Evolution of superposition solitons and breathers in strongly nonlocal nonlinear media

Optik 122 (2011) 1501–1507

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Optik journal homepage: www.elsevier.de/ijleo

Evolution of superposition solitons and breathers in strongly nonlocal nonlinear media Chunfu Huang ∗ , Yaping Cao, Zhifang Zhao, Meidong 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 2 April 2010 Accepted 28 August 2010

PACS: 42.65.Jx 42.65.Tg 42.70.Nq 42.70.Df

a b s t r a c t We analyzed and obtained the condition to form a new kind of two-dimensional solitons with complex structure named as “superposition solitons and breathers” in strongly nonlocal media based on the analytical and numerical investigations. Furthermore we investigated how such solitons and breathers persist and are robust when degree of nonlocality is gradually decreased when taken the fully nonlinear system, then we can find such solitons by using the method. © 2010 Elsevier GmbH. All rights reserved.

Keywords: Superposition solitons and breathers Two-dimensional solitons with complex structure Strongly nonlocal media Propagation

1. Introduction Nonlocal spatial solitons have attracted much attention in recent years both from experimental and theoretical side [1–34]. Many nonlinear optical materials may have the nonlocality of nonlinearity. For example, nematic liquid crystals [1–3] and lead glass [4–6]. Nonlocality can suppress modulational instability [7,8], stabilize fundamental and higher order solitons, such as Laguerre and Hermite soliton cluster [9], dipole solitons and multipole solitons [10–17], vortex solitons [18–20]. In particular, what nonlocality support two-dimensional solitons with complex structure has attracted much interest, then in this paper we will discuss the formation and propagation of such solitons in strongly nonlocal media in another way. It is known that most author obtained new stationary solutions in the model, where approximation of strong nonlocality of nonlinear response is utilized to transform nonlinear Schrödinger equation describing real system dynamics into linear evolution equation with parabolic potential, and the underlying equation describing beam dynamics is linear [21–30]. So, all these solutions are not real nonlinear solitons because there is no nonlinearity

∗ Corresponding author. E-mail address: [email protected] (C. Huang). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.08.026

in the system. It is known that an arbitrary initial field can be expressed as a linear superposition of accessible solitons in the Snyder–Mitchell model [21]. Whileas most author consider linear superposition of more Hermite–Gaussian accessible solitons when they are separated with a distance, the existence condition is irrespective of the modes number of Hermite–Gaussian function when considered only the (1+1) dimensional cases. In this paper we would like to superpose such two-dimensional accessible solitons collinearly and to make the initial field be a new spatial soliton also, and we call them “superposition solitons and breathers”, where “breathers” means oscillating states of solitons. The paper is organized as two parts. Firstly, we analyze and obtain the condition to form two-dimensional solitons with complex structure named as “superposition solitons and breathers” in Snyder–Mitchell linear model. Results show that, when their phase difference between the accessible solitons is a constant, such solitons and breathers can form. Secondly, we investigate numerically the propagation of such solitons and breathers in fully nonlinear system with different degree of nonlocality, and find that the “superposition solitons and breathers” is robust in the strongly nonlocal cases. So such solitons can be termed as the real solutions of the fully nonlinear system when degree of nonlocality is decreased enough, and we can search the two-dimensional solitons with complex structure in strongly nonlocal media based on this method.

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lowing form i

1 1 ∂

+ ⊥ − (X 2 + Y 2 ) = 0 2 2 ∂Z

(3)

where  = P0 /PC , PC = 1/k2 w04 , P0 is the input power. It is known that the linear evolution equation with parabolic potential has Hermite–Gaussian (H–G) soliton solutions,like those discussed in Refs. [21,29,36]. Firstly, we will discuss the condition to form superposition solitons. In particular, the composite beam can be written in the following form (X, Y, Z) =

N 

dmn mn (X, Y, Z)

(4)

m,n

where mn (X, Y, Z) is the accessible H–G soliton solution, dmn is a coefficients, which is similar to that discussed in free space [35], but is different from that discussed in incoherent solitons [36]. So the expression for the light intensity in a transverse cross section of an H–G beam has the following form



I(X, Y, Z) =

N   2 2  dmn  Hm

X W (Z)



 Hn2

Y W (Z)



m,n=0

+

N N      dmn dm n  Hm m,n m ,n

 ×

Y W (Z)



 Hn

Y W (Z)

X W (Z)





 Hm

 mn cos m  n

X W (Z)

Hn



1 W (Z)



2

exp −

(X 2 + Y 2 ) W (Z)



2

(5)

where Fig. 1. Dynamic propagation of the normalized intensity profiles of the superposition soliton based on the Snyder–Mitchell linear model (the mode numbers of Hermite-function are (1,4) and (2,3), respectively). (a) Z = 0; (b) Z = 50; (c) Z = 100.

To start, let us assume that the light beam propagates along the z axis and diffracts both in the x and y directions. Propagation of 

a two-dimensional optical beam with varying amplitude ϕ( r , z) is governed by the following nonlinear Schrödinger equation [21], 1 ∂ϕ k + ⊥ ϕ + ın(I)ϕ = 0 n0 2k ∂z

(1) 

where k = n0 k0 , k0 = 2/, r = (x, y), ın(I), is the lightinduced refractive index. In this paper we consider the nonlocal kerr media, so the refractive index ın(I) = n2 











R( r − r )I( r , z)d r ,







dition, −∞ R( r − r )d r = 1. Just as Refs. [21–30] pointed out, the nonlinear Schrödinger equation can be simplified as the Snyder–Mitchell linear model in the strongly nonlocal cases, i

is a constant, independent of the thus the phase difference propagation distance Z, then the solitons or breathers form. Easy to find the superposition solitons form when P0 = PC and W(Z) = 1, and superposition breathers form when P0 = / PC , while W(Z) = / 1. Next let us consider methods for choosing the mode numbers to form a superposition solitons for more than two accessible solitons. If a pair of modes with numbers (m, n), (m , n ) is choosing, thus the condition (m + n − m − n ) = 0 should satisfy, then the additional modes (m , n ) also should satisfy the condition, (m + n − m − n ) = 0, (m + n − m − n) = 0

(8)

The superposition solitons or breathers would form when their mn ) is independent of the propagation distance phase difference ( m  n Z. Eq. (7) and Eq. (8) are the existence condition for a superposition soliton to form. Similar condition like Eq. (8) should satisfy for even more accessible solitons.



where R( r − r ) is the normalized symmetrical real spatial response function of the media, which satisfy the normalized con-

∞

(7) mn

m  n

2.1. The condition to form superposition solitons and breathers – analytical results

I = | ϕ | 2,

(6)

mn is very sensitive to the mode numbers The phase difference m  n of Hermite-function. If we let

(m + n − m − n ) = 0

2. Superposition solitons and breathers in the Snyder–Mitchell linear model

i

mn  

m  n = dmn − dm n + 2(m + n − m − n )(Z)

1 1 ∂ϕ ⊥ ϕ − k P0 r 2 ϕ = 0 + 2 2k ∂z

(2)

where  is a material constant, and is the material param



eter associated with the response function R( r − r ). Through the normalization given by Z = z/kw02 , X = x/w0 , Y = y/w0 , = kw0 1/2 ϕ, we can rewrite the Snyder–Mitchell model in the fol-

2.2. Numerical investigation on the propagation of superposition solitons and breathers In this session we will give some numerical examples to show their propagation behavior and to prove that the composite beam that we construct above is the solution of the Snyder–Mitchell linear model. First we consider a superposition soliton formed with two accessible H–G solitons. In particular, we take the mode numbers be m = 2, n = 3 and m = 1, n = 4, respectively. Where P0 = PC , W(Z) = 1, then the input field can be written in the following form √ 2 (X, Y, Z = 0) = (9) ( 23 (X, Y, Z = 0) + 14 (X, Y, Z = 0)) 2

C. Huang et al. / Optik 122 (2011) 1501–1507

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Fig. 3. Dynamic propagation of normalized intensity profiles of the superposition soliton when formed with three accessible Hermite–Gaussian solitons (the mode numbers of Hermite-function are (1,5), (2,4) and (3,3), respectively). (a) Z = 0; (b) Z = 50; (c) Z = 100.

Fig. 2. Dynamic propagation of the normalized intensity profiles of the superposition breathers when P0 = 0.6PC (a–d) and P0 = 1.6PC (e–h), respectively (the mode numbers of Hermite-function are (1,4) and (2,3), respectively). (a) Z = 0; (b) Z = 2; (c) Z = 4; (d) Z = 6; (e) Z = 0; (f) Z = 1.25; (g) Z = 2.5; (h) Z = 3.75.

The coefficients dmn is equal to each other. Propagation of such composite beam is shown in Fig. 1(a–c). Stable propagation of such solitons is observed for more than 100, which is more than ten diffraction lengths compared to the transverse scale of the composite beam. Moreover we verify that the composite beam can propagate stably for even longer propagation distance as longer as we can. Diffraction lengths of such superposition soliton are related to the transverse scale of the beam and are very sensitive to the maximum mode numbers of the accessible H–G solitons. Stationary propagation of superposition solitons confirm the existence condition (Eq. (7)) is true. Superposition breathers would form when the input power P0 = / PC , and such breathers will undergo linear harmonic oscillations, i.e., the width and the amplitude of such breathers will oscillate periodically. Without loss of generality we assume the input power P0 = 0.6PC and P0 = 1.6PC , respectively. Typical numerical examples are shown in Fig. 2. Fig. 2(a–d) shows the dynamic propagation of superposition breathers when P0 = 0.6PC . The breathers in Fig. 2(b) is larger than that at Z = 0 (Fig. 2(a)), i.e., the breather expands. Then √ after an another propagation distance about ız = T/2 = /2  ≈ 2 the breathers contract, which is presented in Fig. 2(c). Obviously the intensity profile in Fig. 2(d) is almost the same with Fig. 2(b). Fig. 2(e–h) shows that these

breathers vibrate periodically also when P0 = 1.6PC , and pertinent behavior is very similar to that in Fig. 2(a–d), but these breathers contract first(Fig. √ 2(f)) then expand(Fig. 2(g)), and the period is equal to T = /  ≈ 2.5. For even longer propagation distance, these breathers vibrate periodically also. Next we consider a superposition soliton formed with three accessible Hermite–Gaussian solitons. We take m = 1, n = 5, m = 2, n = 4 and m = 3, n = 3, respectively, where P0 = PC , W(Z) = 1, and the input field be the following form √ 1 2 (X, Y, Z = 0) =

15 (X, Y, Z) + 24 (X, Y, Z = 0) 2 2 +

1

33 (X, Y, Z = 0) 2

(10)

The coefficients dmn is not equal to each other. Stable propagation of this composite beam is shown in Fig. 3(a–c). Stable propagation of such beam is observed for more than 100 propagation length, without changing their shape and intensity, thus we can conclude that the superposition soliton is formed. The composite beam can propagate stably for even longer propagation distance as longer as we can. Finally, instable propagation of the composite beam when their modes number is not satisfied the existence conditions is shown in Figs. 4 and 5. Composite beam composed of modes (2,3) and (2,4) is shown in Fig. 4. Although P0 = PC and W(Z) = 1, the superposition solitons can not form. They cannot propagate stably even in the initial propagation distance Z = 5 (Fig. 4(b)) for they have changed their intensity profile. Another case, we consider a composite beam formed with three Hermite–Gaussian solitons. Let m = 1, n = 5, m = 1, n = 4 and m = 3, n = 3, respectively , and the input field

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Fig. 4. Dynamic propagation of normalized intensity profiles of the composite beam when formed with two accessible Hermite–Gaussian solitons (the mode numbers of Hermite-function are (2,3) and (2,4), respectively). (a) Z = 0; (b) Z = 5.

are similar to Eq. (10). Obviously in this case, (m + n − m − n ) = / 0, / 0. From Fig. 5 we see that the superposition (m + n − m − n ) = solitons cannot form either. They change their intensity profile even in the initial propagation distance Z = 5, then we can conclude that the soliton can not form. All the numerical simulations above show that, when the phase difference is independent of the propagation distance Z, such superposition soliton can form. So we can select the mode number of the Hermite-function to control such two-dimensional solitons with complex structure in strongly nonlocal media.

Fig. 6. Dynamic propagation of the normalized intensity profiles of the superposition soliton in nonlocal media with degree of nonlocality ˛ = 0.2 (the mode numbers of Hermite-function are (1,4) and (2,3), respectively). (a) Z = 0; (b) Z = 3; (c) Z = 3.5.

3. Superposition solitons and breathers in nonlocal nonlinear media In this section we prove that the composite beam that we construct above is the solution of the nonlocal nonlinear model (Eq. (1)) when degree of nonlocality is gradually decreased enough, thus we

Fig. 5. Dynamic propagation of normalized intensity profiles of the composite beam when formed with three accessible Hermite–Gaussian solitons (the mode numbers of Hermite-function are (1,4), (1,5) and (3,3), respectively). (a) Z = 0; (b) Z = 5.

Fig. 7. Dynamic propagation of the normalized intensity profiles of the superposition soliton in nonlocal media with degree of nonlocality ˛ = 0.1 (the mode numbers of Hermite-function are (1,4) and (2,3), respectively). (a) Z = 0; (b) Z = 10; (c) Z = 15.

C. Huang et al. / Optik 122 (2011) 1501–1507

Fig. 8. Dynamic propagation of the normalized intensity profiles of the superposition soliton in nonlocal media with degree of nonlocality ˛ = 0.06 (the mode numbers of Hermite-function are (1,4) and (2,3), respectively). (a) Z = 0; (b) Z = 40; (c) Z = 40, where solitons are perturbed by random perturbations at the input.

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Fig. 10. Dynamic propagation of the normalized intensity profiles of the superposition soliton in nonlocal media with degree of nonlocality ˛ = 0.1 (the mode numbers of Hermite-function are (1,5), (2,4) and (3,3), respectively). (a) Z = 0; (b) Z = 5; (c) Z = 10.

can find the two-dimensional solitons with complex structure in strongly nonlocal media based on the method. As comparison, we assume the material response is Gaussian function, i.e., R(X, Y ) =

Fig. 9. Dynamic propagation of the normalized intensity profiles of the superposition soliton in nonlocal media with degree of nonlocality ˛ = 0.2 (the mode numbers of Hermite-function are (1,5), (2,4) and (3,3), respectively). (a) Z = 0; (b) Z = 3; (c) Z = 3.5.

˛2 exp(−0.5˛2 (X 2 + Y 2 )) 2

(11)

where ˛ denotes the degree of the material nonlocality, the less the ˛ is, the stronger the nonlocality is. Firstly, we consider superposition solitons formed with two accessible H–G solitons, where the mode numbers of Hermitefunction are (1,4) and (2,3), respectively, and the input field are similar to Eq. (9). Figs. 6–8 show the propagation of superposition solitons in nonlocal media with different degree of nonlocality. When ˛ = 0.2, instable propagation of such composite beam is presented in Fig. 6(a–c). While when ˛ = 0.1 (Fig. 7(a–c)), the max propagation distance is about 15. Then for even strongly nonlocal case, see ˛ = 0.06 (Fig. 8(a and b)), they can propagate stably for even longer distance larger than 40 (longer than ten diffraction lengths compared to the transverse scale of the beam). These characteristic feature can be understood and explained in the following way. In a nonlocal media, when the degree of nonlocality ˛ varies from 0.2 to 0.06 or more less, it means that nonlocality changes from weak nonlocality to strong nonlocality, and the characteristic response length of nonlocal media is much broader than the width of the optical beam, thus the wider refractive index distribution can support the formation of spatial soliton with complex transverse shapes, so the superposition solitons can form in this strongly nonlocal media. Furthermore, we investigated the stability of superposition solitons when introduced with different random perturbations. Results show that the superposition solitons can propagate stably and robust also (Fig. 8(c)). Through comparison

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random perturbations are introduced to the system at the input, the superposition solitons can propagate stably and robust also (Fig. 11(c)). Obviously, the stronger the nonlocality, the longer distance they can propagate stably. Finally, when superposition solitons formed with more accessible H–G solitons, the propagation behavior is similar to that formed with two or three H–G solitons, so we can get the similar conclusions from our numerical simulations above. 4. Conclusions In conclusion, we have investigated a new kind of twodimensional solitons with complex structure named as “superposition solitons and breathers” in strongly nonlocal media, and obtained the condition to form such solitons and breathers. Results show that such solitons can form as long as the phase difference is independent of the propagation distance Z, and such solitons state can be controlled by the mode numbers of the Hermite-function. Moreover, we investigated the propagation of such solitons in nonlocal nonlinear media with different degree of nonlocality. Numerical simulations show that the less the degree of nonlocality (˛) is, the stronger the nonlocality is, and the longer distance the superposition solitons can propagate stably, and such solitons can be termed as the real solutions of the fully nonlinear system when ˛ is decreased enough. So we can find the two-dimensional solitons with complex structure in strongly nonlocal media based on the method discussed above. 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. The author thanks the useful discussion with professor Q. Guo , W. Hu and D. Deng in South China Normal University. References

Fig. 11. Dynamic propagation of the normalized intensity profiles of the superposition soliton in nonlocal media with degree of nonlocality ˛ = 0.04 (the mode numbers of Hermite-function are (1,5), (2,4) and (3,3), respectively). (a) Z = 0; (b) Z = 40; (c) Z = 40, where solitons are perturbed by random perturbations at the input.

we found that, the less the degree of nonlocality(˛) is, the longer the superposition solitons can propagate stably. So when the characteristic response length of nonlocal media is much broader than the width of the optical beam (i.e., in the strongly nonlocal cases), the superposition solitons can propagate stably and robust, and such solitons can be termed as the real solutions of the fully nonlinear system when ˛ is decreased enough. Secondly, Propagation of superposition solitons formed with mode numbers (1,5), (2,4) and (3,3) with different degree of nonlocality is shown in Figs. 9–11. Where the input field are similar to Eq. (10). The stable propagation length is less than 3 when ˛ = 0.2 (Fig. 9(b)). While when ˛ = 0.1 (Fig. 10(b)),they can propagate for a longer distance for about Z = 5. When ˛ = 0.04 (Fig. 11(b)),a more longer distance for about Z = 40 is observed, even when different

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