15 September 2001

Optics Communications 197 (2001) 73±81

www.elsevier.com/locate/optcom

Femtosecond-time-resolved highly selective
molecular-vibration excitation using a novel vibrationallysynchronized pumping technique with frequency
dierence resonanceRyuji Morita a,*, Mikio Yamashita a, Akira Suguro b, Hidemi Shigekawa caDepartment of Applied Physics, Hokkaido University, and Core Research for Evolutional Science and Technology,
Japan Science and Technology Corporation, Kita-13, Nishi-8, Kita-ku, Sapporo 060-8628, Japanb
Display Development, Technical Research Laboratory, Citizen Watch Co. Ltd., 840 Shimotomi, Tokorozawa 359-0001, Japanc
Institute of Applied Physics, University of Tsukuba, and Core Research for Evolutional Science and Technology,
Japan Science and Technology Corporation, 1-1-1 Tennodai, Tsukuba 305-8573, Japan
Received 9 August 2000; received in revised form 25 June 2001; accepted 2 July 2001

Abstract
Selective stimulated Raman scattering using a single-color femtosecond N-pulse train of ®nite pulse duration leads toa low excitation eciency in the high frequency region. This low eciency of Raman mode excitation is due to spectral
modulation which stems from the ®nite duration of the pulses in each train. To overcome the low eciency of excitationin the high frequency region, a vibrationally synchronized pumping technique with frequency dierence resonance is
proposed. In this technique, both the pulse-repetition rate and the center frequency dierence of carrier-phase-locked
two-color beams consisting of a femtosecond N-pulse train are chosen to be resonant with the frequency of a speci®cvibrational or rotational mode of interest. In addition, a formula describing the optimum pulse number N for ecient
selective excitation is derived. Ó 2001 Elsevier Science B.V. All rights reserved.
PACS: 42.65.)k; 42.65.Re; 42.65.Dr
Keywords: Stimulated Raman scattering; Selective excitation; Vibrationally synchronized pumping; Spatial light amplitude modulation;
Spatial light phase modulation; Pulse shaping; Femtosecond optical pulse; Coherent control

1. Introduction
The recent development of the ultrafast technology has enabled us to generate optical pulses
whose duration is shorter than the molecular vi*

Corresponding author. Fax: +81-11-706-6706.
E-mail address: morita@eng.hokudai.ac.jp (R. Morita).

bration or rotation period. Typical values of thevibrational or rotational period corresponding to
the Stokes shift in many Raman scattering media
are of the order of 10 13 ±10 14 s. This means thatwith the use of laser pulses whose duration 
10±100 fs, many Raman modes can be excitedin a time-resolved stimulated Raman scattering(SRS) experiment simultaneously [1±3], but not

0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 4 0 9 - 2

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R. Morita et al. / Optics Communications 197 (2001) 73±81

selectively. On the other hand, selective excitationof a speci®c Raman mode or quantum control ofmolecular motion with a high time resolution has
been carried out using femtosecond shaped opticalpulse sequences (consisting of N-pulse trains) whose
repetition rate are equal to their vibrational orrotational frequency [4,5]. In addition, an analysisof this selective excitation with the N-pulse train
has been done only under the assumption that eachconstituent pulse is a d-function in time [6,7]. However, a more realistic analysis of the selective excitation with the N-pulse train consisting of pulses
with a ®nite duration is essential for proper analysis of experimental results.
In this paper, we show that the practicalanalysis of selective excitation of molecular vibration or rotation using an N-pulse train with ®nite
pulse duration leads to a low selectivity or eciency due to the envelope spectral modulation
arising from the ®nite pulse duration. In order toovercome this problem we propose an ultrafast vibrationally synchronized pumping technique withfrequency dierence resonance where not only thepulse repetition rate (T 1 ) but also the center frequency dierence ( x1 x2 =2p) between carrier
phase-locked two-color femtosecond-pulse-train
beams are tuned to the corresponding resonancefrequency (xR;int =2p) of a speci®c vibrational orrotational mode. With this detailed analysis that
accounts for the ®nite pulse duration, it is demonstrated that this novel technique allows ecient
selective excitation.

2. Proposal and fundamental analysis for a femtosecond vibrationally synchronized pumping technique with frequency dierence resonance2.1. Proposal
Here we propose a vibrationally synchronizedpumping technique with frequency dierence resonance for ecient selective molecular vibrational orrotational excitation that can be applied up to thehigh frequency region. The basic concept of thevibrationally synchronized pumping technique withfrequency dierence resonance is that both thecenter frequency dierence and the repetition ratesof two-color N-pulse train beams can be simultaneously tuned to the resonance of a speci®c molecular vibrational or rotational frequency of interest.
For example, this vibrationally synchronized pumping technique with frequency dierence resonancecan be implemented as shown in Fig. 1. After theampli®ed femtosecond optical pulses are spectrally
broadened by self-phase or induced-phase modulation [8±10], they are guided into a two-dimensional (2D) pulse shaper consisting of a 4f-system inwhich a 2D spatial light amplitude- and phasemodulator are located in the Fourier-plane. By useof this 2D-pulse shaper, simultaneous two-color
pulse shaping is carried out with an arbitrary center
frequency (wavelength) selection (x1 and x2 ) andwith arbitrary control of the pulse intervals T (pulse
repetition rates: T 1 ). Thus we generate two-color
shaped N-pulse train beams whose center frequen-Fig. 1. Schematic diagram of the vibrationally synchronized pumping technique with frequency dierence resonance.

R. Morita et al. / Optics Communications 197 (2001) 73±81

cies are x1 and x2 , with the same repetition rates
T 1 as the pump beams (pump1 and pump2) which
are focused into the medium under study. When we
adjust the center frequency dierence of thesebeams x1 x2 to a speci®c molecular vibrationalor rotational frequency of interest xR;int and thepulse repetition rates T 1 to xR;int =2p, we canachieve vibrationally synchronized pumping excitation with frequency dierence resonance. In thisvibrationally synchronized pumping technique withfrequency dierence resonance, the pulse-repetitionrate synchronization with vibration leads to selective
excitation and the center-frequency dierence resonance brings Raman-signal enhancement of theselective excitation. That is, the xR;int -vibrational orrotational mode is selectively excited with a relatively high signal intensity. When the probe pulse
having time delay sD between the pump1 andpump2 is also focused into the medium, we canobserve behavior characteristic of temporally SRSof a selected Raman mode. A local oscillator pulse
LO, derived from a portion of the signal before
detection, is used for optical heterodyne detection(OHD). We term / the relative phase of the pump2
with respect to the pump1 or probe and w the phaseof the local oscillator with respect to the probe. The
phase dierence between the pump1 and the probe
is arbitrary. When one adjusts / and w to the appropriate values, the real and imaginary parts of theRaman response can be obtained.

75

Here, u t is the unit step function, represented by0; if t < 0;
u t 
2
1; if t P 0:
In the case where pump1 and pump2 have thesame pulse-train envelopes and T0  T , as shown
in Fig. 2(a), the factor FfEpu1 tEpu2 tg in Eq. (1)i N 1xT =2
has the form sin N xT =2e
FfjF t=2
T0 j g=W sin xT =2 (see Appendix B). From this,it should be noted that, because of the ®nite pulse
duration, the spectrum of an N-pulse train ismodulated by a factor FfjF t=T0 j2 g. This corresponds to the spectrum of the constituent pulse,
unlike in the delta-function pulse train case. The

2.2. Fundamental analysis
Fundamental analysis of selective excitation
using an N-pulse train with ®nite duration in thetwo-color pump case as well as the one-color
pump case is described in Appendix A. From
Appendix A, the optical-heterodyne-detected coherent Raman scattering signal is expressed by
Shetero ks ; sD  /"Z
1
dt Epr t
2xs Im
1

sD 
(

2

Z 1

dx ixt
e
1 2p

n
o

 F Epu1 tEpu2 t  F expi x1)#
X
rj t expi /  w :j

x2 tu t
1

Fig. 2. (a) An example of an N-pulse train (F t=T0  sech t=T0 , tp  1:763T0  30 fs, N  6 and T  333:56 fs). (b)Its spectrum is represented by the solid line. The dashed line
corresponds to a spectrum of a tp  30 fs single pulse. The wave
number 0 cm 1 corresponds to the center wavelength. (c) The
spectral peak around T 1  100 cm 1 is magni®ed. The full
width at half maximum dx1 of the peak approximately equals
dx2 , which is the dierence between the peak and the ®rst adjacent zero point.

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R. Morita et al. / Optics Communications 197 (2001) 73±81

conventional analysis using an in®nitesimal duration is impractical because it could excite a mode
strongly in an arbitrarily high vibrational frequency
region. As an example, in our analysis here, we
consider F t=T0  as a hyperbolic secant function.
Then
n
o

C x  F j F t=T0 j2  F sech2 t=T0 
 pxT02 cosech pxT0 =2;3as shown in Fig. 2(b). Since spectral ®ltering for
selective excitation at xR;int is done by the peak ofx  xR;int  2p=T , the stimulated Raman signal
for N-pulse train beams with ®nite duration is reduced by a factor of C xR;int =C x  0 comparedto that for delta-function N-pulse train beams. For
the example described in Section 4 (xR;int =2p 
448:2 cm 1 and T0  30=1:763  17:0 fs), the analysis using the N-pulse train with the in®nitesimal duration overestimates the power spectrum of
the coherent Raman scattering signal excited selectively at xR;int =2p  448:2 cm 1 by about afactor of ®ve compared to our analysis using theN-pulse train with ®nite duration. Thus the analysis using the N-pulse train with the ®nite duration is essential for an accurate interpretation of
the experimental results.
3. Optimum pulse number of an N-pulse train
For a practical consideration of the selective
excitation using an N-pulse train with ®nite duration, the condition that the total energy of the Npulse train is constant should be considered. SRS
is a third-order non-linear process governed by thepulse peak intensities. Thus, under the constanttotal-energy condition, it is important to derive theformula that determines the minimum pulse
number for ecient selective excitation in bothcases of one- and two-color N-pulse train beams.
Fig. 2(c) shows the magni®cation of the ®rst positive frequency (or negative-frequency) sub-spectrum of FfEpu1 tEpu2 tg at x  2p=T , which isused for conventional selective excitation. Fromequation in Appendix B, the shape of this subspectrum is expressed mainly through the factor
sin N xT =2=W sin xT =2. Adjacent spectral zero

points are at x  2p 1  1=N =T . Approximating
the shape around the ®rst positive frequency subspectrum to that of an isosceles triangle, we canevaluate the full width at half maximum (FWHM;
dx1 ) of the ®rst positive frequency (or negativefrequency) sub-spectrum to be 2p=NT , which isequal to half of the base width dx2 of the triangle. Considering that only the Raman spectral
component almost within the FWHM of the ®rst
positive frequency (or negative-frequency) subspectrum can be excited eectively, we ®nd that thecondition for ecient selective excitation is that
the FWHM should be smaller than the minimum
interval DxR min of the intervals between the
Raman line xR;int of interest and the lines xR;ni;j(j  1; . . . ; m 1) of no interest. That is,
NJ

2p:
DxR min T

4

This is the criterion for the optimum pulse number
N for ecient selective excitation using an N-pulse
train. It should be noted that the optimum pulse
number does not depend on the shape of eachpulse that constitutes the N-pulse train, because
the width of the ®rst positive frequency (or negative-frequency) sub-spectrum is determined by thefactor sin N xT =2=W sin xT =2.

4. Ecient selective excitation by the vibrationallysynchronized pumping technique with frequency
dierence resonance
As mentioned above, the spectral modulation of
FfEpu1 tEpu2 tg in Eq. (1) leads to a low selective-excitation eciency of a speci®c Raman modein the high frequency region in the case of x1x2  0. In order to avoid this low selective-excitation eciency, due to the factor FfEpu1 t Epu2 tg, we take notice
P of another factor of
Ffexpi x1 x2 tu t j rj tg in Eq. (1). While
the spectral modulation due to FfEpu1 tEpu2 tgdecreases the spectral amplitude at x  xR;int , especially in the high frequency region, the spectral
amplitude at x  0 is not modulated even in the®nite duration case. Moreover, shifting the Raman
spectrum is possible with a factor expi x1 x2 t

R. Morita et al. / Optics Communications 197 (2001) 73±81

77

P

in Ffexpi x1 x2 tu t j rj tg by varying thex1 x2 value. When we adjust x1 x2  xR;int ,we can shift the Raman spectrum by xR;int , thus
we can use the un-modulated spectral component
at x  0 of the N-pulse train for selective Ramanmode excitation. That is, to greatly improve on the
low eciency of selective excitation in the highfrequency region, we employ the vibrationallysynchronized pumping technique with frequency
dierence resonance. We use two-color N-pulse
train beams with a frequency dierence of x1 x2
which corresponds to the Raman frequency xR;int
and for which the pulse interval T satis®es xR;int 
2p=T . These two-color N-pulse train beams, as wellas one-color N-pulse train beams, can selectively
excite the speci®c Raman mode of xR;int . Moreover, when adjusting the center frequencies of thetwo-color beams that satisfy x1 x2  xR;int , wecan shift the spectrum of the Raman responseeectively to make them resonant with the central spectral peak of the N-pump±pulse trains.
Thus, we can perform ecient selective excitationeven in the high frequency region, with complicated Raman lines, using two-color N-pulse train
beams.
As an example, we investigate the ecient selective excitation of tetrachloroethylene that has
Raman modes at 236.6, 346.5, 448.2 and 511.9
cm 1 in the high frequency region [11], as shown in
Fig. 3(a). Here, we consider the selective excitationof the 448.2 cm 1 Raman line. The pulse interval T
for selective excitation at 448.2 cm 1 is 74.4 fs, andthe minimum frequency interval Dxmin =2p is 63.7
cm 1 . To avoid a decrease in power of each constituent pulse for selective excitation under thecondition that the total energy of the N-pulse train
is constant, N should be at least 7 to satisfy theinequality (4).
The power spectrum of Eq. (B.4) in the casewhere Epu1 t  Epu2 t, F t=T0   sech t=T0 , N 
7, T  74:4p fs and every pulse duration tp 
2T0 ln 1  2  30 fs (T0  17.0 fs) for selective
excitation at 448.2 cm 1 is shown in Fig. 3(b). The
height of the ®rst peak at the positive or negative
frequency (x=2p  448:2 cm 1 ) is much lower
than that of the peak at the center of x  0 cm 1 .
As a measure of the selectivity and eciency of
excitation, we de®ne a factor gS , such that

Fig. 3. (a) Raman spectrum of tetrachloroethylene. (b) The
spectrum of an N  7 pulse train for selective excitation at
448.2 cm 1 mode. The wave number 0 cm 1 corresponds to thecenter wavelength.

2e
S hetero ks ; xR;int :gS e
S hetero ks ; xmax 

5

R;ni

The
ratio
of
the
power
spectrum2
S hetero ks ; x atje
S hetero ks ; xR;int j of the SRS signal e
the speci®c frequency of interest xR;int to j e
S hetero ks ;
2
max
xmax
R;ni j at another Raman frequency xR;ni thatmaximizes the spectral intensity among the Raman
frequencies xR;ni;j (j  1; . . . ; m 1; eS hetero ks , x isthe Fourier transform of Shetero ks ; sD ). In Section
4, the ecient selective excitation by the vibrationally synchronized pumping technique withfrequency dierence resonance will be discussed
together with the gS values. Fig. 4 shows the calculated coherent Raman scattering signals as a

78

R. Morita et al. / Optics Communications 197 (2001) 73±81

Fig. 4. Temporal pro®les of heterodyne-detected coherent Raman scattering signals with pump pulses of (a) x1 x2  0 and N  1,(b) x1 x2  0 and N  7, (c) x1 x2 =2p  448:2 cm 1 and N  1, and (d) x1 x2 =2p  448:2 cm 1 and N  7. The comparison of the signals in the (b) and (d) cases is shown in the inset.

function of delay time in the OHD scheme for fourcases ((a) N  1, x1  x2 , (b) N  7, x1  x2 , (c)N  1, x1 x2 =2p  448:2 cm 1 and (d) N  7,x1 x2 =2p  448:2 cm 1 ), on the assumptionthat the damping rates c of all Raman modes in
Eq. (4) are all 20 cm 1 . To obtain maximum signal
intensities in the OHD scheme, we set the /  wvalues to p=2 and 0 for the x1 x2  0 and 448.2
cm 1 cases, respectively. In the N  1 (Fig. 4(a)and (c)) cases, it is found that the signals exhibitcombined oscillation of more than one frequency.
In the N  7 and x1 x2  0 case (Fig. 4(b)), it isseen that the signal oscillates with a speci®c frequency at xR;int but the signal oscillating at xR;int
is weak. In the N  7 and x1 x2 =2p 
448:2 cm 1 (Fig. 4(d)), it is seen that the signal isdominated by a non-oscillating component. This
non-oscillating part corresponds to the selectivelyexcited mode of interest owing to the frequency
shift of x1 x2 =2p  448.2 cm 1 . By considering the integral of the non-oscillating part withrespect to the time delay, the selectively excited
Raman signal at xR;int  448:2 cm 1 mode isfound to be comparatively large. (See the inset of
Fig. 4(d).) To further clarify the selectivity of theexcitation, the corresponding power spectra areshown in Fig. 5. Signal intensities when x1
x2  0 (Fig. 5(a) and (b)) are low owing to thespectrum modulation factor FfjF t=T0 j2 g in Eq.
(1). On the other hand, signal intensities whenx1 x2 =2p  448:2 cm 1 (Fig. 5(c) and (d)) aremuch more enhanced than those when x1 x2 
0. For Fig. 5(c) and (d), the horizontal axis represents wave number corresponding to xx1
x2 =2p. Thus, the peak at 0 cm 1 corresponds tothe vibrational mode of 448.2 cm 1 . That is, by
adjusting x1 x2 =2p to the mode frequency
xR;int =2p of interest, we can eectively shift the
Raman mode frequencies by x1 x2 =2p so thatthe only the signal intensity of interest is strengthened. As a result we can achieve much more ecient excitation at x  0 cm 1 . In addition, the N 
7 pulse train enables the further selective excitationof only the 448.2 cm 1 mode (Fig. 5(d)), while theN  1 pulse excites all the Raman modes simultaneously (Fig. 5(c)). The quantitative comparison

R. Morita et al. / Optics Communications 197 (2001) 73±81

79Fig. 5. Power spectra of heterodyne-detected coherent Raman scattering signals with pump pulses of (a) x1 x2  0 and N  1, (b)x1 x2  0 and N  7, (c) x1 x2 =2p  448:2 cm 1 and N  1, and (d) x1 x2 =2p  448:2 cm 1 and N  7. The selectivity isdetermined by gS , which is de®ned by Eq. (5). This is evaluated to be (a) 0.285, (b) 7.35, (c) 12.2 and (d) 199.

of the selection factor gS de®ned by Eq. (5) makes
it clear that the gS values are 0.285, 7.35, 12.2 and
199 for Fig. 5(a)±(d), respectively. Thus, it is foundthat, even in the high frequency region, ecient
selective excitation can be made with two-color
synchronized shaped-N-pulse train beams whose
frequency dierence is adjusted to the phononmode frequency of interest (Fig. 5(d)). The calculated results show that this vibrationally synchronized pumping technique with frequency dierence
resonance (Fig. 5(d)) has the advantage of exhibiting an enhancement factor about 20 times larger
in the power spectrum of Raman signal for selective excitation than the conventional one-color
pulse shaping technique (Fig. 5(b)).
5. Conclusions
We have analyzed the SRS signal using N-pulse
trains with ®nite duration. It was found thatspectral modulation due to the ®nite duration of
the constituent pulse causes low eciency of selective excitation in the high vibrational frequency

region. To greatly improve on this low eciency inthe high frequency region, the vibrationally synchronized pumping technique with frequency difference resonance is proposed, in which both thepulse-repetition rate and the center-frequency difference of the two-color N-pulse train beams are
made resonant with the frequency of the speci®cvibrational or rotational mode of interest. Fromthe experimental point of view, under the condition that the total pulse energy is constant, theformula describing the optimum pulse number foreective selective excitation has been derived. As
an example, we have calculated the SRS signal of
tetrachloroethylene for (a) single pulses (N  1,x1 x2  0) case, (b) single pulses with frequency
dierence resonance (N  1, x1 x2  xR;int ) case,(c) single frequency N-pulse trains (only repetition rate resonance: N  7, x1 x2  0) caseand (d) N-pulse trains with repetition-rate andfrequency dierence resonances (vibrationally synchronized pumping technique with frequency difference resonance: N  7, x1 x2  xR;int ) case.
In the vibrationally synchronized pumping casewith frequency dierence resonance (d), a speci®c

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R. Morita et al. / Optics Communications 197 (2001) 73±81

vibration mode is selectively excited with a 20times larger power spectrum than in the repetitionrate resonance case (b). Thus, it is shown that thisvibrationally synchronized pumping techniquewith frequency dierence resonance for selective
excitation is useful in the high frequency region as
well as in the low frequency region.
Appendix A. Fundamental analysis of selective
Raman-mode excitation
In order to clarify the advantage of the selective
excitation using two-color pulse trains, we ®rst
describe the analysis of selective excitation by an
N-pulse train with ®nite duration including twocolor pump case as well as one-color pump casehere. Consider a thin medium interacting with an
external electric ®eld:
E r; t  Epu r; t  Epr r; t;Epu r; t  12Epu1 t expi k1  r

A:1

Epr r; t  12Epr t

sD  expi k3  r

x2 t/  c:c:;
A:2

x3 t  c:c:;
A:3

ELO r; t  12ELO t

sD  expi ks  r

xLO t

xeg  x1  1=tp;1 ;xeg  x2  1=tp;2 ;xeg  x3  1=tp;3 ;A:5

where hxeg is the energy dierence between theelectronic excited and ground states, and tp;i (i  1,2, 3) is the pulse duration of pump1, pump2 andthe probe pulses, respectively. In this case, the optical-heterodyne-detected coherent Raman scattering signal in the ks -direction is given by
Shetero ks ; sD  /(Z
1
dtjEpr t
2xs Im1

 Epu2 ts expi x1

sD j
x2 s

w  c:c:Z 1
0ds Epu1 t

s

rj s

j

 expi /  w ;A:6

integrated over all times t, on the assumption that
ELO t / expi ks  r xs t. rj t is the Raman response of the medium exhibiting the jth vibrational mode (j  1; . . . ; m), expressed byrj t  Aj exp

A:4

Here, Epu1 t, Epu2 t, Epr t and ELO denote thetemporal envelopes of pump1, pump2, probe andlocal oscillator pulses with dierent center frequencies xj (j  1, 2, 3 or LO), respectively. The
probe pulse is delayed by the time interval sD withrespect to pump1 and pump2.
Pump1 and pump2 interfere in the medium,
creating a transient grating with a wave vector
k1 k2 . The probe beam then is Bragg-diracted
by this grating, resulting in the scattered beam in
the ks  k1 k2  k3 direction. The center frequency of the scattered signal is xs  x1 x2 
x3 . An experimental setup of the box-CARS geometry for coherent Raman scattering is one of thecases of this con®guration. In addition to thesethree pulses, a fourth pulse that can be used as alocal oscillator for OHD is included here [12±14].

2

X

)

x1 t

 12Epu2 t expi k2  r

We here focus on Raman processes, with oresonant electronic-state excitation such that

xR;j 

cj t=2 sin xR;j t;q
2x2R;jcj =2 ;

A:7
A:8

where Aj ; cj and xR;j are the relative strength, thedamping rate and the natural vibrational frequency of the jth mode, respectively. In accordance with the philosophy of selective excitation,m modes are classi®ed into a mode of interest(xR;int ) and m 1 modes of no particular interest(xR;ni1 ; . . . ; xR;ni m 1 ), hereafter. We do not include
the electronic response function here.
De®ning the Fourier transform of a function
f t asZ 1
Fff tg 
dt eixt f t;
A:9
1

we can rewrite Eq. (A.6) as Eq. (1).

R. Morita et al. / Optics Communications 197 (2001) 73±81

Appendix B. Mode selection
The coherent Raman scattering signal producedby the vibrationally synchronized pumping technique with frequency dierence resonance is analyzed by describing the pro®les of the pump pulses,
using a shaped-pulse train in the form
Epu1 t  Epu2 tNXF t  np
1T =T0 = W ;

B:1

n1

W 

Z 1
1

dt

N
X

2

F t  n

1T =T0  ;

B:2

n1

where F t=T0  denotes the pro®le of each constituent pulse composing an N-pulse train with theconstituent-pulse duration parameter T0 , and T is
the time interval between pulses, as shown by Fig.
2(a). Considering practical cases for selective excitation using ppulse trains, the electric ®eld isnormalized by W under the condition that the
total pulse energy is constant. In the case where
T0 < T ,

Epu1 tEpu2 t '(NX
j F t  n

1T =T0 j2

n1



N 1
X

F t  n

1T =t0 F  t  nT =t0 F t  n

1T =t0 

n1



N
X
n2


 F t  n

81

n
o

F Epu1 tEpu2 t ')2T =t0  =W :B:3When T0  T , the second and the third terms in
Eq. (B.3) can be neglected. Thus, the Fouriertransform of the ®rst term in Eq. (B.3) iso
1 sin N xT =2 i N 1xT =2 ne
F j F t=T0 j2 :W sin xT =2
B:4

The spectrum of Eq. (B.4) is almost discretizedwith the frequency spacing Dx  2p=T , determined by the time interval T of the N-pulse train.
This discretization enables us to excite coherent
phonons selectively, that is, to excite only a speci®cvibrational mode. This is in stark contrast to thesingle-pulse (non-pulse train; N  1) case where
many Raman modes are excited simultaneously.
This selective excitation can be explained throughEq. (1): among the Raman modes represented byrj t, only a speci®c vibrational mode xR;int is ®ltered by the spectrum FfEpu1 tEpu2 tg, whenm Dx  xR;int (m: integer) in Eq. (A.8).
References[1] Y.X. Yan, E.B. Gamble, K. Nelson, J. Chem. Phys. 83(1985) 5391.
[2] J. Herrman, A. Nazarkin, Opt. Lett. 19 (1994) 2065.
[3] E.M. Belenov, A.V. Nazarkin, I.P. Prokopovich, JETP
Lett. 55 (1992) 218.
[4] A.M. Weiner, D.E. Leaird, G.P. Widerrechet, K.A. Nelson, Science 247 (1990) 1317.
[5] A.M. Weiner, D.E. Leaird, G.P. Widerrechet, K.A. Nelson, J. Opt. Soc. Am. B 8 (1991) 1264.
[6] Y.J. Yan, S. Mukamel, J. Chem. Phys. 94 (1991) 997.
[7] S. Mukamel, Principles of Nonlinear Optical Spectroscopy,
Oxford University Press, 1995 (Chapter 14).
[8] L. Xu, N. Karasawa, N. Nakagawa, R. Morita, H.
Shigekawa, M. Yamashita, Opt. Commun. 162 (1999) 256.
[9] N. Karasawa, R. Morita, L. Xu, H. Shigekawa, M.
Yamashita, J. Opt. Soc. Am. B 16 (1999) 662.
[10] N. Karasawa, R. Morita, H. Shigekawa, M. Yamashita,
Opt. Lett. 25 (2000) 183.
[11] G. Herzberg, Molecular Spectra and Molecular Structure,
Krieger, 1991, p. 639.
[12] P. V
ohringer, N.F. Scherer, J. Phys. Chem. 99 (1995) 2684.
[13] Y.J. Chang, P. Cong, J.D. Simon, J. Phys. Chem. 99 (1995)
7857.
[14] S. Matsuo, T. Tahara, Chem. Phys. Lett. 264 (1997) 636.