Vortex solitons in a saturable optical medium

Transcription

1 Tikhonenko et al. Vol. 15, No. 1/January 1998/J. Opt. Soc. Am. B 79 Vortex solitons in a saturable optical medium Vladimir Tikhonenko Australian Photonics Cooperative Research Centre, Research School of Physical Sciences and Engineering, Laser Physics Centre, The Australian National University, Canberra, ACT 0200, Australia Yuri S. Kivshar and Victoria V. Steblina Australian Photonics Cooperative Research Centre, Research School of Physical Sciences and Engineering, Optical Sciences Centre, The Australian National University, Canberra, ACT 0200, Australia Alex A. Zozulya Joint Institute for Laboratory Astrophysics, University of Colorado, Campus Box 440, Boulder, Colorado Received December 10, 1996; revised manuscript received March 3, 1997 Optical vortex solitons in a defocusing saturable medium are analyzed in the framework of the (2 1)-dimensional generalized nonlinear Schrödinger equation. Stationary, radially symmetric localized solutions with nonvanishing asymptotics and a phase singularity (vortex solitons) are found numerically for the varying saturation parameter. Relaxation of some localized initial profiles (e.g., vortex-type structures of an elliptic shape) to a vortex soliton is investigated numerically and then compared with the experimentally measured propagation of the vortex solitons created by a laser beam passed through a rubidium vapor, known as a nonlinear medium with strong saturation of the nonlinear refractive index. Reasonably good agreement is found, supporting the validity of the phenomenological model of the saturable nonlinear medium Optical Society of America [S (97) ] OCIS code: INTRODUCTION Spatiotemporal evolution of light in nonlinear media and stable propagation of self-localized beams or spatial solitons have been subjects of considerable theoretical and experimental research in nonlinear optics. In a defocusing isotropic medium, solitary-wave solutions of nonlinear propagation equations are dark solitons, 1 which exist as dips on a modulationally stable continuous-wave (cw) background field. In a bulk medium the dark solitons can appear as dark-stripe solitary waves, which demonstrate properties similar to those of the (1 1)- dimensional dark solitons. Stripe dark solitons have been observed experimentally 2 ; however, linear stability analysis 3,4 shows that a dark solitary stripe is unstable to transverse modulations. This snake-type transverse instability was recently observed experimentally, 5 7 and it results in the induced generation of optical vortices with alternating polarities (the transverse modulation instability of a bright solitary stripe that is due to a selffocusing nonlinearity has been observed 8 ). The snaketype instability of dark solitons and the process of vortex generation have also been investigated both numerically 7,9,10 and analytically by means of a multiscale asymptotic approach. 11 Nonlinear localized waves that describe the vortex soliton solutions of the nonlinear Schrödinger equation were introduced in the pioneering papers of Ginzburg and Pitaevskii 12 and Pitaevskii 13 as topological excitations of a superfluid. In the context of nonlinear optics they were suggested theoretically by Snyder et al. 14 However, the dynamics of the similar vortex-type structures created by dislocations of wavefront in linear optics was addressed much earlier. 15 Linear propagation of optical vortices was studied experimentally in Ref. 16. Optical vortices, similar to their counterparts in liquid helium, are quantized. The phase shift accumulated in going around the zero-field point is an integer multiple of 2. The integer is frequently referred to as the vortex charge. It is usually believed that single-charged vortex solitons are stable. Most of the previous theoretical work in optics concentrated on studying vortices in the framework of prototypical unsaturated Kerr nonlinear response. However, in several experiments dealing with the observation of vortexlike structures and their evolution (and even decay) the nonlinearity-induced change in the refractive index was usually achieved for a strongly saturable self-defocusing medium. Saturation of optical nonlinearity is one of the typical effects observed for many materials. For example, in the experiments reported in Ref. 18 the nonlinear change of the refractive index was achieved for a rubidium vapor irradiated with a cw Ti:sapphire laser near the D 2 resonance line, for which the estimated saturation intensity is known to be quite low, making the effect produced by the nonlinearity saturation very important. In contrast, the theory of optical vortex solitons developed in local isotropic media has been based on the assumption of a nonsaturable Kerr nonlinearity for which the nonlinearity-induced change of the refractive index is directly proportional to the light intensity. In this paper we develop, for the first time to our knowledge, a theory of optical vortex solitons in a strongly /98/ $ Optical Society of America

2 80 J. Opt. Soc. Am. B/Vol. 15, No. 1/January 1998 Tikhonenko et al. saturable medium with a local nonlinear response and compare our analytical and numerical results with the experimentally measured parameters of optical vortex solitons. The paper is organized as follows. In Section 2 we discuss our model, which is shown to reduce to the generalized nonlinear Schrödinger (GNLS) equation. Then we look for stationary localized solutions of radial symmetry with a phase singularity at the center and nonvanishing asymptotics and find numerically a family of vortex solitons that are characterized by the dimensionless parameter s I 0 /I sat, where I 0 is the background intensity and I sat is the saturation intensity. We investigate also the evolution of localized profiles with a phase variation, which are close to a vortex soliton. In particular, we observe oscillation and rotation of elliptical-profile vortex structures. In Section 3 we present experimental results on the propagation of optical vortex solitons, which are then compared with the prediction of the theory, showing reasonably good qualitative and even quantitative agreement. 2. THEORY A. Model Following the standard approach, we consider propagation of a monochromatic scalar electric field E in a bulk optical medium with an intensity-dependent refractive index n n 0 n NL (I), where n 0 is the uniform refractive index of the linear medium and n NL (I) describes the variation in the index that is due to the field intensity I E 2. The function n NL (I) is usually introduced phenomenologically and, generally speaking, should be characterized by parameters that can be then measured in an experiment, such as the Kerr coefficient n 2 and the maximum change in the refractive index for large intensities n. These values have a simple physical meaning. Indeed, for small intensities, I I sat, where I sat is the characteristic saturation intensity, it is convenient to assume a familiar linear dependence of n NL on I known as the Kerr effect, n NL (I) n 2 I, where n 2 is the Kerr coefficient of an optical material (below we assume a defocusing medium with n 2 0). For larger intensities, I I sat, the refractive index saturates and approaches its maximum value, n NL (I) max n. Here we assume a standard model of the nonlinearity saturation: n 2 I n NL I, (1) 1 I/I sat so n 2 is the Kerr coefficient and n n 2 I sat. Equation (1) describes the refractive-index saturation corresponding to the standard two-level model. Solutions of the governing Maxwell equation can be presented in the form E R, Z; t E R, Z exp i 0 Z i t c.c., (2) where E is the slowly varying wave envelope, c.c. stands for the complex conjugate, is the source frequency, and 0 kn 0 2 n 0 / is the plane-wave propagation constant for the uniform-background medium. Here 2 c/ is the wavelength and c is the speed of light. Further, we assume a (2 1)-dimensional model, so the Z axis is parallel to the direction of propagation and the X and the Y axes are two coordinates in the plane transverse to the propagation direction, R (X, Y). Substituting Eq. (2) into the (2 1)-dimensional scalar wave equation, we obtain the GNLS equation E 2ikn 0 2 Z 2 E X 2 2 E 2n 0 k 2 n NL I E 0. (3) Y We assume that for X, Y the modulus of the field envelope E tends to a nonzero value I 0 and look for a solution in the form E I 0 exp(i NL Z)u, where NL k n 2 I 0 1 I 0 /I sat (4) is the nonlinearity-induced shift of the propagation constant, 0 NL. The equation for the function u(r, z) with the boundary condition u(r ) 1 can be then presented in the following dimensionless form: i u z 1 2 u 2 x u y 1 1 s 1 s 2 1 s u, 1 s u (5) where the parameter s I 0 /I sat characterizes the effect of the nonlinearity saturation. The normalized propagation coordinate z and the transverse coordinates r (x, y) are measured in units of k n 2 I 0 and (n 0 n 2 k 2 I 0 ) 1/2, respectively, which do not depend on the dimensionless saturation parameter s. B. Vortex Solitons A class of radially symmetric stationary (i.e., not depending on z) localized solutions of Eq. (5) with a phase singularity that describe the vortex-type topological structure of the field can be sought in the polar coordinates r and as follows: u r A r exp im, (6) where m is integer that defines the vortex charge or its winding number (m 1, 2,...); is the azimuthal angle, defined as tan y/x; and r r x 2 y 2 is the radial coordinate. Scalar amplitude A(r) satisfies the following ordinary differential equation: d 2 A dr 2 1 da r dr m2 r 2 A 2 s 1 s 1 1 s 1 sa 2 A, with the boundary conditions A(r) 0 for r 0 and A(r) 1 for r. A localized solution of Eq. (7) can be found only numerically. For small r the solution asymptotics is described by (7) A r ar m O r m 2 as r 0. (8) The leading linear dependence of the vortex amplitude on the radius is determined by the next-order term in the left hand side of Eq. (7).

3 Tikhonenko et al. Vol. 15, No. 1/January 1998/J. Opt. Soc. Am. B 81 For large values of radius r the solution asymptotics can be also found analytically: A r 1 m2 4r 2 1 s 2 O r 4 as r. (9) The power-law correction ( r 2 ) to the limiting value A( ) 1 is determined by the r 2 term on the left-hand side of Eq. (7) that dominates for large r. This correction also depends on the saturation parameter s. Exact profiles of the stationary solutions A(r) that describe optical vortex solitons cannot be obtained analytically, even in the simplest case of a nonsaturable Kerr medium s 0, and therefore the subsequent analysis was performed numerically by solution of stationary equation (7) with the corresponding boundary conditions and also by analysis of the dynamics within GNLS equation (5). First we solve Eq. (7) numerically for the function A(r) with the boundary conditions A(0) 0 and A( ) 1. Figures 1(a) and 1(b) present, respectively, profiles of the single- (m 1) and double- (m 2) charged vortex solitons described by the function A(r) as a solution of Eq. (7). Each of the figures consists of three curves that show the vortex soliton shapes for the values of the saturation parameter s equal to 0, 1, and 5. The curve s 0 in Fig. 1(a) is the vortex solution at m 1 discussed by Pitaevskii. 13 The limit s 0 in Fig. 1 recovers the single- and double-charged vortex solitons of the prototypical Kerr Fig. 1. Profiles A(r) of the vortex soliton amplitude defined by Eq. (7) for (a) the single-charged vortex soliton and (b) the double-charged vortex soliton for several values of the dimensionless saturation parameter, s 0, 1, 5. Fig. 2. Diameters of the vortex soliton d e2 and d 05 as functions of the dimensionless saturation parameter s. The dashed curve is the analytical prediction that follows from Eq. (9). nonlinearity. The next value, s 1, corresponds to moderate effects of saturation; s 5 is representative of a relatively strong saturation. Figure 1 demonstrates that at a fixed background intensity A( ) 1 the size of the vortex core is highly sensitive to a variation of the dimensionless saturation parameter s. Because the vortex profile and its diameter can be directly measured in experiment (Section 3 below), we further investigate the vortex structure by introducing two characteristics that describe the vortex core. Figure 2 shows two diameters of the single-charged vortex soliton as functions of the saturation parameter s. The diameters were calculated with the help of the socalled complementary intensity, defined as I c (r) I( ) I(r) 1 I(r). The lowest curve is the diameter d 05 (FWHM width), estimated as the distance between the points with the intensity equal to half of the maximum intensity. The uppermost curve is the diameter d e2, estimated as the distance between the points with the complementary intensity equal to 1/e 2 of the maximum intensity. The dashed curve is the analytical dependence d e2 2e(1 s)m found from the asymptotics [Eq. (9)]. The diameter d 05 characterizes the size of the central part of the solitary wave, whereas the diameter d e2 is more indicative of the size of the whole localized structure including the long-range nonexponential tail. Both diameters are practically linear functions of the saturation parameter s. Increasing the value of s results in an increase of both diameters, and d 05 is approximately twice lower than d e2. This increase is more pronounced for the periphery of the vortex because the slope of the function d 05 is approximately twice lower than that of d e2. C. Evolution of the Vortex-Type Localized Structures Besides establishing the fact of the existence of solitarywave solutions of the vortex type, of crucial importance is the investigation of the dynamics of an input light beam with an externally superimposed vortex structure exp(im ) incident upon a nonlinear medium. Such an analysis is important for experimental verifications when an input beam with exactly the same shape as that of the stationary solution cannot be generated. This analysis allows us to verify that the stationary localized solutions found above are stable and to estimate the speed of con-

4 82 J. Opt. Soc. Am. B/Vol. 15, No. 1/January 1998 Tikhonenko et al. Fig. 3. Spatial evolution of a radially symmetric input vortex structure [Eq. (10)] with s 1 when its initial diameter d is approximately two times smaller than that of the corresponding vortex diameter: (a) diameters d 05 and d e2 as functions of the propagation distance z; (b) intensity profiles u 2 for the values of the propagation distance z equal to 1, 5, 10, and 15. Figure 3(b) presents the variation of the spatial profiles u(r, z) 2 that characterize the distribution of the beam intensity for several values of the longitudinal propagation distance z. As one can see from Fig. 3(b), the vortex at the initial stage of its evolution emits radiation waves starting from its core and propagating outward to infinity. These waves carry away some amount of the input beam energy and allow the vortex profile to relax rapidly to its solitary shape, which corresponds to a balance between diffraction and nonlinearity. Because the initial intensity distribution is narrower than the final vortex profile, the field is pushed out from the core, creating radiation waves with intensities exceeding that of the background. Figure 3(a) demonstrates that the core relaxes to the asymptotic solitary shape slightly faster than the periphery because it takes some time for the radiated waves to reach the peripheral parts of the vortex. Figure 4 demonstrates the spatial evolution of a Gaussian intensity notch [Eq. (10)] that is initially approximately two times wider than the vortex solution. Because the initial intensity distribution is wider than the final stationary profile, the field moves in from the periphery, and the radiation waves correspond mostly to dips in intensity. The discontinuity in the value of the d e2 is due to the passage of such a wave through the region where I(r) 1 1/e 2. Quite fast convergence of both diameters to their stationary values is observed in this case, too, as is verified by the results presented in Fig. 4(a). We also analyzed the dynamics of vortex beams with nonradially symmetric input profiles. Numerical analysis shows that these distributions also converge quite rap- vergence of the beam shape to the corresponding stationary solution. For this analysis we solve Eq. (5) for various input vortex-type profiles and investigate the convergence to the stationary radially symmetric solitary solutions obtained in Subsection 2.B. This analysis will also be required for the subsequent comparison with the experimental data reported in Section 3. First we consider the class of radially symmetric initial vortex profiles with the transverse distribution of the input amplitude u(r ) different from that of the soliton solution A(r). Because the initial distribution of the field in this case is radially symmetric, the beam remains radially symmetric everywhere in the nonlinear medium. First we present the results for an input beam of the form u r, z 0 1 exp r 2 /d 2 1/2 exp i. (10) The input profile [Eq. (10)] corresponds to an infinite plane-wave background that has a Gaussian intensity notch with the characteristic diameter d at the center and the vortex structure of the phase. Other input distributions (e.g., a hyperbolic tangent) have been tried and were found to result in the same qualitative behavior. Figure 3 shows the evolution of the input beam [Eq. (10)] in the moderate saturation regime s 1 when the beam s initial diameter is approximately two times smaller than that of the corresponding solitary solution. Figure 3(a) shows the evolution of the vortex diameters d 05 and d e2 versus the longitudinal propagation distance. Fig. 4. Same as in Fig. 3 but for the case when the input beam diameter d is approximately two times larger than the diameter of the corresponding vortex soliton.

5 Tikhonenko et al. Vol. 15, No. 1/January 1998/J. Opt. Soc. Am. B 83 (11) with m 1. The value of the saturation parameter s is equal to 1. The initial FWHM diameters are d 05, x 3 and d 05, y 6. Figures 5(a), 5(b), 5(c), 5(d), 5(e), and 5(f) correspond to the values of the propagation coordinate z equal to 0, 1, 2, 3, 4, and 5, respectively. The most interesting feature of Fig. 5 is the clockwise rotation of the vortex core with the simultaneous relaxation to the radially symmetric shape. The reason for this rotation is that the nonlinearity couples azimuthal harmonics exp(im ) with different transverse indices m that have different phase velocities. Changing the topological charge of the input vortex, i.e., exp(i ) exp( i ), reverses the direction of the rotation. The relaxation of the input field to the soliton shape is accompanied by the emission of radiative waves in full qualitative analogy with the previously discussed radially symmetric case. These waves propagate from the center of the vortex to its periphery but are no longer radially symmetric and exhibit two lobes. In other words, the spatial relaxation of a general input profile toward the soliton shape is accompanied by a nontrivial evolution of both the phase and the amplitude of the nonlinear wave. The dependence of the characteristic vortex diameters (e.g., along the x and the y axes) on the propagation coordinate is qualitatively similar to that presented in Figs. 3 and 4. Figures 3 5 demonstrate the spatial evolution of a vortex structure that takes place on an infinite plane-wave background. This approach has the advantage of revealing intrinsic vortex dynamics. On the other hand, one usually carries out experimental observation of optical vortices by embedding the vortices in a wider Gaussian carrier beam for the simple reason that an infinite-extent plane-wave background has infinite energy. Linear (diffraction) and nonlinear (defocusing) spreading of the carrier changes some quantitative features of the vortex evolution. Nevertheless, the two approaches result in the same qualitative features of the dynamics. Fig. 5. Spatial evolution of the single-charged ( m 1) elliptical vortex-type input beam [Eq. (11)] for s 1, d 05, x 3, and d 05, y 6. (a) (f) Correspond to the values of the propagation distance z changing from 0 through 5 in increments of 1. idly to the radially symmetric stationary solutions A(r) discussed in Subsection 2B. As an example, Fig. 5 displays the evolution of an input elliptic notch of the form u r, z 0 1 exp x2 y2 2 d x 1/2 d y 2 exp im, 3. EXPERIMENTAL RESULTS AND DISCUSSION An experimental setup is shown in Fig. 6. The source was a cw tunable Ti:sapphire laser with a short-term line width of 50 MHz and a maximum single-mode output power of 500 mw. The laser wavelength was tuned close to the rubidium D 2 line at 780 nm. An expanding telescope imaged the beam waist onto the surface of a singly charged vortex phase mask, which imposed the desired phase structure on the transverse profile of the beam. The first diffraction order of the mask was then projected onto the input face of the nonlinear medium by another telescope, providing an initial condition consisting of a single-charged vortex nested in the center of the Gaussian beam. An image of the output window of the cell was then projected onto a screen, and the resulting intensity pattern was recorded by a CCD camera. The 50-mm-diameter, 200-mm-long cylindrical Pyrex cell contained atomic-rubidium vapor, used as a nonlinear medium. The cell was mounted inside a double-chamber oven, permitting uniform temperatures to within 1 C to be maintained along the surface of the cell. Rubidium metal was held in a sidearm of the cell, enclosed in a separate chamber, and maintained at 30 C below the temperature of the cell chamber, providing a saturated rubidium-vapor concentration N Rb of less than cm 3. No buffer gas was used in the cell. Checks were carried out to ensure that the input and output optical windows of the cell were free from distortions caused by temperature gradients. We obtained resonant enhancement of the nonlinearity by tuning the laser frequency L close to the rubidium atomic D 2 line. With negligible collision broadening, the Fig. 6. Schematic presentation of the experimental setup for the measurement of the vortex parameters.

6 84 J. Opt. Soc. Am. B/Vol. 15, No. 1/January 1998 Tikhonenko et al. resonance lines were inhomogeneously broadened with Doppler width D 0.75 GHz at the highest cell temperatures ( 100 C). Hyperfine splitting of the 85 Rb ground state was substantially greater than the inhomogeneous linewidth, allowing the hyperfine transition 5S 1/2 (F 2) 5P 3/2 (F 3), at frequency res, to be used as the working resonance, with red detunings producing a self-defocusing nonlinearity. We calibrated the laser frequency detuning, L L res, using the absorption spectra of the 85,87 Rb hyperfine transitions. It was possible to manipulate the strength of the nonlinearity and the character of its saturation by varying vapor concentration and changing the laser detuning from the resonance. The power in the beam at the input face of the cell was 120 mw with a 1/e 2 waist of 0.6 mm. Detunings ranged from 0.5 to 1.5 GHz, providing the maximum range of nonlinearity for the least absorption. From previous experiments, 4 these numbers yield a maximum nonlinearity of 10 5 cm 2 /W to be achieved at a metal-vapor concentration of cm 3. We investigated the dynamics of the soliton propagation by changing the parameters responsible for the nonlinear properties of the medium, the concentration of rubidium atoms, or the laser frequency detuning from the resonance and analyzing the output beam profiles at the exit window of the cell. The sequence of binary images in Fig. 7 demonstrates a typical evolution of the beam observed with increasing nonlinearity. Because of the initial ellipticity of the laser beam the input vortex profiles were radially asymmetric, and the experimental beam evolution exhibited the same relaxation dynamics as observed numerically (Fig. 5) with the rotation of the elliptical vortex core up to 40 at the experimental conditions corresponding to Fig. 7. The experimental parameters and the data that we obtained by varying the detuning of the laser at a constant atom concentration N Rb cm 3 and the maximum background intensity I W/cm 2 are shown in Table 1. In this table the first column corresponds to the laser frequency detuning from the resonance. Our estimations of the nonlinearity of the resonance medium were based on the model of a two-level atom with inhomogeneously broadened transitions in the presence of strong hole burning. The nonlinear refraction n NL (I) was calculated for different laser detunings in the conditions of our experiments and interpolated with Eq. (1) by a procedure similar to one described earlier in Ref. 15. We then used the parameters I sat and n 2 obtained by this procedure (Table 1) to calculate the normalization coefficients for the transverse and propagation coordinates and to estimate the dimensionless saturation parameters s defined in Subsection 2.A. To compare our experimental results with the theory we measured the effective diameter at half of the maximum amplitude of the vortex solitons at the output of the cell as a function of the parameters responsible for the nonlinearity of our medium. The effective soliton diameter was defined as d eff d xx05 d yy05, where d xx05 and d yy05 are the major and the minor axes, respectively, of the experimentally registered elliptical soliton core at halfmagnitude, and corresponded to the average diameter of a circularly symmetric vortex soliton with the same core area. The values of the normalized d eff are shown in Table 1 together with the corresponding dimensionless propagation distance for each detuning used in the experiments (for a fixed length of the cell this value varies because of normalization). As can be seen from the data in Table 1, the decrease in the laser detuning from resonance led to a rapid increase in both nonlinearity and saturation parameters, accordingly increasing the propagation distance z(norm), which corresponded to the normalized length of the cell with the nonlinear medium. The experimental and theoretical values for the soliton diameters are compared in Fig. 8. The upper dashed curve in Fig. 8 corresponds to the numerical estimations Fig. 7. Experimentally observed evolution of the output vortex soliton profile (binary images). The experimental parameters in (a) correspond to row 1 of Table 1; those in (b) to row 4; and those in (c) to row 6. L (GHz) Table 1. I sat (W/cm 2 ) Experimental Data n 2 (10 6, W/cm 2 ) s d eff (norm) z (norm)

7 Tikhonenko et al. Vol. 15, No. 1/January 1998/J. Opt. Soc. Am. B 85 measured parameters of the vortex solitons has demonstrated reasonably good agreement with the theory developed. This agreement also justifies the validity of the phenomenological models based on the generalized nonlinear Schrödinger equation for describing nonlinear effects in self-defocusing saturable media. ACKNOWLEDGMENTS A. A. Zozulya acknowledges the support of National Science Foundation grant PHY and of the Optoelectronics Computing Center and the National Science Foundation Engineering Research Center. Fig. 8. Comparison between theory and experiment for the diameters of the vortex soliton versus saturation. Shown are the experimentally measured beam diameter d eff (solid curve) as a function of the dimensionless saturation s. The similar value d 05 obtained from theory is shown as a long-dashed curve. The short-dashed curve displays the variation of the dimensionless propagation distance. of the FWHM diameters of stationary vortex solitons obtained in the procedure described in Subsection 2.B, and the experimental data points for the effective diameters were taken from the Table 1. The lower dashed curve in Fig. 8 shows the variation of the normalized propagation distance. Inasmuch as the vortex solitons in our experiments were formed from initially Gaussian intensity distribution by imposition of a vortex-type phase modulation on the laser beam, the evolution of the experimental soliton profiles corresponded to the situation analyzed in Fig. 3, and those profiles are generally observed to be narrower than the stationary vortex solutions. A good correlation was observed in the region of high saturation parameters where the normalized propagation distances were long enough to reach the condition of the stationary propagation in accordance with our theoretical analysis of the soliton core relaxation dynamics presented in Fig. 3(b). Another factor that affected the evolution of vortices can be attributed to the influence of absorption in the resonance medium, which was been taken into account in our theoretical analysis. 4. CONCLUSIONS In conclusion, we have developed a theory of vortex solitons in optical materials with strong saturation of the nonlinear refractive index. The theory allows us to analyze the modification of the stationary beam profiles that describe vortex solitons for various saturation (or increasing background) intensities. This analysis predicts the change of the soliton parameters, e.g., its two diameters, as the dimensionless saturation varies. 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