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ISSN: 0256-307X 中国物理快报 Chinese Physics Letters Volume 31 Number 7 July 2014 A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/0256-307X http://cpl.iphy.ac.cn C HINESE P HYSICAL S OCIET Y CHIN. PHYS. LETT. Vol. 31, No. 7 (2014) 074210 Novel Dynamics of a Vortex in Three-Dimensional Dissipative Media with an Umbrella-Shaped Potential * LIU Yun-Feng(4云Â), LIU Bin(4Q)**, HE Xing-Dao(何,道), LI Shu-Jing(o淑·) Key Laboratory of Nondestructive Test (Ministry of Education), Nanchang Hangkong University, Nanchang 330063 (Received 21 March 2014) We report the novel dynamic of 3D dissipative vortices supported by an umbrella-shaped potential (USP) in the 3D complex Ginzburg–Landau (CGL) equation with the cubic-quintic nonlinearity. The stable solution of vortices with intrinsic vorticity S=1 and 2 are obtained in the 3D CGL equation. An appropriate USP forces the vortices continuously to throw out fundamental 3D solitons (light bullets) along the folding umbrella. The dynamic regions of the strength of the potential with the changing number of folding umbrella are analyzed, and the rate of throwing increases with the strength of the potential. A weak potential cannot provide vortices with enough force. Then, the vortices will be stretched into polygons. However, a strong potential will destroy the vortices. PACS: 42.65.Tg, 05.45.−a DOI: 10.1088/0256-307X/31/7/074210 Solitons in optical media have attracted much In this Letter, we introduce a 3D CGL model with attention.[1−14] A spatiotemporal soliton is referred to an external umbrella-shaped potential. We consider as a ‘light bullet’ localized in all spatial dimensions the nonlinear dynamic on stable dissipative vortices and in the time dimension. The generation of a light initially placed at the apex of the potential. The ex- bullet might be of importance in soliton-based com- tra force of the potential breaks the original dynam- munication systems. An optical vortex soliton is a ical balance of the central vortices. A series of novel self-localized nonlinear wave, which has a point (sin- dynamics are observed. Especially, light bullets are gularity) of zero intensity, and with a phase that twists periodically thrown out from the central vortex. around that point, with a total phase accumulation of We consider the following 3D CQ CGL equation 2휋S for a closed circuit around the singularity.[15] The in terms of nonlinear optics, as the evolution equation quantity S is an integer number known as the vorticity for the amplitude of an electromagnetic wave in an or topological charge of the solution. active bulk optical medium,[27;36] Complex Ginzburg–Landau (CGL) equations are (︁D )︁ well known as basic models of the pattern forma- iu + 푖훿u + (1=2 − 푖훽)(u + u ) + + 푖훾 u z xx yy 2 tt tion in various nonlinear dissipative media, such + (1 − i")juj2u − (휈 − 푖휇)juj4u = F (x; y)u; (1) as in superconductivity and superfluidity, fluid dy- namics, reaction-diffusion phenomena, nonlinear op- where (x; y) and t are the transverse coordinates and tics, Bose–Einstein condensates, and quantum field temporal coordinate, z is the propagation distance. theories.[16−18] The CGL equation with the cubic- The coefficients of diffraction and cubic self-focusing quintic nonlinearity has been widely used in nonlin- nonlinearity are scaled respectively, to be 1/2 and 1, ear dissipative optics. Among the important appli- D is the group-velocity dispersion (GVD) coefficient. cation are passively mode-locked laser systems and In the following we set D = 1=2 for the anomalous optical transmission lines.[19] A series of nonlinear dispersion propagation regime, 휈 is the quintic self- dynamical reports in this model have been focused defocusing coefficient, 훿 is the coefficient correspond- on the complex stable patterns[20−28] and interac- ing to the linear loss (훿 > 0) or gain (훿 < 0), 휇 > 0 ac- tions of localized pulses.[29−32] Recently, adding ex- counts for the quintic-loss parameter, and " > 0 is the ternal potentials in these models has been a theme cubic-gain coefficient, 훾 > 0 accounts for spectral fil- of extensive studies.[34−36] Desirable patterns of the tering in optics, 훽 is the spatial-diffusion term, which refractive-index modulation in materials described by appears in a model of laser cavities, where it is gener- CGL equation, which may induce the effective poten- ated by the interplay of the dephasing of the local po- tials, can be achieved by means of various techniques, larization in the dielectric medium, cavity loss, and de- such as optics induction[37] and writing patterns by tuning between the cavity’s and atomic frequencies.[39] streams of ultrashort laser pulses.[38] As mentioned above, we will keep 훽 > 0 to secure the *Supported by the National Natural Science Foundation of China under Grant Nos 61205119, 41066001 and 11104128, the Natural Science Foundation of Jiangxi Province under Grant No 20132BAB212001, and the Natural Science Foundation of Jiangxi Province Office of Education under Grant No GJJ13485. **Corresponding author. Email: [email protected] © 2014 Chinese Physical Society and IOP Publishing Ltd 074210-1 CHIN. PHYS. LETT. Vol. 31, No. 7 (2014) 074210 stability of the vortex solitons in the 3D CGL model. as E(S = 1)=415, and E(S = 2) = 719. The vortices The last term on the right-hand side of Eq. (1) in- with S > 2 have been reported in Ref. [32], which can- troduces the umbrella-shaped potential in the trans- not steadily propagate. verse plane. The analytical form of F (x; y) is First, we consider the dynamics of vortices with S = 1, placed at the apex of the USP. A series of novel F (x; y) = − arjcos(푚휃=2)j; m ≥ 3 dynamics are studied by performing a large number of p r = x2 + y2; (2) numerical simulations. The circular vortex is gradually stretched into a where 휃 is the angular coordinate, and a is the strength polygonal one by a weak potential. Figure 2(a) shows of the potential. Integer m stands for the number of that a stable vortex with S = 1 evolves into quadrate folding umbrellas. at a = 0:03. The evolutions of energy at a = 0:02 and 0.03 (shown in Fig. 2(e)) reveal that the size of (a) the polygonal vortex increases with a. 15 1.5 2π t 0 (a) z/ z/ z/ z/ z/ z/ -15 -15 15 0 0 0 0 (b) x 15 -15 y (b) 1.5 2π 15 z/ z/ z/ z/ z/ z/ t 0 -15 (c) -15 15 0 0 0 0 x 15 -15 y z/ z/ z/ z/ z/ z/ Fig. 1. (Color online) The profile (a) and phase (b) of stable 3D vortex solutions with topological charge S=1 (d) and 2, respectively. z/ z/ z/ z/ z/ z/ We have solved Eq. (1) using a split-step Fourier 800 (e) 3000 (f) method with typical transverse and longitudinal step 600 2000 sizes ∆x = ∆y = ∆t = 0:3 and ∆z = 0:1 as described E 400 1000 in the following. The second-order derivative terms in 200 x, y, and z are solved in Fourier space under the peri- 0 0 0 500 1000 1500 2000 0 500 1000 1500 odic boundary conditions. Other linear and nonlinear z z terms in the equation are solved in real space by using Fig. 2. (a)–(d) Isosurface plots of total intensity a fourth-order Runge–Kutta method. ju(x; y; t)j2, evolutions of the central vortex with S = 1 Generic results may be adequately represented by at (a = 0:03, m = 4), (a = 0:08, m = 4)(a = 0:1, m = 4), setting 훿 = 0:4 휇 = 1, " = 2:43, 훾 = 훽 = 0:5, and and (a = 0:16, m = 4). (e) Evolutions of the energy region 휈 = 0:1, corresponding to a physically realistic situ- of a at m = 4 with a = 0:02 (solid line) and 0.03 (dashed line). (f) Evolutions of energy at m = 4 with a = 0:08 ation, and making the evolution relatively fast, thus (dashed line) and 0.1 (solid line). helping to elucidate its salient features.[29;31] Table 1. The relationship between a with m for S = 1. The initial solutions of the vortex in Eq. (1) are set as follows: m Region of a for throwing light bullet from vortices with S=1 2 2 h (︁ r t )︁]︁ 4 0.038≤ a ≤0.142 u(z = 0; r; t) = AjrjS exp − + exp(iS휃); w2 w2 5 0.041≤ a ≤0.128 (3) 6 0.044≤ a ≤0.116 7 0.045≤ a ≤0.106 where A is the amplitude, and w is the width. We dis- 8 0.046≤ a ≤0.098 tinguish the vortex solutions with topological charge S, also called the angular momentum quantum num- For an appropriate potential, the vortices are ber. The stable vortices with S = 1 and 2 are obtained forced to continuously throw out m jet pulses along in the numerical form by an image-step propagation the slopes of the potential. Then the pulses will self method without potential, as shown in Figs. 1(a) and trap into fundamental solitons. The dynamical evo- 1(b). In this case, the 3D vortices with S = 1 and 2 lutions of continuously throwing out four jets of fun- all are stable, characterized by the following values of damental solitons at a = 0:08 and 0.1 are shown in the energy (alias norm) Figs. 2(b) and 2(c), respectively.

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