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B. Intramolecular Vibrations in Carbon Tetrachloride

B. Intramolecular Vibrations in Carbon Tetrachloride

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Fleming et al.

Figure 4 The 1D polarized Raman spectrum from liquid CCl4 at room temperature fit with a sum of four Brownian oscillators.

For the two-dimensional response we will first consider the case

of nonlinear polarizability coupling and simulate the response using

Equation (21) and the Brownian oscillators used to fit the one-dimensional

spectrum in Fig. 4. Figure 5 shows the simulations for the limiting cases

where the system is (a) fully uncoupled, ˛ij2 D υij , and (b) fully coupled,

˛ij2 D 1. For the uncoupled case (Fig. 5a), the response is simply additive in

the four modes. The peaks along the frequency diagonals ω2 D ω4 D š i

are the peaks observed in the one-dimensional spectrum. The only other

features observed are the sum and difference frequency peaks that arise from

the interaction of a vibrational coordinate with itself: the overtones 2ω2 D

ω4 D 2 i and zero-frequency peaks ω2 D š i ; ω4 D 0 . Figure 5b shows

the simulation, including the off-diagonal coupling. The ability of the

radiation field to create coherences between each of the modes leads

to the appearance of various sum and difference frequency peaks, in

addition to the features previously exhibited in the decoupled spectrum.

The placement of these combination peaks in ω4 is correlated with the

coupled fundamentals in ω2 in a manner that makes assignment of the

origin simple.

We can also consider the case of anharmonic coupling between the

modes. This will lead to peaks in the same positions as shown in Fig. 5b, but

the relative amplitudes of the peaks will now be dictated by the magnitude

Copyright © 2001 by Taylor & Francis Group, LLC

Fifth-Order 2D Raman Spectroscopy


Figure 5 Simulations of the 2D Raman spectrum for the Brownian oscillators in

Fig. 4, CCl4 , for nonlinear polarizability coupling [Equation (21)] in the (a) fully

uncoupled, ˛ij2 D υij , and (b) fully coupled, ˛ij2 D 1, limits.

of the cubic anharmonicity, gijk3 . Thus, the different patterns for the different

relative magnitudes between the diagonal, cross-, and difference-frequency

peaks provide a direct indication of the nature and magnitude of the underlying nonlinearity. The different patterns will also depend on the interaction

ordering and phase that lead to the response. We can simulate examples of

Copyright © 2001 by Taylor & Francis Group, LLC


Fleming et al.

different anharmonic couplings using Equations (21) and (36) and the two

highest frequency Brownian oscillators in Fig. 4. Equation (21) is a sum

over the modes with each permutation of the modes generating a separate response weighted by the magnitude of the coupling, gijk3 . Figure 6

shows an example of two different responses labeled by their respective



weighting factors, g411

and g441

, where the subscripts refers to the 1 and

4 vibrational modes. The differences between these two examples, each

Figure 6 Simulations of the 2D Raman spectrum for the anharmonically (fully)


coupled 1 and 4 modes [Equation (20)]. (a) The g411

coupled response. (b) The


g441 coupled response.

Copyright © 2001 by Taylor & Francis Group, LLC

Fifth-Order 2D Raman Spectroscopy


involving a response generated by coupling between the same two vibrational modes, demonstrates the direct sensitivity of the two-dimensional

response to different orderings of the individual field-matter interactions.

We should note that, although there will be different patterns generated for

different interaction time orderings even when there are the same number



of interactions for each vibrational mode, for example, g411

and g141

, if the

coupling is considered time independent the responses must be weighted

equally and therefore they cannot be considered on an individual basis (27).


There have been a number of recent experimental investigations aimed at

measuring the two-dimensional nonresonant Raman response (21–32). One

of the greatest challenges in measuring the direct two-dimensional Raman

spectrum comes from contamination by the third-order cascaded responses

(see Section II.C). At the time of this writing, the potential magnitude

of this contamination in current experimental results had only just become

evident (32). In this section we will present the current state of experimental

progress, and we will discuss the potential for future investigations designed

to measure the unadulterated direct two-dimensional response.

A. Experimental Setup

The results presented here were obtained using a standard Ti:sapphire-based

regeneratively amplified 3 kHz laser system that provided 45–65 fs pulses

(FWHM Gaussian) centered at 800 nm with ắ30 àJ/pulse. A diagram for

the generation of 5 individual pulses is shown in Fig. 7. The initial beam

was first split into three separate pulses with computer-controlled stages

providing the primary time delays, 2 and 4 . Two of the beams were

then split again, resulting in five beams of approximately equal intensity. The power was attenuated using a /2-plate and cube polarizer, and

the polarization of each beam at the sample was individually adjustable

with /2-plates. For all of the spectra presented, the polarization of all the

incoming beams was set parallel. The individual beams were focused into

a room-temperature, 1.0 mm thick liquid sample cell using a 30 cm singlet

lens. After the sample, the signal along a chosen phase matched direction

was selected with an iris and imaged onto a silicon photodiode. Modulating

one of the incoming beams with a mechanical chopper wheel, the signal

was collected using a lockin amplifier.

Copyright © 2001 by Taylor & Francis Group, LLC


Fleming et al.

Figure 7 Schematic diagram of the experimental setup to split the incoming laser

pulse into 5 separate pulses with the two adjustable time delays.

B. Intermolecular Motions in CS2

Experiments on liquid CS2 were conducted using a homodyne detection

configuration. The expression for the overall homodyne detected signal is

given in Equation (33). To ascertain the relative contribution from direct

and cascaded responses in the fifth-order experiments on liquid CS2 , we

employed three different phase matching geometries. These geometries are

shown in Fig. 8a–c. The signals were collected along the phase matched

direction k1 k2 k3 C k4 C k5 . In addition to the overall fifth-order

phase matched direction, we are also concerned with the intermediate

steps involved in both the sequential and parallel cascade processes. There

are two possible intermediate steps for the sequential cascade, along

k1 k2 k3 and k1 k2 C k4 , and there are two possible intermediate

steps for the parallel cascade along k1 k2 C k5 and k3 C k4 C k5 .

The overall signal intensity will be weighted by the magnitude of the

phase matching factor for each individual process [Equation (33)]. Using

Equation (12) we can express the magnitude of the phase matching factor,

F, as sinc(kl/2), where k represents the difference between the incoming

wavevectors and the signal wavevector and 1 is the path length.

Copyright © 2001 by Taylor & Francis Group, LLC

Fifth-Order 2D Raman Spectroscopy


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.

Copyright © 2001 by Taylor & Francis Group, LLC

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B. Intramolecular Vibrations in Carbon Tetrachloride

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