Dynamics of rotating Laguerre-Gaussian soliton arrays

Limin Song; Zhenjun Yang; Shumin Zhang; Xingliang Li

doi:10.1364/OE.27.026331

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]

(1)$$2ik\frac{\partial{A}}{\partial{z}}+\Delta_\bot{A}+2k^2\frac{\delta{n}}{n_0}A=0,$$

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

(2)$$\delta{n}\approx n_{2}P_0\left(R_0+\frac{1}{2}R_0''\textbf{r}^2\right),$$

here the special case $\textbf {r}_\textbf {c}=\textbf {0}$ is taken into account. And Eq. (1) reduces to

(3)$$2ik\frac{\partial A}{\partial z}+\Delta_{\bot}A-k^2\gamma^2\textbf{r}^2P_0A+\frac{2n_2k^2R_0P_0}{n_0}A=0,$$

where $\gamma =\sqrt {-n_{2}R_{0}^{''}/n_0}$ is a material constant associated with the nonlocal effect of the medium and $\gamma ^2>0$ means that the nonlinear medium is a self-focusing medium [ $R_{0}=R({\bf 0})$, $R_{0}^{''}=R^{''}({\bf 0})$, $R_{0}^{''}<0$ because $R_0$ is a maximum of $R(\textbf {r})$ ]. $P_0=\iint |A(\textbf {r},0)|^2d^2\textbf {r}$ is the input power and it is conserved in the process of beam propagation [10]. By introducing the variable transformation

(4)$$A(\textbf{r},z)=\Phi(\textbf{r},z)\exp\left(ik\frac{n_{2}R_{0}P_{0}}{n_0}z\right),$$

Eq. (3) can be simplified as the famous strongly nonlocal nonlinear model

(5)$$2ik\frac{\partial \Phi}{\partial z}+\Delta_{\bot}\Phi-k^2\gamma^2\textbf{r}^2P_0\Phi=0.$$

It should be noted here that in the derivation of Eq. (5), we assume that the response function is symmetric and twice differentiable at $x=y=0$. However, the response function of the actual physical system has no rules to follow, so it cannot be simply transformed into the strongly nonlocal nonlinear model. Luckily, it has been proved that the physical characteristics in SNNM do not intensively depend on the form of the response function as long as the nonlocality of the media is strong enough [46].

The propagation expression of a single LG beam in cylindrical coordinates $(r,\varphi ,z)$ can be obtained as [20]

(6)$$\begin{aligned} \Phi_{nm}&=\frac{C_{nm}}{w(z)}\left[\frac{r}{w(z)}\right]^mL_n^m\left[\frac{r^2}{w^2(z)}\right]\exp{\bigg\{}-\frac{r^2}{2w^2(z)}{\bigg.}\\ &\quad {\bigg.}+i\left[c(z)r^2+(2n+m+1)\theta(z)+m\varphi\right]{\bigg\}}\end{aligned}$$

by using the method of separation of variables, where

(7)$$w(z)=w_0\left(\cos^2z_s+\frac{P_c}{P_0}\sin^2z_s\right)^{1/2}$$

denotes the beam width of Gaussian beam, $w_0=w(z)|_{z=0}$ is the initial beam width of the Gaussian beam. Note that in the process of solving the propagation expression, in order to facilitate calculation, we use the beam width of Gaussian beam to express the beam width of LG beam. The quantitative relationship between the two can be described as

(8)$$w_{nm}(z)=\sqrt{2n+m+1}w(z)$$

according to the definition of the second-order moment beam width.

(9)$$\begin{aligned} c(z)&=\frac{k\left({P_c}/{P_0}-1\right)\sin(2z_s)}{4z_p\left[\cos^2z_s+ ({P_c}/{P_0})\sin^2z_s\right]},\end{aligned}$$

(10)$$\begin{aligned} \theta(z)&={-}\arctan\left(\frac{P_c}{P_0}\tan z_s\right),\end{aligned}$$

denotes the phase-front curvature of the beam and the phase of the complex amplitude, respectively; $r=|\textbf {r}|$ is the radial coordinate; $P_0=\int _0^{2\pi }\int _0^{\infty }|\Phi _{nm}|^2rdr\,d\varphi$ is the input power; $C_{nm}=\{n!P_0/[\pi (n+m)!]\}^{1/2}$ is the normalized coefficient, it conserves the total power of the propagating beam; $P_c=1/(k^2\gamma ^2w_0^4)$ is the critical power for the soliton propagation; $z_s=z/z_p$ is the propagation distance scaled by $z_p=(\gamma ^2P_{0})^{-1/2}$. The stability of the solution with arbitrary perturbation has been tested [20].

The propagation expression form of the even and odd LG beams with radial number $n$ and azimuthal number $m$ can be written as

(11)$$\begin{aligned} \Phi_{nm}^{e,o}&=\frac{C_{nm}^{e,o}}{w(z)}\left[\frac{r}{w(z)}\right]^mL_n^m\left[\frac{r^2}{w^2(z)}\right]\exp{\bigg\{}-\frac{r^2}{2w^2(z)}{\bigg.}\\ &\quad {\bigg.}+i\left[c(z)r^2+(2n+m+1)\theta(z)\right]{\bigg\}} {\bigg[}\begin{array}{l} \cos(m\varphi) \\ \sin(m\varphi) \end{array}{\bigg]}, \end{aligned}$$

where the superscripts $e$ and $o$ refer to even and odd parity, respectively; $C_{nm}^{e,o}=\{2n!P_0/[(1+\delta _{0m})\pi (n+m)!]\}^{1/2}$ is the normalized coefficient which can be obtained by $\int _0^{2\pi }\int _0^{\infty }|\Phi _{nm}^{e,o}|^2rdr\,d\varphi =P_0$; $\delta _{0m}=1$ for $m=0$; $\delta _{0m}=0$ for $m\neq 0$. Both Eqs. (6) and (11) can be used to describe the propagation of LG beam in SNNM, and we only consider its soliton propagation form in this paper, i.e., $P_0=P_c$.

Provided that the center of mass ${\textbf {r}}_c$ of a single soliton traveling in SNNM satisfies the ray equation [1]

(12)$$\frac{d^2\textbf{r}_c(z)}{dz^2}+\frac{\textbf{r}_c(z)}{z_p^2}=0.$$

One can get the solution of Eq. (12) as

(13)$$\textbf{r}_c(z)=\textbf{r}_c(0)\cos z_s+z_p\textbf{r}_c'(0)\sin z_s.$$

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 is

(14)$$\Phi_{{\pm}}({\textbf{r}},z)=\Phi\left({\textbf{r}}\pm{\textbf{r}}_c(z),z\right)\exp\left[{\mp} i{{\boldsymbol{u}}}(z){\textbf{r}}+i\phi(z)\right],$$

where ${{\boldsymbol{u}}}(z)$ and $\phi (z)$ are determined by the following equations

(15)$$\begin{aligned} {{\boldsymbol{u}}}(z)&=k{\textbf{r}}_c^{'}(z),\end{aligned}$$

(16)$$\begin{aligned} \phi^{'}(z)&=\frac{k}{2}\left[\frac{{\textbf{r}}_c^2(z)}{z_p^2}-{\textbf{r}}_c^{'2}(z)\right].\end{aligned}$$

Obviously, the second term on the right side of Eq. (14) represents the corresponding phase shift after the center of mass of the soliton moves.

The combined field of the LG soliton array can be constructed by using the superposition principle

(17)$$\Phi_{nm}(\textbf{r},z)=C_0\sum_{j=1}^{N}\Phi_{nm}^{(j)}(\textbf{r},z),$$

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 obtain

(18)$$\begin{aligned} \Phi_{nm}^{(j)}(\textbf{r},z)&=\Phi_{nm}^{(j)}\left[{\textbf{r}}-{\textbf{r}}_c(z),z\right]\exp\left\{ik\left(-\frac{\textbf{r}_c(0)}{z_p}\sin{z_s}+\textbf{r}'_c(0)\cos{z_s}\right)\cdot{\textbf{r}}\right.\\ &\quad \left.+i\left[\frac{k}{4}\left(\frac{\textbf{r}_c^2(0)}{z_p}-z_p\textbf{r}_c^{'2}(0)\right)\sin(2z_s)-\frac{k}{2}\textbf{r}_c(0)\cdot\textbf{r}_c'(0)\cos(2z_s)\right]\right\},\end{aligned}$$

where ${\textbf {r}}_c=(x_c, y_c)$ and ${\textbf {r}}'_c=(x_c', y_c')$, $\Phi _{nm}^{(j)}\left [{\textbf {r}}-{\textbf {r}}_c(z),z\right ]$ can be substituted by Eqs. (6) or (11). Whereas two-dimensional transverse coordinate vector ${\textbf {r}}=x{\textbf {e}}_x+y{\textbf {e}}_y$ in the Cartesian coordinates system has the transformation relationship ${\textbf {r}}=(x,y)=(r\cos \varphi ,r\sin \varphi )=(r,\varphi )$ in the cylindrical coordinates system, Eq. (18) can be transformed to the Cartesian coordinates system.

Considering the independence of variables in $x$ and $y$ directions and combining the Eqs. (13), (15) and (16), then we obtain

(19)$$\begin{aligned} x_{cj}(z)&=c_{xj}\cos z_s+t_{xj}z_{p}\sin {z_s},\end{aligned}$$

(20)$$\begin{aligned} y_{cj}(z)&=c_{yj}\cos z_s+t_{yj}z_{p}\sin z_s,\end{aligned}$$

which jointly govern the trajectory of the center of mass of each constituent LG soliton, and

(21)$$\begin{aligned} u_{xj}(z)&={-}\frac{kc_{xj}}{z_{p}}\sin{z_s}+kt_{xj}\cos{z_s},\end{aligned}$$

(22)$$\begin{aligned} u_{yj}(z)&={-}\frac{kc_{yj}}{z_{p}}\sin{z_s}+kt_{yj}\cos{z_s},\end{aligned}$$

(23)$$\phi_j(z)=\frac{k}{4}\left[\frac{c_{xj}^{2}+c_{yj}^{2}}{z_{p}}-z_{p}(t_{xj}^2+t_{yj}^2)\right] \sin(2z_s)-\frac{k}{2}\left(c_{xj}t_{xj}+c_{yj}t_{yj}\right)\cos(2z_s).$$

In all of the above equations, $c_{xj}=x_{cj}(z)|_{z=0}$ and $c_{yj}=y_{cj}(z)|_{z=0}$ are the initial position of the center of mass of each soliton, named as “initial displacement,” indicating the distance between the center of mass of each LG soliton and the origin of coordinates; $t_{xj}=x_{cj}'(z)|_{z=0}$ and $t_{yj}=y_{cj}'(z)|_{z=0}$ are the slope of the projection trajectory of each soliton center on the $x$-$z$ plane and $y$-$z$ plane, named as “initial tangential velocity,” indicating the angle between the incident direction and the propagation axis. This is a term that we borrow the concept from classical physics, that is, the propagation distance $z$ is regarded as time $t$. The term of initial tangential velocity brings us great convenience to understand the soliton propagation. Of course, we can also call it initial incident slope directly. Then the angle between the projection of the incident direction vector on the $x$-$z$ (or $y$-$z$) plane and the propagation axis can be expressed as $\Lambda _{xj}=\arctan (t_{xj})$ [or $\Lambda _{yj}=\arctan (t_{yj})$]. We can define the angle $\Lambda$ as initial tilted angle and in the case of strongly nonlocality $\Lambda$ can approach $90$ degrees in theory [47].

Consider 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

(24)$$c_{xj}=r\cos\varphi_{0j},\ \ c_{yj}=r\sin\varphi_{0j},$$

and the initial tangential velocity of each LG soliton can be expressed as

(25)$$t_{xj}={-}\frac{\xi{r}\sin\varphi_{0j}}{z_p},\ \ t_{yj}=\frac{\xi{r}\cos\varphi_{0j}}{z_p},$$

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

(26)$$\Phi_{A,B,C}=\Phi_{nm}^{(j=1,2,3)}(x,y,z;r),$$

where the subscript “$A, B, C$” correspond to $j=1, 2, 3$ respectively, $r$ is the length of initial displacement. Correspondingly,

(27)$$\Phi_{A',B',C'}=\Phi_{nm}^{(j=1,2,3)}\left({-}x,-y,z;\frac{r}{2}\right),$$

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.

Fig. 1. (a)–(e) The evolution of intensity pattern of LG soliton array with tangential velocity parameter $\xi =0$. (f) and (g) The normalized intensity distribution in the transverse plane and $x$-direction, respectively, at $z=0.5T$. The total number of the constituent solitons is $N_t=6$. Parameters: $P_0=P_c$, $r=9w_0$, $n=1$, $m=2$.

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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

(28)$$S_{\triangle{ABC}}(z)=\frac{3\sqrt{3}r^2}{4}\left(\cos^2{z_s}+\xi^2\sin^2{z_s}\right).$$

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)

(29)$$S_{min(max)}=\frac{3\sqrt{3}\xi^2r^2}{4}$$

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.

Fig. 2. The evolution of intensity pattern of LG soliton array with different tangential velocity parameters. $\xi =0.5$ for (a)–(e), $\xi =1$ for (f)–(j), $\xi =2$ for (k)–(o). The beginning of each arrow points to the position of soliton $A$, the end of each arrow indicates the direction of rotation. Common parameters: $P_0=P_c$, $N_t=6$, $r=9w_0$, $n=1$, $m=2$.

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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.

Fig. 3. $C$ (solid line) and $C'$ (dashed line) are taken as examples to illustrate the dynamic changes in the propagation process of multiple solitons. (a) Three-dimensional propagation trajectories of $C$ and $C'$. (b) Projection trajectories of $C$ and $C'$ in the $x$-$z$ plane. (c) Projection trajectories of $C$ and $C'$ in the $y$-$z$ plane. (d) Projection trajectories of $C$ and $C'$ in the $x$-$y$ plane. Parameters: $P_0=P_c$, $\xi =0.5$, $r=9w_0$.

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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

(30)$$x_{cj}(z)=\frac{r}{2}\left[(1-\xi)\cos(\varphi_{0j}-z_s)+(1+\xi)\cos(\varphi_{0j}+z_s)\right]$$

and

(31)$$y_{cj}(z)=\frac{r}{2}\left[(1-\xi)\sin(\varphi_{0j}-z_s)+(1+\xi)\sin(\varphi_{0j}+z_s)\right]$$

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 as

(32)$$(\xi^2\cos^2\varphi_{0j}+\sin^2\varphi_{0j})x^2+(\xi^2\sin^2\varphi_{0j}+\cos^2\varphi_{0j})y^2+[(\xi^2-1)\sin(2\varphi_{0j})]xy=\xi^2r^2.$$

In the spatial case of $\xi =1$, the projection trajectory of each constituent soliton in the $x$-$y$ plane is a circle. Their equations can be written uniformly as

(33) $$x^2+y^2=r^2$$

i.e., the projection trajectories of constituent solitons with equal initial transverse displacements in the $x$-$y$ plane share the same circle, as shown in Fig. 4(f). These trajectory equations show that the propagation path of each soliton is controlled by initial tangential velocity and displacement. For an actual medium with fixed length $L$, the coordinate of each soliton at the exit plane can be separately written as $\left (x_{cj}(L),y_{cj}(L)\right )$. In reverse, the length of the medium can be determined according to the output position of the soliton and the initial incident conditions. In this regard, the model of soliton array have potential applications in optical communication and particle control.

Fig. 4. (a)–(c) The projection trajectories of the rotating LG solitons with velocity parameter $\xi =0.5$. (a) $x$-$z$ plane, (b) $y$-$z$ plane, (c) $x$-$y$ plan. Solid red, green, blue lines represent $A, B, C$, respectively. Dashed red, green, blue lines represent $A', B', C'$, respectively. (d)–(f) and (g)–(i) are the same as (a)–(c) except that the velocity parameter are taken as $\xi =1$ and $\xi =2$, respectively. Parameters: $P_0=P_c$, $r=9w_0$.

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4 Evolution of the rotating states

During propagation, the angle between each constituent soliton and the $x$ axis in transverse plane can be described as

(34)$$\Omega_j(z)=\arctan\left(\frac{\sin\varphi_{0j}\cos{z_s}+\xi\cos\varphi_{0j}\sin{z_s}} {\cos\varphi_{0j}\cos{z_s}-\xi\sin\varphi_{0j}\sin{z_s}}\right)$$

based on Eqs. (19) and (20). $\Omega _j(z)$ is defined as the rotating angle of each constituent LG soliton with initial azimuth angle $\varphi _{0j}$ taken into consideration. Figures 5(a)–5(c) show the continuous change of rotating angle in one complete evolution period. It can be seen that the rotating angle increases every $2\pi$ radians accompanying the soliton array travels $2\pi {z_p}$ distance along the $z$ axis. By borrowing the concept of angular velocity from classical physics, we can obtain the angular velocity of each constituent LG soliton

(35)$$\omega_j(z)=\frac{\xi}{z_p(\cos^2z_s+\xi^2\sin^2z_s)}.$$

The distance of each soliton center to the propagation axis (i.e., the center of mass of the combined filed) is

(36)$$d_{A,B,C}(z)=r\left(\cos^2{z_s}+\xi^2\sin^2{z_s}\right)^{1/2},$$

defined as center distance. Correspondingly, $d_{A',B',C'}=d_{A,B,C}/2$. Obviously, the angular velocity and the center distance kept constant when and only when $\xi =1$ (we call it the critical velocity parameter). Otherwise, the angular velocity and the center distance will vary periodically with the same period $\Delta {z}=\pi {z_p}$. In each period, the smaller (larger) the center distance is, the faster (slower) the angular velocity will be [as shown in Figs. 5(d) and 5(f)]. We can get that the angular velocity is inversely proportional to the square of the center distance based on Eqs. (35) and (36).

Fig. 5. (a)–(c) Evolution of the rotating angle versus propagation distance without taking initial azimuth angle into account. (d)–(f) Evolution of the angular velocity $\omega (z)$ (green dashed line, left ordinate) and the center distance $d(z)$ (blue dash-dotted line, right ordinate) for different tangential velocity parameters.

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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.

Fig. 6. The examples of rotating LG soliton arrays with different negative tangential velocity parameters. (a) $\xi =-0.5$, (b) $\xi =-1$, (c) $\xi =-2$. Surrounded by the dashed circles represent the same soliton at different propagation positions in each row, the arrows indicate the direction of rotation. Parameters: $P_0=P_c$, $N_t=9$, $n=0$, $m=2$.

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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

(37)$$S(z)=\frac{3\sqrt{3}r^2}{2}\left(\cos^2{z_s}+\xi^2\sin^2{z_s}\right).$$

Fig. 7. The examples of rotating LG soliton arrays with the same tangential velocity parameter $\xi =1$. Surrounded by the dashed circles represent the same soliton at different propagation positions in each row, the arrows indicate the direction of rotation. (a) Ring-like soliton array with $n=2$, $m=1$, $N_t=10$; (b) rhombus-like soliton array with $n=4$, $m=2$, $N_t=8$; (c) pentagram-like soliton array with $n=2$, $m=2$, $N_t=10$. The input power is taken as $P_0=P_c$.

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Fig. 8. The examples of rotating LG soliton arrays with different tangential velocity parameters. $\xi =0.7$ for (a), $\xi =1$ for (b), $\xi =1.5$ for (c). Parameters: $P_0=P_c$, $N=6$, $r=12w_0$.

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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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Dynamics of rotating Laguerre-Gaussian soliton arrays

Abstract

1. Introduction

2. Theoretical model

3. Propagation dynamics

1 Evolution of non-rotation case

2 Evolution of the rotating pattern

3 Evolution of the propagation trajectories

4 Evolution of the rotating states

5 Rotating arrays of other shapes

4. Conclusion

Funding

References

Cited By

Figures (8)

Equations (37)

Optics Express