- Email: [email protected]
Three-dimensional spatiotemporal solitary waves in strongly nonlocal media
Optics Communications 283 (2010) 5213–5217
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Discussion
Three-dimensional spatiotemporal solitary waves in strongly nonlocal media Wei-Ping Zhong a,⁎, Milivoj Belić b, Rui-Hua Xie c, Tingwen Huang b, Yiqin Lu d a
Department of Electronic and Information Engineering, Shunde Polytechnic, Guangdong Province, Shunde 528300, China Texas A & M University at Qatar, 23874 Doha, Qatar c Department of Applied Physics, Xi'an Jiaotong University, Xi'an 710049, China d School of Electronic and Information Engineering, South China University of Technology, Guangzhou 510640, China b
a r t i c l e
i n f o
Article history: Received 8 May 2010 Received in revised form 2 August 2010 Accepted 2 August 2010 Keywords: Nolinear optics Spatiotemporal solitary waves Accessible light bullets
a b s t r a c t We construct a class of three-dimensional strongly nonlocal spatiotemporal solitary waves of the nonlocal nonlinear Schrödinger equation, by using superpositions of single accessible solitons as initial conditions. Evolution of such solitary waves, termed the accessible light bullets, is studied numerically by choosing specific values of topological charges and other solitonic parameters. Our numerical results reveal that in strongly nonlocal nonlinear media with a Gaussian response function, different classes of accessible spatiotemporal solitons can be generated and controlled by tailoring different soliton parameters. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Spatial solitons — self-trapped optical beams — have stimulated strong interest, due to their rich potential for applications in photonic switching, all-optical switching and logic gating, as well as all-optical signal processing [1]. Nonlocality is a universal phenomenon, encountered in many physical systems. Recently nonlocal optical solitons have been studied extensively in theory [2–6]. According to these studies, the degree of nonlocality can be roughly classified into four cases: local, weekly nonlocal, generally nonlocal, and strongly nonlocal [2] with respect to the ratio of the characteristic nonlocality length of the nonlinear medium to the beam width. It has been found that the nonlocality can prevent the collapse of self-focusing beams in Kerr-type media [2], suppress azimuthal instabilities of vortex solitons [3], and stabilize Laguerre and Hermite soliton families [4], azimuthons [5], and multipole solitons [6]. Of particular importance is the case, referred to as the strongly nonlocal case, in which the characteristic nonlocal length is much larger than the beam width. It can support solitons with new properties, the so-called “accessible solitons” [7], which are described by the solutions of a linear differential equation, tantamount to the high-dimensional quantum harmonic oscillator [6,8]. Owing to this feature, strong nonlocality strongly affects the dynamics of solutions, as the solutions of linear systems cannot be unstable or chaotic. Experiments have revealed fundamental [9] and vortex optical solitons [10] supported by the strong nonlocality, as well as steering
⁎ Corresponding author. E-mail address: [email protected] (W.-P. Zhong). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.08.004
solitons [11]. In addition, theoretical analyses have predicted the stabilization of other self-trapped modes [12] and partially coherent accessible solitons [13]. He et al. have found the spinning “bearings” solitons [14] and crescent solitons [15] by superposing concentric vortices with widely different topological charges. Optical spatiotemporal solitons, or light bullets, are some of the most exciting entities in nonlinear optics [16]. They are threedimensional (3D) wave packets in which the dispersion and diffraction are simultaneously balanced by the nonlinearity. The search for suitable media for the creation of stable “bullets” is a challenging problem, as multidimensional solitons in Kerr-type focusing media are unstable against collapse (blowup) [17]. A number of settings have been proposed where their stabilization is possible. Thus, stable bullets may form in materials with nonlocal nonlinearities [2,18]. Also, accessible light bullets have been studied in Ref. [19]. Against the recurring concern of stability of multidimensional spatiotemporal solitons there emerged concepts of quasi-stable slowly expanding self-similar light bullets and vortex solutions [20], modulational instability control in media with non-instantaneous Kerr response [21], and localized similaritons in distributed systems that might offer increased stability [22]. In this paper we demonstrate that a class of new 3D strongly nonlocal spatiotemporal solitons (including the stable spinning solitary waves) can be supported by the strongly nonlocal nonlinear media. These solitons are generated by the superposition of two exact accessible solitons with different topological charges which are then numerically evolved in space and time according to the nonlocal nonlinear Schrödinger equation (NNSE). The superposition principle works here because the strong nonlocality makes the system behave like a linear system. In addition, strong nonlocality influences
5214
W.-P. Zhong et al. / Optics Communications 283 (2010) 5213–5217
dynamics of solutions, since it exerts a strong stabilizing effect. Also, multiple accessible light bullets can be generated by the superposition of different single light bullets. The stability of this class of 3D strongly nonlocal solitons is expected through their accessible nature, but a rigorous proof is beyond the scope of this paper. The approach in this paper is geared for systems with nonlocal nonlinearity; more specifically to the strongly nonlocal nonlinear systems. As mentioned, the superposition works there because strong nonlocality effectively makes the system linear. This feature also helps the stability of new solutions, because linear systems cannot be unstable or chaotic. There are not too many areas of nonlinear spatiotemporal dynamics where such nonlocal nonlinear limits naturally occur, but the generation of nematicons (solitons in nematic liquid crystals) is one of the fields to which our method naturally applies [23]. The paper is organized as follows. In Section 2 we introduce the general nonlocal nonlinear model, described by the general NNSE, and note the exact accessible soliton solutions in the strongly nonlocal nonlinear limit. Using superpositions of such accessible solitons with different topological charges as initial states, we construct a new class of accessible light bullets of the full NNSE. In Section 3, we present numerical simulations of such superpositions of light bullet beams for some specific parameters. We find that the new class of 3D strongly nonlocal spatiotemporal solitons displays various forms. In Section 4 we summarize our results. 2. Nonlocal nonlinear model The evolution of an optical spatiotemporal wave in a general nonlocal nonlinear medium can be described by the general NNSE for the wave amplitude u in the following scaled form: i
∂u 1 ∂2 u 2 ∇⊥ u + + 2 ∂z ∂τ2
!
+ NðIÞ → r; τ; z u = 0;
ð1AÞ
where z is the evolution coordinate along the beam, vector → r = ðx; yÞ provides the transverse position coordinates, ∇ 2⊥ is the transverse 2D Laplacian operator, τ is the reduced temporal variable [16], and NðIÞ represents the nonlocal nonlinearity induced by the optical beam intensity I → r ; τ; z = ju → r ; τ; z j2 . We assume N to be of the form: r ′ ; τ; z d→ r ′; NðIÞ → r; τ; z = ∫R → r −→ r′ I →
ð1BÞ
where R is the normalized symmetric real response function of the medium whose characteristic length determines the degree of nonlocality. Note that Eq. (1A) is rather generic; one may assign different physical interpretations to essentially the same type of equation by choosing different physical systems and variables. For example, in a typical quantum mechanical setting one may understand Eq. (1A) as the scaled Schrödinger equation for the wave function of a particle moving in the potential − N: i
∂u 1 2 r; t u = 0; + ∇ u + NðI Þ → 2 ∂t
ð1CÞ
where now the vector → r = ðx; y; zÞ is the 3D position vector, t is time and ∇ 2 is the full 3D Laplacian. The setting chosen in this paper is reminiscent of the X wave generation [24–26]: the wave equations are similar and there exist natural linear — nonlinear regimes of wave packet dynamics. The major difference is that we consider a nonlocal medium with anomalous dispersion. One may also choose different physically relevant forms for R → → ′ in Eq. (1B) [2]. We choose the Gaussian response R r − r ¼ 1 →r −→r ′ . for its relevance in nonlinear optics and ease of manipe πσ 2 −
2
σ2
ulation. The width σ of the Gaussian controls the nonlocality; large values
of σ correspond to the strong nonlocality. There exists no minimum value of σ below which the superposition principle would not be applicable; this also depends on the width of the initial beams. The transition to the strong nonlocality is rather gradual and not threshold-like sharp. Therefore, we can only guess an order of magnitude for σ; for the beams considered here the rule of thumb is that σ=1 is too low and σ=10 is marginal; therefore we choose σ=100. In the case of strong nonlocality, Eq. (1A) can be simplified to the accessible soliton model in cylindrical coordinates [6,7]: i
! 2 2 ∂u 1 1∂ ∂u 1∂ u ∂ u 2 −sρ u = 0; + r + 2 2 + 2 r ∂r ∂z ∂r r ∂φ ∂τ2
ð2Þ
where sðN0Þ is a parameter proportional to the beam power. Note that the beam power is constant, equal to the total input power P0; φ is the azimuthal angle and ρ2 = x2 + y2 + τ2. An exact single accessible light bullet solution to Eq. (2) has been found in [8]:
s unml
! 1 τ2 2 − 2 2w2 kP0 ½cosðmφÞ + iq sin ðmφÞ τ r 0 Hl Wnm = e pffiffiffiffiffiffiffi 2 r w0 w0 w0
+ i
a0 −2n
+ 2l + 1 z w2 0
;
ð3Þ where w0 is the beam sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n! , Wnm are the Whittaker initial width, k = pffiffiffi 2l l! π Γðn + jmj + 1Þ functions, Hl are the Hermite polynomials, n is an integer, and m, a real number, stands for the topological charge. It should be stressed that there exist three kinds of light bullet solitons in the form of Eq. (3): (1) Gaussian solitons (m = 0); (2) Vortex solitons (q = 1 and m ≠ 0); (3) Multipolar solitons (q = 0). S i n ce Eq any super position . (2) is a linear equation, u → r ; τ; z = u1 → r ; τ; z + u2 → r ; τ; z of two single solitary wave solutions of Eq. (3) generates a new exact solution. Thus, according to Eqs. (2) and (3), a class of new light bullets can be constructed as superpositions of single 3D spatiotemporal solitons of the form Eq. (3), with the same P0, w0 and a0, but with different n, l and m. In this paper we maintain P0 = 1, w0 = 1 and a0 = 0. 3. Spatiotemporal solitary waves as superpositions of two single light bullet solutions The superposition of two single accessible light bullets is formed by picking two solutions (3) of the linear Eq. (2) as an initial condition for the integration of Eq. (1A). A new class of different soliton solutions is obtained by applying the direct numerical method [27] to Eq. (1A). Gaussian response function R with σ = 100 is chosen in Eq. (1B), so as to be in the strongly nonlocal region. It should be noted that we choose superpositions of exact accessible solitons in the strongly nonlocal limit, but propagate such solutions in the full nonlocal model, i.e. the nonlocal model with the Gaussian kernel and a finite but large value of σ. The idea is to demonstrate that the solutions so obtained are more stable than the solutions with lower values of σ, which then allows an easy classification of such new solutions to the full model. Although this approach to finding new soliton solutions is more cumbersome, it provides more meaningful and classifiable results. Below we present a few examples. Asymmetric rotating cashew-shaped spatiotemporal solitary waves can be generated by choosing as the initial condition u → r ; τ; 0 = u1 → r ; τ; 0 + u2 → r ; τ; 0 a superposition of two vortex spatiotemporal solitons with m2 = m1 ± 1. Typical examples of the patterns are shown in Fig. 1, where the soliton pairs are used with (a) n = 0, l = 1, m1 = 2, m2 = 1; (b) n = 1, l = 0, m1 = 2, m2 = 1; (c) n = 0, l = 0, m1 = 2 and m2 = 3 (left); m1 = 3 and m2 = 4 (middle), m1 = 4 and m2 = 5 (right). Multilayer asymmetric cashew solitary waves can
W.-P. Zhong et al. / Optics Communications 283 (2010) 5213–5217
5215
Fig. 1. Intensity isosurfaces in the space ðx; y; τÞof spinning asymmetric cashew spatiotemporal solitary waves, generated by the superposition of two vortex solitons with m2 = m1 ± 1. (a) n = 0, l = 1, m1 = 2 and m2 = 1; the evolution intervals are z = 0, 10, 100 from left to right; (b) n = 1, l = 0, m1 = 2 and m2 = 1; z = 0, 10, 100 from left to right; (c) n = 0, l = 0, m1 = 2 and m2 = 3 (left); m1 = 3 and m2 = 4 (middle), m1 = 4 and m2 = 5 (right); z = 100.
be constructed by the superposition of two single solitons with n N 1, see Fig. 1(a) and (b). It is seen in Fig. 1 that the number of layers in the horizontal plane and in the vertical (τ-axis) direction are decided by n and l, respectively. With the increase in m1, the solitary waves gradually shrink into thin waves, see Fig. 1(c). Within the framework of the linear Eq. (2) with constant P0, the cashew solitary waves are obviously stable. Asymmetric necklace solitary waves are the beams in the form of a necklace whose radius is large compared to the size of individual ellipsoidal beads. It has been found that the stable asymmetric rotating necklace spatiotemporal solitons can be generated by an initial condition in the form of a superposition of multipolar soliton pairs with m2 = m1 ± 1. To avoid modulation instability in the azimuthal direction, the intensity is periodically modulated. Typical examples of intensity patterns of asymmetric necklace solitary waves are presented in Fig. 2. Such shapes emerge from an initial superposition of two single multipolar solitons, thus resulting in an interference pattern in the azimuthal direction. In general, evolution of an asymmetric necklace solitary wave is considered stable, although the beams exhibit slow radial expansion as they propagate. The expansion is a result of a net radial force that results from the
azimuthally alternating phase, and is typically much slower than the diffractive expansion (see Fig. 2). In the example presented the expansion is even negligible over the propagation distances considered. When n = 0 there exist monolayer necklace solitary waves in the horizontal plane (see Fig. 2(a)): The intensity is strong on one side and weak on the other. We also note that the ellipsoids are larger on one side and smaller on the other. The reason for such appearance is the interference between single solitons. Multilayer asymmetric necklace solitary waves can be constructed by initial conditions of two superposing multipolar solitons with n ≥ 1 and l ≥ 1 in the horizontal and vertical directions, respectively. In particular, two layers in the horizontal plane, with the larger one embracing the smaller counterpart (see Fig. 2(b)), and two layers in the vertical direction are formed by n = 1 and l = 1, respectively. The stability of these solitary waves can be further tested by adding strong perturbations to the multipolar spatiotemporal solitons that generate them [12]. Our numerical results demonstrate that the two kinds of spatiotemporal solitary waves mentioned above rotate in the transverse (x,y) plane as they evolve. As shown in Figs. 1 and 2, these solitary waves rotate counter-clockwise around the vertical τ
5216
W.-P. Zhong et al. / Optics Communications 283 (2010) 5213–5217
Fig. 2. Typical examples of the asymmetric necklace spatiotemporal solitary waves and of the counter-clockwise rotation with adjacent integer TC. (a) n = 0, l = 1, m1 = 3 and m2 = 1; (b) n = 1, l = 0, m1 = 2 and m2 = 1. The evolution distances are z = 0, 10, 100 from left to right.
axis. Further, we find that the sense of rotation depends on the sign of the topological charge of the second soliton. When m2 N 0, the wave's rotation is counter-clockwise; when m2 b 0, it is clockwise. These intriguing properties motivate us to explore superpositions of multipolar solitary waves in the form of simple Gaussian solitons and vortex solitons. Numerical simulations of such superpositions are presented in Figs. 3 and 4. For the initial condition in the form of a superposition of multipolar spatiotemporal soliton with integer topological charge and a Gaussian spatiotemporal soliton, we see a symmetric solid gear
in the center; see Fig. 3(a). These solitary waves form adjacent alternating big and small ellipsoids along the radial direction except for the central gear (Fig. 3(b)). They consist of n + 2 layers: the outer layer is composed of 2m1 ellipsoids, half of them being big and the other half small. The number of ellipsoids in each layer is determined by m1, and the number of layers is determined by n. For n = 1, the layer at the center is a symmetric solid gear, the layer outside the gear is composed of uniformly distributed ellipsoids with alternating big and small ones along the azimuthal angle. The optical intensity is maximum at the center.
Fig. 3. Intensity distributions of solid gear spatiotemporal solitary waves. The initial condition is in the form of a superposition of a multipolar soliton with different m1 and a fundamental spatiotemporal soliton m2 = 0. (a) n = 0, l = 0, m1 = 3, 4, 5 from left to right; (b) n = 1, l = 0, m1 = 3, 4, 5 from left to right, the evolution distance is z = 100.
W.-P. Zhong et al. / Optics Communications 283 (2010) 5213–5217
5217
Fig. 4. Intensity profiles of the superposition of a multipolar spatiotemporal soliton with different m1 and a vortex spatiotemporal soliton with m2 = 1. The parameters are chosen as n = 0, l = 0,m1 = 3, 4, 5 from left to right, and the evolution distance is z = 100.
One more interesting example is the superposition of a multipolar spatiotemporal soliton and a vortex spatiotemporal soliton with nonzero integer topological charges. One may also obtain symmetric singlelayer and multilayer necklace solitary waves. The solitary wave distributions come from the linear superpositions at z = 0 in Eq. (3). Fig. 4 shows some illustrative examples. These solitary waves possess adjacent alternating big and small ellipsoids along the azimuthal angle. For n = 0 the optical field along the radial direction has one layer. Such superpositions form the symmetric single-layer necklace solitary waves. The number of big and small ellipsoids is 2m1, one-half of these being big and the other half small (see Fig. 4). For n N 0, we further find that it consists of n + 1 layers: the most outside is composed of 2m1 big and small ellipsoids. It should be pointed out that the combination of the nonlinear polarization in the strongly nonlocal nonlinear media has the symmetry of the electric field, due to the strong nonlocality. Under proper conditions the nonlocality leads to a change of the refractive index in the overlapping region, giving rise to the formation of different symmetric and asymmetric spatiotemporal solitary waves. 4. Conclusions In conclusion, we have investigated numerically 3D strongly nonlocal spatiotemporal solitary waves. We have found that by choosing the initial condition in the form of a superposition of solitons with different topological charges and other parameters, it is possible to generate a new class of solitary waves, consisting of asymmetric cashew solitary waves, asymmetric necklace solitary waves and solid gear solitary waves. The superposition principle works here because the strong nonlocality makes the system behave like a linear system. Owing to this feature, strong nonlocality strongly affects the dynamics of solutions, since the solutions of linear systems cannot be unstable or chaotic. Our results reveal that in the nonlocal nonlinear media with Gaussian response function, the shape of the new class of 3D accessible spatiotemporal solitary waves can be controlled by the choice of different parameters. Acknowledgements This work is supported by the Science Research Foundation of Shunde Polytechnic under Grant No. 2008-KJ06, China. Work at the
Texas A&M University at Qatar is supported by the NPRP 25-6-7-2 and NPRP 09-462-1-074 projects with the Qatar National Research Foundation.
References [1] G.I. Stegeman, M. Segev, Science 286 (1999) 1518. [2] O. Bang, W. Krolikowski, J. Wyller, J.J. Rasmussen, Phys. Rev. E 66 (2002) 046619. [3] D. Buccoliero, A.S. Desyatnikov, W. Krolikowski, Y.S. Kivshar, Opt. Lett. 33 (2008) 198. [4] D. Buccoliero, A.S. Desyatnikov, W. Krolikowski, Y.S. Kivshar, Phys. Rev. Lett. 98 (2007) 053901; W.P. Zhong, M. Belic, Phys. Lett. A 373 (2009) 296. [5] S. Lopez-Aguayo, A.S. Desyatnikov, Y.S. Kivshar, Opt. Express 14 (2006) 7903. [6] W.P. Zhong, L. Yi, Phys. Rev. A 75 (2007) 061801(R); W.P. Zhong, M. Belic, Phys. Rev. A79 (2009) 023804. [7] A. Snyder, J. Mitchell, Science 276 (1997) 1538. [8] W.P. Zhong, L. Yi, R.H. Xie, M. Belic, G. Chen, Mol. Opt. Phys. 41 (2008) 025402. [9] C. Conti, M. Peccianti, G. Assanto, Phys. Rev. Lett. 92 (2004) 113902. [10] C. Rotschild, O. Cohen, O. Manela, M. Segev, T. Carmon, Phys. Rev. Lett. 95 (2005) 213904. [11] B. Alfassi, C. Rotschild, O. Manela, M. Segev, D.N. Christodoulides, Opt. Lett. 32 (2007) 154. [12] S. Lopez-Aguayo, J.C. Gutiérrez-Vega, Phys. Rev. A 76 (2007) 023814. [13] M. Shen, Q. Wang, J. Shi, P. Hou, Q. Kong, Phys. Rev. E 73 (2006) 056602. [14] Y.J. He, B.A. Malomed, D. Mihalache, H.Z. Wang, Phys. Rev. A 77 (2008) 043826. [15] Y.J. He, B.A. Malomed, D. Mihalache, H.Z. Wang, Phys. Rev. A 78 (2008) 023824. [16] B.A. Malomed, D. Mihalache, F. Wise, L. Torner, J. Opt. B7 (2005) R53. [17] L. Berge, Phys. Rep. 303 (1998) 260. [18] D. Mihalache, D. Mazilu, F. Lederer, B.A. Malomed, Y.V. Kartashov, L.-C. Crasovan, L. Torner, Phys. Rev. E73 (2006) 025601(R). [19] I.B. Burgess, M. Peccianti, G. Assanto, R. Morandotti, Phys. Rev. Lett. 102 (2009) 203903. [20] S. Chen, J.M. Dudley, Phys. Rev. Lett. 102 (2009) 233903. [21] L. Zhang, S. Wen, X. Fu, J. Deng, J. Zhang, D. Fan, Opt. Commun. 283 (2010) 2251. [22] C. Dai, Y. Wang, J. Zhang, Opt. Lett. 35 (2010) 1437. [23] G. Assanto, M. Peccianti, C. Conti, Nematicons, Opt. Phot. News, February Issue, 2003, p. 45. [24] C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trulli, Phys. Rev. Lett. 90 (2003) 170406. [25] Y. Kominis, N. Moshonas, P. Papagiannis, K. Hizanidis, D.N. Christodoulides, Opt. Lett. 30 (2005) 2924. [26] D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J.V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, P. Di Trapani, Opt. Express 15 (2007) 13077. [27] W.P. Zhong, R.H. Xie, M. Belic, G. Chen, Phys. Rev. A 78 (2008) 013826.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Discussion
Three-dimensional spatiotemporal solitary waves in strongly nonlocal media Wei-Ping Zhong a,⁎, Milivoj Belić b, Rui-Hua Xie c, Tingwen Huang b, Yiqin Lu d a
Department of Electronic and Information Engineering, Shunde Polytechnic, Guangdong Province, Shunde 528300, China Texas A & M University at Qatar, 23874 Doha, Qatar c Department of Applied Physics, Xi'an Jiaotong University, Xi'an 710049, China d School of Electronic and Information Engineering, South China University of Technology, Guangzhou 510640, China b
a r t i c l e
i n f o
Article history: Received 8 May 2010 Received in revised form 2 August 2010 Accepted 2 August 2010 Keywords: Nolinear optics Spatiotemporal solitary waves Accessible light bullets
a b s t r a c t We construct a class of three-dimensional strongly nonlocal spatiotemporal solitary waves of the nonlocal nonlinear Schrödinger equation, by using superpositions of single accessible solitons as initial conditions. Evolution of such solitary waves, termed the accessible light bullets, is studied numerically by choosing specific values of topological charges and other solitonic parameters. Our numerical results reveal that in strongly nonlocal nonlinear media with a Gaussian response function, different classes of accessible spatiotemporal solitons can be generated and controlled by tailoring different soliton parameters. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Spatial solitons — self-trapped optical beams — have stimulated strong interest, due to their rich potential for applications in photonic switching, all-optical switching and logic gating, as well as all-optical signal processing [1]. Nonlocality is a universal phenomenon, encountered in many physical systems. Recently nonlocal optical solitons have been studied extensively in theory [2–6]. According to these studies, the degree of nonlocality can be roughly classified into four cases: local, weekly nonlocal, generally nonlocal, and strongly nonlocal [2] with respect to the ratio of the characteristic nonlocality length of the nonlinear medium to the beam width. It has been found that the nonlocality can prevent the collapse of self-focusing beams in Kerr-type media [2], suppress azimuthal instabilities of vortex solitons [3], and stabilize Laguerre and Hermite soliton families [4], azimuthons [5], and multipole solitons [6]. Of particular importance is the case, referred to as the strongly nonlocal case, in which the characteristic nonlocal length is much larger than the beam width. It can support solitons with new properties, the so-called “accessible solitons” [7], which are described by the solutions of a linear differential equation, tantamount to the high-dimensional quantum harmonic oscillator [6,8]. Owing to this feature, strong nonlocality strongly affects the dynamics of solutions, as the solutions of linear systems cannot be unstable or chaotic. Experiments have revealed fundamental [9] and vortex optical solitons [10] supported by the strong nonlocality, as well as steering
⁎ Corresponding author. E-mail address: [email protected] (W.-P. Zhong). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.08.004
solitons [11]. In addition, theoretical analyses have predicted the stabilization of other self-trapped modes [12] and partially coherent accessible solitons [13]. He et al. have found the spinning “bearings” solitons [14] and crescent solitons [15] by superposing concentric vortices with widely different topological charges. Optical spatiotemporal solitons, or light bullets, are some of the most exciting entities in nonlinear optics [16]. They are threedimensional (3D) wave packets in which the dispersion and diffraction are simultaneously balanced by the nonlinearity. The search for suitable media for the creation of stable “bullets” is a challenging problem, as multidimensional solitons in Kerr-type focusing media are unstable against collapse (blowup) [17]. A number of settings have been proposed where their stabilization is possible. Thus, stable bullets may form in materials with nonlocal nonlinearities [2,18]. Also, accessible light bullets have been studied in Ref. [19]. Against the recurring concern of stability of multidimensional spatiotemporal solitons there emerged concepts of quasi-stable slowly expanding self-similar light bullets and vortex solutions [20], modulational instability control in media with non-instantaneous Kerr response [21], and localized similaritons in distributed systems that might offer increased stability [22]. In this paper we demonstrate that a class of new 3D strongly nonlocal spatiotemporal solitons (including the stable spinning solitary waves) can be supported by the strongly nonlocal nonlinear media. These solitons are generated by the superposition of two exact accessible solitons with different topological charges which are then numerically evolved in space and time according to the nonlocal nonlinear Schrödinger equation (NNSE). The superposition principle works here because the strong nonlocality makes the system behave like a linear system. In addition, strong nonlocality influences
5214
W.-P. Zhong et al. / Optics Communications 283 (2010) 5213–5217
dynamics of solutions, since it exerts a strong stabilizing effect. Also, multiple accessible light bullets can be generated by the superposition of different single light bullets. The stability of this class of 3D strongly nonlocal solitons is expected through their accessible nature, but a rigorous proof is beyond the scope of this paper. The approach in this paper is geared for systems with nonlocal nonlinearity; more specifically to the strongly nonlocal nonlinear systems. As mentioned, the superposition works there because strong nonlocality effectively makes the system linear. This feature also helps the stability of new solutions, because linear systems cannot be unstable or chaotic. There are not too many areas of nonlinear spatiotemporal dynamics where such nonlocal nonlinear limits naturally occur, but the generation of nematicons (solitons in nematic liquid crystals) is one of the fields to which our method naturally applies [23]. The paper is organized as follows. In Section 2 we introduce the general nonlocal nonlinear model, described by the general NNSE, and note the exact accessible soliton solutions in the strongly nonlocal nonlinear limit. Using superpositions of such accessible solitons with different topological charges as initial states, we construct a new class of accessible light bullets of the full NNSE. In Section 3, we present numerical simulations of such superpositions of light bullet beams for some specific parameters. We find that the new class of 3D strongly nonlocal spatiotemporal solitons displays various forms. In Section 4 we summarize our results. 2. Nonlocal nonlinear model The evolution of an optical spatiotemporal wave in a general nonlocal nonlinear medium can be described by the general NNSE for the wave amplitude u in the following scaled form: i
∂u 1 ∂2 u 2 ∇⊥ u + + 2 ∂z ∂τ2
!
+ NðIÞ → r; τ; z u = 0;
ð1AÞ
where z is the evolution coordinate along the beam, vector → r = ðx; yÞ provides the transverse position coordinates, ∇ 2⊥ is the transverse 2D Laplacian operator, τ is the reduced temporal variable [16], and NðIÞ represents the nonlocal nonlinearity induced by the optical beam intensity I → r ; τ; z = ju → r ; τ; z j2 . We assume N to be of the form: r ′ ; τ; z d→ r ′; NðIÞ → r; τ; z = ∫R → r −→ r′ I →
ð1BÞ
where R is the normalized symmetric real response function of the medium whose characteristic length determines the degree of nonlocality. Note that Eq. (1A) is rather generic; one may assign different physical interpretations to essentially the same type of equation by choosing different physical systems and variables. For example, in a typical quantum mechanical setting one may understand Eq. (1A) as the scaled Schrödinger equation for the wave function of a particle moving in the potential − N: i
∂u 1 2 r; t u = 0; + ∇ u + NðI Þ → 2 ∂t
ð1CÞ
where now the vector → r = ðx; y; zÞ is the 3D position vector, t is time and ∇ 2 is the full 3D Laplacian. The setting chosen in this paper is reminiscent of the X wave generation [24–26]: the wave equations are similar and there exist natural linear — nonlinear regimes of wave packet dynamics. The major difference is that we consider a nonlocal medium with anomalous dispersion. One may also choose different physically relevant forms for R → → ′ in Eq. (1B) [2]. We choose the Gaussian response R r − r ¼ 1 →r −→r ′ . for its relevance in nonlinear optics and ease of manipe πσ 2 −
2
σ2
ulation. The width σ of the Gaussian controls the nonlocality; large values
of σ correspond to the strong nonlocality. There exists no minimum value of σ below which the superposition principle would not be applicable; this also depends on the width of the initial beams. The transition to the strong nonlocality is rather gradual and not threshold-like sharp. Therefore, we can only guess an order of magnitude for σ; for the beams considered here the rule of thumb is that σ=1 is too low and σ=10 is marginal; therefore we choose σ=100. In the case of strong nonlocality, Eq. (1A) can be simplified to the accessible soliton model in cylindrical coordinates [6,7]: i
! 2 2 ∂u 1 1∂ ∂u 1∂ u ∂ u 2 −sρ u = 0; + r + 2 2 + 2 r ∂r ∂z ∂r r ∂φ ∂τ2
ð2Þ
where sðN0Þ is a parameter proportional to the beam power. Note that the beam power is constant, equal to the total input power P0; φ is the azimuthal angle and ρ2 = x2 + y2 + τ2. An exact single accessible light bullet solution to Eq. (2) has been found in [8]:
s unml
! 1 τ2 2 − 2 2w2 kP0 ½cosðmφÞ + iq sin ðmφÞ τ r 0 Hl Wnm = e pffiffiffiffiffiffiffi 2 r w0 w0 w0
+ i
a0 −2n
+ 2l + 1 z w2 0
;
ð3Þ where w0 is the beam sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n! , Wnm are the Whittaker initial width, k = pffiffiffi 2l l! π Γðn + jmj + 1Þ functions, Hl are the Hermite polynomials, n is an integer, and m, a real number, stands for the topological charge. It should be stressed that there exist three kinds of light bullet solitons in the form of Eq. (3): (1) Gaussian solitons (m = 0); (2) Vortex solitons (q = 1 and m ≠ 0); (3) Multipolar solitons (q = 0). S i n ce Eq any super position . (2) is a linear equation, u → r ; τ; z = u1 → r ; τ; z + u2 → r ; τ; z of two single solitary wave solutions of Eq. (3) generates a new exact solution. Thus, according to Eqs. (2) and (3), a class of new light bullets can be constructed as superpositions of single 3D spatiotemporal solitons of the form Eq. (3), with the same P0, w0 and a0, but with different n, l and m. In this paper we maintain P0 = 1, w0 = 1 and a0 = 0. 3. Spatiotemporal solitary waves as superpositions of two single light bullet solutions The superposition of two single accessible light bullets is formed by picking two solutions (3) of the linear Eq. (2) as an initial condition for the integration of Eq. (1A). A new class of different soliton solutions is obtained by applying the direct numerical method [27] to Eq. (1A). Gaussian response function R with σ = 100 is chosen in Eq. (1B), so as to be in the strongly nonlocal region. It should be noted that we choose superpositions of exact accessible solitons in the strongly nonlocal limit, but propagate such solutions in the full nonlocal model, i.e. the nonlocal model with the Gaussian kernel and a finite but large value of σ. The idea is to demonstrate that the solutions so obtained are more stable than the solutions with lower values of σ, which then allows an easy classification of such new solutions to the full model. Although this approach to finding new soliton solutions is more cumbersome, it provides more meaningful and classifiable results. Below we present a few examples. Asymmetric rotating cashew-shaped spatiotemporal solitary waves can be generated by choosing as the initial condition u → r ; τ; 0 = u1 → r ; τ; 0 + u2 → r ; τ; 0 a superposition of two vortex spatiotemporal solitons with m2 = m1 ± 1. Typical examples of the patterns are shown in Fig. 1, where the soliton pairs are used with (a) n = 0, l = 1, m1 = 2, m2 = 1; (b) n = 1, l = 0, m1 = 2, m2 = 1; (c) n = 0, l = 0, m1 = 2 and m2 = 3 (left); m1 = 3 and m2 = 4 (middle), m1 = 4 and m2 = 5 (right). Multilayer asymmetric cashew solitary waves can
W.-P. Zhong et al. / Optics Communications 283 (2010) 5213–5217
5215
Fig. 1. Intensity isosurfaces in the space ðx; y; τÞof spinning asymmetric cashew spatiotemporal solitary waves, generated by the superposition of two vortex solitons with m2 = m1 ± 1. (a) n = 0, l = 1, m1 = 2 and m2 = 1; the evolution intervals are z = 0, 10, 100 from left to right; (b) n = 1, l = 0, m1 = 2 and m2 = 1; z = 0, 10, 100 from left to right; (c) n = 0, l = 0, m1 = 2 and m2 = 3 (left); m1 = 3 and m2 = 4 (middle), m1 = 4 and m2 = 5 (right); z = 100.
be constructed by the superposition of two single solitons with n N 1, see Fig. 1(a) and (b). It is seen in Fig. 1 that the number of layers in the horizontal plane and in the vertical (τ-axis) direction are decided by n and l, respectively. With the increase in m1, the solitary waves gradually shrink into thin waves, see Fig. 1(c). Within the framework of the linear Eq. (2) with constant P0, the cashew solitary waves are obviously stable. Asymmetric necklace solitary waves are the beams in the form of a necklace whose radius is large compared to the size of individual ellipsoidal beads. It has been found that the stable asymmetric rotating necklace spatiotemporal solitons can be generated by an initial condition in the form of a superposition of multipolar soliton pairs with m2 = m1 ± 1. To avoid modulation instability in the azimuthal direction, the intensity is periodically modulated. Typical examples of intensity patterns of asymmetric necklace solitary waves are presented in Fig. 2. Such shapes emerge from an initial superposition of two single multipolar solitons, thus resulting in an interference pattern in the azimuthal direction. In general, evolution of an asymmetric necklace solitary wave is considered stable, although the beams exhibit slow radial expansion as they propagate. The expansion is a result of a net radial force that results from the
azimuthally alternating phase, and is typically much slower than the diffractive expansion (see Fig. 2). In the example presented the expansion is even negligible over the propagation distances considered. When n = 0 there exist monolayer necklace solitary waves in the horizontal plane (see Fig. 2(a)): The intensity is strong on one side and weak on the other. We also note that the ellipsoids are larger on one side and smaller on the other. The reason for such appearance is the interference between single solitons. Multilayer asymmetric necklace solitary waves can be constructed by initial conditions of two superposing multipolar solitons with n ≥ 1 and l ≥ 1 in the horizontal and vertical directions, respectively. In particular, two layers in the horizontal plane, with the larger one embracing the smaller counterpart (see Fig. 2(b)), and two layers in the vertical direction are formed by n = 1 and l = 1, respectively. The stability of these solitary waves can be further tested by adding strong perturbations to the multipolar spatiotemporal solitons that generate them [12]. Our numerical results demonstrate that the two kinds of spatiotemporal solitary waves mentioned above rotate in the transverse (x,y) plane as they evolve. As shown in Figs. 1 and 2, these solitary waves rotate counter-clockwise around the vertical τ
5216
W.-P. Zhong et al. / Optics Communications 283 (2010) 5213–5217
Fig. 2. Typical examples of the asymmetric necklace spatiotemporal solitary waves and of the counter-clockwise rotation with adjacent integer TC. (a) n = 0, l = 1, m1 = 3 and m2 = 1; (b) n = 1, l = 0, m1 = 2 and m2 = 1. The evolution distances are z = 0, 10, 100 from left to right.
axis. Further, we find that the sense of rotation depends on the sign of the topological charge of the second soliton. When m2 N 0, the wave's rotation is counter-clockwise; when m2 b 0, it is clockwise. These intriguing properties motivate us to explore superpositions of multipolar solitary waves in the form of simple Gaussian solitons and vortex solitons. Numerical simulations of such superpositions are presented in Figs. 3 and 4. For the initial condition in the form of a superposition of multipolar spatiotemporal soliton with integer topological charge and a Gaussian spatiotemporal soliton, we see a symmetric solid gear
in the center; see Fig. 3(a). These solitary waves form adjacent alternating big and small ellipsoids along the radial direction except for the central gear (Fig. 3(b)). They consist of n + 2 layers: the outer layer is composed of 2m1 ellipsoids, half of them being big and the other half small. The number of ellipsoids in each layer is determined by m1, and the number of layers is determined by n. For n = 1, the layer at the center is a symmetric solid gear, the layer outside the gear is composed of uniformly distributed ellipsoids with alternating big and small ones along the azimuthal angle. The optical intensity is maximum at the center.
Fig. 3. Intensity distributions of solid gear spatiotemporal solitary waves. The initial condition is in the form of a superposition of a multipolar soliton with different m1 and a fundamental spatiotemporal soliton m2 = 0. (a) n = 0, l = 0, m1 = 3, 4, 5 from left to right; (b) n = 1, l = 0, m1 = 3, 4, 5 from left to right, the evolution distance is z = 100.
W.-P. Zhong et al. / Optics Communications 283 (2010) 5213–5217
5217
Fig. 4. Intensity profiles of the superposition of a multipolar spatiotemporal soliton with different m1 and a vortex spatiotemporal soliton with m2 = 1. The parameters are chosen as n = 0, l = 0,m1 = 3, 4, 5 from left to right, and the evolution distance is z = 100.
One more interesting example is the superposition of a multipolar spatiotemporal soliton and a vortex spatiotemporal soliton with nonzero integer topological charges. One may also obtain symmetric singlelayer and multilayer necklace solitary waves. The solitary wave distributions come from the linear superpositions at z = 0 in Eq. (3). Fig. 4 shows some illustrative examples. These solitary waves possess adjacent alternating big and small ellipsoids along the azimuthal angle. For n = 0 the optical field along the radial direction has one layer. Such superpositions form the symmetric single-layer necklace solitary waves. The number of big and small ellipsoids is 2m1, one-half of these being big and the other half small (see Fig. 4). For n N 0, we further find that it consists of n + 1 layers: the most outside is composed of 2m1 big and small ellipsoids. It should be pointed out that the combination of the nonlinear polarization in the strongly nonlocal nonlinear media has the symmetry of the electric field, due to the strong nonlocality. Under proper conditions the nonlocality leads to a change of the refractive index in the overlapping region, giving rise to the formation of different symmetric and asymmetric spatiotemporal solitary waves. 4. Conclusions In conclusion, we have investigated numerically 3D strongly nonlocal spatiotemporal solitary waves. We have found that by choosing the initial condition in the form of a superposition of solitons with different topological charges and other parameters, it is possible to generate a new class of solitary waves, consisting of asymmetric cashew solitary waves, asymmetric necklace solitary waves and solid gear solitary waves. The superposition principle works here because the strong nonlocality makes the system behave like a linear system. Owing to this feature, strong nonlocality strongly affects the dynamics of solutions, since the solutions of linear systems cannot be unstable or chaotic. Our results reveal that in the nonlocal nonlinear media with Gaussian response function, the shape of the new class of 3D accessible spatiotemporal solitary waves can be controlled by the choice of different parameters. Acknowledgements This work is supported by the Science Research Foundation of Shunde Polytechnic under Grant No. 2008-KJ06, China. Work at the
Texas A&M University at Qatar is supported by the NPRP 25-6-7-2 and NPRP 09-462-1-074 projects with the Qatar National Research Foundation.
References [1] G.I. Stegeman, M. Segev, Science 286 (1999) 1518. [2] O. Bang, W. Krolikowski, J. Wyller, J.J. Rasmussen, Phys. Rev. E 66 (2002) 046619. [3] D. Buccoliero, A.S. Desyatnikov, W. Krolikowski, Y.S. Kivshar, Opt. Lett. 33 (2008) 198. [4] D. Buccoliero, A.S. Desyatnikov, W. Krolikowski, Y.S. Kivshar, Phys. Rev. Lett. 98 (2007) 053901; W.P. Zhong, M. Belic, Phys. Lett. A 373 (2009) 296. [5] S. Lopez-Aguayo, A.S. Desyatnikov, Y.S. Kivshar, Opt. Express 14 (2006) 7903. [6] W.P. Zhong, L. Yi, Phys. Rev. A 75 (2007) 061801(R); W.P. Zhong, M. Belic, Phys. Rev. A79 (2009) 023804. [7] A. Snyder, J. Mitchell, Science 276 (1997) 1538. [8] W.P. Zhong, L. Yi, R.H. Xie, M. Belic, G. Chen, Mol. Opt. Phys. 41 (2008) 025402. [9] C. Conti, M. Peccianti, G. Assanto, Phys. Rev. Lett. 92 (2004) 113902. [10] C. Rotschild, O. Cohen, O. Manela, M. Segev, T. Carmon, Phys. Rev. Lett. 95 (2005) 213904. [11] B. Alfassi, C. Rotschild, O. Manela, M. Segev, D.N. Christodoulides, Opt. Lett. 32 (2007) 154. [12] S. Lopez-Aguayo, J.C. Gutiérrez-Vega, Phys. Rev. A 76 (2007) 023814. [13] M. Shen, Q. Wang, J. Shi, P. Hou, Q. Kong, Phys. Rev. E 73 (2006) 056602. [14] Y.J. He, B.A. Malomed, D. Mihalache, H.Z. Wang, Phys. Rev. A 77 (2008) 043826. [15] Y.J. He, B.A. Malomed, D. Mihalache, H.Z. Wang, Phys. Rev. A 78 (2008) 023824. [16] B.A. Malomed, D. Mihalache, F. Wise, L. Torner, J. Opt. B7 (2005) R53. [17] L. Berge, Phys. Rep. 303 (1998) 260. [18] D. Mihalache, D. Mazilu, F. Lederer, B.A. Malomed, Y.V. Kartashov, L.-C. Crasovan, L. Torner, Phys. Rev. E73 (2006) 025601(R). [19] I.B. Burgess, M. Peccianti, G. Assanto, R. Morandotti, Phys. Rev. Lett. 102 (2009) 203903. [20] S. Chen, J.M. Dudley, Phys. Rev. Lett. 102 (2009) 233903. [21] L. Zhang, S. Wen, X. Fu, J. Deng, J. Zhang, D. Fan, Opt. Commun. 283 (2010) 2251. [22] C. Dai, Y. Wang, J. Zhang, Opt. Lett. 35 (2010) 1437. [23] G. Assanto, M. Peccianti, C. Conti, Nematicons, Opt. Phot. News, February Issue, 2003, p. 45. [24] C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trulli, Phys. Rev. Lett. 90 (2003) 170406. [25] Y. Kominis, N. Moshonas, P. Papagiannis, K. Hizanidis, D.N. Christodoulides, Opt. Lett. 30 (2005) 2924. [26] D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J.V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, P. Di Trapani, Opt. Express 15 (2007) 13077. [27] W.P. Zhong, R.H. Xie, M. Belic, G. Chen, Phys. Rev. A 78 (2008) 013826.