Abstract
Trajectory control of spatial solitons is an important subject in optical transmission field. Here we investigate the propagation dynamics of Laguerre-Gaussian soliton arrays in nonlinear media with a strong nonlocality and introduce two parameters, which we refer to as initial tangential velocity and displacement, to control the propagation path. The general analytical expression for the evolution of the soliton array is derived and the propagation properties, such as the intensity distribution, the propagation trajectory, the center distance, and the angular velocity are analyzed. It is found that the initial tangential velocity and displacement make the solitons sinusoidally oscillate in the $x$ and $y$ directions, and each constituent soliton undergoes elliptically or circularly spiral trajectory during propagation. A series of numerical examples is exhibited to graphically illustrate these typical propagation properties. Our results may provide a new perspective and stimulate further active investigations of multisoliton interaction and may be applied in optical communication and particle control.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since Snyder and Mitchell opened the prelude to the systematic study of nonlocal spatial optical solitons in 1997 [1], the propagation and interaction of spatial soliton in nonlocal nonlinear media have attracted extensive interest and been widely investigated in recent years both in theory and in experiment [2–7]. Spatial nonlocality means that the response of a point in the medium to the light field is related not only to the light field of that point but also to the light field of other points in the space [1,8]. According to the relative scales of beam width and characteristic length of nonlinear response of medium, nonlocality can be divided into four situations: local, weak nonlocal, general nonlocal, and strong nonlocal [9]. Strong nonlocality refers to the situation that the beam width is much smaller than the characteristic length of the nonlinear response of the medium [10], which provides a favorable condition for researchers to investigate the spatial solitons. Various kinds of solitons are found in strongly nonlocal nonlinear media (SNNM), such as multipole solitons [11], surface-wave solitons [12], ring dark and antidark solitons [13], and vortex solitons [14,15]. In experiment, it has been proved that nematic liquid crystals [16,17] and lead glasses [18] have large nonlinear response characteristic length so that strong nonlocal transmission conditions can be easily realized. They are derived from the reorientation mechanism and thermally nonlinear mechanism of liquid crystal molecules respectively [16–18]. In general, it is an effective way to understand the formation of spatial soliton by making an analogy between the spatial effects induced by the diffraction for the beam and the temporal effects induced by the group-velocity dispersion for the pulse [19,20]. The study of optical solitons has always been the research frontier of nonlinear optics and has very important applications in all-optical networks, optical communications and optical logic devices [21–26].
As early as 1997, the theory of the motion of spiraling solitons has been proposed [27,28], i.e., the competition between centrifugal force and mutual attraction of in-phase solitons leads to stable spiralling. The interaction between solitons was found to be universal and diverse later [29]. The long-range three-dimensional interactions between two optical solitons has been implemented in 2006 [2], which suggests that the construction of novel model systems for studying the behavior of complex nonlinear networks is possible. Just recently, it is found that the spiraling propagation behavior can be achieved in linear media with external harmonic potentials [30]. And many interesting effects, such as steerable output position [30,31], periodic focusing and reversion [32,33], and the anharmonic oscillation [33,34], can be realized by introducing external potentials. In spite of this, the manipulation of optical solitons is an important subject in optical transmission field.
Laguerre-Gaussian (LG) mode has been investigated widely because it possess several special properties, including shape-invariant during propagation and carrying orbital angular momentum [3,20,35,36]. In this paper, we carry out the basic principles for constructing the so-called rotating LG soliton arrays to explore the propagation dynamics of multiple solitons in SNNM. Two parameters, which we refer to as initial tangential velocity and displacement, are introduced to control the propagation path of the arrays. The exact LG soliton solution with initial tangential velocity and displacement for the 1+2D simplified Snyder-Mitchell model is presented. It is found that the propagation and interaction of the LG solitons in SNNM is deeply affected by the initial tangential velocity and displacement, which can cause the rotating behavior of LG solitons, and lead to the occurrence of proximity and separation during the interaction of multiple LG solitons. The results in strongly nonlocal nonlinear optical system can be extended to many similar systems, such as optical fractional Fourier transform system [37], quadratic nonlinear system [38], linear system with harmonic potentials [30] and so on. In addition, these results may have potential applications in gravitational system in that there is a one-to-one correspondence between the physical quantities in optical system (soliton power, the transverse components of wave vector, nonlinear refractive index, propagation direction) and the physical quantities in mechanical system (mass, velocity, gravitational potential, time) [39], even in Fresnel diffraction system [40] and free space [41]. It is useful for further understanding the propagation and interaction of optical solitons in SNNM and may be applied in optical communication and particle control.
The rest of this paper is organized as follows. In Sec. 2., beginning with nonlocal nonlinear Schrödinger equation (NNLSE), we describe the evolution of laser beams in a general nonlocal medium. A more specific version of the NNLSE in SNNM is given by using the Taylor’s expansion of response function and the technique of variable transformation. Then we gave the analytical solution of a single LG beam. After this, the trajectory equation (which describes the motion of LG soliton center) is given explicitly and the propagation and interaction model of multiple LG solitons, i.e. LG soliton array, has been constructed. In Sec. 3., the propagation and interaction properties of LG solitons are analyzed and discussed. In Sec. 4., we conclude this work and point out its potential applications.
2. Theoretical model
The evolution dynamics of paraxial laser beams in nonlocal nonlinear media is ruled phenomenologically by the NNLSE [1,10,42–45]
where $A$ is the complex amplitude of the input laser beams; $k=\omega {n_0}/c$, with $\omega$ the circular frequency, is the wave number in the media without nonlinearity; $\Delta _{\bot }=\frac {\partial ^2}{\partial {r^2}}+\frac {1}{r}\frac {\partial }{\partial {r}}+\frac {1}{r^2}\frac {\partial ^2}{\partial {\varphi ^2}}$ is two-dimensional transverse Laplacian operator; $n_0$ is the linear part of the refractive index of the media; $\delta {n}=n_{2}\iint {R({\textbf {r}}-{\textbf {r}}_{c})}|A({\textbf {r}}_{c},z)|^2d^2{\textbf {r}}_{c}$ is the nonlinear perturbation of the refractive index caused by the beam ($|\delta {n}|\ll {n_0}$); $n_{2}$ is the nonlinear index coefficient; ${\textbf {r}}$ and ${\textbf {r}}_c$ represent two-dimensional transverse coordinate vectors; $R({\textbf {r}})$ is the normalized symmetrical real spatial response function of the media and it satisfies $\iint {R({\textbf {r}})}d^2\textbf {r}=1$. Equation (1) describes the evolution of a spatial beam trapped in an effective parabolic graded index channel with the profile given by the nonlocal response function $R({\textbf {r}})$, which usually be taken as Gaussian function form $R({\textbf {r}})=1/(2\pi w_R^2)\exp [-\textbf {r}^2/(2w_R^2)]$, where $w_R$ is the characteristic width of $R({\textbf {r}})$ [20,43].In the physical setting of strongly nonlocal nonlinearity, $R({\textbf {r}})$ can be expanded in Taylor’s series [10]. If we only expand the response $R(\textbf {r}-\textbf {r}_\textbf {c})$ with respect to $\textbf {r}_\textbf {c}$ to the second-order term, then we obtain
here the special case $\textbf {r}_\textbf {c}=\textbf {0}$ is taken into account. And Eq. (1) reduces toThe propagation expression of a single LG beam in cylindrical coordinates $(r,\varphi ,z)$ can be obtained as [20]
The propagation expression form of the even and odd LG beams with radial number $n$ and azimuthal number $m$ can be written as
Provided that the center of mass ${\textbf {r}}_c$ of a single soliton traveling in SNNM satisfies the ray equation [1]
One can get the solution of Eq. (12) as Here, the superscript “ $'$ ” represents the first derivative, and it can be proved that if $\Phi (\textbf {r},z)$ is a solution of Eq. (5) then so isThe combined field of the LG soliton array can be constructed by using the superposition principle
where $C_0$ is the normalized amplitude of the total input field, $\Phi _{nm}^{(j)}$ is the optical field of the $j$th LG soliton. After considering that each constituent soliton has been applied an initial displacement, one can obtainConsidering the independence of variables in $x$ and $y$ directions and combining the Eqs. (13), (15) and (16), then we obtain
which jointly govern the trajectory of the center of mass of each constituent LG soliton, andConsider a simple configuration in which the center of mass of each constituent soliton is equidistant from the origin of coordinates. Then the position of each constituent LG soliton center can be taken as
and the initial tangential velocity of each LG soliton can be expressed as where $r$ is the assumed initial displacement length, $\varphi _{0j}=2j\pi /N$ ($j=1,2,\ldots ,N$) is the angle between the center of the corresponding LG soliton and the $x$ axis at the source plane (initial azimuth angle), and $\xi$ is defined as “tangential velocity parameter.” If $\xi =0$, it is the general case and the initial incident direction of each constituent LG soliton is perpendicular to the source plane. If $\xi \neq 0$, there is an angle between the initial incident direction of each constituent soliton and the source plane, i.e., every interacting LG soliton is launched with a twisted trajectory.3. Propagation dynamics
Based on analytical propagation expression, we can study the propagation dynamics of the rotating LG soliton array in this section. We construct a triangle LG soliton array as an example first. The propagation expression of each constituent soliton in the triangle LG soliton array shown in Fig. 1(a) is recorded as
where the subscript “$A, B, C$” correspond to $j=1, 2, 3$ respectively, $r$ is the length of initial displacement. Correspondingly, which means that soliton $A'$, $B'$, $C'$ and soliton $A$, $B$, $C$ are centrosymmetric with respect to the origin of coordinates but the length of initial displacement is $r/2$. Since there is a certain quantitative relationship between $\Phi _{A,B,C}$ and $\Phi _{A',B',C'}$, we will only discuss $\Phi _{A,B,C}$ as an example in the following if there is no special explanation.1 Evolution of non-rotation case
To give the preliminaries, we briefly introduce the propagation properties of the combined field without rotation first. Figure 1 displays the general propagation case that $\xi =0$. One can obtain that a complete evolution period is $\Delta {z}=2T=2\pi {z_p}$ based on Eq. (18). If we divide each complete evolution period (from $z=0$ to $z=2T$) into two half-periods, then the trailing half-period (from $z=T$ to $z=2T$) is a reverse process of the leading half-period (from $z=0$ to $z=T$), just as the propagation state from $z=0.5T$ to $z=T$ is a reverse process of that from $z=0$ to $z=0.5T$. Therefore, only the first half-period is given in Fig. 1. In this respect, it is similar to the evolution of a single beam [44,48] or a high-order temporal soliton in nonlinear fiber [19], which can recover into its initial pattern at the end of each evolution period. In other words, it is revivable periodically. Of course, the combined light field itself can also be viewed as a generalized high-order single beam. Each constituent soliton has a linear harmonic oscillation centered on the propagation axis. At $z=0.5T$, the light intensity is the most concentrated and the spot size is the smallest. Many bright intensity peaks appear in the interference domain, which results from constructive interference. Its normalized intensity distribution details are shown in Figs. 1(f) and 1(g). It can be seen that at this time it has the similar distribution profile as a single LG soliton, namely one dark ring and two dark diameters.
2 Evolution of the rotating pattern
In this section, we are going to discuss the quintessential propagation properties of the rotating LG soliton arrays with different tangential velocity parameters ($\xi \neq 0$). Figure 2 shows the evolution of the intensity patterns. The initial incident position of each constituent soliton is the same as that in Fig. 1(a). According to the geometric relation, we can easily obtain the area of the triangle enclosed by the centers of the three solitons $A$, $B$ and $C$ such that
One sees that if $\xi =1$ [see Figs. 2(f)–2(j)], the size of the LG soliton array remain invariant intuitively, though it rotates while propagating. This can be verified by Eq. (28), when $\xi =1$, $S_{\triangle {ABC}}\equiv {const}$. If $0\,<\,\xi\,<\,1$ ($\xi\,>\,1$), the size of the LG soliton array evolves to be smaller (larger) first and then larger (smaller) periodically during propagation with the period $\Delta {z}=T$, and the minimum (maximum) appears at $z=(p+1/2)\pi {z_p}$ ($p=0,1,2,\ldots$). Therefore, by borrowing concepts from geometric mathematics, the similarity ratio of the optical field range between extreme position and initial position is $\xi$. Each constituent LG soliton spirals inward and outward simultaneously, as shown in Figs. 2(a)–2(e) and 2(k)–2(o). Intuitively, it is the competition between centrifugal force and mutual attraction of the constituent LG solitons leads to the stable rotation. The initial tangential velocity provides an initial angular momentum and the centrifugal force for the rotating LG soliton array system, while in SNNM the solitons are always attract each other no matter how the initial separation distances. When the initial tangential velocity is small, the centrifugal force is initially weaker than the attractive force, the soliton array has a spiraling centripetal motion; with the decrease of the soliton spacing, the centrifugal force becomes stronger than the attractive force, the soliton array has a spiraling centrifugal motion; then it repeats this process. When the initial tangential velocity is large or moderate, the change of the soliton array is the opposite or critical state of the above motion process. It can also be regarded as the constituent solitons that cross induce a spiral waveguide that, in turn, guides the soliton array propagation in SNNM. The three ranges ($0\,<\,\xi\,<\,1$, $\xi >1$, $\xi =1$) of the tangential velocity parameter correspond to the three states (shrink, expansion and size-invariant) of soliton array propagation.3 Evolution of the propagation trajectories
For clarity, only the trajectories of $C$ and $C'$ are given in Figs. 3(a)–3(d), other constituent LG solitons are similar to this. Figures 3(b), 3(c) and 3(d) are the main view, the left view, and the top view of Fig. 3(a), respectively. One sees that when $0\,<\,\xi\,<\,1$, the solitons $C$ and $C'$ propagate forward in a helical structure, which projection trajectories in the $x$-$y$ plane are concentric ellipses as shown in Fig. 3(d), where the arrows indicate the rotating direction. They are sinusoidally oscillate in $x$ and $y$ directions. The maximum and minimum deviations from $z$ axis are the long and short half-axis lengths of elliptic trajectories, respectively.
Figure 4 displays all the projection trajectories of the constituent LG solitons in Fig. 2. The corresponding projection trajectories in the $x$-$z$ plane and $y$-$z$ plane can be described as
and respectively. This indicates that whatever the value of $\xi$, the projection trajectories of each constituent soliton in $x$-$z$ plane and $y$-$z$ plane always change sinusoidally. Every interacting soliton undergoes a twisted trajectory that oscillates around the propagation axis. If the constituent solitons do not propagate along the $z$ axis, instead, they are confined in the source plane, then the rotation state of them is similar to that of multiple celestial objects or multiple moving charged particles. The corresponding projection trajectory in the $x$-$y$ plane can be described as4 Evolution of the rotating states
During propagation, the angle between each constituent soliton and the $x$ axis in transverse plane can be described as
5 Rotating arrays of other shapes
In this section, we will present some other forms of LG soliton array to further understand its propagation. In Fig. 6, we show the $3\times 3$ (0, 2)-mode LG “soliton matrix,” the tangential velocity parameters are taken as negative values. Interestingly, the rotating pattern is shape-invariant during propagation but the direction of rotation is opposite compared with the situation in Fig. 2. The reason is that the negative tangential velocity parameter provides an opposite angular velocity for the rotation of the array based on Eq. (35). When $-1\,<\,\xi\,<\,0$ and $\xi <-1$, each constituent soliton undergoes alternating transformation of dispersion and aggregation as discussed earlier, while when $\xi =-1$, the size of the rotating pattern remain invariant. The propagation states shown in Figs. 6(a) and 6(c) can be viewed as array breathers and in Fig. 6(b) can be viewed as array soliton.
In Fig. 7, we show the ring-like, rhombus-like, and pentagram-like rotating LG soliton arrays with the same tangential velocity parameter $\xi =1$ and the different radial and azimuthal numbers, the initial displacement of each constituent soliton is properly designed. Figures 7(a) and 7(b) are based on Eq. (11), Fig. 7(c) is based on Eq. (6). One sees that the rotating LG soliton array is size-invariant during propagation under the case of $\xi =1$. In Fig. 8, we show the orthohexagonal LG soliton array with different tangential velocity parameters, each constituent LG soliton is chosen as different transverse modes. The propagation properties this time are the same as that in Fig. 2, not tired in words here. The variation of the array’s size in Fig. 8 satisfies
By changing the initial incident conditions, we can artificially interfere with the trajectory of soliton interaction. The theoretical model proposed in this paper can be used as a general method to construct various size-invariant or size-variant rotating soliton arrays, which may has potential applications in particle control. The SNNM provides a convenient way for controlling the propagation properties of soliton arrays. No more soliton array’s forms are given here, but the interested readers can construct them according to the approach provided in this paper.
4. Conclusion
In summary, the propagation dynamics of rotating LG soliton arrays in SNNM are investigated systematically. The general analytical expressions for the evolution and interaction of LG solitons are derived, and the propagation properties are analyzed in detail. It is found that the initial tangential velocity and displacement play key roles in the propagation and interaction of LG solitons. They make the solitons sinusoidally oscillate in the $x$ and $y$ directions, each constituent soliton undergoes elliptically or circularly spiral trajectory during propagation. Depending on the tangential velocity parameter, the soliton array can rotate clockwise or counterclockwise. Moreover, the propagation path can easily be controlled by a proper choice of initial incident parameters. Our study suggests that the construction of novel model systems for studying the behavior of complex soliton interactions is possible. The results in this paper can be extended to many similar systems of strongly nonlocal nonlinear optical system, such as optical fractional Fourier transform system, quadratic nonlinear system, linear system with external harmonic potentials, and gravitational system [30,37–39]. While the results are obtained only by studying LG solitons, it can provide references to investigate other types of solitons mentioned above. Our results may provide new insight into the interaction of multisoliton and may be applied in optical communication and particle control, as well as routing light in optical information processing field.
Funding
National Natural Science Foundation of China (61308016, 11374089, 61605040); Chunhui Plan of Ministry of Education of China (Z2017020); Natural Science Foundation of Hebei Province (F2017205060, F2017205162, F2016205124); Technology Key Project of Colleges and Universities of Hebei Province (ZD2018081); Science Fund for Distinguished Young Scholars of Hebei Normal University (L2017J02); Innovation Funding Project for Graduate Students of Hebei Normal University CXZZSS2019069.
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