Abstract

We report the observation of multi-component dipole and vortex vector solitons composed of eight coexisting four-wave mixing (FWM) signals in two-level atomic system. The formation and stability of the multi-component dipole and vortex vector solitons are observed via changing the experiment parameters, including the frequency detuning, powers, and spatial configuration of the involved beams and the temperature of the medium. The transformation between modulated vortex solitons and rotating dipole solitons is observed at different frequency detunings. The interaction forces between different components of vector solitons are also investigated.

© 2012 OSA

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 [69]. 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 [2024]. 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(3S1/2) and |1(3P3/2) 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 Ω (Fig. 1(a) ). Laser beams are spatially aligned in the configuration shown in Fig. 1(b). Two laser beams E1 (with frequency ω1, wavevector k1, and Rabi frequency G1) and E1 (ω1, k1, G1), with a small angle θ10.3 between them, propagate in the opposite direction of the weak probe beam E3 (ω3, k3, G3). 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 Δ1=ω1Ω, and the wave-vectors of them are all nearly in the x-o-z plane. Other three laser beams E2 (ω2, k2, G2), E2 (ω2, k2, G2) and E3 (ω3,k3,G3) are from another dye laser DL2 (which has the same characteristics as the DL1) with a frequency detuning Δ2=ω2Ω. Among them, E2 co-propagates with E1; E3 co-propagates with E3, while E2 propagates with a small angle θ20.3 between E2 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: EF1(ω1, kF1=k1k1+k3),EF2(ω2, kF2=k1k1+k3),EF5(ω2, kF5=k2k1+k3),EF6 (2ω2ω1, kF6=k2k1+k3), EF3(ω1,kF3=k2k2+k3), EF4(ω2, kF4=k2k2+k3),EF7(2ω1ω2, kF7=k1k2+k3),EF8 (ω1, kF8=k1k2+k3). According to the phase-matching conditions, EF1,2,5,6 (EF3,4,7,8) propagate in the opposite direction of k1(k2) with very small angles among them. The stronger dressing beams E1,2 are approximately 10 times stronger than the pump beams E1,2, and 1000 times stronger than the weak probe beam E3, E3 and the generated FWM beams. The coherence length of the FWM signals is LFc=π/|Δk|, where Δk is the mismatching in the wave mixing process. According to the phase-matching conditions, we can obtain the coherence lengths of these FWM signals: LF1c=2πcω1/[n1ω1|ω1ω2|θ2], LF2c=2πcω1/[n1ω1|ω1ω2|θ2]=4.58km, LF3c=2πcω1/[n1ω1|ω1ω2|θ2]=3.1cm,LF4c=2πcω2/[n1ω1|ω1ω2|θ2], LF5c=2πc(2ω1ω2)/n1|(ω1ω2)(8ω14ω2+ω1θ2)|=3.1cm,LF6c=LF7c=πc/[2n1|ω1ω2|]=3.14cm, LF8c=2πc(2ω2ω1)/n1|(ω2ω1)(8ω24ω1+ω2θ2)|=4.58km. The diffraction length of the generated FWM beams is defined as LD=kFw02, where w0 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 n2I) in the atomic medium [12]. The sodium atomic density needs to reach 2.9×1013cm-3 (T = 250C), which can produce the needed variation in the nonlinear index of Δn=1.94×104. The strong beams E1 and E1 (E2 and E2) 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 E1 and E1 (E2 and E2) 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 Λi=λi/θi (i = 1 for EIG1, and 2 for EIG2). The periodically modulated total linear and nonlinear refractive index is given by the expression n(ζ)=n0+δn1cos(2πξ/Λi)+δn2cos(4πξ/Λi), where n0=n01+n02=(1+Ga)GF2Nμ10/(ε0E3Δ1) is the spatial uniform refractive index; δn1=Ga(n0121)/4n01+(Ga2/2+Ga)(n0221)/2n02 and δn2=(n0221)Ga2/16n02 are the coefficients of the spatially varying terms in the modulated index, and Ga=G22/[(Δ1+Δ2)Δ1]. 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 ΔΩ=2Ω0n2I/πn1, where Ω0 is the center frequency of it. If Δ1(or Δ2) 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 E3 or E3 will experience intensive Bragg reflection. The FWM signals EF1,2 and EF3,4 can be considered as the results of the electromagnetically-induced diffraction (EID) of the probe beam E3 or E3 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 EF1,2 and EF3,4 will be produced. Moreover, the beams E2 and E1 (or beams E1 and E2) can also induce their moving grating EIG3 (or EIG4). The wave vector of the grating is kg=k2-k1 (or kg=k1-k2) with the phase velocity v=Δω/|kg|, where Δω=ω2ω1 is the frequency difference between the two induced beams [29]. Similar to the signals EF1,2 and EF3,4, the FWM signals EF5,6 and EF7,8 are the EID results of the probe beam E3 or E3 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 EFp=AFp(ζ)exp(ikFpz)p=1,2,5,6 andEFq=AFq(ζ)exp(ikFqz)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:

p=1,2,5,6EFpzi2EFp2kFp=p=1,2,5,6ikFpn1(n2Sp|EFp|2+2n2Fp)EFp+η1E1(E1)*q=3,4,7,8EFq
q=3,4,7,8EFqzi2EFq2kFq=q=3,4,7,8ikFqn1(n2Sq|EFq|2+2n2Fq)EFq+η2E2(E2)*p=1,2,5,6EFp
Where n2Sp,Sq are self-Kerr nonlinear coefficients of EFq and EFq, n2Fp=k=16Δn2Xpk=n2Xp1|E3|2+n2Xp2|E1|2+n2Xp3|E2|2+n2Xp4|E1|2+n2Xp5|E2|2+n2Xp6|E3|2 and n2Fq=k=16Δn2Xqk=n2Xq1|E3|2+n2Xq2|E1|2+n2Xq3|E2|2+n2Xq4|E1|2+n2Xq5|E2|2+n2Xq6|E3|2,n2Xp1Xp6 and n2Xq1Xq6 are cross-Kerr nonlinear coefficients of E1,2,3 and E1,2,3 to EFq and EFq, respectively. All the Kerr nonlinear coefficients are generally defined as n2=Reχ(3)/(ε0cn1). The dressed third-order nonlinear susceptibility is χ(3)=Dρ10(3), where D=Nμ104/(3ε0GFp,FqGk2) (GFp,Fq are Rabi frequencies of EFp,Fq, respectively). N is the atomic density and μ10 is the dipole-matrix element between |0 and |1. The density matrix element ρ10(3) can be obtained by solving the density-matrix equations of the two-level atomic system. η1=ξμ105Fa and η2=ξμ215Fb with ξ=i4πω1N/ch4, whereFa and Fb related to the Rabi frequencies of the involved fields, the frequency detuning Δ1 (Δ2) and the relaxation rates of the system [30]. The differences in the last terms in the right side of the Eqs. (1a) and (1b) shows the signals EF1,2,5,6 mainly suffer the cross-Kerr effect from the dressing field E1 and EIG1, while EF3,4,7,8 suffer that from E2 and EIG2. The in-phase dipole mode of a FWM signal can be created by its origin EIG, and the two poles in the dipole are trapped jointly in neighbor photonic fringes in the EIG.

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 θc=(nmaxnmin)/nmax [31] plays a key role, here nmax and nmin are the maximum and minimum values of the nonlinear refractive index induced by the soliton, respectively. When the collision angle θ is less than θc, 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 EF1,2,5,6 (with topological chargemF1,2,5,6=1) generates a new horizontally oriented dipole-mode soliton (four-component dipole-mode soliton) with total amplitude pEFp (p = 1,2,5,6). Similarly, the superposition of four vertically-aligned dipole components EF3,4,7,8 (mF3,4,7,8=1) generates a vertically oriented four-component soliton with amplitude qEFq (q = 3,4,7,8). Two nodeless probe beams E3 and E3 (mk3,k3=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 mk3,k3+mF1,2,5,6+mF3,4,7,8=0 makes the structure stable. Therefore, the total intensity (I=|Ek3,k3|2+|pEFp|2+|qEFp|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 (E1,2,3 and E1,2,3) 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]:

p=1,2,5,6EFpzi2kFp(1rEFpr+2EFpr2+1r22EFpφ2)=p=1,2,5,6ikFpn1(n2Sp|EFp|2+2n2Fp)EFp
q=3,4,7,8EFqzi2kFq(1rEFqr+2EFqr2+1r22EFqφ2)=q=3,4,7,8ikFqn1(n2Sq|EFq|2+2n2Fq)EFq

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, EFp=upsech[up(kFpn2Sp/n0)1/2(rrp)]cos(Mφ/2)exp(imFpφ+iϕp)exp(ikFpz)and EFq=uqsech[uq(kFqn2Sq/n0)1/2(rrq)]cos(Mφ/2)exp(imFqφ+iϕq)exp(ikFqz), where up,q are soliton amplitudes; rp,q are initial peak positions; ϕp,q=2kFp,qn2zI2,1er2/2/(n0IFp,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, M3. Furthermore, if mFφ+ϕ=0, the angular velocity of the modulated soliton becomes ω=0 and the rotation of the soliton cancels. The topological charge m of the vortex soliton is determined by δr,iδr=arctan(i=1nEinikiysinTψi/i=1nEinikixsinTψi), δi=arctan(i=1nEinikiycosTψi/i=1nEinikixcosTψi), with n being the number of laser beams which create the spiral phase plate and ni being the nonlinear refractive index; Tψi=ψi0+ni(kixTx+kiyTy),with (Tx,Ty) being the coordinates of the singularity point. When δr>δi (δr<δi),m=1 (m=1), which means the phase changes clockwise (anticlockwise). To observe vortex soliton, the ideal temperature is around 265C(atomic density needs to reach 5.6×1013cm3). 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 E1 and E1 (or E2 and E1) are horizontally aligned in the x-z plane, the grating EIG1 and EIG3 have horizontal orientation (Fig. 1(b)). Therefore, dipole components EF1,2,5,6 induced by EIG1 and EIG3 have their two humps horizontally along x-axis. For the same reason, the dipole components EF3,4,7,8 induced by vertically oriented grating EIG2 and EIG4 have their two humps along the vertical y-axis. The probe beam E3 (or E3) is deviated from the FWM signals with a small angle (θ10.3), 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 EF1,2,5,6 and (0,1) for EF3,4,7,8. When the frequency detuning Δ2 is scanned from the large negative values to the resonant frequency, except EF1, 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 n2 (Fig. 3(d) ) and the nonlinear phase shift ϕp,q=2kFp,qn2zI2,1er2/2/(n0IFp,q), which determines the spatial splitting of the FWM signal. When Δ2 is set at resonance or the large negative detuning, n2 and ϕp,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 EF1 always keeps dipole-like pattern because its cross Kerr coefficient is not affected by E2, E2 and E3. The right panels of Fig. 2 describe the horizontal (vertical) size of the beams EF1,2,5,6(EF3,4,7,8). It is shown that the beam sizes of these eight dipole components keep constant when the frequency detuning Δ2 is changed from 40GHz to 10GHZ. This indicates that the dipole soliton is formed in this region.

 

Fig. 2 (a) Images (left) and horizontal-sizes (right) of FWM signals EF1,2,5,6 at different detuning Δ2, EF1(squares), EF2(circles), EF5(triangles) and EF6(reverse triangles). (b) Images (left) and vertical-sizes (right) of FWM signals EF3,4,7,8 at different detuning Δ2, EF3 (squares), EF4(circles), EF7(triangles) and EF8(reverse triangles).

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Fig. 3 (a1,2) Images of EF7 at different powers of (a1) E1, (a2) E2. (a3) Images of EF1 for different spatial configuration of E1 and E1. From left to right, the intersection position of E1 and E1 is adjusted to change from the front to the back of the heat-pipe oven. (b) Beam vertical-size of EF7 versus P1 (Squares) and P2 (circles). (c) Beam vertical-size (triangles) and horizontal-size (asterisks) of EF1 versus the intersection position of E1 and E1. Here the distance between the intersection position and the oven centre is denoted as l. (d) Nonlinear refractive index n2 of EF4. (e) Images of the pure FWM signal EF4 and coexisting signals EF4+EF3, EF4+EF8, EF1+EF2 and EF1+EF5, 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 EF7 versus the input powers of E1 and E2. The spatial splitting of signal EF7 is determined by the nonlinear phase shift introduced by the EIG4 which created by the beams E1 and E2. At low power I1,2, the beam EF7 presents a nodeless spot. With increasing I1,2, EF7 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 EF7 remains unchanged in a specific region. Further with increasing I1,2, the nonlinear phase shift ϕ7 becomes large, so the splitting of EF7 become significant, and the vertical-size of EF7 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 E1 and E1 is adjusted from the front to the back of the heat-pipe oven. We can see that EF1 is split in x direction when E1 and E1 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 EF1 in this transformation. This transformation can be explained by the different spatial position of E1 overlapped with EF1.

Figure 3(e) shows the splitting of the pure FWM signal EF4 and coexisting signals EF4+EF3 and EF4+EF8 when Δ2 is changed. It is theoretically obtained in Fig. 3(d) that the Kerr coefficient is negative (positive) with Δ2>0 (Δ2<0), which will lead to the defocusing (focusing) of the FWM signal. It can be seen that the beam EF4 splits along vertical direction in the self-focusing region, but it becomes horizontal-splitting in the self-defocusing region with small Δ2. Further increasing Δ2, the beam spot becomes large and presents three spots. Subsequently, it becomes vertical dipole-mode and decays into a nodeless spot when Δ2 far away from resonance. Such evolution can be explained by the follows. As mentioned above, in the self-focusing region, the dipole component EF4 induced by vertically-aligned EIG2 splits along the y-axis. But in Δ2>0 region, the FWM EF4 is shifted along the down-right direction due to the repulsion effect (n2<0) of the cross Kerr nonlinearity of the strong beam E2, and thus, the beam EF4 splits in x-direction [33]. The spatial shift of the FWM signal is proportional to |n2|. At a proper value, horizontal- and vertical-splitting appear simultaneously. With further increasing of Δ2 (|n2| deceases), the spatial shift of EF4 becomes small. Therefore, the beam EF4 gets back to the unshifted position and suffers from vertical-splitting. It decays into single spot when |n2| is close to zero.

As the probe beam E3 added, there exist two dipole-mode components EF3 and EF4 (mF3,4=1). They co-propagate in the opposite direction of k2with 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=[C20σp1v201(r,φ)rdrdφσp2v202(r,φ)rdrdφ]/R7, where v20i(r,φ)=C(2r/ω0s)202er2/ω0s2cos2φ(i=1,2) is the intensity distribution profile of one dipole soliton, ω0s the waist radius of Gaussian beams, σpi the average power in unit area of soliton beam, C20 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 EF3 and EF4 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 E3 is on. This indicates that the field E3 could suppress the spatial shift and thus the horizontal-splitting of EF4. When the pump beam E1 is turned on, there exist two incoherent dipole-mode components EF4 and EF8 (mF4,8=1). They also can fuse to form a dipole-mode vector soliton. Figure 3(e) also presents the coexisting signals EF1+EF2 and EF1+EF5 which have topological charge mF=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 E1,2,3 and E1,3 are turned on (dressing field E2 is blocked) at the same time, there are four horizontal-splitting dipole-mode components EF1,2,5,6 co-propagating in the opposite direction of k1 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 pEFp. The beam profile of pEFp keeps horizontal-splitting. Similarly, when E1,2,3 and E2,3 are turned on (dressing field E1 is blocked), the vertically-aligned four-component dipole-mode soliton qEFq is obtained with the interaction among four vertical-splitting dipole components EF3,4,7,8. When six laser beams are all turned on, pEFp and qEFq can coexist in the same atomic systems and be influenced by the dressing field E2 and E1, respectively. Compared with the case of absence of the dressing fields, the intensities of pEFp and qEFq become weak. It is attributed to the suppression effect in the dressing of E1 or E2 (with Δ1+Δ20satisfied). Moreover, the dipole-like splitting of beams pEFp and qEFq 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 E3 and E3 co-propagating in the same direction. They fuse to form a new fundamental soliton (E0=E3+E3) with circular cross section (as seen in Fig. 4(b)). The fundamental component E0 and two mutually perpendicular dipole components pEFp and qEFq co-propagate with a small angle (θ10.3) between E0 and pEFp (or E0 and qEFq). The interaction force between fundamental soliton and dipole soliton can be written as F=[C12σp1v001(r,φ)rdrdφσp2v202(r,φ)rdrdφ]/R4, where C12 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 (F1/R7) of two dipole solitons decays more quickly than the force (F1/R4) 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 E0 and two dipole components pEFp and qEFq constitute a new ten-component dipole-mode vector soliton.

 

Fig. 4 (a) Images of the four-component dipole-mode soliton iEFi and jEFj at different Δ2 with and without dressing field E2 and E1, respectively. (b) Experimental (left) and numerical (right) results of ten-component dipole-mode soliton with the propagation length Z=0,15.9mmat Δ2=15GHZ

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The atomic density N0 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=aLh=15.9mm, where Lh=19.4mm 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 (LD=1.5mm) of the FWM signal or the probe beam. We can see (in Fig. 4(b)) the spatial profiles of the beams pEFp and qEFq 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 (E1,2,3 and E1,2,3) 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 265C),suitable detunings and configurations of input laser beams. Figure 5 presents the images of the FWM signals EF1,2,5,6 versus the detuning Δ1. In the Δ1>0 region, four FWM signals all have self-defocusing character and show vortex patterns, then they decay into a fundamental spots when Δ1=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 EF1, the interference between E1, E1 and E3 forms a spiral phase plate. In this case, The parameters are δr=arctanE1n1k1ysinTψ1+E1n1k1ysinTψ1+E3n3k3ysinTψ3E1n1k1xsinTψ1+E1n1k1xsinTψ1+E3n3k3xsinTψ3,δi=arctanE1n1k1ycosTψ1+E1n1k1ycosTψ1+E3n3k3ycosTψ3E1n1k1xcosTψ1+E1n1k1xcosTψ1+E3n3k3xcosTψ3. For nondegenerate FWM signals EF2,5,6, we can see that three nearly degenerate frequency waves also can create spiral phase plate. For EF2, the spiral phase plate is formed by E1, E1 and E3, so δr=arctanE1n1k1ysinTψ1+E1n1k1ysinTψ1+E3n3k3ysinTψ3E1n1k1xsinTψ1+E1n1k1xsinTψ1+E3n3k3xsinTψ3,δi=arctanE1n1k1ycosTψ1+E1n1k1ycosTψ1+E3n3k3ycosTψ3E1n1k1xcosTψ1+E1n1k1xcosTψ1+E3n3k3xcosTψ3. At the same time, the cross-phase modulation of the strong field E1 [33] separates the FWM beam into three spots along a ring, forming a modulated vortex soliton. The modulated vortex beam has angular momentum MmP, where P=|E|2dr is the total power of vortex.

 

Fig. 5 Images (the five top rows) of the pure FWM signals EF1,2,5,6 and E2 dressed signal EF6 versus Δ1. The two bottom rows give the numerical results of EF6 with and without dressed field E2, respectively

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In the Δ1<0 region, vortex solitons become instability due to the self-focusing nature of nonlinearity. We can see the modulated vortex pattern of EF1,2,5,6 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 E2 is turned on, the incoherent coupling between the fundamental beam E2 and the vortex beam EF6 can lead the vortex pattern of EF6 to appear in the self-focusing region. It is attributed to the enhanced cross-Kerr nonlinear modulation by E2. EF6 is deflected closer to E2 due to the attraction of E2. The attraction force between the fundamental beam E2 and the vortex beam EF6 isF=[C30σp1v001(r,φ)rdrdφσp2v302(r,φ)rdrdφ]/R7, where v30i(r,φ)=C(2r/ω0s)303er2/ω0s2cos3φ is the intensity distribution profile of one vortex soliton, and C30 is the interaction coefficient of vortex and fundamental solution. More importantly, the interaction between E2 and EF6 could counterbalance the self-focusing effect of EF6 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 E1,2,3 and E1 are turned on, a stronger degenerate FWM signal EF1 and a weak nondegenerate signal EF5 are generated and co-propagate with a very small angle. The experimental results detected in k1 direction can be regard as the superposition of two mutually-incoherent vortex beams. If the intensity of the signal EF1 is far larger than that of EF5, the vortex soliton EF1 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 EF5 tends to that of EF1, 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 Δ1. (a) Two-component vortex solitons EF1+EF5 and EF1+EF2, respectively. The two bottom rows give the numerical results of EF1+EF2 and E2 dressed EF1+EF2. (b) Four-component vortex solitons iEFi and jEFj.

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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 EF1+EF2. However, when a very strong dressing field E2 is turned on, vortex vector soliton can be stabilized in the self-focusing region. As discussed above, the enhanced cross-Kerr nonlinearity by E2 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 EF1 (or EF4) and three weak vortex beams EF2,5,6 (or EF3,7,8) also can fuse to form a new four-components vortex beam pEFp (or qEFq). 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.

Acknowledgments

This work was supported by the 973 Program (2012CB921804), NSFC (10974151, 61078002, 61078020, 11104214, 61108017, 11104216), NCET (08-0431), RFDP (20110201110006, 20110201120005, 20100201120031), and CDPXJTU (xjj2011084, xjj20100100, xjj20100151, xjj2012080).

References and links

1. S. Trillo and W. Torruella, Spatial Solitons (Springer-Verlag, Berlin, 2001).

2. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286(5444), 1518–1523 (1999). [CrossRef]   [PubMed]  

3. V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996). [CrossRef]   [PubMed]  

4. A. S. Desyatnikov, Y. S. Kivshar, K. Motzek, F. Kaiser, C. Weilnau, and C. Denz, “Multicomponent dipole-mode spatial solitons,” Opt. Lett. 27(8), 634–636 (2002). [CrossRef]   [PubMed]  

5. K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun. 209(4-6), 501–506 (2002). [CrossRef]  

6. L. J. Ge, Q. Wang, M. Shen, and J. L. Shi, “Dipole solitons in nonlocal nonlinear media with anisotropy,” Opt. Commun. 284(9), 2351–2356 (2011). [CrossRef]  

7. M. Shen, X. Chen, J. L. Shi, Q. Wang, and W. Krolikowski, “Incoherently coupled vector dipole soliton pairs in nonlocal media,” Opt. Commun. 282(24), 4805–4809 (2009). [CrossRef]  

8. F. W. Ye, Y. V. Kartashov, and L. Torner, “Stabilization of dipole solitons in nonlocal nonlinear media,” Phys. Rev. A 77(4), 043821 (2008). [CrossRef]  

9. C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nature 2, 769–774 (2006).

10. J. K. Yang, I. Makasyuk, A. Bezryadina, and Z. G. Chen, “Dipole solitons in optically induced two-dimensional photonic lattices,” Opt. Lett. 29(14), 1662–1664 (2004). [CrossRef]   [PubMed]  

11. N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003). [CrossRef]   [PubMed]  

12. Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011). [CrossRef]   [PubMed]  

13. A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005). [CrossRef]  

14. A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88(5), 053901 (2002). [CrossRef]   [PubMed]  

15. A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005). [CrossRef]   [PubMed]  

16. A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of optical soliton azimuthons,” Opt. Express 17(26), 23610–23616 (2009). [CrossRef]   [PubMed]  

17. W. J. Firth and D. V. Skryabin, “Optical solitons carrying Orbital angular momentum,” Phys. Rev. Lett. 79(13), 2450–2453 (1997). [CrossRef]  

18. J. K. Yang and D. E. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(1), 016608 (2003). [CrossRef]   [PubMed]  

19. A. S. Desyatnikov and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 87(3), 033901 (2001). [CrossRef]   [PubMed]  

20. J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056613 (2004). [CrossRef]   [PubMed]  

21. A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001). [CrossRef]   [PubMed]  

22. C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92(4), 043904 (2004). [CrossRef]   [PubMed]  

23. A. S. Desyatnikov, D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Stable counter-rotating vortex pairs in saturable media,” Phys. Lett. A 364(3-4), 231–234 (2007). [CrossRef]  

24. Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Multicomponent two-dimensional solitons carrying topological charges,” Opt. Lett. 25(1), 61–63 (2000). [CrossRef]   [PubMed]  

25. C. Anastassiou, M. Soljačić, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85(23), 4888–4891 (2000). [CrossRef]   [PubMed]  

26. Z. H. Musslimani, M. Soljacić, M. Segev, and D. N. Christodoulides, “Interactions between two-dimensional composite vector solitons carrying topological charges,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(6), 066608 (2001). [CrossRef]   [PubMed]  

27. C. Pigier, R. Uzdin, T. Carmon, M. Segev, A. Nepomnyaschchy, and Z. H. Musslimani, “Collisions between (2+1)D rotating propeller solitons,” Opt. Lett. 26(20), 1577–1579 (2001). [CrossRef]   [PubMed]  

28. Y. V. Izdebskaya, J. Rebling, A. S. Desyatnikov, and Y. S. Kivshar, “Observation of vector solitons with hidden vorticity,” Opt. Lett. 37(5), 767–769 (2012). [CrossRef]   [PubMed]  

29. Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010). [CrossRef]   [PubMed]  

30. H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87(7), 073601 (2001). [CrossRef]   [PubMed]  

31. A. W. Snyder and A. P. Sheppard, “Collisions, steering, and guidance with spatial solitons,” Opt. Lett. 18(7), 482–484 (1993). [CrossRef]   [PubMed]  

32. Y. P. Zhang, Z. Q. Nie, Y. Zhao, C. B. Li, R. M. Wang, J. H. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express 18(11), 10963–10972 (2010). [CrossRef]   [PubMed]  

33. Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A 80(5), 055804 (2009). [CrossRef]  

34. M. Marinescu and A. Dalgarno, “Dispersion forces and long-range electronic transition dipole moments of alkali-metal dimer excited states,” Phys. Rev. A 52(1), 311–328 (1995). [CrossRef]   [PubMed]  

35. K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atoms pairs correlated to the ns-ns, np-np and nd-nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005). [CrossRef]  

36. S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31(8), 1100–1102 (2006). [CrossRef]  

37. Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Dipole azimuthons and vortex charge flipping in nematic liquid crystals,” Opt. Express 19(22), 21457–21466 (2011). [CrossRef]   [PubMed]  

38. A. H. Carlsson, J. N. Malmberg, D. Anderson, M. Lisak, E. A. Ostrovskaya, T. J. Alexander, and Y. S. Kivshar, “Linear and nonlinear waveguides induced by optical vortex solitons,” Opt. Lett. 25(9), 660–662 (2000). [CrossRef]   [PubMed]  

39. A. P. Sheppard and M. Haelterman, “Polarization-domain solitary waves of circular symmetry in Kerr media,” Opt. Lett. 19(12), 859–861 (1994). [CrossRef]   [PubMed]  

40. C. T. Law, X. Zhang, and G. A. Swartzlander Jr., “Waveguiding properties of optical vortex solitons,” Opt. Lett. 25(1), 55–57 (2000). [CrossRef]   [PubMed]  

References

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  1. S. Trillo and W. Torruella, Spatial Solitons (Springer-Verlag, Berlin, 2001).
  2. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286(5444), 1518–1523 (1999).
    [CrossRef] [PubMed]
  3. V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996).
    [CrossRef] [PubMed]
  4. A. S. Desyatnikov, Y. S. Kivshar, K. Motzek, F. Kaiser, C. Weilnau, and C. Denz, “Multicomponent dipole-mode spatial solitons,” Opt. Lett. 27(8), 634–636 (2002).
    [CrossRef] [PubMed]
  5. K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun. 209(4-6), 501–506 (2002).
    [CrossRef]
  6. L. J. Ge, Q. Wang, M. Shen, and J. L. Shi, “Dipole solitons in nonlocal nonlinear media with anisotropy,” Opt. Commun. 284(9), 2351–2356 (2011).
    [CrossRef]
  7. M. Shen, X. Chen, J. L. Shi, Q. Wang, and W. Krolikowski, “Incoherently coupled vector dipole soliton pairs in nonlocal media,” Opt. Commun. 282(24), 4805–4809 (2009).
    [CrossRef]
  8. F. W. Ye, Y. V. Kartashov, and L. Torner, “Stabilization of dipole solitons in nonlocal nonlinear media,” Phys. Rev. A 77(4), 043821 (2008).
    [CrossRef]
  9. C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nature 2, 769–774 (2006).
  10. J. K. Yang, I. Makasyuk, A. Bezryadina, and Z. G. Chen, “Dipole solitons in optically induced two-dimensional photonic lattices,” Opt. Lett. 29(14), 1662–1664 (2004).
    [CrossRef] [PubMed]
  11. N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003).
    [CrossRef] [PubMed]
  12. Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
    [CrossRef] [PubMed]
  13. A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005).
    [CrossRef]
  14. A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88(5), 053901 (2002).
    [CrossRef] [PubMed]
  15. A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005).
    [CrossRef] [PubMed]
  16. A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of optical soliton azimuthons,” Opt. Express 17(26), 23610–23616 (2009).
    [CrossRef] [PubMed]
  17. W. J. Firth and D. V. Skryabin, “Optical solitons carrying Orbital angular momentum,” Phys. Rev. Lett. 79(13), 2450–2453 (1997).
    [CrossRef]
  18. J. K. Yang and D. E. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(1), 016608 (2003).
    [CrossRef] [PubMed]
  19. A. S. Desyatnikov and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 87(3), 033901 (2001).
    [CrossRef] [PubMed]
  20. J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056613 (2004).
    [CrossRef] [PubMed]
  21. A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
    [CrossRef] [PubMed]
  22. C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92(4), 043904 (2004).
    [CrossRef] [PubMed]
  23. A. S. Desyatnikov, D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Stable counter-rotating vortex pairs in saturable media,” Phys. Lett. A 364(3-4), 231–234 (2007).
    [CrossRef]
  24. Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Multicomponent two-dimensional solitons carrying topological charges,” Opt. Lett. 25(1), 61–63 (2000).
    [CrossRef] [PubMed]
  25. C. Anastassiou, M. Soljačić, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85(23), 4888–4891 (2000).
    [CrossRef] [PubMed]
  26. Z. H. Musslimani, M. Soljacić, M. Segev, and D. N. Christodoulides, “Interactions between two-dimensional composite vector solitons carrying topological charges,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(6), 066608 (2001).
    [CrossRef] [PubMed]
  27. C. Pigier, R. Uzdin, T. Carmon, M. Segev, A. Nepomnyaschchy, and Z. H. Musslimani, “Collisions between (2+1)D rotating propeller solitons,” Opt. Lett. 26(20), 1577–1579 (2001).
    [CrossRef] [PubMed]
  28. Y. V. Izdebskaya, J. Rebling, A. S. Desyatnikov, and Y. S. Kivshar, “Observation of vector solitons with hidden vorticity,” Opt. Lett. 37(5), 767–769 (2012).
    [CrossRef] [PubMed]
  29. Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010).
    [CrossRef] [PubMed]
  30. H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87(7), 073601 (2001).
    [CrossRef] [PubMed]
  31. A. W. Snyder and A. P. Sheppard, “Collisions, steering, and guidance with spatial solitons,” Opt. Lett. 18(7), 482–484 (1993).
    [CrossRef] [PubMed]
  32. Y. P. Zhang, Z. Q. Nie, Y. Zhao, C. B. Li, R. M. Wang, J. H. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express 18(11), 10963–10972 (2010).
    [CrossRef] [PubMed]
  33. Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A 80(5), 055804 (2009).
    [CrossRef]
  34. M. Marinescu and A. Dalgarno, “Dispersion forces and long-range electronic transition dipole moments of alkali-metal dimer excited states,” Phys. Rev. A 52(1), 311–328 (1995).
    [CrossRef] [PubMed]
  35. K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atoms pairs correlated to the ns-ns, np-np and nd-nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005).
    [CrossRef]
  36. S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31(8), 1100–1102 (2006).
    [CrossRef]
  37. Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Dipole azimuthons and vortex charge flipping in nematic liquid crystals,” Opt. Express 19(22), 21457–21466 (2011).
    [CrossRef] [PubMed]
  38. A. H. Carlsson, J. N. Malmberg, D. Anderson, M. Lisak, E. A. Ostrovskaya, T. J. Alexander, and Y. S. Kivshar, “Linear and nonlinear waveguides induced by optical vortex solitons,” Opt. Lett. 25(9), 660–662 (2000).
    [CrossRef] [PubMed]
  39. A. P. Sheppard and M. Haelterman, “Polarization-domain solitary waves of circular symmetry in Kerr media,” Opt. Lett. 19(12), 859–861 (1994).
    [CrossRef] [PubMed]
  40. C. T. Law, X. Zhang, and G. A. Swartzlander., “Waveguiding properties of optical vortex solitons,” Opt. Lett. 25(1), 55–57 (2000).
    [CrossRef] [PubMed]

2012 (1)

2011 (3)

Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Dipole azimuthons and vortex charge flipping in nematic liquid crystals,” Opt. Express 19(22), 21457–21466 (2011).
[CrossRef] [PubMed]

L. J. Ge, Q. Wang, M. Shen, and J. L. Shi, “Dipole solitons in nonlocal nonlinear media with anisotropy,” Opt. Commun. 284(9), 2351–2356 (2011).
[CrossRef]

Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[CrossRef] [PubMed]

2010 (2)

Y. P. Zhang, Z. Q. Nie, Y. Zhao, C. B. Li, R. M. Wang, J. H. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express 18(11), 10963–10972 (2010).
[CrossRef] [PubMed]

Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010).
[CrossRef] [PubMed]

2009 (3)

Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A 80(5), 055804 (2009).
[CrossRef]

A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of optical soliton azimuthons,” Opt. Express 17(26), 23610–23616 (2009).
[CrossRef] [PubMed]

M. Shen, X. Chen, J. L. Shi, Q. Wang, and W. Krolikowski, “Incoherently coupled vector dipole soliton pairs in nonlocal media,” Opt. Commun. 282(24), 4805–4809 (2009).
[CrossRef]

2008 (1)

F. W. Ye, Y. V. Kartashov, and L. Torner, “Stabilization of dipole solitons in nonlocal nonlinear media,” Phys. Rev. A 77(4), 043821 (2008).
[CrossRef]

2007 (1)

A. S. Desyatnikov, D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Stable counter-rotating vortex pairs in saturable media,” Phys. Lett. A 364(3-4), 231–234 (2007).
[CrossRef]

2006 (2)

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31(8), 1100–1102 (2006).
[CrossRef]

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nature 2, 769–774 (2006).

2005 (3)

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005).
[CrossRef]

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atoms pairs correlated to the ns-ns, np-np and nd-nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005).
[CrossRef]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005).
[CrossRef] [PubMed]

2004 (3)

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92(4), 043904 (2004).
[CrossRef] [PubMed]

J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056613 (2004).
[CrossRef] [PubMed]

J. K. Yang, I. Makasyuk, A. Bezryadina, and Z. G. Chen, “Dipole solitons in optically induced two-dimensional photonic lattices,” Opt. Lett. 29(14), 1662–1664 (2004).
[CrossRef] [PubMed]

2003 (2)

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003).
[CrossRef] [PubMed]

J. K. Yang and D. E. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(1), 016608 (2003).
[CrossRef] [PubMed]

2002 (3)

A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88(5), 053901 (2002).
[CrossRef] [PubMed]

A. S. Desyatnikov, Y. S. Kivshar, K. Motzek, F. Kaiser, C. Weilnau, and C. Denz, “Multicomponent dipole-mode spatial solitons,” Opt. Lett. 27(8), 634–636 (2002).
[CrossRef] [PubMed]

K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun. 209(4-6), 501–506 (2002).
[CrossRef]

2001 (5)

A. S. Desyatnikov and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 87(3), 033901 (2001).
[CrossRef] [PubMed]

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
[CrossRef] [PubMed]

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87(7), 073601 (2001).
[CrossRef] [PubMed]

Z. H. Musslimani, M. Soljacić, M. Segev, and D. N. Christodoulides, “Interactions between two-dimensional composite vector solitons carrying topological charges,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(6), 066608 (2001).
[CrossRef] [PubMed]

C. Pigier, R. Uzdin, T. Carmon, M. Segev, A. Nepomnyaschchy, and Z. H. Musslimani, “Collisions between (2+1)D rotating propeller solitons,” Opt. Lett. 26(20), 1577–1579 (2001).
[CrossRef] [PubMed]

2000 (4)

1999 (1)

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286(5444), 1518–1523 (1999).
[CrossRef] [PubMed]

1997 (1)

W. J. Firth and D. V. Skryabin, “Optical solitons carrying Orbital angular momentum,” Phys. Rev. Lett. 79(13), 2450–2453 (1997).
[CrossRef]

1996 (1)

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996).
[CrossRef] [PubMed]

1995 (1)

M. Marinescu and A. Dalgarno, “Dispersion forces and long-range electronic transition dipole moments of alkali-metal dimer excited states,” Phys. Rev. A 52(1), 311–328 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

Alexander, T. J.

Alfassi, B.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nature 2, 769–774 (2006).

Anastassiou, C.

C. Anastassiou, M. Soljačić, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85(23), 4888–4891 (2000).
[CrossRef] [PubMed]

Anderson, D.

Assanto, G.

Bang, O.

Bezryadina, A.

Carlsson, A. H.

Carmon, T.

Chang, H.

Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A 80(5), 055804 (2009).
[CrossRef]

Chen, H. X.

Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[CrossRef] [PubMed]

Chen, S.

Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010).
[CrossRef] [PubMed]

Chen, X.

M. Shen, X. Chen, J. L. Shi, Q. Wang, and W. Krolikowski, “Incoherently coupled vector dipole soliton pairs in nonlocal media,” Opt. Commun. 282(24), 4805–4809 (2009).
[CrossRef]

Chen, Z. G.

Christodoulides, D. N.

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003).
[CrossRef] [PubMed]

Z. H. Musslimani, M. Soljacić, M. Segev, and D. N. Christodoulides, “Interactions between two-dimensional composite vector solitons carrying topological charges,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(6), 066608 (2001).
[CrossRef] [PubMed]

C. Anastassiou, M. Soljačić, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85(23), 4888–4891 (2000).
[CrossRef] [PubMed]

Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Multicomponent two-dimensional solitons carrying topological charges,” Opt. Lett. 25(1), 61–63 (2000).
[CrossRef] [PubMed]

Christou, J.

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996).
[CrossRef] [PubMed]

Cohen, O.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nature 2, 769–774 (2006).

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003).
[CrossRef] [PubMed]

Côté, R.

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atoms pairs correlated to the ns-ns, np-np and nd-nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005).
[CrossRef]

Dalgarno, A.

M. Marinescu and A. Dalgarno, “Dispersion forces and long-range electronic transition dipole moments of alkali-metal dimer excited states,” Phys. Rev. A 52(1), 311–328 (1995).
[CrossRef] [PubMed]

Denz, C.

A. S. Desyatnikov, Y. S. Kivshar, K. Motzek, F. Kaiser, C. Weilnau, and C. Denz, “Multicomponent dipole-mode spatial solitons,” Opt. Lett. 27(8), 634–636 (2002).
[CrossRef] [PubMed]

K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun. 209(4-6), 501–506 (2002).
[CrossRef]

Desyatnikov, A.

K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun. 209(4-6), 501–506 (2002).
[CrossRef]

Desyatnikov, A. S.

Y. V. Izdebskaya, J. Rebling, A. S. Desyatnikov, and Y. S. Kivshar, “Observation of vector solitons with hidden vorticity,” Opt. Lett. 37(5), 767–769 (2012).
[CrossRef] [PubMed]

Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Dipole azimuthons and vortex charge flipping in nematic liquid crystals,” Opt. Express 19(22), 21457–21466 (2011).
[CrossRef] [PubMed]

A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of optical soliton azimuthons,” Opt. Express 17(26), 23610–23616 (2009).
[CrossRef] [PubMed]

A. S. Desyatnikov, D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Stable counter-rotating vortex pairs in saturable media,” Phys. Lett. A 364(3-4), 231–234 (2007).
[CrossRef]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31(8), 1100–1102 (2006).
[CrossRef]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005).
[CrossRef]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005).
[CrossRef] [PubMed]

A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88(5), 053901 (2002).
[CrossRef] [PubMed]

A. S. Desyatnikov, Y. S. Kivshar, K. Motzek, F. Kaiser, C. Weilnau, and C. Denz, “Multicomponent dipole-mode spatial solitons,” Opt. Lett. 27(8), 634–636 (2002).
[CrossRef] [PubMed]

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
[CrossRef] [PubMed]

A. S. Desyatnikov and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 87(3), 033901 (2001).
[CrossRef] [PubMed]

Durand, M.

Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010).
[CrossRef] [PubMed]

Efremidis, N. K.

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003).
[CrossRef] [PubMed]

Eugenieva, E. D.

C. Anastassiou, M. Soljačić, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85(23), 4888–4891 (2000).
[CrossRef] [PubMed]

Firth, W. J.

W. J. Firth and D. V. Skryabin, “Optical solitons carrying Orbital angular momentum,” Phys. Rev. Lett. 79(13), 2450–2453 (1997).
[CrossRef]

Fleischer, J. W.

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003).
[CrossRef] [PubMed]

Forestier, B.

Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010).
[CrossRef] [PubMed]

García-Ripoll, J. J.

Ge, L. J.

L. J. Ge, Q. Wang, M. Shen, and J. L. Shi, “Dipole solitons in nonlocal nonlinear media with anisotropy,” Opt. Commun. 284(9), 2351–2356 (2011).
[CrossRef]

Goorskey, D.

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87(7), 073601 (2001).
[CrossRef] [PubMed]

Haelterman, M.

Houard, A.

Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010).
[CrossRef] [PubMed]

Hudock, J.

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003).
[CrossRef] [PubMed]

Izdebskaya, Y. V.

Jeng, C. C.

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92(4), 043904 (2004).
[CrossRef] [PubMed]

Kaiser, F.

K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun. 209(4-6), 501–506 (2002).
[CrossRef]

A. S. Desyatnikov, Y. S. Kivshar, K. Motzek, F. Kaiser, C. Weilnau, and C. Denz, “Multicomponent dipole-mode spatial solitons,” Opt. Lett. 27(8), 634–636 (2002).
[CrossRef] [PubMed]

Kartashov, Y. V.

F. W. Ye, Y. V. Kartashov, and L. Torner, “Stabilization of dipole solitons in nonlocal nonlinear media,” Phys. Rev. A 77(4), 043821 (2008).
[CrossRef]

Kip, D.

C. Anastassiou, M. Soljačić, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85(23), 4888–4891 (2000).
[CrossRef] [PubMed]

Kivshar, Y.

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92(4), 043904 (2004).
[CrossRef] [PubMed]

Kivshar, Y. S.

Y. V. Izdebskaya, J. Rebling, A. S. Desyatnikov, and Y. S. Kivshar, “Observation of vector solitons with hidden vorticity,” Opt. Lett. 37(5), 767–769 (2012).
[CrossRef] [PubMed]

Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Dipole azimuthons and vortex charge flipping in nematic liquid crystals,” Opt. Express 19(22), 21457–21466 (2011).
[CrossRef] [PubMed]

A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of optical soliton azimuthons,” Opt. Express 17(26), 23610–23616 (2009).
[CrossRef] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31(8), 1100–1102 (2006).
[CrossRef]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005).
[CrossRef]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005).
[CrossRef] [PubMed]

J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056613 (2004).
[CrossRef] [PubMed]

A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88(5), 053901 (2002).
[CrossRef] [PubMed]

K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun. 209(4-6), 501–506 (2002).
[CrossRef]

A. S. Desyatnikov, Y. S. Kivshar, K. Motzek, F. Kaiser, C. Weilnau, and C. Denz, “Multicomponent dipole-mode spatial solitons,” Opt. Lett. 27(8), 634–636 (2002).
[CrossRef] [PubMed]

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
[CrossRef] [PubMed]

A. S. Desyatnikov and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 87(3), 033901 (2001).
[CrossRef] [PubMed]

A. H. Carlsson, J. N. Malmberg, D. Anderson, M. Lisak, E. A. Ostrovskaya, T. J. Alexander, and Y. S. Kivshar, “Linear and nonlinear waveguides induced by optical vortex solitons,” Opt. Lett. 25(9), 660–662 (2000).
[CrossRef] [PubMed]

Krolikowski, W.

Law, C. T.

Lederer, F.

A. S. Desyatnikov, D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Stable counter-rotating vortex pairs in saturable media,” Phys. Lett. A 364(3-4), 231–234 (2007).
[CrossRef]

Li, C. B.

Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[CrossRef] [PubMed]

Y. P. Zhang, Z. Q. Nie, Y. Zhao, C. B. Li, R. M. Wang, J. H. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express 18(11), 10963–10972 (2010).
[CrossRef] [PubMed]

Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A 80(5), 055804 (2009).
[CrossRef]

Lisak, M.

Liu, Y.

Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010).
[CrossRef] [PubMed]

Lopez-Aguayo, S.

Lu, K. Q.

Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[CrossRef] [PubMed]

Luther-Davies, B.

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996).
[CrossRef] [PubMed]

Makasyuk, I.

Malmberg, J. N.

Malomed, B. A.

A. S. Desyatnikov, D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Stable counter-rotating vortex pairs in saturable media,” Phys. Lett. A 364(3-4), 231–234 (2007).
[CrossRef]

Marinescu, M.

M. Marinescu and A. Dalgarno, “Dispersion forces and long-range electronic transition dipole moments of alkali-metal dimer excited states,” Phys. Rev. A 52(1), 311–328 (1995).
[CrossRef] [PubMed]

Mazilu, D.

A. S. Desyatnikov, D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Stable counter-rotating vortex pairs in saturable media,” Phys. Lett. A 364(3-4), 231–234 (2007).
[CrossRef]

McCarthy, G.

K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun. 209(4-6), 501–506 (2002).
[CrossRef]

Mihalache, D.

A. S. Desyatnikov, D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Stable counter-rotating vortex pairs in saturable media,” Phys. Lett. A 364(3-4), 231–234 (2007).
[CrossRef]

Minovich, A.

Motzek, K.

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92(4), 043904 (2004).
[CrossRef] [PubMed]

K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun. 209(4-6), 501–506 (2002).
[CrossRef]

A. S. Desyatnikov, Y. S. Kivshar, K. Motzek, F. Kaiser, C. Weilnau, and C. Denz, “Multicomponent dipole-mode spatial solitons,” Opt. Lett. 27(8), 634–636 (2002).
[CrossRef] [PubMed]

Musslimani, Z. H.

C. Pigier, R. Uzdin, T. Carmon, M. Segev, A. Nepomnyaschchy, and Z. H. Musslimani, “Collisions between (2+1)D rotating propeller solitons,” Opt. Lett. 26(20), 1577–1579 (2001).
[CrossRef] [PubMed]

Z. H. Musslimani, M. Soljacić, M. Segev, and D. N. Christodoulides, “Interactions between two-dimensional composite vector solitons carrying topological charges,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(6), 066608 (2001).
[CrossRef] [PubMed]

C. Anastassiou, M. Soljačić, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85(23), 4888–4891 (2000).
[CrossRef] [PubMed]

Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Multicomponent two-dimensional solitons carrying topological charges,” Opt. Lett. 25(1), 61–63 (2000).
[CrossRef] [PubMed]

Mysyrowicz, A.

Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010).
[CrossRef] [PubMed]

Nepomnyaschchy, A.

Neshev, D.

Neshev, D. N.

Nie, Z. Q.

Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[CrossRef] [PubMed]

Y. P. Zhang, Z. Q. Nie, Y. Zhao, C. B. Li, R. M. Wang, J. H. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express 18(11), 10963–10972 (2010).
[CrossRef] [PubMed]

Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A 80(5), 055804 (2009).
[CrossRef]

Ostrovskaya, E. A.

Pelinovsky, D. E.

J. K. Yang and D. E. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(1), 016608 (2003).
[CrossRef] [PubMed]

Pérez-García, V. M.

Pigier, C.

Prade, B.

Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010).
[CrossRef] [PubMed]

Rebling, J.

Rotschild, C.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nature 2, 769–774 (2006).

Salgueiro, J. R.

J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056613 (2004).
[CrossRef] [PubMed]

Segev, M.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nature 2, 769–774 (2006).

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003).
[CrossRef] [PubMed]

Z. H. Musslimani, M. Soljacić, M. Segev, and D. N. Christodoulides, “Interactions between two-dimensional composite vector solitons carrying topological charges,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(6), 066608 (2001).
[CrossRef] [PubMed]

C. Pigier, R. Uzdin, T. Carmon, M. Segev, A. Nepomnyaschchy, and Z. H. Musslimani, “Collisions between (2+1)D rotating propeller solitons,” Opt. Lett. 26(20), 1577–1579 (2001).
[CrossRef] [PubMed]

Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Multicomponent two-dimensional solitons carrying topological charges,” Opt. Lett. 25(1), 61–63 (2000).
[CrossRef] [PubMed]

C. Anastassiou, M. Soljačić, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85(23), 4888–4891 (2000).
[CrossRef] [PubMed]

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286(5444), 1518–1523 (1999).
[CrossRef] [PubMed]

Shen, M.

L. J. Ge, Q. Wang, M. Shen, and J. L. Shi, “Dipole solitons in nonlocal nonlinear media with anisotropy,” Opt. Commun. 284(9), 2351–2356 (2011).
[CrossRef]

M. Shen, X. Chen, J. L. Shi, Q. Wang, and W. Krolikowski, “Incoherently coupled vector dipole soliton pairs in nonlocal media,” Opt. Commun. 282(24), 4805–4809 (2009).
[CrossRef]

Sheppard, A. P.

Shi, J. L.

L. J. Ge, Q. Wang, M. Shen, and J. L. Shi, “Dipole solitons in nonlocal nonlinear media with anisotropy,” Opt. Commun. 284(9), 2351–2356 (2011).
[CrossRef]

M. Shen, X. Chen, J. L. Shi, Q. Wang, and W. Krolikowski, “Incoherently coupled vector dipole soliton pairs in nonlocal media,” Opt. Commun. 282(24), 4805–4809 (2009).
[CrossRef]

Shih, M. F.

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92(4), 043904 (2004).
[CrossRef] [PubMed]

Si, J. H.

Singer, K.

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atoms pairs correlated to the ns-ns, np-np and nd-nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005).
[CrossRef]

Skryabin, D. V.

W. J. Firth and D. V. Skryabin, “Optical solitons carrying Orbital angular momentum,” Phys. Rev. Lett. 79(13), 2450–2453 (1997).
[CrossRef]

Skupin, S.

Snyder, A. W.

Soljacic, M.

Z. H. Musslimani, M. Soljacić, M. Segev, and D. N. Christodoulides, “Interactions between two-dimensional composite vector solitons carrying topological charges,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(6), 066608 (2001).
[CrossRef] [PubMed]

C. Anastassiou, M. Soljačić, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85(23), 4888–4891 (2000).
[CrossRef] [PubMed]

Song, J. P.

Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A 80(5), 055804 (2009).
[CrossRef]

Stanojevic, J.

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atoms pairs correlated to the ns-ns, np-np and nd-nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286(5444), 1518–1523 (1999).
[CrossRef] [PubMed]

Sukhorukov, A. A.

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005).
[CrossRef] [PubMed]

Swartzlander, G. A.

Tikhonenko, V.

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996).
[CrossRef] [PubMed]

Torner, L.

F. W. Ye, Y. V. Kartashov, and L. Torner, “Stabilization of dipole solitons in nonlocal nonlinear media,” Phys. Rev. A 77(4), 043821 (2008).
[CrossRef]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005).
[CrossRef]

Torres, J. P.

C. Anastassiou, M. Soljačić, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85(23), 4888–4891 (2000).
[CrossRef] [PubMed]

Uzdin, R.

Wang, H.

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87(7), 073601 (2001).
[CrossRef] [PubMed]

Wang, Q.

L. J. Ge, Q. Wang, M. Shen, and J. L. Shi, “Dipole solitons in nonlocal nonlinear media with anisotropy,” Opt. Commun. 284(9), 2351–2356 (2011).
[CrossRef]

M. Shen, X. Chen, J. L. Shi, Q. Wang, and W. Krolikowski, “Incoherently coupled vector dipole soliton pairs in nonlocal media,” Opt. Commun. 282(24), 4805–4809 (2009).
[CrossRef]

Wang, R. M.

Wang, Z. G.

Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[CrossRef] [PubMed]

Weidemüller, M.

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atoms pairs correlated to the ns-ns, np-np and nd-nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005).
[CrossRef]

Weilnau, C.

A. S. Desyatnikov, Y. S. Kivshar, K. Motzek, F. Kaiser, C. Weilnau, and C. Denz, “Multicomponent dipole-mode spatial solitons,” Opt. Lett. 27(8), 634–636 (2002).
[CrossRef] [PubMed]

K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun. 209(4-6), 501–506 (2002).
[CrossRef]

Xiao, M.

Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[CrossRef] [PubMed]

Y. P. Zhang, Z. Q. Nie, Y. Zhao, C. B. Li, R. M. Wang, J. H. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express 18(11), 10963–10972 (2010).
[CrossRef] [PubMed]

Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A 80(5), 055804 (2009).
[CrossRef]

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87(7), 073601 (2001).
[CrossRef] [PubMed]

Yang, J. K.

J. K. Yang, I. Makasyuk, A. Bezryadina, and Z. G. Chen, “Dipole solitons in optically induced two-dimensional photonic lattices,” Opt. Lett. 29(14), 1662–1664 (2004).
[CrossRef] [PubMed]

J. K. Yang and D. E. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(1), 016608 (2003).
[CrossRef] [PubMed]

Ye, F. W.

F. W. Ye, Y. V. Kartashov, and L. Torner, “Stabilization of dipole solitons in nonlocal nonlinear media,” Phys. Rev. A 77(4), 043821 (2008).
[CrossRef]

Zhang, X.

Zhang, Y. P.

Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[CrossRef] [PubMed]

Y. P. Zhang, Z. Q. Nie, Y. Zhao, C. B. Li, R. M. Wang, J. H. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express 18(11), 10963–10972 (2010).
[CrossRef] [PubMed]

Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A 80(5), 055804 (2009).
[CrossRef]

Zhao, Y.

Zheng, H. B.

Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A 80(5), 055804 (2009).
[CrossRef]

Zuo, C. C.

Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A 80(5), 055804 (2009).
[CrossRef]

J. Phys. At. Mol. Opt. Phys. (1)

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atoms pairs correlated to the ns-ns, np-np and nd-nd asymptotes,” J. Phys. At. Mol. Opt. Phys. 38(2), S295–S307 (2005).
[CrossRef]

Nature (1)

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nature 2, 769–774 (2006).

Opt. Commun. (3)

K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun. 209(4-6), 501–506 (2002).
[CrossRef]

L. J. Ge, Q. Wang, M. Shen, and J. L. Shi, “Dipole solitons in nonlocal nonlinear media with anisotropy,” Opt. Commun. 284(9), 2351–2356 (2011).
[CrossRef]

M. Shen, X. Chen, J. L. Shi, Q. Wang, and W. Krolikowski, “Incoherently coupled vector dipole soliton pairs in nonlocal media,” Opt. Commun. 282(24), 4805–4809 (2009).
[CrossRef]

Opt. Express (3)

Opt. Lett. (11)

A. H. Carlsson, J. N. Malmberg, D. Anderson, M. Lisak, E. A. Ostrovskaya, T. J. Alexander, and Y. S. Kivshar, “Linear and nonlinear waveguides induced by optical vortex solitons,” Opt. Lett. 25(9), 660–662 (2000).
[CrossRef] [PubMed]

A. P. Sheppard and M. Haelterman, “Polarization-domain solitary waves of circular symmetry in Kerr media,” Opt. Lett. 19(12), 859–861 (1994).
[CrossRef] [PubMed]

C. T. Law, X. Zhang, and G. A. Swartzlander., “Waveguiding properties of optical vortex solitons,” Opt. Lett. 25(1), 55–57 (2000).
[CrossRef] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31(8), 1100–1102 (2006).
[CrossRef]

A. W. Snyder and A. P. Sheppard, “Collisions, steering, and guidance with spatial solitons,” Opt. Lett. 18(7), 482–484 (1993).
[CrossRef] [PubMed]

C. Pigier, R. Uzdin, T. Carmon, M. Segev, A. Nepomnyaschchy, and Z. H. Musslimani, “Collisions between (2+1)D rotating propeller solitons,” Opt. Lett. 26(20), 1577–1579 (2001).
[CrossRef] [PubMed]

Y. V. Izdebskaya, J. Rebling, A. S. Desyatnikov, and Y. S. Kivshar, “Observation of vector solitons with hidden vorticity,” Opt. Lett. 37(5), 767–769 (2012).
[CrossRef] [PubMed]

A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. 26(7), 435–437 (2001).
[CrossRef] [PubMed]

Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Multicomponent two-dimensional solitons carrying topological charges,” Opt. Lett. 25(1), 61–63 (2000).
[CrossRef] [PubMed]

J. K. Yang, I. Makasyuk, A. Bezryadina, and Z. G. Chen, “Dipole solitons in optically induced two-dimensional photonic lattices,” Opt. Lett. 29(14), 1662–1664 (2004).
[CrossRef] [PubMed]

A. S. Desyatnikov, Y. S. Kivshar, K. Motzek, F. Kaiser, C. Weilnau, and C. Denz, “Multicomponent dipole-mode spatial solitons,” Opt. Lett. 27(8), 634–636 (2002).
[CrossRef] [PubMed]

Phys. Lett. A (1)

A. S. Desyatnikov, D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Stable counter-rotating vortex pairs in saturable media,” Phys. Lett. A 364(3-4), 231–234 (2007).
[CrossRef]

Phys. Rev. A (3)

Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A 80(5), 055804 (2009).
[CrossRef]

M. Marinescu and A. Dalgarno, “Dispersion forces and long-range electronic transition dipole moments of alkali-metal dimer excited states,” Phys. Rev. A 52(1), 311–328 (1995).
[CrossRef] [PubMed]

F. W. Ye, Y. V. Kartashov, and L. Torner, “Stabilization of dipole solitons in nonlocal nonlinear media,” Phys. Rev. A 77(4), 043821 (2008).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (3)

Z. H. Musslimani, M. Soljacić, M. Segev, and D. N. Christodoulides, “Interactions between two-dimensional composite vector solitons carrying topological charges,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(6), 066608 (2001).
[CrossRef] [PubMed]

J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056613 (2004).
[CrossRef] [PubMed]

J. K. Yang and D. E. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(1), 016608 (2003).
[CrossRef] [PubMed]

Phys. Rev. Lett. (11)

A. S. Desyatnikov and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 87(3), 033901 (2001).
[CrossRef] [PubMed]

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92(4), 043904 (2004).
[CrossRef] [PubMed]

C. Anastassiou, M. Soljačić, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. 85(23), 4888–4891 (2000).
[CrossRef] [PubMed]

Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. 105(5), 055003 (2010).
[CrossRef] [PubMed]

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87(7), 073601 (2001).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996).
[CrossRef] [PubMed]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003).
[CrossRef] [PubMed]

Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[CrossRef] [PubMed]

W. J. Firth and D. V. Skryabin, “Optical solitons carrying Orbital angular momentum,” Phys. Rev. Lett. 79(13), 2450–2453 (1997).
[CrossRef]

A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88(5), 053901 (2002).
[CrossRef] [PubMed]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005).
[CrossRef] [PubMed]

Prog. Opt. (1)

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005).
[CrossRef]

Science (1)

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286(5444), 1518–1523 (1999).
[CrossRef] [PubMed]

Other (1)

S. Trillo and W. Torruella, Spatial Solitons (Springer-Verlag, Berlin, 2001).

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Figures (6)

Fig. 1
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.

Fig. 2
Fig. 2

(a) Images (left) and horizontal-sizes (right) of FWM signals E F1,2,5,6 at different detuning Δ 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 Δ 2 , E F3 (squares), E F4 (circles), E F7 (triangles) and E F8 (reverse triangles).

Fig. 3
Fig. 3

(a1,2) Images of E F7 at different powers of (a1) E 1 , (a2) E 2 . (a3) Images of E F1 for different spatial configuration of E 1 and E 1 . From left to right, the intersection position of E 1 and E 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 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.

Fig. 4
Fig. 4

(a) Images of the four-component dipole-mode soliton i E Fi and j E Fj at different Δ 2 with and without dressing field E 2 and E 1 , respectively. (b) Experimental (left) and numerical (right) results of ten-component dipole-mode soliton with the propagation length Z=0, 15.9mm at Δ 2 =15GHZ

Fig. 5
Fig. 5

Images (the five top rows) of the pure FWM signals E F1,2,5,6 and E 2 dressed signal E F6 versus Δ 1 . The two bottom rows give the numerical results of E F6 with and without dressed field E 2 , respectively

Fig. 6
Fig. 6

Images of the superposition vortex signals at different Δ 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 2 dressed E F1 + E F2 . (b) Four-component vortex solitons i E Fi and j E Fj .

Equations (4)

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p=1,2,5,6 E Fp z i 2 E Fp 2 k Fp = p=1,2,5,6 i k Fp n 1 ( n 2 Sp | E Fp | 2 +2 n 2 Fp ) E Fp + η 1 E 1 ( E 1 ) * q=3,4,7,8 E Fq
q=3,4,7,8 E Fq z i 2 E Fq 2 k Fq = q=3,4,7,8 i k Fq n 1 ( n 2 Sq | E Fq | 2 +2 n 2 Fq ) E Fq + η 2 E 2 ( E 2 ) * p=1,2,5,6 E Fp
p=1,2,5,6 E Fp z i 2 k Fp ( 1 r E Fp r + 2 E Fp r 2 + 1 r 2 2 E Fp φ 2 )= p=1,2,5,6 i k Fp n 1 ( n 2 Sp | E Fp | 2 +2 n 2 Fp ) E Fp
q=3,4,7,8 E Fq z i 2 k Fq ( 1 r E Fq r + 2 E Fq r 2 + 1 r 2 2 E Fq φ 2 )= q=3,4,7,8 i k Fq n 1 ( n 2 Sq | E Fq | 2 +2 n 2 Fq ) E Fq

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