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Solitons in liquid crystals with competing nonlinearities
Optics Communications 450 (2019) 78–86
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Solitons in liquid crystals with competing nonlinearities Shaozhi Pu ∗, Ming Chen, Yingjia Li, Liuyang Zhang Department of Optical Information Science and Technology, Harbin University of Science and Technology, Harbin 150080, China
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Keywords: Thermal nonlinearity Reorientational nonlinearity Nonlocality Optical solitons
ABSTRACT We employ a model which is proposed by P. S. Jung et al. [see P. S. Jung et al. Opt. Express 25 (2017) 23893– 23897] to analytically demonstrate properties of individual bright solitons in liquid crystals with competing nonlinearities. These results are confirmed by the numerical simulations. In addition, we numerically find that the competition between thermal and reorientational nonlinearities can lead to the formation of bright or dark-like solitons whose formation depends on the intensity profiles of the input beams. We show that the competing nonlinearities drastically affect the interaction of out-of-phase bright solitons, such as the change of attraction of out-of-phase bright solitons into repulsion force at certain initial separations of the input solitons.
1. Introduction Nonlocal nonlinearity is one of the most fascinating phenomena appearing in nonlinear systems. As is well known, in the nonlocal media, the refractive index changes in a particular location depend on the light intensity in a certain neighborhood of this location. In the past decade, attentions have been focused on the nonlocal solitons, because they have unique properties compared with local solitons [1–4]. In particular, the nonlocality may arrest the collapse of multidimensional soliton and affect the interaction of solitons [5–11]. Nematic liquid crystals have been an ideal material to investigate nonlocal spatial solitons because of their unique merits, such as large electro-optical tunability and nonlocal nonlinearity which can be adjusted by low-frequency electric field and light beams since the excellent work by G. Assanto [12,13]. So far, most of the investigations of nematicons have been focused on the nematic liquid crystals which consist of elongated rod-like molecules and belong to positive uniaxial materials. It is found that an extraordinarily polarized optical beam in this material will result in molecular orientational effects and thermal effects [14]. Usually, the former (orientational effects) leads to the self-focusing effect and the latter (orientational effects) yields to the self-defocusing effect for the extraordinary wave. So far, many research works have shown that the formation of the nematicon is due to the reorientational nonlinearity [12,13,15–17]. Recently, the thermal solitons in nematic liquid crystal are observed in the experiments by M. Warenghem et al. [14]. In recent years, nonlocal solitons in media with competing nonlinearities have been well investigated [18–24]. It is found that the competition between two types of nonlocal nonlinearities may lead to the formation of many complex soliton structures which are unstable in nonlocal media with only one type of nonlocal nonlinearity, such as ∗
solitons of even and odd parities [25], gap solitons [26], and higher order vortex solitons [27]. Especially, Du et al. have numerically found that there is the repulsive force between two in-phase bright solitons due to the competing nonlocal nonlinearities [28]. Thereafter, Esbensen et al. have reached similar conclusions and a more detailed analysis is provided by a variational approach [29]. Recently, this phenomenon has been confirmed experimentally by Cyprych et al. in dye-doped nematic liquid crystals which possess thermal and reorientational nonlinearities [30]. Very recently, P. S. Jung et al. analytically investigated the solitons in nematic liquid crystals involving thermal effects and molecular orientation effects [31]. In this case, they propose a simple analytical model to investigate the solitons in nonlocal medium with competing nonlinearities, and they also find the formation of supermode spatial solitons based on this model [32]. So far, the research works on spatial solitons in media with competing nonlinearities are mainly focused on two types of nonlinearities which are independent with each other. However, in reality, these two effects are closely related. In this case, the light propagation in media with competing nonlinearities obeys the model proposed by P. S. Jung et al. [32]. So far, to our knowledge, a detailed investigation on solitons and their interactions by this model is still needed. In this work, we employ the model which is developed by P. S. Jung et al. to analytically and numerically investigate bright solitons in media with competing nonlinearities. The bright soliton solutions and their characteristics are obtained by a variational approach. In addition, for the first time to our knowledge, dark-like soliton is found numerically by this model. Moreover, we study the effect of competing nonlocal nonlinearity on the interaction of out-of-phase bright solitons.
Corresponding author. E-mail address: [email protected] (S. Pu).
https://doi.org/10.1016/j.optcom.2019.05.065 Received 16 February 2019; Received in revised form 25 May 2019; Accepted 29 May 2019 Available online 31 May 2019 0030-4018/© 2019 Published by Elsevier B.V.
S. Pu, M. Chen, Y. Li et al.
Optics Communications 450 (2019) 78–86
2. Model and analytical solutions We consider the light propagation in media with competing nonlinearities, described by the following nonlocal nonlinear Schrödinger equation: 𝑖
𝜕𝜓 1 𝜕2 𝜓 + 𝛥𝑛𝜓 = 0, + 𝜕𝑧 2 𝜕𝑥2
(1)
where x and z stand for the transverse and longitudinal coordinates normalized to diffraction length and beam width, respectively, 𝜓 is the electric field amplitude, and 𝛥n is the refractive index change of the medium induced by the incident beam. Here, the nonlinear change in index can be written as 𝛥n=𝛥𝑛2 𝛥𝑛1 in nematic liquid crystals, where 𝛥𝑛2 and 𝛥𝑛1 represent the opt-thermal nonlinearity and reorientational nonlinearity, respectively. As we know, the reorientational nonlinearity is closely related to the optical anisotropy 𝛥𝜀 = 𝑛2e –𝑛2o . Here, 𝑛e and 𝑛o are ordinary and extraordinary refractive indices respectively, both of which are functions of temperature. In this case, it is natural to introduce a figure of nonlinearities 𝛥𝑛2 and 𝛥𝑛1 as [ ] +∞ 𝛥𝑛2 = 1 − 𝛾 𝑅(𝑥 − 𝜉) |𝜓(𝜉, 𝑧)|2 𝑑𝜉 , ∫−∞ (2) +∞ 𝛥𝑛1 = 𝑅(𝑥 − 𝜂) |𝜓(𝜂, 𝑧)|2 𝑑𝜂, ∫−∞
Fig. 1. The soliton power versus the propagation constant. Solid line is the solution by the variational analysis, dash line is the numerical solutions of Eq. (1). The medium parameters are 𝜎 = 5 and 𝛾 = 0.1.
solution of the bright solitons can be obtained by numerical simulations with the initial input form 𝜓=w(x)exp(ibz), where w and b represent the field amplitude and the propagation constant, respectively. To compare the theoretical analysis with the numerical simulations, the is chosen in this part. In addition, the beam propagation constant 𝑏 = 𝑑𝜃 𝑑𝑧 ∞ power 𝑃 = ∫−∞ |𝜓|2 𝑑𝑥 is used in the numerical simulations. In Fig. 1, we illustrate the relation between soliton power and the propagation constant at 𝜎 = 5 and 𝛾 = 0.1. It is shown that the power increases with an increment of the propagation constant, and this tendency is in good agreement with the numerical simulations based on Eq. (1). The monotonic increase of this function indicates that the solitons are stable. √ It can be seen from Eq. (8) that the wave energy 𝑃0 = 𝜋𝐴2 𝑊 is a conserved quantity. Next, one can obtain from Eqs. (7) and (9) the following equation
where 𝛾 represents the strength of the thermal nonlinearity, √ and we call it the opt-thermal nonlinearity coefficient, where 𝑅 = ( 𝜋𝜎)−1 exp (−𝑥2 ∕𝜎 2 ) is the nonlocal response function in one transverse dimension. Here, 𝜎 denotes the nonlocality of the reorientational and thermal nonlinearity. Eq. (1) can be considered as the Euler–Lagrange variational equation corresponding to the Lagrangian density, ∞ |𝜓|2 𝑖 ∗ 1 2 (𝜓 𝜓𝑧 − 𝜓𝜓𝑧∗ ) − ||𝜓𝑥 || + 𝑅(𝑥 − 𝜉) |𝜓(𝜉)|2 𝑑𝜉 ∫ 2 2 2 −∞ [ ] , ∞ ∞ |𝜓|2 −𝛾 𝑅(𝑥 − 𝜉)|𝜓(𝜉)|2 𝑑𝜉 𝑅(𝑥 − 𝜂)|𝜓(𝜂)|2 𝑑𝜂 ∫−∞ 2 ∫−∞
=
(3)
where the asterisk denotes the complex conjugate. To find the soliton solutions of Eq. (1), we use the following Gaussian ansatz: 𝑥2 ], (4) 2𝑊 2 where A is the amplitude of the beam, 𝜃 and 𝑐 are the phase and phasefront curvature of the complex amplitude of 𝜓, respectively, W is the width. Substituting trial function (4) into the system Lagrangian 𝐿 = +∞ ∫−∞ 𝑑𝑥 and integrating over x, we obtain the averaged Lagrangian, √ √ √ √ 𝑑𝜃 𝜋 𝑑𝑐 𝐴2 𝜋 𝐿 = −𝐴2 𝑊 𝜋 − 𝐴2 𝑊 3 − 𝐴2 𝑐 2 𝑊 3 𝜋 − 𝑑𝑧 2 𝑑𝑧 4𝑊 √ 4 2 √ (5) 𝜋𝐴 𝑊 𝛾 𝜋𝐴6 𝑊 3 + √ − √ . 2 𝜎 2 + 2𝑊 2 2 4𝜎 2 𝑊 2 + 3𝑊 4 + 𝜎 4 𝜓(𝑥, 𝑧) = 𝐴(𝑧) exp[𝑖𝜃(𝑧) + 𝑖𝑐(𝑧)𝑥2 −
2𝛾𝑃02 𝑊 (2𝜎 2 + 3𝑊 2 ) 2𝑃0 𝑊 𝑑2𝑊 1 = −√ + , 2 3 3∕2 2 2 𝑑𝑧 𝑊 𝜋(4𝜎 2 𝑊 2 + 3𝑊 4 + 𝜎 4 )3∕2 𝜋(2𝑊 + 𝜎 )
where 𝑃0 is the input power. The Eq. (10) is equivalent to Newton’s second law in classical mechanics for the motion of a one-dimensional particle with the equivalent mass 1 acted by the equivalent force F, i.e. 𝐹 =
(6)
𝛾𝐴4 𝑊 2 (2𝜎 2 + 3𝑊 2 ) 𝑑𝑐 1 𝐴2 𝑊 = −2𝑐 2 + − + , 4 3∕2 2 2 𝑑𝑧 2𝑊 (2𝑊 + 𝜎 ) (4𝜎 2 𝑊 2 + 3𝑊 4 + 𝜎 4 )3∕2
(7)
𝑑 (𝐴2 𝑊 ) = 0, 𝑑𝑧
(8)
2𝛾𝑃02 𝑊 (2𝜎 2 + 3𝑊 2 ) 2𝑃0 𝑊 1 −√ + . 3 𝑊 𝜋(2𝑊 2 + 𝜎 2 )3∕2 𝜋(4𝜎 2 𝑊 2 + 3𝑊 4 + 𝜎 4 )3∕2
(11)
It can be seen that F is a conservative force, the critical power 𝑃c of the fundamental solitons could be obtained when 𝐹 = 0. In addition, the equivalent potential V (W ) is given by
Taking variations of the averaged Lagrangian with respect to the parameters A, c, 𝜃 and W, we obtain that 𝐴2 𝑊 𝑑𝜃 1 𝐴2 𝑊 3 =− +√ + 2 + 𝜎 2 )3∕2 𝑑𝑧 2𝑊 2 2(2𝑊 2 2 2𝑊 + 𝜎 , 3𝛾𝐴2 𝑊 2 𝛾𝐴2 𝑊 4 (2𝜎 2 + 3𝑊 2 ) − √ − 2 2 4 4 3∕2 2 4𝜎 2 𝑊 2 + 3𝑊 4 + 𝜎 4 2(4𝜎 𝑊 + 3𝑊 + 𝜎 )
(10)
2
𝑉 =
𝛾𝑃0 𝑃 1 −√ √ 0 + √ . 2𝑊 2 2 2 2 𝜋 2𝑊 + 𝜎 𝜋 4𝜎 𝑊 2 + 3𝑊 4 + 𝜎 4
(12)
Indeed, the potential V described by Eq. (12) does predict the bright solitons in media with competing nonlocal nonlinearity. As is well 𝑑𝑉 known, 𝐹 = − 𝑑𝑊 = 0 corresponds to the soliton state. It can be seen that the width of the solitons is closely related to the input power as shown in Fig. 2, that is, 𝑊 = 1 when input power 𝑃 equal to critical power 𝑃𝑐 , 𝑊 < 1 when 𝑃 > 𝑃𝑐 and 𝑊 > 1 when 𝑃 < 𝑃𝑐 . The reason is that the self-focusing effect increases with the increase of the incident light power. Eventually, a narrower beam would be self-confined with bigger input power, vise verse. In addition, one can obtain the evolution of the beam width versus the propagation distance z from Eq. (10). To check the correctness of the analytical results, we first focus on the condition of 𝛾 = 0, which
𝑑𝑊 = 2𝑊 𝑐. (9) 𝑑𝑧 Eqs. (6)–(9) represent analytical relations for the parameters of bright solitons in media with competing nonlocal nonlinearity. Usually, the 79
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Fig. 4. Evolution of beam width W (z). The medium parameters are 𝜎 = 5, 𝑃 = 0.8𝑃c and 𝛾 = 0.1.
Fig. 2. Shapes of potential V for various incident powers. The medium parameters are 𝜎 = 10 and 𝛾 = 0.1.
interactions of solitons based on Eq. (1). So, we will use numerical methods to deal with these problems. Usually, the nonlocality of the reorientational and thermal nonlinearity is not equal. In this case, it is natural to introduce a figure of nonlinearities 𝛥𝑛2 and 𝛥𝑛1 as [ ] +∞ 𝛥𝑛2 = 1 − 𝛾 𝑅2 (𝑥 − 𝜉) |𝜓(𝜉, 𝑧)|2 𝑑𝜉 , ∫−∞ (13) +∞ 𝛥𝑛1 = 𝑅1 (𝑥 − 𝜂) |𝜓(𝜂, 𝑧)|2 𝑑𝜂, ∫−∞ here, 𝜎1 and 𝜎2 denote the nonlocality of the reorientational and thermal nonlinearity respectively, and 𝜎2 is usually bigger than 𝜎1 . To numerically obtain the soliton solutions, we will search for soliton solutions in the form 𝜓=w(x)exp(ibz), where w and b represent the field amplitude and the propagation constant, respectively. In the following part, we first find soliton solution of Eq. (1) by the modified squared-operator iteration method [33], and then use these solutions as the initial conditions in Eq. (1), which is numerically simulated by a split-step Fourier scheme.
Fig. 3. Evolution of beam width W (z). The medium parameters are 𝜎 = 5, 𝑃 = 0.8𝑃c and 𝛾 = 0.
3.1. Stationary solitons To numerically obtain the stationary solutions, the initial condition is taken as w(x)=2exp(−𝑥2 /2). In this part, we mainly investigate the impact of thermal nonlinear on the formation of stationary solutions. Under weakly nonlocal conditions, parameters 𝜎1 = 1, 𝜎2 = 1 and 𝑏 = 1 are used in the simulations. In this case, the refractive index change 𝛥n significantly decreases with 𝛾 when 𝛾<0.3, while the intensity is almost unaffected by the thermal nonlinearity as shown in Fig. 5(a). It is interesting to note that it is almost impossible to obtain the stationary solitons solutions with the initial condition w(x)=2exp(−𝑥2 /2) when 𝛾>0.3. So, such initial condition w(x)=tanh(x) is used in the simulations when 𝛾>0.3, which represents the dark soliton solution in local self-defocusing nonlinear medium. One of the interesting results of this work is that the dark-like soliton is formed when 𝛾 = 0.4, because of the competition between thermal and reorientational nonlinearity as shown in Fig. 5(b). To investigate the propagation of the stationary solitons in nonlinear media with competing nonlinearity, we perform numerical simulations of Eq. (1) with the input profiles of stationary soliton described in Fig. 5. Fig. 6(a) shows the propagation of stable bright solitons in nonlocal media without thermal nonlinearity, which is formed due to the balance between diffraction and nonlinearity. With the presence of thermal nonlinearity, the width and peak intensity of the beams exhibit periodic oscillation along the z axis as shown in Fig. 6(b) and (c).
means that the medium has only one nonlocal nonlinearity. We choose the parameters as input power 𝑃 = 0.8𝑃c , and 𝜎 = 5. Moreover, the initial beam width at 𝑧 = 0 is defined as 𝑊0 , and all the beam width in this part is divided by 𝑊0 . In this case, the width of the beams exhibit periodic oscillation along the z axis as shown in Fig. 3. The solid line is obtained by a split-step Fourier scheme, and the initial condition is the same as that used in the analytical study. Here, the integral width √ ∞
of the soliton is defined as 𝑊 = ∫−∞ (𝑥2 𝜓 2 ∕𝜓 2 )𝑑𝑥. The dashed line is obtained from the analytical Eq. (10). It can be seen that the variational results are in perfect agreement with the numerical simulations. When the parameter 𝛾≠0, there are great differences between analytical results and numerical simulations as shown in Fig. 4. The discrepancy between the variational results and the numerical results is due to the inadequacy of the ansatz when the appearance of the competing nonlinearities. 3. Numerical solutions In the above section, the stationary soliton solution of Eq. (1) is obtained by the variational approach. Due to the complexity of the governing propagation equation (1), it is indeed a difficult task to analytically investigate the solutions of higher-order solitons and the 80
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Fig. 5. Intensity and nonlinear refractive index change of stationary solitons with (a) 𝛾 = 0 and 𝛾 = 0.2, (b) 𝛾 = 0.4. The parameters are 𝜎1 = 1, 𝜎2 = 1 and 𝑏 = 1.
Fig. 6. Trajectories of stationary solitons in nonlinear media with (a) 𝛾 = 0, (b) 𝛾 = 0.1, (c) 𝛾 = 0.2 and (d) 𝛾 = 0.4. The parameters are 𝜎1 = 1, 𝜎2 = 1 and 𝑏 = 1.
Usually, this wave is known as breather. The formation of the breather is due to the unbalance between diffraction and nonlinearity as the presence of the thermal nonlinearity. When the parameter 𝛾 is equal to 0.4, it is found that dark-like soliton propagate in such media is accompanied with small energy radiation as shown in Fig. 6(c). In what follows, we will investigate the stationary solitons in nonlinear media with large thermal and reorientational nonlocality. Usually, the nonlocality of the thermal nonlinearity is bigger than the reorientational nonlocality. Hence, we choose the parameters (𝜎1 = 5, 𝜎2 = 10 and 𝑏 = 1) to further study the influence of thermal nonlinearity on
stationary soliton dynamics. In this case, although the thermal nonlinearity has less effect on the intensity profile of bright soliton, it can lead to obvious reduction of the refractive index induced by the bright soliton. Interestingly, for the first time to our knowledge, the interplay of the thermal and reorientational nonlocality yields interesting results which are shown in Fig. 7. Here, the profiles of the bright and darklike solitons are obtained by different initial input conditions in the simulations. From this figure, one can see that two distinct soliton families are formed in the same media under the same parameters. Especially, we can also say that the profile of the soliton not only 81
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Fig. 7. Intensity and nonlinear refractive index change of (a) bright soliton and (b) dark soliton. The parameters are 𝜎1 = 5, 𝜎2 = 10, 𝛾 = 0.3 and 𝑏 = 1.
Fig. 8. Trajectories of (a) bright and (b) dark solitons in nonlinear media with competing nonlinearity. The parameters are 𝜎1 = 5, 𝜎2 = 10, 𝛾 = 0.3 and 𝑏 = 1.
shown in Fig. 8(a). In Fig. 8(b), dark-like soliton can exhibit stable propagation in media with competing nonlinearities. 3.2. Dipole solitons One may wonder whether dipole solitons can exist in the system described by Eq. (1). To find the dipole solitons, the initial solution w(x)=2xexp(−𝑥2 /2) is used in our numerical simulations. Preliminary numerical calculations show that it is difficult to find the dipole solitons under weakly nonlocal condition, such as in the case of 𝜎1 = 1 and 𝜎2 = 1. So, our numerical simulations are focused on the strongly nonlocal condition. Here, the parameters 𝑏 = 1, 𝜎1 = 5 and 𝜎2 = 10 are used in the simulations. Under this condition, it is found that the profiles of dipole soliton with different 𝛾 are exactly the same as shown in Fig. 9. In addition, from this figure, one can see that the refractive index change 𝛥n significantly decreases with 𝛾 when other parameters are fixed. To test whether there is stable dipole soliton in nonlinear media with competing nonlinearity, we perform numerical simulations of Eq. (1) with the input profiles described in Fig. 9. In the absence of thermal nonlinearity, the dipole solitons can propagate stably in nonlocal medium, even in the presence of random perturbations of the initial condition as shown in Fig. 10(a). However, it can be seen that the bound state which is found by numerical simulation is no longer able to keep the separation between individual solitons as constant when 𝛾 = 0.1, as shown in Fig. 10(b). The main reason is that the thermal effect leads to the decrease of the refractive index change 𝛥n which plays an important role in the formation of the higher-order solitons. The important result is that it is impossible to form stable
Fig. 9. Profiles of dipole soliton and nonlinear refractive index change with different 𝛾. The parameters are 𝜎1 = 5, 𝜎2 = 10 and 𝑏 = 1.
depend on the competing nonlinearities of the media but also on the spatial profiles of the input beam. We would also like to stress that this is a rather counterintuitive result. To test the stability of the two soliton families, we perform numerical simulations of Eq. (1) with the input profiles which are described in Fig. 7(a) and (b). In this case, it is found that the width and peak intensity of the input beams exhibit periodic oscillation along the z axis in the presence of thermal nonlinearity as 82
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Fig. 10. Trajectories of dipole solitons in nonlinear media with (a) 𝛾 = 0 and (b) 𝛾 = 0.1. The parameters are 𝜎1 = 5, 𝜎2 = 10 and 𝑏 = 1.
Fig. 11. Trajectories of two out-of-phase bright soliton in nonlinear media with 𝜎1 = 1, 𝜎2 = 1, 𝑏 = 1. The initial separation of solitons is (a) 𝑥0 = 2.5, (b) 𝑥0 = 2.7, (c) 𝑥0 = 3 and (d) 𝑥0 = 4.
dipole soliton in media with thermal and orientational nonlinearity
nonlocality smooths out the refractive index profile thereby leading to
unless the parameters 𝜎1 and 𝜎2 are large enough. That is because the
a reduction of thermal effects. 83
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Fig. 12. Trajectories of two out-of-phase bright soliton in nonlinear media with competing nonlinearity for 𝜎1 = 1, 𝜎2 = 1, 𝑏 = 1 and 𝛾 = 0.1. The initial separation of solitons is (a) 𝑥0 = 2.5, (b) 𝑥0 = 3, (c) 𝑥0 = 3.3 and (d) 𝑥0 = 4.
Fig. 13. Trajectories of two out-of-phase bright soliton in nonlinear media with competing nonlinearity for 𝜎1 = 10, 𝜎2 = 10, 𝑏 = 1 and 𝛾 = 0.1. The initial separation of solitons is (a) 𝑥0 = 2.1, (b) 𝑥0 = 2.4, (c) 𝑥0 = 4, (d) 𝑥0 = 7, (e) 𝑥0 = 10 and (f) 𝑥0 = 14.
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formation of bright or dark-like solitons whose formation depends on the intensity profiles of the input beams, and their stabilities are demonstrated by a beam propagation method. We also numerically investigate the effect of thermal nonlinearity on the interaction of outof-phase bright solitons. We find that the interaction of out-of-phase bright solitons is closely related to the separation of input beams, opt-thermal effects and the nonlocality of thermal and reorientational nonlinearities.
3.3. Out-of-phase bright soliton interaction It is well known that two out-of-phase bright solitons can form bound states in nonlocal media, whose formation is determined not only by the phase difference, but also by the separation between solitons. So, an open question is: under what conditions, can dipole soliton be formed in nonlocal media with competing nonlinearities? To study this issue, we adopt the split-step Fourier method to perform simulations of Eqs. (1) with input conditions 𝜓(x, 𝑧 = 0)=w(x−𝑥0 )+w(x+𝑥0 ) exp(i𝜋) which corresponds to two out-of-phase bright solitons separated by 2𝑥0 . First of all, we investigate the interaction between out-of-phase bright solitons in weakly nonlocal media. In this case, the parameters 𝜎1 = 1, 𝜎2 = 1, 𝑏 = 1 and 𝛾 = 0 are used in the simulations. Fig. 11 presents the interaction of out-of-phase bright soliton under these conditions, which is closely related to the separation of the initial solitons. It can be seen that the dipole soliton can be composed by two out-of-phase bright solitons whose separation is equal to 5.4 (2𝑥0 = 5.4) as shown in Fig. 11(b). From Fig. 11, one also can see that the distance of the solitons exhibits periodic oscillations along the z axis when 𝑥0 is unequal to 2.7. Usually, the distance of the solitons increases first and then decreases along the z axis when 𝑥0 is less than 2.7, as shown in Fig. 11(a), and vice versa, when 𝑥0 is bigger than 2.7, as shown in Fig. 11(c) and (d). Next, we will address the impact of thermal nonlinearities on the interaction of out-of-phase bright solitons under weakly nonlocal conditions. For comparison, the same parameters 𝜎1 = 1, 𝜎2 = 1 and 𝑏 = 1 are used, but the difference is that the parameter 𝛾 is unequal to zero in the simulations. It is found that, for small separations, out-of-phase bright solitons exhibit strong repulsion rather than attraction when there is the thermal effect. This is quite different with the interaction in nonlocal media, in which there is always attraction between outof-phase bright solitons under the same initial separations unless the separations are small enough. Thus, one can see that two bright solitons repeal each other and their separation is monotonically increasing with transmission distance z, as shown in Fig. 12(a). As the separation of the input solitons increases, the repel force between the solitons gradually decreases, so that, an oscillatory bound state is formed when 𝑥0 = 3 as shown in Fig. 12(b). Interesting, the dipole-like state is formed when 𝑥0 = 3.3 and 𝛾 = 0.1 as shown in Fig. 12(c), which is unable to obtain by the direct simulations. Note that another oscillatory bound state is formed when 𝑥0 = 4 as shown in Fig. 12(d), in which the separation of bright solitons first decreases and then increases along the z axis. In the following, for the fixed parameters: b=1, 𝜎1 = 10, 𝜎2 = 10 and 𝛾 = 0.1, we will do a series of simulations of the interactions of two out-of-phase bright solitons by adjusting their initial separations, as shown in Fig. 13. In the case of small initial separations, two outof-phase bright solitons could form the oscillatory bound states whose separations exhibit periodic oscillations, as depicted in Fig. 13(a). Interestingly, it can be seen that the stable dipole solitons can be formed at 𝑥0 = 2.4 as shown in Fig. 13(b). As the increase of the initial separations, we still find that two out-of-phase bright solitons can form an oscillatory bound state, provided that the initial separations are much larger than the width of single soliton as shown in Fig. 13(c)– (e). Although, it can be seen that the oscillatory bound states could be formed at different initial separations, and the oscillation period decreases with the increase of the initial separations. These findings are almost consistent with those in the nonlocal media.
Acknowledgment This research was supported by the Natural Science Foundation of Heilongjiang, China (QC2015086). References [1] A.W. Snyder, D.J. Mitchell, Accessible solitons, Science 276 (1997) 1538–1541. [2] O. Bang, W. Krolikowski, J. Wyller, J.J. Rasmussen, Modulational instability in nonlocal nonlinear Kerr media, Phys. Rev. E 66 (2002) 046619. [3] C. Rotschild, T. Schwartz, O. Cohen, M. Segev, Incoherent spatial soliton in effectively instantaneous nonlinear media, Nature Photon. 2 (2008) 371–376. [4] C. Rotschild, O. Cohen, O. Manela, M. Segv, T. Carmon, Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons, Phys. Rev. Lett. 95 (2005) 213904. [5] O. Bang, W. Krolikowski, J. Wyller, J.J. Rasmussen, Collapse arrest and soliton stabilization in nonlocal nonlinear media, Phys. Rev. E 66 (2002) 046619. [6] G.I. Stegeman, M. Segev, Optical spatial solitons and their interactions: university and diversity, Science 286 (1999) 1518–1523. [7] N.I. Nikolov, D. Neshev, W. Królikowski, O. Bang, J.J. Rasmussen, P.L. Christiansen, Attraction of nonlocal dark optical solitons, Opt. Lett. 29 (2004) 286–288. [8] M. Peccianti, C. Conti, G. Assanto, Interplay between nonlocality and nonlinearity in nematic liquid crystals, Opt. Lett. 27 (2002) 415–417. [9] W. Hu, T. Zhang, Q. Guo, L. Xuan, S. Lan, Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals, Appl. Phys. Lett. 89 (2006) 071111. [10] Q. Kong, Q. Wang, O. Bang, W. Królikowski, Analytical theory for the darksoliton interaction in nonlocal nonlinear materials with an arbitrary degree of nonlocality, Phys. Rev. A 82 (2010) 013826. [11] F.W. Ye, Y.V. Kartashov, L.I. Torner, Enhanced soliton interactions by inhomogeneous nonlocality and nonlinearity, Phys. Rev. A 76 (2007) 033812. [12] C. Conti, M. Peccianti, G. Assanto, Route to nonlocality and observation of accessible solitons, Phys. Rev. Lett. 91 (2003) 073901. [13] C. Conti, M. Peccianti, G. Assanto, Observation of optical spatial solitons in a highly nonlocal medium, Phys. Rev. Lett. 92 (2004) 113902. [14] M. Warenghem, J. Blach, J. Henninot, Thermo-nematicon: an unnatural coexistence of solitons in liquid crystals, J. Opt. Soc. Amer. B 25 (2008) 1882–1887. [15] M. Peccianti, C. Conti, G. Assanto, A.D. Luca, C. Umeton, All-optical switching and logic gating with spatial solitons in liquid crystals, Appl. Phys. Lett. 81 (2002) 3335–3337. [16] A. Piccardi, M. Peccianti, G. Assanto, A. Dyadyusha, M. Kacamarek, Voltage-driven in-plane steering of nematicons, Appl. Phys. Lett. 94 (2009) 091106. [17] J. Beeckman, H. Azarinia, M. Haelterman, Countering spatial soliton breakdown in nematic liquid crystals, Opt. Lett. 34 (2009) 1900–1902. [18] B.K. Esbensen, A. Wlotaka, M. Bache, O. Bang, W. Królikowski, Modulational instability and solitons in nonlocal media with competing nonlinearities, Phys. Rev. A 84 (2011) 053854. [19] B.K. Essbensen, M. Bache, O. Bang, W. Królikowski, Anomalous interaction of nonlocal solitons in media with competing nonlinearities, Phys. Rev. A 86 (2012) 033838. [20] L. Chen, Q. Wang, M. Shen, H. Zhao, Y.Y. Lin, C.C. Jeng, R.K. Lee, W. Królikowski, Nonlocal dark solitons under competing cubic-quintic nonlinearities, Opt. Lett. 38 (2013) 13–15. [21] Q. Kong, M. Shen, Z.Y. Chen, Q. Wang, R.K. Lee, W. Królikowski, Dark solitons in nonlocal media with competing nonlinearities, Phys. Rev. A 87 (2013) 063832. [22] W. Chen, M. Shen, Q. Kong, J.L. Shi, Q. Wang, W. Królikowski, Interactions of nonlocal dark solitons under competing cubic–quintic nonlinearities, Opt. Lett. 39 (2014) 1764–1767. [23] M. Shen, H.W. Zhao, B.L. Li, J.L. Shi, Q. Wang, W. Królikowski, Stabilization of vortex solitons by combining competing cubic-quintic nonlinearities with a finite degree of nonlocality, Phys. Rev. A 89 (2014) 025804. [24] I.B. Burgess, M. Peccianti, G. Assanto, R. Morandotti, Accessible light bullets via synergetic nonlinearities, Phys. Rev. Lett. 102 (2009) 203903. [25] D. Mihalache, D. Mazilu, F. Lederer, L.C. Crasovan, Y.V. Kartashov, L. Torner, B.A. Malomed, Stable solitons of even and odd parities supported by competing nonlocal nonlinearities, Phys. Rev. E 74 (2006) 066614.
4. Conclusion In summary, we analytically investigate the formation of bright solitons in liquid crystals with competing thermal and reorientational nonlinearities. The analytical results are confirmed by the numerical simulations. In particular, we numerically find that the competition between thermal and reorientational nonlinearities can lead to the 85
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Optics Communications 450 (2019) 78–86 [30] K. Cyprych, P.S. Jung, Y. Izdebskaya, V. Shvedov, D.N. Christodoulides, W. Królikowski, Anomalous interaction of spatial solitons in nematic liquid crystals, Opt. Lett. 44 (2019) 267–270. [31] P.S. Jung, W. Krolikowski, U.A. Laudyn, M. Trippenbach, M.A. Karpierz, Supermode spatial optical solitons in liquid crystals with competing nonlinearities, Phys. Rev. A 95 (2017) 023820. [32] P.S. Jung, W. Krolikowski, U.A. Laudyn, M.A. Karpierz, M. Trippenbach, Semianalytical approach to supermode spatial solitons formation in nematic liquid crystals, Opt. Express 25 (2017) 23893–23897. [33] J.K. Yang, T.I. Lakoba, Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations, Stud. Appl. Math. 118 (2007) 153–197.
[26] K.H. Kuo, Y.Y. Lin, R.K. Lee, B.A. Malomed, Gap solitons under competing local and nonlocal nonlinearities, Phys. Rev. A 83 (2011) 053838. [27] Y.V. Kartashov, V.A. Vysloukh, L. Torner, Stabilization of higher-order vortices and multihump solitons in media with synthetic nonlocal nonlinearities, Phys. Rev. A 79 (2009) 013803. [28] Y.W. Du, Z.X. Zhou, H. Tian, D.J. Liu, Bright solitons and repulsive in-phase interaction in media with competing nonlocal kerr nonlinearities, J. Opt. 13 (2011) 015201. [29] B.K. Esbensen, M. Bache, O. Bang, W. Królikowski, Anomalous interaction of nonlocal solitons in media with competing nonlinearities, Phys. Rev. A 86 (2012) 033838.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Solitons in liquid crystals with competing nonlinearities Shaozhi Pu ∗, Ming Chen, Yingjia Li, Liuyang Zhang Department of Optical Information Science and Technology, Harbin University of Science and Technology, Harbin 150080, China
ARTICLE
INFO
Keywords: Thermal nonlinearity Reorientational nonlinearity Nonlocality Optical solitons
ABSTRACT We employ a model which is proposed by P. S. Jung et al. [see P. S. Jung et al. Opt. Express 25 (2017) 23893– 23897] to analytically demonstrate properties of individual bright solitons in liquid crystals with competing nonlinearities. These results are confirmed by the numerical simulations. In addition, we numerically find that the competition between thermal and reorientational nonlinearities can lead to the formation of bright or dark-like solitons whose formation depends on the intensity profiles of the input beams. We show that the competing nonlinearities drastically affect the interaction of out-of-phase bright solitons, such as the change of attraction of out-of-phase bright solitons into repulsion force at certain initial separations of the input solitons.
1. Introduction Nonlocal nonlinearity is one of the most fascinating phenomena appearing in nonlinear systems. As is well known, in the nonlocal media, the refractive index changes in a particular location depend on the light intensity in a certain neighborhood of this location. In the past decade, attentions have been focused on the nonlocal solitons, because they have unique properties compared with local solitons [1–4]. In particular, the nonlocality may arrest the collapse of multidimensional soliton and affect the interaction of solitons [5–11]. Nematic liquid crystals have been an ideal material to investigate nonlocal spatial solitons because of their unique merits, such as large electro-optical tunability and nonlocal nonlinearity which can be adjusted by low-frequency electric field and light beams since the excellent work by G. Assanto [12,13]. So far, most of the investigations of nematicons have been focused on the nematic liquid crystals which consist of elongated rod-like molecules and belong to positive uniaxial materials. It is found that an extraordinarily polarized optical beam in this material will result in molecular orientational effects and thermal effects [14]. Usually, the former (orientational effects) leads to the self-focusing effect and the latter (orientational effects) yields to the self-defocusing effect for the extraordinary wave. So far, many research works have shown that the formation of the nematicon is due to the reorientational nonlinearity [12,13,15–17]. Recently, the thermal solitons in nematic liquid crystal are observed in the experiments by M. Warenghem et al. [14]. In recent years, nonlocal solitons in media with competing nonlinearities have been well investigated [18–24]. It is found that the competition between two types of nonlocal nonlinearities may lead to the formation of many complex soliton structures which are unstable in nonlocal media with only one type of nonlocal nonlinearity, such as ∗
solitons of even and odd parities [25], gap solitons [26], and higher order vortex solitons [27]. Especially, Du et al. have numerically found that there is the repulsive force between two in-phase bright solitons due to the competing nonlocal nonlinearities [28]. Thereafter, Esbensen et al. have reached similar conclusions and a more detailed analysis is provided by a variational approach [29]. Recently, this phenomenon has been confirmed experimentally by Cyprych et al. in dye-doped nematic liquid crystals which possess thermal and reorientational nonlinearities [30]. Very recently, P. S. Jung et al. analytically investigated the solitons in nematic liquid crystals involving thermal effects and molecular orientation effects [31]. In this case, they propose a simple analytical model to investigate the solitons in nonlocal medium with competing nonlinearities, and they also find the formation of supermode spatial solitons based on this model [32]. So far, the research works on spatial solitons in media with competing nonlinearities are mainly focused on two types of nonlinearities which are independent with each other. However, in reality, these two effects are closely related. In this case, the light propagation in media with competing nonlinearities obeys the model proposed by P. S. Jung et al. [32]. So far, to our knowledge, a detailed investigation on solitons and their interactions by this model is still needed. In this work, we employ the model which is developed by P. S. Jung et al. to analytically and numerically investigate bright solitons in media with competing nonlinearities. The bright soliton solutions and their characteristics are obtained by a variational approach. In addition, for the first time to our knowledge, dark-like soliton is found numerically by this model. Moreover, we study the effect of competing nonlocal nonlinearity on the interaction of out-of-phase bright solitons.
Corresponding author. E-mail address: [email protected] (S. Pu).
https://doi.org/10.1016/j.optcom.2019.05.065 Received 16 February 2019; Received in revised form 25 May 2019; Accepted 29 May 2019 Available online 31 May 2019 0030-4018/© 2019 Published by Elsevier B.V.
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Optics Communications 450 (2019) 78–86
2. Model and analytical solutions We consider the light propagation in media with competing nonlinearities, described by the following nonlocal nonlinear Schrödinger equation: 𝑖
𝜕𝜓 1 𝜕2 𝜓 + 𝛥𝑛𝜓 = 0, + 𝜕𝑧 2 𝜕𝑥2
(1)
where x and z stand for the transverse and longitudinal coordinates normalized to diffraction length and beam width, respectively, 𝜓 is the electric field amplitude, and 𝛥n is the refractive index change of the medium induced by the incident beam. Here, the nonlinear change in index can be written as 𝛥n=𝛥𝑛2 𝛥𝑛1 in nematic liquid crystals, where 𝛥𝑛2 and 𝛥𝑛1 represent the opt-thermal nonlinearity and reorientational nonlinearity, respectively. As we know, the reorientational nonlinearity is closely related to the optical anisotropy 𝛥𝜀 = 𝑛2e –𝑛2o . Here, 𝑛e and 𝑛o are ordinary and extraordinary refractive indices respectively, both of which are functions of temperature. In this case, it is natural to introduce a figure of nonlinearities 𝛥𝑛2 and 𝛥𝑛1 as [ ] +∞ 𝛥𝑛2 = 1 − 𝛾 𝑅(𝑥 − 𝜉) |𝜓(𝜉, 𝑧)|2 𝑑𝜉 , ∫−∞ (2) +∞ 𝛥𝑛1 = 𝑅(𝑥 − 𝜂) |𝜓(𝜂, 𝑧)|2 𝑑𝜂, ∫−∞
Fig. 1. The soliton power versus the propagation constant. Solid line is the solution by the variational analysis, dash line is the numerical solutions of Eq. (1). The medium parameters are 𝜎 = 5 and 𝛾 = 0.1.
solution of the bright solitons can be obtained by numerical simulations with the initial input form 𝜓=w(x)exp(ibz), where w and b represent the field amplitude and the propagation constant, respectively. To compare the theoretical analysis with the numerical simulations, the is chosen in this part. In addition, the beam propagation constant 𝑏 = 𝑑𝜃 𝑑𝑧 ∞ power 𝑃 = ∫−∞ |𝜓|2 𝑑𝑥 is used in the numerical simulations. In Fig. 1, we illustrate the relation between soliton power and the propagation constant at 𝜎 = 5 and 𝛾 = 0.1. It is shown that the power increases with an increment of the propagation constant, and this tendency is in good agreement with the numerical simulations based on Eq. (1). The monotonic increase of this function indicates that the solitons are stable. √ It can be seen from Eq. (8) that the wave energy 𝑃0 = 𝜋𝐴2 𝑊 is a conserved quantity. Next, one can obtain from Eqs. (7) and (9) the following equation
where 𝛾 represents the strength of the thermal nonlinearity, √ and we call it the opt-thermal nonlinearity coefficient, where 𝑅 = ( 𝜋𝜎)−1 exp (−𝑥2 ∕𝜎 2 ) is the nonlocal response function in one transverse dimension. Here, 𝜎 denotes the nonlocality of the reorientational and thermal nonlinearity. Eq. (1) can be considered as the Euler–Lagrange variational equation corresponding to the Lagrangian density, ∞ |𝜓|2 𝑖 ∗ 1 2 (𝜓 𝜓𝑧 − 𝜓𝜓𝑧∗ ) − ||𝜓𝑥 || + 𝑅(𝑥 − 𝜉) |𝜓(𝜉)|2 𝑑𝜉 ∫ 2 2 2 −∞ [ ] , ∞ ∞ |𝜓|2 −𝛾 𝑅(𝑥 − 𝜉)|𝜓(𝜉)|2 𝑑𝜉 𝑅(𝑥 − 𝜂)|𝜓(𝜂)|2 𝑑𝜂 ∫−∞ 2 ∫−∞
=
(3)
where the asterisk denotes the complex conjugate. To find the soliton solutions of Eq. (1), we use the following Gaussian ansatz: 𝑥2 ], (4) 2𝑊 2 where A is the amplitude of the beam, 𝜃 and 𝑐 are the phase and phasefront curvature of the complex amplitude of 𝜓, respectively, W is the width. Substituting trial function (4) into the system Lagrangian 𝐿 = +∞ ∫−∞ 𝑑𝑥 and integrating over x, we obtain the averaged Lagrangian, √ √ √ √ 𝑑𝜃 𝜋 𝑑𝑐 𝐴2 𝜋 𝐿 = −𝐴2 𝑊 𝜋 − 𝐴2 𝑊 3 − 𝐴2 𝑐 2 𝑊 3 𝜋 − 𝑑𝑧 2 𝑑𝑧 4𝑊 √ 4 2 √ (5) 𝜋𝐴 𝑊 𝛾 𝜋𝐴6 𝑊 3 + √ − √ . 2 𝜎 2 + 2𝑊 2 2 4𝜎 2 𝑊 2 + 3𝑊 4 + 𝜎 4 𝜓(𝑥, 𝑧) = 𝐴(𝑧) exp[𝑖𝜃(𝑧) + 𝑖𝑐(𝑧)𝑥2 −
2𝛾𝑃02 𝑊 (2𝜎 2 + 3𝑊 2 ) 2𝑃0 𝑊 𝑑2𝑊 1 = −√ + , 2 3 3∕2 2 2 𝑑𝑧 𝑊 𝜋(4𝜎 2 𝑊 2 + 3𝑊 4 + 𝜎 4 )3∕2 𝜋(2𝑊 + 𝜎 )
where 𝑃0 is the input power. The Eq. (10) is equivalent to Newton’s second law in classical mechanics for the motion of a one-dimensional particle with the equivalent mass 1 acted by the equivalent force F, i.e. 𝐹 =
(6)
𝛾𝐴4 𝑊 2 (2𝜎 2 + 3𝑊 2 ) 𝑑𝑐 1 𝐴2 𝑊 = −2𝑐 2 + − + , 4 3∕2 2 2 𝑑𝑧 2𝑊 (2𝑊 + 𝜎 ) (4𝜎 2 𝑊 2 + 3𝑊 4 + 𝜎 4 )3∕2
(7)
𝑑 (𝐴2 𝑊 ) = 0, 𝑑𝑧
(8)
2𝛾𝑃02 𝑊 (2𝜎 2 + 3𝑊 2 ) 2𝑃0 𝑊 1 −√ + . 3 𝑊 𝜋(2𝑊 2 + 𝜎 2 )3∕2 𝜋(4𝜎 2 𝑊 2 + 3𝑊 4 + 𝜎 4 )3∕2
(11)
It can be seen that F is a conservative force, the critical power 𝑃c of the fundamental solitons could be obtained when 𝐹 = 0. In addition, the equivalent potential V (W ) is given by
Taking variations of the averaged Lagrangian with respect to the parameters A, c, 𝜃 and W, we obtain that 𝐴2 𝑊 𝑑𝜃 1 𝐴2 𝑊 3 =− +√ + 2 + 𝜎 2 )3∕2 𝑑𝑧 2𝑊 2 2(2𝑊 2 2 2𝑊 + 𝜎 , 3𝛾𝐴2 𝑊 2 𝛾𝐴2 𝑊 4 (2𝜎 2 + 3𝑊 2 ) − √ − 2 2 4 4 3∕2 2 4𝜎 2 𝑊 2 + 3𝑊 4 + 𝜎 4 2(4𝜎 𝑊 + 3𝑊 + 𝜎 )
(10)
2
𝑉 =
𝛾𝑃0 𝑃 1 −√ √ 0 + √ . 2𝑊 2 2 2 2 𝜋 2𝑊 + 𝜎 𝜋 4𝜎 𝑊 2 + 3𝑊 4 + 𝜎 4
(12)
Indeed, the potential V described by Eq. (12) does predict the bright solitons in media with competing nonlocal nonlinearity. As is well 𝑑𝑉 known, 𝐹 = − 𝑑𝑊 = 0 corresponds to the soliton state. It can be seen that the width of the solitons is closely related to the input power as shown in Fig. 2, that is, 𝑊 = 1 when input power 𝑃 equal to critical power 𝑃𝑐 , 𝑊 < 1 when 𝑃 > 𝑃𝑐 and 𝑊 > 1 when 𝑃 < 𝑃𝑐 . The reason is that the self-focusing effect increases with the increase of the incident light power. Eventually, a narrower beam would be self-confined with bigger input power, vise verse. In addition, one can obtain the evolution of the beam width versus the propagation distance z from Eq. (10). To check the correctness of the analytical results, we first focus on the condition of 𝛾 = 0, which
𝑑𝑊 = 2𝑊 𝑐. (9) 𝑑𝑧 Eqs. (6)–(9) represent analytical relations for the parameters of bright solitons in media with competing nonlocal nonlinearity. Usually, the 79
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Fig. 4. Evolution of beam width W (z). The medium parameters are 𝜎 = 5, 𝑃 = 0.8𝑃c and 𝛾 = 0.1.
Fig. 2. Shapes of potential V for various incident powers. The medium parameters are 𝜎 = 10 and 𝛾 = 0.1.
interactions of solitons based on Eq. (1). So, we will use numerical methods to deal with these problems. Usually, the nonlocality of the reorientational and thermal nonlinearity is not equal. In this case, it is natural to introduce a figure of nonlinearities 𝛥𝑛2 and 𝛥𝑛1 as [ ] +∞ 𝛥𝑛2 = 1 − 𝛾 𝑅2 (𝑥 − 𝜉) |𝜓(𝜉, 𝑧)|2 𝑑𝜉 , ∫−∞ (13) +∞ 𝛥𝑛1 = 𝑅1 (𝑥 − 𝜂) |𝜓(𝜂, 𝑧)|2 𝑑𝜂, ∫−∞ here, 𝜎1 and 𝜎2 denote the nonlocality of the reorientational and thermal nonlinearity respectively, and 𝜎2 is usually bigger than 𝜎1 . To numerically obtain the soliton solutions, we will search for soliton solutions in the form 𝜓=w(x)exp(ibz), where w and b represent the field amplitude and the propagation constant, respectively. In the following part, we first find soliton solution of Eq. (1) by the modified squared-operator iteration method [33], and then use these solutions as the initial conditions in Eq. (1), which is numerically simulated by a split-step Fourier scheme.
Fig. 3. Evolution of beam width W (z). The medium parameters are 𝜎 = 5, 𝑃 = 0.8𝑃c and 𝛾 = 0.
3.1. Stationary solitons To numerically obtain the stationary solutions, the initial condition is taken as w(x)=2exp(−𝑥2 /2). In this part, we mainly investigate the impact of thermal nonlinear on the formation of stationary solutions. Under weakly nonlocal conditions, parameters 𝜎1 = 1, 𝜎2 = 1 and 𝑏 = 1 are used in the simulations. In this case, the refractive index change 𝛥n significantly decreases with 𝛾 when 𝛾<0.3, while the intensity is almost unaffected by the thermal nonlinearity as shown in Fig. 5(a). It is interesting to note that it is almost impossible to obtain the stationary solitons solutions with the initial condition w(x)=2exp(−𝑥2 /2) when 𝛾>0.3. So, such initial condition w(x)=tanh(x) is used in the simulations when 𝛾>0.3, which represents the dark soliton solution in local self-defocusing nonlinear medium. One of the interesting results of this work is that the dark-like soliton is formed when 𝛾 = 0.4, because of the competition between thermal and reorientational nonlinearity as shown in Fig. 5(b). To investigate the propagation of the stationary solitons in nonlinear media with competing nonlinearity, we perform numerical simulations of Eq. (1) with the input profiles of stationary soliton described in Fig. 5. Fig. 6(a) shows the propagation of stable bright solitons in nonlocal media without thermal nonlinearity, which is formed due to the balance between diffraction and nonlinearity. With the presence of thermal nonlinearity, the width and peak intensity of the beams exhibit periodic oscillation along the z axis as shown in Fig. 6(b) and (c).
means that the medium has only one nonlocal nonlinearity. We choose the parameters as input power 𝑃 = 0.8𝑃c , and 𝜎 = 5. Moreover, the initial beam width at 𝑧 = 0 is defined as 𝑊0 , and all the beam width in this part is divided by 𝑊0 . In this case, the width of the beams exhibit periodic oscillation along the z axis as shown in Fig. 3. The solid line is obtained by a split-step Fourier scheme, and the initial condition is the same as that used in the analytical study. Here, the integral width √ ∞
of the soliton is defined as 𝑊 = ∫−∞ (𝑥2 𝜓 2 ∕𝜓 2 )𝑑𝑥. The dashed line is obtained from the analytical Eq. (10). It can be seen that the variational results are in perfect agreement with the numerical simulations. When the parameter 𝛾≠0, there are great differences between analytical results and numerical simulations as shown in Fig. 4. The discrepancy between the variational results and the numerical results is due to the inadequacy of the ansatz when the appearance of the competing nonlinearities. 3. Numerical solutions In the above section, the stationary soliton solution of Eq. (1) is obtained by the variational approach. Due to the complexity of the governing propagation equation (1), it is indeed a difficult task to analytically investigate the solutions of higher-order solitons and the 80
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Fig. 5. Intensity and nonlinear refractive index change of stationary solitons with (a) 𝛾 = 0 and 𝛾 = 0.2, (b) 𝛾 = 0.4. The parameters are 𝜎1 = 1, 𝜎2 = 1 and 𝑏 = 1.
Fig. 6. Trajectories of stationary solitons in nonlinear media with (a) 𝛾 = 0, (b) 𝛾 = 0.1, (c) 𝛾 = 0.2 and (d) 𝛾 = 0.4. The parameters are 𝜎1 = 1, 𝜎2 = 1 and 𝑏 = 1.
Usually, this wave is known as breather. The formation of the breather is due to the unbalance between diffraction and nonlinearity as the presence of the thermal nonlinearity. When the parameter 𝛾 is equal to 0.4, it is found that dark-like soliton propagate in such media is accompanied with small energy radiation as shown in Fig. 6(c). In what follows, we will investigate the stationary solitons in nonlinear media with large thermal and reorientational nonlocality. Usually, the nonlocality of the thermal nonlinearity is bigger than the reorientational nonlocality. Hence, we choose the parameters (𝜎1 = 5, 𝜎2 = 10 and 𝑏 = 1) to further study the influence of thermal nonlinearity on
stationary soliton dynamics. In this case, although the thermal nonlinearity has less effect on the intensity profile of bright soliton, it can lead to obvious reduction of the refractive index induced by the bright soliton. Interestingly, for the first time to our knowledge, the interplay of the thermal and reorientational nonlocality yields interesting results which are shown in Fig. 7. Here, the profiles of the bright and darklike solitons are obtained by different initial input conditions in the simulations. From this figure, one can see that two distinct soliton families are formed in the same media under the same parameters. Especially, we can also say that the profile of the soliton not only 81
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Fig. 7. Intensity and nonlinear refractive index change of (a) bright soliton and (b) dark soliton. The parameters are 𝜎1 = 5, 𝜎2 = 10, 𝛾 = 0.3 and 𝑏 = 1.
Fig. 8. Trajectories of (a) bright and (b) dark solitons in nonlinear media with competing nonlinearity. The parameters are 𝜎1 = 5, 𝜎2 = 10, 𝛾 = 0.3 and 𝑏 = 1.
shown in Fig. 8(a). In Fig. 8(b), dark-like soliton can exhibit stable propagation in media with competing nonlinearities. 3.2. Dipole solitons One may wonder whether dipole solitons can exist in the system described by Eq. (1). To find the dipole solitons, the initial solution w(x)=2xexp(−𝑥2 /2) is used in our numerical simulations. Preliminary numerical calculations show that it is difficult to find the dipole solitons under weakly nonlocal condition, such as in the case of 𝜎1 = 1 and 𝜎2 = 1. So, our numerical simulations are focused on the strongly nonlocal condition. Here, the parameters 𝑏 = 1, 𝜎1 = 5 and 𝜎2 = 10 are used in the simulations. Under this condition, it is found that the profiles of dipole soliton with different 𝛾 are exactly the same as shown in Fig. 9. In addition, from this figure, one can see that the refractive index change 𝛥n significantly decreases with 𝛾 when other parameters are fixed. To test whether there is stable dipole soliton in nonlinear media with competing nonlinearity, we perform numerical simulations of Eq. (1) with the input profiles described in Fig. 9. In the absence of thermal nonlinearity, the dipole solitons can propagate stably in nonlocal medium, even in the presence of random perturbations of the initial condition as shown in Fig. 10(a). However, it can be seen that the bound state which is found by numerical simulation is no longer able to keep the separation between individual solitons as constant when 𝛾 = 0.1, as shown in Fig. 10(b). The main reason is that the thermal effect leads to the decrease of the refractive index change 𝛥n which plays an important role in the formation of the higher-order solitons. The important result is that it is impossible to form stable
Fig. 9. Profiles of dipole soliton and nonlinear refractive index change with different 𝛾. The parameters are 𝜎1 = 5, 𝜎2 = 10 and 𝑏 = 1.
depend on the competing nonlinearities of the media but also on the spatial profiles of the input beam. We would also like to stress that this is a rather counterintuitive result. To test the stability of the two soliton families, we perform numerical simulations of Eq. (1) with the input profiles which are described in Fig. 7(a) and (b). In this case, it is found that the width and peak intensity of the input beams exhibit periodic oscillation along the z axis in the presence of thermal nonlinearity as 82
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Fig. 10. Trajectories of dipole solitons in nonlinear media with (a) 𝛾 = 0 and (b) 𝛾 = 0.1. The parameters are 𝜎1 = 5, 𝜎2 = 10 and 𝑏 = 1.
Fig. 11. Trajectories of two out-of-phase bright soliton in nonlinear media with 𝜎1 = 1, 𝜎2 = 1, 𝑏 = 1. The initial separation of solitons is (a) 𝑥0 = 2.5, (b) 𝑥0 = 2.7, (c) 𝑥0 = 3 and (d) 𝑥0 = 4.
dipole soliton in media with thermal and orientational nonlinearity
nonlocality smooths out the refractive index profile thereby leading to
unless the parameters 𝜎1 and 𝜎2 are large enough. That is because the
a reduction of thermal effects. 83
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Fig. 12. Trajectories of two out-of-phase bright soliton in nonlinear media with competing nonlinearity for 𝜎1 = 1, 𝜎2 = 1, 𝑏 = 1 and 𝛾 = 0.1. The initial separation of solitons is (a) 𝑥0 = 2.5, (b) 𝑥0 = 3, (c) 𝑥0 = 3.3 and (d) 𝑥0 = 4.
Fig. 13. Trajectories of two out-of-phase bright soliton in nonlinear media with competing nonlinearity for 𝜎1 = 10, 𝜎2 = 10, 𝑏 = 1 and 𝛾 = 0.1. The initial separation of solitons is (a) 𝑥0 = 2.1, (b) 𝑥0 = 2.4, (c) 𝑥0 = 4, (d) 𝑥0 = 7, (e) 𝑥0 = 10 and (f) 𝑥0 = 14.
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formation of bright or dark-like solitons whose formation depends on the intensity profiles of the input beams, and their stabilities are demonstrated by a beam propagation method. We also numerically investigate the effect of thermal nonlinearity on the interaction of outof-phase bright solitons. We find that the interaction of out-of-phase bright solitons is closely related to the separation of input beams, opt-thermal effects and the nonlocality of thermal and reorientational nonlinearities.
3.3. Out-of-phase bright soliton interaction It is well known that two out-of-phase bright solitons can form bound states in nonlocal media, whose formation is determined not only by the phase difference, but also by the separation between solitons. So, an open question is: under what conditions, can dipole soliton be formed in nonlocal media with competing nonlinearities? To study this issue, we adopt the split-step Fourier method to perform simulations of Eqs. (1) with input conditions 𝜓(x, 𝑧 = 0)=w(x−𝑥0 )+w(x+𝑥0 ) exp(i𝜋) which corresponds to two out-of-phase bright solitons separated by 2𝑥0 . First of all, we investigate the interaction between out-of-phase bright solitons in weakly nonlocal media. In this case, the parameters 𝜎1 = 1, 𝜎2 = 1, 𝑏 = 1 and 𝛾 = 0 are used in the simulations. Fig. 11 presents the interaction of out-of-phase bright soliton under these conditions, which is closely related to the separation of the initial solitons. It can be seen that the dipole soliton can be composed by two out-of-phase bright solitons whose separation is equal to 5.4 (2𝑥0 = 5.4) as shown in Fig. 11(b). From Fig. 11, one also can see that the distance of the solitons exhibits periodic oscillations along the z axis when 𝑥0 is unequal to 2.7. Usually, the distance of the solitons increases first and then decreases along the z axis when 𝑥0 is less than 2.7, as shown in Fig. 11(a), and vice versa, when 𝑥0 is bigger than 2.7, as shown in Fig. 11(c) and (d). Next, we will address the impact of thermal nonlinearities on the interaction of out-of-phase bright solitons under weakly nonlocal conditions. For comparison, the same parameters 𝜎1 = 1, 𝜎2 = 1 and 𝑏 = 1 are used, but the difference is that the parameter 𝛾 is unequal to zero in the simulations. It is found that, for small separations, out-of-phase bright solitons exhibit strong repulsion rather than attraction when there is the thermal effect. This is quite different with the interaction in nonlocal media, in which there is always attraction between outof-phase bright solitons under the same initial separations unless the separations are small enough. Thus, one can see that two bright solitons repeal each other and their separation is monotonically increasing with transmission distance z, as shown in Fig. 12(a). As the separation of the input solitons increases, the repel force between the solitons gradually decreases, so that, an oscillatory bound state is formed when 𝑥0 = 3 as shown in Fig. 12(b). Interesting, the dipole-like state is formed when 𝑥0 = 3.3 and 𝛾 = 0.1 as shown in Fig. 12(c), which is unable to obtain by the direct simulations. Note that another oscillatory bound state is formed when 𝑥0 = 4 as shown in Fig. 12(d), in which the separation of bright solitons first decreases and then increases along the z axis. In the following, for the fixed parameters: b=1, 𝜎1 = 10, 𝜎2 = 10 and 𝛾 = 0.1, we will do a series of simulations of the interactions of two out-of-phase bright solitons by adjusting their initial separations, as shown in Fig. 13. In the case of small initial separations, two outof-phase bright solitons could form the oscillatory bound states whose separations exhibit periodic oscillations, as depicted in Fig. 13(a). Interestingly, it can be seen that the stable dipole solitons can be formed at 𝑥0 = 2.4 as shown in Fig. 13(b). As the increase of the initial separations, we still find that two out-of-phase bright solitons can form an oscillatory bound state, provided that the initial separations are much larger than the width of single soliton as shown in Fig. 13(c)– (e). Although, it can be seen that the oscillatory bound states could be formed at different initial separations, and the oscillation period decreases with the increase of the initial separations. These findings are almost consistent with those in the nonlocal media.
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