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Multipole solitons and vortex solitons in nonlocal nonlinear media

S. F. Wang

Open Access Open Access

Abstract

The nonlinear Schrödinger equation (NLSE) under nonlocal nonlinear media (NNM) is described and the approximate analytical solutions of the vector multipole solitons and vortex optical soliton clusters are obtained via the variational method. The results show that the structure of the optical solitons is determined by modulation depth and topological charge. In the propagation process, the spatial soliton has an observable rotation property. Under certain conditions, the rotating space modulated vortex optical solitons degenerate into circular symmetric vortex optical solitons. The results can be extended to other physical systems.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Spatial soliton is an important research content of nonlinear optics, which has more applications in beam diffraction control [1], logic gate design [2], optical switching devices [3] and information transformation [4]. Vortex solitons are gener- ated in photorefractive crystals equipped with photonic lattice. Fixed solitons in Kerr media are always unstable to prevent collapse or attenuation due to the key property of local cubic self-absorption nonlinearity [5]. However, due to the complexity of nonlinear responses in the medium, these studies are almost exclusively conducted by numerical methods, and the analytical solutions of local vector vortex solitons have not been reported. In recent years, due to the development of the soliton theory and the introduction of medium with adjustable refractive index in modern optical materials, the study of vector vortex solitons has become a hot topic in the study of solitons theory [6]. WANG et al [7] analyzed the dynamics and stability of spatially modula ted vortex optical solitons. CAO et al [8] studied the coupled variable-coefficient Lugiato -Lefever equation, the dynamic evolution of vortex solitons and the stability of dipole solitons were discussed. Moreover, the influence of the topological charge on the energy distribution for multipole solitons were investigated in [9]. The soliton pulses and bound states in a dissipative system were explored in [10].The dynamics and spectral analysis of optical waves in isotropic media were analyzed in [11]. The higher-order wave solutions via the generalized Darboux transformation were derived in [12]. The soliton fusion and fission in fiber lasers were studied in [13]. In addition, the literature [14–17] studies rotating-type space modulated vortex optical solitons with whittaker function. They are shown that under certain conditions, the rotating modulated vortex solitons with the whitaker function evolve into LG vortex solitons. They are proposed that the rotational angular velocity of the vortex was determined by both the propagation constant and the modulation parameters. However, the multipole soliton and vortex soliton solutions for the NLSEs in nonlinear media via variational method and the influence of their parameters on the dynamics of the solutions are rarely reported.

Stimulated by the above studies and the literature [18–24], we will deal with the NLSEs by using the variational approach. And its self-similar rotational analytical solutions are obtained. The dynamics and the stability of the soliton solutions are also discussed. The research motivation and innovation of the article are described in following aspects: 1) By the variational method, the self-similar solution of the corresponding equation is derived. 2) The dynamics with different parameters are analyzed and discussed. 3) The significance of the results show this proposed method can quickly approximate the exact solution of the proposed equation. Meanwhile, the results can be extended to other physical systems and their potential applications in all-optical switching design.

The rest structure is presented as: the model of NLSE is described and its variational solutions with the Hypergeometric function is constructed in section 2. In section 3, the spatial distribution and the propagation characteristics of vortex solution are discussed. Section 4 gives the conclusions.

2. Model

The model of NLSE in NNM [18] is described as

$$2i{u_z} + {\nabla ^2}u - \varsigma {P_0}{\rho ^2}u = 0, $$
where
$${\nabla ^2} = \frac{{{\partial ^2}}}{{\partial {x^2}}} + \frac{{{\partial ^2}}}{{\partial {y^2}}}, $$
$u$ denotes the paraxial beam, ${P_0}$ denotes the incident power, $\varsigma $ denotes the material constant, $\rho $ is the radius of the polar.

Using the transformation relation between the Cartesian and cylindrical coordinates, one has

$$\left\{ {\begin{array}{{l}} {x = \rho \cos \phi }\\ {y = \rho \sin \phi }\\ {z = z} \end{array}} \right.$$

Equation (1) can be rewritten as

$$i\frac{{\partial u}}{{\partial z}} + \frac{1}{2}\frac{{{\partial ^2}u}}{{\partial {\rho ^2}}} + \frac{1}{{2\rho }}\frac{{\partial u}}{{\partial \rho }} + \frac{1}{{2{\rho ^2}}}\frac{{{\partial ^2}u}}{{\partial {\phi ^2}}} - \frac{1}{2}\varsigma {P_0}{\rho ^2}u = 0. $$

A self-similar solution of (4) is assumed as the following form:

$$\psi (\rho ,\phi ,z) = V[\rho ,\phi - \Theta (z)[\exp [i\omega (z)]. $$
where $\Theta (z)$ denotes the rotational angular. Let
$$\theta = \phi - \Theta , $$

One has

$$\psi (\rho ,\theta ,z) = V(\rho ,\theta )\exp [i\omega (z)]. $$

Substituting (7) into (4), one has

$$\frac{1}{2}\frac{{{\partial ^2}V}}{{\partial {\rho ^2}}} + \frac{1}{{2\rho }}\frac{{\partial V}}{{\partial \rho }} + \frac{1}{{2{\rho ^2}}}\frac{{{\partial ^2}V}}{{\partial {\theta ^2}}} - i\Theta \frac{{\partial V}}{{\partial \theta }} - \omega V - \frac{1}{2}\varsigma {P_0}{\rho ^2}V = 0. $$

Using variational method [25], one has

$$\delta J = \delta \int_0^\infty {\int_0^{2\pi } {\tilde{s}\left( {V,{V^\ast },\frac{{\partial V}}{{\partial \rho }},\frac{{\partial {V^\ast }}}{{\partial \rho }},\frac{{\partial V}}{{\partial \theta }},\frac{{\partial {V^\ast }}}{{\partial \theta }}} \right)d\rho d\theta } } . $$

The Lagrangian function is defined as the following form:

$$\tilde{s} = \frac{i}{2}\Theta \rho \left( {V\frac{{\partial {V^\ast }}}{{\partial \theta }} - {V^\ast }\frac{{\partial V}}{{\partial \theta }}} \right) - \frac{1}{{2\rho }}{\left|{\frac{{\partial V}}{{\partial \theta }}} \right|^2} - \frac{1}{2}\rho {\left|{\frac{{\partial V}}{{\partial \rho }}} \right|^2} - \frac{1}{2}\varsigma {P_0}{\rho ^3}{|V |^2} - \omega \rho {|V |^2}. $$

Let the trial solution of the variational Eq. (9) as

$$V(\rho ,\theta ) = CH(\rho )[\cos (m\theta ) + iq\sin (m\theta )]. $$
where $C$ denotes the normalized constant, $H(\rho )$ is the radial amplitude, and m denotes the topological charge. $q$ denotes the modulation depth.

For ease of calculation, Let $\varsigma $ is the reciprocal of the incident power ${P_0}$, that is

$$\varsigma = \frac{1}{{{P_0}}}$$

Substituting Eq. (11) into Eq. (9), one has a self-similar solution as follow:

$$\psi (\rho ,\phi ,z) = {C_{\kappa m}}{\rho ^{ - 1}}{W_{\kappa ,\frac{m}{2}}}({\rho ^2}) \times \exp \{ i\omega (z)[\cos m(\phi - \Theta ) + iq\sin m(\phi - \Theta )]\} $$
where ${C_{\kappa m}}$ is the normalization constant. ${W_{\kappa ,\frac{m}{2}}}({\rho ^2})$ is the Whitaker function.

3. Results and discussion

The self-similar solution which expressed by Eq. (11) is simplified by Laguerre polynomials [19,20] as

$$\Psi (\rho ,\phi ,z) = {C_{lm}}{\rho ^m}L_l^m({\rho ^2})\exp ( - {\rho ^2}/2) \times \exp \{ i\omega (z)[\cos m(\phi - \Theta ) + iq\sin m(\phi - \Theta )]\} $$
where
$${C_{lm}} = \sqrt {\frac{{2l!}}{{(m + 1)!}}{P_0}} . $$
and $L_l^m({\rho ^2})$ denotes the Laguerre polynomial which is defined as
$$L_l^m(\xi ) = \frac{{\Gamma (m + 1 + l)}}{{l!\Gamma (m + 1)}}Hypergeom\;( - l,m + 1,\xi ). $$
where m is the topological charge number, $\Gamma (m + 1 + l)$ and $\Gamma (m + 1)$ are the gamma functions. $Hypergeom( - l,m + 1,\xi )$ denotes the hypergeometric distribution function. Equation (14) describes the multipole solitons or vortex beam. When ${P_0} = 2$, $z = 0$, $q = 0$, the plots varying with topological charge $m = 0,1,2,3$ are presented in Fig. 1(a)–(d), respectively. The spatially modulated rotating vortex optical solitons have M = 2 m intensity peaks in the angular direction and $N = l + 1$(m is 0 or a positive integer) or $N = l + m + 1$(m is a negative integer) intensity peak in the radial direction. The rotation of the vortex optical solitons depends on the value of the angular velocity $\Theta $.

 figure: Fig. 1.

Fig. 1. Normalized response ($I = u{u^\ast }$) of the hypergeometric spatial vortex solitons in (14) and the intensity with the topological charge $m = 0,1,2,3$, respectively.

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When $m = 1$, ${P_0} = 5,10,15,20$, $z = 0$, the plots varying with the incident power are depicted in Fig. 2(a)–(d), respectively. From Fig. 2, it can be seen that the field modulus distribution of the beam perpendicular to the propagation direction remains unchanged, that is, the Laguerre-Gaussian vortex optical soliton can be formed. This is the result of the balance between the diffraction effect of the beam and the nonlinear self-focusing effect of the medium when the beam is transmitted in a nonlinear medium.

 figure: Fig. 2.

Fig. 2. Normalized response ($I = u{u^\ast }$) of the hypergeometric spatial vortex solitons in (14) and the intensity with ${P_0} = 5,10,15,20$, $m = 1$, other parameters as those in Fig. 1.

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When ${P_0} = 2$, the beam evolved into a hypergeometric vortex optical soliton. The parameter $q(0 \le q \le 1)$ is a physical quantity introduced to describe the modulated depth of light intensity. When $q \to 1$, the expression $\cos m(\phi - \Theta ) + iq\sin m(\phi - \Theta )$ has the radial symmetry. When $q \to 0$, this expression describes a multipole soliton in a medium. When $m = 1$, $q = 0,1/3,2/3,1$, the other parameters are the same as those in Fig. 2, the plots varying with different modu lation depths are depicted in Fig. 3. It can be seen that when q = 0, m = 1, the solution behaves as a vector dipole soliton. The number of azimuth lobes of a multipole soliton is determined by the value of m. When q increases, there is an interaction between two sidelobe solitons, and the distance between two sidelobed solitons decreases continuously. And then, two orthogonal vector dipole solitons are also incoherently superimposed to produce the vortex solitons.

 figure: Fig. 3.

Fig. 3. Normalized response ($I = u{u^\ast }$) of the hypergeometric spatial vortex solitons in (14) and the intensity with modulation depth $q = 0,1/3,2/3,1$, besides $m = 1$, other parameters as those in Fig. 1.

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When q = 1, m is an integer (m = 0, 1, 2, 3, respectively), the optical soliton and the different vortex soliton clusters are formed in Fig. 4. From Fig. 4, it is observed that when m = 0, a spatial soliton is formed. As m gets bigger, the hypergeometric spatial vortex solitons in (14) can be obtained, and the radius of the soliton ring also gets larger. When q = 1, a ring soliton can be formed. As can be clearly seen from Fig. 4, the solitons consist of a series of alterna ting bright and dark rings with a common center. When $m \ne 0$, the intensity of the ring center is zero. Conversely, when $m = 0$, the maximum light intensity is at the center of the ring. The radial distribution of the intensity is as shown in Fig. 4. The physics of this phenomenon is nonlocal nonlinearity means that the polarization of the medium is not only related to the electric field in the region, but also to the electric field in the surrounding region of the medium. Although the different modulation depth q affects the properties of solitons, the vortex solitons or multi-pole solitons are formed. However, according to Eq. (14), the shape of the soliton along the transmission depends on the transmission power of the pulse, the quantum number m, $l$, and the parameters related to the wave number.

 figure: Fig. 4.

Fig. 4. Normalized response ($I = u{u^\ast }$) of the hypergeometric spatial vortex solitons in (14) and the intensity with $q = 1$, (a)$m = 0$, (b) $m = 1$, (c)$m = 2$, (d)$m = 3$, other parameters as those in Fig. 2.

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When ${P_0} = 2$, $z = 0,40,80,120,160,200,240,280$, respectively, the plots varying with the prop agation distance are presented in Fig. 5. It can be seen that when z = 0 and m = 1, the solution behaves as a vector dipole soliton. As the propagation distance z increases, the bipolar solitons rotate and gradually form vortex solitons. The beam keeps the waveform unchanged and propagates forward. Figure 5 shows the propagation and intensity distribution of 4-pole soliton ($m = 2$, $q = 0$) at different propagation distances $z$. From Fig. 5, it can be seen that the beam remains unchanged during transmission, that is, the strong nonlocal multipole solitons are formed. Such spatial solitons described by Eq. (14) are determined by two parameters m and l. For different $l$ and $m$, the spatial solitons form the soliton clusters.

 figure: Fig. 5.

Fig. 5. Normalized response ($I = u{u^\ast }$) of the hypergeometric vortex solitons in (14) and their intensity with different $z = 0,40,80,120,160,200,240,280$, respectively, and $q = 0$, $m = 2$.

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4. Conclusions

To sum up, the NLSE of the hypergeometric beam in free space was described. The analytical solutions of hypergeometric beam propagation were obtained by deformation reduction and variational method for the proposed model. The results show that the multipole soliton and the vortex soliton can be formed, the low order hypergeometric solitons show a hollow multipole bright ring distribution. This equation has important applications in non linear optics and other physical fields. It can describe the propagation characteristics of optical solitons in optical fibers, and the nonlocal nonlinear response can eliminate the instability of the symmetry breaking of the vortex itself. This is because the nonlocal nonlinear response provides conditions for symmetric stability, so that various kinds of the solitons can be formed. The same type of higher order and different types of spatial solitons can also probe the overall structure of the solution of the equation. In addition, the results can be extended to other physical systems. These new properties of the solitons indicate their potential applications in all-optical signal processing, switching and logic gating.

Disclosures

There was no conflict of interest regarding the publication of this work.

Data availability

No data was used for the research described in the article.

References

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Data availability

No data was used for the research described in the article.

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Figures (5)
Fig. 1.
Fig. 1. Normalized response ($I = u{u^\ast }$) of the hypergeometric spatial vortex solitons in (14) and the intensity with the topological charge $m = 0,1,2,3$, respectively.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Normalized response ($I = u{u^\ast }$) of the hypergeometric spatial vortex solitons in (14) and the intensity with ${P_0} = 5,10,15,20$, $m = 1$, other parameters as those in Fig. 1.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. Normalized response ($I = u{u^\ast }$) of the hypergeometric spatial vortex solitons in (14) and the intensity with modulation depth $q = 0,1/3,2/3,1$, besides $m = 1$, other parameters as those in Fig. 1.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Normalized response ($I = u{u^\ast }$) of the hypergeometric spatial vortex solitons in (14) and the intensity with $q = 1$, (a)$m = 0$, (b) $m = 1$, (c)$m = 2$, (d)$m = 3$, other parameters as those in Fig. 2.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. Normalized response ($I = u{u^\ast }$) of the hypergeometric vortex solitons in (14) and their intensity with different $z = 0,40,80,120,160,200,240,280$, respectively, and $q = 0$, $m = 2$.

View in Article | Download Full Size | PDF

Equations (16)

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2 i u z + ∇ 2 u − ς P 0 ρ 2 u = 0 ,
∇ 2 = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ,
{ x = ρ cos ⁡ ϕ y = ρ sin ⁡ ϕ z = z
i ∂ u ∂ z + 1 2 ∂ 2 u ∂ ρ 2 + 1 2 ρ ∂ u ∂ ρ + 1 2 ρ 2 ∂ 2 u ∂ ϕ 2 − 1 2 ς P 0 ρ 2 u = 0.
ψ ( ρ , ϕ , z ) = V [ ρ , ϕ − Θ ( z ) [ exp ⁡ [ i ω ( z ) ] .
θ = ϕ − Θ ,
ψ ( ρ , θ , z ) = V ( ρ , θ ) exp ⁡ [ i ω ( z ) ] .
1 2 ∂ 2 V ∂ ρ 2 + 1 2 ρ ∂ V ∂ ρ + 1 2 ρ 2 ∂ 2 V ∂ θ 2 − i Θ ∂ V ∂ θ − ω V − 1 2 ς P 0 ρ 2 V = 0.
δ J = δ ∫ 0 ∞ ∫ 0 2 π s ~ ( V , V ∗ , ∂ V ∂ ρ , ∂ V ∗ ∂ ρ , ∂ V ∂ θ , ∂ V ∗ ∂ θ ) d ρ d θ .
s ~ = i 2 Θ ρ ( V ∂ V ∗ ∂ θ − V ∗ ∂ V ∂ θ ) − 1 2 ρ | ∂ V ∂ θ | 2 − 1 2 ρ | ∂ V ∂ ρ | 2 − 1 2 ς P 0 ρ 3 | V | 2 − ω ρ | V | 2 .
V ( ρ , θ ) = C H ( ρ ) [ cos ⁡ ( m θ ) + i q sin ⁡ ( m θ ) ] .
ς = 1 P 0
ψ ( ρ , ϕ , z ) = C κ m ρ − 1 W κ , m 2 ( ρ 2 ) × exp ⁡ { i ω ( z ) [ cos ⁡ m ( ϕ − Θ ) + i q sin ⁡ m ( ϕ − Θ ) ] }
Ψ ( ρ , ϕ , z ) = C l m ρ m L l m ( ρ 2 ) exp ⁡ ( − ρ 2 / 2 ) × exp ⁡ { i ω ( z ) [ cos ⁡ m ( ϕ − Θ ) + i q sin ⁡ m ( ϕ − Θ ) ] }
C l m = 2 l ! ( m + 1 ) ! P 0 .
L l m ( ξ ) = Γ ( m + 1 + l ) l ! Γ ( m + 1 ) H y p e r g e o m ( − l , m + 1 , ξ ) .
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