Servando Lopez-Aguayo and Julio C. Gutierrez-Vega- Elliptically modulated self-trapped singular...

8/3/2019 Servando Lopez-Aguayo and Julio C. Gutierrez-Vega- Elliptically modulated self-trapped singular beams in nonlocal n

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Elliptically modulated self-trapped

singular beams in nonlocal nonlinear

media: ellipticons

Servando Lopez-Aguayo and Julio C. Guti errez-Vega

Photonics and Mathematical Optics Group, Tecnologico de Monterrey, Monterrey, Mexico

64849

[email protected]

Abstract: We introduce a new class of elliptically modulated self-trapped

singular beams in isotropic nonlinear media where nonlocality plays a

crucial role in their existence. The analytical expressions in the highly

nonlocal nonlinear limit of these elliptically shaped self-trapped beams,

or ellipticons, is obtained and their existence in more general nonlocal

nonlinear media is demonstrated. We show that the ellipticons represent

a generalization of several known self-trapped beams, for example vortex

solitons, azimuthons, and the Hermite and Laguerre solitons clusters. For

the limit of the highly nonlocal nonlinear medium, the ellipticons are

described in close form in terms of the InceGauss functions.

2007 Optical Society of America

OCIS codes: (190.4420) Nonlinear optics, transverse effects in; (190.5940) Self-action effects;

(350.5030) Phase

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#89346 - $15.00 USD Received 2 Nov 2007; revised 14 Dec 2007; accepted 16 Dec 2007; published 20 Dec 2007

(C) 2007 OSA 24 December 2007 / Vol. 15, No. 26 / OPTICS EXPRESS 18326


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1. Introduction

Nonlocality has been a phenomenon of intense research over the last years in various nonlin-ear physical systems [1]. Basically, nonlocality extends the effects of localized excitations in a

medium, allowing a certain degree of interconnection among several regions of the medium in

question [2]. Nonlocality can be generated by different mechanisms, such as transport processes

in charge carriers [3], many-body interactions in Bose Einstein condensates or matter waves

[4], and long-range forces in liquid crystals [5]. Particularly in nonlinear optics, the nonlocal-




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ity allows that the refractive index of a material in a particular point can be related with the

beams intensity in all the others material points. Stronger relation means a stronger degree of

nonlocality.

Specifically, it has been shown that nonlocality can provide new physical effects in the field

of nonlinear optics. For example, nonlocality suppresses beam collapse [6], it allows attraction

between dark solitons [7] and it can also help to stabilize differentkind of self-trappednonlinear

beams or spatial solitons- that are known to be unstable in pure local media,like vortex solitons

[8] and rotating dipoles solitons [9]. Some examples of experiments in nonlocal nonlinearmedia

where optical spatial solitons have been already observed include lead glasses exhibiting self-

focusing thermal nonlinearity [10], photorefractive media [11] and nematic liquid crystals [12].

In a previous work [13], it was shown that nonlocality can completely change, in comparison to

a pure local and isotropic medium, the domain of existence of the recently introduced spatially

modulated vortex solitons, i.e. azimuthons [14].

In this paper, we demonstrate the existence of a novel class of elliptical self-trapped beams

in media where nonlocality plays a crucial role, i.e. these stationary beams cannot exist in pure

local and isotropic nonlinear media. These solitons have an inherent elliptical structure, and for

this reason we call them ellipticons. The ellipticons can also be seen as a generalization of a

wide diversity of self-trapped structures in nonlocal nonlinear media, for example, adjusting the

mode indices and an ellipticity parameter, it is possible to produce vortex solitons of different

topological charges and number of rings, dipole or m-pole solitons, some particular cases of

azimuthons, and even the recently introduced Laguerre and Hermite soliton clusters [15]. For

the limit of the highly nonlocal nonlinear (HNN) medium, the ellipticons can be described in

close form in terms of the recently studied InceGauss modes of the paraxial wave equation

[16, 17, 18, 19].

In Sect. 2, we introduce the physical model for the propagation of paraxial beams in nonlocal

nonlinear media, and in Sect. 3 we obtain its elliptically solution in the case of the HNN limit.

Then in Sect. 4 using a variational approach, we show the existence of these ellipticons in more

general nonlocal nonlinear media and we discuss some of the particular characteristics of these

beams outside from the HNN limit. Finally, in Sect. 5 we show that the ellipticons can be an

useful approach to explore the revival phenomena presented by some beams in highly nonlocal

nonlinear media.

2. Nonlocal nonlinear media

We begin the analysis from the nonlinear Schrodinger equation [1], that describes the complex

amplitude E(r,z) of a paraxial beam propagating along the z axis

2ikn0zE(r,z) + n02E(r,z) + 2k

2n(I,z)E(r,z) = 0, (1)

where r = (x,y) = (r,) represents the transverse coordinates, 2 stands for the transverseLaplacian, k is the wave vector in a linear medium, and n 0 is the linear part of the refractive

index. We assume that the nonlinear refractive index n depends on the intensity I = |E(r,z)| 2by the following nonlocal nonlinear relation:

n(I,z) = Rr r

E(r,z)

2dr, (2)

where the response function R(r) is determined by the specific physical process responsible forthe medium nonlocality. While the response function is symmetric, real, positive, definite, and

monotonically decaying, it has been shown that the physical properties do not depend strongly

on the its shape [20]. Throughout this paper, we consider the standard Gaussian nonlocal re-




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sponse function of the form [9, 20, 21]

R(r) =1

2exp

r

2

2

, (3)

where the width parameter determines the degree of nonlocality. In the limit

0 we

recover the Kerr medium, while with we have the highly nonlocal nonlinear (HNN)limit. In the highly nonlocal case, the length of the beam is very narrow in comparison to the

length of response function. Expanding an arbitrary R(r r ) function respect to r aroundr = r, we get

R(r r) = R0 + (r r) R0 + 1/2(r r)

2R0 + ..., (4)

In the particular case of Eq. (3), the symmetry respect to r = 0 cancels the second term ofthe expansion. Considering only solutions whose center of mass is located at r = 0 during theirpropagation, and taking the expansion of the response function until the r 2 term, we get the

typical approximation of the nonlinear refractive index [21]

n(I,z)

P0

2 1r2

2 , (5)where P0 =

|E(r,z)|2 dr is the constant beam power. Using the change of variable E(r,z) =U(r,z) exp(ikP0z/n0

2) and Eq. (5) in Eq. (1), we finally obtain the propagation equation forthe HNN limit [2, 22, 23]

2ikzU(r,z) +2U(r,z) k2a2r2U(r,z) = 0, (6)

where a2 = P0/n0k24.

It has been shown that a pure local isotropic nonlinear medium only allows the formation of

soliton structures with a circular symmetry, such as the fundamental and circular vortex solitons

[24]. Hence, elliptically shaped solitons cannot exist in pure isotropic nonlinear local media.

The latter because the anisotropicdiffraction experimented by an elliptically shaped beam

cannot be balanced by a pure spatially isotropic local nonlinearity. Nevertheless, in the HNNlimit of Eq. (1), from group theory it has also been shown that is possible to obtain stationary

solutions with an elliptical symmetry [25]. The following questions naturally arises then: is

it possible to propagate an elliptically shaped soliton in general isotropic nonlocal nonlinear

media? If so, is this class of soliton stable?

In this work we demonstrate that it is indeed possible to get solitons with an elliptical sym-

metry in general nonlocal nonlinear media, as a certain degree of nonlocality is reached. In

a similar way, we find that it is possible to have stable propagation when a higher degree of

nonlocality is achieved. Recently, Buccoliero et al. [26], have used the generalized Hermite-

Laguerre-Gaussian modes [27] to describe another different complete set of soliton modes in

nonlocal nonlinear media, these modes can also be seen as a link between the Hermite and

Laguerre soliton clusters; however, our ellipticon approach is unique in the sense that it gives a

soliton solution in an appropriate elliptical coordinate system.

3. Ellipticons in highly nonlocal nonlinear media

3.1. Stationary accessible solitons

To get a physical insight into the ellipticons, we begin by discussing their properties in the

HNN limit. Adopting the terminology introduced by Snyder and Mitchell [2], we will denote




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the ellipticons propagating in HNN media as accessible ellipticons, i.e. soliton solutions of

Eq. (6) in elliptic coordinates.

Equation (6) is recognized to be the same as the equation that describes the propagation in a

gradedindex (GRIN) medium whose refractive index varies radially as n (r) = n 0

1a2r2/2.Because the physics of this problem is well understood, it is easy to translate it into the context

of soliton propagation. The accessible soliton solutions of Eq. (6) in elliptic coordinates have

the general form U(r,z) = mp ( ,) exp(iz) and are given by the InceGaussian modes oforder p, degree m, and ellipticity parameter , namely [16, 17, 18, 19]

p,m( ,) =

C Cmp (i ;)C

mp ( ;) iS Smp (i ;)Smp ( ;)

exp

akr

2

2

, (7)

= (p + 1)a, (8)

where Cmp (;) and Smp (;) are the even and odd Ince polynomials of order p and degree

m respectively [16, 28, 29], C and S are normalization constants, and the ellipticity para-

meter is given by = ak f2. The elliptic coordinates are defined by x = fcosh cos, andy = fsinh sin , where [0,) is the radial coordinate and [0,2) is the angular co-ordinate. Elliptic coordinates are dimensionless, and semi-focal parameter f has the dimension

of length.

The transverse distribution of the accessible ellipticons is described by three parameters,

namely, the ellipticity 0


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Fig. 1. (1.3 Mb) (a) Intensity and phase of a stationary accessible ellipticon given by

+5,3( ,) for = 0,1, and . (b) Phase structure around the interfocal line. (c) Inten-

sity and phase distribution of a stationary accessible ellipticon with parameters p = 5 andm = 5 in HNN media. The intensity pattern remain invariant in propagation while the phasefront rotate around the interfocal line.

For m 1, the accessible ellipticon carries an intrinsic orbital angular momentum (OAM)that has a nonlinear dependence with the ellipticity parameter. Within the paraxial regime, the

z component of the OAM per photon in unit length about the origin of a transverse slice of a

beam U(r,z) is given by

Jz = h

r

Im(UU) dxdy

|U|2 dxdy ,(10)

where r = xx +yy is the transverse radius vector. To determine Jz, we evaluated numericallyEq. (10) using a two-dimensional Gauss-Legendre quadrature for a number of combinations of

mode indices (p,m) within the interval [0,60]. The numerical analysis corroborated that theOAM carried by the ellipticon exhibits a nonlinear dependence on the ellipticity parameter, see

Fig. 2(b).

3.2. Rotating accessible ellipticons

Following the procedure outlined by Bekshaev and Soskin [33] to construct spiral beams in

circular symmetries, it is possible to construct accessible ellipticons in HNN media with rotat-

ing intensity on propagation. Consider the field resulting from the following superposition of

fundamental ellipticons:

( ,) = A1+p1,m1

( ,) +A2+p2,m2

( ,), (11)

where A1 and A2 are weight coefficients, and for simplicity, we restrict ourselves to consider

the superposition of only two accessible ellipticons with the same power. As soon as we move


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Fig. 2. (a) Nodal lines for the pure even and the pure odd part of +5,3( ,) for = 1 and respectively. Red and blue lines represent nodal lines of the even and odd components,

respectively. Positive and negative vortices are represented by white and black circles, re-

spectively. (b) Orbital angular momentum carried by +p,m in function of for p = 2n + mand m = {1,2,...,7}.

out of the HNN limit, the nonlinear effect will invalidate the superposition principle; hence it

is expected to observe several changes in the rotating ellipticons in general nonlocal nonlinear

media respect to the HNN limit case. For example, the parameter a will depend of the particularvalues of the weight coefficients and even more, at a low degree of nonlocality and also due

to the nonlinear isotropic nature of the medium, we expect that the rotating ellipticons become

highly unstable. Even there is the possibility that these beams cannot exist under that condition.

There are some important differences of the combination of accessible ellipticons respect

to the spiral beams or also with respect to the so-called nondiffracting beams in linear media

[34]. For example, unlike the beams presented in Refs. [33, 35], the HNN limit provides the

opportunity to produce beams (described by an exact analytical and closed expression) with

two important conditions: finite energy and invariance in the transverse scale upon propagation.

This happens by virtue of the balance among the beam diffraction, the self-focusing effect, and

the nonlocality.

Depending on the combination of indices (p 1,m1,p2,m2), the rotating ellipticons in HNNmedia can be classified in four classes according to their elliptic intensity and phase rotation

(see Fig. 3)

1. When m2 = m1 the field ( ,) does not exhibit rotation or even stationary behavior,but because there is a periodic dependence of the intensity pattern with the longitudi-

nal coordinate z. We illustrate this case in Fig. 3(a). This case presents a self-imaging

phenomenon.

2. When the condition p2 = p1 is presented, the field ( ,) shows an invariant intensitypropagation as can be seen in Fig. 3(b). This happens because the dependence of the

intensity beam on z have been canceled; however, the phase front of the beam exhibits an

elliptical rotation. In this particular case (and with the same power for the both beams),

the HNN limit allows to obtain new solitons where the relation of P0 against is givenagain by Eq. (9).

3. For the condition (p2p1)/(m2m1) < 0, the intensity and the phase of( ,) rotateelliptically in opposite directions. Figure 3(c) shows this case.

4. For the condition (p2 p1)/(m2 m1) > 0 the intensity and the phase ( ,) rotateelliptically in the same direction. Figure 3(d) shows this case.




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Fig. 3. Different scenarios for the propagation of accessible rotating ellipticons given by

the combination of two stationary accessible ellipticons: (a) (1.4 Mb) self-imaging phe-

nomenon in the case of p1 = 6,m1 = 2,p2 = 2, and m2 = 2, (b) (1.3 Mb) stationary be-

havior; p1 = 3,m1 = 1,p2 = 3, and m2 = 3, (c) (1.5 Mb) rotation of the intensity patternand the phase front in opposite directions, here p1 = 8,m1 = 4,p2 = 6,and m2 = 6, and(d) (1.5 Mb) rotation of the intensity pattern and the phase front in the same direction;

p1 = 8,m1 = 4,p2 = 10, and m2 = 10. In all the cases = 1 and L = 2/a.

In these four scenarios there is the possibility to adjust the path of circulation of the rotation of

the intensity by changing the ellipticity parameter of the constituent ellipticons, so it is possible

to propagate a wide variety of self-trapped beams with different symmetries imposed in their

intensity rotation: from self-trapped beams whose intensity rotates in a circular way, passing

through self-trapped beams that have an arbitrary elliptic path of circulation until the possibility

of having a square-like path of circulation of the intensity pattern.

As a generalized solution of self-trapped beams, the ellipticons allow us to obtain a large

variety of different classes of beams. For example, we can recover the spiral beams in the limit

0. In this case, a combination of accessible ellipticons will reduce to a spiral beam if anypair of its members with sets of indices (p,m) and (p ,m) fulfill the relation

pp = (mm), (12)where should be a real constant. The intensity pattern (in the case of 0) rotates with aconstant angular velocity given by

= a. (13)

Unlike to others results presented in Refs. [33, 34, 35, 36], where the angular velocity decreases

on propagation, the angular velocity of the ellipticons remains constant. This difference is be-

cause ellipticons remain self-trapped, while the other beams diffract, leading to a decreasing

rotation rate on propagation [33, 37].

4. Ellipticons in nonlocal nonlinear media

Ellipticons given by Eq. (7) are valid only in the HNN limit. It is natural to ask then if it is

possible to have ellipticons in nonlocal nonlinear media at different degrees of nonlocality.

To strictly show the existence of ellipticons in a general nonlocal nonlinear media, we should

find stationary solutions of Eq. (1) in elliptical cylindrical coordinates, which seems to be a


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Fig. 4. Propagation dynamics of an elliptcon with parameters p = 4, m = 4, = 1, andP0 = 10

3 in an xy box of 2.6 2.6 in a nonlocal nonlinear medium. (a) Using directlythe accessible ellipticon solution the beam diffracts and the maximum normalized intensity

decays [see (b)]. (c) Using the same trial function but modified with the variational ap-

proach (A =1.6066) the beam remain self-trapped and the maximum normalized intensityoscillates remaining within a finite and small range [see (d)].

cumbersome task (for both analytical and numerical methods). Fortunately, this can be per-

formed by applying a variational approach [38]. The use of the beam amplitude and the beam

width as standard variational parameters does not allow to perform all the integrations needed

for the general elliptical case. Nevertheless, it is yet possible to use the ellipticon solution in a

first variational approach by using just the beam amplitude as a single variational parameter. It

can be shown that Eq. (1) can be derived from the Lagragian density given by

L =2kn0|E(r)|2n0 |E(r)|2 + k2 |E(r)|2

2

R(r r) E(r)2 dr, (14)

where= kP0/n02(p+ 1)a. InsertingE=Apm( ,), our ellipticon solution in the HNN

limit, as a trial function into the Lagragian L =Ldr, we obtain the effective Lagrangian

depending only of the parameter A. Finally, finding the value of A from the Euler-Lagrange




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Fig. 5. Intensity and phase of two ellipticons in nonlocal nonlinear media with parameters

(a) (2 Mb) p = 2, m = 2, = 0.75,and P0 = 103 in an xy box of 4.2 4.2, and (b) (2.1 Mb)

p = 3, m = 3, = 1, and P0 = 103 in an xy box of 4.8 4.8.

equations, we arrive to the following expression for the amplitude parameter

A =P20 (p + 1)

n0P

3/20 /k

2 + (2n0/2k2)p,m( ,)2 drp,m( ,)2exp(|r r|2 /2) p,m( , )2 drdr . (15)

We found that under a highly degree of nonlocality, the analytical solutions given by U =A

pm( ,) exp(iz) remain self-trapped during propagation in nonlocal nonlinear media given

described by Eq. (1). It is important to remark that in our model of Eq. (1), we have related the

degree of nonlocality of Eq. (3) in a direct way with the power of the beam and hence P 0 corresponds effectively to the HNN limit.

From our simulations, we have observed that using the variational approach is indeed useful

to find ellipticons in a medium with an high nonlocality. We have corroborated the validity ofour method from values as low as P0 = 103, where similar values of power have been used

before to represent highly nonlocal nonlinear media [39]. As it is expected, when the model

described by Eq. (1) moves towards a lower degree of nonlocality, the variational approach is

not longer valid and hence the elliptic structures do not remain self-trapped anymore. In Fig

4, we show two examples of propagation in a HNN media using directly Eq. (1). In Fig. 4(a)

we apply the accessible ellipticon solution [Eq. (7)] and observe that the beam diffracts and the

maximum normalized intensity decays, see subplot 4(b). In Fig. 4(c) we use the same accessible

ellipticon but now modified by the variational approach. Observe that the beam remain self-

trapped and the maximum normalized intensity oscillates remaining within a finite and small

range, see Fig. 4(d). An important contribution of this work is to show that stable ellipticons

can be obtained outside of the HNN limit.

We present now some examples of propagation of ellipitcons in nonlocal nonlinear media

using directly Eq. (1). For simplicity purposes, in the simulations here presented we have prop-agated ellipticons whose shape is given by a single elliptical ring and hence we have selected

two ellipticons with parameters m = 2, p = 2, and m = 3 and p = 3. We have explored severaldegrees of nonlocality starting from P0 = 10

6 and ending with P0 = 103. In Fig. 4 we show

the intensity and phase distributions of both examples of ellipticons for P0 = 103. As a result

of moving out from the HNN limit, all the propagations of the ellipticons here simulated are


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Fig. 6. (a) Propagation in nonlocal nonlinear media of a vortex soliton of single charge and

double ring (or soliton Laguerre mode L11). (b) Different modes that also coexist in the

propagation before mentioned.

characterized by a rotation of phase, as similar as occurs in the HNN limit case, but now there

is also a rotation in the pattern of intensity. This rotation can be explained due to the particlelike interaction of the beamss modulation [14]. We believe that the rotation observed in the

intensity pattern of the ellipticons can be also seen as a necessary and natural condition to help

to stabilize them, because as it was pointed out before, elliptically shaped self-trapped station-

ary beams cannot exist in a pure local isotropic media, and hence the rotation in the ellipticons

create an average of the anisotropic distribution of the beam intensity that becomes more neces-

sary conform our degree of nonlocality moves closer to the pure local limit case. The existence

of stationary ellipticons in a medium with an arbitrary degree of nonlocality still remains as an

open problem.

Even thought that we cannot claim an accurate stability analysis from our pure numerical

simulations, we do observe in the ellipticon dynamics that for an enough high degree of non-

locality, all the ellipticons are stable during their propagation, as similar to other self-trapped

structures such as soliton vortices [8], rotating dipoles [9], and azimuthons in nonlocal nonlin-

ear media [13]. Therefore we expect to observe these elliptical structures propagating in highlynonlocal nonlinear media in future experimental works.

5. Revivals of ellipticons

The recently introduced Laguerre and Hermite solitons clusters in nonlocal nonlinear media

[15] can experiment revivals [40] due to modulational instability. In this section we provide an

explanation of this effect using the ellipticons approach.

An Ince polynomial can be expressed as a finite series of Laguerre or Hermite functions and

vice versa [17]. Hence in the HNN limit, it is possible to extract the finite number of modes

of solitons that can coexist. For example, consider the Laguerre mode with a double ring and

a single topological charge, i.e. L1,1. It can be shown that this mode can be expressed by a

combination of two accessible ellipticons E3,1 and E3,3 [17]. Similarly, the Laguerre mode L 1,1can also be expressed as a superposition of two accessible Hermite solitons H21 and H12. In

other words, in the Laguerre mode L 1,1 also coexist other four fundamental modes: two based

in our ellipticon approachand other two based on the Hermite functions; in Fig. 6 we show these

modes. Modulational instability leads to a periodic readjust of energy between all the coexisting

modes and their ellipticity, and hence the revival phenomenon case can observed, where in this

particular case two modes that correspond to the accessible ellipticon mode E3,1 and to the




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Fig. 7. (a) (1.4 Mb) Elliptically intensityrotating beam with = 0 close to the HNN limit(P0 = 10

6) in a xy box of 0.8 0.8 produced by two ellipticons with parameters p1 =10,m1 = 10,p2 = 2, and m2 = 2. (b) (1.4 Mb) Intensity rotating beam produced by threeellipticons with parameters p1 = 5,m1 = 5,p2 = 5,m2 = 3,p3 = 5, and m3 = 1 close to theHNN limit (P0 = 10

6) in a xy box of 0.8 0.8, here = 1, for all the three beams.

Hermite soliton mode H21 (or equivalently to the mode H12), predominate periodically in the

propagation.

Even though this mode decomposition is strictly only for the HNN limit, it can be also useful

to explain qualitatively the revivals of the beams in nonlocal nonlinear media, due to the fact

that the revivals are produced in the condition of a high nonlocality. In Ref. [13], we propagated

an azimuthon with two peaks along the ring in its intensity distribution and a topological charge

of three, but interestingly, in its dynamics the original vortex configuration quickly splits into

three elementary vortices, forming then a kind of elliptically ring in the intensity distribution

(as similar as the shown in Fig. 5 (b). We can claim now that this azimuthon decayed into an

ellipticon, which in this particular case seems to be a more stable configuration due to the ini-

tially no azimuthally symmetric distribution of the intensity. This transition from an azimuthon

into an ellipticon is more evident if we observe the hyperbolic lines in the beam phase structure

and also the elliptical structure produced in the intensity pattern, being these symmetries in the

distributions of intensity and phase, an important feature of the ellipticons.

We have also propagated more complex ellipticons in nonlocal nonlinear media, produced

by the anstaz resulting of the combination of two accessible ellipticons in the HNN limit. As

expected, they are more unstable in comparison with a pure ellipticon mode; however, with

an enough high degree of nonlocality, it is indeed possible to stabilize also all these kind of

ellipticons, allowing then the possibility to observe very interesting vortex dynamics, at leastclose to the region given by the HNN limit. As an example, in Fig. 7 we observe a rotating

wheel of vortices using ellipticons with the limit 0. Another example shown in Fig. 7 is arotating ellipticon where we have used three beams (all the ellipticons that have p=5) to produce

the structure.

6. Conclusions

We have demonstrated the existence of a novel class of elliptically shaped solitons that has an

analytical and closed expression in the HNN limit, and also can be propagated in isotropic non-

linear media with an enough degree of nonlocality. These elliptically modulated self-trapped

singular beams, or ellipticons, can be considered as a natural link between the recently intro-

duced Laguerre and Hermite soliton clusters, and besides they can be related to the many other

soliton beams like vortex solitons and the recently introduced azimuthons. Indeed, we have

found that when certain initial symmetry is imposed in the beams, azimuthons can even de-

cay into an elliptical symmetry and hence be transformed in their propagation in an ellipticon.

We have also given an explanation of the revivals found in the propagation of other several

self-trapped structures like the azimuthons and vortex solitons, all this based in the ellipticon

approach. Finally, we believe that ellipticons can be useful to explain many more soliton phe-


http://www.opticsexpress.org/viewmedia.cfm?URI=oe-15-26-18326-11http://www.opticsexpress.org/viewmedia.cfm?URI=oe-15-26-18326-9


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nomena in nonlocal nonlinear media.

Acknowledgments

The authors thank Anton Desyatnikov and Yuri S. Kivshar for useful comments. This research

was supported by Consejo Nacional de Ciencia y Tecnologa Mexico Grant No. 42808 and by

Tecnologico de Monterrey Grant No. CAT007.



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