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We report novel dynamical regimes of “light bullets” supported by an annularly periodic potential in the three-dimensional (3D) complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. This is a model of an active optical medium with respective expanding anti-waveguiding structures with (integer) annularly periodic modulation. If the potentials are strong enough, they give rise to continuous generation of m jets light bullet by an initial light bullet initially placed at the center. The influence of m and diffusivity term (viscosity) β on the corresponding strength of potential is studied. In the case of m = 0 (conical geometry), these are concentric waves expanding in the radial direction.
©2011 Optical Society of America
1. Introduction
Complex Ginzburg-Landau (CGL) equations are well known as basic models of the pattern formation in various nonlinear dissipative media [1–3]. The CGL equation with the cubic-quintic (CQ) nonlinearity has been widely used in nonlinear dissipative optics, due to the clear physical meaning of all its terms in any particular application. Among the important application are passively mode-locked laser systems and optical transmission lines [4]. The reports in this model have been focused on complex stable patterns [5–13] and interactions of localized pulses [14–16] in this model have been reported. Recently, adding external potentials in CGL models has been a theme of extensive studies, expanding the already wide spectrum of relevant application [17–22].
Spatiotemporal solitons (STSs) in optical media have attracted much attention [23–30]. A spatiotemporal soliton is referred to as a “light bullet” localized in all spatial dimensions and in the time dimension. The generation of a “light bullet” might be of importance in soliton-based communication systems, where each soliton represents a bit of information. Recently, the existence of stable light bullets with complicated shaped was demonstrated using the 3D CGL equation for any sign of chromatic dispersion [11,31] and for a wide range of equation parameters. We consider the study on dynamics of light bullet in 3D CGL models with adding the external potentials. Desirable patterns of the refractive-index modulation in materials described by CGL equation, which may induce the effective potentials, can be achieved by means of various techniques, such as optics induction [32] and writing patterns by streams of ultrashort laser pulses [33].
In this paper, we introduce the 3D CGL models with external annularly periodic potentials. We consider the action of these potentials on dissipative light bullets initially placed at the central position (apex of the respective potential). The extra force of potential will break the original dynamical balance of central STS. A novel dynamic that the central STS continuously emits m (annular periods) jets fundamental STS along symmetry directions of the potential is observed. The region of strength of potential by variety of m and diffusion term is obtained by performing large number of numerical simulations. Finally, the potential with m = 0 gives rise to the generation of an array of concentric annular STSs.
2. The model
We consider the following 3D CQ CGL equation in terms of nonlinear optics, as the evolution equation for the amplitude of electromagnetic wave in an active bulk optical medium [12,20]
Where (x,y) and t are the transverse coordinates and temporal coordinate. z is the propagation distance. The coefficients of diffraction and cubic self-focusing nonlinearity are scaled respectively, to be 1/2 and 1. D is group-velocity dispersion (GVD) coefficient. Below, we set D = 1/2 is for the anomalous dispersion propagation regime. ν is the quintic self-defocusing coefficient, δ is the coefficient corresponding to the linear loss (δ>0) or gain (δ<0), m>0 accounts for the quintic-loss parameter, and ε>0 is the cubic-gain coefficient. accounts for spectral filtering in optics, while β>0 is the spatial-diffusion term, appears in a model of laser cavities, where it is generated by the interplay of the dephasing of the local polarization in the dielectric medium, cavity loss, and detuning between the cavity’s and atomic frequencies [34]. Generic results may be adequately represented by setting δ = 0.5, m = 1, ε = 2.498, , and ν = 0.115, which corresponds to a physically realistic situation and, simultaneously, makes the evolution relatively fast, thus helping to elucidate its salient features [16,19].The last term on the right-hand side of Eq. (1) introduces the annularly periodic potential in the transverse plane. The analytical form of is
where is the angular coordinate, and a is the strength of potential. stands for integer periods in angular coordinate.3. Results and analysis
The stable STS solution was obtained in the numerical form by image-step propagation method without the external potential (see Fig. 1 ) with the center placed at the apex of the potential. Then, simulations of Eq. (1) reveal several typical dynamical regimes for appropriate strength a. A novel dynamical regime that the central STS continuously generates m streams of secondary pulses is observed, as shown in Figs. 2(a) and (b) at (m = 4, a = 0.08) and (m = 8, a = 0.07). The gain term in the CQ CGL equation is the source of the energy necessary for continuous generation. The emitted pulses self-trap into fundamental STSs which slide along slopes of the potential. We have calculated the evolution of total amount of energy E that it carries:
as a function of propagation distance z. The evolutions of E [shown in Fig. 2(c)] reveal an obviously periodic generation of m STSs. z1 and z2 stand for the propagation distance of one period at a = 0.08 and 0.09, respectively. The comparison of them demonstrates that the stronger potential provides for a higher emission rate. In addition, the corresponding region of a is shown in Fig. 2(d). The upper critical value reduce with the growth of m, but the lower critical value (a = 0.05) is almost unchanged.
Fig. 2 (Color online) (a) (b) Isosurfaces plot of total intensity , evolutions of the central STS at (a = 0.08, m = 4) and (a = 0.07, m = 8). (c) Evolutions of energy at m = 4 with a = 0.08 (blue line) and 0.09 (black line). (d) Region of a for continuous generation depend on m. (e) Region of a depend on β, at m = 4.
The viscosity term in Eq. (1), ~β, plays an important role in maintaining the position of the central STS. With little or zero viscosity, the central STS is always subject to the drift instability, starting to roll down from the tip of the potential. So, we study the influence of β on the regime of continuous generation. The gray region in Fig. 2(e) shows the region of a by varying β. The lower critical value (a = 0.05) is almost unchanged, but the upper critical value significantly decreases with the growth of β. At β = 0.6, there is no appropriate strength a for attaining continuous generation.
For a weaker potential in , jets of secondary STSs are not generated. Instead, the simulations demonstrate a gradual stretch of the central STS, as shown in Fig. 2(a) with a = 0.04. By comparing the evolution of total energy at (m = 4, a = 0.02) and (m = 4, a = 0.03) [black and blue lines in Fig. 3(c) ], the stretching force increases with a.
Fig. 3 (Color online) (a) (b) Isosurfaces plot of total intensity , evolutions of the central STS at (a = 0.12, m = 4) and (a = 0.04, m = 4). (c) Evolutions of the energy E at m = 4 for different a. The black, blue, red, pink and green lines correspond to a = 0.02, 0.03, 0.15, 0.2 and 1, respectively. (d) Region of a for dynamical regime in Fig. 3(b) depend on m.
For a stronger potential, see a typical example in Fig. 3(b) with m = 4 and a = 0.12, the generated pulses rapidly dissipate, failing to self-trap into secondary STSs, because the potential’s slope exceeds the critical value admitting steady motion of the dissipative solitons [19], while the central STS survives, despite the fact that it sits on the center. The region of a depending on m is shown in Fig. 3(d). The periodic evolutions of E are observed at (a = 0.15, m = 4) and (a = 0.2, m = 4) [red and pink lines in Fig. 3(c)].
But, if a potential stronger than in Fig. 3(d), the evolution of E at (m = 4, a = 1) [green line in Fig. 3(c)] shows that the combination of the sharp tip and steep slopes completely destroys the central STS.
In addition, we consider a conical potential with the circular cross section [m = 0 in Eq. (2)], with the simplified form as:
For, the simulations demonstrate that the central STS continuously generates ring-shaped patterns which expand in the radial direction, see an example in Fig. 4(a) for a = 0.12. The comparison of the evolutions of E in Fig. 4(c) at different a demonstrates that the generation rate also increases with a. The weaker potential in Eq. (4), with , causes the stretching of the central STS, without emitting concentric waves just as the dynamical regime in Fig. 3(a). The evolutions of E at a = 0.02 and 0.025 [shown in Fig. 4(d)] demonstrate that the stretching force also increase with a. On the other hand, a stronger potential, with , transforms the central STS into a single ring expanding in the radial direction, as shown in Fig. 4(b) for a = 0.45. The evolutions of E at a = 0.45 and 0.5 in Fig. 4(e), reveals that a stronger a leads to a rapidly expanding of ring pattern. At , the evolution of E [red line in Fig. 4(e)] reveals the central STS suffers a decay.
Fig. 4 (Color online) (Color online) (a) (b) Isosurfaces plot of total intensity , evolutions of the central STS at a = 0.12 and 0.45. (c) Evolutions of the energy E at a = 0.12 (blue line) and 0.15 (red line). (d) Evolutions of E at a = 0.02 (blue line) and 0.025 (red line). (e) Evolutions of E at a = 0.45 (black line), 0.5 (blue line) and 0.6 (red line).
4. Conclusions
We have introduced an annularly periodic potential into 3D CGL equation with the CQ nonlinearity. For an appropriate strength of potential, the setting gives rise to the continuous generation of arrays of secondary light bullets by an initial STS. The corresponding region of strength of potential decreases with the growth of annular periods m and diffusivity term β. Besides, the rate of generation increases with the strength of potential. If the potential is weak, the stretch of the central STS is observed instead. But for the conical potential (m = 0) readily generates arrays of concentric pulses expanding in the radial direction.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (Grant No. 41066001), Natural Science Foundation of Jiangxi, China (Grant No. 2009GZW0024), and Graduate Innovation Base of Jiangxi Province, and the Doctor Startup Foundation of Nanchang Hangkong University (Grant No. EA201008231).
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