Controllable Gaussian-shaped soliton clusters in strongly nonlocal media

Limin Song; Zhenjun Yang; Xingliang Li; Shumin Zhang

doi:10.1364/OE.26.019182

1. Introduction

Spatial solitons, usually be treated as “atoms of light”, have attracted extensive interest and been widely investigated in past decades (e.g., [1–11], and references therein), both in local and nonlocal media, because of the robust nature that they display in interactions and the potential use in optical communication, information storage and processing. While the theory of spatial optical soliton was proposed as early as 1960s [2], the (1+1)-dimensional (2D) solitons were first observed until 1985 [3], the (2+1)-dimensional (3D) solitons were observed until 1993 in photorefractive media [4]. The universality and diversity of the solitons’ interactions are summarized by Stegeman et al. [5]. In 2002, Desyatnikov et al. [6] first introduced the concept of soliton clusters—multisoliton bound states in a homogeneous bulk optical medium—and revealed a key physical mechanism for their stabilization associated with a staircase-like phase distribution. Up to now, various types of soliton clusters have been found in both 2D and 3D physical settings [12–15].

Here, we briefly recall the history of soliton interactions. In 1997, Snyder and Mitchell [1] simplified the nonlinear Schrödinger equation (NLSE) to investigate the soliton problem in strongly nonlocal nonlinear (SNN) media and obtained an elegant description of soliton collisions, interactions, and deformations by borrowing the idea of the linear harmonic oscillator. In 2006, Rotschild et al. [7] studied the long-range interactions of nonlocal solitons in lead glass and found that there is still obvious interaction when the distance of two solitons is more than 10 times of the soliton width. In 2007, Shigen Ouyang et al. [16] theoretically shown that one can steer lights in SNN media by tuning the incident conditions of coherently interacting beams like the phase difference between beams and their relative amplitude. In 2008, Yuqing Wang et al. [17] studied the dynamics for the ring-like rotating soliton clusters by using the technique of variational approach and found that there exists a critical value of the ring radius for the stationary rotation. In 2009, Daquan Lu et al. [18] studied the multibeam interactions in SNN media by the modified Snyder-Mitchell model and shown that the evolution of each interacting beam can be regarded as the cross-induced fractional Fourier transform in the comoving reference frame; Longgui Cao et al. [19] studied the long-range interactions of solitons in nematic liquid crystals (NLCs) both theoretically and experimentally. It is found that the interaction should still be seen when the spacing between interacting solitons is 43 times of the initial beam width, which is far greater than that in the lead glass. In 2015, Wei Hu group [20] showed that the interactions of two spatial solitons in NLCs are dependent on phase and bias voltage by numerical simulation and experimental tests. In 2017, Siliu Xu et al. [21] obtained stable 3D light bullet pairs by using the self-similarity method. Of course, there are many articles about the interaction of spatial solitons, which are not listed here. To sum up, the factors such as the medium, the input power, the phase, the soliton spacing, the degree of the nonlocality, the bias voltage all have important influence on the interaction of solitons.

Considering so many factors have effects on the interaction of solitons, it is possible to construct a model that allows the motion of solitons to be manipulated. Therefore, we want to control the behaviors of multisoliton by changing the initial incident parameters, such as the initial transverse velocity, the initial position, the input power and so on. The rest of this paper is organized as follows. In Sec. 2, the governing equation of Gaussian-shaped soliton clusters is given. In Sec. 3, the evolution of the intensity pattern, the trajectory, the center distance and the angular velocity of the soliton clusters in SNN media are discussed in detail, and a series of analytical solutions of interacting Gaussian solitons are given to describe these characteristics. In Sec. 4, we summarized the work of this paper and pointed out its significance.

2. Model

The propagation of the optical field in nonlinear media is modeled by the well-known NLSE [1,22–24]

2 i k \frac{\partial Φ}{\partial z} + Δ_{⊥} Φ + 2 k^{2} \frac{Δ n}{n_{0}} Φ = 0,

where Φ is the complex amplitude envelope of the input field; k = 2πn₀/λ, with λ the wavelength of light in vacuum, is the wave number in the media without nonlinearity; z is the longitudinal (propagation direction) coordinate; Δ_⊥ represents two-dimensional transverse Laplacian operator. n₀ is the linear part of the refractive index of the media,

Δ n = n_{2} \iint R (| r - r_{c} |) {| Φ (r_{c}, z) |}^{2} d^{2} r_{c}

is the nonlinear perturbation of the refractive index, n₂ is the nonlinear refractive index coefficient. r = (x, y) and r_c = (x_c, y_c) are the transverse coordinate vectors. R(r) is the normalized symmetric real spatial response function of the nonlocal nonlinear media such that ∬ R(r)d²r = 1. Note that not stated otherwise all integrals in this paper will extend over the whole x–y plane. If R(r) ≡ 0, i.e., the last term on the left of Eq. (1) is ignored, then Eq. (1) reduces to the paraxial wave equation in free space. If R(r) is a delta function, Eq. (1) reduces to the NLSE in local nonlinear media. In previous investigations, the Gaussian function

R (r) = \frac{1}{2 π w_{R}^{2}} exp (- \frac{r^{2}}{2 w_{R}^{2}})

is usually taken as the nonlocal response function [17, 23–27], where w_R is the characteristic width of the response function. If the characteristic scale of the response function, i.e., the characteristic length of the material, is much larger than that of the optical field, this medium can be considered as a strongly nonlocal medium. Then the response function R can be expanded in Taylor’s series. If we only expand the response R(r − r_c) with respect to r_c about r_c = 0 to the second-order term, then Eq. (1) reduces to [1,23]

2 i k \frac{\partial Φ}{\partial z} + Δ_{⊥} Φ - k^{2} γ^{2} P_{0} r^{2} Φ + \frac{2 k^{2} n_{2} R_{0} P_{0}}{n_{0}} Φ = 0,

where

γ = \sqrt{- n_{2} R_{0}^{″} / n_{0}}

is a material constant associated with the nonlocal effect of the medium and γ² > 0 means that the nonlinear medium is a self-focusing medium [ R₀ = R(0), R″₀ = R″(0), R″₀ < 0 because R₀ is a maximum of R(r)]. P = ∬|Φ(r, z)|²d²r is the beam power and P₀ is the input power at z = 0. We could replace P with P₀ in Eq. (4) follows the fact that the beam power P is conserved. By introducing the variable transformation

Φ (r, z) = ψ (r, z) exp (i k \frac{n_{2} R_{0} P_{0}}{n_{0}} z),

Eq. (4) can be simplified as the famous SNN model [1,18]. In the rectangular coordinate system, it can be expressed as

2 i k \frac{\partial ψ}{\partial z} + (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) ψ - k^{2} γ^{2} P_{0} (x^{2} + y^{2}) ψ = 0 .

Equation (6) has a single-beam exact solution with Gaussian function form [23]

ψ (x, y, z) = \frac{\sqrt{P_{0}}}{\sqrt{π} w (z)} exp [- \frac{x^{2} + y^{2}}{2 w^{2} (z)} + i c (z) (x^{2} + y^{2}) + i θ (z)],

where

w (z) = w_{0} {({cos}^{2} α + {sin}^{2} α / η)}^{1 / 2},

c (z) = \frac{k β (1 - η) sin (2 α)}{4 (η {cos}^{2} α + {sin}^{2} α)},

θ (z) = - arctan (tan α / \sqrt{η}),

denotes the beam width, the phase-front curvature of the beam and the phase of the complex amplitude, respectively; w₀ = w(z)|_z=0 is the initial beam width; α = βz,

β = \sqrt{γ^{2} P_{0}}

is propagation constant;

P_{c} = 1 / (k^{2} γ^{2} w_{0}^{4})

is the critical power. Here we define η = P₀/P_c as the power ratio. When η = 1, the nonlinear compression exactly balances diffraction broadening, then the Gaussian beam preserves its width and presents as a soliton during propagation, which is “accessible soliton” named by Snyder and Mitchell; when η ≠ 1, the width of Gaussian beam will fluctuate periodically, which presents as a breather.

If ψ(r, z) is a solution of Eq. (6), then

ψ_{\pm} (r, z) = ψ (r \pm r_{0} (z), z) exp [\mp i u (z) r + i ϕ (z)]

are also solutions of Eq. (6), where r₀(z) satisfies the following equation [1],

r_{0}^{″} (z) + β^{2} r_{0} (z) = 0,

u(z) and ϕ(z) can be solved by

u (z) = k r_{0}^{'} (z),

ϕ^{'} (z) = \frac{k}{2} [β^{2} r_{0}^{2} (z) - r_{0}^{' 2} (z)] .

In Eqs. (11)–(14), r and r₀(z) are two-dimensional transverse coordinate vectors. r₀(z) is the position of the beam center on the cross section. r′₀(z) and r″₀(z) are the first and second derivatives of r₀(z) for z, ϕ′(z) is the first derivative of ϕ(z) for z. Equation (12) is the second order ordinary differential equation. If one solves Eq. (12), two initial conditions are needed, i.e., the initial position r₀(0) and initial transverse velocity r′₀(0). Hence r₀(z) satisfies the following expression

r_{0} (z) = r_{0} (0) cos α + \frac{r_{0}^{'} (0)}{β} sin α .

Equation (6) is a linear differential equation, therefore, the analytical solution of the soliton clusters can be constructed by using the principle of linear superposition. We consider a coherent superposition of N Gaussian solitons, then the soliton clusters can be expressed as

ψ (x, y, z) = G_{0} \sum_{n = 1}^{N} ψ_{n} (x, y, z) (n = 1, 2, \dots, N),

G₀ is the normalized coefficient associated with the input power. In Eq. (16), ψ_n(x, y, z) is the light field of the n-th beam and it is also the solution of the Eq. (6), being of the form

\begin{array}{l} ψ_{n} (x, y, z) & = & \frac{\sqrt{P_{0}}}{\sqrt{π} w (z)} exp [- \frac{{(x - x_{0 n})}^{2} + {(y - y_{0 n})}^{2}}{2 w^{2} (z)} + i θ (z)] \\ \times exp {i c (z) [{(x - x_{0 n})}^{2} + {(y - y_{0 n})}^{2}] + i (u_{x n} \cdot x + u_{y n} \cdot y) + i ϕ_{n}}, \end{array}

where w(z), c(z), and θ(z) are the same as those in Eqs. (8)–(10), and

x_{0 n} (z) = c_{x n} cos α + \frac{v_{x n}}{β} sin α,

y_{0 n} (z) = c_{y n} cos α + \frac{v_{y n}}{β} sin α,

u_{x n} (z) = - c_{x n} k β sin α + k v_{x n} cos α,

u_{y n} (z) = - c_{y n} k β sin α + k v_{y n} cos α,

\begin{array}{l} ϕ_{n} (z) & = & \frac{k}{4} [β (c_{x n}^{2} + c_{y n}^{2}) - \frac{v_{x n}^{2} + v_{y n}^{2}}{β}] sin (2 α) \\ - \frac{k}{2} [c_{x n} v_{y n} + c_{y n} v_{x n}] cos (2 α) . \end{array}

In all of the above, c_xn = x_0n(z)|_z=0 and c_yn = y_0n(z)|_z=0 are the initial transverse coordinates (initial position) of the center of mass of each soliton. v_xn = x′_0n(z)|_z=0 and v_yn = y′_0n(z)|_z=0 are the values of the transverse coordinates of each soliton center on the first derivative of z at z = 0, i.e., the slope of the projection trajectory of each soliton center on the x–z plane and y–z plane with respect to the z axis at initial incident position, which we call the initial transverse velocity and it can be used to represent the incident direction of each soliton [28]. The wave vector of each soliton at z = 0 can be expressed as

\begin{array}{l} k_{n} & = & k_{n x} e_{x} + k_{n y} e_{y} + k_{n z} e_{z} \\ = & k v_{x n} e_{x} + k v_{y n} e_{y} + k e_{z}, \end{array}

where e_x, e_y and e_z are the unit vectors in x, y and z directions, respectively. When v_xn = 0 (v_yn = 0), it means that the incident direction is perpendicular to the x axis (the y axis). When v_xn = v_yn = 0, the incident direction is perpendicular to the x–y plane. When v_xn and v_yn are not zero, it is the general case.

In this paper, we assume that N off-axis Gaussian solitons are uniformly distributed on a ring with the radius being r, i.e., the distance from the center of each soliton and the origin (0,0) of the coordinate. Considering the case of symmetrical incidence, then the initial position of each soliton center can be valued as

c_{x n} = r cos φ_{n}, c_{y n} = r sin φ_{n},

and the initial transverse velocity of each soliton can be expressed as

v_{x n} = - ξ β r sin φ_{n}, v_{y n} = ξ β r cos φ_{n},

where φ_n = 2nπ/N. We name the coefficient ξ as the “initial transverse velocity parameter”, simplified as “velocity parameter”. When ξ = 0, each soliton has a parallel and symmetrical incidence (the initial incident direction of each soliton is perpendicular to the x–y plane); when ξ ≠ 0, each soliton has an oblique symmetrical incidence. Specifically speaking, the wave vector of each soliton is oblique symmetry.

3. Propagation characteristics

3.1. Analytical expression

Here, we adopt N = 6 as an example to illustrate the dynamics of the soliton clusters and the input power of each soliton is the same. The initial optical field of each soliton at the entrance plane z = 0 can be written as

\begin{array}{l} ψ_{n} (x, y, 0) & = & \frac{\sqrt{P_{0}}}{\sqrt{π} w_{0}} exp [- \frac{{(x - r cos φ_{n})}^{2} + {(y - r sin φ_{n})}^{2}}{2 w_{0}^{2}}] \\ \times exp [- i k ξ β r (sin φ_{n} \cdot x - cos φ_{n} \cdot y)] (n = 1, 2, \dots, 6) . \end{array}

Then, the total initial amplitude of the soliton clusters is superimposed by above six amplitudes,

ψ (x, y, 0) = G_{0} \sum_{n = 1}^{6} ψ_{n} (x, y, 0) .

Based on the definition of the input power, one can obtain the normalized coefficient G₀,

\begin{array}{l} G_{0} & = & {6 exp [(- k^{2} ξ^{2} β^{2}) / w_{0}^{2}] [1 + 2 exp (- 3 r^{2} (w_{0}^{4} - k^{2} ξ^{2} β^{2}) / 4 w_{0}^{2}) \\ {+ 2 exp (- r^{2} (w_{0}^{4} - k^{2} ξ^{2} β^{2}) / 4 w_{0}^{2})] / w_{0}^{4}}}^{- 1 / 2} . \end{array}

If the power ratio η = 1, i.e., the soliton state is considered, w(z) becomes a constant independent of z, c(z) ≡ 0, and θ(z) = −α. The propagation expression of Gaussian-shaped soliton clusters in SNN media can be obtained as

\begin{array}{l} ψ (x, y, z) & = & G_{0} \frac{\sqrt{P_{0}} exp (- i α)}{\sqrt{π} w_{0}} \\ \times {exp [- \frac{{[x - r (cos α - \sqrt{3} ξ sin α) / 2]}^{2} + {[y - r (\sqrt{3} cos α - ξ sin α) / 2]}^{2}}{2 w_{0}^{2}} \\ + \frac{i k β r}{2} [(- sin α - \sqrt{3} ξ cos α) \cdot x + (- \sqrt{3} sin α + ξ cos α) \cdot y] \\ - \frac{i k β r^{2} (ξ^{2} - 1)}{4} sin (2 α)] \\ + exp [- \frac{{[x - r (cos α - \sqrt{3} ξ sin α) / 2]}^{2} + {[y - r (\sqrt{3} cos α + ξ sin α) / 2]}^{2}}{2 w_{0}^{2}} \\ + \frac{i k β r}{2} [(sin α - \sqrt{3} ξ cos α) \cdot x + (- \sqrt{3} sin α - ξ cos α) \cdot y] \\ - \frac{i k β r^{2} (ξ^{2} - 1)}{4} sin (2 α)] \\ + exp [- \frac{{(x + r cos α)}^{2} + {(y + ξ r sin α)}^{2}}{2 w_{0}^{2}} + i k β r (sin α \cdot x - ξ cos α \cdot y) \\ - \frac{i k β r^{2} (ξ^{2} - 1)}{4} sin (2 α)] \\ + exp [- \frac{{[x - r (- cos α + \sqrt{3} ξ sin α) / 2]}^{2} + {[y - r (- \sqrt{3} cos α - ξ sin α) / 2]}^{2}}{2 w_{0}^{2}} \\ + \frac{i k β r}{2} [(sin α + \sqrt{3} ξ cos α) \cdot x + (\sqrt{3} sin α - ξ cos α) \cdot y] \\ - \frac{i k β r^{2} (ξ^{2} - 1)}{4} sin (2 α)] \\ + exp [- \frac{{[x - r (cos α + \sqrt{3} ξ sin α) / 2]}^{2} + {[y - r (- \sqrt{3} cos α + ξ sin α) / 2]}^{2}}{2 w_{0}^{2}} \\ + \frac{i k β r}{2} [(- sin α + \sqrt{3} ξ cos α) \cdot x + (\sqrt{3} sin α + ξ cos α) \cdot y] \\ - \frac{i k β r^{2} (ξ^{2} - 1)}{4} sin (2 α)] \\ + exp [- \frac{{(x - r cos α)}^{2} + {(y - r sin α)}^{2}}{2 w_{0}^{2}} - i k β r (sin α \cdot x - ξ cos α \cdot y) \\ - \frac{i k β r^{2} (ξ^{2} - 1)}{4} sin (2 α)]} . \end{array}

If η ≠ 1, the total input power is unequal to the critical power, the propagation expression can be obtained based on Eqs. (16)–(25), which is not given here because it is too long.

3.2. Intensity pattern

Firstly, we discuss the evolution of intensity patterns of Gaussian-shaped soliton clusters for different P₀ and ξ. Then, we give the physical reasons for the evolution of the clusters.

In order to better describe the periodicity of its evolution, we introduce a new parameter z_p,

z_{p} = \frac{1}{β} = \frac{1}{\sqrt{γ^{2} P_{0}}} .

In this section, the propagation distance is scaled by z_p. As can be seen from Eq. (30), z_p is a parameter related to the properties of materials and the input power. That is to say, the actual propagation distance represented by the normalized propagation distance z_p at different input powers is unequal.

For the case of ξ = 0, Fig. 1 shows the evolution of transverse intensity patterns. It can be seen from Fig. 1 that the evolution period of the intensity is Δz_i = πz_p, which can be verified by Eqs. (17) and (29). At the beginning (z = 0), the spacing between solitons is the largest and their optical fields almost do not overlap. The intensity distribution of each soliton keeps its own shape. But, due to the strongly nonlocal effect, when multisoliton are transmitted together, the solitons will attract each other and the energy will gradually converge toward the central region of the combined field. The maximum-intensity of the center appears at z = πz_p/2 [see Figs. 1(a4), 1(b4), and 1(c4)], resulting from constructively interfere. Then with the propagation distance increasing, the central light intensity becomes weaker and weaker, and the intensity distribution of initial position appears again until z = πz_p. At this time, the intensity pattern completes an evolution period, and the second half period (from z = πz_p/2 to z = πz_p) is an inverse process of the first half period (from z = 0 to z = πz_p/2). This is similar to the evolution of interacting vortex solitons in SNN media [29]. In the overlap region of the optical fields of the clusters, some interference fringes appear. During propagation, each constituent soliton has relative movement to both the x and y directions, so the interference pattern presents as a flower, which has sloping fringes [see Figs. 1(a3), 1(b3), 1(c3) and their counterparts].

Fig. 1 Transverse intensity patterns of the Gaussian-shaped soliton clusters in SNN media at different propagation positions shown at the top. Each row is a period for different input powers, and (a): η = 2, (b): η = 1, (c): η = 1/2. Parameters: N = 6, ξ = 0, r = 6w₀.

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For the case of 0 < ξ < 1, we take ξ = 1/3 as an example, the evolution of transverse intensity patterns is shown in Fig. 2. Under this assumption the intensity pattern of the clusters spiral inward and outward periodically, and with the same period as the case ξ = 0. Similarly, the second half period (the intensity pattern appears to rotate clockwise [see Figs. 2(b5) and 2(c5)]) is an inverse process (the intensity pattern appears to rotate counterclockwise [see Figs. 2(b3) and 2(c3)]) of the first half period. It is important to note that in the actual propagation process, the soliton clusters is always going counterclockwise (the viewing direction is from the positive direction of the z axis to its negative direction). The interacting spiral solitons possess a transverse energy flow during propagation. And we can see that with the decrease of the power ratio η (the input power P₀), the central light intensity of the combined field increases obviously at z = πz_p/2 [see Figs. 2(a4), 2(b4) and 2(c4)].

Fig. 2 The same as Fig. 1 except that the velocity parameter ξ = 1/3.

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For the case of ξ = 1, as shown in Fig. 3 that the combined field keeps its original shape (regular hexagon) unchanged as it travels along the z axis. In the cases of η = 2 and η = 1/2, the width of each constituent Gaussian beam varies periodically during propagation [see Figs. 3(a) and 3(c)]. This variation is the result of dynamic balance between the diffraction effect and the nonlinear self-focusing effect. According to Eq. (8), when η > 1, the nonlinear effect of the material will overcome the diffraction effect of the beam at the initial time, thus the beam is compressed and the ratio $w^{2} (z) / w_{0}^{2}$ vibrates between peak value η and valley value 1. Instead, when η < 1, the beam is initially broadened and the ratio $w^{2} (z) / w_{0}^{2}$ vibrates between peak value 1 and valley value η. Therefore, the size of the optical field seems to change slightly for different input powers. However, when ξ = 1 and η = 1, the size of the combined field doesn’t change at all [see Fig. 3(b)]. Strictly speaking, it can be called rotational soliton clusters in this particular case and the evolution period is one third of that for the case η ≠ 1 (Note that each constituent soliton does not overlap with its original position, but only because of the symmetry of the initial distribution of the six identical solitons, when they turn around at an angle of π/3, the intensity distribution of the optical field appears to be the same as the initial distribution). In addition, the soliton clusters rotate an angle of π/6 radians counterclockwise as each of the soliton travels a distance of πz_p/6, that is, the solitons are rotating at angular velocity 1/z_p = β.

Fig. 3 The same as Fig. 1 except that the velocity parameter ξ = 1.

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For the case of ξ > 1, Fig. 4 shows that the Gaussian-shaped soliton clusters keeps its original shape invariant during propagation. But its size, and the spot size of each constituent beam, breathe with a period Δz_i = πz_p. Combined with the analysis in Figs. 1–3 and Eq. (30), we get that the input power plays a key role in the evolution period, and the larger the input power is, the smaller the evolution period will be. It is inversely proportional to the square root of the input power P₀. The whole shape of the combined field undergoes the process of enlarging and shrinking as it propagates down the z axis. It is getting bigger first and then smaller, which is completely different from the case in Fig. 2 or Fig. 9(a). And with the decreasing of the input power, the width of each constituent beam seems to be expanded at the same propagation distance of each period [see Figs. 4(a3), 4(b3) and 4(c3)]. The variation of the beam width is the same as that in Fig. 3.

Fig. 4 The same as Fig. 1 except that the velocity parameter ξ = 3/2.

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The evolution of the beam is essentially determined by the refractive index distribution of the material. Therefore, the above phenomenon can be explained qualitatively by the nonlinear variation of refractive index induced by the light field itself. In nonlocal nonlinear media, it allows the variation of the refractive index at a particular point to be related to the light field in a certain scale (the characteristic scale or relevant scale of the material) [22], which is distinctly different from the conventional local nonlinear media. The greater the nonlocality, the larger the characteristic scale of the material, namely, the larger the range of the light field that affects the refractive index of that point. Under the condition of strongly nonlocality, the characteristic scale of the response function of the media is much larger than the space occupied by the all interacting beams (including the spatial extent of each beam, as well as the distance between them) [1,23]. Although the beams may not overlap at all, they can also interact with each other through characteristic scale that far greater than the scale of their light field. Furthermore, no matter how the distance between the beams varies, they can interact definitely in theory.

Each constituent Gaussian soliton whether rotates or not is determined by the initial transverse spatial momentum M_n [18],

M_{n} = \frac{i}{2 k} \iint (ψ_{n} \nabla_{⊥} ψ_{n}^{*} - ψ_{n}^{*} \nabla_{⊥} ψ_{n}) d x d y,

where ∇_⊥ = e_x∂_x + e_y∂_y,

ψ_{n}^{*}

is a complex conjugate function of ψ_n. M_n can be obtained by substituting Eq. (26) into Eq. (31),

M_{n} = ξ β r P_{0} (- sin φ_{n} e_{x} + cos φ_{n} e_{y}) .

Therefore, if ξ = 0, then M_n = 0, the soliton clusters does not rotate during propagation (see Fig. 1); if ξ ≠ 0, then M_n ≠ 0, and when ξ > 0, it rotates counterclockwise in the propagation process (see Figs. 2–4). The velocity parameter ξ is not taken as negative value in this paper for the reason that when it is taken as negative values, except for the opposite direction of rotation, the other characteristics of the clusters remain unchanged. This rotating motion is launched with tangential velocity and is different from what was studied in [6] and [17], which are due to the net angular momentum induced by the staircase-like phase distribution. M_n happens to have the following relationship with the previous parameters,

\begin{array}{l} M_{n} / P_{0} & = & v_{x n} e_{x} + v_{y n} e_{y} \\ = & (k_{n x} e_{x} + k_{n y} e_{y}) / k \\ = & [u_{x n} (0) e_{x} + u_{y n} (0) e_{y}] / k \\ = & u_{n} (0) / k \\ = & r_{0 n}^{'} (0), \end{array}

so the motion of each soliton satisfies the Ehrenfest’s theorem [16]. In other words, the slope of the projection trajectory of each soliton center on x–z plane and y–z plane with respect to the z axis is determined by M_n/P₀. Moreover, the total initial transverse spatial momentum

M = \sum M_{n} = 0

, and remains invariant. Therefore, the center of the soliton clusters does not deviate from the z axis during propagation.

3.3. Projection trajectory

The projection of the trajectory of each constituent soliton’s center on x–y plane can be obtained by Eqs. (18) and (19),

{(v_{y n} x - v_{x n} y)}^{2} + β^{2} {(c_{y n} x - c_{x n} y)}^{2} = {(c_{x n} v_{y n} - v_{y n} v_{x n})}^{2} .

In general, this equation is a centrosymmetric oblique ellipse, so the solitons are entangled in the propagation process. Meanwhile, when the values of v_xn and v_yn are taken as zero, i.e., each soliton has a parallel incidence, Eq. (34) becomes an equation of a straight line as follow,

c_{y n} \cdot x - c_{x n} \cdot y = 0 .

This indicates that there are no longer intertwining structures among the solitons. Equation (34) is valid for studying the projection trajectories of solitons under arbitrary incidence conditions. The combination of Eqs. (24) and (25) with the Eq. (34) yields

(ξ^{2} {cos}^{2} φ_{n} + {sin}^{2} φ_{n}) x^{2} (ξ^{2} {sin}^{2} φ_{n} + {cos}^{2} φ_{n}) y^{2} + [(ξ^{2} - 1) sin (2 φ_{n})] x y = ξ^{2} r^{2},

a concrete expression for oblique symmetrical incidence cases.

When ξ = 0, Eq. (36) is deduced to an equation of line: sin φ_n · x − cos φ_n · y = 0, φ_n (within the domain 0 < φ_n < π) or (2π − φ_n) (within the domain π ≤ φ_n ≤ 2π) is the slope angle of the line. Under this circumstance, the three soliton pairs (ψ₁ and ψ₄, ψ₂ and ψ₅, ψ₃ and ψ₆) with trajectories that are confined in a single plane respectively [see Figs. 1 and 5(b)].

Fig. 5 (a) Locations of the interacting solitons at initial position. (b)–(e): The projection trajectories of each interacting soliton on x–y plane, and (b): ξ = 0, (c): ξ = 1/3, (d): ξ = 1, (e): ξ = 3/2, respectively. The arrows indicate the direction of motion of the soliton beams. Parameters: N = 6, r = 6w₀.

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When ξ = 1, Eq. (36) becomes an equation of circle: x² + y² = r², the radius of the circle is the very radius of the ring at initial incident position, which means that all trajectories of the constituent Gaussian solitons are concentric circles [see Fig. 5(d)]. The hexagon shape of the clusters and its size remain invariant (see Fig. 3), although it rotates during propagation.

When 0 < ξ < 1 or ξ > 1, Eq. (36) is obviously a general elliptic equation. The standard elliptic equation can be obtained through the rotation of the coordinate system, and it can be written as: ξ²x² + y² = ξ²r². Therefore, all of the ellipses have the same eccentricity, and if 0 < ξ < 1, the eccentricity of the ellipse is $e = \sqrt{1 - ξ^{2}}$ [see Fig. 2 and Fig. 5(c)]; if ξ > 1, the eccentricity is $e = \sqrt{1 - 1 / ξ^{2}}$ [see Figs. 4 and 5(e)]. Namely, the eccentricity of the projection trajectory is only related to the velocity parameter ξ. In the case of oblique symmetrical incidence, the projection trajectories of two centrosymmetric beams are coincident.

3.4. Center distance

We define the distance from the soliton center to the transmission axis (z axis) as the center distance in the propagation process, represented by the letter d. Therefore, the radius r of the soliton clusters at z = 0 is the initial center distance. The center distance of each soliton can be obtained based on Eqs. (18) and (19):

d_{n} (z) = \sqrt{{(c_{x n} cos α + \frac{v_{x n}}{β} sin α)}^{2} + {(c_{y n} cos α + \frac{v_{y n}}{β} sin α)}^{2}} .

Inserting Eqs. (24) and (25) into Eq. (37), we get a simpler expression of the center distance

d_{n} (z) = r \sqrt{{cos}^{2} α + ξ^{2} {sin}^{2} α} .

We define the derivative of Eq. (38) with respect to z as the centrifugal velocity,

Δ d_{n} (z) = \frac{\partial d_{n} (z)}{\partial z} = \frac{(ξ^{2} - 1) β r (sin 2 α)}{\sqrt{2} \sqrt{ξ^{2} + 1 - (ξ^{2} - 1) cos (2 α)}} .

Both the center distance and the centrifugal velocity are used to describe the variation of the distance between each soliton and the transmission axis during propagation.

Another expression of the normalized distance z_p is given below:

z_{p} = z_{R} / \sqrt{η},

where

z_{R} = k w_{0}^{2}

is the Rayleigh length of the Gaussian beam [18,30]. In order to better display the changes caused by different input powers, the propagation distance in Fig. 6 is not normalized by z_p, instead the actual propagation distance z_R, which is power-independent. When the power ratio η decreases, the curve of the center distance looks like it is stretched in z direction but the overall shape doesn’t change. This indicates that the input power affects the evolution period only and not the extreme value of the center distance in each period. Taking Δd_n(z) as zero, one can straightforwardly get that, when 0 < ξ < 1 the center distance d_n(z) arrives its maximum [d_n(z)]_max = r at z = jπz_p(j = 0, 1, 2, · · ·) and arrives its minimum [d_n(z)]_min = ξr at z = (j + 1/2)πz_p, when ξ > 1 it is exactly the opposite. Of course, when ξ = 1, d_n(z) ≡ r, which is in good agreement with Fig. 3 and the analysis in Sec. 3.3 [Fig. 5(d)].

Fig. 6 (a) The center distance of each constituent soliton varies with the propagation distance. (b) The centrifugal velocity, corresponding to the first row. (a1) and (b1): η = 2; (a2) and (b2): η = 1; (a3) and (b3): η = 1/2. Dashed-dot-dot line: ξ = 0, solid line: ξ = 1/3, dashed line: ξ = 1, dashed-dot line: ξ = 3/2. Parameters: N = 6, r = 6w₀.

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As shown in Fig. 6(a), when ξ = 0, the minimum of center distance is zero, so the soliton center are periodic collisions in the form of simple harmonic oscillator (see Fig. 1). When 0 < ξ < 1, the minimum of center distance is not zero, which indicates that the soliton center does not collide in precession, but their optical fields may be overlap (see Fig. 2). The trajectory of each soliton is similar to the “DNA helix” structure. We can come to the conclusion that: when 0 < ξ < 1 or ξ > 1, the center distance changes periodically; when 0 < ξ < 1, the center distance is smaller first and then larger; when ξ > 1, the center distance is larger first and then smaller; and the critical value of the turning point is exactly equal to the initial center distance r. Based on the analysis in Sec. 3.3, the projection trajectory of each soliton center on x–y plane is an ellipse, one can find that the maximum value of the center distance is the semimajor axis of the ellipse, and the minimum value is the semiminor axis. It can be known from Eq. (38) that the period in which the semimajor axis (or semiminor axis) appears is Δz = πz_p, i.e., the period of center distance is Δz_d = πz_p. The semimajor (semiminor) axis appears twice in one period, so the evolution period of the Gaussian-shaped soliton clusters is Δz_t = 2πz_p (or $Δ z_{t} = 2 π z_{R} / \sqrt{η}$ ).

Figure 6(b) is the centrifugal velocity Δd_n(z) varies with the propagation distance. The positive Δd_n(z) means that the solitons are in centrifugal motion (the center distance is getting lager), and Δd_n(z) is negative, which indicates that the solitons are in centripetal motion (the center distance is getting smaller). Δd_n(z) ≡ 0 means that the center distance remain invariant during propagation, and this motion is a “uniform spiral motion”, see Fig. 3(b). The reason is that the repulsion (centrifugal) force is exactly balanced by attraction due to the soliton interaction [31]. Moreover, when ξ = 0, Δd_n(z) has a jump at the middle of each period, which represents a sudden change in the direction of centrifugal velocity. This is consistent with the motion state shown in Fig. 1 and Fig. 5(b).

3.5. Angular velocity

If the solitons in the clusters are regarded as particles, making a vertical line between each soliton and the transmission axis. The angle between this line and the positive direction of the x axis is Ω_n(z), then

Ω_{n} (z) = arctan (\frac{β c_{y n} cos α + v_{y n} sin α}{β c_{x n} cos α + v_{x n} sin α}) .

With the aid of the concept of angular velocity in classical mechanics, we regard the propagation distance z in this paper as the time t, the angular velocity can be obtained as

ω_{n} (z) = \frac{\partial Ω_{n} (z)}{\partial z} = \frac{β^{2} (c_{x n} v_{y n} - c_{y n} v_{x n})}{{(β c_{x n} cos α + v_{x n} sin α)}^{2} + {(β c_{y n} cos α + v_{y n} sin α)}^{2}} .

Substituting Eqs. (24) and (25) into Eq. (42), then the angular velocity can be deduced as

ω_{n} (z) = \frac{ξ β}{{cos}^{2} α + ξ^{2} {sin}^{2} α} .

While there is a subscript n in ω_n(z), the expression in fact is n-independent. Therefore, each and every constituent soliton rotates around the center of the clusters with an equal angular velocity, only at different propagation distances with different angular velocities. The period of angular velocity is also Δz = πz_p. Similarly, with the aid of classical mechanics, we can get the angular acceleration

Δ ω_{n} (z) = \frac{\partial ω_{n} (z)}{\partial x} = \frac{ξ β^{2} (1 - ξ^{2}) sin (2 α)}{{({cos}^{2} α + ξ^{2} {sin}^{2} α)}^{2}} .

Figure 7 shows the angular velocity and the angular acceleration. When 0 < ξ < 1, the angular velocity increases first and then decreases; when ξ > 1, the angular velocity decreases first and then increases. The corresponding angular acceleration changes are shown in Fig. 7(b). What needs to be explained is that, when ξ = 0 or ξ = 1, the angular acceleration is Δω_n(z) = 0, see the dashed-dot-dot line and dashed line in Fig. 7(b) respectively, and the two lines coincide each other. According to Eq. (40), another expression of the propagation constant β can be obtained easily:

β = \sqrt{η} / z_{R},

therefore, ω_n(z) is proportional to the square root of the input power P₀. When the power ratio η decreases, the velocity curve looks like it is not only stretched in z direction but also the overall shape is compressed. This indicates that the input power affects both the period of the angular velocity and the value in each period. This is somewhat different from the influence of the input power on center distance. Taking Δω_n(z) = 0, we get that, when 0 < ξ < 1, the angular velocity ω_n(z) arrives its minimum [ω_n(z)]_min = ξβ at z = jπz_p and arrives its maximum [ω_n(z)]_max = β/ξ at z = (j + 1/2)πz_p; when ξ > 1 it is exactly the opposite. When ξ = 0, one can get ω_n(z) ≡ 0; when ξ = 1, one can get ω_n(z) ≡ β, see the dashed-dot-dot line and dashed line in Fig. 7(a) respectively. This is consistent with the intuitive results in Fig. 1 and Fig. 3.

Fig. 7 (a) The angular velocity of each constituent soliton varies with the propagation distance. (b) The angular acceleration, corresponding to the first row. (a1) and (b1): η = 2; (a2) and (b2): η = 1; (a3) and (b3): η = 1/2. Dashed-dot-dot line: ξ = 0, solid line: ξ = 1/3, dashed line: ξ = 1, dashed-dot line: ξ = 3/2. Parameters: N = 6, r = 6w₀.

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From Eqs. (38) and (43) we get the relationship between center distance and angular velocity,

ω_{n} (z) = \frac{ξ β r^{2}}{d_{n}^{2} (z)} (ξ \neq 0),

i.e., the angular velocity is inversely proportional to the square of the center distance. Figure 7 compares with Fig. 6, we can also obtain that the angular velocity of each soliton decreases with the increasing of center distance. The angular velocity reaches its maximum when the center distance is minimum, and vice versa. Figure 8 shows the constraints between the variables in Eq. (46), i.e., when 0 < ξ < 1, the center distance and the angular velocity can only change within the ranges such that ξr ≤ d_n(z) ≤ r and ξβ ≤ ω_n(z) ≤ β/ξ respectively; when ξ = 1, one can get d_n(z) ≡ r and ω_n(z) ≡ β; when ξ > 1, their ranges become r < d_n(z) ≤ ξr and β/ξ ≤ ω_n(z) ≤ ξβ. Therefore, the evolutions of the angular velocity and the center distance are interdependent.

Fig. 8 Variation relationship between the angular velocity and the center distance. Solid line: η = 2, dashed-dot line: η = 1, dashed line: η = 1/2. Three points from top to bottom represent, respectively, η = 2, η = 1 and η = 1/2. Parameters: N = 6, r = 6w₀.

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Fig. 9 Transverse intensity patterns of Gaussian-shaped soliton clusters in SNN media at different propagation positions shown at the top. Parameters: (a): N = 6, ξ = 1/3, η = 1, r = 12w₀; (b): N = 4, ξ = 1, η = 1, r = 4w₀; (c): N = 8, ξ = 1/3, η = 1, r = 6w₀.

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3.6. Extension

This section is mainly a supplement and extension to the above. Firstly, an example is given to illustrate the effect of the change of the initial center distance, and then it is shown that the laws obtained above are also valid for other N values.

Figure 9(a) is the same as Fig. 2(b) except that the initial center distance r = 12w₀. As we can see, even when the center distance is minimal (at z = πz_p/2), the solitons’ optical fields are still clearly discernible. It is generally believed that if the spacing between adjacent solitons lager than three times of the beam width, there is no overlap between their optical fields. Then the distance between the soliton center should be at least fourfold beam widths. According to geometric relation and sine theorem, it is not difficult to find that the initial center distance must satisfy

r \geq \frac{2 w_{0}}{ξ sin (π / N)} .

For the case of N = 6 and ξ = 1/3, one can get r ≥ 12w₀. Note that the soliton spacing could not arbitrarily large. So far as we know, the largest soliton interaction spacing observed in experiment is 43 times of the beam width [19].

Figure 9(b) is a counterpart of Fig. 3(b), which shows that a bound state of steady rotation of the soliton clusters happens if and only if the initial incident parameters are ξ = 1 and η = 1. The evolution period of the intensity pattern at this time is Δz_i = 2πz_p/N. Here, again the emphasis is placed on the meaning of the period of the intensity pattern mentioned in this paper. In the process of the clusters’ evolution, some intensity patterns are repeated many times, and the time after which they appear twice in succession is simply called “period”, not taking into account other factors. For the case N = 4, we have Δz_i = πz_p/2 [see Figs. 9(b1)–9(b5)]. The parameters in Fig. 9(c) is the same as that in Fig. 2(b) except that N = 8. We can see that their evolution behaviors are the same. If there is almost no overlap of the optical fields at z = πz_p/2, the initial center distance r ≥ 15.7w₀ is a must based on Eq. (47).

4. Conclusion

In conclusion, we have theoretically investigated the Gaussian-shaped soliton clusters in SNN media and especially the effects of different initial conditions on propagation properties. The results show that initial incident parameters (the initial transverse velocity, the input power, the initial position) play important roles in the propagation process. When solitons are transmitted in SNN media, they could present various forms of motion, such as rotate, non-rotate, shape-variant or shape-invariant, etc. And these motion forms can be realized by adjusting the appropriate initial incident parameters. Properly speaking, the initial transverse velocity [Eq. (25)] mainly controls the rotation of the soliton clusters and leads to the change of the size of optical field; the input power P₀ mainly controls the distribution of the intensity by causing the change of the width of each constituent beam; the initial position [Eq. (24)] mainly controls the spacing between each constituent soliton, and it is also a key factor in controlling the size of the combined field. They are interrelated.

It is of great practical significance to realize the controllable multisoliton. So that we can distinguish each soliton at the exit end. Very recently, some articles [32,33] have explored the realization of soliton’s controllability. The laws of soliton clusters’ interaction in this paper can be applied to arrayed waveguide and optical communication. Although the results are obtained only by studying the Gaussian-shaped soliton clusters, it can provide reference to investigate the propagation of other form soliton groups (or multibeam interactions). Additionally, the SNN media is equivalent to optical fractional Fourier transform systems [27] and quadratic nonlinear or χ⁽²⁾ materials [34,35], therefore the results in this paper can be expanded to these equivalent physical systems. These novel properties of the soliton clusters imply their potential applications in all-optical signal processing, switching and logic gating by using this kind of interaction.

Funding

National Natural Science Foundation of China (Grant Nos. 61308016, 11374089, and 61605040); Chunhui Plan of Ministry of Education of China (Grant No. Z2017020); Natural Science Foundation of Hebei Province (Grant Nos. F2017205060, F2017205162, and F2016205124); Technology Key Project of Colleges and Universities of Hebei Province (Grant No. ZD2018081); Science Fund for Distinguished Young Scholars of Hebei Normal University (Grant No. L2017J02).

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Controllable Gaussian-shaped soliton clusters in strongly nonlocal media

Abstract

1. Introduction

2. Model

3. Propagation characteristics

3.1. Analytical expression

3.2. Intensity pattern

3.3. Projection trajectory

3.4. Center distance

3.5. Angular velocity

3.6. Extension

4. Conclusion

Funding

References

Cited By

Figures (9)

Equations (47)

Optics Express