1. Introduction

In recent years, spatial optical solitons and their interactions have attracted considerable attention because of their potential applications in many areas including all-optical switching, optical data storage, distribution, and processing [1]. The interaction between solitons can result in soliton fusion, fission, spiraling [2, 3] and the formation of more complicated localized states. It has been shown that several beams can interact with each other to produce multi-component vector solitons. The three-component dipole vector solitons have been studied theoretically [4] and observed in photorefractive crystals [5]. Dipole solitons in nonlocal nonlinear media have also been investigated [6–9]. It was suggested that the non-locality can significantly modify the interaction between solitons. In the past decade, vector solitons in periodic photonic lattices have become an active field of research [10, 11]. These lattices can be formed by either interfering pairs of optical beams, or by using amplitude masks. It will exhibit more novel propagation phenomena and leads to many novel spatial solitons due to the periodic refractive index in the lattice. Very recently, we have observed four-wave mixing (FWM) three-component dipole vector solitons in laser-induced atomic gratings [12], in which the linear and nonlinear index modulations enhanced by the atomic coherence play important role. The easy controls of experimental parameters in the interaction between the multi-level atom system and multi-beams make the current system can be used to observe the formation of multi-component spatial solitons.

Composite vortex solitons have been widely studied in self-focusing and self-defocusing media [13], such as radially symmetric vortex, rotating soliton clusters [14] and azimuthally modulated vortex solitons [15, 16]. In self-focusing nonlinear media, vortex solitons may undergo azimuthal instability and they decay into fundamental solitons during propagation [17]. Several mechanisms are exploited to overcome such instability. The saturation nonlinearity can arrest the instability [18]. Mutual coupling between the components of vector solitons can suppress the azimuthal instability [19]. Many theoretical studies of vortex vector solitons have been reported [20–24]. Some studies show if the components of a vector soliton are made sufficiently incoherent in the transverse dimension, the instability can be eliminated [22, 25]. Counter-rotating vortex vector solitons can be stable in self-focusing saturable media [23]. Furthermore, the existence of N-component (N>3) vector solitons that carry different topological charges has been predicted theoretically [24]. The study on the interactions of vector solitons [26, 27] allows us to understand the formation of more complicated solitons. However, the experimental observation reports of vortex vector solitons are very seldom till now. The first experimental observation of the two-component vortex vector solitons with hidden vorticity has been reported in nematic liquid crystals recently [28].

In this paper, we report our experimental observations of dipole and vortex multi-component vector solitons in generated four-wave mixing signals in a two-level atomic system. It includes two degenerate and six nondegenerate FWM components. We analyze the interactions between such co-propagating FWM soliton components and the formations of multi-component dipole- and vortex-mode vector solitons. The controllability of these multi-component solitons by the frequency detunings of input laser beams is also investigated.

2. Basic theory and experimental scheme

Our experiments are carried out in a Na atom vapor oven. Energy levels $|0〉(3S_{1/2})$ and $|1〉(3P_{3/2})$ form a two-level atomic system interacting with six laser beams. These laser beams are all adjusted to connect the transition between $|0〉$ and $|1〉$, of which the resonant frequency is $\Omega$ (Fig. 1(a) ). Laser beams are spatially aligned in the configuration shown in Fig. 1(b). Two laser beams $E_1$ (with frequency ${\omega }_1$, wavevector $k_1$, and Rabi frequency $G_1$) and ${E^{\prime }}_1$ (${\omega }_1$, ${k^{\prime }}_1^{}$, ${G^{\prime }}_1^{}$), with a small angle ${\theta }_1\approx {0.3}^{\circ }$ between them, propagate in the opposite direction of the weak probe beam $E_3$ (${\omega }_3$, $k_3$, $G_3$). These three laser beams come from the same dye laser DL1 (10 Hz repetition rate, 5 ns pulse width, and 0.04 cm⁻¹ linewidth) with the frequency detuning ${\Delta }_1={\omega }_1-\Omega$, and the wave-vectors of them are all nearly in the x-o-z plane. Other three laser beams $E_2$ (${\omega }_2$, $k_2$, $G_2$), ${E^{\prime }}_2$ (${\omega }_2$, ${k^{\prime }}_2^{}$, ${G^{\prime }}_2^{}$) and ${E^{\prime }}_3$ (${{\omega }^{\prime }}_3$,${k^{\prime }}_3^{}$,${G^{\prime }}_3^{}$) are from another dye laser DL2 (which has the same characteristics as the DL1) with a frequency detuning ${\Delta }_2={\omega }_2-\Omega$. Among them, $E_2$ co-propagates with $E_1$; ${E^{\prime }}_3$ co-propagates with $E_3$, while ${E^{\prime }}_2$ propagates with a small angle ${\theta }_2\approx {0.3}^{\circ }$ between $E_2$ and its wave-vector in the y-o-z plane. In this case, there will be eight FWM signals coexisting in the same atomic systems: $E_{F1}$(${\omega }_1$, $k_{F\text{1}}=k{}_1-{k^{\prime }}_1+k_3$),$E_{F2}$(${\omega }_2$, $k_{F\text{2}}=k{}_1-{k^{\prime }}_1+{k^{\prime }}_3$),$E_{F5}$(${\omega }_2$, $k_{F\text{5}}=k{}_2-{k^{\prime }}_1+k_3$),$E_{F6}$ ($2{\omega }_2-{\omega }_1$, $k_{F\text{6}}=k{}_2-{k^{\prime }}_1+{k^{\prime }}_3$), $E_{F3}$(${\omega }_1$,$k_{F\text{3}}=k{}_2-{k^{\prime }}_2+k_3$), $E_{F4}$(${\omega }_2$, $k_{F\text{4}}=k{}_2-{k^{\prime }}_2+{k^{\prime }}_3$),$E_{F7}$($2{\omega }_1-{\omega }_2$, $k_{F\text{7}}=k{}_1-{k^{\prime }}_2+k_3$),$E_{F8}$ (${\omega }_1$, $k_{F\text{8}}=k{}_1-{k^{\prime }}_2+{k^{\prime }}_3$). According to the phase-matching conditions, $E_{F1,2,5,6}$ ($E_{F3,4,7,8}$) propagate in the opposite direction of ${k^{\prime }}_1$(${k^{\prime }}_2^{}$) with very small angles among them. The stronger dressing beams ${E^{\prime }}_{1,2}$ are approximately 10 times stronger than the pump beams $E_{1,2}$, and 1000 times stronger than the weak probe beam $E_3$, ${E^{\prime }}_3$ and the generated FWM beams. The coherence length of the FWM signals is $L_{\text{F}}^c=\pi /\left|\Delta k\right|$, where $\Delta k$ is the mismatching in the wave mixing process. According to the phase-matching conditions, we can obtain the coherence lengths of these FWM signals: $L_{\text{F1}}^c=2\pi c{\omega }_1/[n_1{\omega }_1\left|{\omega }_1-{\omega }_2\right|{\theta }^2]\to \infty$, $L_{\text{F2}}^c=2\pi c{\omega }_1/[n_1{\omega }_1\left|{\omega }_1-{\omega }_2\right|{\theta }^2]=4.58\text{km}$, $L_{\text{F3}}^c=2\pi c{\omega }_1/[n_1{\omega }_1\left|{\omega }_1-{\omega }_2\right|{\theta }^2]=3.1\text{cm}$,$L_{\text{F4}}^c=2\pi c{\omega }_2/[n_1{\omega }_1\left|{\omega }_1-{\omega }_2\right|{\theta }^2]\to \infty$, $L_{\text{F5}}^c=2\pi c(2{\omega }_1-{\omega }_2)/n_1\left|({\omega }_1-{\omega }_2)(8{\omega }_1-4{\omega }_2+{\omega }_1{\theta }^2)\right|=3.1\text{cm}$,$L_{\text{F6}}^c=L_{\text{F7}}^c=\pi c/[2n_1\left|{\omega }_1-{\omega }_2\right|]=3.14\text{cm}$, $L_{\text{F8}}^c=2\pi c(2{\omega }_2-{\omega }_1)/n_1\left|({\omega }_2-{\omega }_1)(8{\omega }_2-4{\omega }_1+{\omega }_2{\theta }^2)\right|=4.58\text{km}$. The diffraction length of the generated FWM beams is defined as $L_D=k_F{w}_0^2$, where $w_0$ is the spot size of the FWM beams. In two-level system, the diffraction length of dipole soliton is about 1.5mm.

 

Fig. 1 (a) Two-level system with six laser beams tuned to the same transition. (b) Spatial beam geometry used in the experiment. (c) and (d) The horizontally- and vertically-aligned EIG1 and EIG2, respectively.

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In this system, several laser beams and FWM signals pass through an atomic medium, and therefore the cross-phase modulation (XPM) and self-phase modulation (SPM) can effectively affect the propagation and spatial patterns of the propagating FWM signals. When the spatial diffraction is balanced by the XPM or SPM Kerr nonlinearity, many types of spatial solitons, such as dipole-mode solitons, vortex solitons, will be observed.

In the experiment, the key to observe dipole-mode solitons is to create a laser-induced index gratings with sufficiently high index contrast (via Kerr nonlinearity $n_{2}I$) in the atomic medium [12]. The sodium atomic density needs to reach $2.9\times {10}^{13}c{m}^{-3}$ (T = 250${}^{\circ }C$), which can produce the needed variation in the nonlinear index of $\Delta n=1.94\times {10}^{-4}$. The strong beams $E_1$ and ${E^{\prime }}_1$ ($E_2$ and ${E^{\prime }}_2$) both can induce atomic coherence, which can modify the linear susceptibility and nonlinear Kerr effect significantly. Therefore, the spatial periodic intensity derived from the interference of $E_1$ and ${E^{\prime }}_1$ ($E_2$ and ${E^{\prime }}_2$) leads to periodic variation of the linear and nonlinear refractive index, finally induces an electromagnetic induced gratings EIG1 (EIG2). The fringe spacings of EIG1 and EIG2 are determined by ${\Lambda }_i={\lambda }_i/{\theta }_i$ (i = 1 for EIG1, and 2 for EIG2). The periodically modulated total linear and nonlinear refractive index is given by the expression $n(\zeta )=n_0+\delta n_1\cos (2\pi \xi /{\Lambda }_i)+\delta n_2\cos (4\pi \xi /{\Lambda }_i)$, where $n_0=n_{01}+n_{02}=(1+G_a)G_{F2}N{\mu }_{10}/({\epsilon }_{\text{0}}{E}_{\text{3}}{\Delta }_1)$ is the spatial uniform refractive index; $\delta n_1=G_a(n_{01}^2-1)/4n_{01}+(G_a^2/2+G_a)(n_{02}^2-1)/2n_{02}$ and $\delta n_2=(n_{02}^2-1)G_a^2/16n_{02}$ are the coefficients of the spatially varying terms in the modulated index, and $G_a=G_2^2/[({\Delta }_1+{\Delta }_2){\Delta }_1]$. Similar to the photonic crystal, the spatial periodic index in the grating can lead to photonic band gap. The width of such gap is given by $\Delta \Omega =2{\Omega }_0{n}_{2}I/\pi n_1$, where ${\Omega }_0$ is the center frequency of it. If ${\Delta }_1$(or ${\Delta }_2$) falling within certain range, the Bragg reflection signals of incident beams can be significantly enhanced and the corresponding transmission signals can be suppressed greatly. Therefore, interacting with the gratings above, the probe beam $E_3$ or ${E^{\prime }}_3$ will experience intensive Bragg reflection. The FWM signals $E_{F1,2}$ and $E_{F3,4}$ can be considered as the results of the electromagnetically-induced diffraction (EID) of the probe beam $E_3$ or ${E^{\prime }}_3$ by the horizontally- and vertically-aligned EIG1 and EIG2, respectively. When the diffraction of the FWM signal is balanced by the cross-Kerr nonlinearity in propagation, the dipole-mode soliton generated from $E_{F1,2}$ and $E_{F3,4}$ will be produced. Moreover, the beams $E_2$ and ${E^{\prime }}_1$ (or beams $E_1$ and ${E^{\prime }}_2$) can also induce their moving grating EIG3 (or EIG4). The wave vector of the grating is $k_g=k_2-k_1{}^{\prime }$ (or $k_g=k_1-k_2{}^{\prime }$) with the phase velocity $v=\Delta \omega /\left|k_g\right|$, where $\Delta \omega ={\omega }_2-{\omega }_1$ is the frequency difference between the two induced beams [29]. Similar to the signals $E_{F1,2}$ and $E_{F3,4}$, the FWM signals $E_{F5,6}$ and $E_{F7,8}$ are the EID results of the probe beam $E_3$ or ${E^{\prime }}_3$ by the moving gratings EIG3 and EIG4, respectively.

For simplicity, we take the plane wave assumption and express the two classes of signals induced by the horizontally- and vertically-aligned EIG as $E_{Fp}=A_{Fp}(\zeta )\exp (i{k}_{Fp}z)p=1,2,5,6$ and$E_{Fq}=A_{Fq}(\zeta )\exp (i{k}_{Fq}z)q=3,4,7,8$, respectively. These FWM fields can couple to each other and the propagation of them satisfies the following evolution equations in the medium with Kerr nonlinearity:

The N-component vector soliton can be constructed from simple soliton components. For interactions between soliton components in medium with saturation nonlinearities, the critical angle ${\theta }_c=(n_{\max }-n_{\min })/n_{\max }$ [31] plays a key role, here $n_{\max }$ and $n_{\min }$ are the maximum and minimum values of the nonlinear refractive index induced by the soliton, respectively. When the collision angle $\theta$ is less than ${\theta }_c$, a beam can be coupled into waveguide induced by other beams, so two soliton components can fusion or fission. In our experiment, in each class of FWM signal, four soliton beams co-propagate with very small angles among them, and therefore they can fuse with each other to form a new soliton in certain condition. The superposition of the four horizontally-aligned dipole components $E_{F1,2,5,6}$ (with topological charge$m_{F1,2,5,6}=-1$) generates a new horizontally oriented dipole-mode soliton (four-component dipole-mode soliton) with total amplitude ${\displaystyle \sum _p{E}_{Fp}}$ (p = 1,2,5,6). Similarly, the superposition of four vertically-aligned dipole components $E_{F3,4,7,8}$ ($m_{F3,4,7,8}=1$) generates a vertically oriented four-component soliton with amplitude ${\displaystyle \sum _q{E}_{Fq}}$ (q = 3,4,7,8). Two nodeless probe beams $E_3$ and ${E^{\prime }}_3$ ($m_{k3,k{3}^{\prime }}=0$) act as the fundamental components which provide an attractive force tightly binding both the dipoles and the multi-component ($N=10$) structure. For the vector soliton composed of all these components, the zero total angular momentum $m_{k3,k{3}^{\prime }}+m_{F1,2,5,6}+m_{F3,4,7,8}=0$ makes the structure stable. Therefore, the total intensity ($I={\left|E_{k3,k{3}^{\prime }}\right|}^2+|{\displaystyle \sum _p{E}_{Fp}}{|}^2+|{\displaystyle \sum _q{E}_{Fp}}{|}^2$) reaches a steady state in propagation after a long distance. On the other hand, multi-component structure can also be stabilized by using of optical lattices or induced gratings. The lattice creates an optical waveguide which could prevent the constituents of the multi-component soliton from expansion or contraction.

The vortex soliton can also be created in this two-level system in which six nearly degenerate frequency waves ($E_{1,2,3}$ and ${E^{\prime }}_{1,2,3}$) exist. Specifically, when three or more plane-waves overlap in the medium, the interference patterns can induce vortex-like index modulation with phase singularities. Furthermore, the diffraction of a light beam can be compensated by the nonlinearity and the FWM modulated solitons be created. The propagation equations of vortex solitons in cylindrical coordinate are written as the follows [32]:

The modulated vortex and dipole solitons are characterized by two independent integer numbers, topological charge m and the number of intensity peaks M. In our experiment, they are created jointly by the interference of multiple beams and the cross-phase modulation of the dressing and pump fields. The soliton solutions can be written as, $E_{Fp}=u_p\sec h[u_p{(k_{Fp}{n}_2^{Sp}/n_0)}^{1/2}(r-r_p)]\cos (M\phi /2)\exp (i{m}_{Fp}\phi +i{\varphi }_p)\exp (i{k}_{Fp}z)$and $E_{Fq}=u_q\sec h[u_q{(k_{Fq}{n}_2^{Sq}/n_0)}^{1/2}(r-r_q)]\cos (M\phi /2)\exp (i{m}_{Fq}\phi +i{\varphi }_q)\exp (i{k}_{Fq}z)$, where $u_{p,q}$ are soliton amplitudes; $r_{p,q}$ are initial peak positions; ${\varphi }_{p,q}=2k_{Fp,q}{n}_{2}z{I}_{2,1}{e}^{-r^2/2}/(n_0{I}_{Fp,q})$ are nonlinear phase shifts introduced by the Kerr effect. For rotating dipole solitons, the number of intensity peaks $M=2$. For modulated vortex solitons, $M\ge 3$. Furthermore, if $m_F\phi +{\varphi }_{}=0$, the angular velocity of the modulated soliton becomes $\omega =0$ and the rotation of the soliton cancels. The topological charge m of the vortex soliton is determined by ${\delta }_{r,i}$${\delta }_r=-\mathrm{arc}\tan ({\displaystyle \sum _{i=1}^n{E}_i{n}_i{k}_{iy}\sin T_{\psi i}}/{\displaystyle \sum _{i=1}^n{E}_i{n}_i{k}_{ix}\sin T_{\psi i}})$, ${\delta }_i=-\mathrm{arc}\tan ({\displaystyle \sum _{i=1}^n{E}_i{n}_i{k}_{iy}\cos T_{\psi i}}/{\displaystyle \sum _{i=1}^n{E}_i{n}_i{k}_{ix}\cos T_{\psi i}})$, with n being the number of laser beams which create the spiral phase plate and $n_i$ being the nonlinear refractive index; $T_{\psi i}={\psi }_{i0}+n_i(k_{ix}{T}_x+k_{iy}{T}_y)$,with $(T_x,T_y)$ being the coordinates of the singularity point. When ${\delta }_r>{\delta }_i$ (${\delta }_r<{\delta }_i$),$m=1$ ($m=-1$), which means the phase changes clockwise (anticlockwise). To observe vortex soliton, the ideal temperature is around 265${}^{\circ }C$(atomic density needs to reach $5.6\times {10}^{13}c{m}^{-3}$). In suitable condition, several co-propagating vortex solitons also can fuse to form a new multi-component vector soliton.

3. Experimental observation of multi-component solitons

When a probe beam and two pump beams are turned on, single FWM signal can be obtained. Figure 2 presents the dipole-like patterns of eight FWM signals in the self-focusing region. Because beams $E_1$ and ${E^{\prime }}_1$ (or $E_2$ and ${E^{\prime }}_1$) are horizontally aligned in the x-z plane, the grating EIG1 and EIG3 have horizontal orientation (Fig. 1(b)). Therefore, dipole components $E_{F1,2,5,6}$ induced by EIG1 and EIG3 have their two humps horizontally along x-axis. For the same reason, the dipole components $E_{F3,4,7,8}$ induced by vertically oriented grating EIG2 and EIG4 have their two humps along the vertical y-axis. The probe beam $E_3$ (or ${E^{\prime }}_3$) is deviated from the FWM signals with a small angle (${\theta }_1\approx {0.3}^{\circ }$), so it is employed as the fundamental soliton. The interaction between the nodeless probe beam and the arbitrary dipole-like FWM signal forms a basic dipole vector soliton, denoted as (0,-1) for $E_{F1,2,5,6}$ and (0,1) for $E_{F3,4,7,8}$. When the frequency detuning ${\Delta }_2$ is scanned from the large negative values to the resonant frequency, except $E_{F1}$, other dipole-mode components decay into a nodeless fundamental one at resonance or large frequency detunings. This detuning dependence can be explained by the nonlinear refractive index $n_2$ (Fig. 3(d) ) and the nonlinear phase shift ${\varphi }_{p,q}=2k_{Fp,q}{n}_{2}z{I}_{2,1}{e}^{-r^2/2}/(n_0{I}_{Fp,q})$, which determines the spatial splitting of the FWM signal. When ${\Delta }_2$ is set at resonance or the large negative detuning, $n_2$ and ${\varphi }_{p,q}$ become minimum, so the index contrast of EIGs is not high sufficiently to maintain the dipole-mode pattern of the FWM signal and they decay into a nodeless ones. The field $E_{F1}$ always keeps dipole-like pattern because its cross Kerr coefficient is not affected by $E_2$, ${E^{\prime }}_2$ and ${E^{\prime }}_3$. The right panels of Fig. 2 describe the horizontal (vertical) size of the beams $E_{F1,2,5,6}$($E_{F3,4,7,8}$). It is shown that the beam sizes of these eight dipole components keep constant when the frequency detuning ${\Delta }_2$ is changed from $-40\text{GHz}$ to $-10\text{GHZ}$. This indicates that the dipole soliton is formed in this region.

 

Fig. 2 (a) Images (left) and horizontal-sizes (right) of FWM signals $E_{F1,2,5,6}$ at different detuning ${\Delta }_2$, $E_{F1}$(squares), $E_{F2}$(circles), $E_{F5}$(triangles) and $E_{F6}$(reverse triangles). (b) Images (left) and vertical-sizes (right) of FWM signals $E_{F3,4,7,8}$ at different detuning ${\Delta }_2$, $E_{F3}$ (squares), $E_{F4}$(circles), $E_{F7}$(triangles) and $E_{F8}$(reverse triangles).

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Fig. 3 (a1,2) Images of $E_{F7}$ at different powers of (a1) $E_1$, (a2) ${E^{\prime }}_2$. (a3) Images of $E_{F1}$ for different spatial configuration of $E_1$ and ${E^{\prime }}_1$. From left to right, the intersection position of $E_1$ and ${E^{\prime }}_1$ is adjusted to change from the front to the back of the heat-pipe oven. (b) Beam vertical-size of $E_{F7}$ versus $P_1$ (Squares) and $P_2$ (circles). (c) Beam vertical-size (triangles) and horizontal-size (asterisks) of $E_{F1}$ versus the intersection position of $E_1$ and ${E^{\prime }}_1$. Here the distance between the intersection position and the oven centre is denoted as l. (d) Nonlinear refractive index $n_2$ of $E_{F4}$. (e) Images of the pure FWM signal $E_{F4}$ and coexisting signals $E_{F4}+E_{F3}$, $E_{F4}+E_{F8}$, $E_{F1}+E_{F2}$ and $E_{F1}+E_{F5}$, respectively.

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In order to observe the influence of other experimental conditions on the pattern formation of the FWM signals, the powers, positions, and frequency detunings of the involved laser beams are adjusted. Figure 3(a) gives the beam profiles of the FWM signal $E_{F7}$ versus the input powers of $E_1$ and ${E^{\prime }}_2$. The spatial splitting of signal $E_{F7}$ is determined by the nonlinear phase shift introduced by the EIG4 which created by the beams $E_1$ and ${E^{\prime }}_2$. At low power $I_{1,2}$, the beam $E_{F7}$ presents a nodeless spot. With increasing $I_{1,2}$, $E_{F7}$ splits into two coherent spots. Now the dipole-mode soliton can be formed if the spatial diffraction is balanced by the XPM or SPM Kerr nonlinearity. The vertical-size of beam $E_{F7}$ remains unchanged in a specific region. Further with increasing $I_{1,2}$, the nonlinear phase shift ${\varphi }_7$ becomes large, so the splitting of $E_{F7}$ become significant, and the vertical-size of $E_{F7}$ increases. Besides the pump fields power, spatial configuration of laser beams also can affect the spatial splitting. In Fig. 3(a), the intersection of laser beams $E_1$ and ${E^{\prime }}_1$ is adjusted from the front to the back of the heat-pipe oven. We can see that $E_{F1}$ is split in x direction when $E_1$ and ${E^{\prime }}_1$ set at the front and back of the oven, but in y direction when they are at the middle of the oven. Figure 3(c) shows the variation of the horizontal- and vertical-size of $E_{F1}$ in this transformation. This transformation can be explained by the different spatial position of ${E^{\prime }}_1$ overlapped with $E_{F1}$.

Figure 3(e) shows the splitting of the pure FWM signal $E_{F4}$ and coexisting signals $E_{F4}+E_{F3}$ and $E_{F4}+E_{F8}$ when ${\Delta }_2$ is changed. It is theoretically obtained in Fig. 3(d) that the Kerr coefficient is negative (positive) with ${\Delta }_2>0$ (${\Delta }_2<0$), which will lead to the defocusing (focusing) of the FWM signal. It can be seen that the beam $E_{F4}$ splits along vertical direction in the self-focusing region, but it becomes horizontal-splitting in the self-defocusing region with small ${\Delta }_2$. Further increasing ${\Delta }_2$, the beam spot becomes large and presents three spots. Subsequently, it becomes vertical dipole-mode and decays into a nodeless spot when ${\Delta }_2$ far away from resonance. Such evolution can be explained by the follows. As mentioned above, in the self-focusing region, the dipole component $E_{F4}$ induced by vertically-aligned EIG2 splits along the y-axis. But in ${\Delta }_2>0$ region, the FWM $E_{F4}$ is shifted along the down-right direction due to the repulsion effect ($n_2<0$) of the cross Kerr nonlinearity of the strong beam ${E^{\prime }}_2$, and thus, the beam $E_{F4}$ splits in x-direction [33]. The spatial shift of the FWM signal is proportional to $\left|n_2\right|$. At a proper value, horizontal- and vertical-splitting appear simultaneously. With further increasing of ${\Delta }_2$ ($\left|n_2\right|$ deceases), the spatial shift of $E_{F4}$ becomes small. Therefore, the beam $E_{F4}$ gets back to the unshifted position and suffers from vertical-splitting. It decays into single spot when $\left|n_2\right|$ is close to zero.

As the probe beam $E_3$ added, there exist two dipole-mode components $E_{F3}$ and $E_{F4}$ ($m_{F3,4}=1$). They co-propagate in the opposite direction of ${k^{\prime }}_2^{}$with a very small angle between them. Because they have different frequencies, there exists incoherent attraction force between them [2]. The soliton interaction likes real particles [9], and the nonlinear interaction of the solitons is analogous to the Coulomb interaction. Thus the attraction force between two dipole solitons is electric dipole-like [34, 35], and can be written as $F=[C_{20}{\displaystyle \int {\sigma }_{p1}{v}_{20}^1(r,\phi )}rdrd\phi {\displaystyle \int {\sigma }_{p2}{v}_{20}^2(r,\phi )rdrd\phi }]/R^7$, where $v_{{}_{20}}^{\text{i}}(r,\phi )=C{}_{20}(\sqrt{2}r/{\omega }_{0s}){}^2{e}^{-r^2/{\omega }_{0s}^2}\cos 2\phi$($i=1,2$) is the intensity distribution profile of one dipole soliton, ${\omega }_{0s}$ the waist radius of Gaussian beams, ${\sigma }_{pi}$ the average power in unit area of soliton beam, $C_{20}$ the interaction coefficient of dipole solution, and $R$ distance between the centers of the two dipole components. This attraction force can prevent soliton beams from diverging and makes the beams approach each other. When the distance between the solitons is too small, the two interacting solitons can exchange energy by coupling light into the waveguide induced by each other and eventually fuse. Therefore, the dipole components $E_{F3}$ and $E_{F4}$ fuse to form a new dipole-mode vector soliton which have their two humps along the vertical y-axis. We denote this multi-component soliton structure as (0,1,1). The horizontal-splitting of the FWM signal in self-defocusing region disappears when $E_3$ is on. This indicates that the field $E_3$ could suppress the spatial shift and thus the horizontal-splitting of $E_{F4}$. When the pump beam $E_1$ is turned on, there exist two incoherent dipole-mode components $E_{F4}$ and $E_{F8}$ ($m_{F4,8}=1$). They also can fuse to form a dipole-mode vector soliton. Figure 3(e) also presents the coexisting signals $E_{F1}+E_{F2}$ and $E_{F1}+E_{F5}$ which have topological charge $m_F=-1$. The generated multi-component solitons have their two humps horizontally along x-axis. We denote this multi-component soliton structure as (0,-1,-1).

Figure 4(a) depicts the superposition of four dipole-mode components. When five laser beams $E_{1,2,3}$ and ${E^{\prime }}_{1,3}$ are turned on (dressing field ${E^{\prime }}_2$ is blocked) at the same time, there are four horizontal-splitting dipole-mode components $E_{F1,2,5,6}$ co-propagating in the opposite direction of ${k^{\prime }}_1$ with very small angles among them. We can see four dipole-mode components also can fuse to generate a four-component dipole-mode soliton with total amplitude ${\displaystyle \sum _p{E}_{Fp}}$. The beam profile of ${\displaystyle \sum _p{E}_{Fp}}$ keeps horizontal-splitting. Similarly, when $E_{1,2,3}$ and ${E^{\prime }}_{2,3}$ are turned on (dressing field ${E^{\prime }}_1$ is blocked), the vertically-aligned four-component dipole-mode soliton ${\displaystyle \sum _q{E}_{Fq}}$ is obtained with the interaction among four vertical-splitting dipole components $E_{F3,4,7,8}$. When six laser beams are all turned on, ${\displaystyle \sum _p{E}_{Fp}}$ and ${\displaystyle \sum _q{E}_{Fq}}$ can coexist in the same atomic systems and be influenced by the dressing field ${E^{\prime }}_2$ and ${E^{\prime }}_1$, respectively. Compared with the case of absence of the dressing fields, the intensities of ${\displaystyle \sum _p{E}_{Fp}}$ and ${\displaystyle \sum _q{E}_{Fq}}$ become weak. It is attributed to the suppression effect in the dressing of ${E^{\prime }}_1$ or ${E^{\prime }}_2$ (with ${\Delta }_1+{\Delta }_2\approx 0$satisfied). Moreover, the dipole-like splitting of beams ${\displaystyle \sum _p{E}_{Fp}}$ and ${\displaystyle \sum _q{E}_{Fq}}$ becomes more distinctly because of the increasing of the nonlinear phase shift induced by the dressing fields. At the same time, there are two fundamental components $E_3$ and ${E^{\prime }}_3$ co-propagating in the same direction. They fuse to form a new fundamental soliton (${\displaystyle \sum E_0}=E_3+{E^{\prime }}_3$) with circular cross section (as seen in Fig. 4(b)). The fundamental component ${\displaystyle \sum E_0}$ and two mutually perpendicular dipole components ${\displaystyle \sum _p{E}_{Fp}}$ and ${\displaystyle \sum _q{E}_{Fq}}$ co-propagate with a small angle (${\theta }_1\approx {0.3}^{\circ }$) between ${\displaystyle \sum E_0}$ and ${\displaystyle \sum _p{E}_{Fp}}$ (or ${\displaystyle \sum E_0}$ and ${\displaystyle \sum _q{E}_{Fq}}$). The interaction force between fundamental soliton and dipole soliton can be written as $F=[C_{12}{\displaystyle \int {\sigma }_{p1}{v}_{{}_{00}}^1(r,\phi )}rdrd\phi {\displaystyle \int {\sigma }_{p2}{v}_{{}_{20}}^2(r,\phi )rdrd\phi ]}/R^4$, where $C_{12}$ is the interaction coefficient between fundamental soliton and dipole soliton. From the expressions of interaction forces we can see, as the separation of solitons is increased, that the force ($F\propto 1/R^7$) of two dipole solitons decays more quickly than the force ($F\propto 1/R^4$) between fundamental soliton and dipole soliton. It indicates that the fundamental soliton has more significant long-range constraint on dipole soliton, and plays an important role in the formation of the multi-component vector soliton. Therefore, the fundamental component ${\displaystyle \sum E_0}$ and two dipole components ${\displaystyle \sum _p{E}_{Fp}}$ and ${\displaystyle \sum _q{E}_{Fq}}$ constitute a new ten-component dipole-mode vector soliton.

 

Fig. 4 (a) Images of the four-component dipole-mode soliton ${\displaystyle \sum _i{E}_{Fi}}$ and ${\displaystyle \sum _j{E}_{Fj}}$ at different ${\Delta }_2$ with and without dressing field ${E^{\prime }}_2$ and ${E^{\prime }}_1$, respectively. (b) Experimental (left) and numerical (right) results of ten-component dipole-mode soliton with the propagation length $Z=0,15.9\text{mm}$at ${\Delta }_2=-15\text{GHZ}$

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The atomic density $N_0$ is determined by the temperature of the atomic vapor, the change of which leads to the change of the propagation distance z for the involved beams. In the experiment, with the temperature increasing, the corresponding propagation distance increases about $Z=a{L}_h=15.9\text{mm}$, where $L_h=19.4\text{mm}$ is the half-length of the heat pipe oven and a is the temperature increasing multiple. This propagation distance is 10.6 times longer than the diffraction length ($L_D=1.5\text{mm}$) of the FWM signal or the probe beam. We can see (in Fig. 4(b)) the spatial profiles of the beams ${\displaystyle \sum _p{E}_{Fp}}$ and ${\displaystyle \sum _q{E}_{Fq}}$ change very little in the propagation distance. This indicates that steady propagation of multi-component dipole-mode vector soliton is achieved.

In the two-level system, the input laser beams ($E_{1,2,3}$ and ${E^{\prime }}_{1,2,3}$) have nearly degenerate frequency, so the spatial interference of these beams can create a stationary beam pattern with a phase singularity,resulting in a vortex soliton at proper temperature (around 265${}^{\circ }C$),suitable detunings and configurations of input laser beams. Figure 5 presents the images of the FWM signals $E_{F1,2,5,6}$ versus the detuning ${\Delta }_1$. In the ${\Delta }_1>0$ region, four FWM signals all have self-defocusing character and show vortex patterns, then they decay into a fundamental spots when ${\Delta }_1=0$. In the defocusing media, a diffracting core of an optical vortex may get self-trapped and generate a vortex soliton. Specifically, for degenerate FWM signal $E_{F1}$, the interference between $E_1$, ${E^{\prime }}_1$ and $E_3$ forms a spiral phase plate. In this case, The parameters are ${\delta }_r=-\mathrm{arc}\tan \frac{E_1{n}_1{k}_{1y}\sin T_{\psi 1}+{E^{\prime }}_1{n}_{1^{\prime }}{k^{\prime }}_{1y}\sin T_{\psi 1^{\prime }}+E_3{n}_3{k}_{3y}\sin T_{\psi 3}}{E_1{n}_1{k}_{1x}\sin T_{\psi 1}+{E^{\prime }}_1{n}_{1^{\prime }}{k^{\prime }}_{1x}\sin T_{\psi 1^{\prime }}+E_3{n}_3{k}_{3x}\sin T_{\psi 3}}$,${\delta }_i=-\mathrm{arc}\tan \frac{E_1{n}_1{k}_{1y}\cos T_{\psi 1}+{E^{\prime }}_1{n}_{1^{\prime }}{k^{\prime }}_{1y}\cos T_{\psi 1^{\prime }}+E_3{n}_3{k}_{3y}\cos T_{\psi 3}}{E_1{n}_1{k}_{1x}\cos T_{\psi 1}+{E^{\prime }}_1{n}_{1^{\prime }}{k^{\prime }}_{1x}\cos T_{\psi 1^{\prime }}+E_3{n}_3{k}_{3x}\cos T_{\psi 3}}$. For nondegenerate FWM signals $E_{F2,5,6}$, we can see that three nearly degenerate frequency waves also can create spiral phase plate. For $E_{F2}$, the spiral phase plate is formed by $E_1$, ${E^{\prime }}_1$ and ${E^{\prime }}_3$, so ${\delta }_r=-\mathrm{arc}\tan \frac{E_1{n}_1{k}_{1y}\sin T_{\psi 1}+{E^{\prime }}_1{n}_{1^{\prime }}{k^{\prime }}_{1y}\sin T_{\psi 1^{\prime }}+{E^{\prime }}_3{n}_{3^{\prime }}{k^{\prime }}_{3y}\sin T_{\psi 3^{\prime }}}{E_1{n}_1{k}_{1x}\sin T_{\psi 1}+{E^{\prime }}_1{n}_{1^{\prime }}{k^{\prime }}_{1x}\sin T_{\psi 1^{\prime }}+{E^{\prime }}_3{n}_{3^{\prime }}{k^{\prime }}_{3x}\sin T_{\psi 3^{\prime }}}$,${\delta }_i=-\mathrm{arc}\tan \frac{E_1{n}_1{k}_{1y}\cos T_{\psi 1}+{E^{\prime }}_1{n}_{1^{\prime }}{k^{\prime }}_{1y}\cos T_{\psi 1^{\prime }}+{E^{\prime }}_3{n}_{3^{\prime }}{k^{\prime }}_{3y}\cos T_{\psi 3}}{E_1{n}_1{k}_{1x}\cos T_{\psi 1}+{E^{\prime }}_1{n}_{1^{\prime }}{k^{\prime }}_{1x}\cos T_{\psi 1^{\prime }}+{E^{\prime }}_3{n}_{3^{\prime }}{k^{\prime }}_{3x}\cos T_{\psi 3}}$. At the same time, the cross-phase modulation of the strong field ${E^{\prime }}_1$ [33] separates the FWM beam into three spots along a ring, forming a modulated vortex soliton. The modulated vortex beam has angular momentum $M\propto mP$, where $P={\displaystyle \int {\left|E\right|}^2}dr$ is the total power of vortex.

 

Fig. 5 Images (the five top rows) of the pure FWM signals $E_{F1,2,5,6}$ and ${E^{\prime }}_2$ dressed signal $E_{F6}$ versus ${\Delta }_1$. The two bottom rows give the numerical results of $E_{F6}$ with and without dressed field ${E^{\prime }}_2$, respectively

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In the ${\Delta }_1<0$ region, vortex solitons become instability due to the self-focusing nature of nonlinearity. We can see the modulated vortex pattern of $E_{F1,2,5,6}$ transforms into a rotating dipole-type pattern. The tilted dipole pattern indicates the spatial twist of the beam, which carries nonzero angular momentum. The rotation of the dipole soliton is because the initial angular momentum of the vortex beam transfers to the orbital angular momentum of dipole splinters. The conservation of angular momentum is satisfied in this interaction process. The stable rotating dipole soliton is defined as dipole azimuthon [36], and the astigmatic transformations of vortex beams into spiraling dipole azimuthons also has been observed recently in nematic liquid crystals [37]. When a stronger dressing field ${E^{\prime }}_2$ is turned on, the incoherent coupling between the fundamental beam ${E^{\prime }}_2$ and the vortex beam $E_{F6}$ can lead the vortex pattern of $E_{F6}$ to appear in the self-focusing region. It is attributed to the enhanced cross-Kerr nonlinear modulation by ${E^{\prime }}_2$. $E_{F6}$ is deflected closer to ${E^{\prime }}_2$ due to the attraction of ${E^{\prime }}_2$. The attraction force between the fundamental beam ${E^{\prime }}_2$ and the vortex beam $E_{F6}$ is$F=[C_{30}{\displaystyle \int {\sigma }_{p1}{v}_{{}_{00}}^1(r,\phi )}rdrd\phi {\displaystyle \int {\sigma }_{p2}{v}_{{}_{30}}^2(r,\phi )rdrd\phi }]/R^7$, where $v_{30}^{\text{i}}(r,\phi )=C{}_{30}(\sqrt{2}r/{\omega }_{0s}){}^3{e}^{-r^2/{\omega }_{0s}^2}\cos 3\phi$ is the intensity distribution profile of one vortex soliton, and $C_{30}$ is the interaction coefficient of vortex and fundamental solution. More importantly, the interaction between ${E^{\prime }}_2$ and $E_{F6}$ could counterbalance the self-focusing effect of $E_{F6}$ which resulting in instability of vortex beam. This phenomenon is coincident with the result reported in vortex vector solitons [20], and it provides an effective method to stabilize the vortex beam in a self-focusing medium.

Figure 6 gives the superposition of two or four vortex beams. When $E_{1,2,3}$ and ${E^{\prime }}_1$ are turned on, a stronger degenerate FWM signal $E_{F1}$ and a weak nondegenerate signal $E_{F5}$ are generated and co-propagate with a very small angle. The experimental results detected in ${k^{\prime }}_1$ direction can be regard as the superposition of two mutually-incoherent vortex beams. If the intensity of the signal $E_{F1}$ is far larger than that of $E_{F5}$, the vortex soliton $E_{F1}$ induces a linear waveguide with an effective potential [38]. Moreover, for mutually incoherent vortex solitons in self-defocusing medium, the potential is attractive independently of their relative phase. This waveguide can support both vortex mode ($m=1$) and fundamental mode ($m=0$) [39, 40]. If the intensity of the signal $E_{F5}$ tends to that of $E_{F1}$, where the self-phase modulation and cross-phase modulation effects operate together, the waveguide induced by solitons is nonlinear and gives rise to vortex vector soliton. In Fig. 6(a) we can see, the beam profiles of the superposition signal show vortex-type in the self-defocusing region, it indicates that two vortex components merge together to form a new vortex vector soliton.

 

Fig. 6 Images of the superposition vortex signals at different ${\Delta }_1$. (a) Two-component vortex solitons $E_{F1}+E_{F5}$ and $E_{F1}+E_{F2}$, respectively. The two bottom rows give the numerical results of $E_{F1}+E_{F2}$ and ${E^{\prime }}_2$ dressed $E_{F1}+E_{F2}$. (b) Four-component vortex solitons ${\displaystyle \sum _i{E}_{Fi}}$ and ${\displaystyle \sum _j{E}_{Fj}}$.

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In the self-focusing region, the radial symmetric character of this vortex vector soliton is broken. Similar to the scalar vortex soliton, it transforms into a rotating dipole pattern. It is indicated that two vortex components can couple with each other to form a rotating dipole vector soliton in self-focusing region. Moreover, the orientation of the dipole vector soliton changes with the frequency detuning. Similar results are obtained in the case of $E_{F1}+E_{F2}$. However, when a very strong dressing field ${E^{\prime }}_2$ is turned on, vortex vector soliton can be stabilized in the self-focusing region. As discussed above, the enhanced cross-Kerr nonlinearity by ${E^{\prime }}_2$ could counterbalance the self-focusing effect and suppress the breakup of the vortex soliton. Therefore, the stable vortex vector soliton can be formed in self-focusing region by the enhanced cross-phase modulation. It has been reported that two vortex components with opposite topological charges also can form a stable vortex vector soliton in self-focusing region [28]. Figure 6(b) is the superposition results of four co-propagating vortex beams. In the self-defocusing region, one stronger vortex beam $E_{F1}$ (or $E_{F4}$) and three weak vortex beams $E_{F2,5,6}$ (or $E_{F3,7,8}$) also can fuse to form a new four-components vortex beam ${\displaystyle \sum _p{E}_{Fp}}$ (or ${\displaystyle \sum _q{E}_{Fq}}$). In the self-focusing region, they decay into rotating dipole-type solitons.

4. Conclusion

In conclusion, we have experimentally demonstrated multi-component dipole and vortex solitons generated in four-wave mixing (FWM) in two-level atomic system. The composite dipole solitons contain four horizontal-splitting and four vertical-splitting dipole-mode components. They can propagate stably in the medium. The composite vortex solitons include four co-propagating vortex components. In the self-defocusing region, these vortex components fuse to form a new vortex vector soliton. In the self-focusing region, both scalar and vector vortex solitons transform into rotating dipole-mode solitons. However, if a very strong dressing field is turned on, vortex solitons can be stabilized in the self-focusing region. The interaction forces among components of dipole and vortex vector solitons are also investigated. This study will help us to understand the fundamental mechanisms in vector soliton formations and the interactions between different components, and open the door for the development of the device in all-optical communication and signal processing.