Modulated vortex solitons of four-wave mixing.

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.

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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
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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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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
πω ω 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 ω ω 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
2F1 F1
2
F11
k11
()(2),
2k
S
E
∂
E
∂
EE
ϕ
∂
ii
nEnE
rrrn
∂∂∂∂
−++=+
∂
(1a)

22
2
2 F2
2
F2
z
F2
r
F2
2
F2
2
F2
2 F2F2
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
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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
)
F1F1 2
n
10 F1
sech[ (k/( )]cos(/2)exp()
S
NL
EEEnrRNimi
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
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