysis methods. The substitution of perturbed soliton solutions in the form $q(x,z)=[w(x)+u(x,z)+iv(x,z)]\exp (ibz)$ into Eq. (1), where $u(x,z)$ and $v(x,z)$ are small perturbations and can grow with a complex rate $\lambda$ upon propagation. The linearization of Eq. (1) for these perturbations yields the following linear-stability eigenvalue problem:

$$\lambda u={-}\frac{1}{2}\frac{d^{2} v}{dx^{2}}+bv-pRv-gw^{2}v,$$
$$\lambda v=\frac{1}{2}\frac{d^{2} u}{dx^{2}}-bu+pRu+3gw^{2}u.$$

Equations (2) and (3) can be solved numerically using the finite-difference method. Solitons can propagate stably only when all real parts of $\lambda$ equal zero.

3. Results and discussion

According to the arrangement of Fibonacci waveguides with a series $\{AABAABABAABAABA\}$, the central waveguide is characterized by a higher amplitude $B$, and its two adjacent three waveguides are non-centrosymmetric (i.e., the left one is $ABA$, while the right one is $AAB$). In such waveguide lattices, we first discuss the properties of fundamental solitons. The maximum peak of the fundamental solitons can be located in any of the three middle waveguides ($A,B,A$). Figure 1(a) shows the profiles $w_c$ of fundamental solitons with their maximum peaks localized in the central waveguide $B$. Such nonlinear states originate from the linear modes supported by the Fibonacci lattice. For $p=1$, the linear eigenvalue spectrum and the linear mode with the maximum eigenvalue $b=0.2699$ are shown in Figs. 1(b) and 1(c), respectively. Note that the peak of the linear mode localized at $x=0$ is slightly higher than the peak localized at $x=-2w_d$. With an increase in the propagation constant $b$, the peak value of fundamental soliton at $x=0$ gradually increases and the width of soliton gradually narrows. When the value of $b$ is close to the lower cut-off value $b_{\textrm {co}}\approx 0.2699$, the profile of the fundamental soliton is localized in multiple waveguides and shows an asymmetric distribution.

 figure: Fig. 1.

Fig. 1. Profiles of fundamental solitons with the maximum peak localized in the central waveguide (a) and in a low-amplitude waveguide (d); (b) The system’s linear spectrum and (c) the linear mode with the maximum eigenvalue $b=0.2699$; (e) $U(b)$ curves corresponding to fundamental solitons in (a) and (d); An enlarged graph of the left part of the $U(b)$ curves for $w_l$ and $w_r$ is also included in (e); (f) The most unstable growth rate $\textrm {max}(\lambda _r)$ versus $b$ for fundamental solitons $w_l$ and $w_r$. Subscripts $l$, $c$, and $r$ indicate that the peak of a fundamental soliton falls on the position of the three middle waveguides ($A$, $B$, $A$). To distinguish $w$, the amplitude of $R(x)$ is multiplied by $0.2$ in (a), (c) and (d). $p=1.0$ in all panels.

Download Full Size | PPT Slide | PDF

The profiles of the fundamental solitons with the maximum peak localized in a low-amplitude waveguide $A$ are shown in Fig. 1(d). To distinguish these two profiles, we named them $w_l$ and $w_r$ respectively according to the position of the maximum amplitude of the profiles. For a larger $b$, $w_l$ is almost coincident with $w_r$ after a shift in space, whereas for a smaller $b$, $w_l$ and $w_r$ are spread out broadly and clearly different.

In addition, the existence domains of three types of fundamental solitons ($w_c$, $w_l$, and $w_r$) are different [Fig. 1(e)]. Evidently, the lower cut-off value ($b_{\textrm {co}}\approx 0.2699$) of $w_c$ is much smaller than that of $w_l$ ($b_{\textrm {co}}\approx 0.498$) and $w_r$ ($b_{\textrm {co}}\approx 0.547$). In the case of self-focusing Kerr-type nonlinearity, the power curve $U(b)$ of soliton $w_c$ increases monotonously. Note that for solitons $w_l$ and $w_r$, $U$ increases with $b$, except for a very narrow region near the lower cut-off, where $dU/db<0$. These features are naturally associated with the asymmetric distribution of Fibonacci waveguides.

By solving Eqs. (2) and (3), we can provide a sufficiently general description of the stability of fundamental solitons. It is worth emphasizing that solitons $w_c$ are stable in the entire existence region. Fundamental solitons $w_l$ and $w_r$ are stable in most of their existence domains except for those near the lower cutoff. Representative plots of the relationship between the real part of the growth rate $\lambda _r$ and propagation constant $b$ for solitons $w_l$ and $w_r$ are shown in Fig. 1(f). Although the unstable region of solitons $w_l$ is wider than that of $w_r$, the stable region of $w_l$ is still wider than that of $w_r$.

Furthermore, to verify the results of the above linear stability analysis, we solved Eq. (1) using the beam-propagation method for the input condition $q(x,z=0)=w(x)[1+\rho (x)]$, where $w(x)$ is the profile of the soliton and $\rho (x)$ is a random function with a Gaussian distribution and variance $\sigma ^2_{\textrm {noise}}$. Propagation of three types of fundamental solitons are illustrated in Fig. 2. Note that fundamental solitons $w_c$ can retain their field moduli over indefinitely long distances [Figs. 2(a) and 2(b)]. For solitons $w_l$ ($w_r$) at $dU/db<0$, its profile has small peaks in the high-amplitude waveguide of $BAB$ ($BAA$) segment. Once the soliton propagated over a short distance, its power is quickly coupled into high-amplitude waveguides [Figs. 2(c) and 2(e)]. Representative examples of stable propagation of solitons $w_l$ and $w_r$ for $dU/db>0$ are shown in Figs. 2(d) and 2(f), respectively.

 figure: Fig. 2.

Fig. 2. Propagation of fundamental solitons $w_c$ at $b=0.29$ (a) and $b=1.00$ (b); unstable (c) and stable (d) propagation of fundamental soliton $w_l$ for $b=0.51$ and $b=0.70$, respectively; unstable (e) and stable (f) propagation of fundamental soliton $w_r$ for $b=0.55$ and $b=0.70$, respectively; $p$=1.0 and $\sigma ^2_{\textrm {noise}}=0.01$ in all panels.

Download Full Size | PPT Slide | PDF

Next, we address the properties of dipole solitons in such Fibonacci waveguides. Here, two types of dipole solitons were considered. Typical profiles of the dipole solitons are shown in Figs. 3(a) and 3(b). These figures show that one peak of the dipole soliton resides in the central waveguide, whereas the other peak resides in the left or right low-amplitude waveguide. To distinguish them, they are denoted as $w_{\textrm {dl}}$ and $w_{\textrm {dr}}$, respectively. Compared with the dipole soliton in a strict periodic structure, the modulus $|w_{\textrm {dl}}|$ ($|w_{\textrm {dr}}|$) supported by Fibonacci lattices is asymmetric with respect to $x=-w_d/2$ ($x=+w_d/2$) for $b$ close to its lower cutoff. This characteristic can be explained as follows: a dipole soliton can be composed of two out-of-phase fundamental solitons; similar to the fundamental solitons $w_l$ and $w_r$, the small peaks of the dipole soliton for $b\rightarrow b_{\textrm {co}}$ are confined in the adjacent high-amplitude waveguides.

 figure: Fig. 3.

Fig. 3. Profiles of both types of dipole solitons plotted in (a) and (b) for the same value of $b$; (c) power $U$ of both types of dipole solitons versus propagation constant $b$, and two enlarged graphs near the lower cutoff are given in the inset graphs; (d) the maximum values of the real part of perturbation growth rate $\textrm {max}(\lambda _r)$ versus propagation constant $b$ for solitons $w_{\textrm {dl}}$ and $w_{\textrm {dr}}$. To distinguish $w_{\textrm {dl}}$ and $w_{\textrm {dr}}$, $|w_{\textrm {dl}}|$ and $|w_{\textrm {dr}}|$ are enhanced by 0.1 and the amplitude of $R(x)$ is multiplied by $0.5$ in (a) and (b). $p=1.0$ in all panels.

Download Full Size | PPT Slide | PDF

Note that the existence domain of the dipole solitons $w_{\textrm {dr}}$ is larger than that of $w_{\textrm {dl}}$ [Fig. 3(c)], in contrast with the existence region of fundamental solitons $w_l$ and $w_r$. This is mainly due to the fact that their profiles are confined in different numbers of Fibonacci waveguides. When the propagation constant $b$ is large, both types of dipole solitons share almost the same power $U$. In terms of their stability and dynamics [Figs. 3(d) and 4], dipole solitons in such a structure are all stable when propagation constant $b$ exceeds a certain value (e.g., $b\geq 0.93$ for $w_{\textrm {dr}}$ and $b\geq 1.06$ for $w_{\textrm {dl}}$). In the region near the lower cut-off value, there is a very narrow stable region for each type of dipole soliton. Stable propagation examples of dipole solitons $w_{\textrm {dr}}$ and $w_{\textrm {dl}}$ in such a narrow region are shown in Figs. 4(a) and 4(b), respectively. For the unstable dipole soliton $w_{\textrm {dr}}$ ($w_{\textrm {dl}}$) with $b\in (0.72,0.93)$ ($b\in (0.85,1.06)$) and $\sigma ^2_{\textrm {noise}}=0.01$, the peak in the center waveguide oscillates with propagation distance, whereas the peak in the low-amplitude waveguide is diffracted away after propagating a short distance [Figs. 4(c) and 4(d)]. When the value of $b$ is very close to the lower cutoff, the field distributions of both types of dipole solitons collapse quickly, and part of their power are coupled into the high-amplitude waveguides [Figs. 4(e) and 4(f)].

 figure: Fig. 4.

Fig. 4. Stable (a, b) and unstable (c-f) propagation of two types of dipole solitons. Dipole solitons $w_{\textrm {dr}}$ at $b=0.70$ (a), $b=0.84$ (c), $b=0.509$ (e), and dipole solitons $w_{\textrm {dl}}$ at $b=0.82$ (b), $b=0.95$ (d), and $b=0.69$ (f). $p=1.0$ and $\sigma ^2_{\textrm {noise}}=0.01$ in all panels.

Download Full Size | PPT Slide | PDF

We further studied the stability of multipole solitons in such Fibonacci waveguides. Representative profiles of $3$-pole, $5$-pole, $7$-pole and $9$-pole solitons are shown in Figs. 5(a), 5(b), 5(c) and 5(d), respectively. In these instances, a multipole soliton can be composed of multiple out-of-phase fundamental solitons. The peak amplitudes of the multipole soliton supported by Fibonacci waveguides are not constant, which is in significant contrast with the same peak amplitude of a multipole soliton in periodic structures [31]. The left and right sidelobes of multipole solitons reside on different types of waveguide segments, which leads to different existence domains of multipole solitons [Fig. 5(e)]. Evidently, in the ordered Fibonacci waveguide array, with an increase in the number of poles, the unstable regions of multipole solitons gradually increase [cf. Figures 5(f), 5(g), 5(h), and 5(i)]. Typical examples of unstable and stable spectrum are shown in Figs. 5(j) and 5(k).

 figure: Fig. 5.

Fig. 5. Profiles of $3$-pole (a), $5$-pole (b), $7$-pole (c), and $9$-pole (d) solitons at $b=0.77$; (e) power $U$ of the four types of multipole solitons versus propagation constant $b$; instability growth rate versus $b$ of the $3$-pole (f), $5$-pole (g), $7$-pole (h), and $9$-pole (i) solitons. Unstable (j) and stable (k) spectrum of the linearization operator of the $3$-pole solitons for $b=0.85$ and $b=1.5$, respectively; To distinguish $w$, $|w|$ is enhanced by 0.1 and the amplitude of $R(x)$ is multiplied by $0.5$ in (a-d). $p=1.0$ in all panels.

Download Full Size | PPT Slide | PDF

The propagation characteristics of multipole solitons with a number of poles greater than or equal to $3$ are similar to those of dipole solitons. A stable multipole soliton is responsible for nonlinear effect and provides diffraction compensation [Figs. 6(a) and 6(b)]. The output field distribution of unstable multipole soliton is heavily distorted after maintaining a certain distance [Figs. 6(c) and 6(d)].

 figure: Fig. 6.

Fig. 6. Stable (a, b) and unstable (c, d) propagation of $5$-pole (a, c) and $9$-pole (b, d) solitons in Fibonacci lattices; $b=1.5$ in (a), $b=1.9$ in (b), $b=1.34$ in (c), $b=1.7$ in (d), $p$=1.0 and $\sigma ^2_{\textrm {noise}}=0.01$ in all panels.

Download Full Size | PPT Slide | PDF

4. Conclusion

We have shown that various families of fundamental and multipole solitons can exist in amplitude-modulated Fibonacci lattices. In such lattices, some unique properties have been identified. We found that a soliton near the lower cut-off value widens and falls on several waveguide lattices. The outer lattice distribution affects the existence domain and stability of fundamental solitons. For example, fundamental solitons localized in the central lattice are stable in the entire existence domain, and those residing in the adjacent central lattice are also stable in a considerable part of their existence domain, except for a narrow region near the lower cutoff. We also found that, in such Fibonacci lattices, field moduli of multipole solitons with pole numbers $2$, $3$, $5$, $7$, and $9$ are all asymmetrically distributed. In particular, some stable multipole solitons with a large number of poles were numerically found. The stable region of the multipole soliton gradually shrinks as the number of poles increases. Our results may help to guide the future quest for nonlinear localized states in this or similar quasi-periodic structures. As an extension of this study, it may be interesting to analyze the properties of multipole or multipeak solitons in higher-dimensional Fibonacci lattices.

Funding

Applied Basic Research Program of Shanxi Province (201901D211466); Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (STIP) (2021L505); National Natural Science Foundation of China (11704339, 11805145); National Key Research and Development Program of China (2019JM-307, 2019JQ-089).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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

2. D. Grempel, S. Fishman, and R. Prange, “Localization in an incommensurate potential: An exactly solvable model,” Phys. Rev. Lett. 49(11), 833–836 (1982). [CrossRef]  

3. M. Kohmoto, “Metal-insulator transition and scaling for incommensurate systems,” Phys. Rev. Lett. 51(13), 1198–1201 (1983). [CrossRef]  

4. C. Rockstuhl and F. Lederer, “The effect of disorder on the local density of states in two-dimensional quasi-periodic photonic crystals,” New J. Phys. 8(9), 206 (2006). [CrossRef]  

5. L. Dal Negro and N.-N. Feng, “Spectral gaps and mode localization in fibonacci chains of metal nanoparticles,” Opt. Express 15(22), 14396–14403 (2007). [CrossRef]  

6. M. Modugno, “Exponential localization in one-dimensional quasi-periodic optical lattices,” New J. Phys. 11(3), 033023 (2009). [CrossRef]  

7. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987). [CrossRef]  

8. I. Aviram, “The diffraction spectrum of a general family of linear quasiperiodic arrays,” J. Phys. A: Math. Gen. 19(16), 3299–3312 (1986). [CrossRef]  

9. W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, “Localization of light waves in fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994). [CrossRef]  

10. M. Kaliteevski, V. Nikolaev, R. Abram, and S. Brand, “Bandgap structure of optical fibonacci lattices after light diffraction,” Opt. Spectrosc. 91(1), 109–118 (2001). [CrossRef]  

11. L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332(6037), 1541–1544 (2011). [CrossRef]  

12. N. Lučić, D. J. Savić, A. Piper, D. Grujić, J. Vasiljević, D. Pantelić, B. Jelenković, and D. Timotijević, “Light propagation in quasi-periodic fibonacci waveguide arrays,” JOSA B 32(7), 1510–1513 (2015). [CrossRef]  

13. D. T. Nguyen, D. A. Nolan, and N. F. Borrelli, “Localized quantum walks in quasi-periodic fibonacci arrays of waveguides,” Opt. Express 27(2), 886–898 (2019). [CrossRef]  

14. 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(1), 013901 (2009). [CrossRef]  

15. 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(12), 2758–2761 (2015). [CrossRef]  

16. C. Huang, F. Ye, X. Chen, Y. V. Kartashov, V. V. Konotop, and L. Torner, “Localization-delocalization wavepacket transition in pythagorean aperiodic potentials,” Sci. Rep. 6, 1–8 (2016). [CrossRef]  

17. C. Huang, C. Shang, J. Li, L. Dong, and F. Ye, “Localization and anderson delocalization of light in fractional dimensions with a quasi-periodic lattice,” Opt. Express 27(5), 6259–6267 (2019). [CrossRef]  

18. P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Localization and delocalization of light in photonic moiré lattices,” Nature 577(7788), 42–46 (2020). [CrossRef]  

19. P. Xie, Z.-Q. Zhang, and X. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E 67(2), 026607 (2003). [CrossRef]  

20. H. Sakaguchi and B. A. Malomed, “Gap solitons in quasiperiodic optical lattices,” Phys. Rev. E 74(2), 026601 (2006). [CrossRef]  

21. C. Huang, C. Li, H. Deng, and L. Dong, “Gap solitons in fractional dimensions with a quasi-periodic lattice,” Ann. Phys. 531(9), 1900056 (2019). [CrossRef]  

22. Q. Fu, P. Wang, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Optical soliton formation controlled by angle twisting in photonic moiré lattices,” Nat. Photonics 14(11), 663–668 (2020). [CrossRef]  

23. A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001). [CrossRef]  

24. A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, G. McCarthy, W. Krolikowski, and B. Luther-Davies, “Multipole composite spatial solitons: theory and experiment,” JOSA B 19(3), 586–595 (2002). [CrossRef]  

25. Y. V. Kartashov, R. Carretero-González, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Multipole-mode solitons in bessel optical lattices,” Opt. Express 13(26), 10703–10710 (2005). [CrossRef]  

26. Y. V. Kartashov and L. Torner, “Multipole-mode surface solitons,” Opt. Lett. 31(14), 2172–2174 (2006). [CrossRef]  

27. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Multipole surface solitons in thermal media,” Opt. Lett. 34(3), 283–285 (2009). [CrossRef]  

28. Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, “Multipole vector solitons in nonlocal nonlinear media,” Opt. Lett. 31(10), 1483–1485 (2006). [CrossRef]  

29. L. Dong and F. Ye, “Stability of multipole-mode solitons in thermal nonlinear media,” Phys. Rev. A 81(1), 013815 (2010). [CrossRef]  

30. Y. Kou, F. Ye, and X. Chen, “Multipole plasmonic lattice solitons,” Phys. Rev. A 84(3), 033855 (2011). [CrossRef]  

31. C. Huang, C. Li, and L. Dong, “Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a PT symmetry,” Opt. Express 21(3), 3917–3925 (2013). [CrossRef]  

32. K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(8), 620–625 (2006). [CrossRef]  

33. A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009). [CrossRef]  

References

  • View by:

  1. S. Aubry and G. André, “Analyticity breaking and anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc. 3, 18 (1980).
  2. D. Grempel, S. Fishman, and R. Prange, “Localization in an incommensurate potential: An exactly solvable model,” Phys. Rev. Lett. 49(11), 833–836 (1982).
    [Crossref]
  3. M. Kohmoto, “Metal-insulator transition and scaling for incommensurate systems,” Phys. Rev. Lett. 51(13), 1198–1201 (1983).
    [Crossref]
  4. C. Rockstuhl and F. Lederer, “The effect of disorder on the local density of states in two-dimensional quasi-periodic photonic crystals,” New J. Phys. 8(9), 206 (2006).
    [Crossref]
  5. L. Dal Negro and N.-N. Feng, “Spectral gaps and mode localization in fibonacci chains of metal nanoparticles,” Opt. Express 15(22), 14396–14403 (2007).
    [Crossref]
  6. M. Modugno, “Exponential localization in one-dimensional quasi-periodic optical lattices,” New J. Phys. 11(3), 033023 (2009).
    [Crossref]
  7. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987).
    [Crossref]
  8. I. Aviram, “The diffraction spectrum of a general family of linear quasiperiodic arrays,” J. Phys. A: Math. Gen. 19(16), 3299–3312 (1986).
    [Crossref]
  9. W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, “Localization of light waves in fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
    [Crossref]
  10. M. Kaliteevski, V. Nikolaev, R. Abram, and S. Brand, “Bandgap structure of optical fibonacci lattices after light diffraction,” Opt. Spectrosc. 91(1), 109–118 (2001).
    [Crossref]
  11. L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332(6037), 1541–1544 (2011).
    [Crossref]
  12. N. Lučić, D. J. Savić, A. Piper, D. Grujić, J. Vasiljević, D. Pantelić, B. Jelenković, and D. Timotijević, “Light propagation in quasi-periodic fibonacci waveguide arrays,” JOSA B 32(7), 1510–1513 (2015).
    [Crossref]
  13. D. T. Nguyen, D. A. Nolan, and N. F. Borrelli, “Localized quantum walks in quasi-periodic fibonacci arrays of waveguides,” Opt. Express 27(2), 886–898 (2019).
    [Crossref]
  14. 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(1), 013901 (2009).
    [Crossref]
  15. 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(12), 2758–2761 (2015).
    [Crossref]
  16. C. Huang, F. Ye, X. Chen, Y. V. Kartashov, V. V. Konotop, and L. Torner, “Localization-delocalization wavepacket transition in pythagorean aperiodic potentials,” Sci. Rep. 6, 1–8 (2016).
    [Crossref]
  17. C. Huang, C. Shang, J. Li, L. Dong, and F. Ye, “Localization and anderson delocalization of light in fractional dimensions with a quasi-periodic lattice,” Opt. Express 27(5), 6259–6267 (2019).
    [Crossref]
  18. P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Localization and delocalization of light in photonic moiré lattices,” Nature 577(7788), 42–46 (2020).
    [Crossref]
  19. P. Xie, Z.-Q. Zhang, and X. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E 67(2), 026607 (2003).
    [Crossref]
  20. H. Sakaguchi and B. A. Malomed, “Gap solitons in quasiperiodic optical lattices,” Phys. Rev. E 74(2), 026601 (2006).
    [Crossref]
  21. C. Huang, C. Li, H. Deng, and L. Dong, “Gap solitons in fractional dimensions with a quasi-periodic lattice,” Ann. Phys. 531(9), 1900056 (2019).
    [Crossref]
  22. Q. Fu, P. Wang, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Optical soliton formation controlled by angle twisting in photonic moiré lattices,” Nat. Photonics 14(11), 663–668 (2020).
    [Crossref]
  23. A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
    [Crossref]
  24. A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, G. McCarthy, W. Krolikowski, and B. Luther-Davies, “Multipole composite spatial solitons: theory and experiment,” JOSA B 19(3), 586–595 (2002).
    [Crossref]
  25. Y. V. Kartashov, R. Carretero-González, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Multipole-mode solitons in bessel optical lattices,” Opt. Express 13(26), 10703–10710 (2005).
    [Crossref]
  26. Y. V. Kartashov and L. Torner, “Multipole-mode surface solitons,” Opt. Lett. 31(14), 2172–2174 (2006).
    [Crossref]
  27. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Multipole surface solitons in thermal media,” Opt. Lett. 34(3), 283–285 (2009).
    [Crossref]
  28. Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, “Multipole vector solitons in nonlocal nonlinear media,” Opt. Lett. 31(10), 1483–1485 (2006).
    [Crossref]
  29. L. Dong and F. Ye, “Stability of multipole-mode solitons in thermal nonlinear media,” Phys. Rev. A 81(1), 013815 (2010).
    [Crossref]
  30. Y. Kou, F. Ye, and X. Chen, “Multipole plasmonic lattice solitons,” Phys. Rev. A 84(3), 033855 (2011).
    [Crossref]
  31. C. Huang, C. Li, and L. Dong, “Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a PT symmetry,” Opt. Express 21(3), 3917–3925 (2013).
    [Crossref]
  32. K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(8), 620–625 (2006).
    [Crossref]
  33. A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009).
    [Crossref]

2020 (2)

P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Localization and delocalization of light in photonic moiré lattices,” Nature 577(7788), 42–46 (2020).
[Crossref]

Q. Fu, P. Wang, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Optical soliton formation controlled by angle twisting in photonic moiré lattices,” Nat. Photonics 14(11), 663–668 (2020).
[Crossref]

2019 (3)

2016 (1)

C. Huang, F. Ye, X. Chen, Y. V. Kartashov, V. V. Konotop, and L. Torner, “Localization-delocalization wavepacket transition in pythagorean aperiodic potentials,” Sci. Rep. 6, 1–8 (2016).
[Crossref]

2015 (2)

N. Lučić, D. J. Savić, A. Piper, D. Grujić, J. Vasiljević, D. Pantelić, B. Jelenković, and D. Timotijević, “Light propagation in quasi-periodic fibonacci waveguide arrays,” JOSA B 32(7), 1510–1513 (2015).
[Crossref]

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(12), 2758–2761 (2015).
[Crossref]

2013 (1)

2011 (2)

Y. Kou, F. Ye, and X. Chen, “Multipole plasmonic lattice solitons,” Phys. Rev. A 84(3), 033855 (2011).
[Crossref]

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332(6037), 1541–1544 (2011).
[Crossref]

2010 (1)

L. Dong and F. Ye, “Stability of multipole-mode solitons in thermal nonlinear media,” Phys. Rev. A 81(1), 013815 (2010).
[Crossref]

2009 (4)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Multipole surface solitons in thermal media,” Opt. Lett. 34(3), 283–285 (2009).
[Crossref]

A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009).
[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(1), 013901 (2009).
[Crossref]

M. Modugno, “Exponential localization in one-dimensional quasi-periodic optical lattices,” New J. Phys. 11(3), 033023 (2009).
[Crossref]

2007 (1)

2006 (5)

C. Rockstuhl and F. Lederer, “The effect of disorder on the local density of states in two-dimensional quasi-periodic photonic crystals,” New J. Phys. 8(9), 206 (2006).
[Crossref]

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(8), 620–625 (2006).
[Crossref]

Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, “Multipole vector solitons in nonlocal nonlinear media,” Opt. Lett. 31(10), 1483–1485 (2006).
[Crossref]

Y. V. Kartashov and L. Torner, “Multipole-mode surface solitons,” Opt. Lett. 31(14), 2172–2174 (2006).
[Crossref]

H. Sakaguchi and B. A. Malomed, “Gap solitons in quasiperiodic optical lattices,” Phys. Rev. E 74(2), 026601 (2006).
[Crossref]

2005 (1)

2003 (1)

P. Xie, Z.-Q. Zhang, and X. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E 67(2), 026607 (2003).
[Crossref]

2002 (1)

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, G. McCarthy, W. Krolikowski, and B. Luther-Davies, “Multipole composite spatial solitons: theory and experiment,” JOSA B 19(3), 586–595 (2002).
[Crossref]

2001 (2)

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
[Crossref]

M. Kaliteevski, V. Nikolaev, R. Abram, and S. Brand, “Bandgap structure of optical fibonacci lattices after light diffraction,” Opt. Spectrosc. 91(1), 109–118 (2001).
[Crossref]

1994 (1)

W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, “Localization of light waves in fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
[Crossref]

1987 (1)

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987).
[Crossref]

1986 (1)

I. Aviram, “The diffraction spectrum of a general family of linear quasiperiodic arrays,” J. Phys. A: Math. Gen. 19(16), 3299–3312 (1986).
[Crossref]

1983 (1)

M. Kohmoto, “Metal-insulator transition and scaling for incommensurate systems,” Phys. Rev. Lett. 51(13), 1198–1201 (1983).
[Crossref]

1982 (1)

D. Grempel, S. Fishman, and R. Prange, “Localization in an incommensurate potential: An exactly solvable model,” Phys. Rev. Lett. 49(11), 833–836 (1982).
[Crossref]

1980 (1)

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

Abram, R.

M. Kaliteevski, V. Nikolaev, R. Abram, and S. Brand, “Bandgap structure of optical fibonacci lattices after light diffraction,” Opt. Spectrosc. 91(1), 109–118 (2001).
[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).

Aviram, I.

I. Aviram, “The diffraction spectrum of a general family of linear quasiperiodic arrays,” J. Phys. A: Math. Gen. 19(16), 3299–3312 (1986).
[Crossref]

Borrelli, N. F.

Brand, S.

M. Kaliteevski, V. Nikolaev, R. Abram, and S. Brand, “Bandgap structure of optical fibonacci lattices after light diffraction,” Opt. Spectrosc. 91(1), 109–118 (2001).
[Crossref]

Carretero-González, R.

Chen, X.

P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Localization and delocalization of light in photonic moiré lattices,” Nature 577(7788), 42–46 (2020).
[Crossref]

C. Huang, F. Ye, X. Chen, Y. V. Kartashov, V. V. Konotop, and L. Torner, “Localization-delocalization wavepacket transition in pythagorean aperiodic potentials,” Sci. Rep. 6, 1–8 (2016).
[Crossref]

Y. Kou, F. Ye, and X. Chen, “Multipole plasmonic lattice solitons,” Phys. Rev. A 84(3), 033855 (2011).
[Crossref]

Dal Negro, L.

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(1), 013901 (2009).
[Crossref]

Deng, H.

C. Huang, C. Li, H. Deng, and L. Dong, “Gap solitons in fractional dimensions with a quasi-periodic lattice,” Ann. Phys. 531(9), 1900056 (2019).
[Crossref]

Desyatnikov, A. S.

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, G. McCarthy, W. Krolikowski, and B. Luther-Davies, “Multipole composite spatial solitons: theory and experiment,” JOSA B 19(3), 586–595 (2002).
[Crossref]

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
[Crossref]

Dong, L.

C. Huang, C. Li, H. Deng, and L. Dong, “Gap solitons in fractional dimensions with a quasi-periodic lattice,” Ann. Phys. 531(9), 1900056 (2019).
[Crossref]

C. Huang, C. Shang, J. Li, L. Dong, and F. Ye, “Localization and anderson delocalization of light in fractional dimensions with a quasi-periodic lattice,” Opt. Express 27(5), 6259–6267 (2019).
[Crossref]

C. Huang, C. Li, and L. Dong, “Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a PT symmetry,” Opt. Express 21(3), 3917–3925 (2013).
[Crossref]

L. Dong and F. Ye, “Stability of multipole-mode solitons in thermal nonlinear media,” Phys. Rev. A 81(1), 013815 (2010).
[Crossref]

Dreisow, F.

Feng, N.-N.

Fishman, S.

D. Grempel, S. Fishman, and R. Prange, “Localization in an incommensurate potential: An exactly solvable model,” Phys. Rev. Lett. 49(11), 833–836 (1982).
[Crossref]

Freedman, B.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332(6037), 1541–1544 (2011).
[Crossref]

Fu, Q.

Q. Fu, P. Wang, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Optical soliton formation controlled by angle twisting in photonic moiré lattices,” Nat. Photonics 14(11), 663–668 (2020).
[Crossref]

García-Ripoll, J. J.

Gellermann, W.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, “Localization of light waves in fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
[Crossref]

Grempel, D.

D. Grempel, S. Fishman, and R. Prange, “Localization in an incommensurate potential: An exactly solvable model,” Phys. Rev. Lett. 49(11), 833–836 (1982).
[Crossref]

Grujic, D.

N. Lučić, D. J. Savić, A. Piper, D. Grujić, J. Vasiljević, D. Pantelić, B. Jelenković, and D. Timotijević, “Light propagation in quasi-periodic fibonacci waveguide arrays,” JOSA B 32(7), 1510–1513 (2015).
[Crossref]

Hang, C.

Heinrich, M.

Huang, C.

Q. Fu, P. Wang, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Optical soliton formation controlled by angle twisting in photonic moiré lattices,” Nat. Photonics 14(11), 663–668 (2020).
[Crossref]

P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Localization and delocalization of light in photonic moiré lattices,” Nature 577(7788), 42–46 (2020).
[Crossref]

C. Huang, C. Shang, J. Li, L. Dong, and F. Ye, “Localization and anderson delocalization of light in fractional dimensions with a quasi-periodic lattice,” Opt. Express 27(5), 6259–6267 (2019).
[Crossref]

C. Huang, C. Li, H. Deng, and L. Dong, “Gap solitons in fractional dimensions with a quasi-periodic lattice,” Ann. Phys. 531(9), 1900056 (2019).
[Crossref]

C. Huang, F. Ye, X. Chen, Y. V. Kartashov, V. V. Konotop, and L. Torner, “Localization-delocalization wavepacket transition in pythagorean aperiodic potentials,” Sci. Rep. 6, 1–8 (2016).
[Crossref]

C. Huang, C. Li, and L. Dong, “Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a PT symmetry,” Opt. Express 21(3), 3917–3925 (2013).
[Crossref]

Huang, G.

Iguchi, K.

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987).
[Crossref]

Itoh, K.

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(8), 620–625 (2006).
[Crossref]

Jelenkovic, B.

N. Lučić, D. J. Savić, A. Piper, D. Grujić, J. Vasiljević, D. Pantelić, B. Jelenković, and D. Timotijević, “Light propagation in quasi-periodic fibonacci waveguide arrays,” JOSA B 32(7), 1510–1513 (2015).
[Crossref]

Kaliteevski, M.

M. Kaliteevski, V. Nikolaev, R. Abram, and S. Brand, “Bandgap structure of optical fibonacci lattices after light diffraction,” Opt. Spectrosc. 91(1), 109–118 (2001).
[Crossref]

Kartashov, Y. V.

P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Localization and delocalization of light in photonic moiré lattices,” Nature 577(7788), 42–46 (2020).
[Crossref]

Q. Fu, P. Wang, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Optical soliton formation controlled by angle twisting in photonic moiré lattices,” Nat. Photonics 14(11), 663–668 (2020).
[Crossref]

C. Huang, F. Ye, X. Chen, Y. V. Kartashov, V. V. Konotop, and L. Torner, “Localization-delocalization wavepacket transition in pythagorean aperiodic potentials,” Sci. Rep. 6, 1–8 (2016).
[Crossref]

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(12), 2758–2761 (2015).
[Crossref]

A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Multipole surface solitons in thermal media,” Opt. Lett. 34(3), 283–285 (2009).
[Crossref]

Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, “Multipole vector solitons in nonlocal nonlinear media,” Opt. Lett. 31(10), 1483–1485 (2006).
[Crossref]

Y. V. Kartashov and L. Torner, “Multipole-mode surface solitons,” Opt. Lett. 31(14), 2172–2174 (2006).
[Crossref]

Y. V. Kartashov, R. Carretero-González, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Multipole-mode solitons in bessel optical lattices,” Opt. Express 13(26), 10703–10710 (2005).
[Crossref]

Kivshar, Y. S.

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, G. McCarthy, W. Krolikowski, and B. Luther-Davies, “Multipole composite spatial solitons: theory and experiment,” JOSA B 19(3), 586–595 (2002).
[Crossref]

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
[Crossref]

Kohmoto, M.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, “Localization of light waves in fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
[Crossref]

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987).
[Crossref]

M. Kohmoto, “Metal-insulator transition and scaling for incommensurate systems,” Phys. Rev. Lett. 51(13), 1198–1201 (1983).
[Crossref]

Konotop, V. V.

Q. Fu, P. Wang, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Optical soliton formation controlled by angle twisting in photonic moiré lattices,” Nat. Photonics 14(11), 663–668 (2020).
[Crossref]

P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Localization and delocalization of light in photonic moiré lattices,” Nature 577(7788), 42–46 (2020).
[Crossref]

C. Huang, F. Ye, X. Chen, Y. V. Kartashov, V. V. Konotop, and L. Torner, “Localization-delocalization wavepacket transition in pythagorean aperiodic potentials,” Sci. Rep. 6, 1–8 (2016).
[Crossref]

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(12), 2758–2761 (2015).
[Crossref]

Kou, Y.

Y. Kou, F. Ye, and X. Chen, “Multipole plasmonic lattice solitons,” Phys. Rev. A 84(3), 033855 (2011).
[Crossref]

Krolikowski, W.

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, G. McCarthy, W. Krolikowski, and B. Luther-Davies, “Multipole composite spatial solitons: theory and experiment,” JOSA B 19(3), 586–595 (2002).
[Crossref]

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
[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(1), 013901 (2009).
[Crossref]

Lederer, F.

A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009).
[Crossref]

C. Rockstuhl and F. Lederer, “The effect of disorder on the local density of states in two-dimensional quasi-periodic photonic crystals,” New J. Phys. 8(9), 206 (2006).
[Crossref]

Levi, L.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332(6037), 1541–1544 (2011).
[Crossref]

Li, C.

C. Huang, C. Li, H. Deng, and L. Dong, “Gap solitons in fractional dimensions with a quasi-periodic lattice,” Ann. Phys. 531(9), 1900056 (2019).
[Crossref]

C. Huang, C. Li, and L. Dong, “Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a PT symmetry,” Opt. Express 21(3), 3917–3925 (2013).
[Crossref]

Li, J.

Lucic, N.

N. Lučić, D. J. Savić, A. Piper, D. Grujić, J. Vasiljević, D. Pantelić, B. Jelenković, and D. Timotijević, “Light propagation in quasi-periodic fibonacci waveguide arrays,” JOSA B 32(7), 1510–1513 (2015).
[Crossref]

Luther-Davies, B.

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, G. McCarthy, W. Krolikowski, and B. Luther-Davies, “Multipole composite spatial solitons: theory and experiment,” JOSA B 19(3), 586–595 (2002).
[Crossref]

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
[Crossref]

Malomed, B. A.

Manela, O.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332(6037), 1541–1544 (2011).
[Crossref]

McCarthy, G.

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, G. McCarthy, W. Krolikowski, and B. Luther-Davies, “Multipole composite spatial solitons: theory and experiment,” JOSA B 19(3), 586–595 (2002).
[Crossref]

Mihalache, D.

Modugno, M.

M. Modugno, “Exponential localization in one-dimensional quasi-periodic optical lattices,” New J. Phys. 11(3), 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(1), 013901 (2009).
[Crossref]

Neshev, D.

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, G. McCarthy, W. Krolikowski, and B. Luther-Davies, “Multipole composite spatial solitons: theory and experiment,” JOSA B 19(3), 586–595 (2002).
[Crossref]

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
[Crossref]

Nguyen, D. T.

Nikolaev, V.

M. Kaliteevski, V. Nikolaev, R. Abram, and S. Brand, “Bandgap structure of optical fibonacci lattices after light diffraction,” Opt. Spectrosc. 91(1), 109–118 (2001).
[Crossref]

Nolan, D. A.

Nolte, S.

Ostrovskaya, E. A.

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, G. McCarthy, W. Krolikowski, and B. Luther-Davies, “Multipole composite spatial solitons: theory and experiment,” JOSA B 19(3), 586–595 (2002).
[Crossref]

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
[Crossref]

Pantelic, D.

N. Lučić, D. J. Savić, A. Piper, D. Grujić, J. Vasiljević, D. Pantelić, B. Jelenković, and D. Timotijević, “Light propagation in quasi-periodic fibonacci waveguide arrays,” JOSA B 32(7), 1510–1513 (2015).
[Crossref]

Pérez-García, V. M.

Pertsch, T.

Piper, A.

N. Lučić, D. J. Savić, A. Piper, D. Grujić, J. Vasiljević, D. Pantelić, B. Jelenković, and D. Timotijević, “Light propagation in quasi-periodic fibonacci waveguide arrays,” JOSA B 32(7), 1510–1513 (2015).
[Crossref]

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(1), 013901 (2009).
[Crossref]

Prange, R.

D. Grempel, S. Fishman, and R. Prange, “Localization in an incommensurate potential: An exactly solvable model,” Phys. Rev. Lett. 49(11), 833–836 (1982).
[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(1), 013901 (2009).
[Crossref]

Rechtsman, M.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332(6037), 1541–1544 (2011).
[Crossref]

Rockstuhl, C.

C. Rockstuhl and F. Lederer, “The effect of disorder on the local density of states in two-dimensional quasi-periodic photonic crystals,” New J. Phys. 8(9), 206 (2006).
[Crossref]

Sakaguchi, H.

H. Sakaguchi and B. A. Malomed, “Gap solitons in quasiperiodic optical lattices,” Phys. Rev. E 74(2), 026601 (2006).
[Crossref]

Savic, D. J.

N. Lučić, D. J. Savić, A. Piper, D. Grujić, J. Vasiljević, D. Pantelić, B. Jelenković, and D. Timotijević, “Light propagation in quasi-periodic fibonacci waveguide arrays,” JOSA B 32(7), 1510–1513 (2015).
[Crossref]

Schaffer, C. B.

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(8), 620–625 (2006).
[Crossref]

Schwartz, T.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332(6037), 1541–1544 (2011).
[Crossref]

Segev, M.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332(6037), 1541–1544 (2011).
[Crossref]

Shang, C.

Silberberg, 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(1), 013901 (2009).
[Crossref]

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(1), 013901 (2009).
[Crossref]

Sutherland, B.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, “Localization of light waves in fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
[Crossref]

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987).
[Crossref]

Szameit, A.

Taylor, P.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, “Localization of light waves in fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
[Crossref]

Timotijevic, D.

N. Lučić, D. J. Savić, A. Piper, D. Grujić, J. Vasiljević, D. Pantelić, B. Jelenković, and D. Timotijević, “Light propagation in quasi-periodic fibonacci waveguide arrays,” JOSA B 32(7), 1510–1513 (2015).
[Crossref]

Torner, L.

P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Localization and delocalization of light in photonic moiré lattices,” Nature 577(7788), 42–46 (2020).
[Crossref]

Q. Fu, P. Wang, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Optical soliton formation controlled by angle twisting in photonic moiré lattices,” Nat. Photonics 14(11), 663–668 (2020).
[Crossref]

C. Huang, F. Ye, X. Chen, Y. V. Kartashov, V. V. Konotop, and L. Torner, “Localization-delocalization wavepacket transition in pythagorean aperiodic potentials,” Sci. Rep. 6, 1–8 (2016).
[Crossref]

A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Multipole surface solitons in thermal media,” Opt. Lett. 34(3), 283–285 (2009).
[Crossref]

Y. V. Kartashov and L. Torner, “Multipole-mode surface solitons,” Opt. Lett. 31(14), 2172–2174 (2006).
[Crossref]

Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, “Multipole vector solitons in nonlocal nonlinear media,” Opt. Lett. 31(10), 1483–1485 (2006).
[Crossref]

Y. V. Kartashov, R. Carretero-González, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Multipole-mode solitons in bessel optical lattices,” Opt. Express 13(26), 10703–10710 (2005).
[Crossref]

Tünnermann, A.

Vasiljevic, J.

N. Lučić, D. J. Savić, A. Piper, D. Grujić, J. Vasiljević, D. Pantelić, B. Jelenković, and D. Timotijević, “Light propagation in quasi-periodic fibonacci waveguide arrays,” JOSA B 32(7), 1510–1513 (2015).
[Crossref]

Vysloukh, V. A.

Wang, P.

Q. Fu, P. Wang, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Optical soliton formation controlled by angle twisting in photonic moiré lattices,” Nat. Photonics 14(11), 663–668 (2020).
[Crossref]

P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Localization and delocalization of light in photonic moiré lattices,” Nature 577(7788), 42–46 (2020).
[Crossref]

Watanabe, W.

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(8), 620–625 (2006).
[Crossref]

Xie, P.

P. Xie, Z.-Q. Zhang, and X. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E 67(2), 026607 (2003).
[Crossref]

Ye, F.

P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Localization and delocalization of light in photonic moiré lattices,” Nature 577(7788), 42–46 (2020).
[Crossref]

Q. Fu, P. Wang, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Optical soliton formation controlled by angle twisting in photonic moiré lattices,” Nat. Photonics 14(11), 663–668 (2020).
[Crossref]

C. Huang, C. Shang, J. Li, L. Dong, and F. Ye, “Localization and anderson delocalization of light in fractional dimensions with a quasi-periodic lattice,” Opt. Express 27(5), 6259–6267 (2019).
[Crossref]

C. Huang, F. Ye, X. Chen, Y. V. Kartashov, V. V. Konotop, and L. Torner, “Localization-delocalization wavepacket transition in pythagorean aperiodic potentials,” Sci. Rep. 6, 1–8 (2016).
[Crossref]

Y. Kou, F. Ye, and X. Chen, “Multipole plasmonic lattice solitons,” Phys. Rev. A 84(3), 033855 (2011).
[Crossref]

L. Dong and F. Ye, “Stability of multipole-mode solitons in thermal nonlinear media,” Phys. Rev. A 81(1), 013815 (2010).
[Crossref]

Zhang, X.

P. Xie, Z.-Q. Zhang, and X. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E 67(2), 026607 (2003).
[Crossref]

Zhang, Z.-Q.

P. Xie, Z.-Q. Zhang, and X. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E 67(2), 026607 (2003).
[Crossref]

Zheng, Y.

P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Localization and delocalization of light in photonic moiré lattices,” Nature 577(7788), 42–46 (2020).
[Crossref]

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)

C. Huang, C. Li, H. Deng, and L. Dong, “Gap solitons in fractional dimensions with a quasi-periodic lattice,” Ann. Phys. 531(9), 1900056 (2019).
[Crossref]

J. Phys. A: Math. Gen. (1)

I. Aviram, “The diffraction spectrum of a general family of linear quasiperiodic arrays,” J. Phys. A: Math. Gen. 19(16), 3299–3312 (1986).
[Crossref]

JOSA B (2)

N. Lučić, D. J. Savić, A. Piper, D. Grujić, J. Vasiljević, D. Pantelić, B. Jelenković, and D. Timotijević, “Light propagation in quasi-periodic fibonacci waveguide arrays,” JOSA B 32(7), 1510–1513 (2015).
[Crossref]

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, G. McCarthy, W. Krolikowski, and B. Luther-Davies, “Multipole composite spatial solitons: theory and experiment,” JOSA B 19(3), 586–595 (2002).
[Crossref]

MRS Bull. (1)

K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(8), 620–625 (2006).
[Crossref]

Nat. Photonics (1)

Q. Fu, P. Wang, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Optical soliton formation controlled by angle twisting in photonic moiré lattices,” Nat. Photonics 14(11), 663–668 (2020).
[Crossref]

Nature (1)

P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, and F. Ye, “Localization and delocalization of light in photonic moiré lattices,” Nature 577(7788), 42–46 (2020).
[Crossref]

New J. Phys. (2)

M. Modugno, “Exponential localization in one-dimensional quasi-periodic optical lattices,” New J. Phys. 11(3), 033023 (2009).
[Crossref]

C. Rockstuhl and F. Lederer, “The effect of disorder on the local density of states in two-dimensional quasi-periodic photonic crystals,” New J. Phys. 8(9), 206 (2006).
[Crossref]

Opt. Express (5)

Opt. Lett. (6)

Opt. Spectrosc. (1)

M. Kaliteevski, V. Nikolaev, R. Abram, and S. Brand, “Bandgap structure of optical fibonacci lattices after light diffraction,” Opt. Spectrosc. 91(1), 109–118 (2001).
[Crossref]

Phys. Rev. A (2)

L. Dong and F. Ye, “Stability of multipole-mode solitons in thermal nonlinear media,” Phys. Rev. A 81(1), 013815 (2010).
[Crossref]

Y. Kou, F. Ye, and X. Chen, “Multipole plasmonic lattice solitons,” Phys. Rev. A 84(3), 033855 (2011).
[Crossref]

Phys. Rev. E (2)

P. Xie, Z.-Q. Zhang, and X. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E 67(2), 026607 (2003).
[Crossref]

H. Sakaguchi and B. A. Malomed, “Gap solitons in quasiperiodic optical lattices,” Phys. Rev. E 74(2), 026601 (2006).
[Crossref]

Phys. Rev. Lett. (5)

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(1), 013901 (2009).
[Crossref]

D. Grempel, S. Fishman, and R. Prange, “Localization in an incommensurate potential: An exactly solvable model,” Phys. Rev. Lett. 49(11), 833–836 (1982).
[Crossref]

M. Kohmoto, “Metal-insulator transition and scaling for incommensurate systems,” Phys. Rev. Lett. 51(13), 1198–1201 (1983).
[Crossref]

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987).
[Crossref]

W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, “Localization of light waves in fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
[Crossref]

Sci. Rep. (1)

C. Huang, F. Ye, X. Chen, Y. V. Kartashov, V. V. Konotop, and L. Torner, “Localization-delocalization wavepacket transition in pythagorean aperiodic potentials,” Sci. Rep. 6, 1–8 (2016).
[Crossref]

Science (1)

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science 332(6037), 1541–1544 (2011).
[Crossref]

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1. Profiles of fundamental solitons with the maximum peak localized in the central waveguide (a) and in a low-amplitude waveguide (d); (b) The system’s linear spectrum and (c) the linear mode with the maximum eigenvalue $b=0.2699$; (e) $U(b)$ curves corresponding to fundamental solitons in (a) and (d); An enlarged graph of the left part of the $U(b)$ curves for $w_l$ and $w_r$ is also included in (e); (f) The most unstable growth rate $\textrm {max}(\lambda _r)$ versus $b$ for fundamental solitons $w_l$ and $w_r$. Subscripts $l$, $c$, and $r$ indicate that the peak of a fundamental soliton falls on the position of the three middle waveguides ($A$, $B$, $A$). To distinguish $w$, the amplitude of $R(x)$ is multiplied by $0.2$ in (a), (c) and (d). $p=1.0$ in all panels.
Fig. 2.
Fig. 2. Propagation of fundamental solitons $w_c$ at $b=0.29$ (a) and $b=1.00$ (b); unstable (c) and stable (d) propagation of fundamental soliton $w_l$ for $b=0.51$ and $b=0.70$, respectively; unstable (e) and stable (f) propagation of fundamental soliton $w_r$ for $b=0.55$ and $b=0.70$, respectively; $p$=1.0 and $\sigma ^2_{\textrm {noise}}=0.01$ in all panels.
Fig. 3.
Fig. 3. Profiles of both types of dipole solitons plotted in (a) and (b) for the same value of $b$; (c) power $U$ of both types of dipole solitons versus propagation constant $b$, and two enlarged graphs near the lower cutoff are given in the inset graphs; (d) the maximum values of the real part of perturbation growth rate $\textrm {max}(\lambda _r)$ versus propagation constant $b$ for solitons $w_{\textrm {dl}}$ and $w_{\textrm {dr}}$. To distinguish $w_{\textrm {dl}}$ and $w_{\textrm {dr}}$, $|w_{\textrm {dl}}|$ and $|w_{\textrm {dr}}|$ are enhanced by 0.1 and the amplitude of $R(x)$ is multiplied by $0.5$ in (a) and (b). $p=1.0$ in all panels.
Fig. 4.
Fig. 4. Stable (a, b) and unstable (c-f) propagation of two types of dipole solitons. Dipole solitons $w_{\textrm {dr}}$ at $b=0.70$ (a), $b=0.84$ (c), $b=0.509$ (e), and dipole solitons $w_{\textrm {dl}}$ at $b=0.82$ (b), $b=0.95$ (d), and $b=0.69$ (f). $p=1.0$ and $\sigma ^2_{\textrm {noise}}=0.01$ in all panels.
Fig. 5.
Fig. 5. Profiles of $3$-pole (a), $5$-pole (b), $7$-pole (c), and $9$-pole (d) solitons at $b=0.77$; (e) power $U$ of the four types of multipole solitons versus propagation constant $b$; instability growth rate versus $b$ of the $3$-pole (f), $5$-pole (g), $7$-pole (h), and $9$-pole (i) solitons. Unstable (j) and stable (k) spectrum of the linearization operator of the $3$-pole solitons for $b=0.85$ and $b=1.5$, respectively; To distinguish $w$, $|w|$ is enhanced by 0.1 and the amplitude of $R(x)$ is multiplied by $0.5$ in (a-d). $p=1.0$ in all panels.
Fig. 6.
Fig. 6. Stable (a, b) and unstable (c, d) propagation of $5$-pole (a, c) and $9$-pole (b, d) solitons in Fibonacci lattices; $b=1.5$ in (a), $b=1.9$ in (b), $b=1.34$ in (c), $b=1.7$ in (d), $p$=1.0 and $\sigma ^2_{\textrm {noise}}=0.01$ in all panels.

Equations (3)

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

i q z = 1 2 2 q x 2 g | q | 2 q p R ( x ) q .
λ u = 1 2 d 2 v d x 2 + b v p R v g w 2 v ,
λ v = 1 2 d 2 u d x 2 b u + p R u + 3 g w 2 u .