Abstract

We experimentally investigate the soliton formation and dynamics in the nonlinear propagation of the generated signal and probe beams in four-wave mixing (FWM) process with atomic coherence in a three-level atomic system, under the competition between focusing and defocusing nonlinearities, as well as between gain and dissipation, due to the third- and fifth-order nonlinear susceptibilities with opposite signs. With multi-parameter controllability and nonlinear competition in the system, fundamental, dipole, and azimuthally-modulated vortex FWM solitons can transform mutually from one to the other. Such investigations have potential applications in optical pattern formation and control, and all-optical communication.

© 2015 Optical Society of America

1. Introduction

The response of a central symmetric medium to incident light fields can often be expanded as P=Σm0χ(2m+1)E(2m+1), in which E(2m+1) (m=1,2,...) can be a mixing product of various light fields [1]. Typically, only the linear (m=0) and third-order (m=1) terms are considered, since the magnitude of χ(2m+1) decreases rapidly with larger m. During the past decade, many techniques were exploited to enhance higher-order nonlinearities, such as by making use of the effect of electromagnetically induced transparency (EIT) [2]. As a fascinating phenomenon in nonlinear systems, optical solitons have been investigated intensively, and different types of solitons with different forms of nonlinearities have been demonstrated [3–8]. It is worth noting that the multi-dimensional spatial solitons are not stable just with third-order nonlinearity because of the catastrophic self-focusing effect [3,4]. In order to avoid this undesirable effect, several types of nonlinearities [9,10] or elliptic beam with orbital angular momentum [11] were proposed to overcome such instability. One of the schemes is to consider the focusing third-order and defocusing fifth-order nonlinear effects simultaneously with relatively high beam intensity, i.e., employ the competing cubic-quintic (CQ) nonlinearities [12], which was experimentally reported in photoconductivity and degenerate four-wave-mixing (FWM) measurements [13,14], π-conjugated polymer [15], chalcogenide glasses [16], and organic materials [17]. Unfortunately, such nonlinearity is always accompanied by significant higher-order multiphoton processes [6] that will affect the beam propagation. In [19,20], the multi-dimensional solitons and light condensates were theoretically studied in atomic system with giant CQ nonlinearity enhanced by EIT. Nevertheless, no further experimental investigation was reported so far.

In this letter, we first experimentally demonstrate controllable soliton formation and dynamics of the FWM signal and probe beams with competing CQ nonlinearities, in which the competitions between nonlinear cubic gain and linear as well as quintic absorptions are also involved. Comparing with pervious literatures, the main novelties of this letter can be summarized as the following two points: (I) The CQ nonlinear competition not only shows up in the real parts (refraction) but also in imaginary parts (absorption/gain). What’s more, the competitive coefficients can be controlled with multiple parameters (e.g., the frequency detunings, powers of pump fields, etc.) in the chosen atomic medium. (II) The EIT-enhanced competitions are investigated in stabilizing fundamental soliton, and high-order dipole and azimuthally-modulated vortex (AMV) soliton of the FWM beam by controlling experiment configuration. Especially for the AMV case with the configuration being fixed, it is found that the CQ competition mechanism also can cause the transformation among fundamental, dipole and AMV solitons, which can be stabilized by using the EIT-enhanced fifth-order nonlinearity with relatively low intensity.

The paper is organized as follows. In section 2, we present the theoretical model and experiment scheme. In Section 3, the competition between CQ nonlinear in soliton formation and dynamics of the FWM signal and probe beams are investigated. In Section 4, we conclude the paper.

2. Basic theory and experimental scheme

The experiments are carried out in a Na atomic vapor oven. As shown in Fig. 1(a), a three-level atomic system is formed by energy levels |0 (3S1/2), |1 (3P3/2) and |2 (4D3/2) and five laser beams are adopted. The laser beams E1, E1 and Ep (E2 and E2) connect the transition between levels |0 and |1 (|2) with the atomic resonant frequency Ω1 (Ω2). The setup is shown in Fig. 1(b), and the configuration of the laser beams is given by the Fig. 1(c). The pump beams E1 and E1 with wavevectors k1 and k1 (with nearly the same power P1, frequency ω1) propagate in the opposite direction to the weak probe beam Ep (kp,P3, frequency ω3) with small angles (θ1=0.3o) between them. The three beams come from the same dye laser DL1 (10 Hz repetition rate, 5 ns pulse width, and 0.04cm1 linewidth) with frequency detuning Δ1=ω1Ω1, and their wave-vectors are in the x-o-z plane. The other two pump beams E2 and E2' (k2 and k2, P2 and P2, ω2) from another dye laser DL2 (which has the same characteristics as the DL1) with Δ2=ω2Ω2 also propagate along the opposite direction of Ep with small angles (θ2=0.3o) between them, and their wave-vectors are in the y-o-z plane. In this system, there will be one FWM process, that the one-photon resonant degenerate FWM process (Fig. 1(a)) satisfies the phase-matching condition kF=kp+k1k1 with the generated signal field EF propagating nearly in the opposite direction to the beam E1. In the experiments, Ep and EF are split by beam-splitter mirrors and recorded by charge coupled device (CCD) cameras and photo-multiply tubes (PMTs) simultaneously, as shown in Fig. 1(b).

 

Fig. 1 (a) The cascade three-level atomic system and the involved light fields used in the experiment. (b) Experimental setup. HR (high reflective mirror), BS (beam splitter50%), PMT (photomultiplier tube), CCD (charge coupled device). (c) Spatial beam geometric drawing used in the experiment.

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In such experimental arrangement, several input laser beams with spatially non-uniform intensity and generated FWM signals interact with each other in a large spatial region, therefore spatially-varying phase modulation can affect the propagation and spatial patterns of the probe and FWM signals, which are relatively weak compared to the pump beams. The propagations of the probe and FWM signals can be described by the set of coupled equations [8]:

Epzi2kp2Ep=ikp2[χp(1)+χp(3)|E|2+xp(5)|E|4]Ep+m=1,2ηEm(Em)EF,
EFzi2kF2EF=ikF2[χF(1)+χF(3)|E|2+χF(5)|E|4]EF+m=1,2ηEm(Em)Ep,
with the items in the first bracket on the right-hand side defined as
χp(n)|E|n1=χp(n)Sp|Ep|n1+2j=1,2xp(n)Xj(|Ej|n1+|Ej|n1),
χF(n)|E|n1=χF(n)SF|EF|n1+2j=1,2χF(n)Xj(|Ej|n1+|Ej|n1),
Where p and F are corresponding to the probe and EF signals, respectively. The second term in the right-hand side of Eq. (1a) and Eq. (1b) describes the conversion between the probe and FWM signals. In Eq. (1a)-(2b), xp,F(1) represents the linear susceptibility of the probe or FWM signals, χp,F(n)Sp,F (n=3,5) the nth-order self-phase modulation (SPM) nonlinear susceptibility, and χp,F(n)Xj the nth-order cross-phase modulation (XPM) nonlinear susceptibility excited by the pump field Ej (Ej). All these susceptibilities can be obtained by solving the density-matrix equations for the three-level atomic medium [22]. Here, we just show three typical expressions, which are involved linear, third-order, and fifth-order nonlinear susceptibilities in Eq. (1a) and Eq. (1b) as following:
χp,F(1)=ηK
χp,F(3)|E|2=ηK2(|Gp,F|2Γ11+|G1|2Γ11+|G1|2Γ11+|G2|2d2+|G2|2d2),
χp,F(5)|E|4=ηK3(|Gp,F|4Γ112+|G1|4Γ112+|G1|4Γ112+|G1|2|G1|2Γ112+|G2|4d22+|G2|4d22+|G2|2|G2|2d22),
With η=iNμ102/(є0), K=d1G22/d2, d1=Γ10+iΔ1, and d2=Γ20+i(Δ1+Δ2). Gp=μ10Ep/ is the Rabi frequency of the probe field, G1 (G1) and G2 (G2) are the Rabi frequency of the couple filed E1 (E1) and E2 (E2), respectively. Further, N is the atomic density, Γij denotes the population decay rate between the corresponding energy levels |i and |j, and μ10 is the electric dipole moment. Ej (Ej) is the electrical field amplitude of the corresponding field, which is spatial dependence as Ej(r) (Ej(r)), with r being the transverse dimensions perpendicular to the propagation direction z. Combining Eqs. (3a)-(3c), the total susceptibility can be written as χp,F=χp,F(1)+χp,F(3)|E|2+χp,F(5)|E|4.

In Fig. 2(a), we present the theoretical nonlinear refractive index of the system versus Δ2 and the total pump intensity. In Figs. 2(b) and (c), the dependences of the real and imaginary parts of the linear term χp,F(1), third-order one χp,F(3)X2(|E2|2+|E2|2) and fifth-order one χp,F(5)X2(|E2|4+|E2|4+|E2|2|E2|2) on the total pump intensity (|E2|2+|E2|2) are shown. It is obvious that χp,F(1) keeps constant with varying G2 (G2). Determined by the real parts of the third- and fifth-order susceptibilities, respectively, the focusing and defocusing nonlinearities follow cubic- and quintic-dependent laws, but the former case is positive and the latter case negative. The total refractive index composed of the linear part and the XPM nonlinear parts due to E2 and E2 can be further obtained as Δn=n1+n2X2(I2+I2)+n4X2(I22+I22), where n2X2=Re[χp,F(3)X2]/(є0cn1)>0 and n4X2={4Re[χp,F(5)X2]є02c2(n2X2)2}/(2є02c2n1)<0 [22] are the corresponding Kerr nonlinear coefficients, respectively, with I2=є0c|E2|2/2 (I2=є0c|E2|2/2) being the intensity determined by the power P2 (P2). So when I2 (I2) is not large enough, the refractive index Δn will increase with I2 (I2) due to the dominant positive third-order nonlinearity increasing in cubic law, but as I2 (I2) exceeds a threshold value and gets into high-power region, it turns to decrease because that the negative fifth-order nonlinearity will play a dominant role on Δn. As shown in Fig. 2(a), this competition between different nonlinearities can be controlled by both the pump field detuning Δ2 and I2 (I2). Thus, in the propagation of the probe or FWM beams, the diffraction will accompany competing nonlinearities between focusing and defocusing for certain pump field intensity, which is crucial for yielding high-order stable spatial solitons. The imaginary parts of the susceptibilities that offer gain or loss to the probe or FWM signals also have a competition which can influence the propagation of these beams. As presented in Fig. 2(c), when the pump intensities are small to satisfy |Im{χp,F(1)}||Im{χp,F(3)X2|E|2}||Im{χp,F(5)X2|E|4}|, the linear dissipation will be the dominant one. However, with I2 (I2) increasing, the satisfied conditions of |Im{χp,F(3)X2|E|2}||Im{χp,F(1)}| and Im{χp,F(3)X2|E|2}<0 can make the third-order nonlinear gain be dominant. If I2 (I2) further increases, Im{χp,F(5)X2|E|4}>0 will make the fifth-order nonlinear dissipation be dominant. So, competitions exist not only among the real parts but also among the imaginary parts of the third- and fifth-order nonlinearities.

 

Fig. 2 (a)The theoretically calculated nonlinear refractive index as a function of Δ2 and the total pump intensity. The theoretically calculated real (b) and imaginary (c) parts of χp,F(1), χp,F(3)X2(|E2|2+|E2|2), χp,F(5)X2(|E2|4+|E2|4+|E2|2|E2|2) and their sum versus the total pump intensity.

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In the experiment, the power of E1,2 is approximately 2 times stronger than that of E1,2, and 10 times stronger than that of Ep,F, therefore the SPM due to Ep,F can be neglected. Furthermore, neglecting diffraction, we can obtain a set of solutions of Ep,F, in which the nonlinear phase shifts introduced in the propagation are ϕp,F=2kp,FzΣm=1,2Σj=1,2[n2mXj(Ijm+Imm)]/(n0Ip,F). The additional transverse propagation wave-vector can be obtained as δkx=(ϕp,F/x)x^, where x^ is the unit vector along the horizontal axes. The direction of δkx determines the horizontal propagation of the beam, while the phase-front curvature 2ϕp,F/x2<0 (2ϕp,F/x2>0) lead to local focusing (defocusing) of the beams imposed on the weak field by the strong filed. Under balanced diffraction and XPM, the probe and FWM signals can form solitons with the electrical field intensity profile generally expressed as

E(R,ф)=αh[cos(γф)+isin(γф+Kπ/2)]exp[i(lф+фNL)],
where 2γ is the hump number, 0K1 is the modulation depth and l the topological charge. If the angles between E1,(E2)E1,(E2) and Ep are all sufficiently small, γ=0, l=0 and fundamental mode solitons of Ep,F are obtained. If the angle between Ep and E1 (E2) is small, but the one between E1 (E2) and E1 (E2) is sufficiently large to introduce the so-called EIG because of interference [8], γ=1, l=1, K=1 and dipole mode soliton are then obtained. However, as the angles are all sufficiently large, γ3/2, l=1, 0K1 and azimuthally-modulated vortex (AMV) solitons appear. Besides by the switch among different beams configuration, such transformations among different types of soliton also can be acquired by competing CQ nonlinearities, which will be shown in the Fig. 4.

3. Experimental observation of cubic-quintic solitons

First, the images of the transmission of the probe beam versus Δ2, captured by CCD cameras with different beams or combinations turned on, are shown in Figs. 3(a1)-(a4), and the corresponding intensities versus Δ2 recorded by PMT are shown in Fig. 3(a5). When one of or both of E2 and E2 are on, it is obvious that the probe suffers more significant focusing around the two-photon resonant (TPR) point (Δ1+Δ2=0) than in other regions along the Δ2-axis, since the large XPM nonlinearity from E2 and E2 can bring strong focusing. The total nonlinear refractive index ΔnX2=Δn2X2+Δn4X2, i.e., the sum of third- and fifth-order nonlinear refractive indices, is positive and very large when Δ2є[3050]GHz, which undergoes sharp decrease when Δ2 is far away from TPR. This explanation can be further verified by the images with E2 and E2 both off, which shows no variation of the probe transmission images versus Δ2 due to the absence of Δ2-dependent Kerr nonlinearity (Fig. 3(a4)). It is interesting that the image focusing is enhanced in Fig. 3(a2) compared with Figs. 3(a1), and the focusing in Fig. 3(a3) is almost the same as that in Fig. 3(a1), though the total pump power in Fig. 3(a1) with E2 and E2 both turned on is larger than those in Fig. 3(a3) with one of the beams turned on. To explain this abnormal phenomenon, the fifth-order nonlinearity should be considered, i.e., the competition between CQ nonlinearities mentioned above. Specifically, in Fig. 3(a1), with both E2 and E2 turned on, the third-order nonlinear refractive index Δn2X2=n2X2(I2+I2) can be effectively weakened by the fifth-order one Δn4X2=n4X2(I22+I22). Compared to the case in Fig. 3(a1), we find Δn4X2 suffers more reduction than Δn2X2 when E2 is turned off (Fig. 3(a2)), which weakens the compensative effect of Δn4X2 on Δn2X2, thus ΔnX2 increases and results in more significant focusing. In Fig. 3(a3), |Δn2X2|=|n2X2I2| and |Δn4X2|=|n4X2I22| have nearly equal reductions as compared to the ones in Fig. 3(a1), so ΔnX2 has nearly no variation and the nonlinear focusing keeps. The three dots on the solid curve in Fig. 2(b) can elucidate the changing of focusing strength in Figs. 3(a1)-(a3). This explanation can be further verified by the more significant defocusing effect around TPR when p2 is further increased to a higher value as shown in Fig. 3(b) from bottom to top. Satisfying the quintic-law, the decreasing factor of Δn4X2=n4X2(I22+I22) is larger than that of Δn2X2=n2X2(I22+I2), so ΔnX2 increases, which leads to the more significant focusing as shown in Figs. 3(b1)-(b6).

 

Fig. 3 (a) The probe images versus Δ2. (b) The probe images with P2 decreasing from top to bottom. (c) and (d) are similar to (a) and (b), respectively, but show images of EF. (e) The iso-surface plot of the beam with increasing P2, and the xtick labels mark the cases shown in (a). (f) The iso-surface of the beam versus Δ2. Images from (a1) to (a4) as well as from (c1) to (c4) correspond to no beam blocked, E2 blocked, E2 blocked, and E2 &E2 blocked, respectively, with the corresponding intensity curves recorded by PMT shown in (a5) and (c5). For (b), (d), and (f), no beam is blocked. For all the cases, Δ1=40GHz.

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The intensity curves also reveal anomaly as shown in Fig. 3(a5), in which the EIT peak with E2 or E2 individually on is higher than that with both E2 and E2 on, though the total pump fields intensity is lower in the former case than in the latter case. The reason is that according to Eq. (3c) and the fifth-order curve (the dashdot curve) in Fig. 2(c), the fifth-order nonlinear loss Im{χp,F(5)X2|E|4} is larger with both E2 and E2 on than with E2 or E2 individually on, which can effectively lower the transmission peak. When both E2 and E2 are off, there is only the linear susceptibility and the nonlinear absorption nearly is eliminated, so the EIT peak disappears.

Also, the spatial shift of probe beam due to the cross nonlinear modulation of E2 and E2 can be observed. In Fig. 3(a1), when E2 and E2 are both on, the nonlinear phase shift introduced by the two fields is ϕp=2ΔnX2kpzer2/2/(n0Ip) and the transverse nonlinear wave-vector δr=rϕp is the gradient of nonlinear phase shift [23,24]. Because the probe beam spot is within the wings of E2 and E2 along the y-axis, the δr can be approximated by δr=2ΔnX2kprzer2/2/(n0Ip) which leads to the upward shift of probe spots, due to positive ΔnX2. When E2 is turned off (Fig. 3(a2)), similar to the nonlinear focusing mentioned above, ΔnX2 is larger than that in Fig. 3(a1), so the upward shift displacement becomes larger. With E2 off in Fig. 3(a3), ΔnX2 is almost the same as that in Fig. 3(a1), but smaller than that in Fig. 3(a2), so the shift displacement is similar to that in Fig. 3(a1). When E2 and E2 are both off, δy=0 for the XPM effect supplied by strong sufficiently pump fields diminishes and the spatial shift disappears (Fig. 3(a4)). Also, in the decreasing of P2 from Figs. 3(b1)-(b6), the spatial shift displacement gradually increases due to the increasing ΔnX2.

Next, we adjust the beam configuration to make the FWM signals form dipole solitons and investigate their dynamics under competing third- and fifth-order nonlinearities. When Δ2 is scanned in the region far away from TPR, the nonlinearity induced by E2 and E2 is very weak, so the dipole modulation effect of electromagnetically induced grating (EIG) induced by E1 and E1 is dominant and EF gets dipole pattern [8]. This is verified by the almost unchanged dipole soliton patterns of EF when Δ2 is within the regions of [8010]GHz and [70160]GHz in Figs. 3(c1)-(c4). When Δ2 is set within the region of [1070]GHz, i.e., around TPR, the nonlinear refractive index induced by E2 and E2 on the FWM signal is positive, which causes the dipole mode of EF to collapse into a single spot due to the strong focusing effect. When E2 and E2 are blocked (Fig. 3(c4)), EF remains to be in dipole mode even around TPR due to the vanishing nonlinear focusing effect. Compared to the nonlinear focusing in Fig. 3(c1), the focusing in Fig. 3(c2) with E2 blocked is stronger and that in Fig. 3(c3) with E2 blocked weaker. The reason is also that the total refractive index with E2 off is smaller than that with E2 off, but larger than that with both beams on. Due to the enhancement effect of the pump fields, the intensity of the FWM signal shows a peak higher than the background when E2 and E2 are blocked. With E2 off, the enhancement effect is weakened because of the decreasing of the pump field intensity, so the peak becomes lower; when E2 is off, the enhancement effect is weaker than that with E2 and E2 both on, but stronger than that with E2 off, however the peak keeps almost the same height with that when E2 is off. The main reason is that the magnitude of fifth-order nonlinear loss with E2 off is much larger than that with E2 off, which can bring considerable nonlinear loss and effectively limit the rising of peak. This variation reveals that the competition between the enhancement and nonlinear absorption plays an important role on the final received FWM signal.

With E2 and E2 both on, we investigate the images of the FWM signal with an increasing power P2. The experimental results are summarized in Fig. 3(d). It is obvious that generated FWM beam is characterized by the dipole mode and single-spot at far away from TPR and around the resonance remain, respectively. But we find that the single-spot focuses more significantly with P2 increasing (Fig. 3(d)), though the magnitude of Δn4X2 increases more rapidly than Δn2X2, which will decrease the spot focusing. The reason for this contradiction is that the imaginary parts of χ(3) and χ(1),(5) are negative and positive, respectively. The nonlinearity of E2 is strong around TPR, and the positive imaginary part of susceptibilities will lead to a considerable dissipation of the FWM signal, which will be easily affected by the nonlinearity according to ϕF=2ΔnX2kFzer2/2/(n0IF). In Fig. 3(d), the total imaginary part of the linear and nonlinear responses is positive and increases with increasing P2 because the initial P2 is set to achieve |Im[χF(5)|E2|4]|>|Im[χF(3)|E2|2]|, which will lead to the decrease of the FWM signal intensity. Thus, IF decreases due to increasing loss with increasing P2, and the decreasing rate is larger than the increasing rate of ΔnX2, which leads to a larger ϕF that the FWM signal around TPR exhibit more significant nonlinear focusing as shown in Figs. 3(d1)-(d7). The iso-surfaces shown in Figs. 3(e) and (f) display the fundamental soliton changing with blockage of different beams and the transformation between dipole and fundamental solitons versus Δ2 based on Eq. (4) due to the competitions of nonlinearities and gain/loss mechanisms.

The influence of nonlinear competition on the transformation among fundamental, dipole and AMV solitons is also investigated. If we adjust the spatial configuration of the beams, the AMV mode of the EF signal can be obtained. The formation such AMV mode is due to the spatial modulation of the interference patterns [25] among E1, E1 and Ep. The AMV FWM field can be rewritten from Eq. (4) as

E(R,ϕ)=(α+βR)exp(R2/R0)cos(γmϕ)+isin(γmϕ+βπ/2)exp[i(lϕ+ϕNL)],
in which α and β are amplitude coefficients introduced by the nonlinearities due to E2 and E2, K in Eq. (4) is replaced by β in Eq. (5), and γ is replaced by γm which is also determined by the nonlinearities due to E2 and E2. Specifically, the coefficients α, β, and γm can be expressed as α=(n2X2I2+n4X2I22)+|(Δ1+Δ2)Δ1|(n2X2I2+n4X2I22), β=(n2X2I2+n4X2I22)+|(Δ1+Δ2)/Δ1|[n2X1(I1+I1)+(n2X2I2+n4X2I22)], and γm=m+exp[n2X2(I2+I2)n4X2(I22+I22)], respectively. It is easy to see that α0 if E2 is turned off around TPR or far away from TPR (Δn2X1 decreases rather sharply), β0 if E2 is blocked around TPR, and m in γm is 0 for the dipole mode modulation (Fig. 3(c)) and 1 for modulated vortex mode modulation (Fig. 4). So when E2 is blocked in Fig. 4(a1), fundamental mode soliton is obtained around TPR due to sufficiently strong XPM focusing of E2. If Δ2 is far away from TPR, α0 and β should be considered because Δn2X1 does not change, so the non-focusing AMV modes appear as shown in the cases of Figs. 4(a2)-(a5). In other word, EF keeps the vortex pattern as compression effect of the XPM nonlinearity from E2 and E2 is very weak. When E2 is blocked in Fig. 4(a2), α0 but β0 around TPR. Since the second term in γm is very small because of the large n2X2I2+n4X2I22, we observe a dipole-like mode around TPR, which is due to the weak focusing effect of E2. When no beam is blocked, as shown in Fig. 4(a3), the total intensity from E2 and E2 is large enough to balance the competing CQ nonlinearities. In this case, α<β because of a higher E2 field as shown by the middle dot in Fig. 2(b) and γm2, so we observe a vortex mode versus different Δ2. The essential reason for the transformation among the three types of modes is the competition between third- and fifth-order nonlinearities.

 

Fig. 4 (a) The images of EF versus Δ2 with (a1) E2 blocked, (a2) E2 blocked, (a3) no beam blocked and the powers of pumping fields set as I210I2, (a4) no beam blocked and I220I2, (a5) no beam blocked and I250I2. (b) The 2D plots of EF around two-photon resonant (TPR) point corresponding to (a1)-(a3). For all the cases, Δ1=30GHz.

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In Figs. 4(a4) and (a5), we increase the power P2 with E2 fixed, and find that the higher total intensity breaks the balance between the cubic and quintic nonlinearities, and the latter one defocuses the beam. This becomes more obvious around TPR because the magnitude of Δn4X2 increases much more significantly than that of Δn2X2, and therefore the defocusing effect exceeds the focusing effect. The detailed explanations are similar to the ones used for Figs. 3(a1)-(a3) and (c1)-(c3). Figs. 4(b1)-(b3) obtained from Eq. (5), which correspond to Figs. 4(a1)-(a3), visually display that the modulated FWM transform vortex modes into fundamental mode or dipole mode around TPR. In Fig. 4, it is worth noting that the vortex versus Δ2 formed here is significantly different from those exhibited in [5,6,7], i.e., the pattern does not rotate compared to the usual vortex solitons. Such stability is due to the counteraction between nonlinear phase shift ϕF and the intrinsic phase from nonzero topology charge and azimuthal modulation in Eq. (5).

4. Conclusion

In conclusion, we have experimentally investigated the multi-parameter controllable solitons under competing χ(3) and χ(5) nonlinear susceptibilities enhanced by atomic coherence or EIT in a three-level atomic system. Both in the spatial images and spectral intensity curves of the probe transmission and FWM beams, we have found that the nonlinear propagation processes can be significantly affected by the fifth-order nonlinearity if the pump field intensities are high. In addition, we have demonstrated transformations among fundamental, dipole and AMV solitons by controlling the pump beam geometric configuration and/or competing CQ nonlinearities. The experimental and theoretical results agree with each other very well. Such spatial pattern formation and propagation control can have important applications in all-optical image storage, processing and communication.

Acknowledgments

This work was supported by the National Basic Research Program of China (2012CB921804) and NSFC (10974151, 61078002, 61078020, 11104214, 61108017), CX12189WL02, and CX12189WL03.

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11. A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 104, 053902 (2010). [CrossRef]   [PubMed]  

12. Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89(27), 273902 (2002). [CrossRef]   [PubMed]  

13. J.-S. Lee, M. V. Kovalenko, J. Huang, D. S. Chung, and D. V. Talapin, “Band-like transport, high electron mobility and high photoconductivity in all-inorganic nanocrystal arrays,” Nat. Nanotechnol. 6(6), 348–352 (2011). [CrossRef]   [PubMed]  

14. S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11(1), 34–38 (2011). [CrossRef]   [PubMed]  

15. J. M. Nunzi and F. Charra, “Picosecond two-photon absorption effects on index variation gratings in polydiacetylenes,” Nonlinear Opt. 2, 131 (1992).

16. B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, “Identification and role of two-photon excited states in a pi -conjugated polymer,” Phys. Rev. Lett. 73(4), 597–600 (1994). [CrossRef]   [PubMed]  

17. G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1–6), 427–433 (2003). [CrossRef]  

18. S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009). [CrossRef]   [PubMed]  

19. H. Michinel, M. J. Paz-Alonso, and V. M. Pérez-García, “Turning light into a liquid via atomic coherence,” Phys. Rev. Lett. 96(2), 023903 (2006). [CrossRef]   [PubMed]  

20. Á. Paredes, D. Feijoo, and H. Michinel, “Coherent cavitation in the liquid of light,” Phys. Rev. Lett. 112(17), 173901 (2014). [CrossRef]   [PubMed]  

21. M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006). [CrossRef]   [PubMed]  

22. Y. Zhang and M. Xiao, Multi-Wave Mixing Processes: From Ultrafast Polarization Beats To Electromagnetically Induced Transparency (H & Springer, 2009).

23. Y. Zhang, Z. Nie, H. Zheng, C. Li, J. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013835 (2009). [CrossRef]  

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

25. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004). [CrossRef]   [PubMed]  

References

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  1. R. W. Boyd, Nonlinear Optics (Academic Press, 2008), 3rd ed.
  2. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
    [Crossref]
  3. M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387(6636), 858–883 (1997).
    [Crossref] [PubMed]
  4. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullet,” Nat. Photonics 4(2), 103–106 (2010).
    [Crossref]
  5. A. S. Desyatnikov and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 87(3), 033901 (2001).
    [Crossref] [PubMed]
  6. A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
    [Crossref] [PubMed]
  7. S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, “Azimuthons in nonlocal nonlinear media,” Opt. Express 14(17), 7903–7908 (2006).
    [Crossref] [PubMed]
  8. Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
    [Crossref] [PubMed]
  9. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).
  10. C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
    [Crossref] [PubMed]
  11. A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 104, 053902 (2010).
    [Crossref] [PubMed]
  12. Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89(27), 273902 (2002).
    [Crossref] [PubMed]
  13. J.-S. Lee, M. V. Kovalenko, J. Huang, D. S. Chung, and D. V. Talapin, “Band-like transport, high electron mobility and high photoconductivity in all-inorganic nanocrystal arrays,” Nat. Nanotechnol. 6(6), 348–352 (2011).
    [Crossref] [PubMed]
  14. S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11(1), 34–38 (2011).
    [Crossref] [PubMed]
  15. J. M. Nunzi and F. Charra, “Picosecond two-photon absorption effects on index variation gratings in polydiacetylenes,” Nonlinear Opt. 2, 131 (1992).
  16. B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, “Identification and role of two-photon excited states in a pi -conjugated polymer,” Phys. Rev. Lett. 73(4), 597–600 (1994).
    [Crossref] [PubMed]
  17. G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1–6), 427–433 (2003).
    [Crossref]
  18. S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
    [Crossref] [PubMed]
  19. H. Michinel, M. J. Paz-Alonso, and V. M. Pérez-García, “Turning light into a liquid via atomic coherence,” Phys. Rev. Lett. 96(2), 023903 (2006).
    [Crossref] [PubMed]
  20. Á. Paredes, D. Feijoo, and H. Michinel, “Coherent cavitation in the liquid of light,” Phys. Rev. Lett. 112(17), 173901 (2014).
    [Crossref] [PubMed]
  21. M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006).
    [Crossref] [PubMed]
  22. Y. Zhang and M. Xiao, Multi-Wave Mixing Processes: From Ultrafast Polarization Beats To Electromagnetically Induced Transparency (H & Springer, 2009).
  23. Y. Zhang, Z. Nie, H. Zheng, C. Li, J. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013835 (2009).
    [Crossref]
  24. Y. Zhang, Z. Nie, Y. Zhao, C. Li, R. Wang, J. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express 18(11), 10963–10972 (2010).
    [Crossref] [PubMed]
  25. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
    [Crossref] [PubMed]

2014 (1)

Á. Paredes, D. Feijoo, and H. Michinel, “Coherent cavitation in the liquid of light,” Phys. Rev. Lett. 112(17), 173901 (2014).
[Crossref] [PubMed]

2011 (3)

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

J.-S. Lee, M. V. Kovalenko, J. Huang, D. S. Chung, and D. V. Talapin, “Band-like transport, high electron mobility and high photoconductivity in all-inorganic nanocrystal arrays,” Nat. Nanotechnol. 6(6), 348–352 (2011).
[Crossref] [PubMed]

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11(1), 34–38 (2011).
[Crossref] [PubMed]

2010 (4)

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 104, 053902 (2010).
[Crossref] [PubMed]

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullet,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

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

2009 (2)

Y. Zhang, Z. Nie, H. Zheng, C. Li, J. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013835 (2009).
[Crossref]

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[Crossref] [PubMed]

2006 (3)

H. Michinel, M. J. Paz-Alonso, and V. M. Pérez-García, “Turning light into a liquid via atomic coherence,” Phys. Rev. Lett. 96(2), 023903 (2006).
[Crossref] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, “Azimuthons in nonlocal nonlinear media,” Opt. Express 14(17), 7903–7908 (2006).
[Crossref] [PubMed]

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006).
[Crossref] [PubMed]

2005 (1)

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

2004 (1)

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[Crossref] [PubMed]

2003 (1)

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1–6), 427–433 (2003).
[Crossref]

2002 (1)

Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89(27), 273902 (2002).
[Crossref] [PubMed]

2001 (1)

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

1997 (2)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387(6636), 858–883 (1997).
[Crossref] [PubMed]

1994 (1)

B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, “Identification and role of two-photon excited states in a pi -conjugated polymer,” Phys. Rev. Lett. 73(4), 597–600 (1994).
[Crossref] [PubMed]

1992 (1)

J. M. Nunzi and F. Charra, “Picosecond two-photon absorption effects on index variation gratings in polydiacetylenes,” Nonlinear Opt. 2, 131 (1992).

Artoni, M.

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006).
[Crossref] [PubMed]

Baker, G.

B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, “Identification and role of two-photon excited states in a pi -conjugated polymer,” Phys. Rev. Lett. 73(4), 597–600 (1994).
[Crossref] [PubMed]

Bartal, G.

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11(1), 34–38 (2011).
[Crossref] [PubMed]

Boudebs, G.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1–6), 427–433 (2003).
[Crossref]

Buccoliero, D.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 104, 053902 (2010).
[Crossref] [PubMed]

Carmon, T.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Cha, M.

B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, “Identification and role of two-photon excited states in a pi -conjugated polymer,” Phys. Rev. Lett. 73(4), 597–600 (1994).
[Crossref] [PubMed]

Charra, F.

J. M. Nunzi and F. Charra, “Picosecond two-photon absorption effects on index variation gratings in polydiacetylenes,” Nonlinear Opt. 2, 131 (1992).

Chen, H.

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

Cherukulappurath, S.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1–6), 427–433 (2003).
[Crossref]

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullet,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Christodoulides, D. N.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullet,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Chung, D. S.

J.-S. Lee, M. V. Kovalenko, J. Huang, D. S. Chung, and D. V. Talapin, “Band-like transport, high electron mobility and high photoconductivity in all-inorganic nanocrystal arrays,” Nat. Nanotechnol. 6(6), 348–352 (2011).
[Crossref] [PubMed]

Cohen, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Courtial, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[Crossref] [PubMed]

Crasovan, L. C.

Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89(27), 273902 (2002).
[Crossref] [PubMed]

Dennis, M. R.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 104, 053902 (2010).
[Crossref] [PubMed]

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[Crossref] [PubMed]

Desyatnikov, A. S.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 104, 053902 (2010).
[Crossref] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, “Azimuthons in nonlocal nonlinear media,” Opt. Express 14(17), 7903–7908 (2006).
[Crossref] [PubMed]

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

Etemad, S.

B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, “Identification and role of two-photon excited states in a pi -conjugated polymer,” Phys. Rev. Lett. 73(4), 597–600 (1994).
[Crossref] [PubMed]

Feijoo, D.

Á. Paredes, D. Feijoo, and H. Michinel, “Coherent cavitation in the liquid of light,” Phys. Rev. Lett. 112(17), 173901 (2014).
[Crossref] [PubMed]

Harris, S. E.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

Huang, J.

J.-S. Lee, M. V. Kovalenko, J. Huang, D. S. Chung, and D. V. Talapin, “Band-like transport, high electron mobility and high photoconductivity in all-inorganic nanocrystal arrays,” Nat. Nanotechnol. 6(6), 348–352 (2011).
[Crossref] [PubMed]

Kartashov, Y. V.

Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89(27), 273902 (2002).
[Crossref] [PubMed]

Kivshar, Y. S.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 104, 053902 (2010).
[Crossref] [PubMed]

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, “Azimuthons in nonlocal nonlinear media,” Opt. Express 14(17), 7903–7908 (2006).
[Crossref] [PubMed]

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

Kovalenko, M. V.

J.-S. Lee, M. V. Kovalenko, J. Huang, D. S. Chung, and D. V. Talapin, “Band-like transport, high electron mobility and high photoconductivity in all-inorganic nanocrystal arrays,” Nat. Nanotechnol. 6(6), 348–352 (2011).
[Crossref] [PubMed]

La Rocca, G. C.

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006).
[Crossref] [PubMed]

Lawrence, B.

B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, “Identification and role of two-photon excited states in a pi -conjugated polymer,” Phys. Rev. Lett. 73(4), 597–600 (1994).
[Crossref] [PubMed]

Leach, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[Crossref] [PubMed]

Leblond, H.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1–6), 427–433 (2003).
[Crossref]

Lee, J.-S.

J.-S. Lee, M. V. Kovalenko, J. Huang, D. S. Chung, and D. V. Talapin, “Band-like transport, high electron mobility and high photoconductivity in all-inorganic nanocrystal arrays,” Nat. Nanotechnol. 6(6), 348–352 (2011).
[Crossref] [PubMed]

Leo, K.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[Crossref] [PubMed]

Li, C.

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

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

Y. Zhang, Z. Nie, H. Zheng, C. Li, J. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013835 (2009).
[Crossref]

Lindner, F.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[Crossref] [PubMed]

Lopez-Aguayo, S.

Lu, K.

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

Lüssem, B.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[Crossref] [PubMed]

Manela, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Meth, J.

B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, “Identification and role of two-photon excited states in a pi -conjugated polymer,” Phys. Rev. Lett. 73(4), 597–600 (1994).
[Crossref] [PubMed]

Michinel, H.

Á. Paredes, D. Feijoo, and H. Michinel, “Coherent cavitation in the liquid of light,” Phys. Rev. Lett. 112(17), 173901 (2014).
[Crossref] [PubMed]

H. Michinel, M. J. Paz-Alonso, and V. M. Pérez-García, “Turning light into a liquid via atomic coherence,” Phys. Rev. Lett. 96(2), 023903 (2006).
[Crossref] [PubMed]

Mihalache, D.

Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89(27), 273902 (2002).
[Crossref] [PubMed]

Mitchell, M.

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387(6636), 858–883 (1997).
[Crossref] [PubMed]

Nie, Z.

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

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

Y. Zhang, Z. Nie, H. Zheng, C. Li, J. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013835 (2009).
[Crossref]

Nunzi, J. M.

J. M. Nunzi and F. Charra, “Picosecond two-photon absorption effects on index variation gratings in polydiacetylenes,” Nonlinear Opt. 2, 131 (1992).

Padgett, M. J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[Crossref] [PubMed]

Palomba, S.

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11(1), 34–38 (2011).
[Crossref] [PubMed]

Paredes, Á.

Á. Paredes, D. Feijoo, and H. Michinel, “Coherent cavitation in the liquid of light,” Phys. Rev. Lett. 112(17), 173901 (2014).
[Crossref] [PubMed]

Park, Y.

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11(1), 34–38 (2011).
[Crossref] [PubMed]

Paz-Alonso, M. J.

H. Michinel, M. J. Paz-Alonso, and V. M. Pérez-García, “Turning light into a liquid via atomic coherence,” Phys. Rev. Lett. 96(2), 023903 (2006).
[Crossref] [PubMed]

Pérez-García, V. M.

H. Michinel, M. J. Paz-Alonso, and V. M. Pérez-García, “Turning light into a liquid via atomic coherence,” Phys. Rev. Lett. 96(2), 023903 (2006).
[Crossref] [PubMed]

Reineke, S.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[Crossref] [PubMed]

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullet,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Rotschild, C.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Sanchez, F.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1–6), 427–433 (2003).
[Crossref]

Schwartz, G.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[Crossref] [PubMed]

Segev, M.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387(6636), 858–883 (1997).
[Crossref] [PubMed]

Seidler, N.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[Crossref] [PubMed]

Si, J.

Smektala, F.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1–6), 427–433 (2003).
[Crossref]

Song, J.

Y. Zhang, Z. Nie, H. Zheng, C. Li, J. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013835 (2009).
[Crossref]

Stegeman, G. I.

B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, “Identification and role of two-photon excited states in a pi -conjugated polymer,” Phys. Rev. Lett. 73(4), 597–600 (1994).
[Crossref] [PubMed]

Sundheimer, M. L.

B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, “Identification and role of two-photon excited states in a pi -conjugated polymer,” Phys. Rev. Lett. 73(4), 597–600 (1994).
[Crossref] [PubMed]

Talapin, D. V.

J.-S. Lee, M. V. Kovalenko, J. Huang, D. S. Chung, and D. V. Talapin, “Band-like transport, high electron mobility and high photoconductivity in all-inorganic nanocrystal arrays,” Nat. Nanotechnol. 6(6), 348–352 (2011).
[Crossref] [PubMed]

Torner, L.

Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89(27), 273902 (2002).
[Crossref] [PubMed]

Torruellas, W. E.

B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, “Identification and role of two-photon excited states in a pi -conjugated polymer,” Phys. Rev. Lett. 73(4), 597–600 (1994).
[Crossref] [PubMed]

Troles, J.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1–6), 427–433 (2003).
[Crossref]

Walzer, K.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[Crossref] [PubMed]

Wang, R.

Wang, Z.

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

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullet,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Xiao, M.

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

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

Y. Zhang, Z. Nie, H. Zheng, C. Li, J. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013835 (2009).
[Crossref]

Yin, X.

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11(1), 34–38 (2011).
[Crossref] [PubMed]

Zhang, S.

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11(1), 34–38 (2011).
[Crossref] [PubMed]

Zhang, X.

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11(1), 34–38 (2011).
[Crossref] [PubMed]

Zhang, Y.

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

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

Y. Zhang, Z. Nie, H. Zheng, C. Li, J. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013835 (2009).
[Crossref]

Zhao, Y.

Zheng, H.

Y. Zhang, Z. Nie, H. Zheng, C. Li, J. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013835 (2009).
[Crossref]

Nat. Mater. (1)

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11(1), 34–38 (2011).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

J.-S. Lee, M. V. Kovalenko, J. Huang, D. S. Chung, and D. V. Talapin, “Band-like transport, high electron mobility and high photoconductivity in all-inorganic nanocrystal arrays,” Nat. Nanotechnol. 6(6), 348–352 (2011).
[Crossref] [PubMed]

Nat. Photonics (1)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullet,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Nature (3)

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387(6636), 858–883 (1997).
[Crossref] [PubMed]

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[Crossref] [PubMed]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[Crossref] [PubMed]

Nonlinear Opt. (1)

J. M. Nunzi and F. Charra, “Picosecond two-photon absorption effects on index variation gratings in polydiacetylenes,” Nonlinear Opt. 2, 131 (1992).

Opt. Commun. (1)

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1–6), 427–433 (2003).
[Crossref]

Opt. Express (2)

Phys. Rev. A (1)

Y. Zhang, Z. Nie, H. Zheng, C. Li, J. Song, and M. Xiao, “Electromagnetically induced spatial nonlinear dispersion of four-wave mixing,” Phys. Rev. A 80(1), 013835 (2009).
[Crossref]

Phys. Rev. Lett. (10)

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

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 104, 053902 (2010).
[Crossref] [PubMed]

Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89(27), 273902 (2002).
[Crossref] [PubMed]

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

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, “Identification and role of two-photon excited states in a pi -conjugated polymer,” Phys. Rev. Lett. 73(4), 597–600 (1994).
[Crossref] [PubMed]

H. Michinel, M. J. Paz-Alonso, and V. M. Pérez-García, “Turning light into a liquid via atomic coherence,” Phys. Rev. Lett. 96(2), 023903 (2006).
[Crossref] [PubMed]

Á. Paredes, D. Feijoo, and H. Michinel, “Coherent cavitation in the liquid of light,” Phys. Rev. Lett. 112(17), 173901 (2014).
[Crossref] [PubMed]

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006).
[Crossref] [PubMed]

Phys. Today (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

Other (3)

R. W. Boyd, Nonlinear Optics (Academic Press, 2008), 3rd ed.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).

Y. Zhang and M. Xiao, Multi-Wave Mixing Processes: From Ultrafast Polarization Beats To Electromagnetically Induced Transparency (H & Springer, 2009).

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Figures (4)

Fig. 1
Fig. 1 (a) The cascade three-level atomic system and the involved light fields used in the experiment. (b) Experimental setup. HR (high reflective mirror), BS (beam splitter 50% ), PMT (photomultiplier tube), CCD (charge coupled device). (c) Spatial beam geometric drawing used in the experiment.
Fig. 2
Fig. 2 (a)The theoretically calculated nonlinear refractive index as a function of Δ 2 and the total pump intensity. The theoretically calculated real (b) and imaginary (c) parts of χ p,F (1) , χ p,F (3) X 2 ( | E 2 | 2 + | E 2 | 2 ) , χ p,F (5) X 2 ( | E 2 | 4 + | E 2 | 4 + | E 2 | 2 | E 2 | 2 ) and their sum versus the total pump intensity.
Fig. 3
Fig. 3 (a) The probe images versus Δ 2 . (b) The probe images with P 2 decreasing from top to bottom. (c) and (d) are similar to (a) and (b), respectively, but show images of E F . (e) The iso-surface plot of the beam with increasing P 2 , and the xtick labels mark the cases shown in (a). (f) The iso-surface of the beam versus Δ 2 . Images from (a1) to (a4) as well as from (c1) to (c4) correspond to no beam blocked, E 2 blocked, E 2 blocked, and E 2 & E 2 blocked, respectively, with the corresponding intensity curves recorded by PMT shown in (a5) and (c5). For (b), (d), and (f), no beam is blocked. For all the cases, Δ 1 =40 GHz .
Fig. 4
Fig. 4 (a) The images of E F versus Δ 2 with (a1) E 2 blocked, (a2) E 2 blocked, (a3) no beam blocked and the powers of pumping fields set as I 2 10 I 2 , (a4) no beam blocked and I 2 20 I 2 , (a5) no beam blocked and I 2 50 I 2 . (b) The 2D plots of E F around two-photon resonant (TPR) point corresponding to (a1)-(a3). For all the cases, Δ 1 =30 GHz .

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

E p z i 2 k p 2 E p =i k p 2 [ χ p (1) + χ p (3) | E | 2 + x p (5) | E | 4 ] E p + m=1,2 η E m ( E m ) E F ,
E F z i 2 k F 2 E F =i k F 2 [ χ F (1) + χ F (3) | E | 2 + χ F (5) | E | 4 ] E F + m=1,2 η E m ( E m ) E p ,
χ p (n) | E | n1 = χ p (n) S p | E p | n1 +2 j=1,2 x p (n) X j ( | E j | n1 + | E j | n1 ) ,
χ F (n) | E | n1 = χ F (n) S F | E F | n1 +2 j=1,2 χ F (n) X j ( | E j | n1 + | E j | n1 ) ,
χ p,F (1) = η K
χ p,F (3) | E | 2 = η K 2 ( | G p,F | 2 Γ 11 + | G 1 | 2 Γ 11 + | G 1 | 2 Γ 11 + | G 2 | 2 d 2 + | G 2 | 2 d 2 ),
χ p,F (5) | E | 4 = η K 3 ( | G p,F | 4 Γ 11 2 + | G 1 | 4 Γ 11 2 + | G 1 | 4 Γ 11 2 + | G 1 | 2 | G 1 | 2 Γ 11 2 + | G 2 | 4 d 2 2 + | G 2 | 4 d 2 2 + | G 2 | 2 | G 2 | 2 d 2 2 ),
E (R,ф)= α h [cos(γф)+isin(γф+Kπ/2)]exp[i(lф+ ф NL )],
E (R,ϕ)=(α+βR)exp( R 2 / R 0 )cos( γ m ϕ)+isin( γ m ϕ+βπ/2)exp[i(lϕ+ϕNL)],

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