Incoherently coupled Hermite–Gaussian breather and soliton pairs in strongly nonlocal nonlinear media

Optics Communications 282 (2009) 4423–4430

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Incoherently coupled Hermite–Gaussian breather and soliton pairs in strongly nonlocal nonlinear media Yajian Zheng, Daquan Lu, Rui Zheng, Wei Hu *, Qi Guo Laboratory of Light Transmission Optics, South China Normal University, Guangzhou 510631, PR China

a r t i c l e

i n f o

Article history: Received 9 March 2009 Received in revised form 23 July 2009 Accepted 28 July 2009

PACS: 42.65.Tg 42.65.Jx

a b s t r a c t We theoretically investigate the propagation of incoherently coupled Hermite–Gaussian breather and soliton pairs in strongly nonlocal nonlinear media. It is found that multipole-mode soliton pairs with arbitrary different orders of Hermite–Gaussian shape can exist when the total power of two beams equals the critical power and the ratio of the beam widths for the Gaussian part is inversely proportional to the square root of the ratio of the wave numbers. When the total power does not equal the critical power, the Hermite–Gaussian breather pair exists and their beam widths evolve analogously. For general cases where the ratio of the beam widths is arbitrary, soliton–breather pairs or breather–breather pairs can be formed and their beam widths evolve synchronously in-phase or out-of-phase. Numerical simulations directly based on the nonlocal nonlinear Schrödinger equation are conducted for comparison with our theoretical predictions. The numerical stability analysis shows the higher-order Hermite–Gaussian solitons can not be stable for small nonlocality or for some media like liquid crystals. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction The strongly nonlocal optical spatial soliton has significant properties and has been of increasing interest in the last decade [1–5]. A linear model for the strongly nonlocal soliton was developed by Snyder and Mitchell [1], and an accurate Gaussian shaped solution called an accessible soliton was found. Subsequently, analytical anticipation [2] and experimental observation [3] of the accessible soliton was turned into reality in nematic liquid crystal (NLC), which was found to be the first strongly nonlocal medium [2,3,6,7]. Some other strongly nonlocal media have been found experimentally, such as lead glass [4,5], thermal nonlinear liquids [8], and nonlinear ion gases [9]. Nonlocality was also found in photo-refractive crystals [10,11], dipolar Bose–Einstein condensates [12,13], and quadratic nonlinear media [14]. Beside the fundamental order soliton, higher-order multipole-mode solitons can be sustained in NLC, which was predicted by Mclaughlin et al. [15], and then reported experimentally by Hutsebaut et al. [16]. In recent years, higher-order multipole-mode solitons were studied further [17–19]. An accurate analytical solution withthe Hermite–Gaussian in strongly nonlocal nonlinear media was obtained by Deng et al. [20]. A vector soliton occurs when more than one field constitutes the waveguide mode. It was first suggested by Manakov in Kerr

* Corresponding author. Tel.: +86 2039310417; fax: +86 2039310083. E-mail address: [email protected] (W. Hu). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.07.065

media [21] and observed by Chen et al. experimentally [22]. An analysis by Christodoulides et al. showed that incoherently coupled soliton pairs are possible in biased photo-refractive crystals, under steady-state conditions [23]. Recently, the two-color vector soliton pairs was observed and studied numerically and analytically in NLC [24–26]. The incoherently coupled two-color Manakov vector solitons in nonlocal media were studied theoretically by Shen et al. They found the fundamental Gaussian-shaped breather and soliton pairs [27]. The multipole-mode vector solitons were also studied numerically by Torner’s group [28,29]. In this paper, we investigate the propagation of incoherently coupled higher-order multipole-mode Hermite–Gaussian breather and soliton pairs in strongly nonlocal nonlinear media. Based on the Snyder and Mitchell model (S–M model), the accurate analytical solutions for incoherently coupled higher-order multipolemode Hermite–Gaussian breather and soliton pairs are obtained. The soliton–soliton pair, soliton–breather pair, and breather– breather pair are found to exist depending on their total incident powers being equal or unequal to the critical power. The beam widths evolve analogously or synchronously in-phase or out-ofphase, depending on their initial beam width ratio. Numerical simulations and stability analysis based on the nonlocal nonlinear Schrödinger equation are made for comparison. The structure of the paper develops as follows. The evolution equations for the Gaussian beams in nonlocal media and their accurate analytical solution are reviewed in Section 2. We then present analytical and numerical discussions in Section 3. Section 4 contains the conclusion.

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2. Mode and solutions We consider two mutually incoherent optical fields with different wave numbers (k1, k2) and frequencies (x1 , x2 ), which are expressed as usual in terms of slowly varying envelopes (a1, a2), i.e. E1 ¼ a1 ðX; Y; ZÞ expðik1 Z  ix1 tÞ þ c:c: and E2 ¼ a2 ðX; Y; ZÞ expðik2 Z  ix2 tÞ þ c:c:, propagating along the Z axis within a strongly nonlocal Kerr-type nonlinear medium. In that case, it can be readily shown that the (a1, a2) envelopes obey the following evolution equations [30,31]:

oa1 1 Dn D? a1 þ k1 a1 ¼ 0; þ n01 oZ 2k1 oa2 1 Dn D? a2 þ k2 a2 ¼ 0; þ i n02 oZ 2k2 Z 1    0 0 Dn ¼ n2 R X  X 0 ; Y  Y 0 I X 0 ; Y 0 ; Z dX dY ; i

ð1aÞ ð1bÞ ð1cÞ

1

I ¼ ja1 ðX; Y; Z Þj2 þ ja2 ðX; Y; Z Þj2 ;

ð1dÞ

where D? is the transverse Laplacian operator, n01 and n02 are the linear refraction indexes, Dn is the nonlinear perturbation of the refraction index, n2 is the nonlinear index coefficient, R is the normalized symmetrical real spatial nonlocal response function of the medium, and I is the total intensity of the two beams. The norR1 malized condition, 1 Rðx; yÞdxdy ¼ 1, is chosen physically to make n2 have the same dimensions as that of the local media. For the strongly nonlocal case, and in the case of circular symmetry Rðx; yÞ, Eqs. (1a) and (1b) can be deduced as the S–M model, by expending the perturbation Dn to second order about the transverse coordinates [1,32,27]. Introducing the normalization

where x ¼ X=w10 , y ¼ Y=w10 , z ¼ Z=2k1 w210 , l ¼ k1 =k2 , D ¼ 2 k1 n2 cw410 P 0 =n0 , and assuming n0 ¼ n01  n02 , then we have the dimensionless system:

   oa1  2 þ ox þ o2y a1  D x2 þ y2 a1 ¼ 0; oz    oa2 1  2 ox þ o2y a2  lD x2 þ y2 a2 ¼ 0; i þ oz l

i

    C m1 n1 x y exp ihm1 n1 Hm1 H n1 w1 w1 w1

2    x þ y2 þ ic1 x2 þ y2 ;  exp  2 2w1     C m2 n2 x y exp ihm2 n2 Hm2 a2 ¼ H n2 w2 w2 w2

2    x þ y2 þ ic2 x2 þ y2 ;  exp  2 2w2

a1 ¼

w1, w2

w1, w2

1.2 1 0.8

0.8

0.6 0.6

0

2

4

6

8

w1, w2

1.2

w1, w2

w1 w2

1 0.8

4

6

0

2

4

6

8

10

w1 w2

1.2

10

0.8

0

2

4

6

2

1 0.8

8

10

w1(λ=0.8) w2(Λ=0.8) w2(Λ=1.4)

1.5

w1, w2

1.4

8

w1(λ=1.2) w2(Λ=0.8) w2(Λ=1.2)

1.2

0.4

1.6

w1, w2

2

1.6

1.4

0.6

0

10

1.6

0.6

ð3bÞ

w1(λ=1) w2(Λ=0.8) w2(Λ=1.2)

1.4

1

ð3aÞ

where hm1 n1 and hm2 n2 are the phases of the complex amplitude of the solution, w1 ðzÞ and w2 ðzÞ are the beam widths, and c1 ðzÞ and c2 ðzÞ represent the phase-front curvatures of the beams, and they are all allowed to vary with the propagation distance z. By inserting the solutions above into Eq. (2), we obtain:

1.6

w1 w2

ð2bÞ

where w10 is the incident beam width of a1, c ¼ Rð0Þ00 is the second-order differential at the origin point, and R P0 ¼ ðja1 ðz ¼ 0Þj2 þ ja2 ðz ¼ 0Þj2 Þdxdy ¼ P1 þ P2 is the total power of the two beams. Eq. (2) are our model for the solutions. We look for solutions to Eq. (2) in the Hermite–Gauss form [20]:

1.4 1.2

ð2aÞ

1 0.5 0

0

2

4

z

6

8

10

Fig. 1. Evolution of beam width of the Hermite–Gaussian breather and soliton pairs for analogous cases: (a) k ¼ 1; K ¼ 1; (b) k ¼ 1:2, K ¼ 1; (c) k ¼ 0:8, and K ¼ 1.

2

4

z

6

8

10

Fig. 2. Evolution of beam width of the Hermite–Gaussian breather and soliton pairs for synchronous cases: (a) k ¼ 1, K ¼ 0:8; 1:2; (b) k ¼ 1:2, K ¼ 0:8; 1:2; and (c) k ¼ 0:8, K ¼ 0:8; 1:4.

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Y. Zheng et al. / Optics Communications 282 (2009) 4423–4430

Synchronous solutions

Analogous solutions 1.8

α=3 α=10 α=30

1.6

1

1.4

w

1.2

α=3 α=10 α=30

1.1

1.2 1

1

0.8 0.6

0

2

4

6

8

1.6

10

2

4

6

8

10

α =3 α =10 α =30

1.4 1.2

w

2

1.2

0

1.6

α =3 α =10 α =30

1.4

0.9

1

1

0.8

0.8

0.6

0

2

4

z

6

8

10

0.6

0

2

4

6

8

10

z

Fig. 3. Comparison of numerical simulation results between different degrees of locality with a Gaussian form response when P 1 =P 2 ¼ 1. The left column corresponds to the analogous solutions (k ¼ 0:8, K ¼ 1) and the right column corresponds to the synchronous solutions (k ¼ 1, K ¼ 0:8).

Γ1/Γ2

a

w1c ¼

1

w2c

0.5

Γ1/Γ2

b

1 0.5

10

wm

20

30

Fig. 4. Comparison of analytical solution with numerical simulation for the ratio of oscillation period with a Gaussian form response when P 1 =P 2 ¼ 1. (a) The analogous solutions (k ¼ 0:8, K ¼ 1). (b) The synchronous solutions (k ¼ 1, K ¼ 0:8).

dwi 4ci wi ¼ 0;  dz li 

4c2 dci 1  i  li D ¼ 0; þ 4 dz li wi li

dhmi ni 2ðmi þ ni þ 1Þ ¼ 0; þ dz li w2i   i ¼ 1; 2; l1 ¼ 1; l2 ¼ l :

ð4aÞ ð4bÞ ð4cÞ

Combining Eqs. (4a) and (4b) gives: 2

d w1 2

dz

¼

2

d w2 2

dz

¼

4  4w1 D; w31 4

l2 w32

 4w2 D:

ð5aÞ ð5bÞ

Eq. (5) describe the evolution of the beam width for each beam. It is obvious that when w1 ðzÞ and w2 ðzÞ remain unchanged, we have:

 14 1 ¼ D 

1 ¼ l2 D

!14

n0 2

k1 n2 cw410 P 0

14 ¼

ð6aÞ

;

n0

l2 k21 n2 cw410 P0

!14 :

ð6bÞ

For a certain total incident power, there is a corresponding critical beam width for each beam. Note that the nonlinearity in Eqs. (1) and (2) is determined by the total power, but not by the power of each beam. On the other hand, the diffraction is determined by the width and wavelength of each beam. Therefore, the definitions of the critical power for each beam are useless for our problem. It is convenient for us to define the critical beam width for each beam, as Eq. (6), which depends only on the total incident power. For each beam, when the incident beam width equals the critical beam width, the beam will be a soliton and its width remains unchanged, and is unaffected by the width of other beams. When the incident beam width is not equal to the critical beam width, then the beam will undergo periodic oscillations, forming a so-called breather. When the incident beam width is less than the critical beam width, the beam width will first increase and then decrease, or contrarily, when the incident beam width is greater than the critical beam width, then the beam width will first decrease and then increase. When both incident beam widths are equal to their critical widths, we get the soliton–soliton pairs, and the total incident power in this case is defined as the critical power for beam pairs, which is dependent on both incident beam widths. The ratio of the critical beam widths is inversely proportional to the ratio of the wave numbers for two beams:

w1c pffiffiffiffi ¼ l; w2c which was obtained in Ref. [27].

ð7Þ

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Y. Zheng et al. / Optics Communications 282 (2009) 4423–4430

Analogous solutions

2

α =3,P1/P2 =1

1.8

w1

Synchronous solutions 1.8

α=3,P1/P2=10

1.6

α=10,P1/P2=1

1.4

α=10,P1/P2=10

1

0.8 0

2

4

1.8

w2

α=10,P1/P2=10

1.2

1

6

8

10

α=3,P1/P2=10

1.4

α=10,P1/P2=1

0.8

0.8 4

z

6

4

8

10

0.6

6

8

10

α =3,P1/P2 =1 α=3,P1/P2=10 α=10,P1/P2=1 α=10,P1/P2=10

1.2 1

2

2

1.4

1

0

0

1.6

α=10,P1/P2=10

1.2

0.8 1.8

α =3,P1/P2 =1

1.6

0.6

α=3,P1/P2=10 α=10,P1/P2=1

1.4

1.2

0.6

α =3,P1/P2 =1

1.6

0

2

4

z

6

8

10

Fig. 5. Comparison of numerical simulation results between different power ratios with a Gaussian form response. The left column corresponds to the analogous solutions (k ¼ 0:8, K ¼ 1) and the right column corresponds to the synchronous solutions (k ¼ 1, K ¼ 0:8).

It is also worth noting that the conclusions we have obtained in the foregoing are not dependent on the mi and ni values. In other words, arbitrary combination of the Hermite–Gaussian modes can form incoherently coupled Hermite–Gaussian breather and soliton pairs, in which the numbers of peaks can be different in each component [29]. Solving the Eq. (5), we have [32]:

h

2

i

w2i ¼ w2i0 cos2 ðbzÞ þ K2i sin ðbzÞ ;

ð8Þ

pffiffiffiffiffiffiffi where b ¼ 2D, Ki ¼ w2ic =w2i0 , and wi0 is incident beam width. Two conclusions can be drawn from the Eq. (8). The first is that two beam breathers have an equal period, no matter what the wavelengths and beam widths are, given by

sffiffiffiffiffiffiffiffiffiffiffiffiffi p 2p2 n0 : C¼ ¼ b n2 cP0

ð9Þ

pffiffiffiffi The other is that when w10 =w20 ¼ l, we obtain K1 ¼ K2 , and pffiffiffiffi w1 ðzÞ=w2 ðzÞ ¼ l. This means that the two beams propagate with analogous evolutions of their beam widths, which we call analogous breather pairs. It is noted that the Rayleigh lengths, (also the diffraction length) of each beam, i.e. zR ¼ ki w2i0 =2 are equal. pffiffiffiffi When w10 =w20 – l and K1 –K2 , the beam width wi evolves independently, but with equal periods, which we call synchronous breather pairs. The substitution of Eq. (8) into Eq. (4a) and Eq. (4c) yields,

  bli K2i  1 sin ð2bzÞ i; ci ¼ h 2 8 cos2 ðbzÞ þ K2i sin ðbzÞ hi ¼ 

2ðmi þ ni þ 1Þ arctan ½Ki tan ðbzÞ: li w2i0 bKi

The normalized coefficient jai j2 dxdy ¼ P i , given by 1

R1

ð10Þ

C mn

ð11Þ can

be

obtain

from

C mi ni ¼

Pi : 2mi þni mi !ni !p

ð12Þ

By substitution of Eqs. (8), (10), (11), and (12) into Eq. (3), the accurate solutions for Eq. (2) can be obtained. 3. Discussion of solutions The propagation of incoherently coupled Hermite–Gaussian breather and soliton pairs has a relationship with the ratio of incident beam width wi0 and the critical beam width wic . We assume pffiffiffiffi that w10 ¼ kw1c , w20 ¼ Kw10 = l, and Eq. (8) can be rewritten as

 1 2 w21 ¼ k2 w21c cos2 ðbzÞ þ 4 sin ðbzÞ ; k  2 2

k w 1 2 2 1c w22 ¼ K cos2 ðbzÞ þ 4 4 sin ðbzÞ : l k K

ð13aÞ ð13bÞ

The propagation of such breather and soliton pairs depends not only on the k parameters, but also on the K parameters. Note that in the following detailed discussion, we maintain a value of l ¼ 1:2. 3.1. Analogous pair solutions and synchronous pair solutions When k ¼ K ¼ 1, the beams preserve their widths, as they travel in the straight path along the z axis, forming an analogous Hermite– Gaussian soliton pair (see Fig. 1a). When K ¼ 1, but k – 1, then the two beams will undergo periodic oscillations, analogously, forming an analogous Hermite–Gaussian breather pair as shown in Fig. 1b and c. When k > 1, the beam widths may first increase and then decrease. Conversely, widths may first decrease and then increase. These cases are same as for the results in Ref. [27], except that the arbitrary combination of the Hermite–Gaussian, mi and ni , are arbitrary here.

Y. Zheng et al. / Optics Communications 282 (2009) 4423–4430

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Fig. 6. Comparison of analytical solution with numerical simulation results for the propagation of higher-order Hermite–Gaussian beams a1 and a2 with a Gaussian form response up to a distance of two periods. All of the numerical simulation results are under conditions of P 1 ¼ P 2 , k ¼ 0:8, and K ¼ 1.

When K–1, then the two beams will undergo periodic oscillations with the same period. When k ¼ 1 and K–1, a1 preserves its width and a2 undergoes periodic oscillations, forming a synchronous Hermite–Gaussian soliton–breather pair (see Fig. 2a). A more general situation is where k – 1, and K – 1, and the two beams will undergo periodic oscillations simultaneously, forming a synchronous Hermite–Gaussian breather–breather pair(see Fig. 2b and c). It is interesting to note that when k > 1 and kK > 1 or k < 1 and kK < 1, the beams will oscillate in-phase, but when k > 1 and kK < 1 or k < 1 and kK > 1, the beams will oscillate out-of-phase. Note that the real beam size is not equal to the value of wi for a higher-order Hermite–Gaussian beam. Generally speaking, the secR1 ond-order moment of the intensity, i.e. hw2ix i ¼ 1 x2 jai j2 dxdy=Pi , gives good information on the beam width. For the Hermite–Gaussian breather or soliton pairs obtained above, the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi beam sizes are gived by hwix i ¼ 2mi þ 1wi , hwiy i ¼ 2ni þ 1wi [20]. Therefore, for the analogous Hermite–Gaussian breather and soliton pairs to exist, the real beam size of the beams must satisfy:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hw1x i 2m1 þ 1 l; ¼ 2m2 þ 1 hw2x i

 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w1y 2n1 þ 1

¼ l: 2n2 þ 1 w2y

Nevertheless, despite the change in size, the trend of evolution has not changed. 3.2. Numerical simulation and stability analysis The solutions we obtained above are the accurate solutions to Eq. (2), but are approximations to Eq. (1) for the strongly nonlocal case. In this subsection, we use a numerical simulation method to investigate the propagation of incoherently coupled Hermite– Gaussian breather and soliton pairs for Eq. (1). Suppose that the response function has a Gaussian form [2,33]

Rðx; yÞ ¼

 2 1 x þ y2 ; exp  2pw2m 2w2m

ð14Þ

where wm is the characteristic length of the nonlocal nonlinear response of the medium. We define the degree of nonlocality as a ¼ wm =w10 , and thus larger the value of a, the stronger the nonlocality will become. To simulate the propagation, we first used an input Gaussian beam:

ai ¼

 2 Pi x þ y2 exp  : pw2i0 2w2i0

ð15Þ

Let us first consider the influence of the degree of nonlocality on the evolution of the beam widths w1 ðzÞ and w2 ðzÞ. As an example, we choose the cases of k ¼ 0:8, K ¼ 1 and k ¼ 1, K ¼ 0:8, which

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Y. Zheng et al. / Optics Communications 282 (2009) 4423–4430

Fig. 7. Hermite–Gaussian soliton pairs propagating in the medium with Gaussian form response over long distances (z = 500) with the difference values of nonlocal degrees a. The top and mid rows are for case 1, i.e. (m1 ¼ 0; n1 ¼ 1) and (m2 ¼ 1; n2 ¼ 1); the bottom row is for case 2, i.e. (m1 ¼ 0; n1 ¼ 1) and (m2 ¼ 2; n2 ¼ 2), for which profiles of a1 are as same as that in case 1.

correspond to the cases of analogous pair solutions, and synchronous pair solutions, respectively. Fig. 3 shows the evolution of beam width for different values of a. It is obvious that, due to the weak nonlocality, the periods of beam width oscillation tend to expand for both analogous solutions and synchronous solutions. The smaller the a value is, then the stronger the trend will be. It is also obvious that under the strong nonlocality conditions the beam widths oscillate harmonically in the form of a cosine function, but do not oscillate harmonically for a small value of a. We want to illustrate that for the larger a ¼ 30, the numerical solutions are so good agreement with our analytical solutions. We do not show the analytical results in figures because they can not be distinguished. In Fig. 4, the ratio of the breather period for each beam is shown as function of the degree of nonlocality. When a > 4, the breather periods are almost equal to 1 and the evolution of the breather pairs remains synchronous. For a < 4, the period C1 is bigger than C2 . For the examples selected, the incident beam width w10 is larger than w20 , and the degree of nonlocality for a1 is smaller than that for a2. This will result in a stronger expansion of the oscillation period for a1 than for a2, i.e. C1 > C2 . It is evident from Fig. 4 that numerical simulations are in agreement with our analytical expectations. In the numerical simulation above, we maintain the value of the power ratio parameter P 1 =P 2 ¼ 1. Hence it is necessary to analyze the influence of different power ratios on the propagation of the beams. Fig. 5 shows the numerical results for different values of P1 =P2 . It is shown that the influence of the power ratio can be neglected. We know from the previous theoretical analysis that the real beam size of the beams is decided by the mode of the Hermite– Gaussian beam. The higher the order of the beam, then the weaker the nonlocality will be [20]. In Fig. 6, we show a breather–soliton pair for higher-order Hermite–Gaussian modes, in which m1 ¼ 0, n1 ¼ 1 for the soliton beam a1, and m2 ¼ 2, n2 ¼ 2 for the breather beam a2. The analytical results and numerical results for different degrees of nonlocality (a ¼ 20 and a ¼ 3) are shown for compari-

son. Our analytical results are in good agreement with the numerical results for the strongly nonlocal case (a ¼ 20). For a smaller degree of nonlocality such as a ¼ 3, obvious distortion different from the analytical prediction can be seen in the last two rows in Fig. 6. Now we perform the stability analysis by simulating the propagation of the coupled solitons with perturbations. The perturbed solitons was given as A  ½1 þ rrðx; yÞ, where A is the amplitude of these Hermite–Gaussian solitons, rðx; yÞ is the random function whose range is changes from 0.5 to 0.5, and r is the perturbation parameter chosen as r ¼ 0:01. Also starting from the Eq. (1), we have simulated the propagation of the beams for different values of parameters a = 10, 20, 30, 50, respectively. we have chosen two cases of beams compounding, the first case is (m1 ¼ 0; n1 ¼ 1) and (m2 ¼ 1; n2 ¼ 1), and another is (m1 ¼ 0; n1 ¼ 1) and (m2 ¼ 2; n2 ¼ 2). Fig. 7 presents the comparisons of the output intensity distribution when z ¼ 500. It is easy to find that, solitons can propagating stability for each values of parameters a in the case 1. In the case 2, the higher-order component a2 (m2 ¼ 2; n2 ¼ 2) can only propagating stability for considerable strong nonlocal degree (a ¼ 50), and it will encountered bizarre transformation when a ¼ 10; 20. a ¼ 30 looks like an threshold for the stability. It is noted that a1 components for case 2 are stable during propagation. More simulations show higher-order coupled soliton pairs can be stable if we choose the value of nonlocal degree big enough. In above, the Gaussian form response function, i.e. Eq. (14), has been used. In the nematic liquid crystals, the higher-order solitons could not be stable even for every large a [17,28]. The nonlinear response function can be described by

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rðx; yÞ ¼ K 0 ð x2 þ y2 =wm Þ=2pw2m ;

ð16Þ

where K0 is the zero order bessel function. The simulations are carried out for a ¼ 30 and r ¼ 0:1. We chosen the values of parameters and (m2 ¼ 1; n2 ¼ 1), for two cases, (m1 ¼ 0; n1 ¼ 1) and (m1 ¼ 0; n1 ¼ 1) and (m2 ¼ 1; n2 ¼ 0). For the first case, i.e.

Y. Zheng et al. / Optics Communications 282 (2009) 4423–4430

4429

Fig. 8. Hermite–Gaussian solitons propagating in the nematic liquid crystals for (a)–(f) the case 1 (m1 ¼ 0; n1 ¼ 1) and (m2 ¼ 1; n2 ¼ 1); and (g)–(j) the case 2 (m1 ¼ 0; n1 ¼ 1) and (m2 ¼ 1; n2 ¼ 0).

(m1 ¼ 0; n1 ¼ 1) and (m2 ¼ 1; n2 ¼ 1), as shown in Fig. 8a–f, Hermite– Gaussian patterns remain when z < 100, and then distort after z > 100. It is noted that similar solitons pairs in Ref. [28] are stable during propagation in liquid crystals, but here for our case it is not stable. The difference is that the initial profile of our beams are in Hermite–Gaussian form, which is not exact for the solitons in liquid crystals. For the second case, i.e. (m1 ¼ 0; n1 ¼ 1) and (m2 ¼ 1; n2 ¼ 0), the lower-order Hermite–Gaussian pairs are stable for a large distance(z ¼ 500), as shown in Fig. 8g–j. 4. Conclusion In conclusion, the propagation of incoherently coupled higherorders multpole-mode Hermite–Gaussian breather and soliton pairs in strongly nonlocal nonlinear media has bean studied theoretically. The propagation of breather and soliton pairs depends on the incident beam widths and their ratio. The soliton–soliton pair, soliton–breather pair, and breather–breather pair are found to exist depending on their total incident powers being equal or unequal to the critical power. The beam widths evolve analogously or synchronously in-phase or out-of-phase, depending on their initial beam width ratio. Numerical simulations for strongly nonlocal media with Gaussian response function are conducted for comparison. For strongly nonlocal media while the Snyder–Mitchell’s modal is valid, arbitrary orders combination of the multipole-mode Hermite–Gaussian soliton pairs can exist. However, a threshold number for the orders of Hermite–Gaussian modes exist in the real nonlocal medium [17,28,29], such as nematic liquid crystal and lead glass.

Acknowledgements This research was supported by the National Natural Science Foundation of China (Grants No. 10804033), the Program for Innovative Research Team of Higher Education in Guangdong (Grant No. 06CXTD005), and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 200805740002). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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