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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 11 — May. 24, 2010
  • pp: 10963–10972
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Modulated vortex solitons of four-wave mixing

Yanpeng Zhang, Zhiqiang Nie, Yan Zhao, Changbiao Li, Ruimin Wang, Jinhai Si, and Min Xiao  »View Author Affiliations


Optics Express, Vol. 18, Issue 11, pp. 10963-10972 (2010)
http://dx.doi.org/10.1364/OE.18.010963


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Abstract

We experimentally demonstrate the vortex solitons of four-wave mixing (FWM) in multi-level atomic media created by the interference patterns with superposing three or more waves. The modulation effect of the vortex solitons is induced by the cross-Kerr nonlinear dispersion due to atomic coherence in the multi-level atomic system. These FWM vortex patterns are explained via the three-, four- and five-wave interference topologies.

© 2010 OSA

1. Introduction

With the self-phase modulation, spatial bright soliton in self-focusing medium or dark soliton in self-defocusing medium can be created [1

1. Y. S. Kivshar, and G. P. Agrawal, Optical solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).

]. Focusing effect can also be induced by cross-phase modulation (XPM) in a self-defocusing nonlinear medium [10

10. G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. 64(21), 2487–2490 (1990). [CrossRef] [PubMed]

]. In such case, the spatial soliton can form by balancing the spatial diffraction with the XPM-induced focusing [11

11. D. Bortman-Arbiv, A. D. Wilson-Gordon, and H. Friedmann, “Induced optical spatial solitons,” Phys. Rev. A 58(5), R3403–R3406 (1998). [CrossRef]

]. Moreover, when three or more plane waves overlap in the medium, complete destructive interference patterns can give rise to phase singularities or optical vortices [12

12. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. Malos, and N. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997). [CrossRef]

15

15. W. Jiang, Q. F. Chen, Y. S. Zhang, and G.-C. Guo, “Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A 74(4), 043811 (2006). [CrossRef]

], which are associated with zeros in the modulated light intensity patterns and can be recognized by specific helical wavefronts.

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

16. 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]

,17

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

].

2. Theoretical model and experimental scheme

Two relevant experimental systems are shown in Figs. 1(a)
Fig. 1 Two FWM processes kF1 and kF2 with five beams k1, k1, k2, k2, k3, and k3 in (a) two-level and (b) cascade three-level atomic systems, respectively, dressed by two beams k1 and k2. (c) Spatial beam geometry used in the experiment.
and 1(b). Three energy levels from Na atoms (the atomic vapor is heated with an atomic density of 5.6×1013cm-3 and a refractive index contrast of Δn=n2I=4.85×104 approximately, where n2 is the cross-Kerr nonlinear coefficient and I is the beam intensity) are involved in the experimental schemes. In Fig. 1(b), energy levels |0 (3S1/2), |1 (3P3/2) and |2 (4D3/2,5/2) form a three-level cascade atomic system. When the energy level |2 is not used, the system reduces into a two-level one [Fig. 1(a)]. The laser beams are aligned spatially as shown in Fig. 1(c), with two dressing beams (E1 and E2) and two pump beams (E1 and E2) propagating through the atomic medium in the same direction with small angles (θ=0.3) between them in a square-box pattern. The probe beams (E3 and E3) propagate in the opposite direction with a small angle as shown in Fig. 1(c). Three laser beams (E1, E1, and E3, with Rabi frequencies G1, G1 and G3, connecting transition |0 to |1) have the same frequency ω1 (from the same dye laser with a 10 Hz repetition rate, 5 ns pulse-width and 0.04 cm-1 line-width), and generate an efficient degenerate FWM signal EF1 (kF1=k1k1+k3) [Fig. 1(a)] in the direction shown at the lower right corner of Fig. 1(c). These beams E2, E2, and E3 (with Rabi frequencies G2, G2 and G3, and connecting the same transition |0 to |1 in the two-level system) are from another near-transform-limited dye laser of frequency ω2, and produce a nondegenerate FWM signal EF2 (kF2=k2k2+k3) [Fig. 1(a)]. All laser beams are horizontally polarized. The diameters of the laser beams are about 25μm. When the six laser beams are all on, there also exist other two FWM processes kF3=k1k1+k3 and kF4=k2k2+k3. However, the coexisting EF1 and EF2 are the dominant ones in the experiment due to phase-matching and chosen beam intensities [17

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

,18

18. Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and Spatial Interference between Four-Wave Mixing and Six-Wave Mixing Channels,” Phys. Rev. Lett. 102(1), 013601 (2009). [CrossRef] [PubMed]

]. According to these FWM phase matching conditions, we can obtain the coherence lengths in the two-level system as LF1c=2cπω1/[n1ω1|ω1ω1|θ2] for EF1, LF2c=2cπω1/[n1ω2|Δ1Δ2|θ2]1.8×103m for EF2, LF3c=2cπω2/[n1ω1|Δ2Δ1|θ2]1.8×103m for EF3, LF4c=2cπω2/[n1ω2|ω2ω2|θ2]for EF4, where n1 is the linear refractive index and Δ1 (Δ2=0) is the detuning of the fields E1,3 and E1 (E2 and E2,3) from the atomic transition.

When E2 and E2 are tuned to the |1|2 transition, the system becomes a cascade three-level system [Fig. 1(b)], which generates a two-photon resonant nondegenerate FWM process EF2 [17

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

,18

18. Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and Spatial Interference between Four-Wave Mixing and Six-Wave Mixing Channels,” Phys. Rev. Lett. 102(1), 013601 (2009). [CrossRef] [PubMed]

]. In this system, the coherence lengths are LF1c=2cπω1/[n1ω1|ω1ω1|θ2] for EF1 and LF2c=2cπω1/[n1ω2|ω2ω1|θ2]0.6m for EF2, respectively.

The mathematical description of the two generated (dominant) FWM beams (including the self- and cross-Kerr nonlinearities) can be obtained by numerically solving the following propagation equations in cylindrical coordinate:
EF1zi2kF1(1rEF1r+2EF1r2+1r22EF1φ2)=ikF1n1(n2S1|EF1|2+2n2F1)EF1,
(1a)
EF2zi2kF2(1rEF2r+2EF2r2+1r22EF2φ2)=ikF2n1(n2S2|EF2|2+2n2F2)EF2,
(1b)
where n2F1=i=15Δn2Xi=n2X1|E3|2+n2X2|E1|2+n2X3|E2|2+n2X4|E1|2+n2X5|E2|2 and n2F2=i=610Δn2Xi=n2X6|E3|2+n2X7|E1|2+n2X8|E2|2+n2X9|E1|2+n2X10|E2|2. n2S1S2 are the self-Kerr nonlinear coefficients of EF1,2, and n2X1X10 are the cross-Kerr nonlinear coefficients due to the fields E1,2,3 and E1,2, respectively. The Kerr nonlinear coefficients are defined as n2=Reχ(3)/(ε0cn1). The third-order nonlinear susceptibility is given by χ(3)=N0μ104ρ10(3)/(3ε0G3,F1,F2Gi2). N0 is atomic density. μij is the dipole matrix element between transition |i> to |j>. We can obtain these Kerr nonlinear coefficients of the FWM beams EF1,2 by calculating the density-matrix element ρ10(3) [16

16. 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]

,17

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

]. In addition, the Doppler effect and power broadening effect are considered in calculating these Kerr nonlinear coefficients.

Solving the propagation equations in the cylindrical coordinate, we demonstrate that the modulated vortex solitons with a screw-type dislocation phase can be characterized by two independent integer numbers [1

1. Y. S. Kivshar, and G. P. Agrawal, Optical solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).

,8

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

] (i.e. the topological charge m and the number of intensity peaks N), and parametrized by the rotating angular velocity (i.e., energy flow velocity) w. We can obtain the stationary transverse solution of the modulated vortex soliton as [8

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

,9

9. Y. J. He, H. Z. Wang, and B. A. Malomed, “Fusion of necklace-ring patterns into vortex and fundamental solitons in dissipative media,” Opt. Express 15(26), 17502–17508 (2007). [CrossRef] [PubMed]

] EF1Esech[E(kF1n2S1/n1)1/2(rR0)]cos(Nφ/2)exp(imF1φ+iϕNL) with an initial radius R0. Moreover, we have w1=ϕNL(r,z)/z=2kF1n2I2er2/2/n1, w2=ϕNL(r,z)/n2=2kF1I2er2/2z/n1 and w3=ϕNL(r,z)/I2=2kF1n2er2/2z/n1.

The dressing beams E1,2 are approximately 10 times stronger than the beams E1,2, 102 times stronger than the weak probe beams E3 and E3, and 104 times stronger than the two generated FWM beams EF1,2. The generated weak beam EF1 (or EF2) partly overlaps with the strong beam E1 (or E2), and other stronger beams (E1,2,3, E3) lie around them [Fig. 1(c)]. As a result, the same frequency waves can interfere to construct polarization ellipse, create phase singularity [13

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

,14

14. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express 14(7), 3039–3044 (2006). [CrossRef] [PubMed]

], and induce local changes of the refractive index. The interference induces a vortex pattern with the superposed n2F1=i=15Δn2Xi and n2F2=i=610Δn2Xi (the center of such vortex lies in the minimum of n2F1,2), and the horizontally- and vertically-aligned dressing fields E1 and E2 modulate a circular-type splitting, with three or four parts around the ellipse. Note that E1 (or E2) is the dominant dressing field of EF1 (or EF2). Such two contributions induce the vortices and splittings of EF1 (or EF2), and finally form the modulated vortex solitons in the two- and three-level atomic systems, as shown in Figs. 2
Fig. 2 (a) Images of EF1 versus Δ1 with G2=14.7GHz at 265C. (b) Images of EF1 at Δ1=8GHz with different values of G2 at 265C. Upper and lower panels are for experimental and simulated results, respectively. (c) Images of EF1 at Δ1=8GHz with different temperatures from 200C to 300C. G2=9.5GHz. Top and bottom rows are the cross sections in y and x directions, respectively. The parameters are G1=12.7GHz, G1=1.8GHz, and G3=0.2GHz in the two-level system.
5
Fig. 5 Under different temperatures, (a) the rotating EF2 solitons (w1=1.1 rad/m) with E2,3 and E1,2 on, and (b) stationary nonrotating EF1 solitons (w1=0) with E1,3 and E1,2 on. Lower images are simulation (N=3,mF1,F2=1) results. (c) Optical vortices of EF1 formed by the interferences of three, four, five, and six waves at 265C, respectively. The parameters are G2=19.7GHz, G1=12.7GHz, G1=1.8GHz, G2=1.1GHz, G3=G3=0.2GHz, and Δ1=6GHz in the two-level system.
below.

In the FWM process in the two-level system, the conservation of the topological charges must be fulfilled, so the topological charges of the FWM signals are determined by mF1=m1m1+m3, mF2=m2m2+m3, mF3=m1m1+m3, and mF4=m2m2+m3, where mF1, mF2, mF2, mF2 are the topological charges of the FWM beams EF1, EF2, EF3, EF4, respectively. and m1, m1, m2, m2, m3, m3 are those of the beams E1, E1, E2, E2, E3, E3, respectively. The topological charges of two FWM signals EF1,EF2 in the cascade three-level system obey the same conservation rules.

3. Modulated vortex solitons

Figure 2(a) presents the effects of spatial dispersion on the FWM signal EF1 in the two-level system, which shows the splitting in the self-focusing region (Δ1<0) and formation of vortex solitons in the self-defocusing region (Δ1>0). In the self-focusing side, while the nonlinear refractive index n2 increases from left to right, EF1 beam breaks up from one to three parts via ϕNL(n2X4), with one large and two small pieces. Thus, the EF1 beam propagates with discrete diffraction in the self-focusing side. By contrast, in the Δ1>0 region, the strong dressing fields E1,2 separate the EF1 beam into three spots along a ring (N=3). Then these spots propagate through the induced spiral phase polarization ellipse. Such screw dislocations create a stationary beam structure with a phase singularity. The interference among the four beams (E1,3 and E1,2) induced a modulated vortex pattern with n2F1=i=1,i35Δn2Xi. Finally, the EF1 beam spot decays into a modulated vortex soliton due to the balanced interaction between the spatial diffraction and the cross-Kerr nonlinearity. There are energy exchanges among three the spots, which rotate around the point of phase singularity. However, when n2 is very small with large detuning or Δ1=0, the phase singularity disappears and the three spots fuse together into a stable fundamental spot.

Figure 2(b) shows the modulated vortex solitons of EF1 for different intensities of the dressing field E2 in self-defocusing regime. With increasing E2 intensity, the spiral phase of EF1 changes into jumping phase between two parts, and the modulated vortex soliton of EF1 decays into dipole-mode soliton at high intensity, which is created by the horizontally aligned beam E1. Specifically, EF1 is circularly modulated by the horizontally-aligned E1 and vertically-aligned E2 beams. With E2 getting stronger, EF1 is shifted away from E2 and then splits into two parts by E1. The dominant phase of EF1 is changed gradually from a spiral phase evolution to a jumping phase (i.e., from interference among four beams E1,3, E1,2 to the dressing of E1).

Figure 2(c) shows EF1 soliton cluster with different temperatures between 200C and 300C in the two-level system. EF1 beam is a single spot at both low and high temperature sides. The single spot breaks up into several fragments (soliton cluster) as the temperature increases from 200C to 240C, the nonlinear phase ϕNL gets larger as the temperature (equivalent to propagation distance z) rises, which leads to several splitting parts with weak absorption. As the temperature gets higher with an increased absorption, the beam intensity decreases. ϕNL (proportional to both beam intensity and propagation distance z) reaches its optimal value at 250C. Moreover, the soliton cluster of EF1 results from two contributions in the two-level system: (i) the interference among the four waves (E1,3, E1,2) with the same frequency induces an interference pattern with n2F1=i=1,i35Δn2Xi, and (ii) E1,2 induce a beam splitting via ϕNL(n2X4) and ϕNL(n2X5). As temperature gets even higher, the dressing beams are significantly absorbed by the hot atoms, so their intensities are reduced and the cross Kerr nonlinear effects are gradually weakened too. Under such condition, the spots merge into a single spot due to strong absorption. So the ideal temperature for the modulated vortex soliton is around 265C for the given experimental conditions (i.e., the modulated vortex soliton can be obtained at a certain propagation distance).

In the cascade three-level system with five laser beams (E1,2,3, E1,2) on, the interference (n2F2=i=6,i89Δn2Xi) among three beams E1, E1, and E3 induces a rotating vortex. With the incident beams having topological charges m3=1 and m1=m1=m2=m2=0, the topological charges of the generated FWM signals EF1, EF2 are mF1=m1m1+m3=1 and mF2=m2m2+m3=1, respectively. Figures 3(a)
Fig. 3 The rotating EF2 solitons with (a) Ω2>0 and (b) Ω2<0 versus Δ1. (c) Stationary nonrotating EF1 solitons (w2=0) versus Δ1. Lower images are the simulated results (N=3, mF1,F2=1). The parameters are G2=19.7GHz, G1=12.7GHz, G1=1.8GHz, G2=1.1GHz, and G3=0.2GHz in the cascade three-level system at 265C.
and 3(b) show the rotating vortices of the FWM beams with three spots (N=3) for different frequency detunings. Here, the ellipse orientation α approaches to zero and TX/TY1.1. From Icos2(Nφ/2)cos2(mφ+Ω2|n2|) with Ω2=sgn[n2]w2 (I2=51W/cm2, r=0.25mm), EF2 circumvolves anticlockwise with n2>0 and Ω2=1.63×103W/cm2>0 in the self-focusing regime [Fig. 3(a)], while moves clockwise with n2<0 and Ω2=1.63×103W/cm2<0 in the self-defocusing regime [Fig. 3(b)].

Figure 3(c) presents the stationary solitons with w2=0 in the cascade three-level system. n2X4 can be a positive value with resonant dressing of E1. When n2X1 and n2X2 have negative values under the self-defoucsing condition, the superposed n2F1=i=1,i34Δn2Xi is close to zero (or ϕNL0). Under this condition, a uniform energy flow exists along the ring, and nonrotating (w2=0) spatially-localized multihump structures can be obtained.

Comparing to the three-level system, there exist five nearly degenerate frequency waves (E1,2,3, E1,2) in the two-level system. Figure 4(a)
Fig. 4 (a) Anticlockwise rotating (Ω2=1.53×103W/cm2) EF2 solitons and (b) stationary nonrotating EF1 solitons with four spots versus Δ1 at 265C in the two-level system. Lower images are simulated (N=4,mF1,F2=1) results. The parameters are G2=15GHz, G1=12.7GHz, G1=1.8GHz, G2=1.1GHz, and G3=0.2GHz.
shows the rotating vortices of the FWM beam with four spots (N=4) for different frequency detunings. With the probe beam E3 carrying topological charge m3=1, the topological charges of the generated FWM signals EF1, EF2 are mF1=m1m1+m3=1 and mF2=m2m2+m3=1, respectively, where m1, m1, m2, and m2 are all zeros. The modulated vortex pattern (N=4) of EF2 is induced by the interference of five waves, and the nonresonant dressing field E2 induces a splitting via ϕNL(n2X10). For EF2, E2 is the nonresonant dressing field, energy flow exists along the ring of spots unequally, inducing a modulated vortex [Fig. 4(a)].

In Fig. 4(b), there exists a stationary, four-spot modulated vortex soliton with N=4 and non-uniform energy distribution for different detunings in the two-level system. The vortex pattern (horizontally-oriented polarization ellipse with α=0 and TX/TY1.5) is induced by the interference of the five beams. With the nonresonant dressing of E2, all terms in n2F2 have the same negative sign [Fig. 4(a)], but the positive n2X4 in n2F1=i=15Δn2Xi has opposite sign with the other terms n2X1,2,3,5 in n2F1 due to the resonant dressing of E1, so one can get n2F10. Therefore, there exist four spots in the stationary (ϕNL0) modulated vortex soliton of EF1 with N=4 and energy is mainly stored in one diagonal pair of spots [Fig. 4(b)], which results from the resonant dressing of E1.

The radially symmetric vortex solitons (mF1,F2=1) in a self-defocusing medium are depicted in Figs. 5(a) and 5(b), which separately demonstrate vortices and steady crescent FWM vortex solitons under different temperatures (atomic densities). The effective propagation distance z increases with the temperature. According to the solution of Eq. (1) EF2Esech[E(kF2n2S2/n1)1/2(rR0)]cos(Nφ/2)exp(imF2φ+iw1z), the spots rotate with an angular velocity w1. In the two-level system with n2F2=i=6,i710Δn2Xi, the vortex pattern of EF2 is induced by the interference of E2,3, E1,2 while the dressing fields E1,2 generate the circular splittings. Here, EF2 forms a crescent FWM modulated vortex soliton with an anticlockwise rotation [Fig. 5(a)]. Moreover, w1z changes 300° from 245C to 275C. When setting z=1 at 245C, we obtain w1=1.1 rad/m, which is close to the theoretical value of w1=1.03 rad/m. However, in Fig. 5(b) with the resonant dressing E1 and n2F1=i=1,i35Δn2Xi0, the propagation length of EF1 is about 27 times longer than the diffraction length(LD0.67cm), which are both much shorter than the coherence length of EF1 described above (LF1c). So we can conclude that EF1 beam becomes a stationary vortex soliton at certain I1 and z values.

Last, we let all six beams on, and set the E1,2,3 and E3 beams just 10 times weaker than the dressing beams E1,2. Figure 5(c) shows the optical vortices created by the interferences of three, four, five, six waves (and the dressing fields) in the two-level system, respectively. Initially, there are three beams E1,3, E1 on, which create the image 3 in Fig. 5(c) As fourth interference beam E2 is added, the split spots change from two to three [image 4 in Fig. 5(c)]. Similarly, as beams E2 and E3 are added gradually, the interference beams increase from four to five (image 5), and then to six (image 6), the split spots in the vortex patterns of EF1 then change from three to four, and then to six, respectively, along the ring, and the shape of the vortex ellipse tends to become more circular. The final superposition nonlinear index is n2F1=i=15Δn2Xi+n2X11|E3|2, where n2X11 is the nonlinear index induced by E3. The FWM modulated vortex solitons are created jointly by the effects of the complex patterns induced by the multiple interference waves [13

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

,14

14. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express 14(7), 3039–3044 (2006). [CrossRef] [PubMed]

] and the cross-Kerr nonlinear dispersions induced by the dressing field [11

11. D. Bortman-Arbiv, A. D. Wilson-Gordon, and H. Friedmann, “Induced optical spatial solitons,” Phys. Rev. A 58(5), R3403–R3406 (1998). [CrossRef]

].

4. Conclusion

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

Acknowledgments

This work was supported by NSFC (No. 10974151, No. 60678005), FANEDD (No. 200339), RFDP (No. 20050698017), FYTEFYTIHEC (No. 101061), NCET (No. 08-0431), 2009xjtujc08.

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8.

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

9.

Y. J. He, H. Z. Wang, and B. A. Malomed, “Fusion of necklace-ring patterns into vortex and fundamental solitons in dissipative media,” Opt. Express 15(26), 17502–17508 (2007). [CrossRef] [PubMed]

10.

G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. 64(21), 2487–2490 (1990). [CrossRef] [PubMed]

11.

D. Bortman-Arbiv, A. D. Wilson-Gordon, and H. Friedmann, “Induced optical spatial solitons,” Phys. Rev. A 58(5), R3403–R3406 (1998). [CrossRef]

12.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. Malos, and N. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997). [CrossRef]

13.

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

14.

K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express 14(7), 3039–3044 (2006). [CrossRef] [PubMed]

15.

W. Jiang, Q. F. Chen, Y. S. Zhang, and G.-C. Guo, “Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A 74(4), 043811 (2006). [CrossRef]

16.

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]

17.

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

18.

Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and Spatial Interference between Four-Wave Mixing and Six-Wave Mixing Channels,” Phys. Rev. Lett. 102(1), 013601 (2009). [CrossRef] [PubMed]

OCIS Codes
(190.3270) Nonlinear optics : Kerr effect
(190.4180) Nonlinear optics : Multiphoton processes
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(270.1670) Quantum optics : Coherent optical effects
(080.4865) Geometric optics : Optical vortices
(190.6135) Nonlinear optics : Spatial solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: February 25, 2010
Revised Manuscript: April 9, 2010
Manuscript Accepted: April 13, 2010
Published: May 10, 2010

Citation
Yanpeng Zhang, Zhiqiang Nie, Yan Zhao, Changbiao Li, Ruimin Wang, Jinhai Si, and Min Xiao, "Modulated vortex solitons of four-wave mixing," Opt. Express 18, 10963-10972 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-10963


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References

  1. Y. S. Kivshar, and G. P. Agrawal, Optical solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).
  2. G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69(17), 2503–2506 (1992). [CrossRef] [PubMed]
  3. B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose-Einstein condensate,” Phys. Rev. Lett. 86(14), 2926–2929 (2001). [CrossRef] [PubMed]
  4. M. J. Holland and J. E. Williams, “Preparing topological states of a Bose-Einstein condensate,” Nature 401(6753), 568–572 (1999). [CrossRef]
  5. M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose-Einstein Condensate,” Phys. Rev. Lett. 83(13), 2498–2501 (1999). [CrossRef]
  6. A. V. Gorbach, D. V. Skryabin, and C. N. Harvey, “Vortex solitons in an off-resonant Raman medium,” Phys. Rev. A 77(6), 063810 (2008). [CrossRef]
  7. A. V. Gorbach and D. V. Skryabin, “Cascaded generation of multiply charged optical vortices and spatiotemporal helical beams in a Raman medium,” Phys. Rev. Lett. 98(24), 243601 (2007). [CrossRef] [PubMed]
  8. A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005). [CrossRef] [PubMed]
  9. Y. J. He, H. Z. Wang, and B. A. Malomed, “Fusion of necklace-ring patterns into vortex and fundamental solitons in dissipative media,” Opt. Express 15(26), 17502–17508 (2007). [CrossRef] [PubMed]
  10. G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. 64(21), 2487–2490 (1990). [CrossRef] [PubMed]
  11. D. Bortman-Arbiv, A. D. Wilson-Gordon, and H. Friedmann, “Induced optical spatial solitons,” Phys. Rev. A 58(5), R3403–R3406 (1998). [CrossRef]
  12. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. Malos, and N. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997). [CrossRef]
  13. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004). [CrossRef] [PubMed]
  14. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express 14(7), 3039–3044 (2006). [CrossRef] [PubMed]
  15. W. Jiang, Q. F. Chen, Y. S. Zhang, and G.-C. Guo, “Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A 74(4), 043811 (2006). [CrossRef]
  16. 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]
  17. Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A 80(5), 055804 (2009). [CrossRef]
  18. Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and Spatial Interference between Four-Wave Mixing and Six-Wave Mixing Channels,” Phys. Rev. Lett. 102(1), 013601 (2009). [CrossRef] [PubMed]

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