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Abstract
We survey the propagation properties of the out-of-phase (OOP) dipole solitons and the single-charged vortex (SCV) soliton in a periodic photonic moiré lattice with $\theta = \arctan ({3 / 4})$ under self-focusing nonlinearity media. Since the rotation angle, periodic photonic moiré lattices have peculiar energy band structures, with highly flat bands and the bandgaps being much more extensive, which is very favorable for the realization and stability of the solitons. When exciting a single point on-site with the OOP dipole beam, its evolution shows a periodic rollover around the lattice axis. Whereas, when exciting a single point on-site with the SCV beam, it transmits counterclockwise rotating periodically. Both the OOP dipole solitons and the SVC soliton maintain the local state, but their phase exhibits different variations. The phase of the OOP dipole solitons is flipped, while that of the SCV is rotated counterclockwise. Our work further complements the exploration of solitons in photonic moiré lattice with nonlinearity.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since the discovery of unconventional superconductivity in rotating bilayer magic-angle graphene in 2018, the peculiar electronic properties of graphene have once again attracted widespread attention. Rotational operations not only break the original symmetry of the lattice, which results in the formation of long-range order periodic moiré stripes at characteristic angles [1,2] but also affect the energy band structure of the lattice, moiré flat bands will appear in the lattice when meeting certain corner conditions [3]. The existence of many-body correlations in the flat bands of magic-angle graphene provides an accessible platform for investigating flat-band systems while facilitating an explanation of unconventional superconductivity mechanisms in some substances. Inspired by rotating graphene, many researchers have also focused their attention on rotating photonic lattices. In 2019, Ye's group [4] used two square lattices with the same period, both of which rotate to form a photonic moiré lattice, and explored experimentally and simulatively the evolution for the first time of light dynamics in photonic moiré lattice. The structure allows us to explore the conversion of different lattice structures by adjusting and controlling the rotation angle to achieve localization to delocalization transitions. The photonic moiré lattice provides us with a simple and easy platform to explore the behavior of light, and its highly tunable nature also provides lessons for featuring moiré lattices in two-dimensional materials and cold atomic systems.
A spatial optical soliton is a beam in which the diffraction and nonlinearity effects of light balance each other so that its waveform remains constant during propagation [5], which has been a hot topic of interest for researchers and are considered being one of the most critical nonlinear phenomena [6–14]. On the other hand, multipole solitons [15–17] in periodic lattice structures have attracted much attention from many research workers, especially dipole solitons and quadrupole solitons [8,18–23]. Because of the phase difference, we can classify dipole solitons and quadrupole solitons as in-phase or out-of-phase. The existence of dipole solitons was first predicted theoretically by Garcia et al. [24] and first observed experimentally by Krolikowski et al. [25]. In addition, the vortex beam, whose beam is distributed in a spiral in the in-phase plane, has orbital angular momentum and is centered at the point of phase distortion. It has been widely studied because it exhibits many exciting phenomena in the photonic lattice. Hence, vortex solitons in periodic structures have intensively dominated research in recent years [7,9,13,14,26–30]. When disrupting the periodicity of the medium, however, there will be some exciting phenomena. In 2017, localization and oscillation of light beams could be achieved in a quasi-periodic moiré lattice consisting of two different periodic lattices, and slow light and Zitterbewegung (ZB) effects have been discovered [31,32]. However, examinations on the formation of solitons in the periodic to aperiodic transition are scarce due to the lack of a suitable experimental platform. The photonic moiré lattice proffers a powerful platform for this purpose and points us in a new direction to clarify the transport behavior of optical solitons in the nonlinear region. Thus, the construction of photonic moiré light fields is also a new area worth exploring [33]. When considering the nonlinear region, research on nonlinearity photonic moiré lattice was first in the delving of spatial solitons in moiré lattices, which evidences the photonic moiré lattice exhibits a localization-delocalization transition related to the amplitude and rotation of the sub-lattice [4,34]. In 2020, the group of Ye [35] employed the photonic moiré lattice to interpret the formation of a smooth transfer of solitons from a fully periodic to a non-periodic geometry under nonlinear conditions and predicted that a similar phenomenon would also exist in moiré lattice formed by the superposition of other Bravais lattices, and in other physical systems with flat bands. In 2021, Mahmut's group explored the dynamics of solitons in moiré lattice by introducing a mean-field into the nonlinear Schrödinger equation. They probed that the effect of quadratic nonlinearity on the stability of solitons in aperiodic composite lattice structures and found that the region of existence of moiré solitons in parameter space is independent of the rotation angle, however, the stability depends on the rotation angle [36]. In the same year, Chen's group revealed for the first time that the effect of the moiré angle-independent leap of light could be extended to higher-order vortex optical fields with introducing frequency tuning, and earliest predicted the localization and delocalization of vortex optical fields in photonic moiré lattices [37].
In conclusion, the localization-delocalization transition of light in the photonic moiré lattice has attracted extensive attention [38–43]. However, the research of multipole solitons and vortex solitons in nonlinear photonic moiré lattices is also worth further investigation. This paper focuses on evaluating the transport properties of multipole solitons and vortex solitons in periodic photonic moiré lattices with saturated self-focusing nonlinearity. The main objective is to outline the evolution of the out-of-phase (OOP) dipole solitons and the single-charged vortex (SCV) soliton in the photonic moiré lattices with rotation angles $\theta = \arctan ({3 / 4})$(only consider periodic moiré lattices) which is obtained by rotational superposition of square lattices of the same period. Our results show that both the OOP dipole solitons and the SCV undergo periodic oscillation in a photonic moiré lattice under nonlinear conditions. The OOP dipole beams can form an out-of-phase soliton, and its phase occurs flipping, while the SCV beams can also evolve into a vortex soliton with a counterclockwise phase rotating.
2. Theoretical models
A photonic moiré lattice can be formed by the rotational superposition of two square or triangular lattices of the same period (the period is 1:1), optically induced by self-focusing nonlinear light, the expression of which is described as
Fig. 1. Intensity distribution and energy band structure of the periodic photonic moiré lattice with $\theta = \arctan ({3 / 4})$. (a1) Lattice intensity distribution; (a2) The primitive lattice cells; (b) Periodic photonic moiré lattice with $\theta = \arctan (3/4)$ pattern optically induced in the homogeneous SBN crystal.
Assuming that the light beam propagates along the z-axis, the dimensionless nonlinear Schrödinger equation under the condition of parallel axis approximation govern the propagation of light in a photorefractive crystal [35], which is,
The energy band relationship of the periodic lattice can be obtained by the plane wave expansion method, unlike the tight-binding model, considering the coupling effect of adjoining lattice points. Therefore, neglecting the nonlinear term in Eq. (2), one obtains the linear Eq. (3) that,
Assuming a plane wave of $\Psi ({x,y,z} )= u({x,y} ){e^{i\beta \textrm{ }z}}$ and substituting into Eq. (3), we can have the linear eigenvalue equation for the lattice as,
Fig. 2. Energy band structures of periodic photonic moiré lattice with $\theta = \arctan ({3 / 4})$. (a) Three-dimension view. (b) Dispersion relationship along high symmetry points $\varGamma $. From top to bottom: ${\beta _{1 \sim 6}}$.
From Fig. 2(a), we capture that the first energy band is distinctly flat, as demonstrated more clearly in the view of Fig. 2(b), and more notably, the higher-order energy bands fuse to form a new flat band. We speculate that when two sub-lattices are rotated and superimposed, specific lattice points overlap, resulting in higher energy at the overlap point than at the no overlapped point in the lattice, and the mutual compensation of energy thus leads to the fusion of the energy bands, which manifests as flat bands. In addition, note that the band gaps between the energy bands open entirely and tend to flatten. Moreover, the bandgap of this lattice is significantly larger than that of the single square lattices, which is undoubtedly advantageous for bandgap solitons, even higher-order bandgap solitons.
3. Numerical simulation and analysis
3.1 Self-trapping and flipping of the OOP dipole solitons
In the single square lattices, the OOP dipole solitons present self-trapping, and they can also rotate around the particular axis of the lattices [44]. Whether the same phenomenon occurs in the photonic moiré lattice, which deserves to investigating in depth. In this section, we employ periodic photonic moiré lattice with $\theta = \arctan ({3 / 4})$ to describe the dynamic evolution of the OOP dipole beams. The mathematical model of the OOP dipole beams can be expressed as follows,
Fig. 3. The position of the OOP dipole beams incident into the periodic moiré lattice with $\theta = \arctan ({3 / 4})$. (a) the position of the OOP input (shown as red dashed circles); (b) the intensity distribution of the probe beam, the right corner shows phase plane; (c) the profile of the lattices and the input probe beam (along the white dashed line).
Under the nonlinear condition of ${V_0} = 30V$, we investigate the transport properties of the OOP dipole beams in the periodic photonic moiré lattice with $\theta = \arctan ({3 / 4})$. In order to accurately represent the dynamic evolution of the OOP dipole beams in this structure, we present the transmission variation of the beam in Fig. 4, obtaining the intensity patterns and phase at intervals of 0.8 units length. The OOP dipole beam maintains this alternating pattern in the transmission process, characterized by an “exhale and inhale” pattern but always localized. We notice that during the dynamic evolution of the OOP dipole beams, they still maintain the out-of-phase local state, but their size changes regularly, for comparing z =1.6 with z =2.4 in Fig. 4. When the beams propagate to z =2.4, z =4.8, and z =7.2, respectively, their sizes become more prominent, while when reach z =9.6, they revert to their initial incidence state, and the phase reverse. Secondly, we observe the behavior of the phase, find that every 0.8 units length, the phase between the neighbors exhibits a $\pi $ degree flip, for example, z =1.6, z =2.4 and z =7.2, z =8.0.
Fig. 4. Dynamical evolution of intensity and phase of the OOP dipole beams in the periodic photonic moiré lattices with $\theta = \arctan ({3 / 4})$ in different propagation distance.
Furthermore, by analyzing the output states in the side-view from Fig. 5 (a), we are convinced that the OOP dipole beams maintain their local state transmission in the periodic photonic moiré lattice with $\theta = \arctan ({3 / 4})$. Figure 5(b1)-(b4) shows the output intensity and phase corresponding to the different unit lengths at z =2.4, z =4.8, z =7.2 and z =9.6, respectively. Comparing Fig. 5(a) and Fig. 5(b), both have essentially the same beam scale and opposite phase, as Fig. 5(b2) and Fig. 5(b4). We then demonstrate from the interferogram in Fig. 5(c) that the local state always maintains the OOP structure of the out-of-phase dipole beam (as marked by the white dashed line). Therefore, we conjecture that energy is always conserved during the transmission of the out-of-phase dipole beam, despite the variation in beam range. The energies of the two spots compensate each other, thus keeping the overall energy constant, and there is no diffraction of energy into neighboring lattice points. Also, from the profile of the OOP dipole beams at different propagation distances in Fig. 6(a), we can easily observe they maintain the same shape with periodic, but with a discrepancy in peak intensity. Moreover, especially separated the input and output profiles of the OOP dipole beam in the lattices for comparison, as shown in Fig. 6(b), the output is symmetrical to the input and has an identical profile with exactly opposite phase. We conclude that the OOP dipole solitons can be formed when the OOP dipole beams are excited at a single point in the periodic photonic moiré lattice with $\theta = \arctan ({3 / 4})$, but they may not be stable.
Fig. 5. Simulated nonlinear transmission of the OOP dipole beams in the periodic photonic moiré lattices with $\theta = \arctan ({3 / 4})$. (a) side view; (b1)-(b4) nonlinear output light intensity map and its phase diagram (inset in the upper right corner) at z =2.4, z =4.8, z =7.2, z =9.6; (c1)-(c4) Interferogram corresponding to (b1)-(b4) (white dotted lines suggesting out-of-phase).
Fig. 6. (a) The profile of the OOP dipole beams in z =2.4, 4.8, 7.2, and 9.6, respectively. (b) The input and output profiles of the OOP dipole beams.
To demonstrate whether the OOP dipole solitons can be steadily transmitted in periodic moiré lattice structure with $\theta = \arctan ({3 / 4})$, we simulate energy flow (white arrows) and the peak intensity variation of the OOP dipole solitons, as shown in Fig. 7. As the energy flow diagram in Fig. 7(a1)–(a4), the energy of the OOP dipole switches between the two peaks, which verifies the change in phase. Moreover, we perceive that the peak intensity of the OOP dipole beam exhibits periodic variations as the transmission distance increase. Every 4.8 units lengths, the peak intensity exhibits a similar variation, and first oscillation decreases then oscillation increases, as shown in Fig. 7(b). A step further, within these 2.4 units lengths, the peak intensity shows a sinusoidal-like variation. The periodic exchange of energy between the two components leads to dynamic inversion.
Fig. 7. Energy flow (white arrows) and peak intensity variation of the OOP dipole beam. (a1) z =2.4; (a2) z =4.8; (a3) z =7.2; (a4) z =9.6, the black dashed circle represents the energy concentrations. (b) Peak intensity profile of OOP dipole beams along the propagation direction.
3.2 Self-trapping and rotation of the SCV
A single-charged discrete vortex soliton, consists of four elementary discrete solitons in terms of light intensity, and the phase difference between two adjacent solitons is ${\pi / 2}$. In the single square lattices, the SCV can self-trap as a localized gap vortex soliton mode residing in the first Bragg reflection gap, for which the vortex is nested in the center of a rotating square-like envelope and there are always four peaks on the four corners [30]. Whether the same variation exists in periodic photonic moiré lattices is worth contemplating in the following. The expression for the single-charged vortex beam in the polar coordinate system can be described as follows,
where $r,\phi $ represent polar coordinates, and $A,\sigma $ denote the maximum amplitude and width magnitude of the input vortex beam, respectively. $m$ denotes the end of the vortex light. In this paper, we take $m ={+} 1$ to indicate that the incident beam is the single-charged vortex beam, and take the beam width $\sigma = {1 / 3}$ to be comparable to the size of a lattice point. In this simulation, we set $A = 1$. Figure 8(a) represents the schematic diagram giving the excitation way: the SCV is focused on exciting a single lattice point of the periodic photonic moiré lattices with $\theta = \arctan ({3 / 4})$, Fig. 8(b) shows the single-charge vortex beam, with its phase in the lower right corner. On the other hand, we derive the lattice incidence profile, showing that the SCV beam is thoroughly excited on-site, see Fig. 8(c).Fig. 8. The position of the SCV incident into the periodic moiré lattice with $\theta = \arctan ({3 / 4})$. (a) the position of the SCV input (shown as red dashed circles); (b) the intensity distribution of the probe beam, the right corner shows phase plane; (c) the profile of the lattices and the input probe beam (along the white dashed line).
We dissect the transport properties of the single-charged vortex light in the periodic photonic moiré lattice structure with $\theta = \arctan ({3 / 4})$. Setting the nonlinearity condition as ${V_0} = 30V$, the dynamic evolution of SCV is shown in Fig. 9. Upon viewing the entire transmission of the SCV beam, we find it still retains its local vortex state. There is no evolutionary behavior of the square envelope, nor does it turn into the OOP quadrupole solitons. Meanwhile, referring to the phase evolution in Fig. 9 (shown in the upper right-hand corner), we extrapolate that the phase becomes $2\pi $ in a circle around its vortex center, while the phase difference between adjoining phases remains ${\pi / 2}$. The SCV rotates counterclockwise by $\pi $ per 0.8 units length of forwarding transmission.
Fig. 9. Dynamical evolution of intensity and phase of the SCV in the periodic photonic moiré lattices with $\theta = \arctan ({3 / 4})$ in different propagation distance.
As shown in Fig. 10(a), the transverse interface diagram of the SCV shows that the SCV beam transmits diffraction-free and still retains stable propagation in the local state. Then, we take out the optical intensity cross-sectional patterns and the phase patterns for each of 2.4 units length, as shown in Fig. 10(b1)–(b4). The comparison shows that the SCV always maintains a first-order vortex state, the topological charge is still 1, and there is no tendency to spread in all directions, but the phase is rotated counterclockwise. For every 2.4 units length of transmission, the phase rotates counterclockwise by ${\pi / 2}$, and when it reaches 9.6 units length, the phase is the same as at z =2.4, with a phase difference of approximately ${\pi / 2}$ from the initial phase. Purpose of verifying the phase structure, Fig. 10(c1)-(c4) shows the corresponding interferograms, which clearly shows that the phase keeps out-of-phase features in the SCV localized state as well as the charge invariance during the dynamic evolution.
Fig. 10. Simulated nonlinear transmission of the SCV in the periodic photonic moiré lattice structure with $\theta = \arctan ({3 / 4})$. (a) Side-view; (b1)-(b4) nonlinear outgoing light intensity map and its phase map (inset in the upper right corner) in z =2.4, z =4.8, z =7.2, and z =9.6, respectively; (c1)-(c4) Interferogram corresponding to (b1)-(b4) (black dashed circle indicates the center of the SCV).
To further illustrate the periodic variation of the SCV, Fig. 11(a) depicts the profile of the SCV beam with different propagation lengths, and Fig. 11(b) the input and output profile of the SCV beam in the periodic photonic moiré lattice with $\theta = \arctan ({3 / 4})$, which interprets that the SCV beam is transmitted in a breath-like manner without loss of energy in essence. On the other hand, by comparing the intensity profiles of the input and output, they essentially overlap, except for a slight height difference, which allows us to ensure the SCV beam can form vortex solitons in periodic photonic moiré lattice with $\theta = \arctan ({3 / 4})$.
Fig. 11. (a) The profile of the SCV in z =2.4, 4.8, 7.2, and 9.6, respectively. (b) The input and output profiles of the SCV dipole beams.
In order to prove the stable propagation of the SCV soliton, Energy flow (white arrows) and the peak variation of the single-charged vortex light during transmission are shown in Fig. 12, where we surface the peak intensity exhibits a periodic variation with increasing transmission length. The period of variation is 2.4 units lengths, and within this one period, the peak intensity exhibits a cosine of sorts variation behavior, but with relatively small variations between amplitudes. We speculate that this periodic oscillation in peak intensity is associated with the conservation of angular momentum and that when the rotation reaches its limit, the SCV reverses to its initial state and move in circles. As shown in Fig. 12(a1)(a4), for every 2.4 units propagated, the energy flow aggregation position is rotated counterclockwise by ${\pi / 2}$, which agrees with the phase change. Although the periodic photonic moiré lattice with $\theta = \arctan ({3 / 4})$ has a fundamentally similar structure to square lattices, the behavior of the SCV is quite different.
Fig. 12. Energy flow (white arrows) and peak intensity variation of the SCV. (a1) z =2.4; (a2) z =4.8; (a3) z =7.2; (a4) z =9.6, the black dashed circle represents the location of the energy gathering. (b) Peak intensity profile of the SCV along the propagation direction.
4. Conclusion
We employed the periodic photonic moiré lattice with $\theta = \arctan ({3 / 4})$ to feature the self-trapping and periodic oscillation of the out-of-phase dipole solitons and the single-charged vortex soliton. Both the OOP dipole beam and the SCV excite on-site the periodic photonic moiré lattice at a single point, and all show a periodically stable variation. The OOP dipole solitons vary in a self-trapping state, with a phase flipping, while the SCV soliton rotates counterclockwise periodically. Our research is an attempt to illustrate the transport properties of multipole and vortex solitons in the photonic moiré lattice with nonlinearity. However, these investigations disregard whether the solitons originate from band gaps or form flat bands, so that the subsequent work will focus on this aspect. Spatial optical solitons have unique applications in the information process, all-optical routing, and optical manipulation. Furthermore, the photonic moiré lattice presents a new design platform for spatial modulation of various optical fields. We are confident that the combination of the two will bring us more surprising phenomena.
Acknowledgments
Statistical support was provided by National Natural Science Foundation of China (61675184, 61275124).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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