Modulated vortex solitons of four-wave mixing

doi:10.1364/OE.18.010963

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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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▸ Article Outline
– Abstract
1. Introduction
2. Theoretical model and experimental scheme
3. Modulated vortex solitons
4. Conclusion
– References and links

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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.

1. Introduction

Vortices play important roles in many branches of physics [1

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

]. 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

G. A. Swartzlander Jr and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69(17), 2503–2506 (1992). [CrossRef] [PubMed]

]. 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

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]

–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]

]. The topological states of a Bose–Einstein condensate can be prepared experimentally [4

M. J. Holland and J. E. Williams, “Preparing topological states of a Bose-Einstein condensate,” Nature 401(6753), 568–572 (1999). [CrossRef]

]. 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

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]

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]

]. Spatially modulated vortex solitons (azimuthons) have been theoretically considered in self-focusing nonlinear media [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]

]. 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

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]

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

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

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

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

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

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]

2. Theoretical model and experimental scheme

Two relevant experimental systems are shown in Figs. 1(a) and 1(b). Three energy levels from Na atoms (the atomic vapor is heated with an atomic density of

5.6 \times 10^{13} c m^{- 3}

and a refractive index contrast of

Δ n = n_{2} I = 4.85 \times 10^{- 4}

approximately, where

n_{2}

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 〉

(

3 S_{1 / 2}

{| 1 〉}^{}

(

3 P_{3 / 2}

) and

| 2 〉

(

4 D_{3 / 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 (

E_{1}^{'}

and

E_{2}^{'}

) and two pump beams (

E_{1}

and

E_{2}

) propagating through the atomic medium in the same direction with small angles (

θ = {0.3}^{\circ}

) between them in a square-box pattern. The probe beams (

E_{3}

and

E_{3}^{'}

) propagate in the opposite direction with a small angle as shown in Fig. 1(c). Three laser beams (

E_{1}

E_{1}^{'}

, and

E_{3}

, with Rabi frequencies

G_{1}

G_{1}^{'}

and

G_{3}

, connecting transition

| 0 〉

| 1 〉

) have the same frequency

ω_{1}

(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_{F 1}^{}

(

k_{F 1} = k_{1} - k_{1}^{'} + k_{3}

) [Fig. 1(a)] in the direction shown at the lower right corner of Fig. 1(c). These beams

E_{2}

E_{2}^{'}

, and

E_{3}^{'}

(with Rabi frequencies

G_{2}

G_{2}^{'}

and

G_{3}^{'}

, and connecting the same transition

{| 0 〉}^{}

| 1 〉

in the two-level system) are from another near-transform-limited dye laser of frequency

ω_{2}

, and produce a nondegenerate FWM signal

E_{F 2}

(

k_{F 2} = k_{2} - k_{2}^{'} + k_{3}

) [Fig. 1(a)]. All laser beams are horizontally polarized. The diameters of the laser beams are about 25

μ m

. When the six laser beams are all on, there also exist other two FWM processes

k_{F 3} = k_{1} - k_{1}^{'} + k_{3}^{'}

and

k_{F 4} = k_{2} - k_{2}^{'} + k_{3}^{'}

. However, the coexisting

E_{F 1}

and

E_{F 2}

are the dominant ones in the experiment due to phase-matching and chosen beam intensities [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]

]. According to these FWM phase matching conditions, we can obtain the coherence lengths in the two-level system as

L_{F 1}^{c} = 2 c π ω_{1} / [n_{1} ω_{1} | ω_{1} - ω_{1} | θ^{2}] \to \infty

for

E_{F 1}

L_{F 2}^{c} = 2 c π ω_{1} / [n_{1} ω_{2} | Δ_{1} - Δ_{2} | θ^{2}] \approx 1.8 \times 10^{3} m

for

E_{F 2}

L_{F 3}^{c} = 2 c π ω_{2} / [n_{1} ω_{1} | Δ_{2} - Δ_{1} | θ^{2}] \approx 1.8 \times 10^{3} m

for

E_{F 3}

L_{F 4}^{c} = 2 c π ω_{2} / [n_{1} ω_{2} | ω_{2} - ω_{2} | θ^{2}] \to \infty

for

E_{F 4}

, where

n_{1}

is the linear refractive index and

Δ_{1}

(

Δ_{2} = 0

) is the detuning of the fields

E_{1, 3}

and

E_{1}^{'}

(

E_{2}

and

E_{2, 3}^{'}

) from the atomic transition.

Fig. 1 Two FWM processes

k_{F 1}

and

k_{F 2}

with five beams

k_{1}

k_{1}^{'}

k_{2}^{}

k_{2}^{'}

k_{3}^{}

, and

k_{3}^{'}

in (a) two-level and (b) cascade three-level atomic systems, respectively, dressed by two beams

k_{1}^{'}

and

k_{2}^{'}

. (c) Spatial beam geometry used in the experiment.

When

E_{2}

and

E_{2}^{'}

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

E_{F 2}

[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

]. In this system, the coherence lengths are

L_{F 1}^{c} = 2 c π ω_{1} / [n_{1} ω_{1} | ω_{1} - ω_{1} | θ^{2}] \to \infty

for

E_{F 1}

and

L_{F 2}^{c} = 2 c π ω_{1} / [n_{1} ω_{2} | ω_{2} - ω_{1} | θ^{2}] \approx 0.6 m

for

E_{F 2}

, 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:

\frac{\partial E_{F 1}}{\partial z} - \frac{i}{2 k_{F 1}} (\frac{1}{r} \frac{\partial E_{F 1}}{\partial r} + \frac{\partial^{2} E_{F 1}}{\partial r^{2}} + \frac{1}{r^{2}} \frac{\partial^{2} E_{F 1}}{\partial φ^{2}}) = \frac{i k_{F 1}}{n_{1}} (n_{2}^{S 1} {| E_{F 1} |}^{2} + 2 n_{2}^{F 1}) E_{F 1},

(1a)

\frac{\partial E_{F 2}}{\partial z} - \frac{i}{2 k_{F 2}} (\frac{1}{r} \frac{\partial E_{F 2}}{\partial r} + \frac{\partial^{2} E_{F 2}}{\partial r^{2}} + \frac{1}{r^{2}} \frac{\partial^{2} E_{F 2}}{\partial φ^{2}}) = \frac{i k_{F 2}}{n_{1}} (n_{2}^{S 2} {| E_{F 2} |}^{2} + 2 n_{2}^{F 2}) E_{F 2},

(1b)

where

n_{2}^{F 1} = \sum_{i = 1}^{5} Δ n_{2}^{X i} = n_{2}^{X 1} {| E_{3} |}^{2} + n_{2}^{X 2} {| E_{1} |}^{2} + n_{2}^{X 3} {| E_{2} |}^{2} + n_{2}^{X 4} {| E_{1}^{'} |}^{2} + n_{2}^{X 5} {| E_{2}^{'} |}^{2}

and

n_{2}^{F 2} = \sum_{i = 6}^{10} Δ n_{2}^{X i} = n_{2}^{X 6} {| E_{3} |}^{2} + n_{2}^{X 7} {| E_{1} |}^{2} + n_{2}^{X 8} {| E_{2} |}^{2} + n_{2}^{X 9} {| E_{1}^{'} |}^{2} + n_{2}^{X 10} {| E_{2}^{'} |}^{2}

n_{2}^{S 1 - S 2}

are the self-Kerr nonlinear coefficients of

E_{F 1, 2}

, and

n_{2}^{X 1 - X 10}

are the cross-Kerr nonlinear coefficients due to the fields

E_{1, 2, 3}

and

E_{1, 2}^{'}

, respectively. The Kerr nonlinear coefficients are defined as

n_{2} = Re χ^{(3)} / (ε_{0} c n_{1})

. The third-order nonlinear susceptibility is given by

χ^{(3)} = N_{0} μ_{10}^{4} ρ_{10}^{(3)} / (ℏ^{3} ε_{0} G_{3, F 1, F 2} G_{i}^{2})

N_{0}

is atomic density.

μ_{i j}

is the dipole matrix element between transition

| i >

| j >

. We can obtain these Kerr nonlinear coefficients of the FWM beams

E_{F 1, 2}

by calculating the density-matrix element

ρ_{10}^{(3)}

[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]

]. 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

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

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

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

]

E_{F 1} \propto E \sec h [E {(k_{F 1} n_{2}^{S 1} / n_{1})}^{1 / 2} (r - R_{0})] \cos (N φ / 2) \exp (i m_{F 1} φ + i ϕ_{N L})

with an initial radius

R_{0}

. Moreover, we have

w_{1} = ϕ_{N L} (r, z) / z = 2 k_{F 1} n_{2} I_{2} e^{- r^{2} / 2} / n_{1}

w_{2} = ϕ_{N L} (r, z) / n_{2} = 2 k_{F 1} I_{2} e^{- r^{2} / 2} z / n_{1}

and

w_{3} = ϕ_{N L} (r, z) / I_{2} = 2 k_{F 1} n_{2} e^{- r^{2} / 2} z / n_{1}

The spatial interference patterns are formed by superposing three or more waves (

E_{1, 2, 3}

and

E_{1, 2, 3}^{'}

) 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

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]

]. 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

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

].) 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 (

C_{X}

C_{Y}

) of the interfering plane waves [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

]. The local structures of the optical vortices are given by the polarization ellipse relation

C_{X}^{2} / (T_{X}^{2} + T_{Y}^{2}) \sin^{2} (β + α) + C_{Y}^{2} / (T_{X}^{2} + T_{Y}^{2}) \cos^{2} (β + α) = 1

, where

β = \arctan (T_{X} / T_{Y})

and α is the ellipse orientation. The ellipse axes

T_{X}

T_{Y}

are related to the spatial configuration (including the incident beam directions, phase differences between beams etc.) and beam intensities.

The dressing beams

E_{1, 2}^{'}

are approximately 10 times stronger than the beams

E_{1, 2}

10^{2}

times stronger than the weak probe beams

E_{3}

and

E_{3}^{'}

, and

10^{4}

times stronger than the two generated FWM beams

E_{F 1, 2}

. The generated weak beam

E_{F 1}

(or

E_{F 2}

) partly overlaps with the strong beam

E_{1}^{'}

(or

E_{2}^{'}

), and other stronger beams (

E_{1, 2, 3}

E_{3}^{'}

) lie around them [Fig. 1(c)]. As a result, the same frequency waves can interfere to construct polarization ellipse, create phase singularity [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

], and induce local changes of the refractive index. The interference induces a vortex pattern with the superposed

n_{2}^{F 1} = \sum_{i = 1}^{5} Δ n_{2}^{X i}

and

n_{2}^{F 2} = \sum_{i = 6}^{10} Δ n_{2}^{X i}

(the center of such vortex lies in the minimum of

n_{2}^{F 1, 2}

), and the horizontally- and vertically-aligned dressing fields

E_{1}^{'}

and

E_{2}^{'}

modulate a circular-type splitting, with three or four parts around the ellipse. Note that

E_{1}^{'}

(or

E_{2}^{'}

) is the dominant dressing field of

E_{F 1}

(or

E_{F 2}

). Such two contributions induce the vortices and splittings of

E_{F 1}

(or

E_{F 2}

), and finally form the modulated vortex solitons in the two- and three-level atomic systems, as shown in Figs. 2 –5 below.

Fig. 2 (a) Images of

E_{F 1}

versus

Δ_{1}

with

G_{2}^{'} = 14.7 GHz

at 265

^{\circ} C

. (b) Images of

E_{F 1}

Δ_{1} = 8 GHz

with different values of

G_{2}^{'}

at 265

^{\circ} C

. Upper and lower panels are for experimental and simulated results, respectively. (c) Images of

E_{F 1}

Δ_{1} = 8 GHz

with different temperatures from 200

^{\circ} C

to 300

^{\circ} C

G_{2}^{'} = 9.5 GHz

. Top and bottom rows are the cross sections in y and x directions, respectively. The parameters are

G_{1}^{'} = 12.7 GHz

G_{1} = 1.8 GHz

, and

G_{3} = 0.2 GHz

in the two-level system.

Fig. 5 Under different temperatures, (a) the rotating

E_{F 2}

solitons (

w_{1} = - 1.1

rad/m) with

E_{2, 3}

and

E_{1, 2}^{'}

on, and (b) stationary nonrotating

E_{F 1}

solitons (

w_{1} = 0

) with

E_{1, 3}

and

E_{1, 2}^{'}

on. Lower images are simulation (

N = 3

m_{F 1, F 2} = 1

) results. (c) Optical vortices of

E_{F 1}

formed by the interferences of three, four, five, and six waves at

265^{\circ} C

, respectively. The parameters are

G_{2}^{'} = 19.7 GHz

G_{1}^{'} = 12.7 GHz

G_{1} = 1.8 GHz

G_{2} = 1.1 GHz

G_{3} = G_{3}^{'} = 0.2 GHz

, and

Δ_{1} = 6 GHz

in the two-level system.

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

m_{F 1} = m_{1} - m_{1}^{'} + m_{3}

m_{F 2} = m_{2} - m_{2}^{'} + m_{3}

m_{F 3} = m_{1} - m_{1}^{'} + m_{3}^{'}

, and

m_{F 4} = m_{2} - m_{2}^{'} + m_{3}^{'}

, where

m_{F 1}

m_{F 2}

m_{F 2}

m_{F 2}

are the topological charges of the FWM beams

E_{F 1}

E_{F 2}

E_{F 3}

E_{F 4}

, respectively. and

m_{1}

m_{1}^{'}

m_{2}

m_{2}^{'}

m_{3}

m_{3}^{'}

are those of the beams

E_{1}

E_{1}^{'}

E_{2}

E_{2}^{'}

E_{3}

E_{3}^{'}

, respectively. The topological charges of two FWM signals

E_{F 1}

E_{F 2}

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

E_{F1}

in the two-level system, which shows the splitting in the self-focusing region (

Δ_{1} < 0

) and formation of vortex solitons in the self-defocusing region (

Δ_{1} > 0

). In the self-focusing side, while the nonlinear refractive index

n_{2}

increases from left to right,

E_{F 1}

beam breaks up from one to three parts via

ϕ_{N L} (n_{2}^{X 4})

, with one large and two small pieces. Thus, the

E_{F 1}

beam propagates with discrete diffraction in the self-focusing side. By contrast, in the

Δ_{1} > 0

region, the strong dressing fields

E_{1, 2}^{'}

separate the

E_{F 1}

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 (

E_{1, 3}

and

E_{1, 2}^{'}

) induced a modulated vortex pattern with

n_{2}^{F 1} = \sum_{i = 1, i \neq 3}^{5} Δ n_{2}^{X i}

. Finally, the

E_{F 1}

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

n_{2}

is very small with large detuning or

Δ_{1} = 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

E_{F 1}

for different intensities of the dressing field

E_{2}^{'}

in self-defocusing regime. With increasing

E_{2}^{'}

intensity, the spiral phase of

E_{F 1}

changes into jumping phase between two parts, and the modulated vortex soliton of

E_{F 1}

decays into dipole-mode soliton at high intensity, which is created by the horizontally aligned beam

E_{1}^{'}

. Specifically,

E_{F 1}

is circularly modulated by the horizontally-aligned

E_{1}^{'}

and vertically-aligned

E_{2}^{'}

beams. With

E_{2}^{'}

getting stronger,

E_{F 1}

is shifted away from

E_{2}^{'}

and then splits into two parts by

E_{1}^{'}

. The dominant phase of

E_{F 1}

is changed gradually from a spiral phase evolution to a jumping phase (i.e., from interference among four beams

E_{1, 3}

E_{1, 2}^{'}

to the dressing of

E_{1}^{'}

Figure 2(c) shows

E_{F 1}

soliton cluster with different temperatures between

200^{\circ} C

and

300^{\circ} C

in the two-level system.

E_{F 1}

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^{\circ} C

240^{\circ} C

, the nonlinear phase

ϕ_{N L}

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.

ϕ_{N L}

(proportional to both beam intensity and propagation distance z) reaches its optimal value at

250^{\circ} C

. Moreover, the soliton cluster of

E_{F 1}

results from two contributions in the two-level system: (i) the interference among the four waves (

E_{1, 3}

E_{1, 2}^{'}

) with the same frequency induces an interference pattern with

n_{2}^{F 1} = \sum_{i = 1, i \neq 3}^{5} Δ n_{2}^{X i}

, and (ii)

E_{1, 2}^{'}

induce a beam splitting via

ϕ_{N L} (n_{2}^{X 4})

and

ϕ_{N L} (n_{2}^{X 5})

. 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^{\circ} 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 (

E_{1, 2, 3}

E_{1, 2}^{'}

) on, the interference (

n_{2}^{F 2} = \sum_{i = 6, i \neq 8}^{9} Δ n_{2}^{X i}

) among three beams

E_{1}

E_{1}^{'}

, and

E_{3}

induces a rotating vortex. With the incident beams having topological charges

m_{3} = 1

and

m_{1} = m_{1}^{'} = m_{2} = m_{2}^{'} = 0

, the topological charges of the generated FWM signals

E_{F 1}

E_{F 2}

are

m_{F 1} = m_{1} - m_{1}^{'} + m_{3} = 1

and

m_{F 2} = m_{2} - m_{2}^{'} + m_{3} = 1

, respectively. Figures 3(a) 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

T_{X} / T_{Y} \approx 1.1

. From

I \propto \cos^{2} (N φ / 2) \cos^{2} (m φ + Ω_{2} | n_{2} |)

with

Ω_{2} = sgn [n_{2}] w_{2}

(

I_{2} = 51 W / c m^{2}

r = 0.25 m m

E_{F 2}

circumvolves anticlockwise with

n_{2} > 0

and

Ω_{2} = 1.63 \times 10^{3} {W/cm}^{2} > 0

in the self-focusing regime [Fig. 3(a)], while moves clockwise with

n_{2} < 0

and

Ω_{2} = - 1.63 \times 10^{3} {W/cm}^{2} < 0

in the self-defocusing regime [Fig. 3(b)].

Fig. 3 The rotating

E_{F 2}

solitons with (a)

Ω_{2} > 0

and (b)

Ω_{2} < 0

versus

Δ_{1}

. (c) Stationary nonrotating

E_{F1}_{}

solitons (

w_{2} = 0

) versus

Δ_{1}

. Lower images are the simulated results (

N = 3

m_{F 1, F 2} = 1

). The parameters are

G_{2}^{'} = 19.7 GHz

G_{1}^{'} = 12.7 GHz

G_{1} = 1.8 GHz

G_{2} = 1.1 GHz

, and

G_{3} = 0.2 GHz

in the cascade three-level system at

265^{\circ} C

Figure 3(c) presents the stationary solitons with

w_{2} = 0_{}

in the cascade three-level system.

n_{2}^{X 4}

can be a positive value with resonant dressing of

E_{1}^{'}

. When

n_{2}^{X 1}

and

n_{2}^{X 2}

have negative values under the self-defoucsing condition, the superposed

n_{2}^{F 1} = \sum_{i = 1, i \neq 3}^{4} Δ n_{2}^{X i}

is close to zero (or

ϕ_{N L} \approx 0

). Under this condition, a uniform energy flow exists along the ring, and nonrotating (

w_{2} = 0

) spatially-localized multihump structures can be obtained.

Comparing to the three-level system, there exist five nearly degenerate frequency waves (

E_{1, 2, 3}

E_{1, 2}^{'}

) in the two-level system. Figure 4(a) shows the rotating vortices of the FWM beam with four spots (

N = 4_{}

) for different frequency detunings. With the probe beam

E_{3}

carrying topological charge

m_{3} = 1

, the topological charges of the generated FWM signals

E_{F 1}

E_{F 2}

are

m_{F 1} = m_{1} - m_{1}^{'} + m_{3} = 1

and

m_{F 2} = m_{2} - m_{2}^{'} + m_{3} = 1

, respectively, where

m_{1}

m_{1}^{'}

m_{2}

, and

m_{2}^{'}

are all zeros. The modulated vortex pattern (

N = 4

) of

E_{F 2}

is induced by the interference of five waves, and the nonresonant dressing field

E_{2}^{'}

induces a splitting via

ϕ_{N L} (n_{2}^{X 10})

. For

E_{F 2}

E_{2}^{'}

is the nonresonant dressing field, energy flow exists along the ring of spots unequally, inducing a modulated vortex [Fig. 4(a)].

Fig. 4 (a) Anticlockwise rotating (

Ω_{2} = - 1.53 \times 10^{3} {W/cm}^{2}

)

E_{F 2}

solitons and (b) stationary nonrotating

E_{F 1}

solitons with four spots versus

Δ_{1}

265^{\circ} C

in the two-level system. Lower images are simulated (

N = 4

m_{F 1, F 2} = 1

) results. The parameters are

G_{2}^{'} = 15 GHz

G_{1}^{'} = 12.7 GHz

G_{1} = 1.8 GHz

G_{2} = 1.1 GHz

, and

G_{3} = 0.2 GHz

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

T_{X} / T_{Y} \approx 1.5

) is induced by the interference of the five beams. With the nonresonant dressing of

E_{2}^{'}

, all terms in

n_{2}^{F 2}

have the same negative sign [Fig. 4(a)], but the positive

n_{2}^{X 4}

n_{2}^{F 1} = \sum_{i = 1}^{5} Δ n_{2}^{X i}

has opposite sign with the other terms

n_{2}^{X 1, 2, 3, 5}

n_{2}^{F 1}

due to the resonant dressing of

E_{1}^{'}

, so one can get

n_{2}^{F 1} \approx 0

. Therefore, there exist four spots in the stationary (

ϕ_{N L} \approx 0

) modulated vortex soliton of

E_{F 1}

with

N = 4

and energy is mainly stored in one diagonal pair of spots [Fig. 4(b)], which results from the resonant dressing of

E_{1}^{'}

The radially symmetric vortex solitons (

m_{F 1, F 2} = 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)

E_{F 2} \propto E \sec h [E {(k_{F 2} n_{2}^{S 2} / n_{1})}^{1 / 2} (r - R_{0})] \cos (N φ / 2) \exp (i m_{F 2} φ + i w_{1} z)

, the spots rotate with an angular velocity

w_{1}

. In the two-level system with

n_{2}^{F 2} = \sum_{i = 6, i \neq 7}^{10} Δ n_{2}^{X i}

, the vortex pattern of

E_{F 2}

is induced by the interference of

E_{2, 3}

E_{1, 2}^{'}

while the dressing fields

E_{1, 2}^{'}

generate the circular splittings. Here,

E_{F 2}

forms a crescent FWM modulated vortex soliton with an anticlockwise rotation [Fig. 5(a)]. Moreover,

w_{1} z

changes 300° from

245^{\circ} C

275^{\circ} C

. When setting

z = 1

245^{\circ} C

, we obtain

w_{1} = - 1.1

rad/m, which is close to the theoretical value of

w_{1} = - 1.03

rad/m. However, in Fig. 5(b) with the resonant dressing

E_{1}^{'}

and

n_{2}^{F 1} = \sum_{i = 1, i \neq 3}^{5} Δ n_{2}^{X i} \approx 0

, the propagation length of

E_{F 1}

is about 27 times longer than the diffraction length(

L_{D} \approx 0.67 c m

), which are both much shorter than the coherence length of

E_{F 1}

described above (

L_{F 1}^{c} \to \infty

). So we can conclude that

E_{F 1}

beam becomes a stationary vortex soliton at certain

I_{1}

and z values.

Last, we let all six beams on, and set the

E_{1, 2, 3}

and

E_{3}^{'}

beams just 10 times weaker than the dressing beams

E_{1, 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

E_{1, 3}

E_{1}^{'}

on, which create the image 3 in Fig. 5(c) As fourth interference beam

E_{2}^{'}

is added, the split spots change from two to three [image 4 in Fig. 5(c)]. Similarly, as beams

E_{2}

and

E_{3}^{'}

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

E_{F 1}

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

n_{2}^{F 1} = \sum_{i = 1}^{5} Δ n_{2}^{X i} + n_{2}^{X 11} {| E_{3}^{'} |}^{2}

, where

n_{2}^{X 11}

is the nonlinear index induced by

E_{3}^{'}

. The FWM modulated vortex solitons are created jointly by the effects of the complex patterns induced by the multiple interference waves [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

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

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). [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]

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

Y. S. Kivshar, and G. P. Agrawal, Optical solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).
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]
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]
M. J. Holland and J. E. Williams, “Preparing topological states of a Bose-Einstein condensate,” Nature 401(6753), 568–572 (1999). [CrossRef]
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]
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]
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]
A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005). [CrossRef] [PubMed]
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]
G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. 64(21), 2487–2490 (1990). [CrossRef] [PubMed]
D. Bortman-Arbiv, A. D. Wilson-Gordon, and H. Friedmann, “Induced optical spatial solitons,” Phys. Rev. A 58(5), R3403–R3406 (1998). [CrossRef]
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]
J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004). [CrossRef] [PubMed]
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]
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]
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]
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]
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]

Agrawal, G. P.
Anderson, B.
Anderson, B. P.
Bortman-Arbiv, D.
Chang, H.
Chen, Q. F.
Clark, C. W.
Collins, L. A.
Cornell, E. A.
Courtial, J.
Dennis, M. R.
Desyatnikov, A. S.
Feder, D. L.
Friedmann, H.
Goorskey, D.
Gorbach, A. V.
Gorshkov, V. N.
Guo, G.-C.
Haljan, P. C.
Hall, D. S.
Harvey, C. N.
He, Y. J.
Heckenberg, N.
Holland, M. J.
Jiang, W.
Khadka, U.
Kivshar, Y. S.
Law, C. T.
Leach, J.
Li, C.
Malomed, B. A.
Malos, J.
Matthews, M. R.
Nie, Z.
O’Holleran, K.
Padgett, M. J.
Regal, C. A.
Skryabin, D. V.
Song, J.
Soskin, M. S.
Sukhorukov, A. A.
Swartzlander, G. A.
Vasnetsov, M. V.
Wang, H.
Wang, H. Z.
Wieman, C. E.
Williams, J. E.
Wilson-Gordon, A. D.
Xiao, M.
Zhang, Y. P.
Zhang, Y. S.
Zheng, H. B.
Zuo, C. C.

Nature

M. J. Holland and J. E. Williams, “Preparing topological states of a Bose-Einstein condensate,” Nature 401(6753), 568–572 (1999). [CrossRef]
J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004). [CrossRef] [PubMed]

Opt. Express

Phys. Rev. A

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

Phys. Rev. Lett.

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]
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]
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]
A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005). [CrossRef] [PubMed]
G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. 64(21), 2487–2490 (1990). [CrossRef] [PubMed]
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]
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]
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]

Other

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

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