I. COHERENT ANTI-STOKES RAMAN SPECTROSCOPY OF SIMPLE LIQUIDS
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in addition to conventional infrared and Raman spectroscopy. To study
the dephasing properties of vibrational transitions, nonlinear Raman
spectroscopies have been developed, representing special versions of the
pump-probe technique, e.g., coherent anti-Stokes Raman scattering (CARS)
and coherent Stokes Raman scattering (CSRS) (21). The ﬁrst liquid
examples were stretching vibrations of carbon tetrachloride, ethanol at room
temperature (3), and the fundamental mode of liquid nitrogen (22), while
phonon modes of calcite (23,24) and diamond (25) were addressed in the
early solid-state investigations. Over the past decades a variety of gases and
liquid and solid state systems have been studied using time-domain CARS
(26–28). Higher-order Raman techniques were also demonstrated (29,30).
B. General Considerations
The CARS and CSRS processes are generally described as four-wave
mixing (31,32); in the time domain spectroscopy with delayed pump and
probe ﬁelds the elementary scattering mechanism is split into a two-step
two-wave interaction (21). For excitation two laser pulses are applied, i.e.,
two coherent electromagnetic waves with appropriate frequency difference
interact with the molecular ensemble and drive a speciﬁc vibrational mode
with transition frequency ω0 resonantly (or close to resonance); “Raman”
is used here as a synonym for “frequency difference resonance.” The
same interaction is involved in the stimulated Raman effect, so that the
latter process was applied in early measurements for the excitation process.
The probing process is coherent scattering of the additional interrogation
pulse off the phase-correlated vibrational excitation, i.e., classical scattering
involving the induced polarization of the molecular ensemble and producing
side bands ωP š ω0 (Stokes and anti-Stokes) of the probe frequency ωP . The
process is the optical phonon analog for light scattering of coherent acoustic
phonons in ultrasonics (e.g., Debye-Sears effect). The two-step interaction
is illustrated in Fig. 1. The pumping process is represented by the simple
energy level scheme of Fig. 1a with the ground and the ﬁrst excited levels
of the considered vibration; the vertical arrows represent the involved pump
photons with frequencies ωL (“laser”) and ωS (“Stokes”), respectively. The
wave vector diagram is also depicted in Fig. 1a; an off-axis beam geometry
is assumed for the input ﬁelds represented by wave vectors kL and kS . The
resulting vector kv represents the spatial phase relation of the vibrational
excitation imposed on the molecular ensemble.
The coherent anti-Stokes scattering of a probing pulse generating
radiation with frequency ωA D ωP C ω0 and wave vector kA is depicted in
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Figure 1 Schematic representation of a time-resolved coherent Raman
experiment. (a) The excitation of the vibrational level is accomplished by a
two-photon process; the laser (L) and Stokes (S) photons are represented by vertical
arrows. The wave vectors of the two pump ﬁelds determine the wave vector of the
coherent excitation, kV . (b) At a later time the coherent probing process involving
again two photons takes place; the probe pulse and the anti-Stokes scattering are
denoted by subscripts P and A, respectively. The scattering signal emitted under
phase-matching conditions is a measure of the coherent excitation at the probing
time. (c) Four-photon interaction scheme for the generation of coherent anti-Stokes
Raman scattering of the vibrational transition.
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Fig. 1b. Obviously ωP may be equal to or differ from ωL . The latter case is
termed three-color CARS and avoids undesirable frequency coincidences
with the secondary processes of the excitation step (22,33). Repeating the
measurement with different time delays between pump and probing pulses,
the loss of coherent vibrational excitation may be observed from the decay
of the scattering signal. The generated anti-Stokes emission is highly collimated and occurs in the direction of wave vector kA as shown by the wave
vector diagram in the ﬁgure. In general, the physical situation is more
complex, since the mentioned four-wave mixing also provides a nonresonant component for temporal overlap of pump and probing pulses. A
corresponding level scheme is indicated in Fig. 1c.
In the electric dipole approximation, one may write for the induced
polarization of the medium the following:
P D N[∂˛/∂q]iso hqiE C N[∂˛/∂q]aniso hqiE C
3
nr
EEE
(1)
The vector character of P and E is omitted here for simplicity. N denotes the
number density of molecules. Equation (1) shows that the coherent Raman
scattering consists of three components: two resonant terms, which are
proportional to the coherent vibrational amplitude hqi and to the change of
the molecular polarizability with nuclear coordinate, ∂˛/∂q (21). hqi is the
ensemble-averaged quantum mechanical expectation value of the normal
mode operator. The coupling ∂˛/∂q is split into an isotropic (iso) and an
anisotropic (aniso) part. We recall that ∂˛/∂q is a tensor generally deﬁned
in the molecular frame and that the isotropic and anisotropic contributions have to be transformed into the laboratory frame. The third term in
Equation (1) represents the nonresonant nonlinear contribution, which may
be quite strong in liquid mixtures and solutions and exists only for temporal
overlap of the excitation and probing pulses (34). The following expressions can be derived for the three scattering components in the resonant
case, ωL ωS D ω0 (35):
Piso t D Fiso EP t
Paniso t D Faniso EP t
Pnr t D Fnr EP t
t
tD
tD
1
t
vib t
1
vib t
tD EL t ES t
Ł
t0 EL t0 ES t0
t0 or t
Ł
dt0
t0 EL t0 ES t0
2
Ł
dt0
3
4
EP , EL , and ES denote the electric ﬁeld amplitudes of the three input
pulses (probe, laser, and Stokes). tD is the delay time of the probe ﬁeld
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relative to the coincident excitation components. It is important to note
that the relative contributions of P depend on the orientations of the electric ﬁeld vectors, i.e., chosen polarization geometry. The latter effect is
described in Equations (2)–(4) by the time-independent prefactors F that
are explicitly known (see below). The F’s also contain the different coupling
elements of ∂˛/∂q and nr [Equation (1)]. vib and or , respectively, represent the vibrational and orientational autocorrelation functions of individual
molecules and enter Equations (2)–(4) in various ways; the resulting differences in temporal behavior of the scattering parts are signiﬁcant. The
equations above refer to moderate pulse intensities so that stimulated ampliﬁcation of the Stokes pulse and depletion of the laser pulse can be ignored.
The measured CARS signal Scoh is proportional to the time integral
over the absolute value squared of the total third-order polarization, P D
Piso C Paniso C Pnr , because of the slow intensity response of the detector:
Scoh tD D const ð
1
1
jP t, tD j2 dt
(5)
The signal Scoh represents a convolution integral of the intensity of the
probing pulse / jEP t tD j2 with the molecular response; the latter is
governed by the autocorrelation functions vib and or . Numerical solutions
of Equations (2)–(5) are readily computed and will be discussed in the
context of experimental results.
vib and or also show up in the theory of spontaneous Raman spectroscopy describing ﬂuctuations of the molecular system. The functions
enter the CARS interaction involving vibrational excitation with subsequent
dissipation as a consequence of the dissipation-ﬂuctuation theorem and
further approximations (21). Equations (2)–(5) refer to a simpliﬁed picture;
a collective, delocalized character of the vibrational mode is not included
in the theoretical treatment. It is also assumed that vibrational and reorientational relaxation are statistically independent. On the other hand, any
speciﬁc assumption as to the time evolution of vib (or or ), e.g., if exponential or nonexponential, is made unnecessary by the present approach.
Homogeneous or inhomogeneous dephasing are included as special cases. It
is the primary goal of time-domain CARS to determine the autocorrelation
functions directly from experimental data.
Regarding the relationship between CARS and conventional Raman
spectroscopy, as is evident from the equations above, the scattered antiStokes ﬁeld amplitude (proportional to P) depends linearly on the autocorrelation functions. With respect to molecular dynamics and disregarding the
minor point that the ﬁeld amplitude is not directly measured, CARS is a
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linear spectroscopy and cannot provide more information than is available
from conventional Raman spectroscopy. On the level of present theoretical
approaches, both methods are simply related by Fourier transformation and
deliver the same information. This is of course only true in principle, not in
practice for real measurements, because of the different role of experimental
accuracy in the two techniques. For example, the asymptotic exponential
decay of vib was observed over more than three orders of magnitude, while
the Raman bandshape could not be measured with similar precision because
of the contributions of neighboring lines, especially in congested parts of
the spectrum. In short, coherent experiments can provide dephasing data
of superior accuracy. On the other hand, conventional Raman spectroscopy
is well suited for measuring frequency positions or shifts. The time- and
frequency-domain versions of vibrational spectroscopy are complementary,
and the combination of the respective results is particularly rewarding.
As far as CARS distinguishing between homogeneous and inhomogeneous broadening mechanisms, some investigators supported the idea that
CARS as a linear technique with respect to molecular response does not
do this (36). The present authors question that opinion; in fact, examples
will be discussed below in which dephasing in the homogeneous, intermediate, or inhomogeneous case was distinguished on the basis of femtosecond
CARS data. On the other hand, it is generally accept that higher-order techniques like infrared echo or Raman echo measurements can more directly
differentiate between homogeneous and inhomogeneous dephasing mechanisms (37).
Two important improvements in time-domain CARS spectroscopy
have been made in recent years and will be brieﬂy discussed in the following
areas:
High-precision CARS (38)
CARS with magic polarization geometry (35,39)
C. Experimental Aspects
In the early days of time-resolved CARS it was often convenient to use
laser and probing pulses at the same frequency position, leading to twocolor CARS ωP D ωL . The approach has the disadvantage that secondary
interaction processes of the excitation pulses also generate emission at the
anti-Stokes frequency position ωL C ω0 , representing an undesirable background (not depending on delay time) for the detection of the coherent
probe scattering at ωP C ω0 . In more advanced approaches, therefore, the
frequency coincidence is avoided (22,38). The latter version, three-color
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CARS, can provide more accurate data because of its higher sensitivity
and lower intensity level of the excitation pulses. The preferred frequency
position of the probing pulse, in general, is between the laser and Stokes
components, ωL > ωP > ωS . We mention here that phase matching arguments for anti-Stokes scattering (21) would suggest a frequency position
close to the Stokes frequency, but the ﬁnite bandwidth of ultrashort pulses
makes a signiﬁcant frequency shift necessary between the (intense) laser
pump and (weak) anti-Stokes scattering at ωP C ω0 .
As an example the experimental apparatus used by the authors’ group
is brieﬂy discussed. The system is based on femtosecond dye laser technology and depicted schematically in Fig. 2b (38,40). Using an ampliﬁed and frequency-doubled, modelocked Nd-YLF laser with repetition rate
50 Hz for synchronous pumping, a hybrid modelocked dye-laser oscillator
is operated. After multipass dye ampliﬁcation of a single pulse, part of the
laser radiation is directed to a quartz plate for continuum generation. Out of
the produced spectral broadening, two frequency bands are selected by pairs
of interference ﬁlters and ampliﬁed in two additional dye ampliﬁers for the
generation of the Stokes and probe pulses. Together with the second part
of the laser pulse that also passes narrow-band ﬁlters, three different input
pulses of approximately 250 fs duration and 50–70 cm 1 width are accomplished. For a given set of three pairs of interference ﬁlters and ampliﬁer
dyes, tuning ranges of the three pulses are accomplished by angle variation
of the ﬁlters (565–571 nm, 675–689 nm, and 605–619 nm for L, S, and P,
respectively). A nonlinear absorber cell (NA) in the probe beam in front of
the sample improves the pulse contrast and helps to increase the dynamical
range of the CARS scattering signal.
Applying /2 plates and a Glan polarizer (Pol1), parallel linear
polarization of the input laser and Stokes pulses is adjusted. For reasons
discussed below the polarization plane of the probe pulse (Pol2) is inclined
by an angle ÂP D 60° with respect to the pump polarization, while in earlier
work an angle of 90° was used. High-quality polarization optics including a
2 mm sample cell practically free of stress birefringence are used. An offaxis beam geometry is adopted providing phasematching for the anti-Stokes
scattering of the probe pulse, as calculated from refractive index data.
The coherent Raman scattering is measured behind an analyzing
polarizer (Pol3) transmitting radiation with the polarization plane oriented
at angle ÂA relative to the vertical pump polarization. A small aperture (AP)
deﬁnes the solid angle of acceptance (³10 5 sr) along the phasematching
direction. The scattering is detected at the proper anti-Stokes frequency
position, using dielectric ﬁlters (IF) with a bandwidth of 80 cm 1 , variable
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Figure 2 (a) Polarization geometries for the suppression of the nonresonant (˛), resonant-isotropic (ˇ),
and resonant-anisotropic ( ) CARS components. Constant polarization of the input ﬁelds EL , ES , and EP ;
magic angles ÂA for the orientation of the detected anti-Stokes ﬁeld EA . (b) Schematic diagram of the
experimental system for three-color CARS with magic polarization conditions. NA, nonlinear absorber;
VD, variable delay; Pol1-Pol3, polarizers; A, aperture; F, calibrated neutral ﬁlters; IF, interference ﬁlters;
PM, photomultiplier.
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neutral ﬁlters (F), and a photomultiplier (PM). The input pulse energies
are also monitored and used to correct the signal amplitude for the single
shot ﬂuctuations <20% of the input pulses. The instrumental response
function, determined by a measurement of the nonresonant CARS signal
of carbon tetrachloride [compare Equation (4)] decays exponentially over
an accessible dynamical range of 106 , suggesting exponential wings of the
input pulses. From the decay of the curve with a slope of 1/60 fs 1 , the
available experimental time resolution is deduced. In earlier applications
of the experimental setup a slightly different time resolution of 80 fs was
achieved. An example is shown in Fig. 3a (open circles, dashed curve). For
the adjusted frequency difference in wavenumber units of ωL ωS /2 c D
2925 cm 1 in CCl4 , off-resonance CARS via the nonresonant part nr of
the third-order nonlinear susceptibility is measured and plotted in the ﬁgure
on a logarithmic scale. The signal maximum is normalized to 1, while its
abscissa position deﬁnes zero delay. The observed steep signal decay by a
factor of 106 within 1 ps is noteworthy.
1. High Precision fs-CARS
For a demonstration of the performance of the instrumental system, some
results for neat acetone at room temperature are depicted in Fig. 3a (38).
The symmetrical CH3 stretching mode at 2925 cm 1 is resonantly excited.
The anti-Stokes scattering signal of the probing pulse with perpendicular
polarization plane relative to the pump beams is plotted versus delay time
(full points, logarithmic scale). The maximum scattering signal (exceeding
the off-resonance scattering of CCl4 by two orders of magnitude) is normalized to unity and displays a small delay relative to the instrumental response
function. For tD > 0.5 ps the signal transient decreases exponentially over
a factor >106 corresponding to a linear dependence in the semi-log plot.
From the slope of the decay curve the time constant T2 /2 D 304 š 3 fs is
directly deduced. For long delays a weak background signal shows up. The
solid curve in Fig. 3a is calculated from Equations (2)–(4). The relevant
ﬁtting parameter for the resonant CARS signal is the dephasing time T2 .
The accuracy of the data is illustrated by Fig. 3b. The ratio of the
signal amplitudes of the experimental points to that of the calculated signal
curve of Fig. 3a is plotted. It is interesting to see the minor scatter of the
data with approximately constant experimental error (Ä10%) in spite of the
signal variation over many orders of magnitude. Each experimental point
represents the average of approximately 400 individual measurements. The
reproducibility of the slope of the signal decay is better than š3 ð 10 3 .
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Figure 3 Femtosecond nondegenerate CARS in liquids: (a) Coherent probe scattering signal versus delay time; open circles, dashed curve: nonresonant scattering
of CCl4 yielding the instrumental response function and the experimental time resolution of 80 fs; full points, solid line: resonant CARS signal from the CH3 -mode of
acetone at 2925 cm 1 , obtaining T2 /2 D 304 š 3 fs. (b) Ratio of experimental and
calculated scattered data of (a) for acetone versus delay time; the small experimental
error of the data points extending over 6 orders of magnitude is noteworthy.
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Taking into account a possible calibration error of the neutral ﬁlters used
to detect the CARS signal, an experimental accuracy of š1% is estimated
for the T2 measurement of Fig. 3.
2. Magic Polarization Conditions
Early work on time-domain CARS was devoted to the measurement of
the vibrational dephasing time T2 , i.e., the time constant accounting for
the asymptotic signal decay. In the general case (not fully depolarized
vibrational transition, sufﬁciently short pulses), the latter originates from
the isotropic component of the nonlinear polarization P, since the other
parts decrease more rapidly. The nonresonant contribution responds almost
instantaneously and follows the wings of the input pulses. The decay of
the anisotropic part is accelerated by the additional effect of reorientational
motion compared to the purely vibrational relaxation of the isotropic scattering [Equations (2), (3)]. The remaining problem for the spectroscopist,
of course, is to recognize when the signal transient has reached the asymptotic behavior. For more information on molecular dynamics, it is highly
desirable to separate the three scattering contributions.
A remedy obviously should be available using polarization tricks. In
conventional Raman spectroscopy, the isotropic and anisotropic components are deduced from linear combinations of the “polarized” and “depolarized” spectra, while a nonresonant part is not clearly recognized (41).
In frequency-domain CARS it is known how to suppress the nonresonant contribution and solely measure resonant scattering (isotropic plus
anisotropic part) (42). In time-domain CARS, polarization interference can
do an even better job with three “magic” cases (derived in Refs. 35,39).
These authors derived explicit expressions for the coupling factors F in
Equations (2)–(4):
Fiso D iÄ˛2 cos ÂP ÂA
Faniso D iÄ2/45 ð 2 [2 cos ÂP cos ÂA sin ÂP sin ÂA ]
Fnr D nr /2 ð [3 cos ÂP cos ÂA C sin ÂP sin ÂA ]
6
7
8
Ä combines several material parameters. ˛ and denote the isotropic and
anisotropic parts of the Raman polarizability tensor ∂˛/∂q. nr represents
here the xxxx element of the nonresonant third-order susceptibility. The
above equations refer to the parallel pump polarization depicted in Fig. 2b.
The above expressions show that for the polarization geometry often
adopted in earlier investigations with ÂP D ÂA D 90° , the isotropic contribution is maximal but the two other components are also present. It is more
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attractive to choose ÂP 6D ÂA and consider situations with constant ÂP D 60°
and variable ÂA . Three magic values are found, where one of the coupling
factors alternatingly vanishes, Fi ÂA D 0:
p
(no anisotropic contribution)
ÂA D tan 2/ 3 ' 49.1°
ÂA D 30°
(no isotropic component)
ÂA D 60°
(no nonresonant contribution)
Simply adjusting these values for the analyzer orientation, different signal
transients are measured where the CARS signal contains only two contributions. The magic polarization geometries are depicted in Fig. 2a. The
theoretical results were veriﬁed experimentally (35,39). Reduction of the
suppressed components by several orders of magnitude was accomplished.
A set of measurements with the three magic angles allows one to
determine the three scattering components with different time dependencies
separately. Examples are presented in the next section. The following pieces
of information become accessible in this way:
Isotropic scattering: In addition to the dephasing time T2 , the correlation time c of the purely vibrational relaxation process can be
measured, providing quantitative information on the question of
homogeneous/inhomogeneous line broadening.
Anisotropic part: The reorientational relaxation of the vibrating
molecular subgroup becomes directly experimentally accessible.
Nonresonant part: Instrumental response function and zero setting of
delay time scale are provided.
Peak amplitudes: The relative magnitudes of the coupling parameters
˛, , and nr can be determined.
The mechanism selecting two scattering components out of three is polarization interference. The polarization of each scattering contribution (for
sufﬁciently weak, linearly polarized input ﬁelds) is linear but with tilted
polarization planes. The isotropic scattering, for example, occurs in the
plane of the incident probing ﬁeld. Blocking of this component simply
requires a crossed analyzer with ÂA D ÂP 90° .
The polarization dependence of the individual contributions can be
measured in special cases when the presence of the other two can be
excluded. Figure 4a presents results for the nonresonant CARS of neat
carbon tetrachloride excited for ωL ωS /2 c D 2925 cm 1 while a resonant vibrational mode does not exist; i.e., resonant scattering is absent. The
time evolution of the signal curve was presented in Fig. 3a (open circles).
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