## Abstract

We show that, in (2+1)-D case, both bright and dark solitons can exist in optical Kerr media when the optical beams are cylindrically symmetric and almost circularly polarized. We characterize the dependence of the properties associated with these solitons, such as their spatial width and intensity profiles, on their normalized intensity and the non-paraxial degree.

©2005 Optical Society of America

## 1. Introduction

Monochromatic optical propagation in a transparent medium with Kerr nonlinearity is usually described by the scalar parabolic equation, namely, nonlinear Shrödinger (NLS) equation [1]. It is derived from the Helmhotz equation based on the scalar and paraxial approximations, which is valid only if the typical transverse scale (such as the beam width *w*
_{0}) is much longer than the light wavelength *λ*. As the beam size decreases because of self-focusing, both of these approximations begin to break down [2]. Therefore, efforts should be taken to carry out more accurate models [3–5]. Fuente *et al*. [6] presented a vector model derived by appropriately simplifying the exact equations in the spatial frequency domain. Ciattoni *et al*. [7] derived a propagation equation that describes forward propagation of highly focused beams in Kerr media to all order non-paraxial corrections, preserving the fully vectorial nature of the light field. Ferrando *et al*. [8] demonstrated the existence of vortex soliton solutions in photonic crystal fibers in the non-paraxial regime. In all these models, the essential question is how to use complete Maxwell’s equations to estimate the importance of the effects not included within the scalar NLS equation.

Usually, the Maxwell’s equations have some z-independent envelope solutions. Each of them corresponds to a non-paraxial spatial soliton because they satisfy the definition of the non-paraxial solitons. Non-paraxial (1+1)-D bright and dark solitons have been proved to exist in Kerr media to the first significant order in *λ*/*w*
_{0} [9, 10]. The modulation instability and coherent interactions of optical beams in this regime have also been developed [11]. In addition, stable elliptically polarized solitons [12] and mixed polarized solitons [13] have also been found as solutions for Maxwell’s equations in the (1+1)-D case. The question naturally arises: Can (2+1)-D non-paraxial solitons exist? The answer is that Maxwell’s equations permit solutions in the form of cylindrically symmetric solitons. For example, Ciattoni *et. al* have proved the existence of azimuthally polarized, spatial, dark soliton solutions of Maxwell’s equations, while exact linearly polarized (2+1)-D solitons do not exist [14]. However, whether circularly polarized non-paraxial solitons can exist in the presence of vectorial Kerr effect still remains an open question. In this paper, we show that non-paraxial (2+1)-D bright- as well as dark- solitons with circular polarization and cylindrical symmetry can exist in optical Kerr media.

## 2. Propagation equation for circularly polarized beams

The propagation of a monochromatic optical field **E**exp[*i*(*kz*-*ωt*)] and **B**exp[*i*(*kz*-*ωt*)] of frequency *ω* in a non-resonant isotropic and homogeneous Kerr medium is governed by the Maxwell’s equations:

where *k*=*n*
_{0}
*ω*/*c*, *n*
_{0} is the medium’s linear refraction index, *c* is the speed of light in vacuum, *μ*
_{0} is the vacuum magnetic permeability, and the vectorial polarizability **P**
_{nl}
is given by [15]

where *n*
_{2} is the nonlinear refractive index coefficient. Eliminating **B** from Eq. (1), we get

The div equation of electric field can be written as

It is important to stress that Eq. (3) and (4) describe the exact propagation of a monochromatic plane wave in a Kerr medium to any orders of non-paraxiality. However, it is too hard to find the analytic soliton solutions of this equation because of its complexity. Fortunately, it can be simplified when some extremely small terms are neglected [see Eqs. (6)–(8)]. For the convenience of estimating these terms and for the demonstration of soliton solutions, we rescale the variables with

where *r*
_{0} is the soliton width parameter and **U**=(*U*
_{1},*U*
_{2},*U*
_{3}) represents the dimensionless electric field vector. In addition, we define a key dimensionless non-paraxial parameter *a*=(*kr*
_{0})^{-1}, which shows the competing relation between the wavelength and the beam width. Let us first consider an ideal left-circularly polarized cylindrically symmetric input beam, i.e., *U*
_{+}(*s*,*t*,0) = ${U}_{+}^{0}$ (*ρ*), *U*
_{-}(*s*,*t*,0) = ${U}_{-}^{0}$=0, where
$\rho =\sqrt{{s}^{2}+{t}^{2}}$, and *U*
_{±} = (*U*
_{1}±*iU*
_{2})/√2 are the left (+) and right (-) circular components. Since such an input beam has no preferred direction in the (*s*, *t*) plane, according to Eq. (3), the cylindrically symmetric input beam will remain its shape during the propagation in the Kerr medium. However, the right circular component and the longitude component should be taken into account because they will not remain zero even if they are initially zero. In fact, an input beam is never perfectly circularly polarized. So, we consider an input beam which is almost left-circularly polarized, i.e., *U*
_{-}/*U*
_{+} ~ *a*
^{2}. Substituting Eq. (5) to Eqs. (3) and (4), and dropping the higher orders than *a*
^{2}, we find that the left, right circular component and the longitude component satisfy, respectively, the following dimensionless equations:

$$=-\frac{2\gamma}{3}{a}^{2}\left[4\left({\mid \frac{\partial {U}_{+}}{\partial s}\mid}^{2}+{\mid \frac{\partial {U}_{+}}{\partial t}\mid}^{2}\right){U}_{+}+\left[{\left(\frac{\partial {U}_{+}}{\partial s}\right)}^{2}+{\left(\frac{\partial {U}_{+}}{\partial t}\right)}^{2}\right]{U}_{+}^{*}+{\mid {U}_{+}\mid}^{2}{\nabla}_{\perp}{U}_{+}+{U}_{+}^{2}{\nabla}_{\perp}{U}_{+}^{*}\right],$$

where *γ*=|*n*
_{2}|/*n*
_{2}=*sign*(*n*
_{2}). Eq.(6) is the non-paraxial propagation equation of the left circular component including the contributions depending on the longitudinal component of the electric field. It shows that both the non-paraxial effect and the contributions from the longitudinal component are of the order *a*
^{2}. The contribution from the right circular component is of the order *a*
^{4}, and thus is negligible. If *a*=0, Eq. (6) reduces to the standard paraxial NLS equation. On the other hand, when *a*
^{2} is large (for example, *r*
_{0}=*λ* results in *a*
^{2}=0.025), the non-paraxial effect becomes important. Without losing the generality, we choose *a*
^{2}=0.015 for our calculations. Eqs. (7) and (8) show that both the right circular component and the longitude component remain higher order infinitesimal of the left circular component over several diffraction lengths. Therefore, we conclude that the beam would remain almost left circularly polarized during propagation. Because the beams are cylindrically symmetric, we hereafter use cylindrical coordinates for our analysis.

## 3. Bright solitons

It is well-known that bright solitons only exist in self-focusing media, i.e., *n*
_{2}>0 and thus *γ*=+1. In the following, we will look for a cylindrical symmetric soliton solution of Eq. (6) in the form

thus getting

where a prime stands for a derivative with respect to *ρ*.

For a bright soliton, the field envelop should have a vanishing background, i.e., *u*(*ρ*→±∞)=0, and all the derivatives of *u*(*ρ*) vanish at infinity. Furthermore, the continuity of *u* at *ρ*=0 forces the boundary conditions *u*(0)=*u*
_{0} and *u*'(0)=0. The asymptotic behaviour of *u*(*ρ*) may be established directly from Eq. (10):

Our strategy for finding bright solitons of Eq. (10) is as follows. For a fixed *u*
_{0}, we vary *β* in the asymptotic solution of Eq. (11). At each *β*, we integrate the nonlinear equation (10) starting from the asymptotic solution (10) at *ρ*=0 toward infinity. If the function *∂u*/*∂ρ* vanishes at large values of *ρ*, then these *β* values give soliton solutions. Following this strategy, we find that for each *u*
_{0}>0, there are a positive and a negative *β* satisfying this condition, which respectively refer to forward and backward traveling solitons [See Eq.(9)]. For each pair of *β*, these two solitons have the same field amplitude *u*(*ρ*)and therefore the same intensity profile. The intensity profiles of bright solitons with different *u*
_{0} are plotted in Fig. 1, and the existence curve of these non-paraxial solitons is shown in Fig. 2. Obviously, the soliton FWHM decreses monotonically with the intensity maximum ${{u}_{0}}^{2}$. Interestingly, for a large *u*
_{0}, the width of the soliton is seen to exhibit a very slow dependence on ${{u}_{0}}^{2}$ and it is indeed independent of ${{u}_{0}}^{2}$ for extremely large *u*
_{0}, which is unlike its paraxial counterpart. In addition, the FWHM is found to be a slightly monotonically increasing function of the parameter *a* (not shown in this figure).

If the power is very high, the self-focusing is always accompanied by additional nonlinear phenomena such as multiple filamentation (MF), i.e., beam breakup into several long and narrow filaments. Recently, Fibich and Ilan [16] found that small ellipticity of the input polarization is unlikely to lead to MF of circularly polarized beams, whereas large input beam noise can lead to MF. Therefore, MF of circularly polarized beams can be suppressed by producing a clean cylindrically symmetric input beam, rather than by producing perfect circular polarization. The detailed analyses of the stability of these solitons require further numerical examinations when the input beams have some deviations from clean cylindrical symmetry and/or perfect circular polarization, which are beyond the scope of the present paper.

## 4. Dark solitons

Since focusing media (*γ*=+1, i.e., *n*
_{2}>0) are not able to support dark solitons, we consider hereafter defocusing media (*γ*=-1, i.e., *n*
_{2}<0). In order to find dark soliton solutions, a vortex phase profile should be introduced. In cylindrical coordinates, we look for dark soliton solution in the form

where *m* is the so-called topological charge. After substituting Eq. (12) into the propagation equation, i.e., Eq. (6), we find that the modulus function *u*(*ρ*) satisfies the following normalized boundary value problem,

for positive *ρ* and the boundary conditions,

The continuity of *u*(*ρ*) at *ρ*=0 forces the first boundary condition, while other conditions are consistent with a locally uniform state as *ρ*→∞. And therefore we can establish the asymptotic behavior of *u*(*ρ*) at *ρ*=0 directly from Eq. (13):

Eq. (13) implies, together with the boundary conditions at infinity,

where the dual sign of *β* refer respectively to forward and backward traveling solitons with the same amplitude *u*(*ρ*). Eq. (16) clearly shows the existence of an upper threshold for the soliton asymptotic amplitude

since, otherwise, *β* would become imaginary. A numerical integration of Eq. (13) can be carried out with the boundary condition (14). Unlike the method used in the case of bright solitons, our strategy to find dark solitons is to vary *δ* in the asymptotic expression (15) for a fixed *u*
_{∞}. For each *δ*, a numerical integration of Eq. (13) can be carried out with the boundary conditions (14) starting from the center toward infinity. If *u*(*ρ*) satisfies the boundary conditions at infinity, then these *δ* values give soliton solutions. The results of simulations confirm the existence of dark solitons in the range of field amplitudes 0<*u*
_{∞}<3/(4*a*
^{2}). Fig. 3 depicts the intensity profiles of dark solitons for different *u*
_{∞}. The region in the vicinity of *ρ*=0, where *u*(*ρ*) is significantly less than 1, is called vortex core. Comparing Fig. 3(a) with Fig. 3(b), we can see that the FWHM of the circular-symmetric double charged vortex solitons (*m*=2) is larger than that of the single charged one (*m*=1) for the same intensity. It is also evident in the existence curves of the dark vortex solitons, which are shown in Fig. 4. Like their bright counterparts, the FWHM of non-paraxial vortex solitons is also a monotonically decreasing function of the intensity. In addition, when the non-paraxial parameter *a* changes from 0 to 0.16, the FWHM increases slightly and monotonically with the parameter *a* (not shown in this figure).

## 5. Conclusion

In conclusion, we have found, in (2+1)-D cases, both bright and dark circularly polarized solitons with cylindrical symmetry can exist in the optical Kerr media beyond paraxial approximation. We have also numerically evaluated the existence curve relating the soliton width to the intensity. It is found that for bright solitons, the soliton width turns out to be practically independent of the peak intensity ${{u}_{0}}^{2}$ for a large *u*
_{0}, while for dark vortex solitons, there exist upper values of the maximum field amplitude for different topological charges.

## Acknowledgments

The authors acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 10374121) and the Natural Science Foundation of Guangdong Province, China (Grant No. 031567).

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