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Numerical simulation of attraction of partially spatially incoherent dark solitons
Optik 123 (2012) 140–143
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Numerical simulation of attraction of partially spatially incoherent dark solitons Chunfu Huang Physics and Electronic Information College, Tianjin Normal University, Tianjin 300387, China
a r t i c l e
i n f o
Article history: Received 13 October 2010 Accepted 26 February 2011
PACS: 42.65.Tg 42.25. Kb 42.65.Jx
a b s t r a c t We investigate numerically the interactions of partially spatially incoherent odd dark solitons in a defocusing kerr nonlinear media based on the coherent density approach, and find that a pair of partially spatially incoherent dark solitons may be changed from repulsion to attraction with an appropriate incoherence parameter. Moreover, a pair of partially incoherent gray soliton may form under appropriate conditions. Some numerical examples are provided to illustrate their interaction behavior further. © 2011 Elsevier GmbH. All rights reserved.
Keywords: Interaction Incoherence Dark solitons
1. Introduction Since the first observation of solitons made of quasimonochromatic spatially incoherent lights and of incoherent white light beams, the properties of incoherent spatial solitons and their interactions have drawn considerable attention [1–18]. It is known that to support these incoherent solitons, the respond time of the non-instantaneous nonlinearity should be much longer than the characteristic beam fluctuation time, and the non-instantaneous nonlinearity of the medium response only to the envelop of the beam. To date, there are four theoretical approaches to describe this incoherent spatial solitons, which are the coherent density approach [3–6], the self-consistent modal theory [7,8], the mutual coherence function theory [9,10], and the Wigner transform method [11]. A statistical approach based on the Wigner transform is proposed to descript the propagation of partially incoherent light beam [12]. Equivalence of the first three theories was shown in Ref. [13]. Lisak et al. demonstrated the relations between the four approaches in Ref. [14]. Here we use the coherent density approach to investigate the interaction of partially spatial incoherent odd dark solitons in this short communications. First let us introduce the progress of incoherent spatial solitons. Recently, spontaneous pattern formation with white light [15] and white-light optical vortex solitons [16], nonlocal incoherent spatial solitons [17–19], random-phase solitons in nonlinear
E-mail address: [email protected] 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.02.025
periodic lattices [20], incoherent-white solitons in periodic lattices [21], random-phase surface-wave solitons in nonlocal nonlinear media [22], incoherent surface solitons in effectively instantaneous nonlocal nonlinear media [23] have been studied. Meanwhile, the properties of interactions of incoherent spatial solitons have been investigated experimentally and theoretically. For example, Lu et al. have observed experimentally the attraction between two white-light photovoltaic dark spatial solitons in a photorefractive LiNbO3 :Fe crystal [24]. Huang et al. have investigated the interactions of partially incoherent bright spatial solitons and incoherent white-light bright solitons in a nonlinear media with logarithmic saturable nonlinear, respectively [25,26]. Cohen et al. have investigated the interaction of incoherent bright solitons in instantaneous nonlocal nonlinear media [27]. Most interesting of all, Ku et al. have demonstrated and observed experimentally a novel type of soliton interaction, which can be controlled by the total partial incoherence, and showed that the interaction may be changed from attractive to repulsive, or vice versa [28]. However, among all the studies mentioned above, the properties of interaction of dark soliton pairs made of quasimonochromatic partially incoherent light beams have not been explored yet. Following Ku’s work in Ref. [28], we would like to investigate numerically the soliton interactions when a pair of closed spaced dark spatial solitons as a whole is made partially spatially incoherent. It is known that there are three factors affect the properties of interactions of spatial solitons, which are the phase difference [29], nonlocal nonlinearity [30–32], and incoherence [28]. In particular, in nonlocal nonlinear media, two dark spatial soliton may attract
C. Huang / Optik 123 (2012) 140–143
141
[31,32]. As mentioned above, incoherence plays an important role in the soliton interactions, and one can show that incoherence always provides an attractive force between them. Furthermore incoherence may provide strong destabilization of solitons [33,34], suppression of modulation instabilities [35] and transverse instabilities [36]. Initiated by the pioneers work, here we present what we believe to be the first theoretically investigation of attraction of partially incoherent odd dark solitons. We show that the incoherence parameter may drastically control the interaction between them, and show that the interaction may be changed from repulsion to attraction with an appropriate incoherence parameter. Moreover we addressed the physical features exhibited by the interaction of partially incoherent dark spatial solitons. 2. The model To start, let us assume that the light beam propagates along the Z axis and diffracts only in the X direction, and consider the nonlinear medium with kerr type with de-focusing nonlinearity. After a simple normalization, propagation of a one-dimensional quasi-monochromatic partially incoherent beam in a de-focusing kerr nonlinear medium is governed by the following nonlinear Schrödinger equation [3–5],
i
∂f ∂f + ∂Z ∂X
∞
I(X, Z) =
+
1 ∂2 f − I(X, Z)f = 0 2 ∂X 2
f (X, , Z)2 d
(1)
(2)
−∞
Eq. (1) describes partially incoherent beam propagation along the Z direction of a noninstantaneous de-focusing kerr nonlinear medium, which is the evolution equation of the coherent density function. Here f represent the so-called coherent density function for partially incoherent beam. is an normalized angle with respect to the Z axis. I(X, Z) denotes the normalized time-averaged total intensity. We can express the normalized coherent density in the following way at Z = 0, 1/2
f (X, Z = 0, ) = GN ()0 (X)
(3)
GN () is the normalized angular power spectrum of the incoherent sources. Now assume that the incoherent angular power spectrum √ is Gaussian, i.e., GN () = exp(− 2 /02 )/ 0 . The coherence of the partially incoherent light beam is determined by the parameter 0 ; i.e., less coherence means larger 0 . 0 (X) is the input spatial modulation function. In this letter, we assume that they have the coherent soliton profile at the input. The input spatial modulation function is 0 (X, Z = 0) = tanh(X + d)
(4a)
where −∞< X < 0, and 0 (X, Z = 0) = −tanh(X − d)
(4b)
where 0 ≤ X < + ∞. 2d is the soliton separation between the two input beams. 3. Results of numerical simulations Next we consider the interactions of dark solitons. In the simulations below, we set d = 1.5. The range of X axis is taken as 100, and the propagation length is 80. As an example we investigate numerically how the two dark solitons may be changed from repulsion to attraction with an appropriate incoherent parameter 0 . Fig. 1(a) shows the repulsion of dark solitons when 0 → 0. It behaves as
Fig. 1. Interactions of two partially incoherent odd dark solitons for different incoherence ( 0 ). (a) 0 → 0; (b) 0 = 0.28; (c) 0 = 0.40; (d) 0 = 0.45 when d = 1.5; (e) bounded partially incoherent gray soliton pair when 0 = 0.40 and d = 2.2.
142
C. Huang / Optik 123 (2012) 140–143
4. Conclusions
Fig. 2. Schematic representation of the force between the two incoherent gray soliton pair.
coherent dark soliton interaction and repels each other. As the incoherence parameter 0 is increased, for example, 0 = 0.28, the repulsion force between the two dark solitons is decreased, so that the separation at the output decreased (Fig. 1(b)). From Fig. 1(a) and (b) we see that, when the incoherent parameter 0 is small, these interactions are very similar to that of coherent soliton interactions, and they repel each other. Then we increase the incoherence parameter 0 , for example, let 0 = 0.40. From Fig. 1(c) we see that separation at the output decreased further, so it confirms that they have a tendency to attract. With an appropriate incoherence parameter, for example, 0 = 0.45, they attract each other, and the separation at the output almost equal to that at the input (Fig. 1(d)). We have performed additional simulations to verify the interactions of partially incoherent dark solitons and found that they may be changed from repulsion to attraction with an appropriate incoherence parameter. Moreover, we find that, under appropriate conditions, i.e., the soliton width, the incoherence parameter and the separation between them, they may form a bound incoherent dark soliton pairs, which is depicted in Fig. 1(e) for 0 = 0.40 and d = 2.2. We discuss the results of our numerical simulations as follows: the partially incoherent dark soliton is a gray soliton. After a certain distance of propagation, an initially odd beam becomes a graylike incoherent quasi-soliton whose FWHM and grayness oscillate around constant values [5,6]. The two graylike quasi-solitons are qualitatively depicted in Fig. 2. When the incoherence parameter 0 is small, the partially incoherent gray solitons are almost coherent dark solitons, thus these interactions are similar to that of coherent soliton interactions, they repulse each other, i.e., the bounded bright solitons are “less brightness”, thus the attraction between them is very weak. With the increase of incoherence parameter 0 , the partially incoherent gray soliton are very “grayer”, and the bounded bright soliton become “brighter” enough, thus the attraction between the bounded bright soliton is stronger enough than the repulsion of the dark incoherent solitons, as a result of fact, the interactions can be changed from repulsion to attraction with an appropriate incoherence parameter 0 . When the force between the dark solitons and the bounded bright solitons are balanced, the bounded incoherent gray soliton pairs are formed (Fig. 1(e)). The attraction effect of partially spatially incoherent dark solitons is qualitatively explained in the above section. Moreover, as was shown above, the incoherence plays an important role in the solitons interactions, just as Ref. [28] pointed out, for even larger incoherence, the solitons become less coherent and may attract. When incoherence parameter 0 increase larger, the incoherent interaction become greater, thus the attraction force is greater. For even larger incoherence, the incoherent interaction is greater enough and force them to attract each other, so as it does to the dark solitons.
In this short communications, we present our numerical results of the interactions of partially incoherent odd dark solitons in a nonlinear media with kerr type media based on the coherent density approach. Numerical simulations show that a pair of partially incoherent dark solitons may be changed from repulsion to attraction with an appropriate incoherence parameter. The incoherence plays an important role in the soliton interaction. When the incoherence is larger enough, the attraction between the dark solitons is greater enough, thus force them to attract each other. When the force between the dark solitons and the bounded bright solitons are balanced, the bounded incoherent gray soliton pairs are formed. We believe that interactions of this partially incoherent dark solitons can be found in other nonlinear media and other fields.
Acknowledgements This research was supported by the Foundation for Young Teachers in Tianjin Normal University under Grant No. 5RL074, also supported by the National Natural Science Foundation of China under Grant Nos. 61078059, 11005080.
References [1] M. Mitchell, Z. Chen, M. Shih, M. Segev, Self-trapping of partially spatially incoherent light, Phys. Rev. Lett. 77 (1997) 490–493. [2] M. Mitchell, M. Segev, Self-trapping of incoherent white light, Nature 387 (1997) 880–882; Z. Chen, M. Segev, T.H. Coskun, D.N. Christodoulides, Self-trapping of dark incoherent light beam, Science 280 (1998) 889–892. [3] D.N. Christodoulides, T.H. Coskun, M. Mitchell, M. Segev, Theory of incoherent self-focusing in biased photorefractive media, Phys. Rev. Lett. 78 (1997) 646–649. [4] D.N. Christodoulides, T.H. Coskun, R.I. Joseph, Incoherent spatial solitons in saturable nonlinear media, Opt. Lett. 22 (1997) 1080–1082. [5] T.H. Coskun, D.N. Christodoulides, M. Mitchell, Z. Chen, M. Segev, Dynamics of incoherent bright and dark self-trapped beams and their coherence properties in photorefractive crystals, Opt. Lett. 23 (1998) 418–420. [6] T.H. Coskun, D.N. Christodoulides, Z. Chen, M. Segev, Dark incoherent soliton splitting and “phase-memory” effects: theory and experiment, Phys. Rev. E 59 (1999) R4777–R4780. [7] M. Mitchell, M. Segev, T.H. Coskun, D.N. Christodoulides, Theory of self-trapped spatially incoherent light beams, Phys. Rev. Lett. 79 (1997) 4990–4993. [8] D.N. Christodoulides, T.H. Coskun, M. Mitchell, M. Segev, Multimode incoherent spatial solitons in logarithmically saturable nonlinear media, Phys. Rev. Lett. 80 (1998) 2310–2313. [9] V.V. Shkunov, D. Anderson, Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media, Phys. Rev. Lett. 81 (1998) 2683–2686. [10] W. Królikowski, D. Edmundson, O. Bang, Unified model for partially coherent solitons in logarithmically nonlinear media, Phys. Rev. E 61 (2000) 3122–3126. [11] L. Helczynski, D. Anderson, R. Fedele, B. Hall, M. Lisak, Propagation of partially incoherent light in nonlinear media via the Wigner transform method, IEEE J. Sel. Top. Quantum Electron. 8 (2002) 408–412. [12] B. Hall, M. Lisak, D. Anderson, R. Fedele, V.E. Semenov, Statistical theory for incoherent light propagation in nonlinear media, Phys. Rev. E 65 (2002), 035602(R)-1-4. [13] D.N. Christodoulides, E.D. Eugenieva, T.H. Coskun, M. Segev, M. Mitchell, Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media, Phys. Rev. E 63 (2001), 035601(R)-1–035601(R)-4. [14] M. Lisak, L. Helczynski, D. Anderson, Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media, Opt. Commun. 220 (2003) 321–323. [15] T. Schwartz, T. Carmon, H. Buljan, M. Segev, Spontaneous pattern formation with incoherent white light, Phys. Rev. Lett. 93 (2005), 223901-1–223901-4. [16] (a) V. Shvedov, W. Królikowski, A. Volyar, et al., Focusing and correlation properties of white-light optical vortices, Opt. Express. 13 (2005) 7393–7398; (b) O.V. Angelsky, A.P. Maksimyak, P.P. Maksimyak, et al., Interference diagnostics of white-light vortices, Opt. Express. 13 (2005) 8179–8183. [17] W. Królikowski, O. Bang, J. Wyller, Nonlocal incoherent solitons, Phys. Rev. E 70 (2004), 036617-1–036617-5. [18] K.G. Makris, H. Sarkissian, D.N. Christodoulides, G. Assanto, Nonlocal incoherent spatial solitons in liquid crystals, J. Opt. Soc. Am. B 22 (2005) 1371–1377. [19] M. Shen, Q. Wang, J.L. Shi, P. Hou, Q. Kong, Partially coherent accessible solitons in strongly nonlocal media, Phys. Rev. E 73 (2006), 056602-1–056602-6. [20] H. Buljan, O. Cohen, J.W. Fleischer, et al., Random-phase solitons in nonlinear periodic lattices, Phys. Rev. Lett. 92 (2004), 223901-1–223901-4.
C. Huang / Optik 123 (2012) 140–143 [21] R. Pezer, H. Buljan, G. Bartal, et al., Incoherent white-light solitons in nonlinear periodic lattices, Phys. Rev. E 73 (2006), 056608-1–056608-9. [22] A. Barak, C. Rotschild, B. Alfassi, M. Segev, D.N. Christodoulides, Randomphase surface-wave solitons in nonlocal nonlinear media, Opt. Lett. 32 (2007) 2450–2452. [23] B. Alfassi, C. Rotschild, M. Segev, Incoherent surface solitons in effectively instantaneous nonlocal nonlinear media, Phys. Rev. E 80 (2009), 0418081–041808-4. [24] Y. Lu, S.M. Liu, G.Q. Zhang, R. Guo, et al., Waveguides and directional coupler induced by white-light photovoltaic dark spatial solitons, J. Opt. Soc. Am. B 21 (2004) 1674–1678. [25] R. Guo, C.F. Huang, S.M. Liu, Interaction of partially incoherent spatial solitons in logarithmic saturable nonlinear media, J. Opt. A: Pure Appl. Opt. 8 (2006) 695–702. [26] C.F. Huang, R. Guo, S.M. Liu, Z.H. Liu, Interaction of incoherent white light solitons, Opt. Commun. 269 (2007) 174–177. [27] O. Cohen, H. Buljan, T. Schwartz, Incoherent solitons in instantaneous nonlocal nonlinear media, Phys. Rev. E 73 (2006), 015601(R)-1–015601(R)-4.
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[28] T.S. Ku, M.F. Shih, A.A. Sukhorukov, Y.S. Kivshar, Coherence controlled soliton interactions, Phys. Rev. Lett. 94 (2005), 063904-1–063904-4. [29] M. Segev, G.I. Stegeman, Optical spatial solitons and their interactions: universality and diversity, Science 286 (1999) 1518–1523. [30] P.D. Rasmussen, O. Bang, W. Królikowski, Theory of nonlocal soliton interaction in nematic liquid crystals, Phys. Rev. E 72 (2005), 066611-1–066611-7. [31] N.I. Nikolov, D. Neshev, W. Królikowski, et al., attraction of nonlocal dark optical solitons, Opt. Lett. 29 (2004) 286–288. [32] D. Dreischuh, D.N. Neshev, D.E. Petersen, et al., Observation of attraction between dark solitons, Phys. Rev. Lett. 96 (2006), 043901-1–043901-4. [33] C.C. Jeng, M.F. Shih, K. Motzek, et al., Partially incoherent optical vortices in self-focusing nonlinear media, Phys. Rev. Lett. 92 (2004), 043904-1–043904-4. [34] K. Motzek, F. Kaiser, J.R. Salgueiro, et al., Incoherent vector vortex-mode solitons in self-focusing nonlinear media, Opt. Lett. 29 (2004) 2285–2287. [35] M. Soljacic, et al., Modulation instability of incoherent beams in noninstantaneous nonlinear media, Phys. Rev. Lett. 84 (2000) 467–470. [36] K. Motzek, F. Kaiser, W.H. Chu, et al., Soliton transverse instabilities in anisotropic nonlocal self-focusing media, Opt. Lett. 29 (2004) 280–282.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Numerical simulation of attraction of partially spatially incoherent dark solitons Chunfu Huang Physics and Electronic Information College, Tianjin Normal University, Tianjin 300387, China
a r t i c l e
i n f o
Article history: Received 13 October 2010 Accepted 26 February 2011
PACS: 42.65.Tg 42.25. Kb 42.65.Jx
a b s t r a c t We investigate numerically the interactions of partially spatially incoherent odd dark solitons in a defocusing kerr nonlinear media based on the coherent density approach, and find that a pair of partially spatially incoherent dark solitons may be changed from repulsion to attraction with an appropriate incoherence parameter. Moreover, a pair of partially incoherent gray soliton may form under appropriate conditions. Some numerical examples are provided to illustrate their interaction behavior further. © 2011 Elsevier GmbH. All rights reserved.
Keywords: Interaction Incoherence Dark solitons
1. Introduction Since the first observation of solitons made of quasimonochromatic spatially incoherent lights and of incoherent white light beams, the properties of incoherent spatial solitons and their interactions have drawn considerable attention [1–18]. It is known that to support these incoherent solitons, the respond time of the non-instantaneous nonlinearity should be much longer than the characteristic beam fluctuation time, and the non-instantaneous nonlinearity of the medium response only to the envelop of the beam. To date, there are four theoretical approaches to describe this incoherent spatial solitons, which are the coherent density approach [3–6], the self-consistent modal theory [7,8], the mutual coherence function theory [9,10], and the Wigner transform method [11]. A statistical approach based on the Wigner transform is proposed to descript the propagation of partially incoherent light beam [12]. Equivalence of the first three theories was shown in Ref. [13]. Lisak et al. demonstrated the relations between the four approaches in Ref. [14]. Here we use the coherent density approach to investigate the interaction of partially spatial incoherent odd dark solitons in this short communications. First let us introduce the progress of incoherent spatial solitons. Recently, spontaneous pattern formation with white light [15] and white-light optical vortex solitons [16], nonlocal incoherent spatial solitons [17–19], random-phase solitons in nonlinear
E-mail address: [email protected] 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.02.025
periodic lattices [20], incoherent-white solitons in periodic lattices [21], random-phase surface-wave solitons in nonlocal nonlinear media [22], incoherent surface solitons in effectively instantaneous nonlocal nonlinear media [23] have been studied. Meanwhile, the properties of interactions of incoherent spatial solitons have been investigated experimentally and theoretically. For example, Lu et al. have observed experimentally the attraction between two white-light photovoltaic dark spatial solitons in a photorefractive LiNbO3 :Fe crystal [24]. Huang et al. have investigated the interactions of partially incoherent bright spatial solitons and incoherent white-light bright solitons in a nonlinear media with logarithmic saturable nonlinear, respectively [25,26]. Cohen et al. have investigated the interaction of incoherent bright solitons in instantaneous nonlocal nonlinear media [27]. Most interesting of all, Ku et al. have demonstrated and observed experimentally a novel type of soliton interaction, which can be controlled by the total partial incoherence, and showed that the interaction may be changed from attractive to repulsive, or vice versa [28]. However, among all the studies mentioned above, the properties of interaction of dark soliton pairs made of quasimonochromatic partially incoherent light beams have not been explored yet. Following Ku’s work in Ref. [28], we would like to investigate numerically the soliton interactions when a pair of closed spaced dark spatial solitons as a whole is made partially spatially incoherent. It is known that there are three factors affect the properties of interactions of spatial solitons, which are the phase difference [29], nonlocal nonlinearity [30–32], and incoherence [28]. In particular, in nonlocal nonlinear media, two dark spatial soliton may attract
C. Huang / Optik 123 (2012) 140–143
141
[31,32]. As mentioned above, incoherence plays an important role in the soliton interactions, and one can show that incoherence always provides an attractive force between them. Furthermore incoherence may provide strong destabilization of solitons [33,34], suppression of modulation instabilities [35] and transverse instabilities [36]. Initiated by the pioneers work, here we present what we believe to be the first theoretically investigation of attraction of partially incoherent odd dark solitons. We show that the incoherence parameter may drastically control the interaction between them, and show that the interaction may be changed from repulsion to attraction with an appropriate incoherence parameter. Moreover we addressed the physical features exhibited by the interaction of partially incoherent dark spatial solitons. 2. The model To start, let us assume that the light beam propagates along the Z axis and diffracts only in the X direction, and consider the nonlinear medium with kerr type with de-focusing nonlinearity. After a simple normalization, propagation of a one-dimensional quasi-monochromatic partially incoherent beam in a de-focusing kerr nonlinear medium is governed by the following nonlinear Schrödinger equation [3–5],
i
∂f ∂f + ∂Z ∂X
∞
I(X, Z) =
+
1 ∂2 f − I(X, Z)f = 0 2 ∂X 2
f (X, , Z)2 d
(1)
(2)
−∞
Eq. (1) describes partially incoherent beam propagation along the Z direction of a noninstantaneous de-focusing kerr nonlinear medium, which is the evolution equation of the coherent density function. Here f represent the so-called coherent density function for partially incoherent beam. is an normalized angle with respect to the Z axis. I(X, Z) denotes the normalized time-averaged total intensity. We can express the normalized coherent density in the following way at Z = 0, 1/2
f (X, Z = 0, ) = GN ()0 (X)
(3)
GN () is the normalized angular power spectrum of the incoherent sources. Now assume that the incoherent angular power spectrum √ is Gaussian, i.e., GN () = exp(− 2 /02 )/ 0 . The coherence of the partially incoherent light beam is determined by the parameter 0 ; i.e., less coherence means larger 0 . 0 (X) is the input spatial modulation function. In this letter, we assume that they have the coherent soliton profile at the input. The input spatial modulation function is 0 (X, Z = 0) = tanh(X + d)
(4a)
where −∞< X < 0, and 0 (X, Z = 0) = −tanh(X − d)
(4b)
where 0 ≤ X < + ∞. 2d is the soliton separation between the two input beams. 3. Results of numerical simulations Next we consider the interactions of dark solitons. In the simulations below, we set d = 1.5. The range of X axis is taken as 100, and the propagation length is 80. As an example we investigate numerically how the two dark solitons may be changed from repulsion to attraction with an appropriate incoherent parameter 0 . Fig. 1(a) shows the repulsion of dark solitons when 0 → 0. It behaves as
Fig. 1. Interactions of two partially incoherent odd dark solitons for different incoherence ( 0 ). (a) 0 → 0; (b) 0 = 0.28; (c) 0 = 0.40; (d) 0 = 0.45 when d = 1.5; (e) bounded partially incoherent gray soliton pair when 0 = 0.40 and d = 2.2.
142
C. Huang / Optik 123 (2012) 140–143
4. Conclusions
Fig. 2. Schematic representation of the force between the two incoherent gray soliton pair.
coherent dark soliton interaction and repels each other. As the incoherence parameter 0 is increased, for example, 0 = 0.28, the repulsion force between the two dark solitons is decreased, so that the separation at the output decreased (Fig. 1(b)). From Fig. 1(a) and (b) we see that, when the incoherent parameter 0 is small, these interactions are very similar to that of coherent soliton interactions, and they repel each other. Then we increase the incoherence parameter 0 , for example, let 0 = 0.40. From Fig. 1(c) we see that separation at the output decreased further, so it confirms that they have a tendency to attract. With an appropriate incoherence parameter, for example, 0 = 0.45, they attract each other, and the separation at the output almost equal to that at the input (Fig. 1(d)). We have performed additional simulations to verify the interactions of partially incoherent dark solitons and found that they may be changed from repulsion to attraction with an appropriate incoherence parameter. Moreover, we find that, under appropriate conditions, i.e., the soliton width, the incoherence parameter and the separation between them, they may form a bound incoherent dark soliton pairs, which is depicted in Fig. 1(e) for 0 = 0.40 and d = 2.2. We discuss the results of our numerical simulations as follows: the partially incoherent dark soliton is a gray soliton. After a certain distance of propagation, an initially odd beam becomes a graylike incoherent quasi-soliton whose FWHM and grayness oscillate around constant values [5,6]. The two graylike quasi-solitons are qualitatively depicted in Fig. 2. When the incoherence parameter 0 is small, the partially incoherent gray solitons are almost coherent dark solitons, thus these interactions are similar to that of coherent soliton interactions, they repulse each other, i.e., the bounded bright solitons are “less brightness”, thus the attraction between them is very weak. With the increase of incoherence parameter 0 , the partially incoherent gray soliton are very “grayer”, and the bounded bright soliton become “brighter” enough, thus the attraction between the bounded bright soliton is stronger enough than the repulsion of the dark incoherent solitons, as a result of fact, the interactions can be changed from repulsion to attraction with an appropriate incoherence parameter 0 . When the force between the dark solitons and the bounded bright solitons are balanced, the bounded incoherent gray soliton pairs are formed (Fig. 1(e)). The attraction effect of partially spatially incoherent dark solitons is qualitatively explained in the above section. Moreover, as was shown above, the incoherence plays an important role in the solitons interactions, just as Ref. [28] pointed out, for even larger incoherence, the solitons become less coherent and may attract. When incoherence parameter 0 increase larger, the incoherent interaction become greater, thus the attraction force is greater. For even larger incoherence, the incoherent interaction is greater enough and force them to attract each other, so as it does to the dark solitons.
In this short communications, we present our numerical results of the interactions of partially incoherent odd dark solitons in a nonlinear media with kerr type media based on the coherent density approach. Numerical simulations show that a pair of partially incoherent dark solitons may be changed from repulsion to attraction with an appropriate incoherence parameter. The incoherence plays an important role in the soliton interaction. When the incoherence is larger enough, the attraction between the dark solitons is greater enough, thus force them to attract each other. When the force between the dark solitons and the bounded bright solitons are balanced, the bounded incoherent gray soliton pairs are formed. We believe that interactions of this partially incoherent dark solitons can be found in other nonlinear media and other fields.
Acknowledgements This research was supported by the Foundation for Young Teachers in Tianjin Normal University under Grant No. 5RL074, also supported by the National Natural Science Foundation of China under Grant Nos. 61078059, 11005080.
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