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Modulated vortex solitons of four-wave mixing

Yanpeng Zhang,1,3 Zhiqiang Nie,1 Yan Zhao,1 Changbiao Li,1 Ruimin Wang,1 Jinhai Si,1

and Min Xiao2,4

1Key Laboratory for Physical Electronics and Devices of the Ministry of Education & Shaanxi Key Lab of Information

Photonic Technique, Xi’an Jiaotong University, Xi’an 710049, China

2Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA

3ypzhang@mail.xjtu.edu.cn

4mxiao@uark.edu

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 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).

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Received 25 Feb 2010; revised 9 Apr 2010; accepted 13 Apr 2010; published 10 May 2010

24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 10963

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1. 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 [3–5]. 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 [12–15], 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].

2. Theoretical model and experimental scheme

Fig. 1. Two FWM processes

F1

k

and

F2

k

with five beams

1

k ,

1′ k ,

2

k

,

2′ k

,

3

k

, and

3′ k

beams

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

1′ k and

2′ k

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

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

index contrast of

2

4.85 10

n n I

∆ ==×

approximately, where

coefficient and I is the beam intensity) are involved in the experimental schemes. In Fig. 1(b),

energy levels 0 (

1/2

3S

), 1 (

3/2

3P ) and 2 (

13-3

5.6 10 cm

×

2n is the cross-Kerr nonlinear

and a refractive

4

−

3/2,5/2

4D

) form a three-level cascade atomic

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(C) 2010 OSA

Received 25 Feb 2010; revised 9 Apr 2010; accepted 13 Apr 2010; published 10 May 2010

24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 10964

Page 3

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

θ =

2

E ) propagating through the atomic medium in the

same direction with small angles (

beams (

3

E and

E ) propagate in the opposite direction with a small angle as shown in Fig.

1(c). Three laser beams (

1

E ,

E , and

3

E , with Rabi frequencies

transition 0 to 1 ) have the same frequency

0.3

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

3′

1′

1

G,

1

G′ and

3

G , connecting

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

F1

E

(

F1113

=

′

−+

kkkk ) [Fig. 1(a)] in the direction shown at the lower right

corner of Fig. 1(c). These beams

2

E ,

E , and

and connecting the same transition 0 to 1 in the two-level system) are from another

ω , and produce a nondegenerate FWM signal

′

−+

kkkk ) [Fig. 1(a)]. All laser beams are horizontally polarized. The diameters

µ

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

two FWM processes

F3113

=

′′

−+

kkkk and

F4

k

and

F2

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

2′

3′

E (with Rabi frequencies

2

G ,

2

G′ and

3

G′,

near-transform-limited dye laser of frequency

2

F2

E (

F2223

=

of the laser beams are about 25 m

22

′

3

′

=

−+

kkk . However, the coexisting

F1

E

2

1111

11

2 /[]

c

F

,

Lcn

πω ω ωω θ

=− → ∞ for

F1

E

,

23

1212

21

2 /[] 1.8 10 m

×

c

F

Lcn

πωωθ

=∆ − ∆≈

for

F2

E

23

2121

31

2 /[] 1.8 10 m

×

c

F

Lcn

πωωθ

=∆ − ∆≈

for

F3

E ,

∆ (

2

2222

41

2/[]

c

F

L

∆ = ) is the detuning of the fields

cn

πωω ωω θ

=−→ ∞ for

F4

E , where

1n is the linear refractive index

E (

2

E and

E ) from the atomic and

12

0

1,3

E and

1′

2,3′

transition.

When

2

E and

2′

E are tuned to the 12

−

transition, the system becomes a cascade

three-level system [Fig. 1(b)], which generates a two-photon resonant nondegenerate FWM

process

F2

E

[17,18]. In this system,

πωω ω ω θ

=− → ∞ for

F1

E

and

E , respectively.

the coherence lengths are

2

1111

11

2/[]

c

F

Lcn

2

1221

21

2/[]m

c

F

Lcn

πωω ωω θ

=−≈ 0. 6

for

F2

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:

22

2

1F1

2

F1

z

F1

r

F1

2

F1

2

F1

2F1F1

2

F11

k11

()(2),

2k

S

E

∂

E

∂

EE

ϕ

∂

ii

nEnE

rrrn

∂∂∂∂

−++=+

∂

(1a)

22

2

2F2

2

F2

z

F2

r

F2

2

F2

2

F2

2 F2 F2

2

F21

k11

()(2),

2k

S

E

∂

E

∂

EE

ϕ

∂

ii

nEnE

rrrn

∂∂∂∂

−++=+

∂

(1b)

where

5

22222

F1

2

i12345

223212221

′

22

′

1

XXXXXX

i

nnnEnEnEnEnE

=

=∆=++++

∑

and

10

∑

22222

F2

2

i678910

223212221

′

22

′

6

XXXXXX

i

nnnEnEnEnEnE

=

=∆=++++

.

12

2

SS

n

−

are the self-Kerr

nonlinear coefficients of

F1,2

E

, and

110

2

XX

n

−

are the cross-Kerr nonlinear coefficients due to the

fields

1,2,3

E

and

1,2′

E

, respectively. The Kerr nonlinear coefficients are defined as

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Received 25 Feb 2010; revised 9 Apr 2010; accepted 13 Apr 2010; published 10 May 2010

24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 10965

Page 4

(3)

201

Re/()

n cn

χε

=

. The third-order nonlinear susceptibility is given by

(3) 4

10

(3)

10

32

i

00 3,F1,F2

G

/()

NG

χ

transition |i > to | j >. We can obtain these Kerr nonlinear coefficients of the FWM beams

by calculating the density-matrix element

µ ρε

=

ℏ

.

0

N is atomic density.

ij

µ is the dipole matrix element between

F1,2

E

(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]

ϕ

∝−

1 1/2

)

F1 F1 2

n

10 F1

sech[ (k/( )]cos(/2)exp()

S

NL

EEEnrRN imi

ϕφ

+

with an initial radius

0 R .

and Moreover, we have

2/2

1 F1 2 2

n I e

1

( , )/

r z

.

2k/

r

NL

wzn

φ

−

==

,

2/2

22F1 2

I e

1

( , )/

r z

2k/

r

NL

wn z n

φ

−

==

2/2

32F1 2

n e

1

( , )/

r z

2k/

r

NL

wIz n

φ

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

−

==

1,2,3

E

and

1,2,3

′

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 (

waves [13,14]. The local structures of the optical vortices are given by the polarization ellipse

relation

/()sin () /(

XXYYXY

CTTCTT

βα

++++

and α is the ellipse orientation. The ellipse axes

X

C ,

Y

C ) of the interfering plane

22222222

)cos ()1

βα

+= , where

arctan(/)

XY

TT

β =

X

T ,

YT are related to the spatial

configuration (including the incident beam directions, phase differences between beams etc.)

and beam intensities.

The dressing beams

E are approximately 10 times stronger than the beams

1,2′

1,2

E ,

2

10

times stronger than the weak probe beams

3

E and

3′

E , and

4

10 times stronger than the two

generated FWM beams

F1,2

E

. The generated weak beam

F1

E (or

F2

E ) partly overlaps with the

strong beam

1′

E (or

2′

E ), and other stronger beams (

1,2,3

E

,

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

5

F1

22

1

i

=

lies in the minimum of

2n

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

and

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

vortex pattern with the superposed

i

X

nn

=∆

∑

and

10

∑

F2

2

i

2

6

X

i

nn

=

=∆

(the center of such vortex

F1,2

1′

E

2′

1′

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.

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

mmmm

′

=−+

,

F2223

mmmm

=−+

,

m

F1113

′

F3113

mmm

′′

=−+

, and

F4223

mmmm

′′

=−+

,

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Received 25 Feb 2010; revised 9 Apr 2010; accepted 13 Apr 2010; published 10 May 2010

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Page 5

where

F1

m ,

F2

m ,

F2

m ,

F2

m ,

m are the topological charges of the FWM beams

m′,

2

m ,

23

m ,

3

E , respectively. The topological charges of two FWM signals

F1

E ,

F2

E ,

E ,

F3

E ,

E ,

F4

E , respectively. and

11

m′ ,

m′ are those of the beams

1

E ,

1′

E ,

22′

3

E ,

3′

F1

E ,

F2

E in the cascade

three-level system obey the same conservation rules.

3. Modulated vortex solitons

Fig. 2. (a) Images of

F1

E versus

1

∆ with

2

14.7GHz

G′ =

at 265 C

?

. (b) Images of

F1

E at

1

8GHz

∆ =

with different values of

2

G′ at 265 C

?

. Upper and lower panels are for

∆ =

experimental and simulated results, respectively. (c) Images of

F1

E at

1

8GHz

with

different temperatures from 200 C

?

to 300 C

?

.

2

9.5GHz

G′ =

. Top and bottom rows are the

G′ =

cross sections in y and x directions, respectively. The parameters are

G =

, and

3

0.2GHz

G =

in the two-level system.

1

12.7GHz

,

1

1.8GHz

Figure 2(a) presents the effects of spatial dispersion on the FWM signal

F1

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

φ

2 n increases from left to right,

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

F1

E beam breaks up from one to three parts

via

4

2

()

X

NLn

F1

∆ > region, the strong

N = ). Then these

E beam propagates with

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

1

0

dressing fields

1,2′

E separate the

F1

E beam into three spots along a ring (

3

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

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Received 25 Feb 2010; revised 9 Apr 2010; accepted 13 Apr 2010; published 10 May 2010

24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 10967