Lagrangian approach for dark soliton in nonlocal nonlinear media

Optics Communications 285 (2012) 3631–3635

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Lagrangian approach for dark soliton in nonlocal nonlinear media Shaozhi Pu a,b,n, Chunfeng Hou a, Kaiyun Zhan a, Chengxun Yuan a, Yanwei Du a a b

Department of Physics, Harbin Institute of Technology, Harbin 150001, China Department of Optical Information Science and Technology, Harbin University of Science and Technology, Harbin 150080, China

a r t i c l e i n f o

abstract

Article history: Received 17 November 2011 Received in revised form 22 April 2012 Accepted 23 April 2012 Available online 9 May 2012

We theoretically investigate the dynamics of dark solitons as well as their interaction in nonlocal media. Approximate equations describing the evolution of the beams are obtained via suitable trial functions of amplitude u and refractive index n in an averaged Lagrangian. Our results reveal that outof-phase dark solitons can evolve into stable bound states in nonlocal materials. Moreover, it is found that the separations in the bound state monotonically increase with the degree of nonlocality in nonlocal limit. These results are in excellent agreement with the numerical simulations. & 2012 Elsevier B.V. All rights reserved.

Keywords: Dark solitons Nonlocal nonlinearity Lagrangian approach

1. Introduction Since the pioneering work of Snyder and Mitchell [1] in 1997, nonlocal spatial solitons have attracted much special interest, not only for their various forms, such as vortex solitons [2–4], multi-pole solitons [5,6], Laguerre–Gaussian and Hermite–Gaussian solitons [7,8], Ince–Gaussian solitons [9], but also for their novel features compared with local spatial solitons, e.g., the attraction between outof-phase bright solitons [10–12] as well as out-of-phase dark solitons [13–15], the suppression of modulational instability of plane waves [16–18], the existence of incoherent solitons and spatiotemporal solitons [19–22]. Typically, nonlocality has a profoundly stabilizing effect on light beams [23–25]. The aforementioned novel features of nonlocal solitions are attributed to the nonlocal nature of the materials which acts as some sort of spatial averaging of the nonlinear response of the medium. Indeed, nonlocal materials are ubiquitous in nature, such as photorefractive crystals [26], thermal materials [27], atomic vapors [28], Bose–Einstein condensates [29] and nematic liquid crystals [30]. In addition, it has been found that the materials with quadratic nonlinearity also display the nonlocal nonlinearity [31–34]. Thus far, a wealth of theoretical and experimental efforts have been devoted to investigate bright nonlocal solitons [35–40]. Among these investigations, the most interesting phenomenon is the dynamics of the bright solitons in nonlocal materials, which can be steered by a variation of nonlocal degree, and these voltage-controllable nonlocal solitons are potentially useful for beam steering and optical interconnectes [41–43]. At the same time, dark solitons in nonlocal materials have also been received a great deal of attention, not only for their

n Corresponding author at: Department of Physics, Harbin Institute of Technology, Harbin 150001, China. E-mail address: [email protected] (S. Pu).

0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.04.030

fundamental interest but also for their potential application as self-induced optical waveguides for all-optical switching and steering devices [13–15,44–46]. Due to the complex relation between solitonlike beam and the nonlocal materials, to our knowledge, the analytical solutions of dark solitons in nonlocal materials are nearly unexplored, except for some special cases and models, such as the weakly and strongly nonlocal limit [17,36,43,47]. Very recently, Kong et al. analytically investigated the dark solitons as well as their interaction in nonlocal materials with an arbitrary degree of nonlocality based on the rectangular profile for the nonlocal response function [43,48]. However, to our knowledge, there is no known physical system which could be described by a rectangular response function. Thus, a physical investigation of dark solitons and dark-soliton interaction in nonlocal materials are still needed. In this paper, the evolution of single dark soliton and their interaction in nonlocal media are investigated by the Lagrangian approach. Different from the previous investigations by the use of the specific form of the response function, which is very important in both analytical and numerical investigation of the nonlocal bright solitons as well as dark solitons, our analysis relies on suitable trial functions of amplitude u and refractive index n. Our results predict the soliton bound state based on the interaction potential which is obtained by the analysis approach. Moreover, we also verify the theoretical predictions by numerical simulations.

2. Theoretical model and equations We consider the light propagation in nonlocal media with cubic nonlinearities, described by the following coupled equations for the light field amplitude u and refractive index n

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S. Pu et al. / Optics Communications 285 (2012) 3631–3635

we obtain the averaged Lagrangian    Z 1 dx0 B 2 2a2 8 L¼  B2 D þ da2 b AB þ tan1 £ dx ¼ 2 þ A 3 15 dz 3b 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

[5,6,13,24,42]. 2

i

@u 1 @ u þ nu ¼ 0, @z 2 @x2

ð1Þ

@2 n n þ juj2 ¼ 0, @x2

ð2Þ

d

aB2

where x and z stand for the transverse and the longitudinal coordinates, respectively, the negative sign in Eq. (1) corresponds to a defocusing nonlinearily, d characterizes the nonlocal length of the cubic nonlinearities. Without pffiffiffi loss of generality, the degree of nonlocality is defined as s ¼ d [5,13]. The coupled equations for the light field amplitude u and refractive index n have the following modified Lagrangian formulation [49–52]        i @u @un 1 1 @u2 1 @n2 1 2 £¼ 1 2    nð9u9 1Þ þ d  þ n2 , un u @z 2 @z 2 @x 2 @x 2 juj ð3Þ

A2 þB2 ¼ 1:

dx0 D a þ ¼A 3B 2B dz 4a 16 dabB2 þ 3b 15

ð9Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!

p

C 2 D2 þF 2 b

2

,

ð10Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p

C 2 D2 þ F 2 b

2

¼ 0,

ð11Þ

8 2a pffiffiffiffi 2 da 2 þ pB2 F 2 bðC 2 D2 þF 2 b Þ3=2 ¼ 0: 15 3b

ð5Þ

1

1.6

6

1.5

Refractive index width

1.4 B2 = 0.9

1.2

B2 = 1 B2 = 0.8

5

B2 = 0.8

1.3 B2 = 1

4 3 2

1.1 1 1.0 0

1

2 3 nonlocality σ

4

ð12Þ

Eqs. (8)–(12) are the equations governing the evolution of the single dark solitons in nonlocal materials. Based on these equations, the soliton width and the width of soliton-induced refractive index are obtained. Some interesting results can be seen in Fig. 1(a). In this graph, whatever the parameter B is, the soliton width first decreases with the increase of the degree of nonlocality, and then reaches a minimum, as the degree of nonlocality evolves, the soliton width will monotonically increase with s. These results are in good qualitative agreement with the result presented in Ref. [43]. This is due to the fact that the nonlocality causes the nonlinear index change to advance towards regions of lower light intensity in the weakly nonlocal limit. In the case of nonlocal limit, the refractive index waveguide which is induced by the light will become weaker and broader with the increase of s. Eventually, only a broader beam would be confined in this waveguide. For a fixed s, the soliton width and the width of soliton-induced refractive index are also affected by the parameter B. That is to say, the smaller the parameter B, the bigger the soliton width and the width of soliton-induced refractive index are. Interestingly, the width of soliton-induced refractive index is monotonically increasing with parameter s as shown in Fig. 1(b). Also note that the width of soliton-induced refractive index distribution greatly exceeds the width of the soliton beam in nonlocal limit.

For the sake of simplicity, the idea of an ‘‘equivalent Gaussian’’ is used to evaluate the cross integral [49,50], that is to say sec h2D(x x0) and sech2b(x  x0) are replaced by exp½ðxx0 Þ2 C 2 D2  2 and exp½ðxx0 Þ2 F 2 b , respectively. The coefficients C and F are then determined by matching the Taylor series of the cross integral with the ‘‘equivalent Gaussian’’ in the nonlocal limit. ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi That is, C ¼ p=2 and F ¼ ðp=8ÞI22 , with I22 ¼ 13 ln 2 þ 16. Finally,

Solitons width

ð8Þ

paDC 2 ðC 2 D2 þF 2 b2 Þ3=2 23 ¼ 0,

where the parameters A, B, D, a, b and x0 are functions of the propagation variable z. Substituting trial functions (4) and (5) into R1 the system Lagrangian L ¼ 1 £ dx and integrating over x. All the resulting integrals can be simply evaluated, except for the cross integral Z 1 Z 1 nðjuj2 1Þdx ¼  aB2 sech2 bðxx0 Þsech2 Dðxx0 Þdx: ð6Þ 1

ð7Þ

pffiffiffiffi

ð4Þ

2

:

dB ¼ 0, dz

Based on the modified Lagrangian formulation, the trial function for the refractive index n is chosen in the following form: n ¼ a½tanh bðxx0 Þ1,

2

Taking variations of the averaged Lagrangian with respect to the parameters x0, D, B, a and b. We obtain that

where the asterisk denotes the complex conjugate. For the sake of simplicity, we only consider the coupled equations for one transverse dimension x. In what follows, we will use the modified Lagrangian formulation (3) to study the evolution of single dark soliton and their interaction in nonlocal materials. First, let us consider one-soliton solution. In this case, the trial function for the electric field u is taken in the following form [48,52]: u ¼ B tanh½Dðxx0 Þ þiA,

p

C 2 D2 þF 2 b

5

0

1

2 3 nonlocality σ

4

5

Fig. 1. (a) Soliton width versus the nonlocal parameter s for different B and (b) refractive index width versus parameter s for different B.

S. Pu et al. / Optics Communications 285 (2012) 3631–3635

Then, to analytically investigate the interaction of dark solitons in nonlocal nonlinear media, the trial function for the electric field u is taken in the following form [48,52]: u ¼ ðB tanh z þ iAÞðB tanh z þ iAÞ,

A2 þ B2 ¼ 1,

ð13Þ

þ

2

ð14Þ

where the parameters A, B, D, a, b and x0 are functions of the propagation variable z. Substituting trial functions (13) and (14) into R1 the system Lagrangian L ¼ 1 £ dx and integrating over x. Considering the case of weakly overlapping dark solitons, 15Dx0, 15 bx0, all the resulting integrals can be simply evaluated, except for the cross R1 2 2 integral, i.e., 1 sech Dðx þ x0 Þsech bðx þ x0 Þdx. In a way similar to 2 Ref. [44], the functions sech D(xþx0) and sech2b(xþx0) are replaced 2 by exp½ðx þ x0 Þ2 C 2 D2  and exp½ðx þ x0 Þ2 F 2 b , respectively. Finally, we obtain the averaged Lagrangian L¼

Z

2

1

£ dx ¼ 2L0 þ

1

"

2

dB 4B 3D B D  þ 16B2 e4x0 D 2D2 x0  dz AD tanhð2x0 DÞ 4 6

#

ð15Þ where pffiffiffiffi    dx0 B 2 aB2 p 8 2 2a2  B2 D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ AB þtan1 a bd þ : A 3 15 dz 3b 2 D2 þ b ð16Þ Taking variations of the averaged Lagrangian with respect to the parameters D,a,b,A,B and x0. We obtain that pffiffiffiffi 2 aD p dB 2  þ 2  3 ðD þ b2 Þ3=2 dz AD2 tanhð2x0 DÞ ! ! 2B2 Dx0 3 B2 8e4x0 D 8D2 x20 7Dx0  þ þ 3 6 4 " # pffiffiffiffi 2 2 2 2 2 2 a p D þ 8Db x0 8D3 b x0 2 ¼ 0, þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eð4D b x0 Þ=ðD þ b Þ  ð17Þ 2 2 2 D2 þ b ðD2 þ b Þ2 D2 þ b

pffiffiffi B2 ba p 2 ðD2 þ b Þ3=2

h pffiffiffi 2 B2 p ffi ð4D2 b2 x20 Þ=ðD2 þ b2 Þ b þ 8D2 bx0 8 2 þ 15 a s 2a2 þ paffiffiffiffiffiffiffiffiffiffiffi  e 2 2 2 2 3b

D þb

D þb

2

3

8D2 b x0 2 ðD2 þ b Þ2

b

where V(x0) represents an interaction potential, and is given by Vðx0 Þ ¼ V 1 ðx0 Þ þ V 2 ðx0 Þ þ V 3 ðx0 Þ

ð24Þ

with   46 7 2 e4x0 b 16b x0 d þ bd þ1 þ 4x0 3 2b B ! 3BD B3 D 4x0 D 2  e þ 4BD x0  2 3 30 1 pffiffiffiffi pffiffiffiffi aB p a p C ð4D2 b2 x20 Þ=ðD2 þ b2 Þ 7B D  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 5@ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA, 3B 2 2 4 D2 þ b 2B D2 þ b   D , V 2 ðx0 Þ ¼ e8Dx0 16D4 x20 d þ 24D3 x0  16 V 3 ðx0 Þ ¼



a2

( 2D2 x0 D2 x0 4D2 x0 2 þ þ ð4a2 b da2 bÞ 2 Dþb 4ðD þ bÞ ðDþ bÞ ðD þ bÞ3 " !# 2 1 13 323a2 b d 2 Dð64a2 b d16a2 bÞ þ 8D2 þ 14a2 þ 2 3 3 16ðD þ bÞ 2



13D 323a2 b d þ 14a2 12ðD þ bÞ 3

!)

e4ðD þ bÞx0 :

ð25Þ

Indeed, the potential (24) does predict a bound state of out-ofphase dark solitons in nonlocal materials as shown in Fig. 2, ×10−3

i

ð19Þ

2

1

0

ð20Þ -1

1

" # pffiffiffiffi a p 4A 4x0 D 3D B2 D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC e  2D2 x0  Aþ B 4 3 2 2B D2 þ b pffiffiffiffi 2 2 2 2 2 Aa p dA 1  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eð4D b x0 Þ=ðD þ b Þ þ dz 2AD tanhð2x0 DÞ 2 2 2B D þ b

dx0 BD A@ þ 3B dz

ð23Þ

3

dx0 dB B þ ¼ 0, dz A2 D tanhð2x0 DÞ dz 0

2

d x0 dVðx0 Þ ¼ , dx0 dz2

ð18Þ

  140bx0 23 þ þ 8a2 de4x0 b 32bx20  3 3 ! 10 40x 0 þ a2 e4x0 b þ 64x20 ¼ 0, 2

b

ð22Þ

Here we will use these equations to predict the bound state of two out-of-phase dark solitons. In the case of weakly interaction and almost black solitons, we obtain from Eqs. (21) and (22) the following equation for the soliton coordinate x0

4

pffiffiffiffi pffiffiffiffi 2 2 2 2 2 B2 p 16 4a B2 p  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ abd þ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eð4D b x0 Þ=ðD þ b Þ 15 3b 2 2 2 2 D þb D þb   23 4a 4x0 b 8bx0  þ16abde4x0 b e ð58bx0 Þ ¼ 0, 3 b

ð21Þ

  2a2 b 4x0 b 116 2a2 4x0 b 2 32b x0  b  ð78bx0 Þ: de e 3 B B

V 1 ðx0 Þ ¼

pffiffiffiffi   4D2 b2 x2 0 2aB2 p  23 a2 8bx0  e4x0 b ð2032bx0 Þ,  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e D2 þ b2 þ 16a2 bde4x0 b 3 b 2 2 D þb

L0 ¼ 2

þ

Effective Potential

2

dD 1 ¼ 0, dz 2D2 tanhð2x0 DÞ

" # pffiffiffiffi 2 dA 2B2 D2 2aBD2 b x0 p ð4D2 b2 x2 Þ=ðD2 þ b2 Þ 0 e ¼ 2Be4x0 D 8D3 x0 þ 5D2 þ þ 2 3=2 2 dz 3 ðD þ b Þ

where z 7 ¼D(x7x0) and 2x0 denotes the separation between solitons. Based on the modified Lagrangian formulation, the trial function for refractive index n is chosen in the following form: n ¼ ½a sech bðx þ x0 Þ1½a sech bðxx0 Þ11,

3633

-2 0

2

4

6

8

10 x0

12

14

16

18

Fig. 2. Dark soliton interaction potential V(x0) for s ¼ 4 and B ¼1.

20

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S. Pu et al. / Optics Communications 285 (2012) 3631–3635

interesting results can be seen in Fig. 3. In the case of small initial separations, whatever the degree of nonlocaltiy is, two out-ofphase dark solitons could trap each other in an oscillatory transversal motion as shown in Fig. 3(a), (e) and (i). In this case, their separations first increase and then decrease. For large initial separations, the solitons could also trap each other in an oscillatory transversal motion, but their separations first decrease and then increase as shown in Fig. 3(c), (g) and (k). It is important to emphasize that two out-of-phase dark solitons can form a bound state at proper separation in nonlocal medium as clearly shown in Fig. 3(b), (f) and (j). Interestingly, we numerically find that the separation of the bound state first decreases and then increases with the increase of parameter s as shown in Fig. 3(b), (f) and (j). In Fig. 4 separations of the bound states which are obtained by numerical solutions of Eq. (26) are plotted together with the approximations found using the potential (24). One can see that the bound state, which is predicted by Eq. (24), is in good agreement with the numerical simulation based on Eq. (26) in nonlocal limit, but shows great discrepancy with the numerical simulation in the weakly nonlocal limit as shown in Fig. 4. This discrepancy is attributed to the weakly interaction condition which is valid for sufficiently large 2x0. In the analytical theory, the attraction is too weak to balance the repulsion in weakly nonlocal limit. So the separation of the bound state tends to infinity in the weakly limit as shown in Fig. 4 by the

where the interaction potential (24) is plotted as a function of peak separation for s ¼4. Here, we take the following parameter B ¼1, other parameters D, a and b are obtained based on Eqs. (17)–(19) in the weakly interaction condition. We can see the potential that it first decreases and then increases with the increase of the peak separation between the solitons, which indicate the potential is attractive for large separation and repulsive for small separation. It is noteworthy that, Fig. 2 shows dVðx0 Þ=dx0 ¼ 0 at x0 ¼2.31, which corresponds to the bound state for s ¼ 4. In this case, the attraction is balanced by the repulsion between two out-of-phase dark solitons. Our results will now be confirmed by full numerical solutions of Eqs. (1) and (2). Without loss of generality, the two coupled Eqs. (1) and (2) could be written as one equation Z þ1 @u 1 @2 u i þ u RðxxÞjuj2 dx ¼ 0, ð26Þ @z 2 @x2 1

100

100

80

80

80

80

60

60

40

40

40

20

20

20

20

0

0 -15

-10

-5 0 5 10 Spatial coordinate

15

-15

-10

-5 0 5 Spatial coordinate

10

0 -15

15

-10

-5 0 5 Spatial coordinate

10

15

-15

100

100

80

80

80

80

60

60

Distance

100

Distance

100

Distance

60

40

40

40

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20

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20

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0

0 -15

-10

-5 0 5 10 Spatial coordinate

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-15

-10

-5 0 5 Spatial coordinate

10

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80

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60

40

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20

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20

-10

-5

0

5

Spatial coordinate

10

15

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15

-15

-10

-5

0

5

Spatial coordinate

10

15

0 5 10 Spatial coordinate

15

-15

-10

-5

0 5 10 Spatial coordinate

15

-10

-5

100

80

60

40

20

0

0

0 -15

-5 0 5 Spatial coordinate

60

40

0

-10

Distance

Distance

100

60

-5

0 -15

15

-10

60

40

Distance

Distance

60

40

0

Distance

60

Distance

100

Distance

100

Distance

Distance

where the response function RðxÞ ¼ ð2sÞ1 expðjxj=sÞ [5,13]. In this part, we first find dark soliton solution of Eq. (26) by the Newton iterative method, and then use this solution as the initial soliton profiles in the simulations. In the following, for a fixed parameter s, we will do a series of simulations of the interactions of two out-of-phase dark solitons by adjusting their initial separations, as shown in Fig. 3(a)–(d) [(e)–(h) and (i)–(l)]. Some

-15

-10

-5

0

5

Spatial coordinate

10

15

-15

0

5

10

15

Spatial coordinate

Fig. 3. Simulated interaction of dark solitons in nonlocal medium for (a) s ¼0.5, x0 ¼1.4, (b) s ¼0.5, x0 ¼1.6, (c) s ¼0.5, x0 ¼2, (d) s ¼0.5, x0 ¼3.15, (e) s ¼1, x0 ¼ 1.3, (f) s ¼ 1, x0 ¼1.5, (g) s ¼ 1, x0 ¼ 3, (h) s ¼1, x0 ¼ 3.3, (i) s ¼ 4, x0 ¼1.6, (j) s ¼ 4, x0 ¼ 2.2, (k) s ¼4, x0 ¼4 and (l) s ¼ 4, x0 ¼ 5.4.

S. Pu et al. / Optics Communications 285 (2012) 3631–3635

14 varational analysis numerical simulations

12

Soliton separation

10

8

6

4

2

0 0

1

2 nonlocality σ

3

4

Fig. 4. Separation between solitons in a bound state as a function of the degree of nonlocality s for B¼1. Solid line is the solution by the variational analysis; squares are the numerical solution of Eq. (25).

solid line. From this figure, one can see that the separation in the bound state monotonically increases with the degree of nonlocality in nonlocal limit. The reason is that the attraction between the solitons increases with the increase of s, which leads to two dark solitons could feel each other even at a large separation.

3. Conclusions In summary, we analytically studied the dynamics of the single dark soliton and soliton pairs in nonlocal media using the Lagrangian approach. We find that, for one-soliton solution, the soliton width is drastically affected by the nonlocality and the parameter B. We also find that out-of-phase dark solitons can form stable bound states in nonlocal medium, and analytically predict that the separation in the bound state is monotonically increasing with the degree of nonlocality. Our analytical results agree well with the exact numerical simulation in nonlocal limit. These results pave the way for better understanding of single dark soliton as well as their interaction in nonlocal medium and also suggest a way to steer soliton beam in nonlocal medium, by controlling the strength of nonlocality.

Acknowledgments This work was supported by the program of excellent Team in Harbin Institute of Technology and the Natural Science Foundation of China (Grant no. 60508005). References [1] A. Snyder, D. Mitchell, Science 276 (1997) 1538. [2] A.I. Yakimenko, Y.A. Zaliznyak, Y. Kivshar, Physical Review E 71 (2005) 065603. [3] D. Briedis, D.E. Petersen, D. Edmundson, W. Krolikowsk, O. Bang, Optics Express 13 (2005) 435.

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