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
Nonlocality has been a phenomenon of intense research over the last years in various nonlinear physical systems . 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 . Nonlocality can be generated by different mechanisms, such as transport processes in charge carriers , many-body interactions in Bose Einstein condensates or matter waves , and long-range forces in liquid crystals . Particularly in nonlinear optics, the nonlocality allows that the refractive index of a material in a particular point can be related with the beam’s 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 , it allows attraction between dark solitons  and it can also help to stabilize different kind of self-trapped nonlinear beams-or spatial solitons-that are known to be unstable in pure local media, like vortex solitons  and rotating dipoles solitons . Some examples of experiments in nonlocal nonlinear media where optical spatial solitons have been already observed include lead glasses exhibiting selffocusing thermal nonlinearity , photorefractive media  and nematic liquid crystals . In a previous work , 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 .
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 . For the limit of the highly nonlocal nonlinear (HNN) medium, the ellipticons can be described in close form in terms of the recently studied Ince-Gauss 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 Schrödinger equation , that describes the complex amplitude E(r, z) of a paraxial beam propagating along the z axis
where r=(x,y)=(r,θ) represents the transverse coordinates, ∇2 stands for the transverse Laplacian, 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)|2 by the following nonlocal nonlinear relation:
where the response function R(r) is determined by the specific physical process responsible for the 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 . Throughout this paper, we consider the standard Gaussian nonlocal response function of the form [9, 20, 21]
where the width parameter σ determines the degree of nonlocality. In the limit σ→0 we recover the Kerr medium, while with σ→∞ 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′ around r′=r, we get
In the particular case of Eq. (3), the symmetry respect to r=0 cancels the second term of the expansion. Considering only solutions whose center of mass is located at r=0 during their propagation, and taking the expansion of the response function until the r 2 term, we get the typical approximation of the nonlinear refractive index 
where P 0=∫∫||E(r, z)|2 dr is the constant beam power. Using the change of variable E(r, z)=U(r, z)exp(ikP0z/n0πσ2) and Eq. (5) in Eq. (1), we finally obtain the propagation equation for the HNN limit [2, 22, 23]
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 . Hence, elliptically shaped solitons cannot exist in pure isotropic nonlinear local media. The latter because the “anisotropic”diffraction experimented by an elliptically shaped beam cannot be balanced by a pure spatially isotropic local nonlinearity. Nevertheless, in the HNN limit of Eq. (1), from group theory it has also been shown that is possible to obtain stationary solutions with an elliptical symmetry . 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 symmetry 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. , have used the generalized Hermite-Laguerre-Gaussian modes  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 , we will denote 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 graded-index (GRIN) medium whose refractive index varies radially as n(r)=n 0(1-a 2 r 2/2). 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)=Ψm p(ξ,η)exp(iβz) and are given by the Ince-Gaussian modes of order p, degree m, and ellipticity parameter ε, namely [16, 17, 18, 19]
where Cmp (η;ε) and Smp (η;ε) 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 parameter is given by ε=akf2. The elliptic coordinates are defined by x=f coshξ cosη, and y=f sinhξ sinη, where ξ ∈ [0,∞) is the radial coordinate and η ∈ [0,2π) is the angular coordinate. 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≤ε<∞, and two integer indices: the order p and the degree m, where 0≤m≤p for even Ince polynomials, and 1≤m≤p for odd Ince polynomials . In both cases, the indices (p,m) must have the same parity, i.e. (-1)(p-m)=1 and only products of functions of the same parity in ξ and η satisfy continuity in the whole space [16, 17, 18, 19]. Ince-Gaussian modes have been generated in free-space from stable laser resonators  and liquid crystal displays .
Equation (7) not only describes accessible ellipticons in elliptic coordinates. If the ellipticity parameter goes to infinity then the ellipticon tends to the soliton solutions based on Hermite-Gaussian functions presented originally by Snyder and Mitchell using Cartesian coordinates , see Fig. 1(a). On the other hand, if ε→0, the elliptic coordinate system becomes a circular cylindrical system and then the ellipticon reduces to a circularly symmetrical soliton described in terms of Laguerre-Gaussian functions . Recently, these two limit solutions, called Hermite and Laguerre soliton clusters, were studied in general nonlocal nonlinear media by D. Buccoliero et al. . The expression for the accessible Hermite and Laguerre solitons in HNN media is directly obtained from Eq. (7) with the appropriate limits. Because the accessible ellipticons are stationary solutions of the NLSE in the HNN limit, the intensity pattern of these structures do remain invariant on propagation, however due to the peculiar elliptical phase distribution [see Fig. 1(b)], their phase-fronts rotate elliptically around the interfocal line (ξ=0) as shown in the animation included in Fig. 1(c).
It is important to remark that in the HNN limit, the parameters p, m and ε are independent of the beam power. The HNN limit also allows to obtain an analytical and close relation of P 0 against β given by
In the HNN limit, the ellipticons are characterized by N=1+(p-m)/2 confocal elliptic rings with foci at x=±f, whose eccentricity is controlled by the parameter ε . The vortices (i.e phase singularities) are located at the crosses of the zero-intensity lines of the real and imaginary parts of the field, see Fig. 2(a). The total number of vortices in an ellipticon is [2(p-m)+1]m, and the total topological charge is simply equals to degree m.
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
where r=xx̂+yŷ is the transverse radius vector. To determine Jz, we evaluated numerically Eq. (10) using a two-dimensional Gauss-Legendre quadrature for a number of combinations of mode indices (p,m) within the interval ε ∈ [0,60]. The numerical analysis corroborated that the OAM 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  to construct spiral beams in circular symmetries, it is possible to construct accessible ellipticons in HNN media with rotating intensity on propagation. Consider the field resulting from the following superposition of fundamental ellipticons:
where A 1 and A 2 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 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 particular values 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 . 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 (p1,m1, p2,m2), the rotating ellipticons in HNN media can be classified in four classes according to their elliptic intensity and phase rotation (see Fig. 3)
1. When m 2=m 1 the field Φ(ξ,η) does not exhibit rotation or even stationary behavior, but because there is a periodic dependence of the intensity pattern with the longitudinal coordinate z. We illustrate this case in Fig. 3(a). This case presents a self-imaging phenomenon.
2. When the condition p 2=p 1 is presented, the field Φ(ξ,η) shows an invariant intensity propagation 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 P 0 against β is given again by Eq. (9).
3. For the condition (p 2-p 1)/(m 2-m 1)<0, the intensity and the phase of Φ(ξ,η) rotate elliptically in opposite directions. Figure 3(c) shows this case.
4. For the condition (p 2-p 1)/(m 2-m 1)>0 the intensity and the phase Φ(ξ,η) rotate elliptically in the same direction. Figure 3(d) shows this case.
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 ε→0. In this case, a combination of accessible ellipticons will reduce to a spiral beam if any pair of its members with sets of indices (p,m) and (p′,m′) fulfill the relation
where τ should be a real constant. The intensity pattern (in the case of ε→0) rotates with a constant angular velocity given by
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 because 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 cumbersome task (for both analytical and numerical methods). Fortunately, this can be performed by applying a variational approach . 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
where Γ=kP0/n0πσ 2-(p+1)a. Inserting E=AΨpm(ξ,η), our ellipticon solution in the HNN limit, as a trial function into the Lagragian L=∫Ld r, we obtain the effective Lagrangian depending only of the parameter A. Finally, finding the value of A from the Euler-Lagrange equations, we arrive to the following expression for the amplitude parameter
We found that under a highly degree of nonlocality, the analytical solutions given by U=AΨpm(ξ,η)exp(iβz) 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 of our method from values as low as P 0=103, where similar values of power have been used before to represent highly nonlocal nonlinear media . 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 selftrapped 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 propagated 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 several degrees of nonlocality starting from P 0=106 and ending with P 0=103. In Fig. 4 we show the intensity and phase distributions of both examples of ellipticons for P 0=103. As a result of moving out from the HNN limit, all the propagations of the ellipticons here simulated are 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 particle like interaction of the beams’s modulation . 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 stationary 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 necessary 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 nonlocality, all the ellipticons are stable during their propagation, as similar to other self-trapped structures such as soliton vortices , rotating dipoles , and azimuthons in nonlocal nonlinear media . Therefore we expect to observe these elliptical structures propagating in highly nonlocal nonlinear media in future experimental works.
5. Revivals of ellipticons
The recently introduced Laguerre and Hermite solitons clusters in nonlocal nonlinear media  can experiment revivals  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 . 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. L 1,1. It can be shown that this mode can be expressed by a combination of two accessible ellipticons E 3,1 and E 3,3 . Similarly, the Laguerre mode L 1,1 can also be expressed as a superposition of two accessible Hermite solitons H 21 and H 12. In other words, in the Laguerre mode L 1,1 also coexist other four fundamental modes: two based in our ellipticon approach and 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 E 3,1 and to the Hermite soliton mode H 21 (or equivalently to the mode H 12), 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. , 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 initially 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 least close 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 ε→0. Another example shown in Fig. 7 is a rotating ellipticon where we have used three beams (all the ellipticons that have p=5) to produce the structure.
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 nonlinear 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 introduced 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 decay 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 phenomena in nonlocal nonlinear media.
The authors thank Anton Desyatnikov and Yuri S. Kivshar for useful comments. This research was supported by Consejo Nacional de Ciencia y Tecnología México Grant No. 42808 and by Tecnológico de Monterrey Grant No. CAT007.
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