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III. IMPLEMENTING COHERENT RAMAN EXPERIMENTS

III. IMPLEMENTING COHERENT RAMAN EXPERIMENTS

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398



Berg



Figure 3 A ladder diagram illustrating the sequence of laser pulses in time and

their interactions with the ground j0i and first excited j1i vibrational states.

Light pulse frequencies: L D laser; S D Strokes; AS D anti-Stokes. The vibrational

density matrix element

during each time interval

is indicated below the

energy levels. Several light-matter interactions (arrows) occur during each laser

pulse (I, II, or III). (a) In the FID, the vibration evolves in the same coherence 01

for the entire experiment. (b) In the Raman echo, an additional set of interactions

(II) splits the evolution between coherences of opposite phase ( 01 and 10 ), causing

cancellation of the dephasing caused by long-lived perturbations.



and small angles are assumed. The frequencies of the pulses are given by

the i ’s and the corresponding indices-of-refraction by the ni ’s and the mean

index-of-refraction by n. The index-of-refraction mismatch is most easily

found by optimizing a two-pulse CARS experiment, where there is only a

single angle to scan (Fig. 4c). In terms of the optimum CARS angle ,

υD



L S



2



L



2

S



Copyright © 2001 by Taylor & Francis Group, LLC



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Vibrational Dephasing in Liquids



399



Figure 4 Phase-matching configurations used for several coherent Raman experiments. All angles are positive as shown. (Adapted from Ref. 6.)



Competing processes are another concern in real experiments. These

processes result from interactions with different time orderings of the pulses

and with perturbation-theory pathways proceeding through nonresonant

states. They correspond to the constant nonresonant background seen

in CARS and other frequency-domain spectroscopies. These nonresonant

interactions are only possible when the excitation and probe pulses are

overlapped in time, so they add an instantaneous component to the total

material response function

MFID t D



R CFID



t C



NR υ



t



26



where R and NR are the amplitudes of the resonant and nonresonant

contributions, respectively, and υ t is the Dirac delta function. When the

effect of nonzero pulse widths is added, the FID signal is

IFID



/



1

1



1



dt ELII t



2



dt1 MFID t1 FI t



t1



27



0



F t D EL t ES t .



28



The nonresonant term leads to an extra peak at t D 0 with a width determined by the pulse widths (109,111). Because the early time data are

essential to determining the shape of CFID t , this nonresonant term is

important in fitting real experimental data taken with finite pulse widths.



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B. Raman Echo



The light-matter interactions of the Raman echo are illustrated in Fig. 3b.

The first excitation pulse (I) and the coherent probing (III) are essentially identical to the corresponding processes in the Raman FID. The

Raman echo differs by the introduction of an intermediate set of pulses

(II), which divide the total time interval into two portions 1 and 3 . The

net effect of the interactions caused by these pulses is to reverse the sign

of the coherence, i.e., to transfer density from 01 to 10 . The change

in sign of the coherence leads to the change in the sign of the integral

over time in Equation (10). As a result, the dephasing during 1 can be

canceled during 3 if the dephasing results from perturbations that remain

constant during the entire time interval 1 C 3 . As 1 and 3 are varied, the

extent of cancellation changes. The characteristic patterns of cancellation

provide information on the vibrational frequency-modulation rates (Figs. 1

and 2).

A total of seven interactions with input fields are required, so the

Raman echo is a 7 process. However, in practice, a single laser pulse is

used to generate both the LII and LII0 interactions, and similarly, a single

pulse generates both SII and SII0 . Thus, five real pulses are used.

In principle, the II and II0 interactions could be performed by separate

pulses, in which case the time separation of these pulses would become a

third time variable 2 . This experiment would be the stimulated Raman echo

(SRE). During 2 , the vibrator would be in a pure state ( 11 in Fig. 3, 00

in other pathways), and the dynamics would be governed by the population

relaxation decay time T1 . Thus the SRE is a three-dimensional experiment

that measures the correlation function

CSRE



1, 2, 3



D



1C 2C 3



exp i



1C 2



υω t dt



2



T1



1



i



υω t dt



0



29

Although there is no fundamental barrier to performing the SRE, no one

has yet attempted this experiment.

An essential point in the quantitative interpretation of the Raman

echo is that the two-dimensional Raman echo is really a reduction of the

SRE experiment, which is fundamentally three dimensional. In the idealized Raman echo correlation function presented earlier [Equation (10)], the

reduction occurs by fixing 2 D 0 in Equation (29). However, in real experiments using pulses with nonzero duration, interactions II and II0 can occur

at different times within the pulses. Although the observed Raman signal



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Vibrational Dephasing in Liquids



401



intensity IRE approximates the ideal correlation function CRE in the limit

of infinitely short pulses, IRE should be derived from CSRE when the pulses

have finite duration,

IRE



1, 3

1



ð



/



1

1



1



dt ELIII t



1



1



dt1



3



dt2



0



dt3 CSRE t1 , t2 , t3 ð FI t



t3



0



t2



t1



0

2



ð FII t



t3



t2



1



FII t



t3



1



30



where EL and ES are the electric field envelopes of the Laser and Stokes

pulses, respectively. In linear spectroscopy, pulse-width effects are included

by using a one-dimensional convolution integral. For nonlinear experiments

like the Raman echo, fourfold integrals like Equation (30) are typically

needed to include the pulse-width broadening.

This is a convenient point in the discussion to compare the Raman

echo to other “echo” spectroscopies. All echo spectroscopies originate from

the same three-dimensional stimulated-echo correlation function defined in

Equation (29). However, the full three-dimensional experiments needed to

completely map out CSRE t1 , t2 , t3 are often too complex to be practical.

Thus, the dimensionality of the experiment needs to be reduced and a

strategy for efficient sampling the remaining dimensions needs to be defined

to simplify both the experiment and its analysis. A variety of different strategies have been implemented, resulting in a bewildering set of variations

on echo spectroscopy (18,117–124). In the Raman echo, the second time

dimension is fixed near zero to reduce the dimensionality. The remaining

two dimensions are sampled by scanning 3 at various fixed values of 1

(see Section IV).

The infrared echo is also used to measure vibrational dynamics but

in the standard implementation involves a further reduction in dimension

(35,36,41,42). The excitation interactions I and II are strictly analogous to

those in the Raman echo; the Raman interaction is simply replaced by a

direct absorption (Fig. 3, dashed arrows). However, whereas the Raman

echo time resolves the signal during 3 , the infrared echo integrates the

signal during this time period. In this way, the infrared echo reduces the

correlation function to one dimension. The standard, two-pulse photon echo

is reduced to one dimension in much the same way. Because the infrared

echo derives from the same basic correlation function as the Raman echo,



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the infrared echo has the same fundamental sensitivity to the modulation

rate of the vibrational frequency. However, the reduction in dimension does

reduce its ability to distinguish between some models, especially when

intermediate modulation rates are involved (6).

The three-pulse-echo peak shift is another two-dimensional echo technique, so far applied only to electronic transitions (122,123). It integrates

over 3 and keeps 1 and 2 as the time variables. The data are reduced

by tracing the maximum in 1 as a function of 2 , resulting in a onedimensional decay curve. Although the implementation of this type of echo

spectroscopy is quite different, the essential information content is much

the same as in the Raman echo approach.

Another type of echo that is receiving much attention are fifth-order

echoes, which are also based on Raman interactions (56–60). The majority

of studies have looked at intermolecular interactions (11,43–54), but a

few studies have looked at an intramolecular vibrational overtone (47,48).

Compared to the Raman echo, the fifth-order echo replaces a pair of interactions by a single, double-quantum interaction. Although the fifth-order

experiments are formally of lower order than the Raman echo, the doublequantum interaction is forbidden in the harmonic approximation. As a

result, it is not clear that the signal from a fifth-order echo will be stronger

than that from a seventh-order Raman echo.

C. Special Problems of Seventh-Order Spectroscopy



Along with the potential for providing important information, high-order

spectroscopies such as the Raman echo entail a number of unique problems.

The first and most practical of these problems is finding the desired signal.

Using the high-intensity pulses needed to drive a 7 process, as many as

50–100 other signals can be identified coming from the sample resulting

from lower-order 3 and 5 processes. 3 Processes are so strong that

beams representing fourth and fifth harmonics of the basic phase-matching

condition are detectable. Cascaded processes, in which signals from one

nonlinear process act to drive another, also produce readily visible beams.

Fortunately, the majority of these processes produce light at frequencies

well separated from the Raman echo anti-Stokes frequency, so a simple

bandpass filter provides a great simplification of the observed signals.

The other major mechanism for identifying the Raman echo signal is

the phase-matching requirement:

kAS D kLI



kSI



2kLII C 2kSII



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kLIII



31



Vibrational Dephasing in Liquids



403



Although the phase-matching requirement imposes an experimental burden

in arranging the angles of the excitation beams, it also ensures that the

Raman echo signal will occur in a distinct and predictable direction, separated from almost all other competing signals. Spatial filtering of the Raman

echo signal provides the maximum discrimination against scattered light

from competing processes.

With five pulses, there are in principle seven adjustable angles available to satisfy this single phase-matching equation [Equation (31)]. In practice, the problem is simplified by fixing each pair of pulses, LI /SI and

LII /SII , to be collinear and by keeping all pulses coplanar. With these

restrictions, there are only two adjustable input angles remaining (Fig. 4a).

Once a value of ˇ is chosen, the values of Â1 and Â2 are determined by the

same index-of-refraction parameter υ [Equation (24)] that appears in the

phase-matching conditions for the Raman FID and CARS [Equations (22),

(25)] (6):

Â12 D



υ

L



Â2 D Â 1 C



C

S



2



L

L



L



2



2



L



L

S



S



S

2



ˇ2



ˇ



32

33



S



Thus, it is possible to optimize the phase-matching for the easier RamanFID and CARS experiments and use the resulting υ to calculate the correct

excitation geometry for the Raman echo.

Once a putative Raman echo signal is found, its identity must be

confirmed. The most straightforward test is to make sure that the signal

disappears when any of the five input pulses is blocked. All interfering

processes are generated by a subset of the excitation pulses and will remain

when one or more of the excitation pulses is blocked.

A more sophisticated test is to measure the dependence of the signal

size on the input Laser intensity. An example is shown in Fig. 5. The

Raman echo signal has an overall seventh-order dependence on the excitation energies: fourth order on the sum of the Laser intensities and third

order on the sum of the Stokes intensities. Lower-order processes have a

weaker dependence on excitation energy. For example the FID is second

order in total Laser intensity and first order in the Stokes intensity (Fig. 5).

A natural concern with a 7 process is the size of the signal to be

measured. However, under realistic conditions, the absolute intensity of the

Raman echo signal is easy to detect. The problem is scattered light from

unintended FID processes. As Fig. 5 shows, the Raman echo signal grows



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