Annularly and radially phase-modulated spatiotemporal necklace-ring patterns in the Ginzburg–Landau and Swift–Hohenberg equations

Bin Liu; Ying-Ji He; Zhi-Ren Qiu; He-Zhou Wang

doi:10.1364/OE.17.012203

1. Introduction

The complex Ginzburg–Landau (CGL) equation is an important model that occurs in many areas such as in superconductivity and superfluidity, fluid dynamics, reaction-diffusion phenomena, nonlinear optics, Bose–Einstein condensates, and quantum field theories [1,2]. Many dynamical behaviors are achieved in such a model, such as the formation of periodic patterns and dissipative solitons [1,2]. Complex stable patterns have also been investigated in dissipative models based on the CGL equation with cubic-quintic (CQ) nonlinearity, including stable vortices [3–6], stable soliton clusters [7,8], fusions of 2D necklace-ring patterns [9], stable spatiotemporal solitons [10,11], and stable spatiotemporal necklace-ring solitons [12].

Another dissipative system described by the complex Swift–Hohenberg (CSH) equation is derived by adding the four-order diffusion term to the CGL model [13–15]. The CSH model possesses stronger friction force than the CGL mode due to the presence of the higher-order term, which leads to some differences between them in optics. The CSH model also has been widely used to study various localized states, including the formation of complex patterns [16] and localized foundational patterns [13–15,17–19]. Some states of this type have been computed in this equation with quadratic and cubic nonlinearities [20,21]. The CGL and CSH equations are generic ones describing systems near subcritical bifurcations [22].

Spatiotemporal solitons in optical media have attracted much attention [23]. A spatiotemporal soliton is referred to as a “light bullet” localized in all spatial dimensions and in the time dimension. Recently stable spatiotemporal soliton clusters have been reported in Hamiltonian nonlinear systems [24,25] and in the Swift–Hohenberg model [26,27]. The generation of a “light bullet” is of importance in soliton-based communication systems, where each soliton represents a bit of information.

In this work we study spatiotemporal necklace-shaped patterns (SNPs) with annular or radial phase modulation in the 3D CGL and 3D CSH equations. We demonstrate that SNPs with annular phase modulation can fuse into stable fundamental or vortex solitons in both models when the initial radius of the necklace is smaller than a critical value, which is similar to the fusion of 2D necklace-shaped patterns into stable fundamental or vortex solitons in a CGL Eq. (9). We predict that SNPs keep the shape of a necklace by the balance equations of both energy and momentum when their radii exceed a critical value. We find that it is easier to implement the above fusions of solitons in the CSH model than in the CGL model due to the effect of the higher-order diffusion term. When a radial phase modulation is added to SNPs, the modulated “bead” will move towards or off the center of the necklace and rapidly vanish due to strong dissipation.

2. The model

We consider the (3 + 1)D CQ complex GL equation in a general form [4,12]:

i u_{Z} + i δ \cdot u + (1 / 2 - i β) (u_{X X} + u_{Y Y} + u_{T T}) + (1 - i ε) {| u |}^{2} u - (ν - i μ) {| u |}^{4} u = 0,

where ν is the quintic self-defocusing coefficient, δ is the linear loss (

δ < 0

) or gain (

δ > 0

) coefficient,

μ < 0

is the quintic-loss parameter,

ε > 0

is the cubic-gain coefficient, and

β > 0

accounts for effective diffusion (viscosity). The temporal viscosity of Eq. (1) is

β u_{T T}

, accounting for the spectral filtering in optics, while the spatial-diffusion term

β (u_{X X} + u_{Y Y})

is a feature known in specific models of laser cavities [17,18].

The addition of the fourth-order term is(u_XXXX + u_YYYY + u_TTTT) transforms the CGL equation to the CSH Eq. (13)–15]:

i u_{Z} + i δ \cdot u + (1 / 2 - i β) (u_{X X} + u_{Y Y} + u_{T T}) + (1 - i ε) {| u |}^{2} u - (ν - i μ) {| u |}^{4} u - i s (u_{X X X X} + u_{Y Y Y Y} + u_{T T T T}) = 0.

In the case of optical systems, the parameter su_TTTT is related to higher-order spectral filtering, and s(u_XXXX + u_YYYY) is the fourth-order diffraction. CGL Eq. (1) can be considered as the specific case of Eq. (2).

Following [28], the initial necklace-shaped pattern with amplitude A, mean radius R ₀, and width w can be taken (in polar coordinates r and θ) as

u (Z = 0, r, θ) = A \sec h [(r - R_{0}) / w] \cos (N θ) \exp (i M θ + i L r^{2}) .

Here, integer N determines the number of elements (“beads”) in the ring structure, which is 2N, and M and L are coefficients of annular and radial phase modulation, respectively.

The generic case can be adequately represented for parameters δ = 0.5, β = 0.5, ν = 0.11, and m = 1 for both CGL and SCH models and ε = 2.52 (for CGL model) and ε = 2.47 (for CSH model), which corresponds to a physically realistic situation and, simultaneously, makes the evolution relatively fast, thus helping to elucidate its salient features [4,5,11,25]. The typical coefficient of the higher-order term is s = −0.1 [15,22]. For these parameters, the amplitude and width of the individual 3D stable fundamental soliton are A = 1.2 and w = 2.5. The robustness of the SNPs is additionally tested in direct simulations of Eqs. (1) and (2) by multiplying Eq. (3) with [1 + ρ(X,Y,T)], where ρ(X,Y,T) is a Gaussian random function whose maximum is 10% of the soliton’s amplitude.

3. Results and analysis

When $M \neq 0$ and $L = 0$ , only the annular phase modulation is added to the SNPs. In this case, for M = N the SNPs evolve into a fundamental soliton when the initial radius R ₀ of the necklace is smaller than a maximum value, R _max [Fig. 1(a) ]. From Eq. (3), the mean phase shift between adjacent beads is Δϕ = Δϕ₀ + 2πM/(2N) = 0, where Δϕ₀ = -π corresponds to the out-of-phase difference between adjacent “beads” on the necklace with M = 0. Therefore, individual elements in the array, being in-phase, attract each other, which lead to their fusion into a stable, fundamental soliton in the CGL and CSH models, corresponding to Figs. 1(b) and 1(c), respectively. It is also easy to understand the increase of R _max with N [see Fig. 1(a)]. Indeed, the attraction between adjacent “beads” necessary for the fusion into the fundamental soliton is not too weak if the separation between them does not exceed a maximum value. The radius of the necklace grows linearly with N, which shows the roughly linear form of dependence R _max (N) in Fig. 1(a). From Figs. 1(b) and 1(c), the maximum value R _max in the CSH model is larger than that in the CGL model. This is because the effect of the higher-order diffusion in the CSH model produces stronger viscous forces between “beads,” which leads to easy fusion of the SNPs into a fundamental soliton with an even larger initial radius.

Fig. 1 Fusion into a fundamental soliton of the necklace array whose initial radius is not larger than R _max, and M = N. (a) The largest radius, admitting the fusion, versus N. Examples of the fusion of the SNPs are given at the propagation distance (from left to right) z = 0, z = 34, z = 102, and z = 170 with R₀ = 5 and M = N = 3 for (b) CGL and (c) CSH models [the examples (including the figures below) are shown by means of the isosurface plots of the power ${| u (X, Y, T) |}^{2}$ ].

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When radius R ₀ of the SNP is in a certain region [see Fig. 2(a) ], the SNP with annular phase modulation evolves into a stable vortex soliton, provided that M = N-1 or M = N + 1 (Fig. 2). The initial configuration in Eq. (3) may be realized as a vorticity whose absolute topological charge is 1, hence the vorticity component that may survive in the course of the evolution. Indeed, the simulations confirm that the emerging vortex solitons feature precisely this value of the vorticity. Similarly, the SNP evolves into a stable vortex soliton whose absolute topological charge is 2, provided that M = N-2 or M = N + 2 see (Fig. 3 ). The asymptotic form of the vortex at r→ 0 is $c o n s t \cdot r^{S}$ (here $S \equiv | M - N |$ ), which shows that topological charge 1 corresponds to the smaller radius of the inner hole in the vortex soliton than that of topological charge 2 [see Figs. 2(b), 2(c), 3(b), and 3(c)]. But, SNPs cannot evolve into stable vorticities with a larger topological charge than 2 because such vorticities were unstable for the CGL Eq. (4) as well as for the CSH equation in our simulations. The values of the radius of the SNPs in the CSH model are smaller than those in the CGL model from Figs. 2(a) and 3(a). This is the reason that if the radius of the SNP is too small, some solitons will fuse into a soliton as described in [9,12]. The viscidity of the CSH model is stronger than that of the CGL model due to its higher-order diffusion, which allows the SNP with a smaller radius to be transformed into the vortex soliton in the CSH mode.

Fig. 2 Fusion into a stable vortex soliton with topological charge 1. (a) The region of the initial radius admitting the fusion versus N; evolutions of fusion of the SNPs plotted at z = 0, z = 34, z = 102, and z = 170 with R₀ = 6 and M = 4 for (b) CGL and (c) CSH models.

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Fig. 3 Fusion into a stable vortex soliton with topological charge 2. (a) The regions of initial radius admitting the fusion versus N; evolutions of fusion of the SNPs given at z = 0, z = 34, z = 102, and z = 170 with R₀ = 6 and M = 3 for (b) CGL and (c) CSH models.

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If the initial radius exceeds a larger threshold value R _min, the interaction between the “beads” in the necklace array becomes very weak, irrespective of its topological charge M. As follows, we can predict the size of the R _min for the bound SNPs by the balance equations of both energy and momentum. The CGL and CSH equations have no known conserved quantities. Instead, the rate of change of energy and momentum with respect to z are [29]:

\begin{array}{l} \frac{d E [u]}{d z} = 2 \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\int_{- \infty}^{\infty} {- δ {| u |}^{2} + ε {| u |}^{4} - μ {| u |}^{6} - β (| u_{X} |}^{2} + {| u_{Y} |}^{2} + {| u_{T} |}^{2}) \\ + s / 2 [u^{*} (u_{X X X X} + u_{X X X X} + u_{X X X X}) + u (u_{X X X X}^{*} + u_{Y Y Y Y}^{*} + u_{T T T T}^{*})]} d X d Y d T \equiv F [u], \end{array}

\frac{d M [u]}{d z} = i \int_{- \infty}^{\infty} {(- δ + ε {| u |}^{2} - μ {| u |}^{4}) (u_{x}^{} u^{*} - u_{x}^{*} u) + β (u_{x}^{} u_{x x}^{*} - u_{x}^{*} u_{x x}) + s (u_{x}^{} u_{x x x x}^{*} - u_{x}^{*} u_{x x x x})]} d x \equiv J [u] .

By means of two-soliton perturbation theory, two solitons with out-of-phase difference are:

u (X, Y, T) = u_{0} (X - x_{0} / 2, Y, T) \exp (i π) + u_{0} (X + x_{0} / 2, Y, T),

where u₀(X,Y,T) is the stable soliton solution and x ₀ is the minimum separation of two solitons. Thus, the corresponding solutions must satisfy the set of two Eqs. (29): F[u] = 0 and J[u] = 0. The former equation indicates the necessary balance that must exist between loss and gain for any stationary solution, while the latter equation guarantees the balance between the transverse forces acting on solitons. The stationary solution is indicated by intersections A (CSH model) and B (CGL model) in Fig. 4(a) , respectively, corresponding to the minimum separations x ₀ = 10.7 and x ₀ = 11.6. This result can achieve the minimum radius of the SNP for keeping the necklace shape by an approximate relation: x ₀ ≈π R _min/N, which is in agreement with the simulation results as shown in Fig. 4(b).

Fig. 4 (a) Intersections based on energy F[u] = 0 (solid curves) and momentum J[u] = 0 (dotted curve); A and B indicate the bound states as a function the separation x ₀, respectively, corresponding to the CSH (blue curve) and CGL (red curve) models. Note that J[u] = 0 occurs in the CSH and CGL models for bound states. (b) Minimum radius of the necklace achieved by simulations admits the keeping of the necklace-ring shape.

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As a result, the pattern keeps its necklace-like structure and the initial radius, thus taking the form of a stable SNP (see Fig. 5 ). The dependence of the respective minimum radius R _min on modulation number N of initial pattern in Eq. (3) is approximately linear [see Fig. 5(a)]. Note that each individual element in the established SNPs observed in Fig. 5 features an isotropic (circular) shape, unlike the “beads” in the initial pattern. This is explained by the fact that each “bead” evolves into a fundamental soliton.

Fig. 5 The SNPs keep necklace-ring shapes. Evolutions of the SNPs given at z = 0, z = 25, and z = 50 with N = 6, and R0 = 13.6 for (a) CGL model and (b) R0 = 12.5 for CSH model.

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Finally, we radially phase modulate the SNP with R ₀ = 10 by setting $L \neq 0$ and M = 0 in Eq. (3). For L < 0, the modulated “bead” moves toward the center of necklace. And if L is up to a certain value, the moving “bead” rapidly disappears (decays to zero) upon propagation as shown in Fig. 6 , which is the result of the strong dissipative property of the models. The strong dissipation is generated by the diffusion term in Eqs. (1) and (2) (the one proportional to β and s) in the necklace array. Naturally, the diffusive dissipation is stronger for a larger radial phase gradient. Note that the CGL and CSH equations, being dissipative ones, do not have any dynamical invariant, hence the total momentum is not conserved.

Fig. 6 Switching off a “bead” on the necklace by radially phase modulating the “bead” in (a) CGL and (b) CSH models. The evolutions are plotted at z = 0, z = 25, and z = 50. Left column is the phase maps, and the arrow indicates the radial phase modulation with L = −0.5.

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4. Conclusions

We have studied SNPs with annular and radial phase modulating of necklaces in two kinds of models of dissipative optical media based on the 3D CGL and CSH equations with cubic-quintic nonlinearity. In the presence of annular phase modulation, the SNPs in both models can fuse into stable fundamental and vortice solitons provided that the initial radius of the ring R ₀ is not too large. The key parameters that determine the results of the evolution are modulation number N and topological charge M of the initial necklace of Eq. (3). These results are similar the 2D cases in a CGL Eq. (9). When radial phase modulation is added to the SNP, the modulated “beads” will move toward or off the center of the necklace and quickly vanish due to the strong dissipation by which some “beads” can be switching off upon propagation. This offers a potential application to optical switching in signal processing.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NNSFC) grants 10874250 and 10674183, National 973 Project of China grant 2004CB719804, Ph.D. Degrees Foundation of Ministry of Education of China grant 20060558068, and Introductory Programs for Science and Technology Development of Bureau of Science and Technology of Guangzhou Municipality grant 2005Z3-C7451. B. Liu and Y.-J. He equally contributed to this work.

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Annularly and radially phase-modulated spatiotemporal necklace-ring patterns in the Ginzburg–Landau and Swift–Hohenberg equations

Abstract

1. Introduction

2. The model

3. Results and analysis

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (6)

Optics Express