- Email: [email protected]
Femtosecond CARS on H2
27 August 1999
Chemical Physics Letters 310 Ž1999. 65–72 www.elsevier.nlrlocatercplett
Femtosecond CARS on H 2 T. Lang, K.-L. Kompa, M. Motzkus
)
Max-Planck-Institut fur ¨ Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany Received 13 May 1999; in final form 7 July 1999
Abstract Time-resolved non-resonant CARS Žcoherent anti-Stokes Raman spectroscopy. is applied to probe H 2 in the gas phase with femtosecond time resolution. The experimental transients show detailed beating structures, reflecting the rotational dynamics of all thermally populated ground state J levels with high fidelity. Modelling of the experimental data yields temperatures and pressures with high resolution. The good agreement with values determined by frequency-domain experiments demonstrates the suitability of this method for the determination of J-dependent collisional-broadening and line-shifting coefficients in gaseous mixtures at high pressures and temperatures, where frequency-domain CARS spectra are difficult to model. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction The development of laser systems capable of producing ultrashort pulses in the femtosecond time-domain has opened the possibility to study chemical reactions in realtime and directly monitor the fundamental step of bond breaking and the formation of new products w1–4x. The experiments are typically designed as pump–probe scheme where a femtosecond pump pulse initiates the dynamics and a second pulse interrogates the dynamics after selected time delays. Usually probe methods are used with linear detection techniques but recently non-linear optical methods, like coherent anti-Stokes Raman scattering ŽCARS. or degenerate four-wave-mixing ŽDFWM., received much attention in order to study chemical dynamics in liquids or gas phases with femtosecond time resolution w5–15x. A detailed theo) Corresponding author: Fax: q49-89-32905-200; e-mail: [email protected]
retical analysis of these FWM processes is given by Mukamel w16x. Using femtosecond FWM processes in the gas phase, Morgen et al. w10x, for example, have applied the femtosecond Raman-induced polarization spectroscopy ŽRIPS. technique to determine rotational relaxation constants of N2 , CO 2 , and O 2 . Hayden et al. w14x have studied large gas-phase molecules with femtosecond-resolved coherent vibrational Raman scattering. Recently, Motzkus et al. w12x have observed wavepacket dynamics in molecular systems on a femtosecond timescale via degenerate and twocolor four-wave mixing. Resonant femtosecond CARS was employed by Schmitt et al. w13x to obtain information on the molecular dynamics evolving on the electronically excited as well as ground state potential energy surfaces of iodine and bromine in the gas phase. Using non-resonant DFWM, Frey et al. w15x have studied rotational coherences of O 2 , N2 , and CO 2 . These experiments are part of rotational coherence spectroscopy ŽRCS. which has been de-
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 7 8 7 - 3
66
T. Lang et al.r Chemical Physics Letters 310 (1999) 65–72
scribed in a comprehensive review by Felker and Zewail w11x. In addition to the time resolution achieved in these experiments using ultrashort laser pulses, the femtosecond CARS technique features some advantages which makes this technique superior compared to applications of frequency-domain CARS spectroscopy. Frequency-domain CARS is well known for its high sensitivity of probing homonuclear molecules and is used to extract temperature and concentration profiles in combustion processes and flames. Temperatures are typically measured by monitoring the Q-branch in nitrogen or hydrogen molecules w17–19x. A major drawback common to these studies is the growing complexity of the spectra at high pressures and the ensuing difficulty to model them due to non-resonant background and the non-availability of the pertinent broadening coefficients. CARS in the time-domain has several advantages which render this method ideal to extract molecular constants and to perform temperature measurements even at these difficult experimental conditions. Other than in frequency-domain CARS, the non-resonant background appears only at time zero and does not influence the transient. In this Letter, we discuss the application of femtosecond non-resonant CARS as a real time detection method for homonuclear molecules in chemical dynamics and as an alternative to high-resolution frequency-domain CARS spectroscopy. In the following, we focus on H 2 , which is the simplest candidate in this class of molecules and also the product of many reactions, but difficult to probe with common techniques in mixtures or environments like cells, reaction chambers, or flames. The high accuracy in modelling of the experimental CARS transient does not only allow to determine J-dependent broadening coefficients but also to measure high temperatures with a picosecond time resolution.
2. Experiment The basic conceptual scheme of the experiment is shown in Fig. 1. In a first step the Q-branch resonances Ž ÕXX s 0, J . ™ Ž ÕX s 1, J . of the H 2 molecule
Fig. 1. Principle of the femtosecond time-domain CARS experiment. L, pump pulse; S, Stokes pulse, P, probe pulse, A, anti-Stokes pulse Žsignal..
are excited by two femtosecond laser pulses Žpump pulse: l L s 600 nm, D lFWHM s 9 nm; Stokes pulse: lS s 800 nm, D lFWHM s 13 nm.. A subsequent probe pulse Ž lP s 600 nm. generates an anti-Stokes signal described by the interaction with the thirdorder non-linear polarizability of the medium. On the left side of Fig. 1, the CARS process is shown for a single Q-branch transition only. However, the differences between the characteristic energies D E J of the Q-branch transitions are small, and the bandwidth of the exciting laser pulses allows for coherent excitation of the entire thermal population of J levels belonging to Õ s 0 Žright side of Fig. 1.. The interferences between the Q-branch resonances are monitored by scanning the time delay between pump, Stokes and probe pulses. The experimental setup ŽFig. 2. comprises a commercial Ti:Sapphire femtosecond laser system ŽCPA1000, Clark MXR, Inc.., which generates pulses with an energy of 1 mJ at 800 nm and a repetition rate of 1 kHz. The output beam is split into two, one of which serves as Stokes beam while the second pumps an optical parametric amplifier ŽOPA.. The OPA output is tuned to 600 nm Ž100 fs. at 5 mJ per pulse and split into pump and probe pulse. The temporal separation between the pulses can be varied by two computer-controlled delay lines. Pump, Stokes and probe pulses are focused into the H 2 cell placed
T. Lang et al.r Chemical Physics Letters 310 (1999) 65–72
67
Fig. 2. Experimental setup. Explanation given in the text.
inside an oven. Two different types of cells have been used. A sealed-off quartz cell, filled to p H 2 s 500 mbar Žat T s 296 K., allowed to perform CARS experiments at different temperatures up to T s 1100 K. The alternative design, which was used at room temperature only, is constructed as a steel tube fitted with two windows and can be operated up to p s 80 bar. Temperatures were determined by K-type thermocouples, placed inside the oven. lr4 and lr2 plates ensure parallel polarizations of the three beams at the common focus inside the cell. At the focus, each of the three beams has pulse energies between 200 and 300 nJ. We have attenuated one of the three incident beams at a time and found the shape of the CARS transients to be independent on the ratio between the pulse energies. A second lens behind the cell collimates the generated femtosecond CARS signal. The phase-matching condition is satisfied by using the folded BOXCARS configuration. As the CARS signal propagates in a direction which diverges from that of the incoming beams, it can be easily separated by a pinhole. After having travelled a long pathway to reduce straylight from the cell, the signal is passed through a monochromator Žbandpass. and detected with a photomultiplier tube ŽPMT.. The detection limit of our setup is approached at p H 2 f 100 mbar. Using collinear CARS should result
in an improvement of the detection limit by one or two orders of magnitude.
3. CARS in the time-domain The CARS signal is described by a coherent superposition of the ro-vibrational transitions involved. In the following, we will consider the Qbranch Ž D J s 0, DÕ s 1. of the H 2 ground state only; significant contributions from other molecular transitions were not seen in our experiments. Each transition of the Q-branch corresponds to a wave function oscillating with a frequency D E Jr", where D E J is the characteristic energy of the Raman transition Žsee Fig. 1.. Collisions between the molecules induce a J-dependent dephasing described by the time T2 Ž J .. Hence, after introducing the reciprocal dephasing time L J :s T2 Ž J .y1 , the CARS signal is expressed by 2
Jmax
ICARS Ž t . ;
Ý NJ e
Ži D E J r "y L J .t
.
Ž 1.
Js0
The transitions are weighted by coefficients NJ , which account for the relative population of the J levels, as well as the spectral shape of the pump and Stokes pulses.
T. Lang et al.r Chemical Physics Letters 310 (1999) 65–72
68
The squared norm in Eq. Ž1. leads to cosine terms oscillating with beating frequencies Ž D E J y D E J X .: Jmax
ICARS Ž t . s
Ý NJ2 ey2t
Jt
Js0 Jmax
q2
Ý
X
NJ NJ X eyŽ L Jq L J X .t
Js0, J )J
=cos
D E J Ž p,T . s D E J0 q a J Ž T . p q bJ Ž T . p 2 q PPP , Ž 4.
Ž D EJ y D EJ X . t "
The energy D E J and dephasing time T2 Ž J . for each rotational level J depends on experimental parameters such as temperature, pressure, and laser intensity. Since in Eq. Ž2. the transition energies do not appear separately, but only as differences Ž D E J y D E J X ., all parameter dependencies of D E J , which are identical for each J level, may be ignored. In traditional molecular spectroscopy the rotational energy of a vibrational state is given w20x by Erot Ž Õ, J . s BÕ J Ž J q 1 . with
BÕ s Be y a e Ž Õ q 1r2 . .
Ž 2.
Then the different Q-branch transition energies in our experiment are calculated by D E J s Erot Ž ÕXX s 1, J . y Erot Ž ÕX s 1, J . s ya e J Ž J q 1 . ,
about 100 fs and affect all J levels alike. The electronic distortion of the molecule should relax very quickly compared to the time delay between pump and probe pulse Žseveral picoseconds up to 1 ns.. The non-linear dependence of the transition energies on the H 2 pressure p is described by the relation
Ž 3.
where the rotational constant Be disappears. In comparison to typical RCS experiments, where the signal is periodical with TRCS s hr2 Be , in our femtosecond CARS experiment the cycle duration for the signal is TCARS s yhr2 a e . At times t s nTCARS all cosine terms of Eq. Ž2. become 1, and apart from the exponential decay the signal reaches the same amplitude as for t s 0. In the case of H 2 , however, Eq. Ž2. has to be extended to higher powers of Ž Õ q 1r2. and J Ž J q 1., and the duration TCA RS between the prominent recurrences can no longer be assigned to a single molecular constant. In this case, the relation TCARS f yhr2 a e only holds approximately. Instead of using Eq. Ž2., we calculate the energies of the ro-vibrational levels as a double power series according to Dunham w21x and Herzberg w20x. Laser-induced AC Stark shifts w22x of the molecular levels may be neglected in a first approximation because they appear only during the short interval of
where D E J0 is the energy of the QŽ J . transition at p s 0. The coefficients a and b both depend on J and T and were determined experimentally by Rahn et al. w23x for each rotational level up to J s 5, at temperatures 296 K ( T ( 1000 K. At the pressure range covered by our cell experiments Ž0.5 bar ( p H 2 ( 80 bar. the line shapes of the QŽ J . transitions are Lorentzian. Their half-width at half-maximum ŽHWHM. may be expressed in Žsy1 . as:
GJ Ž p,T . s 2 p D 0 Ž T . n J2rp q g J Ž T . p .
Ž 5.
In this equation, D 0 ŽT . s 0.01176T 0.8314 is the optical diffusion coefficient w24x for the H 2 molecule in units of Žcmy1 bar sy1 ., n J is the Raman transition energy in Žcmy1 ., and g J ŽT . the self-broadening coefficient for the rotational level J at temperature T. The J-dependent dephasing in Eq. Ž2. can be calculated by using the relation L J s T2 Ž J .y1 s 2 p GJ . The weighting coefficients NJ in Eqs. Ž1. and Ž2. depend on the relative population of the J levels and the spectral shapes of the laser pulses. The relative population is calculated by using a Boltzmann distribution at temperature T weighted by the degree of degeneracy of the rotational levels and by the statistical weight Ž S J s 1 or 3. of J due to the nuclear spins. The influence due to the spectral shapes of the pump and Stokes pulses is described as a convolution of their respective spectra, I LŽ n . and IS Ž n .: NJ s S J Ž 2 J q 1 . exp
yBe kT
J Ž J q 1.
= I L Ž n . IS Ž n y D E Jr" . d n .
H
Ž 6.
T. Lang et al.r Chemical Physics Letters 310 (1999) 65–72
4. Results and discussion We have simulated and measured CARS transients in the range 0.5 bar ( p H 2 ( 80 bar and 296 ( T ( 1100 K. Neglecting the pressure-dependent energy shift according to Eq. Ž4. confines the range of a faithful simulation of the experimental CARS signal to small time delays only Ž t s 0–10 ps.. The use of a global dephasing time T2 in the calculation already produces good agreement with the experimental data. Applying individual T2 times according to Ref. w24x, however, substantially improves the agreement. Fig. 3 shows the theoretical CARS transient for p s 1 bar, T s 276 K and the corresponding experimental data. As mentioned in the previous section, the duration, TCA RS s 5.66 ps, between the prominent recurrences of the CARS signal can be used to determine the molecular constant a e f yhr2TCARS s y2.95 cmy1 . As the coefficients of higher powers of Ž Õ q 1r2. and J Ž J q 1. are not negligible for H 2 , the quantum beat patterns are shifted slightly at each recurrence. At large time delays between pump, Stokes and probe pulses, the appearance of the beating pattern has changed completely.
69
The excellent agreement between simulation and experiment suggests that Eq. Ž2. be regarded as a theoretical model apt to fit parameters such as pressure or temperature to data obtained from time-resolved CARS experiments. We have implemented a least-squares fit, in which the shifted transition energies D E J , the relative population NJ of the J levels, and individual T2 times for each rotational level are varied independently. The T2 times returned by the algorithm are in good agreement with values determined by inverse Raman spectroscopy w24x in the frequency-domain, and the accuracy of the parameters improves for those J which are highly populated. The same accuracy should be obtainable in gas mixtures at high pressures, since in femtosecond-domain CARS the electronic contribution will decouple from the ro-vibrational response. Self-broadening coefficients can be then determined under conditions where frequency-domain CARS meets severe problems in its interpretation. The theoretical simulation is very sensitive to relative energy shifts between the Raman transitions. ‘Relative’ is to suggest that femtosecond time-domain CARS precludes the absolute measurement of the energy of a single Q-branch transition. Since the
Fig. 3. CARS transient at p H 2 s 1 bar, T s 296 K. Solid line, theory; open squares, experiment.
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T. Lang et al.r Chemical Physics Letters 310 (1999) 65–72
absolute magnitude of any QŽ J . is thus irrelevant, we have fixed the QŽ0.-transition energy constantly to the value D E00 . This implies that all energy values in the calculation must be shifted in first order by w a J ŽT . y a0 ŽT .x p. The relative shifting coefficients Ž a J ŽT . y a0 ŽT .. obtained by fitting them to the experimental data were in good agreement with those by Rahn et al. w23x. Owing to the Dicke effect the T2 times peak at 2.5 bar. The very slow decay of the signal ŽT2 f 1 ns. in this pressure range permits to observe a maximum number of oscillations, greatly enhancing the energy resolution. At p s 1 bar the energy resolution for J s 1 is about 10y3 cmy1 . The uncertainty in the determination of p H 2 is D p s 1 bar within the entire pressure range Ž0 - p H 2 ( 80 bar. accessible by our experiments.
At small time delays Ž t < T2 Ž J .., the CARS signal is obviously not very sensitive to changes of T2 Ž J . or the linewidths GJ , respectively. Neither will minor energy shifts affect the transient at small time delays, because the energy-dependent divergence of the rotational recurrences sums up gradually with increasing delay time. As Eq. Ž2. contains no explicit pressure dependence, p H 2 influences the CARS transient via the transition energies D E J and the linewidths GJ only. As a consequence the shape of the transient is not very sensitive to changes of p H 2 at small time delays. Variations of the temperature T, on the other hand, not only act on D E J and GJ , but they also modify the shape of the CARS transient directly via the weighting coefficients NJ ŽEq. Ž6... The temperature and hence the relative population of the rota-
Fig. 4. Ža. CARS transients at four different temperatures. Solid line, theory; open squares, experiment. Žb. Rotational population of the J levels of electronic and vibrational ground state.
T. Lang et al.r Chemical Physics Letters 310 (1999) 65–72
tional levels determines the shape of the transients at any time delay within the dephasing time. At the onset of the CARS transient, while the dependence on pressure is still weak, its shape is governed by the weighting coefficients NJ only. This fact enables to perform temperature measurements which are not marred by density effects such as line-shifting or line-broadening. Fig. 4 illustrates the strong dependence of the transients on rotational population and thus on temperature. Fig. 4a shows time-resolved CARS transients of H 2 in the temperature range 296 ( T ( 1100 K. Again, the experimental data are in very good agreement with the predictions of theory. The calculations were based on the J level populations shown in Fig. 4b which correspond to Boltzmann distributions at the experimentally determined temperatures. To judge the aptness of time-resolved CARS as a method of temperature measurement, we focus on the the highest peak past time zero Žat t s 0 the data may be distorted by the coherence peak of non-resonant background.. The calculated CARS signals ŽFig. 5. suggest two simple procedures: the first interprets the width of the recurrence peak Žc., which decreases inversely with temperature w11x. The alternate method is to compare the signal intensities at two different
71
time delays. Time delay t 1 s 5.7 ps serves as reference point Žc.; the second point of measurement must be picked such as to provide optimum temperature resolution, e.g., t 2 s 5.1 ps for the range 300 ( T ( 1000 K Žb. and t 2 s 4.8 ps for temperatures T 0 1000 K Ža.. We have also fitted the temperature T to the full transients Ž0–1 ns. recorded in the range 0.5 ( p H 2 ( 1.8 bar, 296 ( T ( 1100 K. The temperature resolution was about 20 K throughout.
5. Conclusions In this Letter, we have screened non-resonant femtosecond CARS with respect to its applicability for detecting homonuclear molecules in the gas phase. The beating structures of the experimental transients truthfully mirror the rotational dynamics in the ground state and hence allow to extract detailed information on molecular constants of higher order, J-dependent collisional-broadening and line-shifting coefficients. A theoretical model which simulates the femtosecond time-resolved CARS signal of H 2 was presented. It depends sensitively on the temperature T, the pressure p, and individual dephasing times T2 Ž J .. Theory and experiment proved to be in excellent agreement. The high degree of time resolution offers the possibility to discriminate against unwanted interferences, as from additional components in gas mixtures, since these will contribute a non-resonant background only at t s 0. This makes the method a promising tool for an experimental determination of pressure, temperature or rotational population as well. We have shown that a determination of temperatures calls for a measurement of only two different time delays.
Acknowledgements Fig. 5. Calculated CARS transients of H 2 : first peak after time zero. The calculation is based on pulses with Gaussian spectral shape. Pump pulse: lP s 598 nm, D lFWHM s8.6 nm; Stokes pulse: lS s800 nm, D lFWHM s12.5 Žfitted to ‘real’ pulses in experiment.. Explanation given in the text.
The authors thank D. Zeidler and D. Proch for many valuable discussions and H. Bauer for his technical assistance.
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T. Lang et al.r Chemical Physics Letters 310 (1999) 65–72
References w1x A.H. Zewail, Femtochemistry: Ultrafast Dynamics of the Chemical Bond, World Scientific, Singapore, 1994. w2x A.H. Zewail, J. Phys. Chem. 100 Ž1996. 12701. w3x J. Manz, L. Woste ¨ ŽEds.., Femtosecond Chemistry, vol. I–II, VCH, Weinheim, 1995. w4x V. Sundstrom ¨ ŽEd.., Femtochemistry and Femtobiology: Ultrafast Reaction Dynamics at Atomic-Scale Resolution, Nobel Symposium 101, Imperial College Press, 1997. w5x W. Zinth, W. Kaiser, Topics in Applied Physics, vol. 60 Ž1988., Springer, Berlin, 1993, p. 235. w6x A. Laubereau, Topics in Applied Physics, vol. 60 Ž1988., Springer, Berlin, 1993, p. 35. w7x M. Fickenscher, A. Laubereau, J. Raman Spectrosc. 21 Ž1990. 857. w8x T. Hofer, ¨ P. Kruck, W. Kaiser, Chem. Phys. Lett. 224 Ž1994. 411. w9x T. Joo, M.A. Dugan, A.C. Albrecht, Chem. Phys. Lett. 177 Ž1991. 4. w10x M. Morgen, W. Price, L. Hunziker, P. Ludowise, M. Blackwell, Y. Chen, Chem. Phys. Lett. 209 Ž1993. 1. w11x P.M. Felker, A.H. Zewail, in: J. Manz, L. Woste ¨ ŽEds.., Femtosecond Chemistry, vol. I, VCH, Weinheim, 1995.
w12x M. Motzkus, S. Petersen, A.H. Zewail, J. Phys. Chem. 100 Ž1996. 5260. w13x M. Schmitt, G. Knopp, A. Materny, W. Kiefer, J. Phys. Chem. A 102 Ž1998. 4059. w14x C.C. Hayden, D.W. Chandler, J. Chem. Phys. 103 Ž1995. 10465. w15x H.-M. Frey, P. Beaud, T. Gerber, B. Mischler, P.P. Radi, A.P. Tzannis, Appl. Phys. B 68 Ž1999. 735. w16x S. Mukamel, Nonlinear Optical Spectroscopy University Press, Oxford, 1995. w17x B. Attal-Tretout, P. Bouchardy, P. Magre, M. Pealat, J.P. ´ ´ Taran, Appl. Phys. B 51 Ž1990. 17. w18x E.M. Castellucci, R. Righini, P. Foggi ŽEds.., Coherent Raman Spectroscopy, World Scientific, 1993. w19x V. Bergmann, W. Stricker, Appl. Phys. B 61 Ž1995. 49. w20x G. Herzberg, Molecular Spectra and Molecular Structure, vol. I, R.E. Krieger, Malabar, 1989. w21x J.L. Dunham, Phys. Rev. 41 Ž1932. 721. w22x L.A. Rahn, R.L. Farrow, M.L. Koszykowski, P.L. Mattern, Phys. Rev. Lett. 45 Ž1980. 620. w23x L.A. Rahn, G.J. Rosasco, Phys. Rev. A 41 Ž1990. 3698. w24x L.A. Rahn, R.L. Farrow, G.J. Rosasco, Phys. Rev. A 43 Ž1991. 6075.
Chemical Physics Letters 310 Ž1999. 65–72 www.elsevier.nlrlocatercplett
Femtosecond CARS on H 2 T. Lang, K.-L. Kompa, M. Motzkus
)
Max-Planck-Institut fur ¨ Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany Received 13 May 1999; in final form 7 July 1999
Abstract Time-resolved non-resonant CARS Žcoherent anti-Stokes Raman spectroscopy. is applied to probe H 2 in the gas phase with femtosecond time resolution. The experimental transients show detailed beating structures, reflecting the rotational dynamics of all thermally populated ground state J levels with high fidelity. Modelling of the experimental data yields temperatures and pressures with high resolution. The good agreement with values determined by frequency-domain experiments demonstrates the suitability of this method for the determination of J-dependent collisional-broadening and line-shifting coefficients in gaseous mixtures at high pressures and temperatures, where frequency-domain CARS spectra are difficult to model. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction The development of laser systems capable of producing ultrashort pulses in the femtosecond time-domain has opened the possibility to study chemical reactions in realtime and directly monitor the fundamental step of bond breaking and the formation of new products w1–4x. The experiments are typically designed as pump–probe scheme where a femtosecond pump pulse initiates the dynamics and a second pulse interrogates the dynamics after selected time delays. Usually probe methods are used with linear detection techniques but recently non-linear optical methods, like coherent anti-Stokes Raman scattering ŽCARS. or degenerate four-wave-mixing ŽDFWM., received much attention in order to study chemical dynamics in liquids or gas phases with femtosecond time resolution w5–15x. A detailed theo) Corresponding author: Fax: q49-89-32905-200; e-mail: [email protected]
retical analysis of these FWM processes is given by Mukamel w16x. Using femtosecond FWM processes in the gas phase, Morgen et al. w10x, for example, have applied the femtosecond Raman-induced polarization spectroscopy ŽRIPS. technique to determine rotational relaxation constants of N2 , CO 2 , and O 2 . Hayden et al. w14x have studied large gas-phase molecules with femtosecond-resolved coherent vibrational Raman scattering. Recently, Motzkus et al. w12x have observed wavepacket dynamics in molecular systems on a femtosecond timescale via degenerate and twocolor four-wave mixing. Resonant femtosecond CARS was employed by Schmitt et al. w13x to obtain information on the molecular dynamics evolving on the electronically excited as well as ground state potential energy surfaces of iodine and bromine in the gas phase. Using non-resonant DFWM, Frey et al. w15x have studied rotational coherences of O 2 , N2 , and CO 2 . These experiments are part of rotational coherence spectroscopy ŽRCS. which has been de-
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 7 8 7 - 3
66
T. Lang et al.r Chemical Physics Letters 310 (1999) 65–72
scribed in a comprehensive review by Felker and Zewail w11x. In addition to the time resolution achieved in these experiments using ultrashort laser pulses, the femtosecond CARS technique features some advantages which makes this technique superior compared to applications of frequency-domain CARS spectroscopy. Frequency-domain CARS is well known for its high sensitivity of probing homonuclear molecules and is used to extract temperature and concentration profiles in combustion processes and flames. Temperatures are typically measured by monitoring the Q-branch in nitrogen or hydrogen molecules w17–19x. A major drawback common to these studies is the growing complexity of the spectra at high pressures and the ensuing difficulty to model them due to non-resonant background and the non-availability of the pertinent broadening coefficients. CARS in the time-domain has several advantages which render this method ideal to extract molecular constants and to perform temperature measurements even at these difficult experimental conditions. Other than in frequency-domain CARS, the non-resonant background appears only at time zero and does not influence the transient. In this Letter, we discuss the application of femtosecond non-resonant CARS as a real time detection method for homonuclear molecules in chemical dynamics and as an alternative to high-resolution frequency-domain CARS spectroscopy. In the following, we focus on H 2 , which is the simplest candidate in this class of molecules and also the product of many reactions, but difficult to probe with common techniques in mixtures or environments like cells, reaction chambers, or flames. The high accuracy in modelling of the experimental CARS transient does not only allow to determine J-dependent broadening coefficients but also to measure high temperatures with a picosecond time resolution.
2. Experiment The basic conceptual scheme of the experiment is shown in Fig. 1. In a first step the Q-branch resonances Ž ÕXX s 0, J . ™ Ž ÕX s 1, J . of the H 2 molecule
Fig. 1. Principle of the femtosecond time-domain CARS experiment. L, pump pulse; S, Stokes pulse, P, probe pulse, A, anti-Stokes pulse Žsignal..
are excited by two femtosecond laser pulses Žpump pulse: l L s 600 nm, D lFWHM s 9 nm; Stokes pulse: lS s 800 nm, D lFWHM s 13 nm.. A subsequent probe pulse Ž lP s 600 nm. generates an anti-Stokes signal described by the interaction with the thirdorder non-linear polarizability of the medium. On the left side of Fig. 1, the CARS process is shown for a single Q-branch transition only. However, the differences between the characteristic energies D E J of the Q-branch transitions are small, and the bandwidth of the exciting laser pulses allows for coherent excitation of the entire thermal population of J levels belonging to Õ s 0 Žright side of Fig. 1.. The interferences between the Q-branch resonances are monitored by scanning the time delay between pump, Stokes and probe pulses. The experimental setup ŽFig. 2. comprises a commercial Ti:Sapphire femtosecond laser system ŽCPA1000, Clark MXR, Inc.., which generates pulses with an energy of 1 mJ at 800 nm and a repetition rate of 1 kHz. The output beam is split into two, one of which serves as Stokes beam while the second pumps an optical parametric amplifier ŽOPA.. The OPA output is tuned to 600 nm Ž100 fs. at 5 mJ per pulse and split into pump and probe pulse. The temporal separation between the pulses can be varied by two computer-controlled delay lines. Pump, Stokes and probe pulses are focused into the H 2 cell placed
T. Lang et al.r Chemical Physics Letters 310 (1999) 65–72
67
Fig. 2. Experimental setup. Explanation given in the text.
inside an oven. Two different types of cells have been used. A sealed-off quartz cell, filled to p H 2 s 500 mbar Žat T s 296 K., allowed to perform CARS experiments at different temperatures up to T s 1100 K. The alternative design, which was used at room temperature only, is constructed as a steel tube fitted with two windows and can be operated up to p s 80 bar. Temperatures were determined by K-type thermocouples, placed inside the oven. lr4 and lr2 plates ensure parallel polarizations of the three beams at the common focus inside the cell. At the focus, each of the three beams has pulse energies between 200 and 300 nJ. We have attenuated one of the three incident beams at a time and found the shape of the CARS transients to be independent on the ratio between the pulse energies. A second lens behind the cell collimates the generated femtosecond CARS signal. The phase-matching condition is satisfied by using the folded BOXCARS configuration. As the CARS signal propagates in a direction which diverges from that of the incoming beams, it can be easily separated by a pinhole. After having travelled a long pathway to reduce straylight from the cell, the signal is passed through a monochromator Žbandpass. and detected with a photomultiplier tube ŽPMT.. The detection limit of our setup is approached at p H 2 f 100 mbar. Using collinear CARS should result
in an improvement of the detection limit by one or two orders of magnitude.
3. CARS in the time-domain The CARS signal is described by a coherent superposition of the ro-vibrational transitions involved. In the following, we will consider the Qbranch Ž D J s 0, DÕ s 1. of the H 2 ground state only; significant contributions from other molecular transitions were not seen in our experiments. Each transition of the Q-branch corresponds to a wave function oscillating with a frequency D E Jr", where D E J is the characteristic energy of the Raman transition Žsee Fig. 1.. Collisions between the molecules induce a J-dependent dephasing described by the time T2 Ž J .. Hence, after introducing the reciprocal dephasing time L J :s T2 Ž J .y1 , the CARS signal is expressed by 2
Jmax
ICARS Ž t . ;
Ý NJ e
Ži D E J r "y L J .t
.
Ž 1.
Js0
The transitions are weighted by coefficients NJ , which account for the relative population of the J levels, as well as the spectral shape of the pump and Stokes pulses.
T. Lang et al.r Chemical Physics Letters 310 (1999) 65–72
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The squared norm in Eq. Ž1. leads to cosine terms oscillating with beating frequencies Ž D E J y D E J X .: Jmax
ICARS Ž t . s
Ý NJ2 ey2t
Jt
Js0 Jmax
q2
Ý
X
NJ NJ X eyŽ L Jq L J X .t
Js0, J )J
=cos
D E J Ž p,T . s D E J0 q a J Ž T . p q bJ Ž T . p 2 q PPP , Ž 4.
Ž D EJ y D EJ X . t "
The energy D E J and dephasing time T2 Ž J . for each rotational level J depends on experimental parameters such as temperature, pressure, and laser intensity. Since in Eq. Ž2. the transition energies do not appear separately, but only as differences Ž D E J y D E J X ., all parameter dependencies of D E J , which are identical for each J level, may be ignored. In traditional molecular spectroscopy the rotational energy of a vibrational state is given w20x by Erot Ž Õ, J . s BÕ J Ž J q 1 . with
BÕ s Be y a e Ž Õ q 1r2 . .
Ž 2.
Then the different Q-branch transition energies in our experiment are calculated by D E J s Erot Ž ÕXX s 1, J . y Erot Ž ÕX s 1, J . s ya e J Ž J q 1 . ,
about 100 fs and affect all J levels alike. The electronic distortion of the molecule should relax very quickly compared to the time delay between pump and probe pulse Žseveral picoseconds up to 1 ns.. The non-linear dependence of the transition energies on the H 2 pressure p is described by the relation
Ž 3.
where the rotational constant Be disappears. In comparison to typical RCS experiments, where the signal is periodical with TRCS s hr2 Be , in our femtosecond CARS experiment the cycle duration for the signal is TCARS s yhr2 a e . At times t s nTCARS all cosine terms of Eq. Ž2. become 1, and apart from the exponential decay the signal reaches the same amplitude as for t s 0. In the case of H 2 , however, Eq. Ž2. has to be extended to higher powers of Ž Õ q 1r2. and J Ž J q 1., and the duration TCA RS between the prominent recurrences can no longer be assigned to a single molecular constant. In this case, the relation TCARS f yhr2 a e only holds approximately. Instead of using Eq. Ž2., we calculate the energies of the ro-vibrational levels as a double power series according to Dunham w21x and Herzberg w20x. Laser-induced AC Stark shifts w22x of the molecular levels may be neglected in a first approximation because they appear only during the short interval of
where D E J0 is the energy of the QŽ J . transition at p s 0. The coefficients a and b both depend on J and T and were determined experimentally by Rahn et al. w23x for each rotational level up to J s 5, at temperatures 296 K ( T ( 1000 K. At the pressure range covered by our cell experiments Ž0.5 bar ( p H 2 ( 80 bar. the line shapes of the QŽ J . transitions are Lorentzian. Their half-width at half-maximum ŽHWHM. may be expressed in Žsy1 . as:
GJ Ž p,T . s 2 p D 0 Ž T . n J2rp q g J Ž T . p .
Ž 5.
In this equation, D 0 ŽT . s 0.01176T 0.8314 is the optical diffusion coefficient w24x for the H 2 molecule in units of Žcmy1 bar sy1 ., n J is the Raman transition energy in Žcmy1 ., and g J ŽT . the self-broadening coefficient for the rotational level J at temperature T. The J-dependent dephasing in Eq. Ž2. can be calculated by using the relation L J s T2 Ž J .y1 s 2 p GJ . The weighting coefficients NJ in Eqs. Ž1. and Ž2. depend on the relative population of the J levels and the spectral shapes of the laser pulses. The relative population is calculated by using a Boltzmann distribution at temperature T weighted by the degree of degeneracy of the rotational levels and by the statistical weight Ž S J s 1 or 3. of J due to the nuclear spins. The influence due to the spectral shapes of the pump and Stokes pulses is described as a convolution of their respective spectra, I LŽ n . and IS Ž n .: NJ s S J Ž 2 J q 1 . exp
yBe kT
J Ž J q 1.
= I L Ž n . IS Ž n y D E Jr" . d n .
H
Ž 6.
T. Lang et al.r Chemical Physics Letters 310 (1999) 65–72
4. Results and discussion We have simulated and measured CARS transients in the range 0.5 bar ( p H 2 ( 80 bar and 296 ( T ( 1100 K. Neglecting the pressure-dependent energy shift according to Eq. Ž4. confines the range of a faithful simulation of the experimental CARS signal to small time delays only Ž t s 0–10 ps.. The use of a global dephasing time T2 in the calculation already produces good agreement with the experimental data. Applying individual T2 times according to Ref. w24x, however, substantially improves the agreement. Fig. 3 shows the theoretical CARS transient for p s 1 bar, T s 276 K and the corresponding experimental data. As mentioned in the previous section, the duration, TCA RS s 5.66 ps, between the prominent recurrences of the CARS signal can be used to determine the molecular constant a e f yhr2TCARS s y2.95 cmy1 . As the coefficients of higher powers of Ž Õ q 1r2. and J Ž J q 1. are not negligible for H 2 , the quantum beat patterns are shifted slightly at each recurrence. At large time delays between pump, Stokes and probe pulses, the appearance of the beating pattern has changed completely.
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The excellent agreement between simulation and experiment suggests that Eq. Ž2. be regarded as a theoretical model apt to fit parameters such as pressure or temperature to data obtained from time-resolved CARS experiments. We have implemented a least-squares fit, in which the shifted transition energies D E J , the relative population NJ of the J levels, and individual T2 times for each rotational level are varied independently. The T2 times returned by the algorithm are in good agreement with values determined by inverse Raman spectroscopy w24x in the frequency-domain, and the accuracy of the parameters improves for those J which are highly populated. The same accuracy should be obtainable in gas mixtures at high pressures, since in femtosecond-domain CARS the electronic contribution will decouple from the ro-vibrational response. Self-broadening coefficients can be then determined under conditions where frequency-domain CARS meets severe problems in its interpretation. The theoretical simulation is very sensitive to relative energy shifts between the Raman transitions. ‘Relative’ is to suggest that femtosecond time-domain CARS precludes the absolute measurement of the energy of a single Q-branch transition. Since the
Fig. 3. CARS transient at p H 2 s 1 bar, T s 296 K. Solid line, theory; open squares, experiment.
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T. Lang et al.r Chemical Physics Letters 310 (1999) 65–72
absolute magnitude of any QŽ J . is thus irrelevant, we have fixed the QŽ0.-transition energy constantly to the value D E00 . This implies that all energy values in the calculation must be shifted in first order by w a J ŽT . y a0 ŽT .x p. The relative shifting coefficients Ž a J ŽT . y a0 ŽT .. obtained by fitting them to the experimental data were in good agreement with those by Rahn et al. w23x. Owing to the Dicke effect the T2 times peak at 2.5 bar. The very slow decay of the signal ŽT2 f 1 ns. in this pressure range permits to observe a maximum number of oscillations, greatly enhancing the energy resolution. At p s 1 bar the energy resolution for J s 1 is about 10y3 cmy1 . The uncertainty in the determination of p H 2 is D p s 1 bar within the entire pressure range Ž0 - p H 2 ( 80 bar. accessible by our experiments.
At small time delays Ž t < T2 Ž J .., the CARS signal is obviously not very sensitive to changes of T2 Ž J . or the linewidths GJ , respectively. Neither will minor energy shifts affect the transient at small time delays, because the energy-dependent divergence of the rotational recurrences sums up gradually with increasing delay time. As Eq. Ž2. contains no explicit pressure dependence, p H 2 influences the CARS transient via the transition energies D E J and the linewidths GJ only. As a consequence the shape of the transient is not very sensitive to changes of p H 2 at small time delays. Variations of the temperature T, on the other hand, not only act on D E J and GJ , but they also modify the shape of the CARS transient directly via the weighting coefficients NJ ŽEq. Ž6... The temperature and hence the relative population of the rota-
Fig. 4. Ža. CARS transients at four different temperatures. Solid line, theory; open squares, experiment. Žb. Rotational population of the J levels of electronic and vibrational ground state.
T. Lang et al.r Chemical Physics Letters 310 (1999) 65–72
tional levels determines the shape of the transients at any time delay within the dephasing time. At the onset of the CARS transient, while the dependence on pressure is still weak, its shape is governed by the weighting coefficients NJ only. This fact enables to perform temperature measurements which are not marred by density effects such as line-shifting or line-broadening. Fig. 4 illustrates the strong dependence of the transients on rotational population and thus on temperature. Fig. 4a shows time-resolved CARS transients of H 2 in the temperature range 296 ( T ( 1100 K. Again, the experimental data are in very good agreement with the predictions of theory. The calculations were based on the J level populations shown in Fig. 4b which correspond to Boltzmann distributions at the experimentally determined temperatures. To judge the aptness of time-resolved CARS as a method of temperature measurement, we focus on the the highest peak past time zero Žat t s 0 the data may be distorted by the coherence peak of non-resonant background.. The calculated CARS signals ŽFig. 5. suggest two simple procedures: the first interprets the width of the recurrence peak Žc., which decreases inversely with temperature w11x. The alternate method is to compare the signal intensities at two different
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time delays. Time delay t 1 s 5.7 ps serves as reference point Žc.; the second point of measurement must be picked such as to provide optimum temperature resolution, e.g., t 2 s 5.1 ps for the range 300 ( T ( 1000 K Žb. and t 2 s 4.8 ps for temperatures T 0 1000 K Ža.. We have also fitted the temperature T to the full transients Ž0–1 ns. recorded in the range 0.5 ( p H 2 ( 1.8 bar, 296 ( T ( 1100 K. The temperature resolution was about 20 K throughout.
5. Conclusions In this Letter, we have screened non-resonant femtosecond CARS with respect to its applicability for detecting homonuclear molecules in the gas phase. The beating structures of the experimental transients truthfully mirror the rotational dynamics in the ground state and hence allow to extract detailed information on molecular constants of higher order, J-dependent collisional-broadening and line-shifting coefficients. A theoretical model which simulates the femtosecond time-resolved CARS signal of H 2 was presented. It depends sensitively on the temperature T, the pressure p, and individual dephasing times T2 Ž J .. Theory and experiment proved to be in excellent agreement. The high degree of time resolution offers the possibility to discriminate against unwanted interferences, as from additional components in gas mixtures, since these will contribute a non-resonant background only at t s 0. This makes the method a promising tool for an experimental determination of pressure, temperature or rotational population as well. We have shown that a determination of temperatures calls for a measurement of only two different time delays.
Acknowledgements Fig. 5. Calculated CARS transients of H 2 : first peak after time zero. The calculation is based on pulses with Gaussian spectral shape. Pump pulse: lP s 598 nm, D lFWHM s8.6 nm; Stokes pulse: lS s800 nm, D lFWHM s12.5 Žfitted to ‘real’ pulses in experiment.. Explanation given in the text.
The authors thank D. Zeidler and D. Proch for many valuable discussions and H. Bauer for his technical assistance.
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