Self-structuring of stable dissipative breathing vortex solitons in a colloidal nanosuspension

V. Skarka; N. B. Aleksić; W. Krolikowski; D. N. Christodoulides; S. Rakotoarimalala; B. N. Aleksić; M. Belić

doi:10.1364/OE.25.010090

1. Introduction

The structuring and controlling of laser light by its propagation through appropriate nonlinear media is the subject of great interest [1–7]. Such manipulation of laser beams and pulses requires engineering of appropriate artificial materials, like colloidal suspensions of nanoparticles exhibiting strong nonlinear response. The medium is modified by the laser propagation, causing in turn the self-organized localization and structuring of the light itself. The main issue in nonlinear propagation of localized optical structures is their stability. Solitons are localized structures self-generated far from thermodynamic equilibrium [8]. In conservative systems, one-dimensional (1D) optical solitons, described by the nonlinear Schrödinger equation (NSE), are self-trapped through the simultaneous balance of diffraction or dispersion by the self-focusing. In two dimensions, the catastrophic collapse of beams occurs that can be prevented by the presence of negative quintic nonlinearity [1,9]. The generation of 2D optical solitons in a cubic-quintic centrosymmetric medium has been recently demonstrated in an experiment [10].

However, in nature losses are always present, causing soliton annihilation in the absence of any gain. Complex Ginzburg-Landau equations that feature both gain and loss adequately describe the self-generation of dissipative solitons in nanophotonics, plasmonics, fluids, plasmas, and electromagnetism, as well as superconductivity, superfluidity, elementary particles, and biological systems [11,12]. In nonlinear media with gain and loss, multidimensional solitons self-stabilize due to the simultaneous balance of diffraction by competing cubic-quintic nonlinearity, as well as by the compensation of loss by gain [13]. Of particular interest for manipulation and tweezing of nanoparticles and nanorods in various colloidal suspensions are vortex solitons [14]. The intensity of vortices displays a volcano shape with zero electrical field in the center, corresponding to a singularity represented by the topological charge S [15–17]. The resulting nonzero angular momentum M of the vortex beam is proportional to the product of the charge and beam power P.

In this paper, we investigate the conditions for self-generation of dissipative vortex solitons in a colloidal suspension of nanoparticles. In particular, we focus on the stability of such structures when laser light redistributes nanoparticles, modifying the spatial distribution of an effective nonlinear susceptibility. The paper is organized as follows. In Sec. 2, theoretical analysis of the colloidal model is provided and an appropriate generalized complex cubic-quintic Ginzburg-Landau equation introduced. In Sec. 3, the variational approach, extended to dissipative systems, is applied to the equation and the stability analysis of obtained solutions carried through. A stability domain is established in the parameter space. In Sec. 4, numerical simulation of the input beams from this domain is carried out, to confirm their stability. Section 5 brings conclusions.

2. Theoretical analysis

We consider here the nonlinear response of a colloidal suspension of nanoparticles, originating from the gradient force exerted by light on the particles. This force causes particles to either agglomerate in high light intensity regions or disperse away from these regions, depending on whether the polarizability of particles is positive or negative. In either case, the effective refractive index seen by the light beam always increases with intensity, leading to self-focusing and ultimately to the formation of spatial solitons [14, 18–24]. The polarizability of particles with refractive index n_p immersed in a transparent liquid with refractive index n_b is given by

α = 3 V_{p} ε_{0} n_{b}^{2} θ,

where V_p is the particle volume, ε₀ is the free space permittivity, and

θ = (n_{p}^{2} / n_{b}^{2} - 1) / (n_{p}^{2} / n_{b}^{2} + 2)

is the relative refractive index. The gradient force exerted by light on particles determines the effective packing fraction of the particles η = ρV_p, where ρ is the concentration of nanoparticles [18–20]. In particular, it has been shown that in the hard sphere approximation for particle-particle interaction, this packing fraction satisfies the following relation:

α {(4 k_{B} T)}^{- 1} {| U (η) |}^{2} = sgn (θ) [g (η) - g (η_{0})],

where |U|² represents the light intensity, g(η) = (3 − η)/(1 − η)³ + ln η, and sgn(θ) denotes the sign function [18–20]. Note that η₀ is the packing fraction in the absence of laser field. In order to follow the evolution of electrical field with amplitude U and to demonstrate the self-generation of stable dissipative solitons, we consider the paraxial approximation of the Helmholtz equation

\nabla^{2} U + k_{0}^{2} n_{eff}^{2} U = 0,

where k₀ is the free space wavevector and n_eff is the effective nonlinear refractive index change caused by nanoparticles moving under the influence of laser beam. Assuming a relatively small index contrast between nanoparticles and the background,

n_{eff}^{2} = n_{b}^{2} (1 + 3 θ η)

. Consequently, the slowly varying electric field envelope E(r) is defined by the expression [19]

U (r) {(3 V_{p} ε_{0} n_{b}^{2} | θ | / 4 k_{B} T)}^{1 / 2} = E (r) exp [i k_{0} n_{b} {(1 + 3 θ η_{0})}^{1 / 2} Z],

where Z stands for the propagation variable. Therefore, the intensity is expressed as

{| E |}^{2} = sgn (θ) [g (η) - g (η_{0})] .

The corresponding paraxial equation reads

i \frac{\partial E}{\partial z} + γ Δ E + sgn (θ) (η - η_{0}) E + i Γ η E = 0,

where z = 3 |θ| k₀n_b Z/2,

γ = 1 / (3 | θ | k_{0}^{2} n_{b}^{2})

and Γ represents the linear scattering losses [19]. In Fig. 1, we illustrate the dependence of sgn (θ) (η − η₀) on the light intensity |E|² for a few values of η₀, which corresponds to the packing fraction in the absence of light. After a careful fitting procedure, this curve is represented as the following polynomial relation:

sgn (θ) (η - η_{0}) = (κ + σ {| E |}^{2} - ν {| E |}^{4})

with appropriate values of the parameters. For instance, when η₀ = 0.01, the coefficients are κ = 0.024722, σ = 0.0035944, and ν = 0.00024943, and the fitting curve is basically indistinguishable from the original one. Recent experimental studies of centrosymmetric colloidal suspensions of metallic particles have confirmed the presence of nonlinear susceptibilities of odd orders, with alternating signs [25]. Indeed, it is well known that the polarization of centrosymmetric media corresponds to the series of only odd order nonlinear susceptibilities [1,11,12]. To allow for the formation of localized structures, the unavoidable presence of losses (having the negative sign) has to be counteracted by gain [1,11–13]. The gain could be introduced, for instance, by adding a dye such as rhodamine B into the colloidal suspension and then pump it with an external light source, to achieve amplification [26]. Consequently, it has become clear that the proper description of self-organization and soliton formation in colloidal suspensions must include a simultaneous balance of diffraction by competing nonlinearities, as well as the compensation of linear and nonlinear losses by gain. All of these conditions are met in the complex cubic-quintic Ginzburg-Landau equation (CQGLE).

Fig. 1 Illustrating the dependence of packing fraction sgn (θ) (η − η₀) on light intensity |E|², for few values of the background packing fraction η₀. Solid squares represent the fitting formula, Eq. (7), for the case η₀ = 0.01

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Therefore, in what follows we investigate the nonlinear beam propagation in colloidal media using as a model the generalized CQGLE that takes into account various aspects of the particle-particle interaction:

i \frac{\partial E}{\partial z} + γ Δ E + (κ + σ {| E |}^{2} - ν {| E |}^{4}) E = - i δ E + i ε {| E |}^{2} E - i μ {| E |}^{4} E + i β Δ E .

Here, positively defined parameters γ, κ, σ, ν, δ, ε, μ, and β characterize respectively diffraction, linear term, cubic self-focusing, quintic self-defocusing, linear loss, cubic gain, quintic loss, as well as field diffusion. They depend of the chosen type of nanoparticles and the background medium. The variational method, extended to dissipative systems, is used to determine the domain of these dissipative parameters where the stable solitons can exist [13].

3. Stability criterion

The generalized complex cubic-quintic Ginzburg-Landau equation describes well the evolution of Gaussian laser beams in colloidal nanosuspensions. However, this equation does not admit exact solutions, except for a particular set of parameters [27,28]. Therefore, it is necessary to resort to numerical simulations. Nevertheless, an analytical approximation that provides clues for finding dissipative solitons in the numerical form has been developed using the variational approach (VA) adapted for dissipative systems [13,16,17,29].

The Lagrangian corresponding to Eq. (8) reads as

L = \frac{i}{2} (\frac{\partial E^{*}}{\partial z} E - \frac{\partial E}{\partial z} E^{*}) + γ {| \nabla E |}^{2} - κ {| E |}^{2} - \frac{σ}{2} {| E |}^{4} + \frac{ν}{3} {| E |}^{6} .

Following an optimization procedure of the extended VA, a trial function in the form of a Gaussian beam is chosen,

E = B A {(\frac{r}{G R})}^{S} exp [- \frac{r^{2}}{G^{2} R^{2}} + i \frac{C}{G^{2}} r^{2} + i S_{φ} + i Ψ],

where the azimuthal coordinate is φ, and A, R, C, S, and Ψ represent the amplitude, radius, wavefront curvature, topological charge and the phase of the beam, respectivelly. To start with, we consider the topological charge S = 1, with the scaling factors G = 16/9 and

B = (9 \sqrt{2} / 8) \sqrt{γ / σ}

Using the VA for dissipative systems introduced in Ref. [13], the four variational parameters, the amplitude, radius, wavefront curvature, and phase – all functions of the propagation variable z – are optimized following Hamilton’s principle. The extremum function E (see Eq. (10)) renders the Lagrangian integral stationary, under the condition that the four Euler-Lagrange equations corresponding to Eq. (8) are valid. The variation of Lagrangian in Eq. (9) with respect to the phase Ψ yields the first Euler-Lagrange equation:

\frac{G^{2}}{4 γ} (R^{2} \frac{d A}{d z} + A R \frac{d R}{d z}) + A [R^{2} (δ_{0} - \frac{2 ε}{σ} A^{2} + \frac{3 μ γ}{2 σ^{2}} A^{4}) + \frac{β}{γ} (1 + 4 C^{2} R^{4})] = 0 .

The optimization of amplitude A gives the second relation:

\frac{G^{2}}{2 γ} \frac{d ψ}{d z} + R^{2} (4 C^{2} + \frac{G^{2}}{γ} \frac{d C}{d z}) + \frac{1}{R^{2}} - κ_{0} - 2 A^{2} + \frac{3}{2} \frac{ν γ}{σ^{2}} A^{4} = 0 .

The variation with respect to the radius R yields the third equation:

\frac{G^{2}}{2 γ} \frac{d ψ}{d z} + 2 R^{2} (4 C^{2} + \frac{G^{2}}{γ} \frac{d C}{d z}) - κ_{0} - A^{2} + \frac{ν γ}{2 σ^{2}} A^{4} + 4 \frac{β}{γ} C = 0 .

The last Euler-Lagrange equation corresponds to the optimized curvature C:

\frac{G^{2}}{γ} (R^{2} \frac{d A}{d z} + 2 A R \frac{d R}{d z}) + A [R^{2} (4 C + 2 δ_{0} - \frac{3 ε}{σ} A^{2} + \frac{2 μ γ}{σ^{2}} A^{4}) + \frac{β}{γ} (1 + 12 C^{2} R^{4})] = 0,

where δ₀ = G²δ/(2γ) and κ₀ = G²κ/(2γ). The laser beam power P = 2π ∫ |u|² rdr = πS!B²G² A²R², angular momentum

M = 2 π \int Im (u^{*} \frac{\partial u}{\partial φ}) d r

, and vortex width

L = \sqrt{(2 π / P) \int {| u |}^{2} r^{3} d r}

are no more conserved in the dissipative systems, as opposed to the conservative ones. Following the variational method for dissipative systems, after some algebra a set of four ordinary differential equations (ODEs) is obtained from Eqs. (11–14) [13,16,30]:

\frac{d A}{d z} = \frac{γ A}{G^{2}} [\frac{5 ε}{σ} A^{2} - 4 (C + \frac{μ γ}{σ^{2}} A^{4} + \frac{β}{γ} R^{2} C^{2}) - \frac{3 β}{γ R^{2}} - 2 δ_{0}],

\frac{d R}{d z} = \frac{γ R}{G^{2}} (4 C - \frac{ε}{σ} A^{2} + \frac{μ γ}{σ^{2}} A^{4} + \frac{β}{γ R^{2}} - 4 \frac{β}{γ} C^{2} R^{2}),

\frac{d C}{d z} = \frac{γ}{G^{2}} (\frac{1}{R^{4}} - 4 C^{2} - \frac{A^{2}}{R^{2}} + \frac{ν γ}{σ^{2}} \frac{A^{4}}{R^{2}} - 4 \frac{β}{γ} \frac{C}{R^{2}}),

\frac{d ψ}{d z} = \frac{γ}{G^{2}} (κ_{0} - \frac{2}{R^{2}} + 3 A^{2} - \frac{5}{2} \frac{ν γ}{σ^{2}} A^{4} + 4 \frac{β}{γ} C) .

To simplify the problem a bit, we consider the approximation β << 1. The stationary solutions of the set of ODEs, Eqs. (15–17), are obtained by solving them for the vanishing derivatives of the amplitude, width, and curvature. The fixed points of these equations correspond to the steady state solutions of Eq. (8) in the variational approximation, with the wavefront curvature C = 0.25[(ε/σ− β/γ)A²−(μγ− βν)A⁴/σ²](1+ β²/γ²)⁻¹ and the radius R = (A² − (νγ/σ²)A⁴)^−1/2. Concerning the amplitude, the family of solutions reduces, in this dissipative case, to a fixed double solution for a given set of dissipative parameters. Indeed, the amplitude in a steady state solution of Eqs. (15–17) has two discrete values A⁺ and A⁻,

A^{\pm} = \sqrt{\frac{(2 ε / σ - β / γ) \pm \sqrt{{(2 ε / σ - β / γ)}^{2} - 2 δ_{0} (3 μ γ / σ^{2} - 2 β ν / σ^{2})}}{(3 μ γ / σ^{2} - 2 β ν / σ^{2})} .}

According to the general principles of the analysis of dissipative systems, A⁺ solution may be stable, while A⁻ is always unstable [1, 12, 13]. The bifurcation point is given by

ε = σ [β / γ + \sqrt{2 δ_{0} / σ^{2} (3 μ γ - 2 β ν)}] / 2

. In Fig. 2, the upper stable branch corresponds to the A⁺ solution, while the lower unstable one represents the A⁻ solution. The former solution satisfies the condition C < 0 that is necessary for the self-organization of dissipative vortex solitons, based on the simultaneous cross-compensation between loss and gain, as well as the competing cubic-quintic nonlinearity that exceeds diffraction [13,29,30]. A steady state solution of the above set of ODEs, Eqs. (15–17) is stable in Lyapunov sense if and only if all the eigenvalues λ of the Jacobian

λ^{3} + α_{1} λ^{2} + α_{2} λ + α_{3} = 0

have negative real parts [13,29,30]. Therefore, the Hurwitz coefficients of Eq. (20) have to be all positive in order for a fixed point to be in the stable dynamic equilibrium,

α_{1} = \frac{γ}{G^{2}} [\frac{6 β}{γ R^{2}} + 16 \frac{μ γ}{σ^{2}} A^{4} - 10 \frac{ε}{σ} A^{2} + 8 C (1 + \frac{β}{γ} {CR}^{2})] > 0,

\begin{array}{l} α_{2} & = & \frac{γ^{2}}{G^{4}} \frac{8}{R^{4}} [1 + \frac{β^{2}}{γ^{2}} - A^{2} R^{2} (1 + 6 \frac{β ε}{γ σ} - 9 \frac{β μ}{σ^{2}} A^{2} - 2 \frac{ν γ}{σ^{2}} A^{2}) \\ - 2 {CA}^{2} R^{4} (\frac{β}{γ} + 5 \frac{ε}{σ} - 8 \frac{μ γ}{σ^{2}} A^{2} - 2 \frac{β ν}{σ^{2}} A^{2}) \\ + 4 C^{2} R^{4} (1 + \frac{β^{2}}{γ^{2}} - 3 \frac{β ε}{γ σ} A^{2} R^{2} + 5 \frac{β μ}{σ^{2}} A^{4} R^{2})] > 0, \end{array}

\begin{array}{l} α_{3} & = & \frac{γ^{3}}{G^{6}} \frac{32 A^{2}}{R^{4}} [\frac{β}{γ} - 2 \frac{ε}{σ} + \frac{β^{2}}{γ^{2}} \frac{ε}{σ} + A^{2} (3 \frac{μ γ}{σ^{2}} + \frac{β^{2} μ}{γ σ^{2}} - 2 \frac{β ν}{σ^{2}}) \\ - 4 {CR}^{2} \frac{β}{γ} (\frac{ε}{σ} - \frac{β}{γ} - 2 \frac{μ γ}{σ^{2}} A^{2} + 2 \frac{β ν}{σ^{2}} A^{2}) \\ - 4 C^{2} R^{4} (2 \frac{ε}{σ} + \frac{β}{γ} + 3 \frac{β^{2}}{γ^{2}} \frac{ε}{σ} - 3 \frac{μ γ}{σ^{2}} A^{2} - 5 \frac{β^{2} μ}{γ σ^{2}} A^{2} - 2 \frac{β ν}{σ^{2}} A^{2})] > 0, \end{array}

and

α_{123} = α_{1} α_{2} - α_{3} > 0 .

Such a stability criterion determines the stable fixed points for a large domain of dissipative parameters. We established that their stability crucially depends on the dimensionless nonlinear gain parameter ε and the nonlinear loss parameter μ. Therefore, the domain of stable steady state solutions is charted as a function of those two parameters in Fig. 3. Following our exhaustive numerical simulations, the variation of the remaining two dissipative coefficients associated with the linear loss (δ) and field diffusion (β) does not entail essential changes. Consequently, main results are adequately represented by fixing these parameters to δ = 0.01, and β = 0.05. Also, the diffusion parameter is fixed at γ = 0.0034 in accordance with the parameters κ, σ and ν fixed earlier. The realistic numerical values of dissipative parameters can be determined experimentally by the choice of laser beam intensity and the active background medium, e.g. by varying the dye concentration. Indeed, in the domain of random lasers, dye-solutions containing TiO₂ or silica nanoparticles were employed for the observation of laser-like emission [26,31]. Up-conversion random laser emission was realized in various colloidal dye solutions [32–34].

Fig. 2 Illustrating the stability of the beam amplitude A as a function of the nonlinear gain ε. The upper (solid line) branch corresponds to the stable A⁺ solution and the lower (dashed line) branch corresponds to the unstable A⁻ solution.

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Fig. 3 Stability domain (in yellow) produced by the VA-generated fixed points, in the plane of the nonlinear-gain strength ε and the nonlinear-loss strength μ (both dimensionless). The stability of vortex solitons is confirmed by direct simulations of Eq. (8) for parameters inside the region delimited by dashed lines.

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4. Numerical self-generation of stable dissipative vortex solitons

In section 3 we demonstrated that, following our variational method extended to dissipative systems, for each set of dissipative parameters, plotted in Fig. 3, only one stable steady state solution exists, as opposed to the family of NLS solitons [12, 13, 16, 29, 30]. Each such fixed point solution can be used as an input for numerical solution of Eq. (8), in order to confirm its solitonic nature [13,16,29,30]. The amplitude A, power P, and angular momentum M are recorded after each propagation step, from z = 0 until z = 21500 (see Fig. 4 for a typical choice of dissipative parameters, ε = 0.38 and μ = 2). Only after a long evolution (about 7000 steps in Figs. 4 and 5) the amplitude, power, and angular momentum start to perform steady state oscillations, while the width L remains practically constant. Hence, a breathing vortex soliton gets self-organized and remains stable during the subsequent evolution. Nanoparticles with positive polarizability are attracted by the laser field of volcano shape. During numerical evolution, nanoparticles are partially filling the crater, as can be seen from Fig. 5. Simultaneously, particles are climbing Gaussian hill from outside, enlarging the plateau of the “millstone” in Fig. 5. The particles concentration growth induces the nonlinear susceptibility increase. Consequently, the self-focusing induced by the nonlinear refractive index causes amplitude growth that would lead to catastrophic collapse in the absence of losses (see the amplitude behavior in Figs. 4 and 6). Linear and nonlinear losses, together with the quintic self-defocusing and diffraction, saturate this growth. This is a slow cyclic process with the breathing period of T = 240 propagation steps. Indeed, after this period the amplitude, power, and angular momentum suddenly start decreasing (see Fig. 6). This would annihilate the soliton without the refluxing effect of the gain that restores the amplitude and consequently, the power and angular momentum, initiating a new breathing cycle. This regular breathing behavior (see Figs. 4–6) is the signature of breathing vortex soliton self-trapping. The regular modulation of the breather over the period T = 240 is shown in Fig. 7 (see also [16,29]). At the amplitude maximum (z ≈ 20160), the vortex resembles a “millstone” with sharp edges. An abrupt fall of the amplitude and quasi-immediate recovery at z = 20170 transforms the “millstone” into a dome. During amplitude saturation (until z ≈ 20220), the edges become sharper. At z = 20400 the vortex resumes its pre-cycle shape. This robust cyclic behavior confirms that such a vortex with constant width is really a breathing soliton. Exhaustive numerical simulations allowed us to establish an area between dashed lines in Fig. 3 where the stability of dissipative vortex solitons is confirmed.

Fig. 4 Amplitude A, power P, and angular momentum M monitored after each propagation step, from z = 0 to z = 21500 (for dissipative parameters ε = 0.38 and μ = 2). The ratio M/P corresponds to the constant value of S = 1.

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Fig. 5 Development of a stable breathing vortex with S = 1. Initial vortex (at z = 0) corresponds to the input Gaussian with parameters β = 0.05, δ = 0.01, μ = 2, and ε = 0.38 from the stability domain. Breathing vortex soliton is self-generated after z = 7000 steps. It remains stable after z = 21000 steps.

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Fig. 6 Regular breathing behavior of the beam characteristics. It corresponds to the vortex soliton self-organization. Different scales are used for amplitude A, power P, angular momentum M, and width L.

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Fig. 7 Robust cyclic behavior with period T = 240 from a “millstone” at z = 21160 through various domes, to the same “millstone” at z = 21400, confirming the stability of a breathing vortex soliton.

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For larger gain parameter ε, the breathing vortex soliton self-organizes more rapidly. For instance, for gain and loss parameters ε = 1.8 and μ = 3 in Fig. 8, the breathing with considerably shorter period T_A = 4 starts already after 500 steps. The breather modulation from a “millstone” into a dome and back is much shorter. Figure 9 displays the perfect correspondence between domes and “millstones” in two cycles, distant for 20 steps. Such a rapid breathing does not alter the remarkable stability of dissipative vortex solitons. However, a new superimposed beat of period T_B ≈ 500 appears at double the breathing period (half the frequency) for the smaller parameter ε (see M evolution in Fig. 8). The same results can be obtained for the negative polarizability nanoparticles that escape laser field and in such a way also increase the nonlinear susceptibility, contributing to the self-organization of dissipative vortex solitons. Such a self-structuring of the laser light in colloidal nanosuspensions can be applied for optical manipulation and selective dynamic tweezing of nanoparticles and nanorods [35,36], as well as for all-optical switching and signal processing [37].

Fig. 8 Similar to Fig. 4 but for larger gain and loss parameters, ε = 1.8 and μ = 3. The amplitude, power, and angular momentum oscillate with a short period T_A = 4 and a superimposed beat period, T_B = 500, as a double breather. The breathing with shorter period T_A = 4 starts already after 500 steps.

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Fig. 9 Perfect correspondence between domes and “millstones” appearing successively after T_A + 1 = 5, in two cycles distant for 20 steps.

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5. Conclusions

We use analytical and numerical approaches to study the localization of light in a model of artificial media. Such models include metal-dielectric nanocomposites formed as colloidal suspensions of either gold or silver nanoparticles in liquids, polymers, and liquid crystals. These media can be suitably engineered, so as to enhance the competing nonlinear susceptibilities by the migration of nanoparticles toward the laser field or away from it. Independent of their polarizability, the nanoparticles move in such a way that the effective nonlinearity inside the beam always increases. We used the model of interacting hard-sphere nanoparticles, as well as experimental results from Ref. [25], to demonstrate higher-order nonlinear susceptibility in nanosuspensions and to corroborate the validity of the CQCGL equation, which describes the self-structuring of laser light into vortex solitons in such media. The corresponding generalized Ginzburg-Landau model exhibits overcompensation of diffraction by the saturating cubic-quintic nonlinearity, expressed analytically and numerically by the negative wave front curvature. This effective self-focusing would cause catastrophic collapse in the absence of losses. The amplification coming from the medium enhances laser field, preventing the soliton annihilation. Consequently, the self-focusing again dominates diffraction, restarting a new self-organization cycle. Therefore, the simultaneous cross-compensation between loss and gain, and the competing cubic-quintic nonlinearity exceeding diffraction, allows self-structuring of laser light into novel practically dissipationless breathing vortex solitons. The established breathers may act as robust dynamic tweezers of nanoparticles, controlling in real time the spatial distribution of effective nonlinear susceptibility in metal-dielectric nanocomposites and allowing for engineering of reconfigurable guides and smart grids.

Funding

National Priorities Research Program Grants No. 6-021-1-005 and No. 9-020-1-006 from the Qatar National Research Fund (a member of Qatar Foundation). Work at the Institute of Physics Belgrade is supported by the Ministry of Education and Science of the Republic of Serbia, under Project OI 171006.

Acknowledgments

This publication was made possible by the National Priorities Research Program Grants No. 6-021-1-005 and No. 9-020-1-006 from the Qatar National Research Fund (a member of Qatar Foundation). MRB acknowledges support from the Al Sraiya Holding Group.

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Self-structuring of stable dissipative breathing vortex solitons in a colloidal nanosuspension

Abstract

1. Introduction

2. Theoretical analysis

3. Stability criterion

4. Numerical self-generation of stable dissipative vortex solitons

5. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (9)

Equations (24)

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