Modulated vortex solitons of four-wave mixing

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-, fourand five-wave interference topologies. ©2010 Optical Society of America OCIS codes: (190.6135) Spatial solitons; (080.4865) Optical vortices; (190.4380) Nonlinear optics, four-wave mixing; (190.3270) Kerr effect; (190.4180) Multiphoton processes; (270.1670) Coherent optical effects. References and links 1. Y. S. Kivshar, and G. P. Agrawal, Optical solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003). 2. G. A. Swartzlander, Jr., and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69(17), 2503–2506 (1992). 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). 4. M. J. Holland, and J. E. Williams, “Preparing topological states of a Bose-Einstein condensate,” Nature 401(6753), 568–572 (1999). 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). 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). 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). 8. A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005). 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). 10. G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. 64(21), 2487–2490 (1990). 11. D. Bortman-Arbiv, A. D. Wilson-Gordon, and H. Friedmann, “Induced optical spatial solitons,” Phys. Rev. A 58(5), R3403–R3406 (1998). 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). 13. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004). 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). 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). 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). 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). 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). #124748 $15.00 USD Received 25 Feb 2010; revised 9 Apr 2010; accepted 13 Apr 2010; published 10 May 2010 (C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 10963


Introduction
Vortices play important roles in many branches of physics [1].The first experimental observation of optical vortex soliton was reported in a self-defocusing medium where the field propagates as a soliton, owing to the counterbalanced effects of diffraction and nonlinear refraction at the phase singularity [2].Such singularity corresponding to vortices can exist in the Bose-Einstein condensates which links the physics of superfluidity, phase transitions, and singularities in nonlinear optics [3][4][5].The topological states of a Bose-Einstein condensate can be prepared experimentally [4].Moreover, several interesting effects including cascade generation of multiple charged optical vortices and helically shaped spatiotemporal solitons in Raman FWM, and coupled vortex solitons supported by cascade FWM in a Raman active medium excited away from the resonance have been investigated [6,7].Spatially modulated vortex solitons (azimuthons) have been theoretically considered in self-focusing nonlinear media [8].Transverse energy flow occurs between the intensity peaks (solitons) associated with the phase structure, which is a staircase-like nonlinear function of the polar angle ϕ .The necklace-ring solitons can merge into vortex and fundamental solitons in dissipative media [9].
With the self-phase modulation, spatial bright soliton in self-focusing medium or dark soliton in self-defocusing medium can be created [1].Focusing effect can also be induced by cross-phase modulation (XPM) in a self-defocusing nonlinear medium [10].In such case, the spatial soliton can form by balancing the spatial diffraction with the XPM-induced focusing [11].Moreover, when three or more plane waves overlap in the medium, complete destructive interference patterns can give rise to phase singularities or optical vortices [12][13][14][15], which are associated with zeros in the modulated light intensity patterns and can be recognized by specific helical wavefronts.
In this letter, we experimentally demonstrate the formations of modulated vortex solitons in two generated four-wave mixing (FWM) waves in a two-level, as well as a cascade three-level, atomic systems.These vortex solitons are created by the interference patterns by superposing three or more waves, and by the greatly enhanced cross-Kerr nonlinear dispersion due to atomic coherence [16,17].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 ( 1 ′ E and 2 ′ E ) and two pump beams ( 1 E and 2 E ) propagating through the atomic medium in the same direction with small angles ( 0.3 θ = ) between them in a square-box pattern.The probe beams ( 3 E and 3 ′ E ) propagate in the opposite direction with a small angle as shown in Fig. 1(c).Three laser beams ( 1 E , 1 ′ E , and 3 E , with Rabi frequencies 1 G , 1 G′ and 3 G , connecting transition 0 to 1 ) have the same frequency 1 ω (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  E are the dominant ones in the experiment due to phase-matching and chosen beam intensities [17,18].According to these FWM phase matching conditions, we can obtain the coherence lengths in the two-level system as for F2 E , 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: where  .The third-order nonlinear susceptibility is given by We can obtain these Kerr nonlinear coefficients of the FWM beams F1,2

E
by calculating the density-matrix element (3)   10 ρ [16,17].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 [1,8] (i.e. the topological charge m and the number of intensity peaks N), and parametrized by the rotating angular velocity (i.e., energy flow velocity) w .We can obtain the stationary transverse solution of the modulated vortex soliton as [8,9] with an initial radius 0 R .Moreover, we have ′ E ) in the medium, as shown in Fig. 1(c).The destructive interference of two waves with similar intensity can result in spatial patterns with zero intensities, which create phase singularities or optical vortices [14].When multi-beam interference occurs, spatial polygon patterns (i.e, closed triangle from three beams, quadrangle from four beams, which gives one vortex point [13,14].)can be formed, with the side lengths being the complex amplitude vectors of the waves.The polygons with more beams will look like a circular shape, and the phase complexity will be enhanced.The complex amplitude vectors can be overlaid at the observation plane and give rise to the total complex amplitude vector ( X C , Y C ) of the interfering plane waves [13,14].The local structures of the optical vortices are given by the polarization ellipse relation and α is the ellipse orientation.The ellipse axes X T , Y T are related to the spatial configuration (including the incident beam directions, phase differences between beams etc.) and beam intensities.
The dressing beams times stronger than the weak probe beams 3 E and 3 ′ E , and 4  10 times stronger than the two generated FWM beams 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    E in the two-level system, which shows the splitting in the self-focusing region ( 1 0 ∆ < ) and formation of vortex solitons in the self-defocusing region ( 1 0 ∆ > ).In the self-focusing side, while the nonlinear refractive index 2 n increases from left to right, F1 E beam breaks up from one to three parts via .Finally, the F1 E 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 2 n is very small with large detuning or 1 0 ∆ = , the phase singularity disappears and the three spots fuse together into a stable fundamental spot.

Modulated vortex solitons
Figure 2 ′ E to the dressing of 1 ′ E ). Figure 2(c) shows F1 E soliton cluster with different temperatures between 200 C and 300 C in the two-level system.F1 E beam is a single spot at both low and high temperature sides.The single spot breaks up into several fragments (soliton cluster) as the temperature increases from 200 C to 240 C , the nonlinear phase NL φ 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.NL φ (proportional to both beam intensity and propagation distance z) reaches its optimal value at 250 C .Moreover, the soliton cluster of F1 E results from two contributions in the two-level system: (i) the interference among the four waves ( .As temperature gets even higher, the dressing beams are significantly absorbed by the hot atoms, so their intensities are reduced and the cross Kerr nonlinear effects are gradually weakened too.Under such condition, the spots merge into a single spot due to strong absorption.So the ideal temperature for the modulated vortex soliton is around 265 C for the given experimental conditions (i.e., the modulated vortex soliton can be obtained at a certain propagation distance).
In the cascade three-level system with five laser beams ( Under this condition, a uniform energy exists along the ring, and nonrotating ( 2 0 w = ) spatially-localized multihump structures can be obtained.
Comparing to the three-level system, there exist five nearly degenerate frequency waves (   E forms a crescent FWM modulated vortex soliton with an anticlockwise rotation [Fig.5(a  in the two-level system. Last, we let all six beams on, and set the ′ E .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 1,3 E , 1 ′ E on, which create the image 3 in Fig. 5(c) As fourth interference beam 2 ′ E is added, the split spots change from two to three [image 4 in Fig. 5(c)].Similarly, as beams 2 and 3 ′ E 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 F1 E 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 is the nonlinear index induced by 3 ′ E .The FWM modulated vortex solitons are created jointly by the effects of the complex patterns induced by the multiple interference waves [13,14] and the cross-Kerr nonlinear dispersions induced by the dressing field [11].

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

F1, 2 E 1 ′E 3 E 2 n 1 ′
. The generated weak beam F1 E (or F2 E ) partly overlaps with the strong beam (or 2 ′ E ), and other stronger beams ( 1,2,, 3 ′ E ) lie around them [Fig.1(c)].As a result, the same frequency waves can interfere to construct polarization ellipse, create phase singularity[13,14], and induce local changes of the refractive index.The interference induces a vortex pattern with the superposed of such vortex lies in the minimum of F1,2 ), and the horizontally-and vertically-aligned dressing fields 1 ′ E and 2 ′ E modulate a circular-type splitting, with three or four parts around the ellipse.Note that E (or 2 ′ E ) is the dominant dressing field of F1 E (or F2 E ).Such two contributions induce the vortices and splittings of F1 E (or F2 E ), and finally form the modulated vortex solitons in the two-and three-level atomic systems, as shown in Figs.2-5 below. and

Figure 2 (
Figure 2(a) presents the effects of spatial dispersion on the FWM signal F1E in the two-level system, which shows the splitting in the self-focusing region ( 1 0 ∆ < ) and formation of vortex solitons in the self-defocusing region ( 1 0 ∆ > ).In the self-focusing side, while the nonlinear refractive index 2 n increases from left to right, F1 E beam breaks up from one to three parts via

2 ′E(
large and two small pieces.Thus, the F1 E beam propagates with discrete diffraction in the self-focusing side.By contrast, separate the F1 E beam into three spots along a ring ( 3 N = ).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 #124748 -$15.00USD Received 25 Feb 2010; revised 9 Apr 2010; accepted 13 Apr 2010; published 10 May 2010

and 3 E 1 m(<
induces a rotating vortex.With the incident beams having topological charges 3 Figures 3(a) and 3(b) show the rotating vortices of the FWM beams with three spots ( 3 N = ) for different frequency detunings.Here, the ellipse orientation α approaches to zero and E circumvolves anticlockwise with #124748 -$15.00USD Received 25 Feb 2010; revised 9 Apr 2010; accepted 13 Apr 2010; published 10 May 2010 in the self-defocusing regime [Fig.3(b)].

Figure 3 ( 2 Xn 1 ′
Figure 3(c) presents the stationary solitons with 2 0 w = in the cascade three-level system.4 2 X n can be a positive value with resonant dressing of 1 ′ E .When 1 2 X n and 2 2 X n have negative

Figure 4 (E carrying topological charge 3 1mF1 2 mE , 2 ′
a) shows the rotating vortices of the FWM beam with four spots ( 4 N = ) for different frequency detunings.With the probe beam 3 = , the topological charges of the generated FWM signals ′ are all zeros.The modulated vortex pattern ( 4 N = ) of F2 E is induced by the interference #124748 -$15.00USD Received 25 Feb 2010; revised 9 Apr 2010; accepted 13 Apr 2010; published 10 May 2010 (C) 2010 OSA of five waves, and the nonresonant dressing field 2 ′ E induces E is the nonresonant dressing field, energy flow exists along the ring of spots unequally, inducing a modulated vortex [Fig.4(a)].

#
124748 -$15.00USD Received 25 Feb 2010; revised 9 Apr 2010; accepted 13 Apr 2010; published 10 May 2010 (C) 2010 OSA the formation and control of complex solitons can lead to potential applications in soliton communications and computations.