1Key Laboratory for Physical Electronics, Devices of the Ministry of Education & Shaanxi Key Lab of InformationPhotonic Technique and School of Science, Xi’an Jiaotong University, Xi’an 710049, China
2Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, [email protected]@mail.xjtu.edu.cn
Received June 23, 2009; accepted July 16, 2009;posted July 22, 2009 (Doc. ID 113174); published August 14, 2009
. INTRODUCTIONhe parametric four-wave mixing (FWM) is a useful pro-ess for generating coherent radiations from vacuum ul-raviolet to infrared wavelengths. The polarization char-cteristics of two-photon resonant FWM processes haveeen investigated in several types of metal vapors gases.sukiyama [1] described polarization properties of theear-infrared FWM signal produced in Kr vapor, Museurt al. [2] studied the polarization dependence of theacuum ultraviolet light generated by a four-wave sum-requency generation process in Hg, and Ishii et al. [3] in-estigated polarization characteristics of FWM in NO gas.esides producing coherent emissions, parametric FWMrocesses can also be used to study interference effects.revious studies have used FWM processes to observe in-erference effects between different atomic polarizations4–6]. Studies of interference effects in multilevel atomicystems have become an active field of research in recentears, which made it possible to coherently control the op-ical properties of atomic media [7,8]. In parametric FWMrocesses, there are multiple quantum paths for each stepf the nonlinear processes, and the probability amplitudesor these different transition paths are intimately relatedo the polarizations of the input laser fields. Therefore, its possible to coherently control the nonlinear processesy manipulating the polarization states of incident lasereams.In this paper, we report both theoretical and experi-ental results on the polarization dependence of theWM signals generated in Na atomic vapor. Both two-
evel and three-level systems in Na atoms are used in
hese studies. The classical, as well as quantum, theoret-cal models have been developed to explain the depen-ence of the FWM signals on the polarization states of thencident laser beams. By comparing the FWM spectra forifferent laser polarization configurations, we can identifyifferent contributions from third-order nonlinear suscep-ibility elements under different conditions. Furthermore,e explain the different contributions by the difference
rom combinations of transition paths for three incidenteam polarization schemes. Moreover, we report the po-arization characteristics of the singly dressed and doublyressed FWM processes in either a two-level or a three-evel atomic system in Na vapor. In the three-level atomicystem the FWM signal generated by a linearly polarizedumping field is greatly suppressed by the dressing field,hile the one generated by a circularly polarized pump-
ng field is only slightly influenced by the dressing field.lso, different change rules are observed when the polar-
zation of the probe field is changed. Different dressing ef-ects of two dressing schemes are used to explain this phe-omenon. The dressing effects, as well as the interferenceetween the two coexisting FWM signals, have been dis-ussed in different laser polarization configurations. In-estigations of the interactions between different FWMrocesses and their polarization properties can help us tonderstand the underlying physical mechanisms and toffectively optimize the generated nonlinear optical sig-als. Controlling nonlinear optical processes can haveany potential applications such as in all-optical
witches [9] and quantum-information processing [10,11].This paper is organized as follows: Section 2 we de-
009 Optical Society of America
stsWws
2Tp2tetatdq|drfbam�lER|sigfTastppbnqesms
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Wang et al. Vol. 26, No. 9 /September 2009 /J. Opt. Soc. Am. B 1711
cribe our experimental setup. In Section 3 we presenthe basic theoretic treatments about Zeeman atomic sub-ystems interacting with arbitrarily polarized laser fields.e will discuss the experimental results and comparisonsith the theoretical calculations in Section 4. Section 6
ummarizes main results and gives conclusions.
. EXPERIMENTAL SETUPhe experiments are carried out in Na vapor (in a heatipe oven), which is heated up to a temperature of about35°C. Three energy levels from Na atoms are involved inhe experimental schemes. As shown in Fig. 1, energy lev-ls �0��3s1/2� and �1��3p3/2� form the two-level atomic sys-em. Two laser beams Ed (�p, kd, and Rabi frequency Gd)nd Ed� (�p, kd�, and Rabi frequency Gd�), connecting theransition between |0� and |1�, propagate in the oppositeirection of the weak probe beam Ep (�p, kp, and Rabi fre-uency Gp), which also connects the transition between0� and |1�. The three laser beams come from the sameye laser DL1 (wavelength of 589.0 nm, 10 Hz repetitionate, 5 ns pulse width, and 0.04 cm−1 linewidth) with therequency detuning �1, pumped by the second-harmoniceam of a Nd:YAG laser. These three laser fields generatedegenerate-FWM (DFWM) process satisfying the phase-atching condition of ks1=kp+kd−kd�. The energy levels
0��3s1/2�– �1��3p3/2�– �2��4d3/2� form the cascade three-evel atomic system. Two additional coupling laser beams
c (�c, kc, and Rabi frequency Gc) and Ec� (�c, kc�, andabi frequency Gc�), connecting the transition between1� to |2� are from another dye laser DL2 (which has theame characteristics as the DL1) with a frequency detun-ng �2. Ec, Ec�, and Ep fields interact with each other andenerate a nondegenerate FWM (NDFWM) signal, satis-ying the phase-matching condition of ks2=kp+kc−kc�.he generated DFWM and NDFWM signals propagatelong slightly different directions due to their differentpatial phase-matching conditions. Two photomultiplierube (PMT) detectors are used to receive the horizontallyolarized component (P polarization) and the verticallyolarized component (S polarization) for one of the signaleams, or horizontally polarized components of both sig-al beams, respectively. A half-wave plate (HWP) and auarter-wave plate (QWP) are selectively used (in differ-nt experiments, respectively) to control the polarizationtates of the incident fields. The generated FWM signalsay pass through another HWP and a polarization beam
plitter (PBS) before being detected by the two PMTs.
(a) (b)ig. 1. Schematic diagrams of the experimental arrangementnd the relevant energy levels in the Na atom.
. BASIC THEORY. Various Nonlinear Susceptibilities for Differentolarization Schemeshe polarization dependence of the FWM signals can bexplained using either a classical or quantum mechanicalescription [12]. Classically, the FWM signal intensity isroportional to the square of the atomic polarization in-uced in the medium. For example, as for phase-onjugated FWM generation in the cascade atomic systemt frequency �s=�c−�c+�p (as shown in Fig. 1, witheams kd and kd� blocked), the nonlinear polarizationlong i �i=x ,y� direction is given by
Pi�3���s� = �0�
jkl�ijkl
�3� �− �s;�c,− �c,�p�Ecj��c�Eck* ��c�Epl��p�,
�1�
here �ijkl�3� �−�s ;�c ,−�c ,�p� is the tensor component of the
hird-order nonlinear susceptibility. For an isotropic me-ium like Na atomic vapor and considering that all the in-ident beams and signals are transverse waves, only fouronzero tensor elements are involved in this system,hich are denoted as �xxxx, �yxxy, �yyxx, �yxyx. Different po-
arization configurations of the incident fields can involveifferent nonlinear susceptibility elements. For example,hen we use a HWP to change the polarization of the Epeld while the other two beams are originally polarized inhe horizontal direction, the probe field will have two per-endicular components: Epx=Ep Cos 2� and EpyEp Sin 2� (� is the rotated angle of the HWP’s axis from
he x axis). Consequently, the polarization has two corre-ponding components, i.e., horizontal component Px
�3���s��0�xxxx�Ec�2�Epx� and perpendicular component Py
�3���s��0�yxxy�Ec�2�Epy�. Then the effective susceptibility ele-ents �x and �y are defined as �x=�xxxx cos 2� and �y�yxxy sin 2�. As for the other two cases with kc and kc�odulated by a HWP. �yyxx and �yxyx become dominant on
enerating FWM signals polarized in the S direction (theignals in the P direction for the three cases are all gen-rated by �xxxx). The microscopic mechanism of nonlinearifferent susceptibilities will be discussed in Subsection.B.If a QWP is used to modulate the incident beams, the
ffective nonlinear susceptibilities will be different whilehe excited susceptibilities are the same as in the corre-ponding cases with HWP modulation. Table 1 presentsll the effective susceptibilities for the three-field polar-zation schemes.
In order to measure the polarization states of the FWMignals, a HWP and a PBS are placed in the path of theignal beam (as shown in Fig. 1). When an arbitrarily po-arized field � Eyei�
Ex � passes through the HWP�PBS combi-ation, the detected intensities are
Ix = cos2 2��Ex�2 + �Ey�2sin2 2� + �Ex��Ey�sin 4� cos �,
1712 J. Opt. Soc. Am. B/Vol. 26, No. 9 /September 2009 Wang et al.
espectively, where � is the rotation angle of the HWProm the x axis and � is the phase difference between thewo polarization [horizontal �x� and vertical �y�] compo-ents of the signal beam.
. Nonlinear Susceptibilities for a Zeeman-Degenerateystem Interacting with Polarized Fieldshe polarization dependence of the FWM signals can alsoe described by the semi-classical treatment. It is basedn the fact that there are different transition paths com-inations consisting of various transitions between Zee-an sublevels for different polarization schemes [as
(a) (b)
(c) (d)
(e) (f)
(g) (h)
p d dG G G�� �
MJ= -3/2 -1/2 1/2 3/2
FG
c cG G��
p d dG G G�� � FG
ig. 2. (Color online) Energy level diagrams and transitionaths at different laser polarization configurations. (a) and (e)chematic diagrams of the P-polarization generation in two-levelnd three-level systems when the waveplates change kp, kd, and
d�. (b)–(d) and (f)–(h) Schematic diagrams of S-polarization gen-ration in two-level and three-level systems when the waveplateshange kp, kd, and kd�, respectively. Dotted, long-dashed, solid,nd short-dashed lines are transitions for the probe, coupling,ressing, and FWM signal fields, respectively.
Table 1. Effective Nonlinear Susceptibilities
kp, kc, kc�, P kp, S
HWP �x=�xxxx Cos 2� �y=�yxxy Sin 2�
QWP �x=�xxxx
��sin4 �+cos4 �
�y=�yxxy
��2�sin � co
hown in Fig. 2]. According to the experimental setup, theaxis is the original polarization direction of all the inci-ent fields, and it is also the quantization axis. We thenecompose the arbitrary field into two components: paral-el to and perpendicular to the x axis, respectively. Whenhis field interacts with atoms, the perpendicular compo-ent can be decomposed into equally left-circularly- andight-circularly-polarized components. Different polariza-ion schemes can excite different transition paths in theeeman-degenerate atomic systems, and so it is necessaryo take into account the Clebsch–Gordan coefficients as-ociated with the various transitions between Zeemanublevels in all pathways when calculating the FWM in-ensities. Figure 2 shows the transition schematic con-gurations for the Zeeman-degenerate two-level andhree-level cascade systems interacting with one arbi-rarily polarized and two horizontally polarized fields.able 2 and 3 list all the perturbation chains for differentases, respectively. By considering the schematic figuresnd the tables, we can obtain the expressions of variousensity matrices corresponding to nonlinear susceptibili-ies for different polarization schemes.
Figure 2(a) shows the configuration of generating theWM signals in P-polarization (represented as �xxxx) in
he two-level system. It contains two sub-two-level sys-ems: �a−1/2�– �b−1/2� and �a1/2�– �b1/2�. The respective per-urbation chains are listed in Table 2 and the total contri-ution of these chains to the density-matrix element thatnduces the FWM signal in the P-polarization direction is
p�3� = − i �
M=±1/2�GdM
0 �2GpM0 � 1
�i�p + bMaM�2 +
1
�p2 + bMaM
2 ��� 1
aMaM
+1
bMbM� . �4�
hen from �xxxx=N�p�3� /�0�Ec�2Ep, we can get �xxxx.
Figure 2(b) presents the configuration for generatinghe S-polarized FWM signals when the waveplate (WP)hanges kp (corresponds susceptibility �yxxy). It containswo right-circularly-polarized V-type subsystems�a−1/2�– �b1/2�– �b−1/2� and �a1/2�– �b3/2�– �b1/2�), two left-ircularly-polarized V-type subsystems�a−1/2�– �b−3/2�– �b−1/2� and �a1/2�– �b−1/2�– �b1/2�), one right-ircularly-polarized reversed V-type subsystema−1/2�– �b1/2�– �a1/2� and one left-circularly-polarized RV-ype subsystem �a1/2�– �b−1/2�– �a−1/2�. Their perturbationhains are listed in Table 2 and the total density-matrixlement including contributions from all the perturbationhains can be written as
ifferent Laser Polarization Configurations
kc, S kc�, S
�y=�yyxx Sin 2� �y=�yxyx Sin 2�
�y=�yyxx
��2�sin � cos ��2�y=�yxyx
��2�sin � cos ��2
for D
s ��2
T
Wang et al. Vol. 26, No. 9 /September 2009 /J. Opt. Soc. Am. B 1713
able 2. Perturbation Chains of the Two-Level System for Different Laser Polarization Configurations
WP changekd (or kd�,
kp)
(I)
aMaM——→
GpM0
bMaM——→�G�dM
0 �*
bMbM——→
GdM0
bMaM, �M= ±1/2�.
(II)
aMaM——→
GpM0
bMaM——→�G�dM
0 �*
bMbM——→
GdM0
bMaM, �M= ±1/2�.
P polarization��xxxx�
(III)
aMaM——→�G�dM
0 �*
aMbM——→
GdM0
aMaM——→
GpM0
bMaM, �M= ±1/2�.
(IV)
aMaM——→�G�dM
0 �*
aMbM——→
GdM0
bMbM——→
GpM0
bMaM, �M= ±1/2�.
(V-R-I)
aMaM——→�G�dM
0 �*
aMbM+1——→
GdM0
aMaM——→
GpM+
bM+1aM, �M= ±1/2�.
WP changekp
(V-R-II)
aMaM——→
GdM0
bMaM——→�G�dM
0 �*
aMaM——→
GpM+
bM+1aM, �M= ±1/2�.
(V-L-I)
aMaM——→
GdM0
bMaM——→�G�dM
0 �*
aMaM——→
GpM−
bM−1aM, �M= ±1/2�.
S polarization��yxxy�
(V-L-II)
aMaM——→
GdM0
bMaM——→�G�dM
0 �*
aMaM——→
GpM−
bM−1aM, �M= ±1/2�.
(V-R)
aMaM——→GpM+1
+
bM+1aM——→
�G�dM+10 �*
aM+1aM——→GdM+1
0
bM+1aM, �M= ±1/2�.
(RV-L)
aMaM——→
GpM−
bM−1aM——→�G�dM
0 �*
aM−1aM——→
GdM0
bM−1aM, �M= ±1/2�.
WP changekd
(I)
aMaM——→
GpM0
bMaM——→�GdM
0 �*
aMaM——→
GdM−
bM−1aM, �M= ±1/2�.
(II)
aMaM——→
GpM0
bMaM——→�GdM
0 �*
aMaM——→
GdM+
bM+1aM, �M= ±1/2�.
S polarization��yyxx�
(III)
a−1/2a−1/2——→Gp−1/2
0
b−1/2a−1/2——→�Gd−1/2
0 �*
a−1/2a−1/2——→Gd−1/2
+
b1/2a−1/2.
(IV)
a1/2a1/2——→
Gp1/20
b1/2a1/2——→�Gd1/2
0 �*
a1/2a1/2——→
Gd1/2−
b−1/2a1/2.
WP changekd�
(I)
a−1/2a−1/2——→Gp−1/2
0
b−1/2a−1/2——→
�G�d−1/2− �*
a−1/2a1/2——→Gd−1/2
0
b1/2a−1/2(II)
a1/2a1/2——→
Gp1/20
b1/2a1/2——→�G�d1/2
+ �*
a−1/2a1/2——→
Gd1/20
b−1/2a1/2S polarization
��yxyx�(III)
a−1/2a−1/2——→Gd−1/2
0
b−1/2a−1/2——→�Gd−1/2
− �*
a−1/2a1/2——→Gp−1/2
0
b1/2a−1/2(IV)
a1/2a1/2——→Gd−1/2
0
b1/2a1/2——→
�G�d−1/2+ �*
a−1/2a1/2——→Gp−1/2
0
b−1/2a1/2
WPscap�sa�tti
T�
FTcm
1714 J. Opt. Soc. Am. B/Vol. 26, No. 9 /September 2009 Wang et al.
s1�3� = − iGp1/2
− �Gd1/2
0 �2
�i�p + b−1/2a1/2�2a−1/2a1/2
+iGp−1/2
+ �Gd−1/2
0 �2
�i�p + b1/2a−1/2�2a1/2a−1/2
− �
M=±1/2
2bMaM�GdM
0 �2
aMaM��p
2 + bMaM
2 � iGpM+
�i�p + bM+1aM�
+iGpM
−
�i�p + bM−1aM� . �5�
When the polarization of the kd field is changed by theP, the subsystems generating FWM signals in the
-polarization direction and their expressions are theame as the ones for changing the kp field. However, theonfiguration of generating the S-polarized FWM signal,s shown in Fig. 2(c), contains two left-circularly-olarized V-type subsystems (�a−1/2�– �b−1/2�– �b−3/2� and
a1/2�– �b1/2�– �b−1/2�), two right-circularly-polarized V-typeubsystems (�a−1/2�– �b−1/2�– �b1/2� and �a1/2�– �b1/2�– �a3/2�),nd one right-circularly-polarized RV-type subsystem�a−1/2�– �a1/2�– �b1/2��, and one left-circularly-polarized RV-ype subsystem ��a−1/2�– �a1/2�– �b1/2��. The total contribu-ion to the third-order nonlinear density-matrix elements
Table 3. Perturbation Chains of the Cascade Sys
WP changekp (orkc, kc�)
P polarization��xxxx�
WP changekp
S polarization��yxxy�
WP changekc
S polarization��yyxx�
WP changekc�
S polarization��yxyx�
s2�r� = −
iGp−1/2
0 Gd−1/2
+ �Gd−1/2
0 �*
a−1/2a−1/2�i�p + b−1/2a−1/2
��i�p + b1/2a−1/2�
−iGp1/2
0 Gd1/2
− �Gd1/2
0 �*
a1/2a1/2�i�p + b1/2a1/2
��i�p + b−1/2a1/2�
− �M=±1/2
iGpM
0 �GdM
0 �*
aMaM�i�p + bMaM
� GdM
−
�i�p + bM−1aM�
+GdM
+
�i�p + bM+1aM� . �6�
hen we can obtain the nonlinear susceptibility elementxyxy.When changing the polarization of kd� field, as shown in
ig. 2(d), there are four perturbation chains as listed inable 2. The total contribution from all the perturbationhains to the third-order nonlinear density-matrix ele-ent can be written as
s3�3� = −
4iGp−1/2
0 Gd1/2
0 �Gd−1/2�− �*
a−1/2a1/2�i�p + b1/2a−1/2
��i�p + b−1/2a−1/2�. �7�
for Different Laser Polarization Configurations
M——→
GpM0
aMbM——→
GcM
aMcM——→�G�cM
0 �*
aMbM, �M= ±1/2�.
M——→
GpM−
aMbM−1——→
GcM0
aMcM−1——→�G�cM
0 �*
aMbM−1, �M= ±1/2�.
M——→
GpM+
aMbM+1——→
GcM0
aMcM+1——→�GcM
0 �*
aMbM+1, �M= ±1/2�.
M——→
GpM0
aMbM——→
GcM−
aMcM−1——→�G�cM
0 �*
aMbM−1, �M= ±1/2�.
M——→
GpM0
aMbM——→
GcM+
aMcM+1——→�G�cM
0 �*
aMbM+1, �M= ±1/2�.
M——→
GpM0
aMbM——→
GcM0
aMcM——→�GcM
− �*
aMbM−1, �M= ±1/2�.
M——→
GpM0
bMaM——→
GcM0
cMaM——→�GcM
+ �*
bM+1aM, �M= ±1/2�.
tem
aMa
(I)
aMa
(II)
aMa
(I)
aMa
(II)
aMa
(I)
aMa
(II)
aMa
Tcm
sifiTn
CoFkF
Ft
OtbefwGw
4Fot
Wang et al. Vol. 26, No. 9 /September 2009 /J. Opt. Soc. Am. B 1715
his expression is simpler due to the symmetry of theonfiguration relative to M=0. Then, we can obtain ele-ent �xxyy.For the three-level cascade-type (C3-type) system, the
chematic charts are shown in Figs. 2(e)–2(h) [with dress-ng fields kd and kd� blocked], which change kp, kc, and kc�elds. The corresponding perturbation chains are listed inable 3. The expressions of the corresponding third-orderonlinear density-matrix elements are
p�3� = − �
M=±1/2
iGpM0 iGcM�GcM
0 �*
�i�p + bMaM�2�i��c + �p� + cMaM
�,
s1�3� = − �
M=±1/2
iGpM− �GcM
0 �2
�i�p + bM−1aM�2��i��c + �p� + cM−1aM
��
− �M=±1/2
iGpM+ �GcM
0 �2
�i�p + bM+1aM�2�i��c + �p� + cM+1aM
�,
d3lsFiiiclotfit
s2�3� = − �
M=±1/2
iGpM0 �GcM�
0 �*
�i�p + bMaM�
� iGcM+
�i�p + bM+1aM��i��c + �p� + cM+1aM
�
+iGcM
−
�i�p + bM−1aM��i��c + �p� + cM−1aM
� ,
s3�3� = − �
M=±1/2
iGpM0 GcM
0
�i�p + bMaM��i��c + �p� + cMaM
�
�� �GcM− �*
i�p + bM−1aM
+�GcM�
+ �*
i�p + bM+1aM
� . �8�
. Third-Order Density-Matrix Elements in the Presencef Dressing Fieldsor the case with a single dressing by the kd field (with
d� blocked), the density-matrix elements of the dressedWM signals (generated in the C3 system) are given by
bMaM
�3�= −
iGpM0
�GcM0
�2
�i��c + �p� + cMaM�
�1
i�p + bMaM+
�GdM�2
i��p − �d� + aMaM+
�GpM0
�2
aMaM
+�GcM
0�2
i��c + �p� + cMaM�2, M = ±
1
2� . �9�
or the case of double dressing (with kd and kd� both on), the third-order nonlinear density-matrix elements can be writ-en as
bMaM
�3�= −
iGpM0
�GcM0
�2
i��c + �p� + cMaM
�1
i�p + bMaM+
2�GdM�2
i��p − �d� + aMaM+
�GpM0
�2
aMaM
+�GcM
0�2
i��c + �p� + cMaM�2, M = ±
1
2� . �10�
ne can easily see that the multidressed fields appear inhe denominator and the dressing effect is mainly causedy the strong dressing field Gd. The other two fields, how-ver, can enhance or suppress such dressing effect. Inact, the coupling field Gc, which is denoted as sequentialith Gd, generally enhances the dressed effect induced byd, while the probe field Gp, which is denoted as nestedith Gd, generally suppresses the dressing effect [13].
. RESULTS AND DISCUSSIONSirst, we use a HWP to modulate the polarization of onef the incident beams, while the other two beams are kepto be in horizontally polarization. In this case, the inci-
ent beams are all linearly polarized. Figures 3(a) and(b) show the relative FWM intensities in the P and S po-arizations, respectively, in the cascade three-level atomicystem with respect to the rotation angle � of the HWP.rom Table 1 we can see that, for the horizontally polar-
zed component [Fig. 3(a)], the dependence of the FWMntensity on � follows �cos 2��2 while the vertically polar-zed component obeys �sin 2��2 (represented by the solidurves in Fig. 3). This means that the FWM signals areinearly polarized. Similar results have been reported inther systems [2,4,6]. From Fig. 3(b), the signal ampli-udes are different for the three laser polarization con-gurations, which can be attributed to different contribu-ions from the third-order nonlinear susceptibility
etcckgtlTft
dQbtlaa2iFav2pnca
wlF
fitngbstk�d
I
I
Iplt�finpecFcpo
s5atfiaissp
FvsTsa
Fne
1716 J. Opt. Soc. Am. B/Vol. 26, No. 9 /September 2009 Wang et al.
lements under different conditions. As discussed in Sec-ion 3, different polarization schemes of incident fieldsan excite different nonlinear susceptibilities, and theyan have different quantum transition paths. When thep field is modulated by the HWP, �yxxy is excited, whichenerates FWM signal in the S-polarization direction. Inhe other two cases, both the kc and kc� fields are modu-ated, so both �yyxx and �yxyx are respectively stimulated.he signal amplitudes indicate different contributions
rom third-order nonlinear susceptibility elements forhree different laser polarization schemes.
Figures 3(c) and 3(d) depict the polarization depen-ences of the FWM signals on the rotation angle of theWP in the two-level DFWM process. First, the probeeam kp is elliptically polarized and the ellipticity is con-rolled by the QWP, while the other beams have linear po-arization along the x axis [the square points in Figs. 3(c)nd 3(d)]. In Figs. 3(c) and 3(d), the experimental resultsre well described by the functions sin4 +cos4 and�sin cos �2 (the solid curves), respectively. If one of thencident beams is elliptically polarized, the generatedWM signal is also elliptically polarized, which is in goodgreement with the theoretical prediction. Comparing theertically polarized intensities of the signal beams [Figs.(b) and 2(d)], there are different ratios of oscillation am-litudes. It indicates that the ratios of the third-orderonlinear susceptibility elements are different for the cas-ade three-level system and the two-level system, whichre well described by Eq. (8).To detect the polarization states of the FWM signals,
e place a HWP+PBS combination as a polarization ana-yzer in the generated FWM signal beam (as shown inig. 1). In fact, when a QWP is used to modulate the kc
(a) (b)
(c) (d)
ig. 3. (Color online) Variations of the relative FWM intensitiesersus the rotation angle of the waveplate. (a) and (b) The FWMignals of the cascade three-level system with the HWP. (c)–(d)he FWM signals of the two-level system with the QWP. Thecattered points are the experimental data and the solid curvesre the theoretical results.
eld’s polarization (or ellipticity �), the polarizations ofhe FWM signals are also changed. Besides, the excitedonlinear susceptibilities of the P and S polarizationsreatly modify the signal’s polarization states, which cane detected by the HWP+PBS combination. Figure 4 pre-ents the detected results. Each curve is obtained by ro-ating the HWP while keeping the QWP in the path of the
c field fixed. In this case, �Ex���xxxx�sin4 �+cos4 � andEy���yyxx�sin2 �cos2 �. According to Eqs. (2) and (3), theetected intensities should be
x = �xxxx2 �cos4 � + sin4 ��cos2 2�
+ �yyxx2 �2 sin2 � cos2 ��sin2 2�
+ �xxxx�yyxx��cos4 � + sin4 ���2 sin2 � cos2 �� sin 4� cos �,
y = �xxxx2 �cos4 � + sin4 ��sin2 2�
+ �yyxx2 �2 sin2 � cos2 ��cos2 2�
− �xxxx�yyxx��cos4 � + sin4 ���2 sin2 � cos2 �� sin 4� cos �.
�11�
f the kc field is linearly polarized (�=0, �=1, the squareoints in Fig. 4), the NDFWM signal is also linearly po-arized, and the horizontal �x� and vertical �y� signal in-ensities obey the relations of �xxxx
2 cos2 2� and
xxxx2 sin2 2�, respectively. When the polarization of the kceld is changed, the other nonlinear susceptibility compo-ents are excited and the FWM signal is then ellipticallyolarized. From Fig. 4 we can see that, if the kc field isither elliptically (�=30°, �=0.5, the triangle points) orircularly (�=45°, �=0, asterisk points) polarized, the ND-WM signals are also elliptically polarized. This can beonfirmed by Eq. (11). When the input kc field is circularlyolarized, the FWM signal can be circularly polarizednce the condition �xxxx=�yyxx is satisfied.
Figure 5 shows the dependence of the dressed FWMignal on the polarization of the kc field. Figures 5(a) and(b) depict the results for the kd singly-dressed and kdnd kd� doubly dressed FWM signals, respectively, forhree different ellipticities of the kc field. The dressingelds kd and kd� are both linearly polarized along the xxis. Comparing to Fig. 4(a), the reduction of the signalntensity is more than 50%. More interestingly, the FWMignals generated by the linearly polarized kc field [thequare points in Fig. 4(a) and Fig. 5(a)] are greatly sup-ressed by the kd dressing field while the FWM signals
ig. 4. (Color online) Dependence of the relative NDFWM sig-al intensity on � for three values of the coupling laser’sllipticity.
gppsodipktsttdcingl
dtepsmtnaFbFkaaApioFsCbw�iu
astttleept
tFhdiEfiwGiqrtGenrtF
Fbo
Fnact
Wang et al. Vol. 26, No. 9 /September 2009 /J. Opt. Soc. Am. B 1717
enerated by the circularly polarized kc field [the asteriskoints in Fig. 4(a) and Fig. 5(a)] are only slightly sup-ressed by the kd dressing field. So, in Fig. 5(a) thequare points are lower than the other curves, which ispposite to the case in Fig. 4(a). Figure 5(c) presents theependences of the pure FWM and dressed FWM signalntensities on the ellipticity of the kc field. The squareoints represent pure FWM case, which decreases as thec’s ellipticity reduces [from 1 to 0 when the QWP is ro-
ated from 0 to 45 degrees]. The dot points represent theingly dressed FWM case, which shows an opposite varia-ion from the pure FWM. This result can be explained byhe expression [Eq. (9)] for the dressed FWM case. In theenominator of Eq. (9), Gd’s sequential dressing field Gcan enhance the dressing effect. When the coupling fields linearly polarized, its Rabi frequency Gc in the denomi-ator is at its maximum, so the dressing effect is stron-est [square points in Fig. 5(a)] and the FWM signal isowest.
Moreover, comparing the singly dressed and doublyressed FWM signals, they have similar suppressed in-ensities when the signals are linearly polarized. How-ver, when the signals are elliptically polarized, the sup-ression in the doubly dressed case is stronger than in theingly dressed case. In order to explain this effect,utual-dressing processes and constructive or destruc-
ive interference between the two coexisting FWM chan-els should be considered [14,15]. According to Eqs. (9)nd (10), if one only considers the dressing effect, theWM intensity should be further suppressed in the dou-ly dressed configuration. However, as can be seen fromig. 1, when five laser beams are all on, the DFWM signals1 and the NDFWM signal ks2 coexist in the experiment,nd these two FWM signals overlap in frequency and thengle between their propagation directions is very small.s mentioned above, if the incident beams are all linearlyolarized, the generated FWM signals are linearly polar-zed also, so constructive or destructive interference canccur in this system. Such interferences between twoWM processes in the two-level and three-level atomicystems can generate entangled photon pairs [16,17].onstructive or destructive interference can be controlledy the phase difference between the two FWM processes,hich can be varied by adjusting the detuning difference��=�1−�2� between the incident laser beams. By vary-
ng the detuning difference � from 0 to very certain val-es, the phase difference between the two FWM processes
(a)
ig. 5. (Color online) Variations of the dressed NDFWM signal ieam kc is modulated by the QWP. (b) Doubly dressed FWM signf the relative FWM intensity versus the rotation angle � of the
lters from in-phase to out-phase, so the interference canwitch back and forth between constructive and destruc-ive values [18–20]. In this case, the observed experimen-al data include two contributions: the dressing effect andhe interference effect. However, when the kc field is el-iptically or circularly polarized, the NDFWM signal ks2 islliptically polarized, but the DFWM signal ks1 is still lin-arly polarized. In this case, the doubly dressed effectlays a dominant role to further suppress the intensity ofhe generated FWM signal.
The dependences of the dressing effects on the polariza-ion of the probe field are shown in Fig. 6. Compared toig. 5(a), the linearly polarized signal (square points) isigher than the other curves even though it is alsoressed. Such opposite behaviors in changing the pump-ng field kc and the probe field kp can be accounted for byq. (9). In the denominator of the equation, the dressingeld Gd and the coupling field Gc are in summation form,hich is called sequential-dressing scheme, and Gd andp are in the nest-dressing scheme [13]. According to the
nteraction properties of the two dressing schemes, the se-uentially dressing Gc field controls the FWM process di-ectly, which can enhance the Gd dressing effect. Whenhe coupling field is linearly polarized, the Rabi frequency
c in the denominator is at its maximum, so the dressingffect is strongest [square points in Fig. 5(a)]. As Gp isested with Gd, it controls the FWM process only indi-ectly and it often suppresses the Gd dressing effect, so forhe linearly polarized kp field, the signal [square points inig. 6(b)] is higher than the other curves.
(b) (c)
ties versus �. (a) Singly dressed FWM signals when the couplingen the coupling beam kc is modulated by the QWP. (c) Variation
(a) (b)
ig. 6. (Color online) Dependence of the relative NDFWM sig-al intensity on � for three values of probe laser’s ellipticity. (a)nd (b) Pure and singly dressed FWM signals, respectively, of theascade three-level system as the probe beam is modulated byhe QWP.
ntensials whQWP.
tsHe�mttbde
tsambgpp�wedh
pDkpFstdcc
5Tptes
dctseFifitbcttfapsesddfwtdplao
ATFttFCFHg(
Fsza�
1718 J. Opt. Soc. Am. B/Vol. 26, No. 9 /September 2009 Wang et al.
Figure 7(a) depicts the dependence of the horizontal in-ensity of the FWM signal on � generated in the two-levelystem when the probe beam kp is modulated by theWP. In this case, the generated FWM signals are lin-
arly polarized ��=0�, with �Ex���xxxx cos 2� and �Ey��yxxy sin 2�. From Fig. 7(a), with a rotating HWP, theaximal values of the curves are shifted, which indicates
hat the polarization of the FWM signal changes withhat of the probe beam. Previous experiment [4] in ru-idium vapor has shown that the polarizations of theriving field and the signal wave are identical and collin-ar.
If the signal and the probe beams are polarized alonghe same direction, a maximum signal intensity is ob-erved when the two HWPs are rotated to be at the samengle [that is, �=�, the solid line in Fig. 7(c)]. The experi-ental results (the scattered points) in Fig. 7(c) present a
ig difference from the solid theoretical line, which sug-ests that the signal and the probe beams have differentolarization directions. According to Eqs. (2) and (3), theolarization of the FWM signal is dependent on the ratioyxxy /�xxxx. If �yxxy=�xxxx, the signal and the probe beamsould have polarization along the same direction. How-ver, our theoretical expressions [Eq. (4) and Eq. (5)], in-icate that �yxxy��xxxx, so the signal and the probe beamsave different polarization directions.Figure 7(b) presents the kc singly dressed (the square
oints) and kc and kc� doubly dressed (the dot points)FWM signals when the input beams (kd, kd�, kp, kc, and
c�) are all horizontally polarized. Compared with theure-FWM signal [the square points in Fig. 7(a)], the twoWM signals dressed by either kc or kc and kc� are bothignificantly suppressed and have similar suppressed in-ensities. As discussed above, for the kc and kc� doublyressed FWM process, the mutual-dressing effect andonstructive or destructive interference should also beonsidered simultaneously.
. CONCLUSIONhe polarization dependences of the DFWM and NDFWMrocesses, and their dressing effects in two-level andhree-level cascade-type atomic systems are investigatedxperimentally and theoretically. The effective nonlinearusceptibilities and the corresponding combinations of
(a)
ig. 7. (Color online) Variations of the DFWM signal intensitieeveral different rotation angles � of the HWP. (b) Singly dressedontally polarized. (c) The rotation angle � of the polarization anre the experimental results, and the solid line represents the ca�=��.
ifferent transition paths for various laser polarizationonfigurations are analyzed in detail. The polarizations ofhe generated FWM signals depend on both the input la-er polarizations and the excited nonlinear susceptibilitylements of the atomic systems. In the singly dressedWM processes, we found that the dependences of dress-
ng effects on the polarizations of the pumping and probeelds are very different. The FWM signal generated byhe linearly polarized pumping field kc is strongly dressedy the kd field while the FWM signal generated by the cir-ularly polarized kc is weakly dressed by the kd field, buthe opposite behavior was seen for changing the polariza-ion of the probe field. Such behavior is attributed to dif-erent dressing effects of the sequential-dressing schemend the nest-dressing scheme. For the doubly dressedrocess, the polarization dependence of the signal inten-ity is also investigated. It is found that when the two co-xisting FWM signals have the same polarization, theuppressed intensities of the singly dressed and doublyressed signals are similar. In this case, both the mutual-ressing effects and the constructive or destructive inter-erence should be considered simultaneously. However,hen the two FWM signals have different polarizations,
he doubly dressed effect plays a dominant role, so theoubly dressed signals are further suppressed. Studyingolarization dependence of the FWM processes in multi-evel atomic systems can be very important in optimizingnd controlling nonlinear optical processes, and for vari-us potential applications in nonlinear signal processing.
CKNOWLEDGMENTShis work was supported by the National Natural Scienceoundation of China (NSFC) (No. 60678005), the Founda-
ion for the Author of National Excellent Doctoral Disser-ation of China (No. 200339), the Specialized Researchund for the Doctoral Program of Higher Education ofhina (No. 20050698017), the Fok Ying-Tong Educationoundation for Young Teachers in the Institutions ofigher Education of China (No. 101061), and the Pro-
ram for New Century Excellent Talents in UniversityNCET-08–0431).
(b) (c)
us � (a) Pure FWM signal of the two-level system versus � foroubly dressed FWM signals when the input beams are all hori-when the maximum intensity is observed. The scattered pointsthe polarization analyzer and polarizer rotating the same angle
s versand d
alyzerse with
R 1
1
1
1
1
1
1
1
1
1
2
Wang et al. Vol. 26, No. 9 /September 2009 /J. Opt. Soc. Am. B 1719
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