Terahertz beats of vibrational modes studied by

femtosecond coherent Raman spectroscopy

R. Leonhardt, W. Holzapfel, W. Zinth, W. Kaiser

To cite this version:

R. Leonhardt, W. Holzapfel, W. Zinth, W. Kaiser. Terahertz beats of vibrational modes studiedby femtosecond coherent Raman spectroscopy. Revue de Physique Appliquee, 1987, 22 (12),pp.1735-1741. <10.1051/rphysap:0198700220120173500>. <jpa-00245733>

HAL Id: jpa-00245733

https://hal.archives-ouvertes.fr/jpa-00245733

Submitted on 1 Jan 1987

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1735

Terahertz beats of vibrational modes studied by femtosecondcoherent Raman spectroscopy

R. Leonhardt, W. Holzapfel, W. Zinth and W. Kaiser

Physik Department Ell der Technischen Universitât München, München, F.R.G.

(Reçu le 9 juin 1987, accepté le 18 septembre 1987)

Résumé. 2014 Une technique par Raman cohérent femtoseconde récemment développée permet de mesurerdes spectres Raman cohérent à transformée de Fourier avec des différences haute-fréquence. L’excitationsimultanée de différents modes de vibration avec une force motrice accordable à large bande conduit à un fortbattement de la diffusion Raman cohérent de la sonde. La haute résolution temporelle du montageexpérimental permet de mesurer des battements de fréquence de plus de 10 THz avec une précision élevée.

Abstract. 2014 A recently developed femtosecond coherent Raman technique allows the measurement ofFourier transform coherent Raman spectra with high-frequency differences. The simultaneous excitation ofdifferent vibrational modes with a broad-band tunable driving force leads to a strong beating of the coherentRaman probe scattering. The high time resolution of the experimental set-up allows one to measure beatfrequencies of more than 10 THz with high precision.

Revue Phys. Appl. 22 (1987) 1735-1741 DÉCEMBRE 1987, PAGE 1735

Classification

Physics Abstracts42.65 - 32.50

1. Introduction.

During the past decade, time-resolved coherentRaman methods have attracted considerable interestfor the study of fast dynamic processes. In particular,time-resolved coherent Raman scattering made it

possible to measure - in the time domain - rapiddephasing processes of molecular vibrations in li-

quids and elucidated various line broadeningmechanisms [1-5]. It is the aim of this paper to focusattention on a more recent application. Althoughtime-resolved Raman scattering collects data in thetime domain, it allows valuable information to beobtained in the frequency domain. There are variousapproaches to relate the time and the frequencydomain [6-11] : (i) Time resolved coherent Ramanscattering, where the spectrum of the coherentlyscattered light is recorded, has been extensivelystudied using picosecond light pulses. It is possible toremove the homogeneous contribution to the transi-tion line-width. One finds spectra with narrowerlines and one obtains transition frequencies withhigh accuracy. This technique has been applied tothe study of congested spectral regions, where thespontaneous Raman spectrum is smooth, but wherethe line-narrowing technique revealed structure dueto distinct transitions [9-11]. (ii) In the second

method the time evolution of the coherent Raman

probe signal is recorded with high time resolution.

The Fourier transformation of the experimental datagives a difference frequency spectrum. The fre-

quency resolution of this technique can be improvedby using numerical filtering procedures prior to theFourier transformation. (iii) A third way to analysethe data is a comparison of the data taken in a hightime resolution experiment with time dependentfunctions. High accuracy can be obtained for thedetermination of difference frequencies.

In this letter we present new experimental data oftime-resolved coherent Raman spectroscopy takenwith a novel femtosecond Raman spectrometer [12,13]. The results of these time-domain experimentsare subsequently numerically analysed. In this waywe obtain frequency differences with high precision.The high time resolution of the femtosecond cohe-rent Raman set-up allows one to measure very highdifference frequencies. After a short theoretical

description of the basic ideas of the technique wegive experimental results. We show that terahertzquantum beats up to 10 THz can be measured givingprecise values for the frequency differences of vibra-tional transitions separated by up to 350 cm-1.

2. Theory.

Time-resolved coherent Raman scattering is com-

monly treated under the following assumptions [1].The light fields are described by Maxwell’s equation

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:0198700220120173500

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and the vibrational transitions are represented bytwo-level systems. Changes in the population of thetwo molecular levels are neglected (weak Ramaninteraction). The expectation value of the vibrationalamplitude, the coherent amplitude ( q), is therelevant quantity for the description of time-resol-ved coherent Raman spectroscopy. First, we discussthe coherent experiment for a single homogeneouslybroadened transition with dephasing time T2. Weuse the ansatz of plane waves for the coherent

amplitude ~q~ = (i/2) Q exp(- iúJq t + ikqx) +c.c. and assume an isotropic Raman tensor for theinvestigated vibration. The coherent amplitude of aRaman active mode at frequency w q is excited viatransient stimulated Raman scattering by a pair oflight pulses, the laser pulse EL and the Stokes pulseEs [1, 14]. The electric fields are considered to beplane waves, e.g.

with the wave vector kL and frequency £0 L for thelaser field. The driving force F (x, t ) for the coherentamplitude Q is proportional to the product of thelaser and the Stokes field, F (x, t ) oc EL Es ; as aresult the frequency of the excitation force is

W L - w S. The linear response theory applied to theexcitation process leads to the following equation forthe coherent amplitude Q :

The constant K contains material parameters suchas Raman cross-section and vibrational frequencywq (see Ref. [1]). T2 stands for the dephasing time ofthe transition. Detuning between the excitation

frequency and the vibrational mode is given byOw = 03C9L-03C9S-03C9q. Under the action of a short

exciting force the coherent amplitude (q) evolves asfollows : ~q~ rises to a maximum with the excitingforce and, subsequently, decays exponentially withthe dephasing time T’2. During the free exponentialdecay the molecules oscillate at the resonance fre-quency w q. Tn the experiment described here thelarge bandwidth of the short driving force allowsseveral vibrations to be excited simultaneously.Since the excitation via stimulated Raman interac-tion is weak, the various modes evolve in time

independent of each other. The different vibrationsmay be treated with the same ansatz introducing theindividual resonance frequencies úJ qj’ phase factors

Oj, amplitudes Qj, and dephasing times T2j.In the time-resolved coherent Raman ex-

periments, the amplitude (q) generated at timezero is monitored by coherent Raman scattering of adelayed probing pulse EL 2. We investigate light on

the anti-Stokes side of the Raman spectrum gener-ated by the nonlinear polarization PNLAS:

The coherent signal consists of two contributions.The first part on the r.h.s. of equation (2) describesthe coherent Raman signal of interest scattered fromthe excited vibrational mode. The constant À con-tains the Raman cross-section of the mode. This partof the coherent signal is emitted at the resonant anti-Stokes frequency ev AS = ev L 2 + ev q and providesinformation on the dynamic properties of the investi-gated mode. The second part on the r.h.s. of

equation (2) is related to the nonresonant nonlinear

susceptibility ~(3)NL resulting from electronic contribu-tions [15]. The nonresonant part of the signal givesan instantaneous response, i.e. it follows the productEL ES EL2 and decays very rapidly with the timeresolution of the experiment. The resonant part, onthe other hand, shows the slow response of theresonantly excited vibrations. The nonresonant partof the coherent signal does not provide any informa-tion on the investigated modes. It has to be separatedfrom the resonant signal. This fact is not possible inthe steady-state coherent anti-Stokes Raman spec-troscopy (CARS). In the time-resolved coherentanti-Stokes experiments described here the separa-tion is possible due to the different time dependenceof the two contributions : the rapidly decayingnonresonant part shows up around time zero, whileat later delay times valuable information on theslower resonant part of the signal is collected.

In the present coherent experiment we measurethe scattered Raman light as a function of the timedelay tp between excitation and probing pulses. Inorder to have optimum temporal resolution, pulsesof very short duration for the exciting and probinglaser fields are required.Under these conditions the nonresonant signal is

concentrated only around time zero and may beneglected at later times. Experimentally, one detectsthe time-integrated coherent signal S (tD ) :

According to equation (3) one measures the timedependence of the coherent amplitude or - moreexactly - one determines the absolute square of thecoherent amplitude, which decays exponentiallywith T,12.When several modes are excited simultaneously,

the signal decay is more complex. It can be shown

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that for very short driving and probing pulses thecoherent signal is equivalent to the absolute squareof the Fourier transform of the spontaneous Ramanspectrum [6, 7]. Here we analyse the coherentRaman signal at late delay times, where the differentvibrational modes oscillate at their resonance fre-

quencies W qj and the amplitudes decay exponentiallywith the individual dephasing times T2j. The cohe-rent anti-Stokes field is the sum of the light fieldgenerated by the individual molecular modes Qj.For short light pulses of duration tp T2,tp 1/ âw ij one obtains :

According to equations (3) and (4) simultaneouslyexcited modes lead to a beating of the coherentsignal at the frequency differences 039403C9ij between thevarious transition frequencies 039403C9ij = lù - lù j. These« quantum beats » of the coherent signal may giveaccurate information on the vibrational spectrum.The effect of different parameters on the coherent

signal is readily seen for the simple example of twovibrational modes. We assume that the two modeswith a frequency difference Aw and dephasing timesT21 and T22 are excited to the coherent amplitude3/o- For equal amplitudes Qlo = Q20 one obtains forthe coherent signal

The signal consists of the two contributions of theindividual transitions decaying with the respectivedephasing times (first two expressions on the r.h.s.of Eq. (5)). The interference between the two

molecular modes gives rise to the third term of

equation (5), which causes strong modulation of thesignal. When the two dephasing times stronglydiffer, the modulation depth of the signal decreaseswith increasing time. At late delay times the modu-lation dies out and a single unmodulated exponentialdecay appears. As an example, the signal curve fortwo vibrational modes at âw = 1013 s- 1

(= 53 cm-1), T2, = 6 ps and T22 = 2 ps, and

Q1 (o )/Q2 (o ) = 0.9 is depicted in figure 1. At earlytimes, when the two coherent amplitudes Q, (t ) andQ2(t) are not yet affected by the exponential decay,the signal is strongly modulated by the frequencydifference Aw. At later times the more rapidly

Fig. 1. - Coherent Raman signal calculated for two

simultaneously excited molecular modes with the fre-

quency difference à w = 1013 s-1 1 (A 53 cm-1 ). The initial

amplitudes have a ratio of Q10IQ20 = 0.9 and dephasingtimes T21 = 6 ps and T22 = 2 ps. The modulation of thesignal decreases with the shorter one of the two dephasingtimes.

decaying coherent amplitude, and with it the modu-lation, vanishes. Data similar to figure 1 allow thedetermination of the two dephasing times and thefrequency difference Aw.

For more complicated situations equations (3) and(4) permit the numerical analysis of the experimentaldata. With a least square-fit procedure one maydeduce the molecular parameters such as dephasingtimes and frequency differences, by minimizing thedifference between the experimental data and thetheoretical curves. As shown below, the molecularquantities may be determined with high precision.An alternative possibility to obtain the frequency-

domain information from the time domain data isthe straight forward application of a Fourier transfor-mation [7]. As discussed in the literature it is advan-tageous to multiply the experimental data by a

correction function, which compensates the expo-nential decay of the signal within the boundaries ofthe experimental time range. In this way the calcu-lated line shape is not Lorentzian, and the accuracyof the line positions is improved.

3. Experimental.

The experimental system is depicted schematically infigure 2. Pulses from a mode-locked argon-ion lasersynchronously pump two dye lasers. The speciallydesigned femtosecond unidirectional ring (UDR)dye laser [16, 17] contains an amplifying jet (dyeRhodamine 6G) and an absorber jet (dye DODCI).The laser emits pulses with a duration of 80 fs atÀ = 625 nm with an average power of 30 mW. The

pulses from this laser provide us with the exciting

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Fig. 2. - Experimental set-up used for Fourier transformcoherent Raman spectroscopy. A mode-locked argon-ionlaser synchronously pumps a femtosecond unidirectionalring dye laser (frequency ú) L) and a tunable picoseconddye laser (frequency ws). Excitation is done with a

synchronized pair of pulses at ú) Land ú) s’ while the

probing pulses at ú) L 2 are produced by a beam splitter BSfrom the femtosecond laser pulses w L. The coherent anti-Stokes light generated via the probing pulse is recorded asa function of the time delay adjusted by the variable delayline.

pulses EL and, via the beam splitter BS, with theprobing pulses EL 2. The second laser is a standardsynchronously mode-locked picosecond dye laserwith amplifying dyes DCM or Pyridine 2 and a three-plate birefringent filter as a tuning element (power30 mW). When tuned through the red and theinfrared part of the spectrum, pulses Es at the Stokesfrequency ws of 6 ps duration are generated. Thefemtosecond pulses from the UDR laser and thepicosecond pulses of the second laser, both withparallel polarization, serve to excite molecular vi-brations with a tuning range between 300 cm-1 and3 000 cm-1. It can be derived from equation (1) thatthe driving force exists only for the time duration ofthe shorter of the two pulses, i.e. for a time of 80 fs.A major advantage of the system is the wide

tunability of the excitation frequency WL - w s withthe help of the birefringent filter of the picosecondlaser while keeping the favorable femtosecond timeresolution. The coherent vibrational excitation ismonitored by probing pulses derived from the fem-tosecond laser. These pulses are polarized perpen-dicular to the excitation pulses. The three beams arefocused into the sample using a geometry appro-

priate for anti-Stokes phase matching. The anti-Stokes radiation EAS generated by the probingprocess passes the second polarizer and is detectedin conjunction with the broad-band spectrometer SPby the cooled photomultiplier PM. The coherentsignal is recorded as a function of the delay time (setby the variable delay line) between the exciting andprobing pulses.

4. Fourier transform cohérent Raman spectroscopyof transitions with terahertz frequency différences.

High-resolution Fourier transform coherent Ramanspectroscopy was first reported by Graener andLaubereau [7], who worked with light pulses of 20 psduration. The authors studied vibrational-rotationaltransitions in CH4 separated by less than 1 cm-1 withhigh frequency resolution of 10- 3 cm-1. The datawere taken over a time interval of 12 ns.In the

experiment described here we extend Fourier trans-form coherent Raman spectroscopy for the first timeto the study of large frequency differences up to350 cm-1. To see the related beating phenomenaone requires high time resolution. For example, afrequency difference of 350 cm-1 produces a beatingpattern with peaks separated by 95 fs. In order toresolve this very rapid phenomenon, one has to

work with femtosecond light pulses. Synchronizedlight pulses of approximately 80 fs duration becameavailable only recently [16, 17] permitting the experi-ments reported here.The first investigation treats vibrations in neat

pyridine and in pyridine/cyclohexane mixtures. Westudy Raman active vibrations of liquid pyridine atfrequencies 991 cm-1

1 and 1030 cm-1 1 assigned totwo A ring modes [18]. Both vibrations have similarspectral width (-.. 2.2 cm-1 ) and similar Raman

cross-sections ; they are separated by 39 cm-1. Inthe Raman excitation process a frequency differenceof (03C9L - ws)/2wc = 1010 cm-1 between the laserand the Stokes frequency is applied. Due to thebroad spectral width of the femtosecond excitingforce of more than 200 cm-’ 1 both pyridine modesare simultaneously excited in the experiment.Figure 3a shows the observed anti-Stokes signalplotted as a function of the time delay betweenexciting and probing pulses. During the excitationprocess at time zero the coherent signal rises to apronounced peak. It subsequently decays quicklyover more than one order of magnitude. Later onthe signal recovers and shows strong oscillations.

The modulation depths exceeds a factor of ten. Twofeatures of the coherent signal are of special interesthere : (i) the period of oscillation is approximately0.85 ps. Consequently, the frequency difference be-tween the two modes is 1.18 THz. (ii) The peaks ofthe oscillation decay exponentially with the decaytime T,12 = 2.55 ± 0.15 ps. The depth of the oscilla-

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Fig. 3. - Time resolved coherent Raman scattering ugingfemtosecond time resolution. (a) Beat pattern of purepyridine after excitation of the two pyridine modes at

991 cm- and 1 030 cm- 1. (b) Beat pattern of a mixture ofpyridine and cyclohexane. Three molecular modes areexcited.

tion slightly decreases for long observation timesindicating a difference between the dephasing timesof the two modes of 10 %.

In figure 3b we present an amazingly complicated,but perfectly reproducible anti-Stokes pattern. Amixture of cyclohexane and pyridine (1 : 1 by vol-ume) was investigated with femtosecond lightpulses. In this case three vibrational modes with

similar T2 values beat together. They are : one

cyclohexane mode at 802 cm-1 [19] and two pyridinemodes at 991 cm-1 1 and 1 030 cm-1. We find a richbeating structure originating from the interferenceof the three modes. The frequency differences of theexcited modes are determined by the followingprocedure. The exponential decay is removed bymultiplying the signal with an exponential risingfunction. An appropriate window function is intro-duced to remove the influence of the boundaries of

the time range. After these arithmetical manipula-tions a Fourier transformation of the time-depen-dent data gives the results shown in figure 4. Three

Fig. 4. - Difference spectrum of pyridine and cyclohex-ane obtained by numerical Fourier transformation of thetime resolved data from figure 3b. Note the strong peaksat 39 cm- 1, 189 cm- 1, and 228 cm- 1, which correspond tothe differences betweeen the three excited modes. The

highest difference frequency is 0 v = 6.84 THz.

sharp and pronounced peaks are found at 39 cm-1,189 cm-1 1 and 228 cm-1, which correspond to thefrequency differences between the three modesexcited in the mixture. The frequency width of thethree different lines is approximately 5 cm-1 ; thiswidth is due to the limited time range of observation

(e.g. 6 ps in Fig. 3). The background noise in

figure 4 reflects the experimental noise of figure 3and the applied window function.A very promising method to evaluate the exper-

imental data is the modelling of the beat patternaccording to equations (3) and (4). An example is

given in figure 5a, where we consider two modes tosimulate an extended beating pattern of neat

pyridine (redrawn in Fig. 5b for ready comparison).Two theoretical curves are calculated for the sameset of experimental parameters. Only the differencefrequency 039403C9 was changed by the small amount of0.4 cm- 1 (i.e. 1 %) from 39.2 cm - 1 (broken curve)to 39.6 cm-1 1 (solid curve). Superposition of theexperimental and calculated curves gives an im-

pressive fit over the entire time range. It is possibleto deduce a very precise frequency difference of39.4 ± 0.2 cm- 1.

Interesting results were obtained in the coherentRaman study of liquid nitrobenzene, where weobserve the highest beat frequencies. With an excita-tion frequency at 1 200 cm - te detect a rich beatingstructure. In figure 6b the coherent signal curve is

plotted on a linear scale for a time interval from0.2 ps to 2.5 ps. An extremely rapid modulation isfound with the shortest time between subsequentpeaks of less than 100 fs. A Fourier transform to thefrequency domain reveals four strong peaks at

16 cm-1, 104 cm-1, 237 cm-1, and 341 cm- 1, and a

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Fig. 5. - Time resolved coherent Raman data for liquidpyridine. (a) Two curves calculated according to

equations (3) and (4) using the dephasing times T21(991 cm-1 ) = 5.1 ps and T22 (1 030 cm-1 ) = 4.7 ps and

the frequency differences of 39.6 cm-1 and 39.2 cm-1.Comparison of the calculated curves with the experimentaldata of figure 5b gives a very accurate number for thefrequency difference between the two modes of àv =

39.4±0.2cm-1.

number of weaker peaks. The value of 341 cm-1 or10.5 THz corresponds to the beating of the twonitrobenzene modes at 1 000 cm-1 1 and 1 341 cm-1.Both modes were excited simultaneously in the

transient stimulated Raman process. In figure 6a weshow a beat curve calculated for the following set ofdata :

Using a least square-fit procedure to adjust thetheoretical parameters a very good agreement of thetheoretical and experimental curves is achieved. Thedata analysing procedure indicates that the frequency

Fig. 6. - Time resolved coherent Raman data for liquidnitrobenzene. (a) The curve (calculated by a least square-fit procedure should be compared with the results shownbelow). The four vibrational components with differencefrequencies 039403BD/c up to 340.8 cm-1 are listed in figure 6b.(b) Experimental curve of liquid nitrobenzene showinghigh-frequency beat phenomena extending to frequenciesof 10.5 THz.

differences da may be determined with a precisionapproaching 10- 3.

5. Summary.

In this paper we have extended Fourier transformcoherent Raman spectroscopy to the frequencyrange of 10 THz. With this technique we are nowable to resolve terahertz phenomena with highaccuracy.Two points are relevant : (i) the frequency resol-

ution is determined by the total time interval of theexperiment. In order to improve spectral resolutionthe coherent signal has to be measured over delaytimes as long as possible. Taking into account theexponential decay of the signal with T2/2 one

immediately finds that the spectral resolution is

directly related to the signal-to-noise ratio of the

experiment. (ii) The duration of the exciting and

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probing pulses does not influence the frequencyresolution of the experiment, but determines thehighest detectable beat frequencies to approximately

039403BDmax ~ 1 /tp. Under the present experimental condi-tions beat frequencies exceeding 10 THz are readilydetected.

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