Coherent
Multidimensional Four Wave Mixing in the Frequency Domain
The coherent multidimensional CARS frequency domain experiments described earlier were based on electronic transitions with uv/visible excitation pulses. This work led to the proposal of creating a new family of multiresonant four wave mixing methods that were based on infrared excitation pulses. By using infrared excitation pulses, this approach could be extended to vibrational transitions which are inherently sharper and offer greater selectivity in analytical measurements. However, vibrational transitions are also inherently weaker and consequently the nonlinear processes involving vibrational states are usually negligible in comparison with electronic states. This difference is inherent in the difference in mass between electrons and nuclei. The feasibility for creating a new family of coherent multiresonant FWM spectroscopies based on vibrational transitions was first demonstrated by showing that changes occurred in the FWM spectra when transitions were vibrationally resonant. Singly vibrationally enhanced (SIVE) FWM was developed to demonstrate that the vibrational contribution could be enhanced by resonance with vibrational transitions so the nuclear nonlinear polarization exceeded the nonresonant electronic polarization. After demonstrating the feasibility of SIVE-FWM, the methods were extended to doubly vibrationally enhanced (DOVE) and triply vibrationally enhanced (TRIVE) FWM by adding in additional vibrational resonances.
Doubly
Vibrationally
Enhanced Four Wave Mixing Spectroscopy (DOVE-FWM)
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to the top]
Figure 1
shows the WMEL and Mukamel diagrams for DOVE-FWM. There are three
pathways that
differ in the time ordering of the pulses and the resonances. There are
two
DOVE-IR pathways that involve excitation of two infrared transitions
from the
ground state followed by a normal nonresonant Raman transition
(although the
latter could be made resonant to achieve higher sensitivities) and one
DOVE-Raman pathway that involves a vibrationally enhanced Raman
transition
followed by a normal nonresonant Raman transition. The nonlinear
polarization
from all pathways add and interfere to create the output signal. Two
tunable
infrared excitation pulses (frequencies labeled as ω1
and ω2)
excite the vibrational coherences and then a third excitation (labeled ω3 and
typically in the
uv/visible) excites the Raman transition between the two vibrational
states.
The three beams are crossed at angles that provide phase
matching, 
.
Scanning ω1
and ω2
while monitoring the output at 
with
a monochromator
creates
a two-dimensional
spectrum.
Figure 2a
shows an example two-dimensional spectrum
for a mixture of CH3CN
and
CD3CN in C6D6.
The two major features
correspond to DOVE-IR resonances where ω1
is tuned to the (C≡N stretch + C-C stretch) combination band
and ω2
is tuned to the C≡N
stretch mode. There are also diagonal features that arise from
DOVE-Raman
resonances where ω1
is
tuned to the (C≡N stretch + C-C stretch) combination band and
ω1-
ω2
is tuned to the C-C stretch mode. There is also a
diagonal CARS feature
where ω1
is nonresonant and ω1-
ω2
is tuned to the C6D6
ring breathing mode. More discussion of this spectrum appears
elsewhere.
The
cross-peaks between vibrational transitions occur only when the modes
are
coupled by intra- or inter-molecular interactions. This coupling
requirement is
the major strength of multidimensional spectroscopy because it isolates
the
spectral features that are associated with interactions. The coupling
requirement is manifested in DOVE-FWM by the requirement that at least
one of
the transitions (2 infrared and 1 Raman) must involve a combination
band that
gets its transition strength from coupling. In the harmonic
approximation,
vibrational transitions require a change of quantum number 
and
with only three
transitions, this
requirement must be violated so a combination band can appear.
The
different pathways for generating the output coherence in nonlinear
spectroscopy result in interference effects between each
pathway’s polarization
and these appear as changes in the peak intensity and shape. These
effects are
best seen in figure 10 as the vanishing of the DOVE-Raman features as
they
merge with the strong DOVE-IR feature on the high energy side and the
enhancement of the features as they re-emerge on the low energy side.
These
effects can be rigorously modeled with simulations that allow one to
extract
the transition moments, coupling information, and dephasing rates for
the
different coherences. Figure 2b shows the simulation for the
experimental data
in figure 2a.
Triply
Vibrationally
Enhanced Four Wave Mixing Spectroscopy (TRIVE-FWM)
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Figure 3
shows the WMEL diagrams for TRIVE-FWM with two lasers of frequency ω1 and ω2. Note
that the subscripts in this section label the
frequencies of the excitation beams, not their time ordering. Three
beams are
created by splitting the ω2
beam into two beams, ω2
and ω2’.
The three beams are crossed
at angles that provide phase matching, 
.
Scanning ω1
and ω2
while monitoring the output at 
with
a monochromator creates
a two-dimensional
spectrum.
There are twelve TRIVE pathways that
differ in the time ordering of the three infrared excitation pulses
(the six
columns) and the resonant states (the two rows). The pathways in the
top row
are parametric processes where the final output coherence emission
returns the
system to the original state before the nonlinear process so no energy
remains
in the system from the interactions. The pathways in the bottom row are
non-parametric processes where the final output coherence emission
returns the
system to a vibrationally excited state so energy is delivered to the
system.
Unlike DOVE-FWM, all of the transitions can be allowed in the harmonic
approximation. Two tunable infrared excitation pulses (frequencies
labeled as ω1
and ω2)
excite the vibrational coherences and then a third
excitation (labeled ω3
and
typically in the uv/visible) excites the Raman transition between the
two
vibrational states. The three beams are crossed at angles that provide
phase
matching, .
Scanning ω1
and ω2
while monitoring the output at 
with
a monochromator creates
a two-dimensional
spectrum.
Figure 4 shows an example spectrum for a mixture of a bis-(triphenylphosphine) dicarbonyl nickel and a (triphenylphosphine) tricarbonyl nickel complex in tetrahydrofuran. The sides of the figure show the infrared spectrum of the mixture. Diagonal peaks correspond to both ω1 and ω2 exciting the same vibrational state while the cross-peaks correspond to ω1 and ω2 exciting two different vibrational states that are coupled. The boxes indicate the diagonal and cross-peaks associated with the Ni(CO)2(PPh3)2 (left-most box) and Ni(CO)3(PPh3) (right-most boxes). The peaks in the inset show that some peaks split because of the frequency domain manifestation of quantum beating that depends on the delay times between pulses.
One can also change the delay times
between pulses in order to measure the dynamics of the coherences and
populations. Changing the delay times can also change the time ordering
of the
pulses and thereby change the coherence pathway. Figure 5 shows a
diagram of
the relationship between the
coherence pathways in figure 3 and the
nonlinear processes
that occur for different delay times 
between the
ω1
and ω2’
and 
between the ω1
and ω2 excitation pulses. The
figure also shows the
relationship of these pathways to the other FWM spectroscopies. For
example,
the stimulated photon echo experiment corresponds to region V while the
normal
photon echo experiment corresponds to boundary between regions III and
V.
An example of the experimental
implementation of the delay scans is shown in figure 6. Here, the
lasers and
monochromator are fixed at positions appropriate for the lower right
cross-peak
in figure 4 and the 
and
delay
times are scanned. The monochromator monitors a
frequency that corresponds to the top row in figure 14. Since ω2
is detuned from the
anharmonically shifted combination band transition, pathways II and IV
are not
important and one observes the dynamics from the other pathways. As
an
example,
a diagonal cross-section of figure 6 in region I shows the dephasing
rate of
the v’g coherence in
pathway I (see
figure 4) while a horizontal cross-section shows the dephasing
rate of
the v’v coherence.
Similarly, a diagonal
cross-section in region V shows the population relaxation rate of the gg and vv
populations. Similar dynamical data can be obtained by fixing
the excitation frequencies and monochromator to emphasize other
pathways.
The cross-peaks in these spectra also require coupling, just as they did for DOVE FWM. In the DOVE-FWM case, the coupling appeared as the intensity of the combination bands. For TRIVE FWM, the transitions are all allowed. The cross-peaks would vanish without coupling because the parametric and nonparametric pathways in figure 4 have opposite signs and cancel. Coupling creates anharmonic shifts in the frequency or changes in the transition moments or relaxation rates so the two interfering pathways are not equivalent and do not cancel.