## Abstract

We find that a surface soliton in nonlocal nonlinear media can be regarded as a half of a bulk soliton with an antisymmetric amplitude distribution. The analytical solutions for surface solitons and breathers in strongly nonlocal media are obtained, and the critical power and breather period are gotten analytically and confirmed by numerical simulations. In addition, the oscillating propagation of nonlocal surface solitons launched away from the stationary position is considered as interaction between the soliton and its out-of-phase image beam. Its trajectory and oscillating period obtained by our model are in good agreement with the numerical simulations.

© 2011 Optical Society of America

## 1. Introduction

The nonlocality of nonlinear response exists in many real physical systems, such as photorefractive crystals [1], nematic liquid crystals [2–5], lead glasses [6, 7], atomic vapors [8], Bose-Einstein condensates [9, 10] etc. In nonlocal nonlinear media, various soliton solutions have been predicted theoretically, such as vortex solitons [11, 12], multi-pole solitons [7, 13–15], Laguerre-Gaussian and Hermite-Gaussian solitons [16–18], Ince-Gaussian solitons [19], and some of these have been observed experimentally [4–7, 20]. For nonlocal solitons, there are many interesting properties, for instance, large phase shift [21], attraction between two bright out-of-phase solitons [5, 22], attraction between two dark solitons [23, 24], self-induced fractional Fourier transform [25], etc.

Recently, nonlocal surface solitons have been investigated numerically and experimentally [26–31]. Nonlocal surface solitons occurring at an interface between a dielectric medium and a nonlocal material exhibit unique properties. Nonlocal multipole surface solitons, vortices, and bound states of vortex solitons, incoherent surface solitons and ring surface waves have been also studied recently [27–30]. But the properties of nonlocal surface solitons mentioned above are all discussed by numerical simulations. In order to get a good understanding of the properties of nonlocal surface solitons, it is essential to present an analytic solution, even an approximate one.

In this paper, based on the propagation equations governing nonlocal surface waves and assuming that all energy of surface solitons resides in nonlocal media, we find that a surface soliton in nonlocal nonlinear media can be regarded as a half of a bulk soliton with an antisymmetric amplitude distribution. The evolution regularities for nonlocal surface waves are discussed both analytically and numerically. The analytical solutions for surface solitons and breathers in strongly nonlocal media are obtained, and the critical power and breather period are gotten analytically and confirmed by numerical simulations. In addition, the oscillating propagation of nonlocal surface solitons launched away from the stationary position is considered as the interaction between the soliton and its out-of-phase image beam. Its trajectory and oscillating period obtained by our model are in good agreement with the numerical simulations.

## 2. Relation between surface solitons and bulk solitons

We consider a (1+1)-D model of a TE polarized surface wave with an envelope *A* propagating in *z* direction near the interface between a nonlocal nonlinear medium and a linear medium [see Fig. 1(a)]. The propagation of surface waves is governed by the nonlocal nonlinear Schrödinger equation (NNLSE), i.e.

- in nonlocal nonlinear media,
*x*≤ 0,$$2i{n}_{0}{k}_{0}\frac{\partial A}{\partial Z}+\frac{{\partial}^{2}A}{\partial {X}^{2}}+2{n}_{0}\Delta N{k}_{0}^{2}A=0,$$$$\Delta N-{W}_{m}^{2}\frac{{\partial}^{2}\Delta N}{\partial {X}^{2}}-{n}_{2}{\left|A\right|}^{2}=0;$$in linear media,*x*> 0,$$2i{n}_{0}{k}_{0}\frac{\partial A}{\partial Z}+\frac{{\partial}^{2}A}{\partial {X}^{2}}-\left({n}_{0}^{2}-{n}_{L}^{2}\right){k}_{0}^{2}A=0,$$where*X,Z*denote, respectively, the transverse and longitudinal coordinates.*n*_{0}and*n*are the unperturbed refractive index of nonlocal medium and the refractive index of less optically dense linear medium, respectively._{L}*k*_{0}= 2*π*/*λ*is the wave number in vacuum, where*λ*is the wavelength.*n*_{2}and Δ*N*are the nonlinear index coefficient and the nonlinear perturbation of refractive index, respectively.*W*is the characteristic length of the nonlocal material response._{m}

Introducing the normalized parameters, *x* = *X/w _{a}*,
$z=Z/\left({n}_{0}{k}_{0}{w}_{a}^{2}\right)$,

*w*=

_{m}*W*, $q={\left({n}_{0}{n}_{2}{k}_{0}^{2}{w}_{a}^{2}\right)}^{1/2}A$, $\Delta n=\left({n}_{0}{k}_{0}^{2}{w}_{a}^{2}\right)\Delta N$, and ${n}_{d}=\left({n}_{0}^{2}-{n}_{L}^{2}\right){k}_{0}^{2}{w}_{a}^{2}/2$, where

_{m}/w_{a}*w*is a characteristic length of beam width, one can get the dimensionless equations

_{a}In fact, Eqs. (4) and (5) denote a NNLSE with an exponential-decay type nonlocal response. The exponential-decay type nonlocal response exists in many real physical systems, for instance, all diffusion-type nonlinearity [32–34], orientational-type nonlinearity [2,22], and the general quadratic nonlinearity describing parametric interaction [35, 36]. Moreover when *w _{m}* → ∞, Eqs. (4) and (5) can be transformed to the forms describing the thermal nonlocal media (for example, lead glasses [6, 26]).

For the TE polarized wave (i.e. polarized along *y*-direction), the boundary conditions are *q*(+0) = *q*(−0), *∂q/∂x| _{x}*

_{=+0}=

*∂q/∂x|*

_{x}_{=−0}, and

*q*(

*x*→ ±∞) = 0 [26]. For the TM polarized waves, the boundary conditions for the transverse magnetic field can be expressed in a similar fashion. Hence, we can deal with the TM polarized surface waves with the similar approach. Following the experimental instance [26], the boundary conditions for the nonlinear perturbation of refractive index are assumed as Δ

*n*(

*x*→ −∞) = 0, and

*∂*Δ

*n/∂x|*

_{x}_{=0}= 0.

In most experiments, we have
${k}_{0}^{2}{w}_{a}^{2}\gg 1$ for any paraxial beams. Therefore, *n _{d}* ≫ 1 can be easily satisfied in an actual physical system. For example,

*n*≈ 280 when

_{d}*n*

_{0}= 1.80,

*n*= 1.79,

_{L}*w*= 10

_{a}*μ*m, and

*λ*= 0.5

*μ*m. References [26–28] indicate if the index difference

*n*is big enough, namely

_{d}*n*≫ 1, the optical energy is almost totally confined in the nonlocal medium. Consequently one can approximately get the relation

_{d}*q*(−0) =

*q*(+0) = 0 and

*q*(

*x*> 0) = 0. Under this approximation, the propagation of surface waves can be solved totally only based on Eqs. (4) and (5) with the boundary conditions (

*x*≤ 0),

We find the solution of a surface soliton under the approximation *q*(0) = 0 is identical with the half part of an antisymmetric soliton in a bulk medium as shown in Fig. 1(b). For the bulk soliton solutions with antisymmetric amplitude distribution which is also governed by Eqs. (4) and (5), one can easily obtain the following relations.

*x*≤ 0) of the antisymmetric bulk solitons. Therefore, a surface soliton can be regarded as a half of a bulk soliton, and all the results of the antisymmetric bulk soliton can be transferred to the surface soliton. On the other hand, the right half of the bulk soliton can be regarded as an image beam of the surface soliton [37]. The interaction between the surface soliton and the interface can be regarded as the interaction between the soliton and its image beam in bulk medium.

In nonlocal bulk media, solitons can have the antisymmetric amplitude distribution [13–15]. Some authors have demonstrated that Hermite-Gaussian function can be applied to describe this type soliton in nonlinear media with several different nonlocal responses, especially for the strongly nonlocal case [15–17]. In addition, it has been discovered that in a nonlinear material with a finite-range nonlocal or a very long-range nonlocal response, the maximal number of peaks in stable multipole bulk solitons is four, and all higher-order soliton bound states are unstable [13, 14]. Analogously, surface solitons only with less than three poles can be stable in the case of thermally nonlocal interface, which is firstly addressed by Kartashov *et al* [27].

In the following, we give some comparisons between bulk solitons and surface solitons to illustrate our conclusions, and some analytical results are given for nonlocal surface solitons.

## 3. Surface solitons and breathers

In strongly nonlocal nonlinear media, there exist Hermite-Gaussian solitons or breathers [15–17]. Because the first-order Hermite-Gaussian beam is antisymmetric about the beam center, it can be used to describe the fundamental nonlocal surface soliton. In bulk nonlocal media, the first-order Hermite-Gaussian trial beams can be expressed as

*a*(

*z*),

*w*(

*z*),

*c*(

*z*),

*θ*(

*z*) represent the amplitude, beam width, phase-front curvature and phase of the beams, respectively. The input power is

*P*

_{0}= ∫|

*q*|

^{2}

*dx*and

*I*= |

*q*|

^{2}is the normalized optical intensity.

In the following, Eqs. (4) and (5) are used for the equivalent bulk case (−∞ < *x* < ∞). Then Eqs. (4) and (5) can be restated as an Euler-Lagrange equation corresponding to a variational problem [16, 38, 39]

*L*is given by

*R*(

*x*) = (1/2

*w*) exp(−|

_{m}*x*|/

*w*). Because the complexity of integration produced by

_{m}*R*(

*x*), the expression of [

*L*] is too difficult to obtain. But for the strongly nonlocal case, i.e.

*w*≫ 1,

_{m}/w*R*(

*x*) can be expanded as

Combining Eq. (9) and Eq. (11), [*L*] can be obtained

#### 3.1. Surface solitons

For the case of solitons, in Eq. (9), *a* and *w* reduce to constants, and *c* = 0, *θ*(*z*) = *βz*, where *β* is the propagation constant. The expression of the trial beams can be rewritten in a simple form

*w*

_{0}is the beam width. One can define

*α*=

*w*

_{m}/w_{0}as the degree of nonlocality.

Based on Eq. (15), we can get the input power, namely the critical power of bulk solitons

and the propagation constantAccording to our conclusion in Sec. 2, the surface soliton can be expressed as

*P*=

_{s}*P*/2. When the optical beam propagates as a surface soliton, we define the soliton position

_{c}*x*as the position where the beam has its maximal intensity, i.e. the distance between the maximal intensity and the interface. According to Eq. (19), the soliton position

_{s}*x*can be obtained as

_{s}Based on Eqs. (4)–(6), some numerical simulations of surface and bulk solitons are carried out by using Eqs. (16) and (19) as incident profiles. In Fig. 2, the propagations using the analytical incident profiles illuminate our analytical results are in good agreement with the actual soliton solutions in strongly nonlocal media. To confirm the stability of surface solitons, we simulate the propagation of surface solitons in the presence of 3% white noise (see Fig. 3). Figure 3 shows the propagations over a very long distance are stable, which is analogous to the stability of fundamental surface solitons in thermal nonlinear medium [27].

In general nonlocal media, the profiles of surface soliton are obtained by numerical iterative method based on Eqs. (4)–(6), and the relations between critical powers and beam width are shown in Fig. 4. For Figs. 4(a) and 4(b) which belong to the strongly nonlocal case, the analytical results are in excellent agreement with the numerical results. As the degree of nonlocality decreases to *α* = 6 in Fig. 4(c), the analytical results are also accordant with the numerical results approximately. For Fig. 4(d), the analytical results begin to deviate from the exact numerical ones with the beam width increasing. Figure 4 confirms that the analytical solutions of surface solitons are valid for the strongly nonlocal case. The relations between critical powers and propagation constants are shown in Fig. 5. It shows that the analytical results from Eqs. (17) and (18) are in good agreement with the numerical results except for small *β* which corresponds to a weak nonlocality.

For the weakly nonlocal case, the analytical solutions can not be obtained because the analytical solutions for multipole bulk solitons can not be found in weakly nonlocal medium. However, in nonlocal bulk media, the diploe solitons are always existent and stable in the entire domain of their existence [13]. According to the theory in Sec. 2, the fundamental surface solitons should be existent for the weakly nonlocal case. Following the method in Ref. [13], we seek the dipole bulk soliton combining two out-of-phase fundamental solitons under weakly nonlocality. As a result, the surface solitons can be also found at the same time, as shown in Fig. 6.

#### 3.2. Surface breathers

Because the breathers exist in nonlocal bulk media, if the input power is not equal to the soliton power, we can find the surface breathers. According to Eq. (15), we can get

*c*

_{0}is a constant. Through expanding the equivalent potential

*V*(

*w*) into the second order at the balance position, the motion of particle is approximated to a harmonic oscillation, and we can obtain the approximate expression of the beam width

According to our theory, the oscillation period of surface breathers is the same as that of bulk breathers. Then, we can obtain the oscillation period of the surface breather,

*P*

_{s}_{0}is the input power of the surface breathers. At the same time, the beam trajectory can be obtained and given by ${x}_{b}=-\sqrt{2}w\left(z\right)/2$. Figure 7 gives the comparison between the surface breathers and the bulk breathers for

*α*= 10. The analytical trajectories are in good agreement with the simulated results.

## 4. Surface solitons launched away from the stationary position

One can learn from Ref. [26] that if the beam launched away from the soliton position, the beam will propagate oscillating about the stationary position. The oscillation induced by the interaction between the soliton and the interface is periodic, and the beam never converges to a straight line trajectory. According to our theory and introducing an out-of-phase image beam [37], the interaction between soliton and interface can be regarded as the interaction between the soliton and its image beam in nonlocal bulk media. Then the oscillating trajectory of surface solitons can be obtained from the trajectories of two interacting out-of-phase solitons in nonlocal bulk media.

For the strongly nonlocal case, the fields of two out-of-phase Gaussian solitons propagating in bulk media can be expressed as

*θ*(

*z*) is the phase of whole beams, ±

*x*(

_{c}*z*) are the mass centers of Gaussian solitons and

*x*(0) =

_{c}*x*

_{c}_{0}.

*u*(

*z*) is the tilting wavefront induced by the attraction between two solitons, and we consider the normally incident case, i.e.

*u*(0) = 0. The amplitude ${a}_{0}^{2}=32\sqrt{2}{w}_{m}^{5}/{w}_{0}^{4}\left[4{w}_{m}\left({w}_{0}^{2}+2{w}_{m}^{2}\right)-\sqrt{\pi}{w}_{0}\left({w}_{0}^{2}+4{w}_{m}^{2}\right)\right]$ is normalized to guarantee that the incident power of each soliton is equal to the critical power, so we can assume that the beam width does not change during propagation.

If the two solitons are separated over 3*w*_{0} each other, their fields almost do not overlap. However because of strong nonlocality, there still exists attractive force between them. The attractive force exerted on one soliton is produced by the other soliton, specifically, by the nonlinear refractive index change induced by the other soliton. Therefore, the trajectory of one soliton (i.e. the left one *q*_{+}) can be determined by the light ray equation,

*n*

_{−}is only induced by the right soliton

*q*

_{−}, as

*R*(

*x*) = (1/2

*w*) exp(−|

_{m}*x*|/

*w*) and Eqs. (25)–(27), one can obtain

_{m}*w*≫ {

_{m}*w*

_{0},

*x*

_{c}_{0}}, we can get a simply approximate expression

*n*′ shown in Fig. 8(e) represents the attractive force between two solitons. Due to the complicated form of Δ

*n*′, Eqs. (28) and (29) are solved numerically using Runge-Kutta methods. The soliton trajectories and the oscillation period are shown in Figs. 8(c) and 8(d). The trajectory of the surface soliton corresponds to the left half of trajectories of two solitons in bulk media, i.e.

*x*= −|

_{sc}*x*|.

_{c}The numerical simulations based on Eqs. (4)–(6) for surface soliton launched away from stationary position and for two ut-of-phase bulk solitons are also shown in Figs. 8(a) and 8(b). The trajectories and the oscillation periods gotten from numerical simulations are compared with that from Eqs. (28) and (29) in Figs. 8(c) and 8(d). These results are approximately coincident with each other. The main reason of disagreement is that Eq. (26) is valid when two beam fields do not overlap. When *x*_{c0} is small, the fields of two beams overlap and we can not distinguish Δ*n*_{+} and Δ*n*_{−}. Then Eq. (26) is invalid, and the simulated results disagree with that obtained by solving Eqs.(28) and (29). When *x*_{c0} is large and the nonlocality is strong enough, the simulated results are in good agreement with the results by numerically solving Eqs. (28) and (29).

## 5. Conclusion

In conclusion, we have studied the nonlocal surface waves numerically and analytically. We find that a surface soliton in nonlocal nonlinear media can be regarded as a half of a bulk soliton with an antisymmetric amplitude distribution. By applying the variational method and taking the first-order Hermite-Gaussian beam as an example, the analytical solutions for the surface solitons and breathers in strongly nonlocal media are obtained, and the critical power and breather period are gotten analytically and confirmed by numerical simulations. In addition, the oscillating propagation of nonlocal surface soliton launched away from the stationary position is considered as the interaction between the soliton and its out-of-phase image beam. We have discussed the oscillation period and the beam trajectory. Its trajectory and oscillating period obtained by our model are in good agreement with the numerical simulations.

## Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant Nos. 10804033 and 11074065), the Program for Innovative Research Team of Higher Education in Guangdong (Grant No. 06CXTD005), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 200805740002), and the Natural Science Foundation of Hebei Province (Grant No. F2009000321).

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