Self-organized Propagation of Spatiotemporal Dissipative Solitons in Saturating Nonlinear Media

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1 PIERS ONLINE, VOL. 4, NO. 6, Self-organized Propagation of Spatiotemporal Dissipative Solitons in Saturating Nonlinear Media V. Skarka 1 and N. B. Aleksić 2 1 Laboratoire POMA, UMR 616 CNRS, Université d Angers 2, Boulevard Lavoisier, Angers 49045, Cedex 1, France 2 Institute of Physics, Pregrevica 118, Belgrade 11000, Serbia Abstract Light bullets as localized self-organized spatiotemporal dissipative structures are studied from the point of view of propagation and stability. Experimental conditions involve consideration of asymmetric inputs. Squeezed input optical pulse evolves toward stable dissipative light bullet, whenever its dissipative parameters belong to the established domain of stability. Such a domain of attraction is found following an extension of our analytical stability criterion confirmed by numerical simulations. Stable and robust dissipative light bullets are generated during a self-organizing propagation due to the cross compensation of various linear and nonlinear effects. Ultra-short optical pulses have a variety of applications in modern physics and technology. Among them are: optical communication systems, solid-state spectroscopy, ultra-cold atom studies, optical memory devices, quantum computers and various security devices. It is desirable that pulses are not only extremely short in length but also have high power. Developing powerful ultrashort pulse generating devices is crucial for all these applications. High-power ultra-short pulses generated in optics using laser systems are self-shaped in saturating nonlinear media due to the compensation of diffraction and dispersion by nonlinearity and essentially are solitons [1]. Optical solitons may soon become the principal carrier in telecommunication in all-optical transmission lines because they propagate long distances without changing shape [2]. Spatiotemporal solitons are good candidates in all-optical signal processing since they are self-guided in bulk media carrying big power for a small dissipated energy []. Stable operation of laser systems, closely related to the issue of dissipative soliton stability, is crucial for practical purposes [4]. Dynamics of dissipative solitons can be described by the complex cubic-quintic Ginzburg-Landau equation (CQGLE) which is nonintegrable and can be solved only numerically [5]. However, an analytical approach, even though approximate, is needed in order to guide simulations and to avoid tedious numerical computations necessary to determine the stability domain point by point [6, 7]. In a recent publication [8] we used the variational method extended to dissipative systems, to establish this stability domain of parameters for spherically symmetric input pulses with radius r = x 2 + y 2 + t 2 imposing a constraint to independent transverse space (x and y) and time (t) variables. In order to meet experimental conditions [9], for instance, a pulse have to be squeezed in order to be injected in optical guide, we study in the present work the generation of dissipative solitons from inputs with no spherical symmetry. Therefore, ( + 1)-dimensional complex cubic-quintic Ginzburg-Landau equations for the normalized field envelope E, describes separately diffractions following x and y coordinates and anomalous group velocity dispersion in time t without such a constraint. i E ( 2 ) E iδe + (1 iβ) z x E y E t 2 + (1 iε) E 2 E + (ν iµ) E 4 E = 0. (1) In order to prevent the wave collapse the saturating nonlinearity is required [6, 7]. As a consequence, cubic and quintic nonlinearity have to have opposite signs, i.e., parameter ν is negative. The stability of the pulse background involves the linear loss, thus the parameter δ must be negative [8]. Parameters ε and µ are associated respectively with cubic gain and quintic loss terms. The parabolic gain (β > 0) is taken also with respect to each transverse coordinate separately. The prerequisite for generation of a completely confined spatiotemporal structure, the dissipative light bullet, is a simultaneous balance of diffraction and dispersion with self-focusing and gain with loss. In order to generalize variational approach to asymmetric conditions, we construct for Eq. (1) the total Lagrangian L = L c + L Q that contains a conservative part L c = i ) (E E E E + E/ x 2 + E/ y 2 + E/ t 2 E 4 2 z z 2 ν E 6, (2)

2 and a dissipative part L Q = iδ E 2 + i ε E 4 2 PIERS ONLINE, VOL. 4, NO. 6, i µ E 6 ( iβ E/ x 2 + E/ y 2 + E/ t 2). () The independent treatment of all three transverse coordinates involves an asymmetric trial function E = A exp [i ( Cx 2 + Gy 2 + St 2 + Ψ ) ] x2 2X 2 y2 2Y 2 t2 2T 2 (4) as functional of amplitude A, temporal (T ) and spatial (X and Y ) pulse widths, anisotropic wave front curvatures C and G, chirp S, and phase Ψ. Each of these functions of the independent variable z is optimized giving one of eight Euler-Lagrange equations. Within variational approximation, to the partial differential CQGLE corresponds a set of eight coupled first order differential equations (FODE) resulting from the variations in unequal widths. dx dz dy dz = 4CX R 2 B 2 X µa4 R 2 B 4 X + β X 4βX C 2 = X, (5) = 4CY R 2 B 2 Y µa4 R 2 B 4 Y + β Y 4βY G 2 = Y, (6) and dt = 4CT dz R 2 B 2 T µa4 R 2 B 4 T + β T 4βT S 2 = T, (7) asymmetric wave front curvatures and amplitude and phase da dz = A dc dz = 4C2 + 1 X 4 1 A 2 R 2 B 2 X 2 dg dz = 4G2 + 1 Y 4 1 A 2 R 2 B 2 Y 2 ν A 4 R 2 B 4 ν A 4 R 2 B 4 X 2 4 β C = C, X2 (8) Y 2 4 β G = G, Y 2 (9) ds dz = 4S2 + 1 T 4 1 A 2 R 2 B 2 T 2 ν A 4 R 2 B 4 T 2 4 β S = S, (10) T 2 ( δ + 7ε 2R 2 B 2 A2 + µ R 2 B 4 A4 β X 2 β Y 2 β ) 2C 2G 2S = A, (11) T 2 dψ dz = 1 X 2 1 Y 2 1 T R 2 B 2 A2 + ν R 2 B 4 A4 + 2βC + 2βG + 2βS = Ω (12) where B = 5/4 /2 7/4 and R = 8/ 5/4. For convenience reasons, all dissipative parameters, considered as small, are divided by δ o = δ : ε o = ε/δ o, µ o = µ/δ o, and β o = β/δ o. The exact steady state solutions are obtained from Eqs. (5) (11) for zero derivatives of amplitude, widths, and curvatures with respect to z. The only possible steady state solutions are symmetric with equal widths X = Y = T and curvatures C = G = S. In the dissipative case the power P = A 2 XY T is no longer a constant [6 8]. However, in steady state the power P o = ( πrb 2) A 1 o (A 2 ob 2 +νa 4 o) /2, the width X o = RB 2 (A 2 ob 2 + νa 4 o) 1/2, and the propagation constant Ω = 2 7/2 A 2 o depend, up to θ = max(δ o, ε, µ, β ), only on the amplitude as in the conservative case [6, 7]. Variationally obtained family of conservative steady state solutions reduces to a fixed double solution for a given set of dissipative parameters. Indeed, the steady state amplitude has two discrete values A + and A. A double solution (A > A + ) exists for a cubic gain (ε o > 0) and a quintic loss (µ o < 0) in the (ε o, µ o )- domain between parabola and straight line in Fig. 1. The striking difference with conservative systems is the negative wave front curvature C 0 = 5/2 2 1 δ o A 2 [ o (εo β o ) B 2 + (µ o νβ o ) A 2 ] o [6 8]. In order to become a soliton a steady state solution must be stable. Our stability criterion based on the variational approach and the method of Lyapunov s exponents, has to be generalized for

3 PIERS ONLINE, VOL. 4, NO. 6, non spherically symmetric conditions [8]. In steady state (st), right hand sides of Eqs. (5) (11) are differentiated with respect to amplitude (e.g., A A ), widths (e.g., S Y ), and curvatures (e.g., X C ) in order to construct a Jacoby matrix. The steady state solutions of seven coupled FODE are stable if and only if the real parts of solutions λ of equation ( λ + aλ 2 + bλ + c ) ( λ 2 + dλ + e ) = 0 are non positive [10]. That is fulfilled when the Hurwitz s conditions are satisfied. The stability criterion for steady state solutions of CQGLE is explicitly expressed up to θ as Figure 1: Stability domain of solutions. b = (A A X X + A A C C + X X C C A C C A C C X A X C C X ) st > 0, (1) c = (A A (X C C X X X C C ) + A X (C C X A C A X C ) + A C (X X C A C X X A )) st > 0, (14) d = (X X + C C ) st > 0, (15) e = (X X C C X C C X ) st > 0, (16) and f = ab c > 0, where a = (A A + X X + C C ) st. As a consequence, in the (ε o, µ o )-domain Figure 2: Upper stable and lower unstable branch of variational (v) and numerical (n) curves. in Fig. 1 only A solution is stable in the shaded region between curves α 2 = 0, and α 6 = 0 (separated by a square), as well as α 4 = 0 obtained from Eqs. (5) (11) solved parametrically. Input pulse chosen in the stable domain of parameters, i.e., on the upper stable branch of the variationally obtained bifurcation curve v in Fig. 2, is not yet a light bullet. This curve is only a good approximation of exact bifurcation curve n obtained by numerical solving of Eq. (1). For illustration, let us take as an input for numerical simulations the stable solution A for the set of

4 PIERS ONLINE, VOL. 4, NO. 6, dissipative parameters δ o = 0.01, ε o = 22, µ o = 6, ν = 1, and β o = 5 labeled by a diamond on the upper branch v which is stable between squares. The triangle labeling the corresponding A + solution is on the lower unstable branch. During the evolution the pulse will self-organize into a dissipative light bullet. Therefore, the diamond will reach the stable branch of the curve n. This exact soliton solution of CQGLE is an attractor for each of input pulses corresponding to the same set of dissipative parameters from established domain of stability. It is worthwhile to stress that even very squeezed input pulse, like the ellipsoid, far from bifurcation curves in Fig. 2, is trapped shrinking towards the stable dissipative soliton. The movie of numerical simulations demonstrates self-stabilizing dynamics of such an ellipsoidal input shown in Fig.. Analytically established domain of attraction (shaded in Fig. 1) is confirmed by numerical simulations of Eq. (1). Establishing of such a large domain of stability can be useful for tuning experiments. Figure : Numerical evolution of an asymmetric input pulse towards a stable dissipative light bullet. Therefore, we demonstrated that light bullets are exceptionally stable and robust dissipative self-organized structures due to the cross compensation between the loss, the gain, and the excess of the saturating nonlinearity that is a consequence of negative curvature. Consequently, light bullets appear to be the best candidate for caring the information that has to be treated in all-optical logic circuits. In order to guide light by light saturating nonlinear materials are needed. Following our recent measurements of organic materials, polydiacetylene para-toluene sulfonate (PTS) exhibits large cubic quintic saturating nonlinearity. As a consequence, either imbedded in porous silicon nanocomposits [11] or as bulk, PTS may be promising for elaboration of reconfigurative guides. Light bullets during their propagation modify nonlinear properties of the sample, inducing a guidable network. Modification of bullets timing and paths reconfigurates the guide altering its geometry in real time. The atmosphere exhibits saturating nonlinearity too. A laser pulse having above critical power is self-focused due to focusing Kerr nonlinearity of the air. Therefore, the intensity of the field is increasing and the plasma is generated due to multiphoton (or tunneling) ionization of neutral molecules. The plasma acts as a defocusing medium suppressing field blow up. A balance between the nonlinear focusing and plasma defocusing results in self-guiding solitons. In contrast to the non singular beams, the dynamics of beams with nonzero topological charge is more robust leading to vortex solitons. A laser beam with phase singularity, i.e., with vortex structure contributes in general, if not to suppress the breakup of ring into modulation instability induced filaments, at least to distribute filaments symmetrically around the singularity in order to conserve the angular momentum. Hence, a vortex structure prevents chaotic filamentation so often observed in powerful laser beams without singularity. We demonstrated the possibility of filaments coalescence that seems to be related with the attraction to the equilibrium vortex soliton state [12]. These observations can be used for controlled triggering of lightning discharge, lightning protection, remote sensing, and long distance air communications. The lightning, is itself a self-organizing phenomenon based on the self-focusing without blow up, hence, it may be related to dissipative solitons. The opportunity to treat in synergy analytically and numerically asymmetric input pulses propagation toward stable and robust dissipative light bullets, opens possibilities for diverse practical applications including experiments.

5 PIERS ONLINE, VOL. 4, NO. 6, ACKNOWLEDGMENT Work at the Institute of Physics is supported by the Ministry of Science of the Republic of Serbia, under the project OI This research has been in part supported by French-Serbian cooperation, CNRS/ MSCI agreement No REFERENCES 1. Kivshar, Yu. S. and B. A. Malomed, Rev. Mod. Phys., Vol. 61, 76, Gabitov, I. and S. K. Turitsyn, Opt. Lett., Vol. 21, 27, Mihalache, D., et al., Phys. Rev. Lett., Vol. 97, 07904, Chen, C.-J., P. K. A. Wai, and C. R. Menyuk, Opt. Lett., Vol. 19, 198, Akhmediev, N. N. and A. A. Ankiewicz, Solitons, Nonlinear Pulses and Beams, Chapman and Hall, London, Skarka, V., V. I. Berezhiani, and R. Miklaszewski, Phys. Rev. E, Vol. 56, 1080, Skarka, V., V. I. Berezhiani, and R. Miklaszewski, Phys. Rev. E, Vol. 59, 1270, Skarka, V. and N. B. Aleksic, Phys. Rev. Lett., Vol. 96, 0190, Liu, X., L. J. Qian, and F. W. Wise, Phys. Rev. Lett., Vol. 82, 461, Nicolis, G. and I. Prigogine, Self-organization in Nonequilibrium Systems, John Wiley and Sons, New York, Simos, C., L. Rodriguez, V. Skarka, X. Nguyen Phu, N. Errien, G. Froyer, T. P. Nguyen, P. Le Rendu, and P. Pirastesh, Phys. Stat. Sol. (C), Vol. 2, 22, Skarka, V., N. Aleksić, and V. I. Berezhiani, Phys. Lett. A, Vol. 19, 17, 200.