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Femtosecond time-resolved four-wave mixing spectroscopy in iodine vapour
5 December 1997
Chemical Physics Letters 280 Ž1997. 339–347
Femtosecond time-resolved four-wave mixing spectroscopy in iodine vapour M. Schmitt, G. Knopp, A. Materny, W. Kiefer
1
Institut fur Am Hubland, D-97074 Wurzburg, Germany ¨ Physikalische Chemie der UniÕersitat ¨ Wurzburg, ¨ ¨ Received 9 July 1997
Abstract Femtosecond time-resolved four-wave mixing ŽFWM. spectroscopy is performed using degenerate laser frequencies in order to investigate molecular dynamics in iodine molecules in the gas phase. It is demonstrated that by varying the timing as well as the wavenumber position for the signal detection different dynamics can be accessed. This is due to a selection of contributions Že.g. described by Feynman diagrams. to the nonlinear polarization of the process. The wave packet dynamics of the electronic excited B as well as the X ground state of iodine can be monitored by degenerate FWM, coherent anti-Stokes Raman scattering or coherent Stokes Raman scattering ŽCSRS. processes all included in the same experiment. q 1997 Elsevier Science B.V.
1. Introduction A new research field has opened due to the development of lasers capable of producing ultrashort pulses. The pulses of a few tens of femtoseconds are shorter than relaxation processes which accompany the act of photochemistry and are even shorter than the elementary intramolecular motions of the molecules involved w1–3x. These pulses allow for a coherent broadband excitation with preparation and detection of vibrational superposition states. The time evolution of these rovibrational wavepackets in the ground as well as electronically excited state give information on the molecular dynamics. The high intensity achieved by femtosecond lasers favours the application of nonlinear methods like four-wave mixing ŽFWM. spectroscopies for the 1
Corresponding author.
study of ultrafast intramolecular dynamical processes. Compared with time-resolved resonance Raman scattering, time-resolved femtosecond coherent anti-Stokes or Stokes Raman scattering ŽCARS and CSRS, respectively. open new possibilities for the study of processes of ultrafast molecular dynamics w4–10x. Also degenerate FWM ŽDFWM. proved to be a valuable tool for the investigation of processes on ultrashort time scales w11–13x. Recently, Motzkus et al. w13x used DFWM to replace the probe pulse in a pump–probe scheme. They demonstrated the advantages of this technique giving access also to systems where a detection of laser induced fluorescence ŽLIF. is not possible. In their work Yang et al. w12x inter alia applied two- and three-pulse time-delayed DFWM to investigate I 2 in the vapour. They used ‘‘incoherent’’ light from a broadband dye laser with nanosecond pulse duration to achieve femtosecond time resolution. By
0009-2614r97r$17.00 q 1997 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 7 . 0 1 1 3 9 - 1
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M. Schmitt et al.r Chemical Physics Letters 280 (1997) 339–347
measuring the intensity of the signal as a function of the delay time between the second and the third pulse, the population as well as the orientational decay time can be determined. During the D t 2 interval, the system may be in either a rovibrational coherence of the ground or the excited electronic state or a vibrational population of either electronic state. For the gas phase the rovibrational coherences should decay much more rapidly than the populations. These coherences are those responsible for the rotational and vibrational quantum beats that have been observed in femtosecond pump–probe experiments w14,2,15–19x. In those experiments, the system interacts with the electric field of the single pump pulse to create a ground- or excited-state rovibrational coherence, which is after a certain delay detected by the probe pulse. In the three pulse experiment performed by Yang et al. w12x, the time D t between the two exciting pulses Žfirst and second pulse. is varied, while for the third laser pulse a constant time delay of about 10 ns is chosen. During this intervall, the system is in an electronic, not a rovibrational coherence. Therefore, the beats seen in their experiments have a different origin compared with the two-pulse pump–probe experiments. For I 2 vapour, the signal intensity as a function of delay time D t exhibited well defined beats with a separation Ž365 fs. corresponding to the vibrational wavenumber spacings in the B state Ž91 cmy1 . at the energy of the exciting laser pulses Ž l s 562 nm.. Here the vibronic transitions ÕX s 19,18 § ÕXX s 0 and ÕX s 21,22 § ÕXX s 1 were involved. It was assumed that the transitions from ÕXX s 2 were relatively insignificant in that excitation region due to small Franck-Condon factors, and transitions from ÕXX G 3 were negligible due to their low thermal populations. From the theoretical predictions it was to be expected that the transient should be significantly asymmetric with respect to D t. Beside the beats belonging to the excited electronic state also weak features were predicted having a spacing corresponding to vibrational energy differences between ground state vibrational levels. However, their experimental transient appears symmetric about D t s 0 and no beating significant for the ground state was observed. In order to correct their theoretical assumptions for their special experimental conditions they assumed that a complete rota-
tional energy randomization had taken place during the waiting time. Rotational energy transfer cross ˚ w21x. sections measured at both 5900 w20x. and 5461 A yield rotational relaxation times of 8–14 ns at low pressures. Vibrational relaxation is about five to ten times slower. The theoretical results obtained under these assumptions gave a signal shape which much closer resembled the experimental results. Very recently, first results were published introducing femtosecond time-resolved CARS spectroscopy applied to the investigation of molecular dynamics in gases w22x. There, it was demonstrated that by this method one can observe the dynamics of a wave packet either evolving on the ground or on the excited state potential energy surface ŽPES.. In order to compare this technique using femtosecond laser pulses with the results obtained by Yang et al. w12x, we repeated the experiments described in Ref. w22x with degenerated laser frequencies. In this letter we will introduce first results and discuss the versatility of this fs time-resolved FWM technique for obtaining information about molecular coherences.
2. Experimental The experimental apparatus has been described in detail elsewhere w22,23,19x. Briefly, a Ti:sapphire laser system in combination with two four pass OPAs was used to create the three femtosecond
Fig. 1. Beam path of the folded BOXCARS arrangement. L1 and L2 are achromatic lenses. The FWM signal is spatially filtered by a pinhole.
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pulses, having all the same center wavelength l0 . The pulses had energies of few mJ and pulse durations of typically 80 fs. The beam geometries employed in these experiments are diagrammed in Fig. 1. The folded BOXCARS configuration was employed in order to separate the signal from the incoming beams. In this geometry the phase-matching condition is fulfilled. The pulses were delayed relative to each other by means of Michelson interferometer arrangements. In the experiments presented in this paper, two laser pulses Ž pu1 and pu 2 . were kept temporally overlapped and fixed. The four-wave mixing ŽFWM. transients were recorded as a function of delay time between the two simultaneous excitation pulses and the third pulse put . The
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FWM signal was detected by a fast photomultiplier tube after it was spectrally filtered by a monochromator, in order to vary the detection wavelength. The sample cell was heated to about 608C which yields a vapour pressure of about 4.2 Torr. A spectral window of about 20 cmy1 for the detection was used for the experiments described below.
3. Results and discussion In the following we describe the basic idea of the femtosecond time-resolved four-wave mixing ŽFWM. experiment. In our experiments three laser fields interact with the ensemble of I 2 molecules. All three
Fig. 2. ŽA. Femtosecond FWM transient for DFWM conditions: detection at the central wavenumber position of the laser pulses Ž n˜det s n˜ 0 .. ŽB. and ŽC. are the FFT spectra obtained for negative and positive delay times, respectively. ŽD. FWM signal for D t s 0 Žfull line. and laser pulse shape Ždashed line. with detection wavenumber position marked. ŽE. and ŽF. are the energy level diagrams corresponding to the processes taking place for negative and positive delay times, respectively.
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laser pulses have the same wavelength l0 s 573 nm Ž n˜ 0 s 17452 cmy1 .. While two pulses Ž pu1 , pu 2 . are coincident in time Žt s 0. the third pulse Ž put . arrives with a variable delay time. Depending on the relative timing of the three pulses as well as the detection wavelength different dynamics in the molecules are probed by one or two photon interaction resulting in the coherent FWM signal. Panels A of Figs. 2–4 show FWM transients as a function of delay time D t between the pulse put and the two time coincident pulses pu1 and pu 2 for I 2 vapour. The coherent FWM signal is detected at different wavelengths Žcompare panels D of Figs. 2–4.. As the spectral widths of the pulses employed in our studies are about 250–300 cmy1 several nonlinear processes may take place at the same time. By
varying the detection wavelength, we are able to distinguish between three cases. 3.1. Degenerate four-waÕe mixing (DFWM) If the detection wavelength equals about the center wavelength of the laser pulses, then the time behaviour which results from a degenerated FWM ŽDFWM. process can be observed. The transient Žpanel A of Fig. 2: ldet f l0 . exhibits well defined beats with a separation of about 350 fs, corresponding to a vibrational energy spacing of about 94 cmy1 . This agrees well with the experimental vapour phase vibrational energy spacings in the excited B state accessed by the 573 nm lasers from the electronic and vibrational ground state.
Fig. 3. Same as Fig. 2 except for having CARS conditions: detection at a wavenumber position anti-Stokes shifted relatively to the center of the pulse shapes Ž n˜det ) n˜ 0 ..
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As already mentioned in the introduction, very recently Yang et al. w12x reported results on time resolved degenerate four-wave mixing experiments on I 2 in the vapour using nanosecond laser pulses. In contrast to our experiments they varied the time between two excitation pulses on a femtosecond time scale while for the probe pulse a constant delay time of more than 10 ns was chosen. They observed a transient signal with beats having a frequency characteristic for the spacing between vibrational levels of the B state near the center laser frequency. The rate of decay of the beats and the symmetry of the signal with respect to D t s 0 were consistent with the assumption that rotational energy randomization
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had taken place during the 10–20 ns waiting time between excitation and probe pulses. These authors also gave a theoretical expression describing the transient which results if neither the vibrational nor the rotational coherences are lost during the waiting time for the third pulse Žwhich is the case in an experiment using femtosecond laser pulses.. The calculated DFWM transient clearly shows beats at the excited state vibrational frequency, as well as weak features at about twice this frequency, which can be assigned to ground state vibrational beats. Furthermore, the calculated signal is significantly asymmetric with respect to D t s 0. The theoretical transient including the rotational
Fig. 4. Same as Fig. 2 except for having CSRS conditions: detection at a wavenumber position Stokes shifted relatively to the center of the pulse shapes Ž n˜det - n˜ 0 .. ŽE. shows two energy level diagrams for positive delay times. The right diagram is given in dashed lines due to minor importance.
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and vibrational coherences Žsee figure 5 of Ref. w12x. has an appearance quite similar to our experimental DFWM result shown in Fig. 2ŽA.. In our experiment we used two pulses, which are coincident in time Ž pu1Ž n˜ 0 . and pu 2 Ž n˜ 0 .. and a third pulse Ž put Ž n˜ 0 .. which is variable in time. The delay times in our experiments range from y8 to 8 ps. For negative delay times D t - 0 the laser pulse put excites the molecular system and the pulses pu1 and pu 2 act as probe. If positive delay times are chosen both pu1 and pu 2 excite the molecules, which afterwards is probed by put . During the short delay times applied in our experiments, a vibrational or rotational energy randomization has not taken place which makes the similarity between the calculations by Yang et al. w12x and our experiments appear reasonable. In order to analyze our experimental data more closely, we show the results of a fast Fourier transform ŽFFT. of the transient given in panel A of Fig. 2, in panels B and C for negative Ž D t - 0. and positive delay times Ž D t ) 0., respectively. For negative delay times the FFT spectrum ŽFig. 2ŽB.. of the DFWM transient ŽFig. 2ŽA.. exhibits a vibrational peak at about 94 cmy1 . As mentioned above, this value agrees with the energy spacings of the vibrational levels accessed in the B state of iodine. One also finds a weak feature at about 189 cmy1 which is the second harmonic belonging to the 94 cmy1 band. For positive delay times ŽFig. 2ŽC.. an additional vibrational peak at about 212 cmy1 occurs in the FFT spectrum. This wavenumber position agrees with the fundamental vibrational transition Ž DÕXX s 1. of iodine in the ground state. In order to understand why different dynamics are probed when the timing of the pulses involved in the nonlinear coherent FWM process changes, the following simple consideration may be applied. The nonlinear polarization responsible for the FWM signal can be theoretically described using a density matrix formalism. An elegant method of calculating the elements of the involved density matrix is using Feynman diagrams Žan application of this method on the calculation of nonlinear polarizations is described e.g. in Ref. w24x.. These diagrams include a time ordering of the possible interactions as well as information about the interactions themselves. The nonlinear susceptibility or polarization is derived by summing up all possible Feynman diagrams. If spec-
troscopical results in the frequency domain are described by this technique, a selection of important diagrams can be performed on the basis of resonance conditions. This means that the choice of laser and detection frequencies together with the molecular transitions determine which diagrams are of importance and which are not. For the femtosecond timeresolved experiments on systems which show coherences much longer than the pulse durations Žas is the case in our experiments. the timing of the laser pulses additionally selects Feynman diagrams which have the right time ordering. Instead of showing Feynman diagrams, we show energy diagrams in this letter affiliated to these diagrams, containing information about resonances as well as timing. For simplicity, we neglect in our discussion diagrams starting with transitions from thermally occupied higher vibrational ground state levels. The FFT spectra Žpanels B and C of Fig. 2. reveal that depending on the relative timing of the pulses different dynamics in iodine are probed. While for negative delay times the DFWM signal reflects the dynamics of the excited B state, for positive delay times the dynamics in the excited B state as well as in the electronic ground state for iodine are probed. The energy level diagrams belonging to the two possible time orderings for the DFWM process are shown in panels E and F for D t - 0 and D t ) 0, respectively. The energy diagram given in Fig. 2ŽE. shows that the interaction of put with the iodine molecule prepares a wave packet within the excited B state. This wave packet then is probed by the laser pulses pu1 and pu 2 resulting in the coherent DFWM signal. It is obvious from this diagram that no ground state contributions should be detected. However, for strong laser pulses it can be shown that a coupling of excited and ground state coherences is possible. Very recently, this was demonstrated for a femtosecond time-resolved coherent anti-Stokes Raman scattering ŽCARS. experiment w23x. For delay times D t ) 0 two possible diagrams are shown Žsee Fig. 2ŽF... The first one shows a two photon interaction preparing a wave packet on the electronic ground state of iodine. This contribution can be seen from the FFT spectrum given in panel C of Fig. 2. The second diagramm shows a strong excitation of the B state giving rise to an intense band at the vibrational frequency of this state accompanied by its second harmonic.
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3.2. Coherent anti-Stokes Raman scattering (CARS) Up to now we only have considered the totally degenerate case where we assumed the wavelengths of the interacting laser pulses as well as the signal at l0 Žcompare panel D of Fig. 2.. However, the laser pulses have a spectral width of about 250-300 cmy1 ŽFWHM. resulting in a FWM signal which also is spectrally broad. The energy diagrams now suggest that besides the DFWM also other FWM processes might be involved in the nonlinear coherent photonmolecule interactions. Without changing the laser wavelengths of the interacting beams Žhere, same l0 . there still remain several parameters which can be changed in order to vary the conditions for the time-dependent experiments. One of these parameters is the detection wavelength ldet which can be easily varied by changing the wavelength position of the monochromator used to disperse the signal light. If we chose a wavelength of ldet s 565 nm Ž n˜det s 17699 cmy1 . for detection, the shape of the resulting transient changes considerably Žpanel A of Fig. 3.. For negative delay times D t - 0 the beat patterns have the frequency of the wave packet in the excited B state of iodine. For positive delay times Ž D t ) 0. the beats have about twice this frequency and can be assigned to the electronic ground state. This becomes also evident from the FFT performed on the transient shown in Fig. 3ŽA.. Panels B and C of Fig. 3 show the FFT spectra for negative and positive delay times, respectively. The FFT spectrum for D t - 0 ŽFig. 3ŽB.. exhibits a line at about 94 cmy1 , which is the vibrational energy spacing in the excited B state. Additionally, the second harmonic of this frequency can be seen at about 189 cmy1 . For D t ) 0 the FFT spectrum ŽFig. 3ŽC.. shows only one peak at about 212 cmy1 which again can be assigned to the fundamental vibrational transition DÕXX s 1 of iodine in the ground state. This behaviour can be easily explained by assuming a FWM process which is favoured by the choice of an anti-Stokes detection wavenumber n˜det ) n˜ 0 Žcompare panel D of Fig. 3.. The corresponding FWM process is CARS. Here, we have put Ž n˜det y D n˜ . as time variable pump pulse. The pulses pu1Ž n˜det y D n˜ . and pu 2 Ž n˜det y 2D n˜ . act as second pump and Stokes lasers, respectively. D n˜ actually can assume any value within the spectral range of
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the involved laser pulses Žstarting from n˜det .. However, the process will be dominated by contributions where the laser intensities at Ž n˜det y D n˜ . and Ž n˜det y 2D n˜ . as well as the transition probabilities between the states involved are high. The energy level diagramms belonging to the time-resolved CARS are shown in panels E and F of Fig. 3 for delay times D t - 0 and D t ) 0, respectively. For negative delay times ŽFig. 3ŽE.. put interacts with the system first, resulting in a coherence in the excited B states of the iodine molecules. The wave packets are probed by the second pump and the Stokes pulse Ž pu1 and pu 2 , respectively. giving rise to the anti-Stokes signal at n˜det . For the process, the energy conservation as well as the phase matching conditions are fulfilled. This of course is also the case for the scheme described in the energy level diagram shown in Fig. 4ŽF.. Here, the time delayed pulse put probes the ground state wave packet which was coherently excited by the interaction of pu1 and pu 2 with the molecular system. As mentioned above, very recently results of femtosecond time resolved CARS spectroscopy for the simultaneous study of ground and excited state dynamics have been published w22,23x. Here, the pulses put and pu1 Ž‘‘pump pulses’’. had the same wavenumber positions, while pu 2 Ž‘‘Stokes pulse’’. was tuned to a lower wavenumber. The frequency difference between Stokes and pump laser pulses was tuned to a vibrational Raman transition in the ground state of the molecules Žno spectral overlap between put and pu1 with pu 2 .. Probing the coherent anti-Stokes signal resulted in transients for D t - 0 and D t ) 0 showing beats characteristic for wave packets prepared on the excited and ground state potential surfaces, respectively. This is in full agreement with the results obtained from the anti-Stokes detection in a FWM experiment using degenerate laser wavenumbers. 3.3. Coherent Stokes Raman scattering (CSRS) Instead of chosing a detection wavelength on the anti-Stokes side of the central laser wavelength l0 s 573 nm Ž n˜ 0 s 17452 cmy1 . it is of course also possible to set the monochromator to a position on the Stokes side. In the following we show results
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obtained for a detection wavelength ldet s 585 nm Ž n˜ 0 s 17094 cmy1 . which lies in the edge of the spectral range of the laser pulses Žcompare panel D of Fig. 4.. The FWM process which is favoured by the Stokes-detection is the coherent Stokes Raman scattering ŽCSRS.. Here, we have put Ž n˜det q D n˜ . as time variable pump pulse. The pulses pu1Ž n˜det q D n˜ . and pu 2 Ž n˜det q 2D n˜ . act as second pump and antiStokes lasers, respectively. As for the CARS process described above, D n˜ can take any value within the spectral range of the involved laser pulses Žstarting from n˜det .. The main contributions will now be determined by maximum laser intensities at Ž n˜det q D n˜ . and Ž n˜det q 2D n˜ . as well as the transition probabilities between the states involved in the CSRS process. The transient obtained under such conditions is shown in panel A of Fig. 4. Now, the transient does show neither signal nor beating structures for negative delay times D t - 0. However, for D t ) 0 intense beat patterns can be found at the excited B state vibrational frequency. This is also demonstrated by the FFT spectra shown in panels B and C of Fig. 4, which were obtained from the transient ŽFig. 4ŽA.. for negative and positive delay times, respectively. While for D t - 0 the FFT does not exhibit any dominating frequency contribution, the frequency spectrum describing the coherence probed for D t ) 0 has an intense band at about 94 cmy1 and possibly a very small contribution at the second harmonic position Žf 189 cmy1 . as well as at the ground state frequency Žf 212 cmy1 .. Again, possible energy level diagrams are shown in Fig. 4. Here, we do not find any diagram which might give considerable contribution to the CSRS signal if put interacts with the system first. This is in accordance to our observations. For positive delay times two diagrams are shown in panel E of Fig. 4. The left energy level diagram shows a coherent interaction where wave packets are prepared by pu1 and pu 2 on the excited B state potential energy surface and afterwards probed by put giving rise to the Stokes-shifted signal. This process is responsible for the bands at about 94 and 189 cmy1 . The right diagram only may give rise to a weak contribution from ground state dynamics if one takes into account the spectral width of the laser pulses involved. We would like to emphasize again that all processes
shown fulfill both the energy conservation and the phase matching conditions.
4. Conclusions In this letter we have demonstrated that femtosecond time-resolved four-wave mixing ŽFWM. spectroscopy with laser pulses having degenerate wavenumber positions is capable of giving information about the wave packet dynamics on both the electronic ground and excited state potential energy surface. The experiments were performed using three laser pulses in a folded BOXCARS configuration. Two of the pulses Ž pu1 and pu 2 . were time coincident while the third laser pulse Ž put . was variable in time. For negative delay times Ž D t - 0. the interaction of put with the molecular system was probed by the two simultaneous pulses pu1 and pu 2 giving rise to a coherent FWM signal. For positive delay times Ž D t ) 0. the interaction of pu1 and pu 2 takes place first and put later probes the excited dynamics. All experiments were performed on the iodine molecule in the gas phase. Besides the timing of the laser pulses also the detection wavenumber was varied giving rise to different transient shapes. Applying a Fourier transform on the transients for the two time ranges D t - 0 and D t ) 0 shows that both timing and detection wavenumber determine which wave packet dynamics is probed. This was interpreted as a selection of Feynman diagrams contributing to the density matrix presentation of the nonlinear polarization of the FWM process. The time ordering as well as resonances involved in the considered nonlinear processes drastically reduce the number of diagrams describing the FWM. Instead of the relatively complicated Feynman diagrams we used energy level diagrams to illustrate the possible interactions. The results can be itemized like follows: Ži. If the signal is detected at about the central wavenumber of the laser pulses Ž n˜det s n˜ 0 . the transient can be explained by diagrams belonging to the degenerate FWM ŽDFWM. process. For D t - 0 the DFWM transient reveals the dynamics of the excited B state of iodine, while for D t ) 0 a superposition of the ground and excited state dynamics is probed. Žii. Chosing a wavenumber on the anti-Stokes
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side of n˜ 0 for detection Ž n˜det ) n˜ 0 . time-resolved coherent anti-Stokes Raman scattering ŽCARS. dominates the transient shape. The transient signal obtained for negative delay times reveals the wave packet dynamics within the excited state. For positive delay times beats can be seen which have a frequency corresponding to the energy difference of vibrational levels in the electronic ground state. Žiii. Stokes-side detection Ž n˜det - n˜ 0 . yields transients which result from a coherent Stokes Raman scattering ŽCSRS. process. Here, no dynamics can be probed for negative delay times. However, for D t ) 0 the excited state dynamics results from the time-dependent coherent Stokes signal. In order to further investigate this interesting experimental technique experiments are in preparation where e.g. also the polarizations of the lasers will be changed. This will give access to different components of the nonlinear susceptibility x Ž3..
Acknowledgements This work was funded by the Deutsche Forschungsgemeinschaft ŽSchwerpunktprogramm ‘‘Femtosekunden-Spektroskopie elementarer Anregungen in Atomen, Molekulen ¨ und Clustern’’, Projekt KI 202r14-1.. Stimulating discussions with Stefan Meyer and Professor Volker Engel are gratefully acknowledged.
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w2x A.H. Zewail, Femtochemistry: ultrafast dynamics of the chemical bond, vols. I and II, World Scientific, Singapore, 1994. w3x J. Manz, L. Woste ¨ ŽEds.., Femtosecond Chemistry, VCH, Weinheim, 1995. w4x W. Zinth, R. Leonhardt, W. Holzapfel, W. Kaiser, IEEE J. Quantum Electron. QE-24 Ž1988. 455. w5x H. Okamoto, K. Yoshihara, J. Opt. Soc. Am. B. 7 Ž1990. 1702. w6x M. Fickenscher, A. Laubereau, J. Raman Spectrosc. 21 Ž1990. 857. w7x T. Joo, M.A. Dugan, A.C. Albrecht, Chem. Phys. Lett. 177 Ž1991. 4. w8x H. Okamoto, K. Yoshihara, Chem. Phys. Lett. 177 Ž1991. 568. w9x H. Okamoto, K. Yoshihara, Chem. Phys. Lett. 202 Ž1993. 161. w10x C.C. Hayden, D.W. Chandler, J. Chem. Phys. 103 Ž1995. 10465. w11x T. Hofer, ¨ P. Kruck, W. Kaiser, Chem. Phys. Lett. 224 Ž1994. 411. w12x T.-S. Yang, R. Zhang, A.B. Myers, J. Chem. Phys. 100 Ž1994. 857. w13x M. Motzkus, S. Pedersen, A.H. Zewail, J. Phys. Chem. 100 Ž1996. 5620. w14x A.H. Zewail, Faraday Discuss. Chem. Soc. 91 Ž1991. 207. w15x N.F. Scherer, D.M. Jonas, G.R. Fleming, J. Chem. Phys. 99 Ž1993. 153. w16x F.W. Wise, M.J. Rosker, C.L. Tang, J. Chem. Phys. 86 Ž1987. 2827. w17x S.L. Dexheimer, Q. Wang, L.A. Peteanu, W.T. Pollard, R.A. Mathies, C.V. Shank, Chem. Phys. Lett. 188 Ž1992. 61. w18x U. Banin, S. Ruhman, J. Chem. Phys. 98 Ž1993. 4391. w19x G. Knopp, M. Schmitt, A. Materny, W. Kiefer, J. Phys. Chem. 101 Ž1997. to be published. w20x J. Derourard, N. Sadeghi, Chem. Phys. Lett. 102 Ž1983. 324. w21x J.I. Steinfeld, W. Klemperer, J. Chem. Phys. 42 Ž1965. 3475. w22x M. Schmitt, G. Knopp, A. Materny, W. Kiefer, Chem. Phys. Lett. 270 Ž1997. 9. w23x M. Schmitt, G. Knopp, A. Materny, W. Kiefer, to be published. w24x M. Weissbluth, Photon–Atom Interactions Academic Press, Boston, 1988.
Chemical Physics Letters 280 Ž1997. 339–347
Femtosecond time-resolved four-wave mixing spectroscopy in iodine vapour M. Schmitt, G. Knopp, A. Materny, W. Kiefer
1
Institut fur Am Hubland, D-97074 Wurzburg, Germany ¨ Physikalische Chemie der UniÕersitat ¨ Wurzburg, ¨ ¨ Received 9 July 1997
Abstract Femtosecond time-resolved four-wave mixing ŽFWM. spectroscopy is performed using degenerate laser frequencies in order to investigate molecular dynamics in iodine molecules in the gas phase. It is demonstrated that by varying the timing as well as the wavenumber position for the signal detection different dynamics can be accessed. This is due to a selection of contributions Že.g. described by Feynman diagrams. to the nonlinear polarization of the process. The wave packet dynamics of the electronic excited B as well as the X ground state of iodine can be monitored by degenerate FWM, coherent anti-Stokes Raman scattering or coherent Stokes Raman scattering ŽCSRS. processes all included in the same experiment. q 1997 Elsevier Science B.V.
1. Introduction A new research field has opened due to the development of lasers capable of producing ultrashort pulses. The pulses of a few tens of femtoseconds are shorter than relaxation processes which accompany the act of photochemistry and are even shorter than the elementary intramolecular motions of the molecules involved w1–3x. These pulses allow for a coherent broadband excitation with preparation and detection of vibrational superposition states. The time evolution of these rovibrational wavepackets in the ground as well as electronically excited state give information on the molecular dynamics. The high intensity achieved by femtosecond lasers favours the application of nonlinear methods like four-wave mixing ŽFWM. spectroscopies for the 1
Corresponding author.
study of ultrafast intramolecular dynamical processes. Compared with time-resolved resonance Raman scattering, time-resolved femtosecond coherent anti-Stokes or Stokes Raman scattering ŽCARS and CSRS, respectively. open new possibilities for the study of processes of ultrafast molecular dynamics w4–10x. Also degenerate FWM ŽDFWM. proved to be a valuable tool for the investigation of processes on ultrashort time scales w11–13x. Recently, Motzkus et al. w13x used DFWM to replace the probe pulse in a pump–probe scheme. They demonstrated the advantages of this technique giving access also to systems where a detection of laser induced fluorescence ŽLIF. is not possible. In their work Yang et al. w12x inter alia applied two- and three-pulse time-delayed DFWM to investigate I 2 in the vapour. They used ‘‘incoherent’’ light from a broadband dye laser with nanosecond pulse duration to achieve femtosecond time resolution. By
0009-2614r97r$17.00 q 1997 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 7 . 0 1 1 3 9 - 1
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measuring the intensity of the signal as a function of the delay time between the second and the third pulse, the population as well as the orientational decay time can be determined. During the D t 2 interval, the system may be in either a rovibrational coherence of the ground or the excited electronic state or a vibrational population of either electronic state. For the gas phase the rovibrational coherences should decay much more rapidly than the populations. These coherences are those responsible for the rotational and vibrational quantum beats that have been observed in femtosecond pump–probe experiments w14,2,15–19x. In those experiments, the system interacts with the electric field of the single pump pulse to create a ground- or excited-state rovibrational coherence, which is after a certain delay detected by the probe pulse. In the three pulse experiment performed by Yang et al. w12x, the time D t between the two exciting pulses Žfirst and second pulse. is varied, while for the third laser pulse a constant time delay of about 10 ns is chosen. During this intervall, the system is in an electronic, not a rovibrational coherence. Therefore, the beats seen in their experiments have a different origin compared with the two-pulse pump–probe experiments. For I 2 vapour, the signal intensity as a function of delay time D t exhibited well defined beats with a separation Ž365 fs. corresponding to the vibrational wavenumber spacings in the B state Ž91 cmy1 . at the energy of the exciting laser pulses Ž l s 562 nm.. Here the vibronic transitions ÕX s 19,18 § ÕXX s 0 and ÕX s 21,22 § ÕXX s 1 were involved. It was assumed that the transitions from ÕXX s 2 were relatively insignificant in that excitation region due to small Franck-Condon factors, and transitions from ÕXX G 3 were negligible due to their low thermal populations. From the theoretical predictions it was to be expected that the transient should be significantly asymmetric with respect to D t. Beside the beats belonging to the excited electronic state also weak features were predicted having a spacing corresponding to vibrational energy differences between ground state vibrational levels. However, their experimental transient appears symmetric about D t s 0 and no beating significant for the ground state was observed. In order to correct their theoretical assumptions for their special experimental conditions they assumed that a complete rota-
tional energy randomization had taken place during the waiting time. Rotational energy transfer cross ˚ w21x. sections measured at both 5900 w20x. and 5461 A yield rotational relaxation times of 8–14 ns at low pressures. Vibrational relaxation is about five to ten times slower. The theoretical results obtained under these assumptions gave a signal shape which much closer resembled the experimental results. Very recently, first results were published introducing femtosecond time-resolved CARS spectroscopy applied to the investigation of molecular dynamics in gases w22x. There, it was demonstrated that by this method one can observe the dynamics of a wave packet either evolving on the ground or on the excited state potential energy surface ŽPES.. In order to compare this technique using femtosecond laser pulses with the results obtained by Yang et al. w12x, we repeated the experiments described in Ref. w22x with degenerated laser frequencies. In this letter we will introduce first results and discuss the versatility of this fs time-resolved FWM technique for obtaining information about molecular coherences.
2. Experimental The experimental apparatus has been described in detail elsewhere w22,23,19x. Briefly, a Ti:sapphire laser system in combination with two four pass OPAs was used to create the three femtosecond
Fig. 1. Beam path of the folded BOXCARS arrangement. L1 and L2 are achromatic lenses. The FWM signal is spatially filtered by a pinhole.
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pulses, having all the same center wavelength l0 . The pulses had energies of few mJ and pulse durations of typically 80 fs. The beam geometries employed in these experiments are diagrammed in Fig. 1. The folded BOXCARS configuration was employed in order to separate the signal from the incoming beams. In this geometry the phase-matching condition is fulfilled. The pulses were delayed relative to each other by means of Michelson interferometer arrangements. In the experiments presented in this paper, two laser pulses Ž pu1 and pu 2 . were kept temporally overlapped and fixed. The four-wave mixing ŽFWM. transients were recorded as a function of delay time between the two simultaneous excitation pulses and the third pulse put . The
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FWM signal was detected by a fast photomultiplier tube after it was spectrally filtered by a monochromator, in order to vary the detection wavelength. The sample cell was heated to about 608C which yields a vapour pressure of about 4.2 Torr. A spectral window of about 20 cmy1 for the detection was used for the experiments described below.
3. Results and discussion In the following we describe the basic idea of the femtosecond time-resolved four-wave mixing ŽFWM. experiment. In our experiments three laser fields interact with the ensemble of I 2 molecules. All three
Fig. 2. ŽA. Femtosecond FWM transient for DFWM conditions: detection at the central wavenumber position of the laser pulses Ž n˜det s n˜ 0 .. ŽB. and ŽC. are the FFT spectra obtained for negative and positive delay times, respectively. ŽD. FWM signal for D t s 0 Žfull line. and laser pulse shape Ždashed line. with detection wavenumber position marked. ŽE. and ŽF. are the energy level diagrams corresponding to the processes taking place for negative and positive delay times, respectively.
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laser pulses have the same wavelength l0 s 573 nm Ž n˜ 0 s 17452 cmy1 .. While two pulses Ž pu1 , pu 2 . are coincident in time Žt s 0. the third pulse Ž put . arrives with a variable delay time. Depending on the relative timing of the three pulses as well as the detection wavelength different dynamics in the molecules are probed by one or two photon interaction resulting in the coherent FWM signal. Panels A of Figs. 2–4 show FWM transients as a function of delay time D t between the pulse put and the two time coincident pulses pu1 and pu 2 for I 2 vapour. The coherent FWM signal is detected at different wavelengths Žcompare panels D of Figs. 2–4.. As the spectral widths of the pulses employed in our studies are about 250–300 cmy1 several nonlinear processes may take place at the same time. By
varying the detection wavelength, we are able to distinguish between three cases. 3.1. Degenerate four-waÕe mixing (DFWM) If the detection wavelength equals about the center wavelength of the laser pulses, then the time behaviour which results from a degenerated FWM ŽDFWM. process can be observed. The transient Žpanel A of Fig. 2: ldet f l0 . exhibits well defined beats with a separation of about 350 fs, corresponding to a vibrational energy spacing of about 94 cmy1 . This agrees well with the experimental vapour phase vibrational energy spacings in the excited B state accessed by the 573 nm lasers from the electronic and vibrational ground state.
Fig. 3. Same as Fig. 2 except for having CARS conditions: detection at a wavenumber position anti-Stokes shifted relatively to the center of the pulse shapes Ž n˜det ) n˜ 0 ..
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As already mentioned in the introduction, very recently Yang et al. w12x reported results on time resolved degenerate four-wave mixing experiments on I 2 in the vapour using nanosecond laser pulses. In contrast to our experiments they varied the time between two excitation pulses on a femtosecond time scale while for the probe pulse a constant delay time of more than 10 ns was chosen. They observed a transient signal with beats having a frequency characteristic for the spacing between vibrational levels of the B state near the center laser frequency. The rate of decay of the beats and the symmetry of the signal with respect to D t s 0 were consistent with the assumption that rotational energy randomization
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had taken place during the 10–20 ns waiting time between excitation and probe pulses. These authors also gave a theoretical expression describing the transient which results if neither the vibrational nor the rotational coherences are lost during the waiting time for the third pulse Žwhich is the case in an experiment using femtosecond laser pulses.. The calculated DFWM transient clearly shows beats at the excited state vibrational frequency, as well as weak features at about twice this frequency, which can be assigned to ground state vibrational beats. Furthermore, the calculated signal is significantly asymmetric with respect to D t s 0. The theoretical transient including the rotational
Fig. 4. Same as Fig. 2 except for having CSRS conditions: detection at a wavenumber position Stokes shifted relatively to the center of the pulse shapes Ž n˜det - n˜ 0 .. ŽE. shows two energy level diagrams for positive delay times. The right diagram is given in dashed lines due to minor importance.
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and vibrational coherences Žsee figure 5 of Ref. w12x. has an appearance quite similar to our experimental DFWM result shown in Fig. 2ŽA.. In our experiment we used two pulses, which are coincident in time Ž pu1Ž n˜ 0 . and pu 2 Ž n˜ 0 .. and a third pulse Ž put Ž n˜ 0 .. which is variable in time. The delay times in our experiments range from y8 to 8 ps. For negative delay times D t - 0 the laser pulse put excites the molecular system and the pulses pu1 and pu 2 act as probe. If positive delay times are chosen both pu1 and pu 2 excite the molecules, which afterwards is probed by put . During the short delay times applied in our experiments, a vibrational or rotational energy randomization has not taken place which makes the similarity between the calculations by Yang et al. w12x and our experiments appear reasonable. In order to analyze our experimental data more closely, we show the results of a fast Fourier transform ŽFFT. of the transient given in panel A of Fig. 2, in panels B and C for negative Ž D t - 0. and positive delay times Ž D t ) 0., respectively. For negative delay times the FFT spectrum ŽFig. 2ŽB.. of the DFWM transient ŽFig. 2ŽA.. exhibits a vibrational peak at about 94 cmy1 . As mentioned above, this value agrees with the energy spacings of the vibrational levels accessed in the B state of iodine. One also finds a weak feature at about 189 cmy1 which is the second harmonic belonging to the 94 cmy1 band. For positive delay times ŽFig. 2ŽC.. an additional vibrational peak at about 212 cmy1 occurs in the FFT spectrum. This wavenumber position agrees with the fundamental vibrational transition Ž DÕXX s 1. of iodine in the ground state. In order to understand why different dynamics are probed when the timing of the pulses involved in the nonlinear coherent FWM process changes, the following simple consideration may be applied. The nonlinear polarization responsible for the FWM signal can be theoretically described using a density matrix formalism. An elegant method of calculating the elements of the involved density matrix is using Feynman diagrams Žan application of this method on the calculation of nonlinear polarizations is described e.g. in Ref. w24x.. These diagrams include a time ordering of the possible interactions as well as information about the interactions themselves. The nonlinear susceptibility or polarization is derived by summing up all possible Feynman diagrams. If spec-
troscopical results in the frequency domain are described by this technique, a selection of important diagrams can be performed on the basis of resonance conditions. This means that the choice of laser and detection frequencies together with the molecular transitions determine which diagrams are of importance and which are not. For the femtosecond timeresolved experiments on systems which show coherences much longer than the pulse durations Žas is the case in our experiments. the timing of the laser pulses additionally selects Feynman diagrams which have the right time ordering. Instead of showing Feynman diagrams, we show energy diagrams in this letter affiliated to these diagrams, containing information about resonances as well as timing. For simplicity, we neglect in our discussion diagrams starting with transitions from thermally occupied higher vibrational ground state levels. The FFT spectra Žpanels B and C of Fig. 2. reveal that depending on the relative timing of the pulses different dynamics in iodine are probed. While for negative delay times the DFWM signal reflects the dynamics of the excited B state, for positive delay times the dynamics in the excited B state as well as in the electronic ground state for iodine are probed. The energy level diagrams belonging to the two possible time orderings for the DFWM process are shown in panels E and F for D t - 0 and D t ) 0, respectively. The energy diagram given in Fig. 2ŽE. shows that the interaction of put with the iodine molecule prepares a wave packet within the excited B state. This wave packet then is probed by the laser pulses pu1 and pu 2 resulting in the coherent DFWM signal. It is obvious from this diagram that no ground state contributions should be detected. However, for strong laser pulses it can be shown that a coupling of excited and ground state coherences is possible. Very recently, this was demonstrated for a femtosecond time-resolved coherent anti-Stokes Raman scattering ŽCARS. experiment w23x. For delay times D t ) 0 two possible diagrams are shown Žsee Fig. 2ŽF... The first one shows a two photon interaction preparing a wave packet on the electronic ground state of iodine. This contribution can be seen from the FFT spectrum given in panel C of Fig. 2. The second diagramm shows a strong excitation of the B state giving rise to an intense band at the vibrational frequency of this state accompanied by its second harmonic.
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3.2. Coherent anti-Stokes Raman scattering (CARS) Up to now we only have considered the totally degenerate case where we assumed the wavelengths of the interacting laser pulses as well as the signal at l0 Žcompare panel D of Fig. 2.. However, the laser pulses have a spectral width of about 250-300 cmy1 ŽFWHM. resulting in a FWM signal which also is spectrally broad. The energy diagrams now suggest that besides the DFWM also other FWM processes might be involved in the nonlinear coherent photonmolecule interactions. Without changing the laser wavelengths of the interacting beams Žhere, same l0 . there still remain several parameters which can be changed in order to vary the conditions for the time-dependent experiments. One of these parameters is the detection wavelength ldet which can be easily varied by changing the wavelength position of the monochromator used to disperse the signal light. If we chose a wavelength of ldet s 565 nm Ž n˜det s 17699 cmy1 . for detection, the shape of the resulting transient changes considerably Žpanel A of Fig. 3.. For negative delay times D t - 0 the beat patterns have the frequency of the wave packet in the excited B state of iodine. For positive delay times Ž D t ) 0. the beats have about twice this frequency and can be assigned to the electronic ground state. This becomes also evident from the FFT performed on the transient shown in Fig. 3ŽA.. Panels B and C of Fig. 3 show the FFT spectra for negative and positive delay times, respectively. The FFT spectrum for D t - 0 ŽFig. 3ŽB.. exhibits a line at about 94 cmy1 , which is the vibrational energy spacing in the excited B state. Additionally, the second harmonic of this frequency can be seen at about 189 cmy1 . For D t ) 0 the FFT spectrum ŽFig. 3ŽC.. shows only one peak at about 212 cmy1 which again can be assigned to the fundamental vibrational transition DÕXX s 1 of iodine in the ground state. This behaviour can be easily explained by assuming a FWM process which is favoured by the choice of an anti-Stokes detection wavenumber n˜det ) n˜ 0 Žcompare panel D of Fig. 3.. The corresponding FWM process is CARS. Here, we have put Ž n˜det y D n˜ . as time variable pump pulse. The pulses pu1Ž n˜det y D n˜ . and pu 2 Ž n˜det y 2D n˜ . act as second pump and Stokes lasers, respectively. D n˜ actually can assume any value within the spectral range of
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the involved laser pulses Žstarting from n˜det .. However, the process will be dominated by contributions where the laser intensities at Ž n˜det y D n˜ . and Ž n˜det y 2D n˜ . as well as the transition probabilities between the states involved are high. The energy level diagramms belonging to the time-resolved CARS are shown in panels E and F of Fig. 3 for delay times D t - 0 and D t ) 0, respectively. For negative delay times ŽFig. 3ŽE.. put interacts with the system first, resulting in a coherence in the excited B states of the iodine molecules. The wave packets are probed by the second pump and the Stokes pulse Ž pu1 and pu 2 , respectively. giving rise to the anti-Stokes signal at n˜det . For the process, the energy conservation as well as the phase matching conditions are fulfilled. This of course is also the case for the scheme described in the energy level diagram shown in Fig. 4ŽF.. Here, the time delayed pulse put probes the ground state wave packet which was coherently excited by the interaction of pu1 and pu 2 with the molecular system. As mentioned above, very recently results of femtosecond time resolved CARS spectroscopy for the simultaneous study of ground and excited state dynamics have been published w22,23x. Here, the pulses put and pu1 Ž‘‘pump pulses’’. had the same wavenumber positions, while pu 2 Ž‘‘Stokes pulse’’. was tuned to a lower wavenumber. The frequency difference between Stokes and pump laser pulses was tuned to a vibrational Raman transition in the ground state of the molecules Žno spectral overlap between put and pu1 with pu 2 .. Probing the coherent anti-Stokes signal resulted in transients for D t - 0 and D t ) 0 showing beats characteristic for wave packets prepared on the excited and ground state potential surfaces, respectively. This is in full agreement with the results obtained from the anti-Stokes detection in a FWM experiment using degenerate laser wavenumbers. 3.3. Coherent Stokes Raman scattering (CSRS) Instead of chosing a detection wavelength on the anti-Stokes side of the central laser wavelength l0 s 573 nm Ž n˜ 0 s 17452 cmy1 . it is of course also possible to set the monochromator to a position on the Stokes side. In the following we show results
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obtained for a detection wavelength ldet s 585 nm Ž n˜ 0 s 17094 cmy1 . which lies in the edge of the spectral range of the laser pulses Žcompare panel D of Fig. 4.. The FWM process which is favoured by the Stokes-detection is the coherent Stokes Raman scattering ŽCSRS.. Here, we have put Ž n˜det q D n˜ . as time variable pump pulse. The pulses pu1Ž n˜det q D n˜ . and pu 2 Ž n˜det q 2D n˜ . act as second pump and antiStokes lasers, respectively. As for the CARS process described above, D n˜ can take any value within the spectral range of the involved laser pulses Žstarting from n˜det .. The main contributions will now be determined by maximum laser intensities at Ž n˜det q D n˜ . and Ž n˜det q 2D n˜ . as well as the transition probabilities between the states involved in the CSRS process. The transient obtained under such conditions is shown in panel A of Fig. 4. Now, the transient does show neither signal nor beating structures for negative delay times D t - 0. However, for D t ) 0 intense beat patterns can be found at the excited B state vibrational frequency. This is also demonstrated by the FFT spectra shown in panels B and C of Fig. 4, which were obtained from the transient ŽFig. 4ŽA.. for negative and positive delay times, respectively. While for D t - 0 the FFT does not exhibit any dominating frequency contribution, the frequency spectrum describing the coherence probed for D t ) 0 has an intense band at about 94 cmy1 and possibly a very small contribution at the second harmonic position Žf 189 cmy1 . as well as at the ground state frequency Žf 212 cmy1 .. Again, possible energy level diagrams are shown in Fig. 4. Here, we do not find any diagram which might give considerable contribution to the CSRS signal if put interacts with the system first. This is in accordance to our observations. For positive delay times two diagrams are shown in panel E of Fig. 4. The left energy level diagram shows a coherent interaction where wave packets are prepared by pu1 and pu 2 on the excited B state potential energy surface and afterwards probed by put giving rise to the Stokes-shifted signal. This process is responsible for the bands at about 94 and 189 cmy1 . The right diagram only may give rise to a weak contribution from ground state dynamics if one takes into account the spectral width of the laser pulses involved. We would like to emphasize again that all processes
shown fulfill both the energy conservation and the phase matching conditions.
4. Conclusions In this letter we have demonstrated that femtosecond time-resolved four-wave mixing ŽFWM. spectroscopy with laser pulses having degenerate wavenumber positions is capable of giving information about the wave packet dynamics on both the electronic ground and excited state potential energy surface. The experiments were performed using three laser pulses in a folded BOXCARS configuration. Two of the pulses Ž pu1 and pu 2 . were time coincident while the third laser pulse Ž put . was variable in time. For negative delay times Ž D t - 0. the interaction of put with the molecular system was probed by the two simultaneous pulses pu1 and pu 2 giving rise to a coherent FWM signal. For positive delay times Ž D t ) 0. the interaction of pu1 and pu 2 takes place first and put later probes the excited dynamics. All experiments were performed on the iodine molecule in the gas phase. Besides the timing of the laser pulses also the detection wavenumber was varied giving rise to different transient shapes. Applying a Fourier transform on the transients for the two time ranges D t - 0 and D t ) 0 shows that both timing and detection wavenumber determine which wave packet dynamics is probed. This was interpreted as a selection of Feynman diagrams contributing to the density matrix presentation of the nonlinear polarization of the FWM process. The time ordering as well as resonances involved in the considered nonlinear processes drastically reduce the number of diagrams describing the FWM. Instead of the relatively complicated Feynman diagrams we used energy level diagrams to illustrate the possible interactions. The results can be itemized like follows: Ži. If the signal is detected at about the central wavenumber of the laser pulses Ž n˜det s n˜ 0 . the transient can be explained by diagrams belonging to the degenerate FWM ŽDFWM. process. For D t - 0 the DFWM transient reveals the dynamics of the excited B state of iodine, while for D t ) 0 a superposition of the ground and excited state dynamics is probed. Žii. Chosing a wavenumber on the anti-Stokes
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side of n˜ 0 for detection Ž n˜det ) n˜ 0 . time-resolved coherent anti-Stokes Raman scattering ŽCARS. dominates the transient shape. The transient signal obtained for negative delay times reveals the wave packet dynamics within the excited state. For positive delay times beats can be seen which have a frequency corresponding to the energy difference of vibrational levels in the electronic ground state. Žiii. Stokes-side detection Ž n˜det - n˜ 0 . yields transients which result from a coherent Stokes Raman scattering ŽCSRS. process. Here, no dynamics can be probed for negative delay times. However, for D t ) 0 the excited state dynamics results from the time-dependent coherent Stokes signal. In order to further investigate this interesting experimental technique experiments are in preparation where e.g. also the polarizations of the lasers will be changed. This will give access to different components of the nonlinear susceptibility x Ž3..
Acknowledgements This work was funded by the Deutsche Forschungsgemeinschaft ŽSchwerpunktprogramm ‘‘Femtosekunden-Spektroskopie elementarer Anregungen in Atomen, Molekulen ¨ und Clustern’’, Projekt KI 202r14-1.. Stimulating discussions with Stefan Meyer and Professor Volker Engel are gratefully acknowledged.
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