B. Intermolecular Motions in CS2
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Fifth-Order 2D Raman Spectroscopy
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Figure 8 Experimental phase matching geometries. The circles represent the ﬁve
incoming laser pulse positions. The open circles with an S inside are the ﬁfth-order
signal positions. The squares are the parallel cascade intermediate positions, and
the diamonds are the sequential cascade intermediates. The phase matching magnitudes and effective path lengths are listed in Table 1. (a)–(c) Homodyne detected
geometries described in the text. (d) Heterodyne detection geometry. Note that the
k1 C k2 C k5 parallel intermediate is located at the same position as the signal,
thus serving as a local oscillator for the heterodyne detected signal.
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Fleming et al.
Although our sample cell has a path length of 1 mm, the actual path
length is determined by the crossing of the incoming beams in the sample.
To estimate the effective path length as determined by the beam crossing,
we calculate the overlap volume of all ﬁve beams as a function of position
in the sample. Our beams had diameters of 3 mm at the focusing lens. The
crossing volume was weighted by a Gaussian transverse beam amplitude
proﬁle with a FWHM of 40% of the beam diameter. We assign the FWHM
of the Gaussian ﬁt to the crossing volume distributions as the effective path
length. The resulting effective path lengths are listed in Table 1 for each of
the phase matching geometries employed along with the resulting values
of the magnitude of the phase matching factors for the overall ﬁfth-order
process and the possible third-order cascade intermediates. The positions
of the cascade intermediates and overall ﬁfth-order signals are shown in
Fig. 8.
The spectra obtained in phase matching geometries 8a and 8b are
shown in Fig. 9a and 9b. The spectrum in Fig. 9a shows a high degree
of asymmetry, while the spectrum in Fig. 9b appears quite symmetrical
in the behavior along the two adjustable time variables. The symmetrical
Table 1 The Magnitude of the Wavevector Matching Factor for the Overall
Fifth-Order Response and the Cascade Intermediates, sinc kl/2
sinc kl/2
Phase matching
k2 k3 C k4 k5
k1 C k2 k3 C k4 k5
k 1 k2 k3
k1 C k2 k3
k1 k2 C k4
k1 C k2 C k4
Parallel
k1 k2 C k5
intermediates
k1 C k2 C k5
k3 C k4 C k5
k3 C k4 C k5
Effective path length (mm)
Overall ﬁfth-order
signal
Sequential
intermediates
k1
8a
8b
8c
8d
0.89
0.46
0.88
1.0
0.20
0.96
0.09
0.77
0.19
0.96
0.09
0.27
0.75
0.09
0.15
1.0
0.75
0.14
0.15
1.0
0.61
0.73
0.73
1.0a
The wavevector matching conditions in parenthesis refer to geometry 8(d); others refer to
geometries 8a–8c. The beam geometries are shown in Fig. 8. The effective path lengths are
the FWHM of the Gaussian ﬁts to the crossing volume distributions.
a The FWHM of the crossing volume distribution for geometry 8d was 1.2 mm, therefore,
the effective path length was set to the smaller sample cell path length of 1.0 mm.
Copyright © 2001 by Taylor & Francis Group, LLC
Fifth-Order 2D Raman Spectroscopy
461
spectrum, Fig. 9b, has the same time-dependent behavior along each of
the two time variables as the one dimension time domain spectrum shown
in Fig. 3. The response at early times rises to a maximum at ¾ 180 fs
and then evolve into a single exponential decay at long times with a time
constant of tc ¾ 0.8 ps. In the asymmetrical spectrum (Fig. 9a), there is a
rapid decay along 2 from a maximum at 2 D 0 that at long times becomes
the 0.8 ps decay seen along both dimensions in Fig. 9b. Along 4 the initial
behavior is similar to the inertial behavior at early times in the third-order
response, but the signal then decays much more rapidly, tc ¾ 0.25 ps, than
along 2 . There is a prominent ridge along both time axes in and along
Figure 9 Measured 2D ﬁfth-order time domain nonresonant Raman spectra for
CS2 at room temperature: (a) using the phase matching geometry shown in Fig. 8a
and (b) using the phase matching geometry shown in Fig. 8b. Figures (c) and
(d) are the cascading ﬁfth-order 2D Raman spectra simulated directly from the
measured 1D Raman spectrum using the phase matching magnitude values listed
in Table 1 for the cascade intermediates and Equation (28) for the same phase
matching geometries used in (a) and (b), respectively.
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462
Fleming et al.
the 4 D 0 axis in Fig. 9a that is pulse width limited in one dimension and
shows nearly identical time-dependent behavior to the third-order response
along the other. Both spectra also contain a strong peak at the time origin
that is pulse width limited in both time dimensions, similar to the electronic
response at the time origin in the third-order spectrum.
While both phase matching geometries 8a and 8b are reasonably well
phase matched for the overall ﬁfth-order response, they are very differently phase matched for the cascade intermediates (Table 1). Geometry 8a
is well phase matched for the parallel cascade intermediates and poorly
phase matched for the sequential cascade intermediates, and geometry 8b
presents the opposite case. If the measured signals were predominantly
generated by the direct ﬁfth-order process, we would expect the response
to be identical between the two experiments. However, the clear difference between the two spectra provides a strong indication that the signals
are dominated by the ﬁfth-order cascading processes. We can simulate the
ﬁfth-order cascaded spectra directly from the measured third-order spectrum in Fig. 3 using Equation (27). Figure 9c shows the simulation for the
parallel cascaded response, and Fig. 9d shows the simulation for the sequential cascaded response. There is excellent agreement between Figs. 9a and
9c and Figs. 9b and 9d consistent with the relative values of the phase
matching factors in Fig. 8.
With the evidence from phase matching geometries 8a and 8b
demonstrating the measured ﬁfth-order signals to be dominated by
cascaded processes, phase matching geometry 8c was designed to provide
a maximum level of discrimination against all four of the cascade
intermediates while leaving the direct overall ﬁfth-order process well phase
matched. Using the same experimental conditions as employed for the data
in Fig. 9, there was no detectable signal with phase matching geometry 8c.
This conﬁrms the domination by cascaded responses in Figs. 9a and 9b.
Although it is always difﬁcult to apply a negative result and quantify the
lack of a signal, consideration of the observed signal levels in ﬁgs. 9(a)
and 9(b), and the failure to detect a signal in geometry 8c, allows us to
estimate an upper limit on the direct ﬁfth-order signal at 2% of the cascaded
response.
In the case of nonlinear polarizability coupling, the ratio of the
cascaded to direct ﬁfth-order response can be expressed in terms of the
ratio of the ﬁrst and second derivative of the polarizability [Equation (32)].
Using instantaneous normal mode simulations, Murry et al. (40) have
calculated the relative ratios of ˛ 1 and ˛ 2 for 5000 intermolecular modes
in CS2 . Although their results show this ratio to be somewhat randomly
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Fifth-Order 2D Raman Spectroscopy
463
distributed among the 5000 modes, the majority of the modes have a ratio
of ˛ 1 :˛ 2 > 100. This result implies that for the homodyne measurement,
where the signal is proportional to jR 5 j2 , the cascaded signal will be
102 –104 times larger than the ﬁfth-order signal from NP coupling. This
is a value well above our lower limit for this ratio of 50. For anharmonic
coupling, even though one might expect a high degree of anharmonicity for
intermolecular motions at room temperature, it is the magnitude, and not
degree, of the cubic anharmonicity that will be reﬂected in the magnitude
of the direct ﬁfth-order signal. The magnitude of the anharmonicity will be
quite small for intermolecular vibrations simply due to the low frequencies
of the modes involved. In addition, Murry and Fourkas (18) have suggested
that a cancellation among the continuous distribution of frequencies for
the anharmonic coupling will result in a much lower contribution to the
ﬁfth-order response than NP coupling.
C. Intramolecular Vibrations in Carbon Tetrachloride and
Chloroform
To provide the additional sensitivity needed to probe the intramolecular
vibrations in liquid CCl4 and CHCl3 , time domain Raman spectra were
taken using an intrinsic heterodyne technique. As a heterodyne detection
technique, the measured signal is the cross-term between a local oscillator ﬁeld and a signal ﬁeld such as the ﬁfth-order signal [Equation (34)].
Intrinsic refers to the fact that the local oscillator ﬁeld is derived from
one of the third-order responses using a phase matching geometry designed
to generate both the third- and ﬁfth-order signals along the same phase
matched direction. The phase matching geometry is shown in Fig. 8d. In
this geometry the overall signal is phase matched along ks5 D k1 C k2
k3 C k4 C k5 . In addition to the ﬁfth-order signal, the third-order signal
ks3 D k1 C k2 C k5 is also generated along the same direction and serves
as the local oscillator. Assuming R 3 × R 5 and using Equation (34), the
resulting signal intensity will be proportional to the cross-term between the
third- and ﬁfth-order responses,
I
2, 4
/ 2 RebR 3
2
C
4
ÐR5
2, 4
c
37
Figure 10 shows the Fourier transformations of intrinsic heterodyne
detected time domain two-dimensional Raman spectra of (a) CCl4 , (b)
CHCl3 , and (c) a 50:50 molar ratio CCl4 :CHCl3 mixture. Only quadrants
I and II are shown, since the other two quadrants are equal by
inversion symmetry. Aside from the diagonal and axial peaks, there
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Fleming et al.
Figure 10 2D ﬁfth-order Raman spectra of (a) CCl4 , (b) CHCl3 , and (c) a 1:1
mole ratio CCl4 :CHCl3 mixture.
Copyright © 2001 by Taylor & Francis Group, LLC