B. Force Correlation Function Approach
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computed. But there are two signiﬁcant problems. First, it is presently
impossible to do the fully quantum mechanical simulation on a system of
any reasonable size. Classical mechanics may be used instead, but then a
“quantum correction” must be introduced (52,60,61) to Equation (4). The
quantum correction may be quite large (orders of magnitude) (52), and
computing it accurately for anharmonic baths is problematic. Second, we
often need to know the VER rate constant for vibrations with frequencies
× ωD . At higher frequencies far above the fundamental characteristic
bath frequency, the Fourier transform is very small. For example, in liquid
O2 , the vibrational frequency D 1552 cm 1 , whereas ωD ¾ 50 cm 1 , so
the Fourier transform is required at a frequency about 30 times greater than
the highest fundamental bath frequency (52). In such cases, numerical errors
in the calculation of the correlation function lead to large inaccuracies in
the Fourier transform.
C. Perturbation Approach
An alternative approach widely used in polyatomic molecule studies is
based on the Golden Rule and a perturbative treatment of the anharmonic
coupling (57,62). This approach is not much used for diatomic molecules.
In the liquid O2 example cited above, the Hamiltonian must be expanded
to 30th order or so to calculate the multiphonon emission rate. But for
vibrations of polyatomic molecules, which can always ﬁnd relatively loworder VER pathways for each VER step, perturbation theory is very useful.
In the perturbation approach, the molecule’s entire ladder of vibrational
excitations is the “system” and the phonons are the “bath.” Only lowerorder processes are ordinarily needed (57) because polyatomic molecules
have many vibrations ranging from higher to lower frequencies and only
a small number of phonons, usually one or two, are excited in each VER
ˆ q ,Q
step. The usual practice is to expand the interaction Hamiltonian V
in Equation (2) in powers of normal coordinates (57,62):
ˆ D
V
˛
C
C
ˆ
∂V
∂Q˛
Q˛ C
fQD0g
1
2!
ˆ
∂3 V
∂Q˛ ∂Q ∂Q∂
˛,
1
3! ˛,
,ˇ
1
4! ˛,
ˆ
∂4 V
∂Q˛ ∂Q ∂Q∂ ∂Qε
,∂,ε
ˆ
∂2 V
∂Q˛ ∂Q
Q˛ Q
fQD0g
Q˛ Q Q∂
fQD0g
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Q˛ Q Q∂ Qε C Ð Ð Ð
fQD0g
5
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549
For polyatomics, usually only the last two terms in Equation (5),
the cubic and quartic anharmonic terms, need be considered (57). The
lowest-order process involving cubic anharmonic coupling involves excited
vibration relaxing by interacting with two other states, say another vibration ω and one phonon ωph , or alternatively a pair of phonons. For example,
the total rate constant for energy loss from for cubic coupling was given
by Fayer and coworkers as (57)
K D
[nω 1 C n
Cω
Cω C Cω
ω
C 1 C nω ˛ C nj
ωj
ω Cj
ωj ]
6
where nω is the thermal occupation number,
nω D eh¯ ω/kT
1
1
7
ω is the density of phonon states at ω, Cω is a product of coupling constants
that contains factors such as h¯ /2 ω and the derivatives of V in Equation (5),
and ˛ D 1 if
> ω or ˛ D 0 if
< ω. When T ! 0, all the thermal
occupation factors in Equation (6) vanish, but the VER rate does not vanish.
VER is then said to occur via spontaneous emission of phonons. As T is
increased, two new thermally activated processes turn on. One involves
stimulated phonon emission and the other phonon absorption. Spontaneous
and stimulated emission processes convert
to a lower energy vibration
ω (down-conversion). Phonon absorption converts
to a higher energy
vibration ω (up-conversion).
Some representative examples of common zero-temperature VER
mechanisms are shown in Fig. 2b–f. Figures 2b,c describe the decay of
the lone vibration of a diatomic molecule or the lowest energy vibrations
in a polyatomic molecule, termed the “doorway vibration” (63), since it
is the doorway from the intramolecular vibrational ladder to the phonon
bath. In Fig. 2b, the excited doorway vibration
lies below ωD , which
can be the case for large molecules or macromolecules. In the language of
Equation (4), ﬂuctuating forces of fundamental excitations of the bath at
frequency are exerted on the molecule, inducing a spontaneous transition
to the vibrational ground state plus excitation of a phonon at ωph D .
The rate of this transition is proportional to the Fourier transform of the
force-force correlation function at frequency , denoted C
.
In Fig. 2c, the vibration
lies well above the phonon cut-off ωD ,
as for example the 379 cm 1 doorway vibration in ACN (46), where ωD
is in the 100–150 cm 1 range. Fluctuating forces exerted by the bath at
frequency
cause the doorway vibration to decay. In the language of
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550
Iwaki et al.
Equation (5), VER involves a higher-order anharmonic coupling matrix
element, which gives rise to decay via simultaneous emission of several
phonons nωph (multiphonon emission). In the ACN case, three phonons
must be emitted simultaneously via quartic anharmonic coupling (or four
phonons via ﬁfth-order coupling, etc.).
In Fig. 2d, is one vibrational fundamental of a polyatomic molecule,
whose relaxation involves exciting ω1 , a lower energy vibration of the same
molecule. In the language of Equation (5), , ω1 , and other intramolecular
vibrations are part of the system and the phonons are the bath. Fluctuating
forces exerted by the bath at frequency ωph D
ω1 induce a transition
from to ω1 via cubic anharmonic coupling. This mechanism, which ought
to be most important in polyatomic molecules where the spacings between
adjacent vibrations are generally less than ωD (4), is the “ladder relaxation”
(57) process mentioned previously. The name derives from the motion of
the vibrational excitation, which hops downward from one vibration to
another, which are the “rungs” of the “ladder” of vibrational fundamentals
of the polyatomic molecule.
In Fig. 2e,
decays by an intramolecular vibrational redistribution
(IVR) process, involving lower energy vibrations ω1 , ω2 , ω3 , . . . , via a
higher-order anharmonic coupling which causes to decay by spontaneous
emission of several lower energy vibrations ω1 , ω2 , ω3 , . . . . In condensed
phases, phonons may play a role in IVR as well, which is analogous
to the role of rotations in gas-phase IVR, by dynamically modulating
and broadening the vibrational energy levels, making it more likely for
a resonance to occur.
D. Vibrational Cascade
The “vibrational cascade” (64) illustrated in Fig. 2f is widely believed to
be a prominent mode of vibrational cooling (VC). The general properties of
vibrational cascades in large molecules at ﬁnite temperature were studied
theoretically by Hill and Dlott (4,5). The vibrational cascade occurs when
the lowest-order ladder processes dominate. In a large molecule, the rungs
of the vibrational ladder are on average closely spaced. Here closely spaced
means the average energy difference is less than ωD , so that a step from
one rung to another can occur with just one phonon. In this case, energy
loss from a polyatomic molecule at zero temperature, with vibration
initially excited, occurs by a sequence of VER processes, each involving the
emission of just one phonon via cubic anharmonic coupling. A vibrational
cascade in an intermediate-size molecule such as ACN or NM might involve
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Ultrafast IR-Raman Spectroscopy
551
some steps where two or three phonons are emitted via quartic or ﬁfthorder coupling. In a vibrational cascade at zero temperature, a vibrational
excitation descends the ladder, losing a small amount of energy in each step.
At the bottom rung of the ladder, the doorway vibration, the ﬁnal VER step
cannot occur by a ladder process since there are no more lower energy rungs
(4). The ﬁnal VER step occurs by a single or multiphonon emission process
(e.g., Fig. 1b or 1c). At ﬁnite temperature, each step along the ladder might
go up or down, depending on whether phonons are absorbed or emitted.
However the net motion of excess vibrational energy is always downward,
to states of lower energy (4,5).
Hill and Dlott (5) illustrated the properties of vibrational cascades in
model calculations of VC in crystalline naphthalene. Naphthalene C10 H8
has 48 normal modes. Forty of these vibrations (all except the eight C–H
stretching vibrations) lie in the frequency range 1627–180 cm 1 . In the
calculation, one unit of excitation is input to the highest vibration in this
range, 1627 cm 1 . The ensemble-averaged population of the ith mode is
determined by a master equation:
dP t
DKÐP t ,
8
dt
where P t is a vector of vibrational populations and K is a matrix of transition rate constants. The elements of the rate matrix were computed using
Equation (6), which assumes that VER occurs solely by cubic anharmonic
coupling. In naphthalene, the phonon density of states ωph is accurately
known from neutron scattering measurements (65,66). The coupling factors
C in Equation (6) were determined using the density of states and VER
lifetimes determined by low-temperature vibrational lineshape measurements (5,67). Where VER measurements were not available, C was taken
to be the average of the known values.
The average energy jump E for every step up or down the ladder in
a large molecule is approximately equal to the average phonon frequency,
Eavg D
h¯ ω
ph
ω dω
ph
ω dω
9
For naphthalene (5), Eavg D 95 cm 1 .
Figure 3 shows the results of a calculation (5) at T D 0 assuming
an initial condition of unit vibrational excitation at 1627 cm 1 . As time
progresses, the center of the vibrational population distribution moves
toward lower energy. The vibrational population distribution spreads out
because the size of each step is distributed over the range 0–¯hωD , with the
average step being 95 cm 1 . Vibrational cooling is essentially complete
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Iwaki et al.
Figure 3 Calculated vibrational cascade in crystalline naphthalene at T D 0, for
initial excitation at 1627 cm 1 . The calculation uses Equation (6), which assumes
that cubic anharmonic coupling dominates. From Ref. 5.
by about 200 ps. When the vibrational density of states is a constant, the
population distribution should eventually approach a Gaussian distribution
(4,5). The peak of the distribution will move toward lower energy at a
constant velocity (the “vibrational velocity,” with units of energy dissipated
per unit time) and the width will increase as the square root of time. The
dashed curves in Fig. 3 are the best ﬁts to this Gaussian distribution.
The temperature dependence of VC discovered by Hill and Dlott (4,5),
as a consequence of the temperature dependence predicted by Equation (6),
is very interesting. Equation (6) shows that the lifetime, that is the rate of
leaving a particular state, decreases with increasing T. Equation (6) has
three parts: temperature-independent spontaneous emission (downward)
and temperature-dependent stimulated emission (down) and absorption
(up). It is the increase in the rates of the latter two temperature-dependent
processes that causes the lifetime to decrease with increasing T. The two
temperature-dependent processes by themselves do not cause vibrationally
hot molecules to cool, since they are as likely to drive an excited
vibration to higher energy states as to lower energy states. Increasing
the temperature only increases the rate of vibrational energy jumping
up and down. Any actual cooling, which is caused by the net motion
of vibrational excitation to lower energy, is driven by the temperatureindependent spontaneous emission processes. Thus the VC process is
largely independent of temperature when only lower order anharmonic
coupling processes dominate. This point is illustrated in Fig. 4, which plots
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553
Figure 4 Average energy of the nonequilibrium vibrational population distribution computed for the vibrational cascade in crystalline naphthalene in Fig. 3. At
T D 0, the peak moves toward lower energy at a roughly constant rate, the “vibrational velocity” of 8.9 cm 1 ps. The initial 1627 cm 1 of vibrational energy is
dissipated in ¾180 ps. The vibrational velocity is the same at 300 K. In the limit
that cubic anharmonic coupling dominates [Equation (6)], increasing the temperature increases the rates of up- and down-conversion processes, but has no effect on
the net downward motion of the population distribution. Although the lifetimes of
individual vibrational levels will decrease with increasing temperature, VC is not
very dependent on temperature in this limit. (From Ref. 5.)
the time dependence of the average energy of the population distribution
after 1627 cm 1 excitation at T D 0 and T D 300 K. At both temperatures
the population distribution moves down at an approximately constant rate,
with a vibrational velocity V0 D 9 cm 1 /ps. That is to say, the average
rate of energy lost from the molecule is 9 cm 1 per picosecond; losing
1627 cm 1 takes about 180 ps. This calculation shows that although the
rates of VER processes may increase dramatically with T, the overall rate
of VC ought not to be much affected by temperature.
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Iwaki et al.
III. THE IR-RAMAN TECHNIQUE
A. The Method
A tunable mid-IR pulse at frequency ωIR pumps vibrational excitations
in a polyatomic liquid (all work discussed here is at ambient temperature
¾295 K). A time-delayed visible probe pulse at frequency ωL generates
incoherent anti-Stokes Raman scattering. For an instantaneous pump pulse
arriving at time t D 0, the change in the anti-Stokes intensity of transition
i, with frequency ωi , the “anti-Stokes transient,” is (44)
IAS
t D constni t gi
i
Ri
ωL C ωi
4
10
where the constant depends on the experimental set up, ni t is the instantaneous change in vibrational population, gi is the degeneracy, and Ri is
the Raman cross section. Equation (10) shows that the intensity of an antiStokes transient is proportional to the population change in the vibrational
transition during the VER process induced by the pump pulse (2).
The IR-Raman experiment is difﬁcult. An ultrashort tunable mid-IR
pulse is needed that produces a substantial number of vibrational excitations. An intense visible pulse is also needed that generates enough
anti-Stokes signal photons despite the small magnitude of the Raman cross
section. Detecting small numbers of anti-Stokes photons is no problem
with today’s 90% quantum efﬁciency CCD detectors. The real problem
arises from optical background, as discussed below. The laser pulses must
be short enough to time-resolve VER processes of interest, but if they
are too short undesirable effects occur (45): (1) the spectral bandwidth of
the pulses becomes too broad to resolve individual vibrational transitions
and (2) the short pulses at ﬂuence levels needed to generate and detect
transient vibrational populations will generate optical background in the
sample via nonlinear optical interactions. We designed our apparatus to
produce an ¾0.8 ps pulse, which is short enough to time-resolve most
VER processes of interest. The transform limited spectral bandwidth of
a 0.8 ps pulse is ¾20 cm 1 , but practical bandwidths in our system are
25–35 cm 1 (45). Figure 5 shows IR and Raman spectra of neat liquid
ACN, obtained using conventional spectrometers with resolution better than
the natural linewidths (46). In Fig. 5 we also show a Stokes Raman spectrum obtained with the ultrafast laser system. Most of the Raman transitions
can be resolved, although the C–H bending modes in particular tend to run
together.
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Ultrafast IR-Raman Spectroscopy
555
Figure 5 Vibrational spectra of neat liquid acetonitrile (ACN): (top) mid-IR spectrum; (bottom) Stokes Raman spectrum using a conventional spectrometer (solid
line). The dashed spectrum obtained with the ultrafast laser system has somewhat
lower resolution (From Ref. [46].)
B. The Laser
Since the early IR-Raman experiments with Nd:glass lasers, Ti:sapphire
lasers with chirped-pulse ampliﬁcation (CPA) (68,69) have revolutionized
ultrafast spectroscopy. Ti:sapphire lasers ordinarily run in femtosecond
mode (pulse duration ¾100 fs) where the spectral bandwidth >140 cm 1
is too large for Raman spectroscopy, so methods have to be found to
lengthen the Ti:sapphire pulses and reduce the spectral bandwidth. That
is a bit ironic, since so much work has been devoted to producing pulses
with ever-shorter duration. Efforts are being made today to produce longerduration, spectrally narrower bandwidth pulses from Ti:sapphire (45) or to
convert femtosecond pulses into spectrally narrower picosecond pulses in a
more efﬁcient manner than simply removing energy with a narrow bandpass
ﬁlter (70).
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556
Iwaki et al.
The experimental setup used at the University of Illinois is
diagrammed in Fig. 6. A CPA laser from Clark-MXR Corp. was modiﬁed
by the manufacturer to run in picosecond mode by substituting more
dispersive gratings (2000 lines/mm for 1400 lines/mm) and by adding
birefringent ﬁlters (71) in the regenerative ampliﬁer cavity. The system
outputs ¾0.8 ps duration pulses with 1.0 mJ energy at a repetition rate of
1 kHz.
The most common technology used today to produce intense pulses
at wavelengths other than 800 nm and its harmonics is optical parametric
ampliﬁcation (OPA) (72). In an OPA, a “signal” laser pulse at frequency ωS
propagates through a nonlinear crystal along with an intense pump pulse at
Figure 6 Block diagram of the two-color optical parametric ampliﬁer (OPA)
and IR-Raman apparatus. CPA D Chirped pulse ampliﬁcation system; Fs
OSC D femtosecond Ti:sapphire oscillator; Stretch D pulse stretcher; Regen D regenerative pulse ampliﬁer; SHGYAG D intracavity frequency-doubled Q-switched
Nd:YAG laser; YAG D diode-pumped, single longitudinal mode, Q-switched
Nd:YAG laser; KTA D potassium titanyl arsenate crystals; BBO D ˇ-barium
borate crystal; PMT D photomultiplier tube; HNF D holographic notch ﬁlter;
IF D narrow-band interference ﬁlter; CCD D charge-coupled device optical array
detector. (From Ref. 96.)
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Ultrafast IR-Raman Spectroscopy
557
ωP . The pump pulse ampliﬁes the signal while simultaneously producing
an “idler” pulse at frequency ωI D ωP ωS . For 800 nm Ti:sapphire
pump pulses, the idler will be tunable throughout the mid-IR (broadly
400–4000 cm 1 ) for signal pulses in the 820–1176 nm range, provided the
nonlinear crystal is transparent in all these spectral regions. We needed a
design that would produce tunable mid-IR pulses, but we wanted the probe
pulses to have a ﬁxed wavelength so that readily available interference
ﬁlters and holographic Raman notch ﬁlters could be used in the detection
setup. In our development phase we took anti-Stokes spectra from various
liquids using ¾0.8 ps pulses at 800 and 400 nm. The longer wavelength
800 nm pulses had poor scattering efﬁciency due to the ω4 dependence in
Equation (10). The 400 nm second harmonic pulses, in principle 16 times
better than the fundamental, produced too much multiphoton ﬂuorescence
and ionization. Believing that a probe pulse in the ¾500 nm region was
optimal, we designed at two-color OPA system (45) as in Fig. 6.
Our OPA is based on potassium titanyl arsenate (KTA) crystals (73).
KTA is quite similar to KTP which is widely used in OPAs, but KTA
has better mid-IR transparency than KTP. The two-color OPA is seeded
(the “seed” is the signal pulse) in a quasi-CW fashion (45,74) by a diodepumped Q-switched Nd:YAG laser running in a single longitudinal mode
at 1.064 àm, which generates ắ10 àJ in a 50 ns pulse (45). Only a ¾1 ps
duration slice of this seed pulse is ampliﬁed in the OPA. The seed power
is about 1 kW. Laser seeding produces narrower bandwidths than the more
usual white-light seeding, and it avoids the problems of generating a seed
supercontinuum with the picosecond pulses (75), which is more difﬁcult
than with femtosecond pulses. With the Nd:YAG seed, the center frequency
of the ampliﬁed signal output will remain pinned at 1.064 µm. The signal
pulses are frequency doubled in a BBO crystal to produce the ﬁxedfrequency Raman probe pulses at 532 nm, where off-the-shelf optics and
ﬁlters are readily available. The mid-IR idler pulses can be tuned over an
¾100 cm 1 range by simply tilting the crystals and over an ¾1000 cm 1
range by tuning the CPA laser. When the CPA pump pulses are tuned in
the 770–820 range, the mid-IR output ranges from 2800–3600 cm 1 . This
mid-IR range allows us to pump almost any molecule containing at least
one hydrogen atom, including common functional groups such as C–H,
O–H, S–H, N–H, etc. (76).
The performance of the two-color OPA (45) is illustrated in Fig. 7.
Mid-IR pulse energies of 40–50 µJ are typically obtained in a nominal
0.8 ps pulse with a 35 cm 1 spectral bandwidth. These are very large midIR energies for a kilohertz laser (45). It is easier to obtain high conversion
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558
Iwaki et al.
Figure 7 Performance of the mid-IR OPA. The insets show a cross-correlation
between mid-IR pulses and 532 nm pulses from the OPA, with FWHM of 0.8 ps,
and the 35 cm 1 FWHM spectrum of mid-IR pulses at 3000 cm 1 . (From Ref. 45.)
efﬁciency with our 0.8 ps pulses than with 100 fs pulses. The higher pump
power of femtosecond pulses is not really an advantage because the power
at the nonlinear crystal must be kept below a critical level, typically a
few hundred GW/cm2 , which is determined mainly by the onset of supercontinuum generation. For a 100 fs pulse with ¾150 cm 1 bandwidth,
our calculations have shown that the KTA crystal length is limited to
<2 mm. The crystal length is ordinarily limited either by group-velocity
mismatch or by the acceptance bandwidth (72). For a 100 fs pulse, the
acceptance bandwidth becomes limiting (i.e., it is ¾150 cm 1 ) at ¾2 mm
thickness, and the group-velocity mismatch becomes signiﬁcantly limiting
at a slightly greater thickness. For 0.7–1.0 ps duration pulses, the bandwidth and the group-velocity mismatch becomes limiting in the 7–10 mm
thickness range. Because the OPA gain in the small-depletion limit is an
exponential function of the usable crystal length (72), the ability to use
longer (7–10 mm) crystals in our picosecond OPA provides us with enormous small-signal gain. With the gain so great, the OPA can be run deeply
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