## Abstract

Radially polarized, circularly symmetric optical vortex solitons are shown to be able to exist in Kerr media beyond paraxial approximation. Unlike those of the paraxial linearly polarized counterparts, the topological charges of these solitons should not be less than 2. The properties associated with these solitons, such as their spatial width, and longitudinal and transverse field profiles, are characterized to depend on their normalized asymptotic intensity ${u}_{\infty}^{2}$ and nonparaxial degree. It is found that the asymptotic intensity ${u}_{\infty}^{2}$ of these solitons cannot exceed a threshold value in correspondence of which their width reaches a minimum value.

©2006 Optical Society of America

## 1. Introduction

The self-focusing effects of an optical beam in Kerr media have been and still are the object of an intense theoretical and experimental investigation [1]. The early description basically hinges upon the use of the nonlinear Schrödinger equation, which is derived from the Helmhotz equation based on the scalar and paraxial approximations. If the beam size *w* is comparable with the wavelength λ, these approximations become invalid and eventually nonparaxial effects should be employed to avoid the nonphysical behaviors (say, catastrophic collapse [2]) in the beam evolution. Although many contributions have been produced in this direction and nonparaxial spatial solitons have also been predicted, most of them are devoted into the case of linear polarization [3–10]. Until very recently, azimuthally polarized spatial dark solitons [10, 11] were found to exist to all nonparaxial orders as exact solutions of Maxwell’s equations in the presence of the vectorial Kerr effect. In addition, the existence of nonparaxial circularly polarized bright and dark solitons [12] was also proved.

On the other hand, there is a growing interest in the formation of laser beams with pure radial polarization recently [13–16]. This polarization configuration can be realized by the coherent summation of two TEM_{01} modes oriented along the *x* and *y* axis [13, 14]. Then, the question naturally arises: Can radially polarized vortex solitons exist in nonlinear Kerr media beyond paraxial approximations? In this letter, we show that vortex solitons with radial polarization and cylindrical symmetry can exist as solutions of Maxwell’s equations. The soliton shape and its existence curve are numerically evaluated.

## 2. Radially polarized vortex solitons

The equation governing the nonlinear evolution of a monochromatic optical field Re [**E**exp(-*iωt*)] and Re[**B**exp (-*iωt*)] in Kerr media is known as Maxwell’s equations:

where **E**=(*E*
_{1},*E*
_{2},*E*
_{3}) is the electric-field vector. *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 [17]

where *n*
_{2} is the nonlinear refractive index coefficient. Since focusing media (*n*
_{2}>0) are not able to support dark solitons, we consider hereafter defocusing media (*n*
_{2}<0). Eliminating B from Eq. (1), we get

The div equation of electric field can be written as

As mentioned above, a radially polarized beam can be viewed as the coherent summation of two orthogonal linearly polarized modes [13, 14]. Therefore, in cylindrical coordinates the field distributions of these two modes can be expressed by

and their simple vector addition yields

For the convenience, we rescale the variables with

where *r*
_{0} is the soliton width parameter. In addition, an important nonparaxial parameter *f*=(*kr*
_{0})^{-1} is introduced in our model, which shows the nonparaxial degree of the beam.
Generally, *f* is very small, i.e., *f*<<1. Even if the beam is focused to *w* ≈ λ, *f* is only approximately 0.16.

Now we go back to Eqs. (3) and (4) to develop the vector theory on radially polarized vortex solitons. We consider an ideal radially polarized cylindrically symmetric beam, i.e., *U*
_{ρ}=*U*
_{1} cos θ + *U*
_{2} sin θ and *U*θ = *U*
_{1} sinθ - *U*
_{2} cos θ = 0. 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. Using Eq. (7) and retaining all terms to the order of *f*
^{2}, we get from Eqs. (3) and (4) that

$$-2{f}^{2}\frac{\partial}{\partial \rho}\left[\frac{1}{\rho}\frac{\partial}{\partial \rho}\left(\rho {\mid {U}_{\rho}\mid}^{2}{U}_{\rho}\right)\right]=0,$$

Eq. (9) clearly shows that the longitude component *U*
_{3} remains higher order infinitesimal of the radial component over several diffraction lengths. If we look for soliton solutions such as

and

where *m* is the so-called topological charge, Eq. (8) becomes

and Eq. (9) becomes

For a vortex soliton, it should satisfy the following boundary conditions:

A localized solution of Eq. (12) can be found only numerically. For small ρ, the asymptotic solution is described by

where δ is a constant to be determined. Eq. (12) also implies, together with the boundary conditions at infinity,

where positive and negative signs in 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

In the actual case, the optical field should be continuous everywhere. Therefore, the continuity of the longitudinal component *u*
_{3} at ρ = 0 should force the boundary condition ∂*u*
_{3}/∂ρ = 0. Substituting this boundary condition into Eq.(13), we reach an important conclusion that single- charged (*m*=1) vortex solitons cannot exist, i.e., *m* should not be less than 2, which is quite different from their paraxial linear polarized counterparts^{1}. By means of the standard shooting technique, Eq. (12) is numerically solved for different values of *m* at *f*
^{2} = 0.015. The strategy is as follows. For a fixed *u*
_{∞}, we vary δ in the asymptotic solution (14). At each δ, we integrate the nonlinear equation (12) starting from the asymptotic solution (15) at ρ=0 toward infinity. If *u*(ρ) satisfies the boundary conditions at infinity, then these δ values give expected soliton solutions. The numerical calculations confirm the existence of dark solitons in the range of field amplitudes 0 < *u* < 1/(√2*f*) ∞. Figures 1(a) and 1(b) show the profiles of transverse and longitudinal component of the double-charged (*m*=2) vortex solitons, respectively. The analogous plots of triple-charged (*m*=3) vortex solitons are shown in Figure 1(c) and (d). Each of these figures consists of three curves that show the vortex soliton shapes for the values of *u*
_{∞} equal to 0.6, 0.8, and 1. From these figures one can see that, as *u*
_{∞} increases, the longitudinal component increases while the soliton width decreases correspondingly. For a given field amplitude *u*
_{∞}, the longitude component of the circular-symmetric triple-charged vortex solitons (*m*=3) is relatively smaller than that of the double-charged one (*m*=2). In Fig. 2 we report the existence curves of double- and triple-charged vortex solitons. It is found that the FWHM of a triple-charged vortex soliton is lager than that of a double-charged soliton. Note that whatever value *m* is, the FWHM grows infinitely for very small *u*
_{∞}, while it approaches a minimum value in correspondence to the threshold value *u*
_{∞} = 1/(√2*f*).

In order to complete our analysis, we also investigate the effect of nonparaxial parameter *f* on the soliton FWHM. It is found that when the non-paraxial parameter *f* varies from 0 to 0.16, the FWHM increases slightly and monotonically (not shown in this figure). Our numerical calculations also show that the FWHM of the radial polarized vortex solitons is an increasing function of the topologic charge *m* for the same intensity.

## 3. Conclusion

In conclusion, we have demonstrated the existence of radially polarized nonparaxial vortex solitons in defocusing Kerr media for the first time, to the best of our knowledge. This kind of solitons can exist provided that the normalized (ρ → ∞) field amplitude *u*
_{∞} is lower than a certain threshold value. Unlike those of their paraxial linearly polarized counterparts, the topological charges of these solitons cannot be less than 2. The existence curve relating the soliton width to *u*
_{∞} has been numerically evaluated, which shows that the soliton width decreases monotonically as *u*
_{∞} increases and eventually it attains a minimum value when *u*
_{∞} approaches the threshold 1/(√2*f*).

## Acknowledgments

The authors acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 10374121 and 10574167).

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