1. Introduction

Vortices play important roles in many branches of physics [1]. The first experimental observation of optical vortex soliton was reported in a self-defocusing medium where the field propagates as a soliton, owing to the counterbalanced effects of diffraction and nonlinear refraction at the phase singularity [2]. Such singularity corresponding to vortices can exist in the Bose–Einstein condensates which links the physics of superfluidity, phase transitions, and singularities in nonlinear optics [3–5]. The topological states of a Bose–Einstein condensate can be prepared experimentally [4]. Moreover, several interesting effects including cascade generation of multiple charged optical vortices and helically shaped spatiotemporal solitons in Raman FWM, and coupled vortex solitons supported by cascade FWM in a Raman active medium excited away from the resonance have been investigated [6,7]. Spatially modulated vortex solitons (azimuthons) have been theoretically considered in self-focusing nonlinear media [8]. Transverse energy flow occurs between the intensity peaks (solitons) associated with the phase structure, which is a staircase-like nonlinear function of the polar angle φ. The necklace-ring solitons can merge into vortex and fundamental solitons in dissipative media [9].

With the self-phase modulation, spatial bright soliton in self-focusing medium or dark soliton in self-defocusing medium can be created [1]. Focusing effect can also be induced by cross-phase modulation (XPM) in a self-defocusing nonlinear medium [10]. In such case, the spatial soliton can form by balancing the spatial diffraction with the XPM-induced focusing [11]. Moreover, when three or more plane waves overlap in the medium, complete destructive interference patterns can give rise to phase singularities or optical vortices [12–15], which are associated with zeros in the modulated light intensity patterns and can be recognized by specific helical wavefronts.

In this letter, we experimentally demonstrate the formations of modulated vortex solitons in two generated four-wave mixing (FWM) waves in a two-level, as well as a cascade three-level, atomic systems. These vortex solitons are created by the interference patterns by superposing three or more waves, and by the greatly enhanced cross-Kerr nonlinear dispersion due to atomic coherence [16,17].

2. Theoretical model and experimental scheme

Two relevant experimental systems are shown in Figs. 1(a) and 1(b). Three energy levels from Na atoms (the atomic vapor is heated with an atomic density of $5.6\times {10}^{13}c{m}^{-3}$ and a refractive index contrast of $\mathrm{\Delta }n=n_{2}I=4.85\times {10}^{-4}$ approximately, where $n_2$ is the cross-Kerr nonlinear coefficient and I is the beam intensity) are involved in the experimental schemes. In Fig. 1(b), energy levels $|0〉$ ($3S_{1/2}$), ${|1〉}^{}$ ($3P_{3/2}$) and $|2〉$ ($4D_{3/2,5/2}$) form a three-level cascade atomic system. When the energy level $|2〉$ is not used, the system reduces into a two-level one [Fig. 1(a)]. The laser beams are aligned spatially as shown in Fig. 1(c), with two dressing beams (${\mathit{E}}_1^{\prime }$ and ${\mathit{E}}_2^{\prime }$) and two pump beams (${\mathit{E}}_1$ and ${\mathit{E}}_2$) propagating through the atomic medium in the same direction with small angles ($\theta ={0.3}^{\circ }$) between them in a square-box pattern. The probe beams (${\mathit{E}}_3$ and ${\mathit{E}}_3^{\prime }$) propagate in the opposite direction with a small angle as shown in Fig. 1(c). Three laser beams (${\mathit{E}}_1$, ${\mathit{E}}_1^{\prime }$, and ${\mathit{E}}_3$, with Rabi frequencies $G_1$, $G_1^{\prime }$ and $G_3$, connecting transition $|0〉$ to $|1〉$) have the same frequency ${\omega }_1$ (from the same dye laser with a 10 Hz repetition rate, 5 ns pulse-width and 0.04 cm-1 line-width), and generate an efficient degenerate FWM signal ${\mathit{E}}_{\mathrm{F}1}{}^{}$ (${\mathbf{k}}_{\mathrm{F}1}={\mathbf{k}}_1-{\mathbf{k}}_1^{\prime }+{\mathbf{k}}_3$) [Fig. 1(a)] in the direction shown at the lower right corner of Fig. 1(c). These beams ${\mathit{E}}_2$, ${\mathit{E}}_2^{\prime }$, and ${\mathit{E}}_3^{\prime }$ (with Rabi frequencies $G_2$, $G_2^{\prime }$ and $G_3^{\prime }$, and connecting the same transition ${|0〉}^{}$ to $|1〉$ in the two-level system) are from another near-transform-limited dye laser of frequency ${\omega }_2$, and produce a nondegenerate FWM signal ${\mathit{E}}_{\mathrm{F}2}$ (${\mathbf{k}}_{\mathrm{F}2}={\mathbf{k}}_2-{\mathbf{k}}_2^{\prime }+{\mathbf{k}}_3$) [Fig. 1(a)]. All laser beams are horizontally polarized. The diameters of the laser beams are about 25$\mu m$. When the six laser beams are all on, there also exist other two FWM processes ${\mathbf{k}}_{\mathrm{F}3}={\mathbf{k}}_1-{\mathbf{k}}_1^{\prime }+{\mathbf{k}}_3^{\prime }$ and ${\mathbf{k}}_{\mathrm{F}4}={\mathbf{k}}_2-{\mathbf{k}}_2^{\prime }+{\mathbf{k}}_3^{\prime }$. However, the coexisting ${\mathit{E}}_{\mathrm{F}1}$ and ${\mathit{E}}_{\mathrm{F}2}$ are the dominant ones in the experiment due to phase-matching and chosen beam intensities [17,18]. According to these FWM phase matching conditions, we can obtain the coherence lengths in the two-level system as $L_{F1}^c=2c\pi {\omega }_1/[n_1{\omega }_1\left|{\omega }_1-{\omega }_1\right|{\theta }^2]\to \infty$ for ${\mathit{E}}_{\mathrm{F}1}$, $L_{F2}^c=2c\pi {\omega }_1/[n_1{\omega }_2\left|{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2\right|{\theta }^2]\approx 1.8\times {10}^3\text{m}$ for ${\mathit{E}}_{\mathrm{F}2}$, $L_{F3}^c=2c\pi {\omega }_2/[n_1{\omega }_1\left|{\mathrm{\Delta }}_2-{\mathrm{\Delta }}_1\right|{\theta }^2]\approx 1.8\times {10}^3\text{m}$ for ${\mathit{E}}_{\mathrm{F}3}$, $L_{F4}^c=2c\pi {\omega }_2/[n_1{\omega }_2\left|{\omega }_2-{\omega }_2\right|{\theta }^2]\to \infty$for ${\mathit{E}}_{\mathrm{F}4}$, where $n_1$ is the linear refractive index and ${\mathrm{\Delta }}_1$ (${\mathrm{\Delta }}_2=0$) is the detuning of the fields ${\mathit{E}}_{1,3}$ and ${\mathit{E}}_1^{\prime }$ (${\mathit{E}}_2$ and ${\mathit{E}}_{2,3}^{\prime }$) from the atomic transition.

 

Fig. 1 Two FWM processes ${\mathbf{k}}_{\text{F}1}$ and ${\mathbf{k}}_{\text{F}2}$ with five beams ${\mathbf{k}}_1$, ${\mathbf{k}}_1^{\prime }$, ${\mathbf{k}}_2{}^{}$, ${\mathbf{k}}_2^{\prime }$, ${\mathbf{k}}_3{}^{}$, and ${\mathbf{k}}_3^{\prime }$ in (a) two-level and (b) cascade three-level atomic systems, respectively, dressed by two beams ${\mathbf{k}}_1^{\prime }$ and ${\mathbf{k}}_2^{\prime }$. (c) Spatial beam geometry used in the experiment.

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When ${\mathit{E}}_2$ and ${\mathit{E}}_2^{\prime }$ are tuned to the $|1〉-|2〉$ transition, the system becomes a cascade three-level system [Fig. 1(b)], which generates a two-photon resonant nondegenerate FWM process ${\mathit{E}}_{\mathrm{F}2}$ [17,18]. In this system, the coherence lengths are $L_{F1}^c=2c\pi {\omega }_1/[n_1{\omega }_1\left|{\omega }_1-{\omega }_1\right|{\theta }^2]\to \infty$ for ${\mathit{E}}_{\mathrm{F}1}$ and $L_{F2}^c=2c\pi {\omega }_1/[n_1{\omega }_2\left|{\omega }_2-{\omega }_1\right|{\theta }^2]\approx 0.6\text{m}$ for ${\mathit{E}}_{\mathrm{F}2}$, respectively.

The mathematical description of the two generated (dominant) FWM beams (including the self- and cross-Kerr nonlinearities) can be obtained by numerically solving the following propagation equations in cylindrical coordinate:

Solving the propagation equations in the cylindrical coordinate, we demonstrate that the modulated vortex solitons with a screw-type dislocation phase can be characterized by two independent integer numbers [1,8] (i.e. the topological charge m and the number of intensity peaks N), and parametrized by the rotating angular velocity (i.e., energy flow velocity) w. We can obtain the stationary transverse solution of the modulated vortex soliton as [8,9] $E_{\mathrm{F}1}\propto E\sec \mathrm{h}[E{({\mathrm{k}}_{\mathrm{F}1}{n}_2^{S1}/n_1)}^{1/2}(r-R_0)]\cos (N\phi /2)\exp (i{m}_{\mathrm{F}1}\phi +i{\varphi }_{NL})$ with an initial radius $R_0$. Moreover, we have $w_1={\varphi }_{NL}(r,z)/z=2{\mathrm{k}}_{\mathrm{F}1}{n}_2{I}_2{e}^{-r^2/2}/n_1$, $w_2={\varphi }_{NL}(r,z)/n_2=2{\mathrm{k}}_{\mathrm{F}1}{I}_2{e}^{-r^2/2}z/n_1$ and $w_3={\varphi }_{NL}(r,z)/I_2=2{\mathrm{k}}_{\mathrm{F}1}{n}_2{e}^{-r^2/2}z/n_1$.

The spatial interference patterns are formed by superposing three or more waves (${\mathit{E}}_{1,2,3}$ and ${\mathit{E}}_{1,2,3}^{\prime }$) in the medium, as shown in Fig. 1(c). The destructive interference of two waves with similar intensity can result in spatial patterns with zero intensities, which create phase singularities or optical vortices [14]. When multi-beam interference occurs, spatial polygon patterns (i.e, closed triangle from three beams, quadrangle from four beams, which gives one vortex point [13,14].) can be formed, with the side lengths being the complex amplitude vectors of the waves. The polygons with more beams will look like a circular shape, and the phase complexity will be enhanced. The complex amplitude vectors can be overlaid at the observation plane and give rise to the total complex amplitude vector ($C_X$,$C_Y$) of the interfering plane waves [13,14]. The local structures of the optical vortices are given by the polarization ellipse relation $C_X{}^2/(T_X{}^2+T_Y{}^2){\sin }^2(\beta +\alpha )+C_Y{}^2/(T_X{}^2+T_Y{}^2){\cos }^2(\beta +\alpha )=1$, where $\beta =\arctan (T_X/T_Y)$ and α is the ellipse orientation. The ellipse axes $T_X$, $T_Y$ are related to the spatial configuration (including the incident beam directions, phase differences between beams etc.) and beam intensities.

The dressing beams ${\mathit{E}}_{1,2}^{\prime }$ are approximately 10 times stronger than the beams ${\mathit{E}}_{1,2}$, ${10}^2$ times stronger than the weak probe beams ${\mathit{E}}_3$ and ${\mathit{E}}_3^{\prime }$, and ${10}^4$ times stronger than the two generated FWM beams ${\mathit{E}}_{\mathrm{F}1,2}$. The generated weak beam ${\mathit{E}}_{\mathrm{F}1}$ (or ${\mathit{E}}_{\mathrm{F}2}$) partly overlaps with the strong beam ${\mathit{E}}_1^{\prime }$ (or ${\mathit{E}}_2^{\prime }$), and other stronger beams (${\mathit{E}}_{1,2,3}$, ${\mathit{E}}_3^{\prime }$) lie around them [Fig. 1(c)]. As a result, the same frequency waves can interfere to construct polarization ellipse, create phase singularity [13,14], and induce local changes of the refractive index. The interference induces a vortex pattern with the superposed $n_2^{\mathrm{F}1}={\displaystyle \sum _{i=1}^5\mathrm{\Delta }{n}_2^{X\text{i}}}$ and $n_2^{\mathrm{F}2}={\displaystyle \sum _{i=6}^{10}\mathrm{\Delta }{n}_2^{X\text{i}}}$ (the center of such vortex lies in the minimum of $n_2^{\mathrm{F}1,2}$), and the horizontally- and vertically-aligned dressing fields ${\mathit{E}}_1^{\prime }$ and ${\mathit{E}}_2^{\prime }$ modulate a circular-type splitting, with three or four parts around the ellipse. Note that ${\mathit{E}}_1^{\prime }$ (or ${\mathit{E}}_2^{\prime }$) is the dominant dressing field of ${\mathit{E}}_{\mathrm{F}1}$ (or ${\mathit{E}}_{\mathrm{F}2}$). Such two contributions induce the vortices and splittings of ${\mathit{E}}_{\mathrm{F}1}$ (or ${\mathit{E}}_{\mathrm{F}2}$), and finally form the modulated vortex solitons in the two- and three-level atomic systems, as shown in Figs. 2 –5 below.

 

Fig. 2 (a) Images of ${\mathit{E}}_{\text{F}1}$ versus ${\mathrm{\Delta }}_1$ with $G_2^{\prime }=14.7\mathrm{GHz}$ at 265${}^{\circ }\text{C}$. (b) Images of ${\mathit{E}}_{\text{F}1}$ at ${\mathrm{\Delta }}_1=8\mathrm{GHz}$ with different values of $G_2^{\prime }$ at 265${}^{\circ }\text{C}$. Upper and lower panels are for experimental and simulated results, respectively. (c) Images of ${\mathit{E}}_{\text{F}1}$ at ${\mathrm{\Delta }}_1=8\mathrm{GHz}$ with different temperatures from 200${}^{\circ }\text{C}$ to 300${}^{\circ }\text{C}$. $G_2^{\prime }=9.5\text{GHz}$. Top and bottom rows are the cross sections in y and x directions, respectively. The parameters are $G_1^{\prime }=12.7\mathrm{GHz}$, $G_1=1.8\mathrm{GHz}$, and $G_3=0.2\mathrm{GHz}$ in the two-level system.

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Fig. 5 Under different temperatures, (a) the rotating ${\mathit{E}}_{\mathrm{F}2}$ solitons ($w_1=-1.1$ rad/m) with ${\mathit{E}}_{2,3}$ and ${\mathit{E}}_{1,2}^{\prime }$ on, and (b) stationary nonrotating ${\mathit{E}}_{\text{F}1}$ solitons ($w_1=0$) with ${\mathit{E}}_{1,3}$ and ${\mathit{E}}_{1,2}^{\prime }$ on. Lower images are simulation ($N=3$,$m_{\text{F}1,\text{F}2}=1$) results. (c) Optical vortices of ${\mathit{E}}_{\text{F}1}$ formed by the interferences of three, four, five, and six waves at ${265}^{\circ }\text{C}$, respectively. The parameters are $G_2^{\prime }=19.7\mathrm{GHz}$, $G_1^{\prime }=12.7\mathrm{GHz}$, $G_1=1.8\mathrm{GHz}$, $G_2=1.1\mathrm{GHz}$, $G_3=G_3^{\prime }=0.2\mathrm{GHz}$, and ${\mathrm{\Delta }}_1=6\mathrm{GHz}$ in the two-level system.

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In the FWM process in the two-level system, the conservation of the topological charges must be fulfilled, so the topological charges of the FWM signals are determined by $m_{\text{F}1}=m_1-m_1^{\prime }+m_3$, $m_{\text{F}2}=m_2-m_2^{\prime }+m_3$, $m_{\text{F}3}=m_1-m_1^{\prime }+m_3^{\prime }$, and $m_{\text{F}4}=m_2-m_2^{\prime }+m_3^{\prime }$, where $m_{\text{F}1}$, $m_{\text{F}2}$, $m_{\text{F}2}$, $m_{\text{F}2}$ are the topological charges of the FWM beams ${\mathit{E}}_{\text{F}1}$, ${\mathit{E}}_{\text{F}2}$, ${\mathit{E}}_{\text{F}3}$, ${\mathit{E}}_{\text{F}4}$, respectively. and $m_1$, $m_1^{\prime }$, $m_2$, $m_2^{\prime }$, $m_3$, $m_3^{\prime }$ are those of the beams ${\mathit{E}}_1$, ${\mathit{E}}_1^{\prime }$, ${\mathit{E}}_2$, ${\mathit{E}}_2^{\prime }$, ${\mathit{E}}_3$, ${\mathit{E}}_3^{\prime }$, respectively. The topological charges of two FWM signals ${\mathit{E}}_{\text{F}1}$,${\mathit{E}}_{\text{F}2}$ in the cascade three-level system obey the same conservation rules.

3. Modulated vortex solitons

Figure 2(a) presents the effects of spatial dispersion on the FWM signal ${\mathit{E}}_{\text{F1}}$ in the two-level system, which shows the splitting in the self-focusing region (${\mathrm{\Delta }}_1<0$) and formation of vortex solitons in the self-defocusing region (${\mathrm{\Delta }}_1>0$). In the self-focusing side, while the nonlinear refractive index $n_2$ increases from left to right, ${\mathit{E}}_{\text{F}1}$ beam breaks up from one to three parts via ${\varphi }_{NL}(n_2^{X4})$, with one large and two small pieces. Thus, the ${\mathit{E}}_{\text{F}1}$ beam propagates with discrete diffraction in the self-focusing side. By contrast, in the ${\mathrm{\Delta }}_1>0$ region, the strong dressing fields ${\mathit{E}}_{1,2}^{\prime }$ separate the ${\mathit{E}}_{\text{F}1}$ beam into three spots along a ring ($N=3$). Then these spots propagate through the induced spiral phase polarization ellipse. Such screw dislocations create a stationary beam structure with a phase singularity. The interference among the four beams (${\mathit{E}}_{1,3}$ and ${\mathit{E}}_{1,2}^{\prime }$) induced a modulated vortex pattern with $n_2^{\text{F}1}={\displaystyle \sum _{i=1,i\ne 3}^5\mathrm{\Delta }{n}_2^{X\text{i}}}$. Finally, the ${\mathit{E}}_{\text{F}1}$ beam spot decays into a modulated vortex soliton due to the balanced interaction between the spatial diffraction and the cross-Kerr nonlinearity. There are energy exchanges among three the spots, which rotate around the point of phase singularity. However, when $n_2$ is very small with large detuning or ${\mathrm{\Delta }}_1=0$, the phase singularity disappears and the three spots fuse together into a stable fundamental spot.

Figure 2(b) shows the modulated vortex solitons of ${\mathit{E}}_{\text{F}1}$ for different intensities of the dressing field ${\mathit{E}}_2^{\prime }$ in self-defocusing regime. With increasing ${\mathit{E}}_2^{\prime }$ intensity, the spiral phase of ${\mathit{E}}_{\text{F}1}$ changes into jumping phase between two parts, and the modulated vortex soliton of ${\mathit{E}}_{\text{F}1}$ decays into dipole-mode soliton at high intensity, which is created by the horizontally aligned beam ${\mathit{E}}_1^{\prime }$. Specifically, ${\mathit{E}}_{\text{F}1}$ is circularly modulated by the horizontally-aligned ${\mathit{E}}_1^{\prime }$ and vertically-aligned ${\mathit{E}}_2^{\prime }$ beams. With ${\mathit{E}}_2^{\prime }$ getting stronger, ${\mathit{E}}_{\text{F}1}$ is shifted away from ${\mathit{E}}_2^{\prime }$ and then splits into two parts by ${\mathit{E}}_1^{\prime }$. The dominant phase of ${\mathit{E}}_{\text{F}1}$ is changed gradually from a spiral phase evolution to a jumping phase (i.e., from interference among four beams ${\mathit{E}}_{1,3}$, ${\mathit{E}}_{1,2}^{\prime }$ to the dressing of ${\mathit{E}}_1^{\prime }$).

Figure 2(c) shows ${\mathit{E}}_{\text{F}1}$ soliton cluster with different temperatures between ${200}^{\circ }\text{C}$ and ${300}^{\circ }\text{C}$ in the two-level system. ${\mathit{E}}_{\text{F}1}$ beam is a single spot at both low and high temperature sides. The single spot breaks up into several fragments (soliton cluster) as the temperature increases from ${200}^{\circ }\text{C}$ to ${240}^{\circ }\text{C}$, the nonlinear phase ${\varphi }_{NL}$ gets larger as the temperature (equivalent to propagation distance z) rises, which leads to several splitting parts with weak absorption. As the temperature gets higher with an increased absorption, the beam intensity decreases. ${\varphi }_{NL}$ (proportional to both beam intensity and propagation distance z) reaches its optimal value at ${250}^{\circ }\text{C}$. Moreover, the soliton cluster of ${\mathit{E}}_{\text{F}1}$ results from two contributions in the two-level system: (i) the interference among the four waves (${\mathit{E}}_{1,3}$, ${\mathit{E}}_{1,2}^{\prime }$) with the same frequency induces an interference pattern with $n_2^{\text{F}1}={\displaystyle \sum _{i=1,i\ne 3}^5\mathrm{\Delta }{n}_2^{X\text{i}}}$, and (ii) ${\mathit{E}}_{1,2}^{\prime }$ induce a beam splitting via ${\varphi }_{NL}(n_2^{X4})$ and ${\varphi }_{NL}(n_2^{X5})$. As temperature gets even higher, the dressing beams are significantly absorbed by the hot atoms, so their intensities are reduced and the cross Kerr nonlinear effects are gradually weakened too. Under such condition, the spots merge into a single spot due to strong absorption. So the ideal temperature for the modulated vortex soliton is around ${265}^{\circ }\text{C}$ for the given experimental conditions (i.e., the modulated vortex soliton can be obtained at a certain propagation distance).

In the cascade three-level system with five laser beams (${\mathit{E}}_{1,2,3}$, ${\mathit{E}}_{1,2}^{\prime }$) on, the interference ($n_2^{\text{F}2}={\displaystyle \sum _{i=6,i\ne 8}^9\mathrm{\Delta }{n}_2^{X\text{i}}}$) among three beams ${\mathit{E}}_1$, ${\mathit{E}}_1^{\prime }$, and ${\mathit{E}}_3$ induces a rotating vortex. With the incident beams having topological charges $m_3=1$ and $m_1=m_1^{\prime }=m_2=m_2^{\prime }=0$, the topological charges of the generated FWM signals ${\mathit{E}}_{\text{F}1}$, ${\mathit{E}}_{\text{F}2}$ are $m_{\text{F}1}=m_1-m_1^{\prime }+m_3=1$ and $m_{\text{F}2}=m_2-m_2^{\prime }+m_3=1$, respectively. Figures 3(a) and 3(b) show the rotating vortices of the FWM beams with three spots ($N=3$) for different frequency detunings. Here, the ellipse orientation α approaches to zero and $T_X/T_Y\approx 1.1$. From $I\propto {\cos }^2(N\phi /2){\cos }^2(m\phi +{\mathrm{\Omega }}_2|n_2|)$ with ${\mathrm{\Omega }}_2=\mathrm{sgn}[n_2]w_2$ ($I_2=51W/c{m}^2$, $r=0.25mm$), ${\mathit{E}}_{\text{F}2}$ circumvolves anticlockwise with $n_2>0$ and ${\mathrm{\Omega }}_2=1.63\times {10}^3{\text{W/cm}}^{\text{2}}>0$ in the self-focusing regime [Fig. 3(a)], while moves clockwise with $n_2<0$ and ${\mathrm{\Omega }}_2=-1.63\times {10}^3{\text{W/cm}}^{\text{2}}<0$ in the self-defocusing regime [Fig. 3(b)].

 

Fig. 3 The rotating ${\mathit{E}}_{\text{F}2}$ solitons with (a) ${\mathrm{\Omega }}_2>0$ and (b) ${\mathrm{\Omega }}_2<0$ versus ${\mathrm{\Delta }}_1$. (c) Stationary nonrotating ${\mathit{E}}_{\text{F1}}{}_{}$ solitons ($w_2=0$) versus ${\mathrm{\Delta }}_1$. Lower images are the simulated results ($N=3$, $m_{\text{F}1,\text{F}2}=1$). The parameters are $G_2^{\prime }=19.7\mathrm{GHz}$, $G_1^{\prime }=12.7\mathrm{GHz}$, $G_1=1.8\mathrm{GHz}$, $G_2=1.1\mathrm{GHz}$, and $G_3=0.2\mathrm{GHz}$ in the cascade three-level system at ${265}^{\circ }\text{C}$.

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Figure 3(c) presents the stationary solitons with $w_2=0_{}$ in the cascade three-level system. $n_2^{X4}$ can be a positive value with resonant dressing of ${\mathit{E}}_1^{\prime }$. When $n_2^{X1}$ and $n_2^{X2}$ have negative values under the self-defoucsing condition, the superposed $n_2^{\text{F}1}={\displaystyle \sum _{i=1,i\ne 3}^4\mathrm{\Delta }{n}_2^{X\text{i}}}$ is close to zero (or ${\varphi }_{NL}\approx 0$). Under this condition, a uniform energy flow exists along the ring, and nonrotating ($w_2=0$) spatially-localized multihump structures can be obtained.

Comparing to the three-level system, there exist five nearly degenerate frequency waves (${\mathit{E}}_{1,2,3}$, ${\mathit{E}}_{1,2}^{\prime }$) in the two-level system. Figure 4(a) shows the rotating vortices of the FWM beam with four spots ($N=4_{}$) for different frequency detunings. With the probe beam ${\mathit{E}}_3$ carrying topological charge $m_3=1$, the topological charges of the generated FWM signals ${\mathit{E}}_{\text{F}1}$, ${\mathit{E}}_{\text{F}2}$ are $m_{\text{F}1}=m_1-m_1^{\prime }+m_3=1$ and $m_{\text{F}2}=m_2-m_2^{\prime }+m_3=1$, respectively, where $m_1$, $m_1^{\prime }$, $m_2$, and $m_2^{\prime }$ are all zeros. The modulated vortex pattern ($N=4$) of ${\mathit{E}}_{\text{F}2}$ is induced by the interference of five waves, and the nonresonant dressing field ${\mathit{E}}_2^{\prime }$ induces a splitting via ${\varphi }_{NL}(n_2^{X10})$. For ${\mathit{E}}_{\text{F}2}$, ${\mathit{E}}_2^{\prime }$ is the nonresonant dressing field, energy flow exists along the ring of spots unequally, inducing a modulated vortex [Fig. 4(a)].

 

Fig. 4 (a) Anticlockwise rotating (${\mathrm{\Omega }}_2=-1.53\times {10}^3{\text{W/cm}}^{\text{2}}$) ${\mathit{E}}_{\text{F}2}$ solitons and (b) stationary nonrotating ${\mathit{E}}_{\text{F}1}$ solitons with four spots versus ${\mathrm{\Delta }}_1$ at ${265}^{\circ }\text{C}$ in the two-level system. Lower images are simulated ($N=4$,$m_{\text{F}1,\text{F}2}=1$) results. The parameters are $G_2^{\prime }=15\mathrm{GHz}$, $G_1^{\prime }=12.7\mathrm{GHz}$, $G_1=1.8\mathrm{GHz}$, $G_2=1.1\mathrm{GHz}$, and $G_3=0.2\mathrm{GHz}$.

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In Fig. 4(b), there exists a stationary, four-spot modulated vortex soliton with $N=4$ and non-uniform energy distribution for different detunings in the two-level system. The vortex pattern (horizontally-oriented polarization ellipse with $\alpha =0$ and $T_X/T_Y\approx 1.5$) is induced by the interference of the five beams. With the nonresonant dressing of ${\mathit{E}}_2^{\prime }$, all terms in $n_2^{\text{F}2}$ have the same negative sign [Fig. 4(a)], but the positive $n_2^{X4}$ in $n_2^{\text{F}1}={\displaystyle \sum _{i=1}^5\mathrm{\Delta }{n}_2^{X\text{i}}}$ has opposite sign with the other terms $n_2^{X1,2,3,5}$ in $n_2^{\text{F}1}$ due to the resonant dressing of ${\mathit{E}}_1^{\prime }$, so one can get $n_2^{\text{F}1}\approx 0$. Therefore, there exist four spots in the stationary (${\varphi }_{NL}\approx 0$) modulated vortex soliton of ${\mathit{E}}_{\text{F}1}$ with $N=4$ and energy is mainly stored in one diagonal pair of spots [Fig. 4(b)], which results from the resonant dressing of ${\mathit{E}}_1^{\prime }$.

The radially symmetric vortex solitons ($m_{\text{F}1,\text{F}2}=1$) in a self-defocusing medium are depicted in Figs. 5(a) and 5(b), which separately demonstrate vortices and steady crescent FWM vortex solitons under different temperatures (atomic densities). The effective propagation distance z increases with the temperature. According to the solution of Eq. (1) $E_{\text{F}2}\propto E\sec \mathrm{h}[E{({\mathrm{k}}_{\text{F}2}{n}_2^{S2}/n_1)}^{1/2}(r-R_0)]\cos (N\phi /2)\exp (i{m}_{\text{F}2}\phi +i{w}_{1}z)$, the spots rotate with an angular velocity $w_1$. In the two-level system with $n_2^{\text{F}2}={\displaystyle \sum _{i=6,i\ne 7}^{10}\mathrm{\Delta }{n}_2^{X\text{i}}}$, the vortex pattern of ${\mathit{E}}_{\text{F}2}$ is induced by the interference of ${\mathit{E}}_{2,3}$, ${\mathit{E}}_{1,2}^{\prime }$ while the dressing fields ${\mathit{E}}_{1,2}^{\prime }$ generate the circular splittings. Here, ${\mathit{E}}_{\text{F}2}$ forms a crescent FWM modulated vortex soliton with an anticlockwise rotation [Fig. 5(a)]. Moreover, $w_{1}z$ changes 300° from ${245}^{\circ }\text{C}$ to ${275}^{\circ }\text{C}$. When setting $z=1$ at ${245}^{\circ }\text{C}$, we obtain $w_1=-1.1$ rad/m, which is close to the theoretical value of $w_1=-1.03$ rad/m. However, in Fig. 5(b) with the resonant dressing ${\mathit{E}}_1^{\prime }$ and $n_2^{\text{F}1}={\displaystyle \sum _{i=1,i\ne 3}^5\mathrm{\Delta }{n}_2^{X\text{i}}}\approx 0$, the propagation length of ${\mathit{E}}_{\text{F}1}$ is about 27 times longer than the diffraction length($L_{\text{D}}\approx 0.67cm$), which are both much shorter than the coherence length of ${\mathit{E}}_{\text{F}1}$ described above ($L_{F1}^c\to \infty$). So we can conclude that ${\mathit{E}}_{\text{F}1}$ beam becomes a stationary vortex soliton at certain $I_1$ and z values.

Last, we let all six beams on, and set the ${\mathit{E}}_{1,2,3}$ and ${\mathit{E}}_3^{\prime }$ beams just 10 times weaker than the dressing beams ${\mathit{E}}_{1,2}^{\prime }$. Figure 5(c) shows the optical vortices created by the interferences of three, four, five, six waves (and the dressing fields) in the two-level system, respectively. Initially, there are three beams ${\mathit{E}}_{1,3}$, ${\mathit{E}}_1^{\prime }$ on, which create the image 3 in Fig. 5(c) As fourth interference beam ${\mathit{E}}_2^{\prime }$ is added, the split spots change from two to three [image 4 in Fig. 5(c)]. Similarly, as beams ${\mathit{E}}_2$ and ${\mathit{E}}_3^{\prime }$ are added gradually, the interference beams increase from four to five (image 5), and then to six (image 6), the split spots in the vortex patterns of ${\mathit{E}}_{\text{F}1}$ then change from three to four, and then to six, respectively, along the ring, and the shape of the vortex ellipse tends to become more circular. The final superposition nonlinear index is $n_2^{\text{F}1}={\displaystyle \sum _{i=1}^5\mathrm{\Delta }{n}_2^{X\text{i}}}+n_2^{X11}{\left|{\mathit{E}}_3^{\prime }\right|}^2$, where $n_2^{X11}$ is the nonlinear index induced by ${\mathit{E}}_3^{\prime }$. The FWM modulated vortex solitons are created jointly by the effects of the complex patterns induced by the multiple interference waves [13,14] and the cross-Kerr nonlinear dispersions induced by the dressing field [11].

4. Conclusion

In conclusion, we have experimentally demonstrated controllable modulated vortex solitons of the degenerate and nondegenerate FWM beams created by the interference patterns via the superposing three or more waves and the cross-Kerr nonlinear dispersion due to atomic coherence in the two-level and cascade three-level atomic systems. The vortex angular velocity and intensity split peaks of the FWM modulated vortex solitons can be controlled by laser intensities, nonlinear dispersion, as well as atomic density. Our theoretical model can explain the observed FWM modulated vortex solitons very well. The current study has opened the door to better understand the formation and dynamics of complex vortex solitons, especially in multi-level atomic media, in which more parameters can be easily controlled. Understanding the formation and control of complex solitons can lead to potential applications in soliton communications and computations.