Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry

Sixth-order correlation on Raman-enhanced polarization beatswith phase-conjugation geometry

Yanpeng Zhang *, Xun Hou, Keqing Lu, Hongcai Wu

Department of Electronic Science and Technology, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China

Received 18 May 2000; accepted 24 July 2000

Abstract

The correlation e�ects of sixth-order on the Raman-enhanced polarization beats (REPB) with phase-conjugation

geometry in carbon disul®de are investigated using chaotic and phase-di�usion models. Based on two types of models,

the cases that the pump beams have either narrow band or broadband linewidth are considered and it is found that the

beat signal oscillates not only temporally with a 50.5 fs period but also spatially. The overall accuracy of using REPB to

measure the resonant frequency of Raman-active vibrational mode is determined by the relaxation rates of the Raman

mode and the molecular-reorientational grating. Moreover, the temporal behavior of the beat signal depends on the

stochastic properties of the lasers and the decay rates of the Raman mode and the molecular-reorientational grating.

Di�erent stochastic models of the laser ®eld only a�ect the sixth- and fourth-order coherence function. Furthermore,

the di�erent roles of the phase ¯uctuation and amplitude ¯uctuation have been pointed out in the time domain. The

di�erence between the REPB and the fourth-order coherence on ultrafast modulation spectroscopy is discussed as well

from a physical viewpoint. Therefore, the sixth-order coherence function theory of chaotic ®eld is of vital importance in

REPB. Ó 2000 Elsevier Science B.V. All rights reserved.

PACS: 42.65.Dr; 42.65.Hw

1. Introduction

The atomic response to stochastic optical ®elds is now largely well understood. Methods exist to cal-culate the ®rst-order moments of the atomic density matrix elements for a wide variety of ®eld statistics,including phase-di�using ®elds, phase-di�using ®elds with colored noise, chaotic ®elds, real Gaussian ®elds,and phase jump ®elds [1±3]. When the laser ®eld is su�ciently intense that many photon interactions occur,the laser spectral bandwidth or spectral shape, obtained from the second-order correlation function, isinadequate to characterize the ®eld. Rather than using higher-order correlation functions explicitly, as inmost discussions of ®nite-bandwidth e�ects we employ soluble models for ¯uctuating light ®elds. Thechaotic ®eld model and the Brownian-motion phase-di�usion model are considered in parallel with a

1 October 2000

Optics Communications 184 (2000) 265±276

www.elsevier.com/locate/optcom

* Corresponding author. Tel.: +86-29-266-8643; fax: +86-29-323-7910.

E-mail address: [email protected] (Y. Zhang).

0030-4018/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S00 3 0-4 0 18 (0 0 )0 09 1 6- 0

discussion on Raman-enhanced polarization beats (REPB). We develop a uni®ed theory which involvessixth-order coherence-function to study the in¯uence of partial-coherence properties of pump beams onpolarization beats. Polarization beats, which originate from the interference between the macroscopicpolarizations, have attracted a lot of attention recently [4±6]. It is closely related to quantum beats, Ramanquantum beats [7], or fourth-order coherence on ultrafast modulation spectroscopy (FOCUMS) [8].

Raman-enhanced nondegenerate four-wave mixing (RENFWM) is a third-order nonlinear phenomenonwith phase-conjugation geometry [9]. It possesses the advantages of nonresonant background suppression,excellent spatial resolution, free choice of interaction volume, and simple optical alignment. They haveperformed time-delayed RENFWM with incoherent light to measure the vibrational dephasing time [10].They also found an enhancement of the ratio between the resonant and nonresonant RENFWM signalintensities as the time delay was increased when the laser had broadband linewidth [11]. One of the relevantproblems is the stationary four wave mixing (FWM) with incoherent light sources, which was proposed byMorita et al. [12] to achieve an ultrafast temporal resolution of relaxation processes. Since they assumedthat laser linewidth is much longer than transverse relaxation rate, their theory cannot be used to study thee�ect of the light bandwidth on the Bragg re¯ection signal. Asaka et al. [13] considered the ®nite linewidthe�ect. However, the constant background contribution has been ignored in their analysis. Our sixth-ordercoherence function theory includes both the ®nite light bandwidth e�ect and constant background con-tribution. This is of vital importance in the REPB.

In this paper, the correlation e�ects of sixth-order on the REPB with phase-conjugation geometry incarbon disul®de are investigated using chaotic and phase-di�usion models. Based on two types of models,the cases that the pump beams have either narrow band or broadband linewidth are considered and it isfound that the beat signal oscillates not only temporally but also spatially. The overall accuracy of usingREPB to measure the resonant frequency of Raman-active vibrational mode is determined by the relax-ation rates of the Raman mode and the molecular-reorientational grating. Moreover, the temporal be-havior of the beat signal depends on the stochastic properties of the lasers and the decay rates of the Ramanmode and the molecular-reorientational grating. Di�erent stochastic models of the laser ®eld only a�ect thesixth- and fourth-order coherence function. Furthermore, the di�erent roles of the phase ¯uctuation andamplitude ¯uctuation have been pointed out in the time domain. The di�erence between the REPB and theFOCUMS is discussed as well from a physical viewpoint. Therefore, the sixth-order coherence functiontheory of chaotic ®eld is of vital importance in REPB.

2. Basic theory

REPB is a third-order nonlinear polarization beat phenomenon. The basic geometry is shown in Fig. 1.Beams 1 and 2 consist of two frequency components, x1 and x3, a small angle exits between them. Beam 3

Fig. 1. Schematic diagram of the geometry of REPB.

266 Y. Zhang et al. / Optics Communications 184 (2000) 265±276

with frequency x3 is almost propagating along the opposite direction of beam 1. In a Kerr medium, thenonlinear interaction of beams 1 and 2 with the medium gives rise to two molecular-reorientationalgratings, i.e., x1 and x3 will induce their own static gratings G1 and G2, respectively. The FWM signal isthe result of the di�raction of beam 3 by G1 and G2.

Now, if x1 ÿ x3j j is near the Raman resonant frequency, XR, a large angle moving grating and two smallangle moving gratings formed by the interference of beams 2, 3 and beams 1, 2, respectively, will excite theRaman-active vibrational mode of the medium and enhance the FWM signal. The beat signal (beam 4) isalong the opposite direction of beam 2 approximately.

The complex electric ®elds of beam 1, EP1, and beam 2, EP2

, can be written as

EP1� E1�~r; t� � E2�~r; t� � A1�~r; t�exp�ÿix1t� � A2�~r; t�exp�ÿix3t�� e1u1�t�exp �i�~k1 �~r ÿ x1t�� e2u3�t�exp �i�~k2 �~r ÿ x3t��; �1�

EP2� E01�~r; t� � E02�~r; t� � A01�~r; t�exp�ÿix1t� � A02�~r; t�exp�ÿix3t�� e01u1�t ÿ s�exp �i�~k01 �~r ÿ x1t � x1s�� e02u3�t ÿ s�exp �i�~k02 �~r ÿ x3t � x3s��: �2�

Here ei,~ki (e0i,~k0i) are the constant ®eld amplitude and the wave vector of frequency components x1 and x3

in beam 1 (beam 2), respectively. ui�t� is a dimensionless statistical factor that contains phase and amplitude¯uctuations. We assume that the x1 (x3) component of EP1

and EP2comes from a single laser source and s is

the time delay of beam 2 with respect to beam 1. On the other hand, the complex electric ®elds of beam 3can be written as

EP3� A3�~r; t�exp�ÿix3t� � e3u3�t�exp �i�~k3 �~r ÿ x3t��: �3�

Here, x3, e3, and ~k3 are the frequency, the ®eld amplitude and the wave vector of the ®eld, respectively.The order parameters Q1 and Q2 of two static gratings induced by beams 1 and 2 satisfy the following

equations [9]:

dQ1

dt� cQ1 � vcE1�~r; t��E01�~r; t��; �4�

dQ2

dt� cQ2 � vcE2�~r; t��E02�~r; t��: �5�

Here c and v are the relaxation rate and the nonlinear susceptibility of the two static gratings, respectively.We consider a large angle moving grating (the order parameter QR1) and two small angle moving

gratings (the order parameters QR2, QR3) formed by the interference of beams 2, 3 and beams 1, 2, re-spectively. The order parameters QR1, QR2, QR3 satisfy the following equations:

dQR1

dt� �cR ÿ iD�QR1 � iaR

4�h�A01�~r; t��A3�~r; t�; �6�

dQR2


4�hA1�~r; t��A02�~r; t��; �7�

dQR3


4�h�A01�~r; t��A2�~r; t�: �8�

Here D � x1 ÿ x3j j ÿ XR; XR and cR are the resonant frequency and the relaxation rate of the Ramanmode, respectively. aR is a parameter denoting the strength of the Raman interaction.

Y. Zhang et al. / Optics Communications 184 (2000) 265±276 267

The induced polarizations which are responsible for the FWM signals are

P1 � Q1�~r; t�E3�~r; t�

� vce1�e01��e3 expfi��~k1 ÿ~k01 �~k3� �~r ÿ x3t ÿ x1s�gZ 1

0

u1�t ÿ t0�u�1�t ÿ t0 ÿ s�u3�t�exp�ÿct0�dt0;

�9�P2 � Q2�~r; t�E3�~r; t�

� vce2�e02��e3 expfi��~k2 ÿ~k02 �~k3� �~r ÿ x3t ÿ x3s�gZ 1

0

u3�t ÿ t0�u�3�t ÿ t0 ÿ s�u3�t�exp�ÿct0�dt0;

�10�

PR1 � 1

2NaRQR1�~r; t�E1�~r; t�exp �i�x1 ÿ x3�t ÿ ix1s�

� ivRcRe1�e01��e3 expfi��~k1 ÿ~k01 �~k3� �~r ÿ x3t ÿ x1s�g�Z 1

0

u1�t�u�1�t ÿ t0 ÿ s�u3�t ÿ t0�exp �ÿ�cR ÿ iD��dt0; �11�

PR2 � 1


� ivRcRe1�e02��e3 expfi��~k1 ÿ~k02 �~k3� �~r ÿ x1t ÿ x3s�g�Z 1

0

u1�t ÿ t0�u�3�t ÿ t0 ÿ s�u3�t�exp �ÿ�cR ÿ iD��dt0; �12�

PR3 � 1


� ivRcR�e01��e2e3 expfi��~k2 ÿ~k01 �~k3� �~r ÿ �2x3 ÿ x1�t ÿ x1s�g�Z 1

0

u�1�t ÿ t0 ÿ s�u3�t ÿ t0�u3�t�exp �ÿ�cR ÿ iD��dt0; �13�

with vR � Na2R=8�hcR and N, the density of molecules.

3. The chaotic ®eld

We have the total polarization P �3� � P1 � P2 � PR1 � PR2 � PR3. The FWM signal is proportional to theaverage of the absolute square of P �3� over the random variable of the stochastic process I�D; s� / h P �3�

�� 2i,which involves sixth-, fourth- and second-order coherence function of ui�t� in phase conjugation geometry.The FWM signal intensity in DeBeerÕs self-di�raction geometry is also related to the sixth-order coherencefunction of the incident ®elds [14]. We ®rst assumed that the laser sources are chaotic ®elds. A chaotic ®eld,which is used to describe a multimode laser source, is characterized by the ¯uctuation of both the amplitudeand the phase of the ®eld. ui�t� has Gaussian statistics with its sixth- and fourth-order coherence functionsatisfying [15]

hui�t1�ui�t2�ui�t3�u�i �t4�u�i �t5�u�i �t6�i � hui�t1�u�i �t4�ihui�t2�ui�t3�u�i �t5�u�i �t6�i � hui�t1�u�i �t5�i� hui�t2�ui�t3�u�i �t4�u�i �t6�i � hui�t1�u�i �t6�i� hui�t2�ui�t3�u�i �t4�u�i �t5�i; i � 1; 3 �14�


and

hui�t1�ui�t2�u�i �t3�u�i �t4�i � hui�t1�u�i �t3�ihui�t2�u�i �t4�i � hui�t1�u�i �t4�ihui�t2�u�i �t3�i: �15�Furthermore assuming that laser sources have Lorentzian line shape, then we have

hui t1� �u�i t2� �i � exp� ÿ ai t1j ÿ t2j�: �16�Here ai � 1

2dxi, dxi is the linewidth of the laser with frequency, xi.

We ®rst consider the situation when the laser linewidths of beams 1, 2 and 3 are broadband (i.e.,a1; a3 � c; cR). In this limit, after performing the tedious integration we obtain for (i) s > 0,

I�D; s� / �1� g21 � g2

2�v2

RcR�a1 � a3��a1 � a3�2 � D2

ÿ vRvcRcD��5a1 � a3��a1 � a3� � D2�

a1��a1 � a3�2 � D2�2 � v2c2a1a3

�a1g21g

22 � a3�

� v2Rc2

R�a1 � a3�a1��a1 � a3�2 � D2�

(ÿ 2vRvcRD

�a1 � a3�2 � D2� v2

)exp�ÿ2a1 sj j� � g2

2

� v2Rc2

R�a1 � a3�a3��a1 � a3�2 � D2�

(� v2g2

1

)exp�ÿ2a3 sj j� � 4g1g2 exp �ÿ�a1 � a3� sj j�

� v2

2

"(ÿ vvRcRD

2��a1 � a3�2 � D2�

#cos �D~k �~r ÿ �x1 ÿ x3�s�

ÿ vvRcR�a1 � a3�2��a1 � a3�2 � D2� sin �D~k �~r ÿ �x1 ÿ x3�s�

); �17�

and for (ii) s < 0,

I�D; s� / �1� g21 � g2

2�v2

RcR�a1 � a3��a1 � a3�2 � D2

ÿ vRvcRcD��5a1 � a3��a1 � a3� � D2�

a1��a1 � a3�2 � D2�2 � v2c2a1a3

�a1g21g

22 � a3�

� 3v2g21g

22c

2

2a23

exp�ÿ2c sj j� � 2v2Rc2

R

a1g22 � a3

�a1 � a3�2 � D2

(� �1� g2

2��a1 � a3�2 ÿ D2��a1 � a3�2 � D2�2

)exp�ÿ2cR sj j�

� exp �ÿ�cR � c� sj j� 4g1g2vvRccR

a3��a1 � a3�2 � D2�2 f�a1 � a3��2cR � c�Dcos �D~k �~r ÿ �x1 ÿ x3�s� D sj j�

ÿ ��a1 � 2a3��a1 � a3�2 � a1D2� sin �D~k �~r ÿ �x1 ÿ x3�s� D sj j�g: �18�

Here, g1 � e2=e1, g2 � e02=e01 and D~k � �~k1 ÿ~k01� ÿ �~k2 ÿ~k02�.

Eq. (17) indicate that when s > 0, the temporal behavior of the beat signal intensity re¯ects mainly thecharacteristic of the lasers, i.e., the frequency x3 ÿ x1 and the damping rate a1 � a3 of the modulation aredetermined by the incident laser beams. If we employ REPB to measure the modulation frequencyxd � x3 ÿ x1, the accuracy can be improved by measuring as many cycles of the modulation as possible.Since the amplitude of the modulation decays with a time constant �a1 � a3�ÿ1

as sj j increases, the maxi-mum sj j should equal approximately 2�a1 � a3�ÿ1

. We obtain the theoretical limit of the uncertainty of themodulation frequency measurement, Dxd, which is Dxd � p�a1 � a3�, i.e., in the modulation frequencymeasurement the theoretical limit of the accuracy is related to the decay time constant of the beat signalmodulation amplitude. In this case, the precision of using REPB to measure the resonant frequencies of theRaman mode is determined by how well x3 ÿ x1 can be tuned to XR. On the other hand, when s < 0,Eq. (18) indicate that beat signal modulates with a frequency �x3 ÿ x1� ÿ D � XR and a damping rate


cR � c as s is varied. We can obtain the resonant frequencies of the Raman vibrational mode with anaccuracy given by p�cR � c� approximately, which is determined by a material property.

Eqs. (17) and (18) indicate that beat signal oscillates not only temporally but also spatially with a period2p=Dk along the direction D~k, which is almost perpendicular to the propagation direction of the beat signal.Physically, the polarization-beat model assumes that the pump beams are plane waves. Therefore FWMsignals, which propagate along~k1 ÿ~k01 �~k3 and~k2 ÿ~k02 �~k3, respectively, are plane waves also. Since FWMsignals propagate along slightly di�erent direction, the interference between them leads to the spatial os-cillation.

We then consider the situation when the laser linewidths of beams 1, 2 and 3 are narrow band (i.e.,a1; a3 � c; cR). In this limit, after performing the tedious integration we obtain

I�D; s� / �1� 2g21 � g2

2�v2

Rc2R

c2R � D2

ÿ 2vRvcRD

c2R � D2

� v2�1� 2g21g

22� �

v2Rc2

R

c2R � D2

ÿ 2vRvcRD

c2R � D2

� v2

!

� exp�ÿ2a1 sj j� � g22

v2Rc2

R

c2R � D2

� 4g2

1v2

!exp�ÿ2a3 sj j� � 4g1g2 exp �ÿ�a1 � a3� sj j� v2

(ÿ vvRcRD

c2R � D2

!

� cos �D~k �~r ÿ �x1 ÿ x3�s� ÿ vvRcR

cR�cR � c�2 � D2�cR � 2c��c2

R � D2��cR � c�2 � D2� sin �D~k �~r ÿ �x1 ÿ x3�s�): �19�

This case when the pump beams have a narrow-band linewidth is similar to Eq. (17).

4. The Raman photon-echo

It is interesting to understand the underlying physics in REPB with incoherent lights. Much attentionhas been paid to the study of various ultrafast phenomena by using incoherent light sources recently[12,13,16]. The REPB with incoherent lights is related to the three-pulse stimulated Raman photon echoes.We now consider the case when the pump beams have a broadband linewidth, then

exp�ÿai t1j ÿ t2j� � 2

aid t1� ÿ t2� i � 1; 3: �20�

When we substitute Eqs. (14)±(16) and Eq. (20) into I�D; s� / h P �3�� 2i, we obtain, for (i) s > 0,

I�D; s� / v2RcR

�1� g21 � g2

2�a1 � a3

�� g2

1cR

a1�cR � a3��ÿ 4vRvcRcD

a1��cR � c� a1 � a3�2 � D2� � v2c1

2a1

�� g2

1g22�3c� a3�

2a3�c� a3��

� v2 exp�ÿ2a1 sj j� � g21g

22v

2 c3 � 5c2a3 � 5ca23 � 2a2

3

a3�c� a3��c� 2a3� exp�ÿ2a3 sj j� � 2g1g2 exp �ÿ�a1 � a3� sj j�

� 2v2�c� a3�c� 2a3

"(ÿ vvRcRcD

c�cR � a1 � a3�2 � cD2ÿ 2vvRcRcD

a3�cR � c� a1 � a3�2 � a3D2

#

� cos �D~k �~r ÿ �x1 ÿ x3�s� ÿ vvRcRc�cR � a1 � a3�c�cR � a1 � a3�2 � cD2

"� 2vvRcRc�c� cR � a1 � a3�

a3�cR � c� a1 � a3�2 � a3D2

#

� sin �D~k �~r ÿ �x1 ÿ x3�s�); �21�

and for (ii) s < 0,


I�D; s� / v2RcR

�1� g21 � g2

2�a1 � a3

�� g2

1cR

a1�cR � a3��ÿ 4vRvcRcD

a1��cR � c� a1 � a3�2 � D2� � v2c1

2a1

�� g2

1g22�3c� a3�

2a3�c� a3��

� 2v2g21g

22c

2

a23

exp�ÿ2c sj j� � 2v2Rc2

R�1� g22�

a1a3

exp�ÿ2cR sj j� � exp �ÿ�cR � c� sj j�

� 8g1g2vvRccR�a1 � a3�a3��cR � cÿ a1 ÿ a3�2 � D2��cR � c� a1 � a3�2 � D2� f2�cR � c�Dcos �D~k �~r ÿ �x1 ÿ x3�s

� D sj j� � ��cR � c�2 ÿ �a1 � a3�2 ÿ D2� sin �D~k �~r ÿ �x1 ÿ x3�s� D sj j�g: �22�

Eqs. (21) and (22) are analogous to Eqs. (17) and (18), respectively. The total polarization (see Eqs. (9)±(13)), which involves the integration of t0 from 0 to1, is the accumulation of the polarization induced at adi�erent time. We consider the case that the pump beams 1, 2 and 3 have a broadband linewidth so that itcan be modeled as a sequence of short, phase-incoherent subpulses of duration sc, where sc is the lasercoherence time [12]. Although grating can be induced by any pair of subpulses in beams 1, 2, 3 only thosepairs that are phase correlated in beams 1 and 2 give rise to the s dependence of the FWM signal. Therefore,the requirement for the existence of a s-dependent FWM signal for s > 0 is that the phase correlatedsubpulses in beams 1 and 2 are overlapped temporally. Since beams 1 and 2 are mutually coherent, thetemporal behavior of the REPB signal for s > 0 should coincide with the case when the pump beams arenearly monochromatic.

5. The phase-di�usion ®eld

We have assumed that the laser sources are chaotic ®eld in the above calculation. A chaotic ®eld, whichis used to describe a multimode laser source, is characterized by the ¯uctuation of both the amplitude andthe phase of the ®eld. Another commonly used stochastic model is phase-di�usion model, which is used todescribe an amplitude-stabilized laser source. This model assumes that the amplitude of the laser ®eld is aconstant, while its phase ¯uctuates as random process. If the lasers have Lorentzian line shape, the sixth-and fourth-order coherence function is [15]

hui�t1�ui�t2�ui�t3�u�i �t4�u�i �t5�u�i �t6�i � exp �ÿai� t1j ÿ t4j � t1j ÿ t5j � t1j ÿ t6j � t2j ÿ t4j � t2j ÿ t5j� t2j ÿ t6j � t3j ÿ t4j � t3j ÿ t5j � t3j ÿ t6j��exp �ai� t1j ÿ t2j� t1j ÿ t3j � t2j ÿ t3j � t4j ÿ t5j � t4j ÿ t6j � t5j ÿ t6j��; �23�

and

hui�t1�ui�t2�u�i �t3�u�i �t4�i � exp �ÿai� t1j ÿ t3j � t1j ÿ t4j � t2j ÿ t3j � t2j ÿ t4j�� exp �ai� t1j ÿ t2j � t3j ÿ t4j��: �24�

We ®rst consider the situation when the laser linewidths of beams 1, 2 and 3 are broadband (i.e.,a1; a3 � c; cR). In this limit, after substituting Eqs. (16), (23) and (24) into I�D; s� / h P �3�

�� 2i, we obtain, for(i) s > 0,


I�D; s� / �1� g21 � g2

2�v2

RcR�a1 � a3��a1 � a3�2 � D2

ÿ vRvcRcD��5a1 � a3��a1 � a3� � D2�

2a1��a1 � a3�2 � D2�2 � v2c2a1a3

�a1g21g

22 � a3�

� 2g1g2 exp �ÿ�a1 � a3� sj j� v2

"(ÿ vvRcRD

�a1 � a3�2 � D2

#cos �D~k �~r ÿ �x1 ÿ x3�s�

ÿ vvRcR�a1 � a3��a1 � a3�2 � D2

sin �D~k �~r ÿ �x1 ÿ x3�s�); �25�

and for (ii) s < 0,

I�D; s� / �1� g21 � g2

2�v2

RcR�a1 � a3��a1 � a3�2 � D2

ÿ vRvcRcD��5a1 � a3��a1 � a3� � D2�

2a1��a1 � a3�2 � D2�2

� v2c2a1a3

�a1g21g

22 � a3� ÿ exp �ÿ�cR � c� sj j�

� 4g1g2vvRccR��a1 � 2a3��a1 � a3�2 � a1D2�

a3��a1 � a3�2 � D2� sin �D~k �~r ÿ �x1 ÿ x3�s� D sj j�; �26�

We then consider the situation when the laser linewidths of beams 1, 2 and 3 are narrow band (i.e.,a1; a3 � c; cR). In this limit, after performing the tedious integration we obtain

I�D; s� / �1� g21 � g2

2�v2

Rc2R

c2R � D2

ÿ vRvcRD

c2R � D2

� v2�1� g21g

22� � 2g1g2 exp �ÿ�a1 � a3� sj j� v2

(ÿ vvRcRD

c2R � D2

!

� cos �D~k �~r ÿ �x1 ÿ x3�s� ÿ vvRc2R

c2R � D2

sin �D~k �~r ÿ �x1 ÿ x3�s�): �27�

Eq. (27) is analogous to Eq. (25), which indicate that the temporal behavior of the beat signal intensityre¯ects mainly the characteristic of the lasers.

After that, based on phase-di�usion model, we consider the Raman photon-echo when the pump beamshave a broadband linewidth. Substituting Eqs. (16), (20), (23) and (24) into I�D; s� / h P �3�

�� 2i, we obtain,for (i) s > 0,

I�D; s� / �1� g21 � g2

2�v2

RcR

a1 � a3

ÿ 4vRvcRcD

a1��cR � c� a1 � a3�2 � D2� �v2c

2a1a3

�a1g21g

22 � a3�

� 2g1g2v2 exp �ÿ�a1 � a3� sj j�cos �D~k �~r ÿ �x1 ÿ x3�s�; �28�

and for (ii) s < 0,

I�D; s� / �1� g21 � g2

2�v2

RcR

a1 � a3

ÿ 4vRvcRcD

a1��cR � c� a1 � a3�2 � D2� �v2c

2a1a3

�a1g21g

22 � a3�

ÿ exp �ÿ�cR � c� sj j� 8g1g2vvRccR

a3�a1 � a3� sin �D~k �~r ÿ �x1 ÿ x3�s� D sj j�; �29�

Eqs. (28) and (29) are analogous to Eqs. (25) and (26), respectively.Di�erent stochastic models of the laser ®eld a�ect only the sixth- and fourth-order coherence function.

The constant terms in relations (17) and (18), which are independent of the relative time-delay betweenbeam 1 and beam 2, originate from the phase ¯uctuation of the chaotic ®eld. The modulation terms in-cluding these factors exp �ÿ�a1 � a3� sj j�, or exp �ÿ�cR � c� sj j� also originate from the phase ¯uctuation ofthe chaotic ®eld. In contrast, the decay terms including these factors exp�ÿa1 sj j�, exp�ÿa3 sj j�,


exp�ÿcR sj j�, or exp�ÿcR sj j� originate from the amplitude ¯uctuation of the chaotic ®eld. On the otherhand, REPB is di�erent for s > 0 and s < 0. However, as sj j ! 1, Eq. (17) is identical to Eq. (18).Physically, when sj j ! 1, beams 1 and 2 are mutually incoherent, therefore whether s is positive ornegative does not a�ect the REPB.

Eqs. (25) and (26) are di�erent from the result based on a chaotic model. The constant terms in Eqs. (25)and (26), which are independent of s, originate from the phase ¯uctuation of the phase-di�usion ®eld. Themodulation terms including these factors exp �ÿ�a1 � a3� sj j�, or exp �ÿ�cR � c� sj j� also originate from thephase ¯uctuation of the phase-di�usion ®eld. But Eqs. (25) and (26) are short of the decay terms includingthese factors exp�ÿa1 sj j�, exp�ÿa3 sj j�; exp�ÿcR sj j�, or exp�ÿcR sj j�. The drastic di�erence of the resultsalso exists in the FOCUMS when these two models are employed. Physically, the chaotic ®eld has theproperty of photon bunching, which can a�ect any multiphoton process when the higher-order correlationfunction of the ®eld plays an important role.

The main purpose of the above discussion is that we reveal an important fact that the amplitude ¯uc-tuation of the laser ®eld plays a critical role in the temporal behavior of REPB signal. Furthermore, thedi�erent roles of the phase ¯uctuation and amplitude ¯uctuation have been pointed out in the time domain.This is quite di�erent from the time delayed FWM with incoherent light in a two-level system [12], where T2

of the system can be deduced. For the latter case, the phase ¯uctuation of the light ®eld is crucial. TheREPB is also di�erent from the FOCUMS. The phase ¯uctuation of the light ®eld is crucial in the FO-CUMS. On the other hand, because of hui�t�i � 0 and hu�i �t�i � 0, the absolute square of the stochasticaverage of the polarization jhP �3�ij2, which involves second-order coherence function of ui�t�, cannot beused to describe the temporal behavior of the REPB [16,17]. Therefore, the sixth-order coherence functiontheory of chaotic ®eld is of vital importance in REPB. We present experimental results for the materialresponse in REPB with phase-conjugation geometry using chaotic ®elds. In the experiment, it is moredi�cult to get a clear picture of physical origins of the e�ects in each type of ¯uctuating ®eld.

6. Experiment and result

We are interested in the temporal behavior of the REPB signal intensity with ®xed frequency detuning.The carbon disul®de (CS2) with 655.7 cmÿ1 vibrational mode was contained in a sample cell with thickness9 mm. The second harmonic of a Quanta-Ray YAG laser was used to pump two dye lasers (DL1 and DL2).DL1 and DL2 had linewidth 0.05 nm, pulse width 5 ns and output energy 1 mJ. The ®rst dye laser DL1 hadwavelength 589 nm. The wavelength of DL2 was approximately 567 nm and could be scanned by acomputer-controlled stepping motor. A beam splitter was used to combine the x1 and x3 componentsderived from DL1 and DL2, respectively, for beams 1 and 2, which intersected in the sample with a smallangle h � 1:5°. The relative time s between beams 1 and 2 could be varied. Beam 3, which propagated alongthe direction opposite to that of beam 1, was derived from DL2. All the incident beams were linearlypolarized in the same direction. The beat signal had the same polarization as the incident beams, propa-gated along a direction almost opposite to that of beam 2. It was detected by a photodiode.

We ®rst performed a nondegenerate FWM experiment in which beams 1 and 2 only consisted of x1

frequency component. We measured the RENFWM spectrum with ®xed time delay (at s � 0 ps) byscanning x3. Our results are shown in Fig. 2, which showed an asymmetrical resonant pro®le due to theinterference between the Raman resonant term and the nonresonant background originating solely fromthe molecular-reorientational grating [9]. From this spectrum x3 was tuned to the resonant frequency (i.e.,D � x1 ÿ x3j j ÿ XR � 0). We then performed the REPB experiment with ®xed frequency detuning bymeasuring the beat signal intensity as a function of the relative time delay when beams 1 and 2 consist ofboth frequencies x1 and x3. Fig. 3 presents the result of the beat experiment. The beat signal intensitymodulates sinusoidally with period 50:5 fs. The modulation frequency can be obtained more directly by


making a Fourier transformation of the REPB data. Fig. 4 presents the Fourier spectrum of the REPB datain which s is varied for a range of 5 ps. Then we obtain the modulation frequency 1:24� 1014 sÿ1 corre-sponding to the resonant frequency of the Raman vibrational mode. Fig. 5 presents the theoretical curve ofthe normalized polarization beat signal intensity versus transverse distance, r, with ®xed time delay andfrequency detuning. Parameters are a1 � 1:36� 1011 sÿ1, a3 � 1:47� 1011 sÿ1, s � 0 ps, h � 2:62� 10ÿ2

rad, k1 � 589 nm, k3 � 567 nm, v=vR � 0:35, cR � 5� 1010 sÿ1, c � 5� 1011 sÿ1, g1 � g2 � 1, and D � 0.The beat signal oscillates spatially with a period 2p=Dk � k1k3= k1 ÿ k3j jh � 0:58 mm. We present a theoryand experimental results for the material response in Raman-enhanced polarization beats with phase-conjugation geometry using chaotic ®elds.

In conclusion, we have employed a sixth-order coherence function theory to study the REPB with phase-conjugation geometry. It is found that, the beat signal oscillates not only temporally with a period of 50.5 fsbut also spatially with a period of 2p=Dk. The measurement accuracy for the resonant frequency of Raman-active vibrational mode is related to the decay time constant of the beat signal modulation amplitude.Moreover, the temporal behavior of the beat signal depends on the stochastic properties of the lasers andthe decay rates of the Raman mode and the molecular-reorientational grating. We also show that thetemporal behavior of the REPB signal is di�erent drastically if the laser sources are described by a

0.0 0.5 1.0 1.5 2.03

6

9

Sig

nalI

nte

nsi

ty(a

rb.units

)

Relative Time Delay (ps)

Fig. 3. Experiment result of the beat signal intensity versus relative time delay.

-10 -5 0 5 100.0

0.5

1.0

Sig

nalI

nte

nsi

ty(a

rb.units

)

∆ (cm-1)

Fig. 2. RENFWM spectrum at s � 0 ps.


Brownian phase-di�usion model. Di�erent stochastic models of the laser ®eld only a�ect the sixth- andfourth-order coherence function. Finally, the di�erence between the REPB and the FOCUMS is discussedas well from a physical viewpoint.

Acknowledgements

This work was supported by the Chinese National Nature Sciences Foundation (grant no. 69978019).

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Fig. 5. Theoretical curve of the normalized polarization beat signal intensity versus transverse distance, r. Parameters are

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Documents

Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry