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C. Special Problems of Seventh-Order Spectroscopy

C. Special Problems of Seventh-Order Spectroscopy

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Vibrational Dephasing in Liquids


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



Â2 D Â 1 C




















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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Figure 5 Log-log plot of the anti-Stokes signal intensity versus the excitation

pulse energy (sym-methyl stretch of CH3 CN at 1 D 3 D 1 ps). In each case, the

energy of all pulses of the same frequency was varied simultaneously. The lines

show the expected power law dependence with exponent a. These measurements

help to confirm that a true 7th-order Raman echo is being measured. (Adapted from

Ref. 5.)

faster than the FID signals as the laser intensity is increased. The echo-toFID ratio improves as the square of the total Laser intensity and the square

of the total Stokes intensity. Thus, applying the maximum intensity to the

sample improves the signal-to-background ratio.

The intensity that can be used is ultimately limited by self-focusing

and self-phase modulation in the sample. An early study concluded that

these problems would preclude Raman echo experiments (31), but careful

experimental design has pushed these limits back. However, the selffocusing problem remains the primary practical limitation in expanding

the application of the Raman echo.

Another unique complication of high-order experiments like the

Raman echo is the presence of combined resonant/nonresonant interactions.

In a third-order experiment, like the Raman FID, there is only one time

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Vibrational Dephasing in Liquids


interval. The desired signal occurs when a resonant interaction causes

the system to occupy a real state during that period. An “artifact” arises

during pulse overlap because nonresonant interactions put the system into

a “virtual” state during that time. In an experiment with multiple time

intervals, new artifacts arise from mixing real states during some time

intervals with virtual states in other intervals (4). As a result, the full

material response function for the Raman echo is

MRE t1 , t2 , t3 D

1 CSRE t1 , t2 , t3


3 υ t1


υ t2 CFID

t2 /T1

υ t3

t3 C 4 υ t1 υ t 2 υ t 3

2 CFID t1



[see Equation (26)]. This more complete response function should be substituted for the more idealized CSRE t1 , t2 , t3 in Equation (30). The final term

in Equation (34) is a fully nonresonant interaction and causes an extra peak

at 1 D 3 D 0. The second and third terms are partially nonresonant and

contribute peaks near, but slightly after, 3 D 0 and 1 D 0, respectively.

These nonresonant interactions significantly complicate the analysis

at short times. As a result, independent measurements the of the 1 D 0 and

3 D 0 points on the time scale are important ingredients in the analysis. If

these parameters are removed from the fitting, the convolution analysis of

the Raman echo signals is stable and unique, even at short times.

The earlier discussion focused on the D 0 and D 1 vibrational

states. However, in a high-order experiment, higher vibrational states must

be considered as well. In fact, if the vibration is purely harmonic, the

Raman echo signal is exactly canceled by additional processes involving the

overtone states (56–60). Thus, anharmonicity in the vibration is essential.

As the difference between the 0 ! 1 and 1 ! 2 transition frequencies

increases beyond the combined linewidths of the two transitions, the echo

signal regains its full strength. However, if the two transitions still lie

within the bandwidth of the excitation pulses, quantum beats arise. This

effect has been most clearly documented by Tokmakoff et al. in the context

of the infrared echo (40). These beats contain information on overtone

transitions, but they also complicate the interpretation of the dephasing

of the fundamental. In all the experimental systems discussed below, the

anharmonicity is sufficient to move the overtone completely out of the

excitation bandwidth, so the simple two-state analysis of Fig. 3 is sufficient.

Recently, there has been a good deal of concern about cascaded loworder processes contaminating fifth-order echo experiments (52–55,125).

In a cascaded process, the signal from a third-order process acts as an

excitation for a second third-order process. Certain cascaded signals mimic

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fifth-order echo signals in frequency, direction, and excitation power dependence. There is a natural concern that analogous processes might interfere

with the seventh-order Raman echo as well. However, all known cascaded

processes involve a switch between an overtone transition and a fundamental transition, relative to the direct process. In a fifth-order echo, a weak

overtone transition can be replaced by a strong fundamental, resulting in a

strong cascaded process (51). A cascaded analog to the Raman echo must

replace a strong fundamental transition with a weak overtone, resulting in

an extremely weak cascaded signal.

The lack of a known cascaded process strong enough to compete with

the Raman echo is reassuring, but the possibility of as-yet-unrecognized

cascaded processes still exists. However, a direct experimental test for

cascaded processes exists. All cascaded processes have a signal intensity

that depends on the fourth power of the concentration of the vibrators,

whereas direct processes depend on the second power (51,125). Interference

between a direct and a cascaded process gives a third-power dependence

on concentration. Measurements of Raman echo signal strength from the

methyl stretch of CH3 CN as it is diluted in CD3 CN are shown in Fig. 6.

The results show the expected second-power dependence, providing direct

experimental confirmation that cascaded processes are not an issue for the

Raman echo.

Figure 6 The concentration dependence of the Raman-echo signal (sym-methyl

stretch of CH3 CN diluted in CD3 CN). The expected quadratic a D 2 dependence

is found. The quartic dependence a D 4 of a cascaded nonlinear process alone or

the cubic dependence a D 3 of a cascade signal interfering with the direct signal

is not found. (Adapted from Ref. 6.)

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Vibrational Dephasing in Liquids


A final question in any dephasing measurement is the potential

involvement of rotational dynamics. It has long been known that infrared

and anisotropic Raman line shapes contain contributions from both

rotational and dephasing dynamics, whereas the isotropic Raman line shape

is immune to rotational dynamics and reflects only dephasing dynamics

(21). The polarization conditions needed to extract analogous rotationfree, isotropic Raman FIDs have been well studied (106,113,126). In

systems where rotation is slow relative to dephasing, the sensitivity of

infrared measurements to rotation presents no problem. However, for smallmolecule liquids, rotation is often as fast or faster than dephasing (Table 1).

For this reason, Raman spectroscopy, rather than infrared, has dominated

the study of dephasing (1,2).

Moving to higher-order echo experiments, it has been shown that

the infrared echo is inescapably affected by rotation as well as dephasing

(38). The full polarization and rotation dependence of the Raman echo has

not been explored. However, if the vibrator has a purely isotropic Raman

cross section, i.e., the depolarization ratio D 0, the analysis is simple.

The Raman echo is completely independent of rotational dynamics. This

fact gives the Raman echo a significant advantage over the infrared echo

for many systems.

All the experiments reported here are on vibrations with near-zero

depolarization ratios (Table 1). In this case, the excitation pulse pairs must

have the same polarization, but the relative polarization of different interactions is unimportant. In practice, we take all excitation polarizations

perpendicular to the plane of the excitation beams and the LIII polarization parallel. In this configuration, the signals with parallel polarization are only generated by scattering from LIII . A polarizer is placed in

the signal beam to provide additional discrimination against competing

nonlinear processes.

D. Experimental Equipment

The choice of laser system is dictated by the considerations outlined above.

High-energy pulses of two different frequencies are needed, and good

temporal synchronization between the pulses is essential. A pulse width

of 0.3–1.0 ps is adequate to resolve most dephasing processes. Good mode

quality is important to minimize self-focusing.

We have performed Raman echo experiments using synchronously

pumped dye lasers, followed by dye amplifiers pumped by a Q-switched

Nd:YAG laser (Fig. 7). These systems are standard (127–129), but several

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Figure 7 Schematic of the laser system used in the Raman FID

and echo experiments. PC D Pulse compressor; AOM D acousto-optic modulator; PD D photodiode; FB D feedback electronics; PBS D polarizing beamsplitter; 3PBF D 3-plate birefringent filter; SDL/LDL D Stokes/Laser dye laser;

P D pellicle; AC D autocorrelator; OC D output coupler; LBO/KDP D doubling

crystals. Final pulses have widths of 0.5–1 ps and energies of 0.3–1 mJ (From

Ref. 6.)

modifications have been made for Raman-echo experiments. Most obviously, the dye laser and amplifier chain have been duplicated to provide

two independently tunable wavelengths. Instead of using the 80–100 ps

pulse from a CW mode-locked Nd:YAG to pump the dye lasers directly, the

Nd:YAG pulse is first compressed to ¾ 2 ps in a fiber-grating compressor

(130). The short pump pulse both shortens the dye laser pulses to the

required 0.5–1 ps duration and markedly improves the synchronization of

the two dye lasers. The synchronization is also improved by stabilizing the

pump power with an acousto-optic attenuator driven by a feedback loop.

Longitudinally pumped amplifiers are chosen for their good mode quality

(128,129). Low-aberration optics, e.g., achromatic lenses, also improve the

beam quality.

Recent advances in solid-state laser systems offer several potential

improvements. In particular, tunable optical parametric amplifiers provide

tunable pulses with excellent synchronization and beam quality. However,

most reported systems produce pulses of <100 fs. Pulses this short are

not needed to measure vibrational dephasing in most systems, and they

unnecessarily exacerbate the problems of self-focusing and excitation of

overtone transitions. A solid-state system optimized for longer pulses has

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Vibrational Dephasing in Liquids


not been used for Raman-echo experiments yet, but it has the potential to

make these experiments considerably easier.

Once appropriate pulses are generated, the experimental setup is

straightforward, if somewhat complex (Fig. 8). The original pulses are split

and recombined to provide the required five pulses with approximately

equal energy at the sample. Polarizer/half-waveplate attenuators allow fine

optimization of the pulse energy. The beams are combined such that the I

and II pulse pairs can be scanned in time as pairs. Each beam is separately

focused, so the spot size at the sample can be controlled by the distance the

sample is placed in front of the focal point. Translation stages containing the

final mirror/lens combination allow the angles of the beams to be adjusted

without changing the spot sizes. A chopper blocks alternate SI pulses,

and software subtracts the resulting measurements to correct for scattered

light. A pinhole at the beam focus and interference filters separate the

Raman echo signal from other beams. A PMT and 16-bit A/D converter

have been found to provide the sensitivity, linearity, and dynamic range


Figure 8 Schematic of the optical system used to perform the Raman FID and

echo experiments. P D Polarizer; (D)BS D (dichroic) beamsplitter; MD D manual

delay line; SD D computer-scanned delay line; CSA D charge sensitive amplifier;

CH D chopper; PH D pinhole; S D sample; F D bandpass and neutral density

filters; PD D photodiode; A/D D analog-to-digital converter; PC D computer;

PMT D photomultiplier; /2 D half-wave plate. (From Ref. 6.)

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A. Concentration Fluctuations in CH3 I:CDCl3

As discussed in Section II.C, dephasing by local concentration fluctuations

has been considered many times in the literature. However, the Raman echo

has proven to be important in fully unraveling this process.

Although Schweizer and Chandler did not explicitly discuss concentration fluctuations (88), their central arguments on the differing effects

of attractive and repulsive interactions can be applied to concentration

fluctuations. If a vibration interacts with the solvent through the longrange attractive portion of the intermolecular potential, it will be sensitive

to the average composition of the first solvation shell. The composition

changes due to diffusion in and out of this shell, which takes 5–10 ps

in typical liquids. This time is characteristically slow compared to the

¾100 fs lifetime expected for short-range repulsive interactions. Schweizer

and Chandler’s analysis predicts that concentration fluctuations should lead

to a slow dephasing process, if and only if the fluctuations are coupled to

the vibration by a long-range attractive interaction. By using the Raman

echo to determine the lifetime of the vibrational frequency fluctuations in

liquid mixture, the validity of this scenario can be tested.

We decided to use this approach on the sym-methyl stretch of methyl

iodide in chloroform-d (4). This system has a large concentration effect

and has been previously studied by Dăoge et al. (131) and by Knapp and

Fischer (87) on the basis of Raman linewidth measurements. We began by

making more detailed measurements of the Raman line shape as a function

of concentration. Figure 9 shows that the peak frequency shifts significantly

as the average environment around the vibrator changes from pure CH3 I

to pure CDCl3 . The shift is large 10.1 cm 1 and linear in concentration.

The linearity implies that there is ideal mixing, i.e., the CH3 I and CDCl3

molecules are positioned randomly throughout the liquid.

Figure 10 shows the corresponding behavior of the Raman linewidth.

The widths of the methyl line in a pure CH3 I environment and in a pure

CDCl3 environment are nearly identical. However, in a mixed environment,

the line is wider and is widest for a 50:50 mixture. This behavior matches

the expected effects of dephasing by local concentration fluctuations.

The FID from the 50:50 mixture is shown in Fig. 11. The decay is

clearly nonexponential, and thus the line shape is not a pure Lorentzian. The

decomposition of the line shape into Lorentzian and Gaussian components

(132) is shown as a function of concentration in Fig. 10. Although there

is some uncertainty in such a decomposition, the major features are clear.

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Vibrational Dephasing in Liquids


Figure 9 Shift in the peak frequency of the isotropic Raman line of the sym-methyl

stretch in CH3 I as a function of concentration in CDCl3 . The shift is relatively and

linear with concentration. The peak frequency in pure CH3 I is 2951 cm 1 . (From

Ref. 4.)

There is a Lorentzian component of about 5.5 cm 1 FWHM, which is

concentration independent. All of the concentration variation is due to a

Gaussian component, which is nearly absent at the ends of the concentration

range but reaches a maximum size of 4.25 cm 1 in the 50:50 mixture. The

Lorentzian shape of the concentration independent component is suggestive

of a fast modulation dephasing. The Gaussian shape of the concentration

dependent component is suggestive of a slow modulation process.

Results of the Raman echo experiment on this system are shown in

Fig. 12. The interpretation of these data proceeds by comparison with a

series of models, which are constrained to be consistent with the linewidth

and FID data. The simplest model assumes that the entire linewidth/FID

is due to fast modulation processes, including the concentration-dependent

process. The predictions are shown as solid curves in Fig. 12. At long

1 and 3 , the predicted signal is substantially smaller than the observed

signal. The enhanced signal results from a rephasing induced by the echo

sequence and indicates that a slow modulation process must be present. On

the other hand, the rephasing is not complete, so there must be a significant

fast-modulation process as well.

The next obvious model is a combination of one fast process

corresponding to the concentration-independent Lorentzian component of

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