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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 iIJ2 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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