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Study of multiphoton excitation of SF6 molecules in two strong IR laser fields
Chemicai Physics63 (1981) 219-226 North-HollandPublishingCompany
STUDY OF MULTIPHOTON EXCITATION IN TWO STRONG PR LASER FLELDS E. BORSELLA,
R. FANTONI,
OF SFs MOLECULES
G. PETROCELLI,
G. SANNA
CNEN, Centro di Frascati, 00044 Fmscati, Roma, Itaiy
and M. CAPITELLI
and M. DELONARDO
Centro di Studio per la Chimica dei Plasmi de1 CNR and Istitrtto di Chbnica Generale de!l*Utlicersirri, 701% Bnri. Iraly Received
22 June
1981
Experiments of multiphoton excitation of SF6 molecules in two strong IR laser fields have been performed as a function of significant physical parameters (fluence and wavelength of the two lasers, time delay between laser pulses). The measured average number of absorbed photons has been compared wirh calculations based on the heat bath feedback model extended IO the case of two-frequency excitation.
I. Introduction Among the processes involving the interaction of laser light with matter, multiphoton absorption and dissociation of polyatomic molecules in a strong IR field is a very interesting one [l-3]. A systematic approach to an understanding of these processes requires correct models for the molecular energy levels and the dynamics of interaction. It is important to distinguish between the initial excitation of the molecules up to few discrete energy levels and the subsequent excitation to the so caIled quasicontinuum region. The two-frequency technique, where the molecules are irradiated by light at two different wavelengths helps to obtain information separately on the two regions [4-73. In these experiments the first laser, tuned on the linear absorption frequency of the studied molecule is responsible of its vibrational excitation up to the discrete levels. The second laser, red or blue detuned with respect to the frequency of the first one, brings the excited molecules through the quasicontinuum. The experiments performed in two 0301-0104/81/0000-0000/$02.75
strong fields require an appropriate theoretical approach which must account for the effects of both lasers. A detailed theoretical analysis has not been performed up to now since much work is, 91 concerned the case where the second laser had such a low intensity to act only as a probe of the effect of the first one. In this paper we present a set of multiphoton absorption measure-. ments performed on SF6 together with a theoretical analysis. The theory is essentially based on the heat bath feed back model recently proposed and successfully appIied by Stone and Goodman [lo] to explain the absorption of SF6 under the action of one IR laser. Here the model has been extended in order to include the effect of the second laser. At this stage of development we cannot expect that the adopted approach will quantitatively reproduce all the experimental results either for the still retained approximations or for the influence of collisions in our measurements which are performed in a cell. However, it comes out that the adopted approach reproduces qualitatively the experimental features, thus allowing for further under-
@ 1981 North-Holland
standing of the process of multiple-photon tation of polyatomic molecules.
exci-
2. Experimentai We have performed the two-frequency multiphoton absorption experiments on SF6 molecules in a reaction cell 30 cm long, kept at room temperature. The experimental apparatus is sketched in fig. 1. Two pulsed TEA CO2 lasers (Lumonics mod. 102 and 2031 have been used as radiation sources. Using a typical CO1 gas mixture, pulse shapes consisted of a ~100 ns peak followed by a tail of about 1 ps as monitored by two fast pyroelectric detectors (Molectron P3-001. i\‘?-lean gas mixtures were used to shorten the pulses by suppressing the tail. Both lasers ii-ere equipped with low jitter units to reduce the jitter to less than 20 ns, thus allowing for a synchronization within 50 ns. When required a variable time delay (O-200 KS) was inserted between the two laser pulses. The laser beams were collimated and completely superimposed in the central region of the cell for 10 cm along the cell optical axis, as monitored on burned patterns of photosensible paper. This geometr) was obtained by means of twin optic lines in which iong focal (f= 1.5 m) spherical mirrors were inserted to correct the divergence of the two counter-propagating laser
beams. The spherical mirrors were rotated of a small angie (2”30’) in order to avoid the laser exit windows to be damaged. Two pyroelectric detectors (20D-Lumonics) were employed to measure the incident pulse energy, after calibration against a disc calorimeter (Scienteck mod 360 2033. Attenuation of the incident energy was accomplished by inserting in the optical lines either an absorption cell filled with propylen at a variable pressure or a couple of NaCl plates mounted at the Brewster angie which could rotate about the optical axis. The stainless steel cell sealed by two ZnSe windows, was equipped by a homemade capacitance microphone for low absorption measurements. The device is able to detect the acoustic pulse originated after V-T relaxation of the vibrationally excited molecules. In ref. [lla] it was shown that the maximum of the opto-acoustic signal is proportional to the average vibrational energy deposited in the molecules. A calibration of the microphone response is accomplished by means of conventional measurements of the incident and transmitted laser intensity in the case of appreciably high absorption. After processing [lib], the maximum of the optoacoustic signal is stored and averaged over a selected number of laser shots together with the signals coming from the two pyroelectrics.
3. Theoretical
analysis
The heat bath feed-back model proposed by Stone and Goodman [lo] for explaining the single laser absorption measurements has been extended in this work to two frequency absorption measurements. According to the model the first few vibrational levels t‘~; of the pumped ~a mode (uva< 3) are treated by Bloch equations (the notation is the same as used in ref. [12]): dpi//dt =ih-‘[p,
H];i-iKz1
+K,~I
)pij
+Ki\::.jpj+l.j+l +KjT,iP
j-l.j-tr
(05zis2)
(1)
dpJdt
= ih-‘[ p, H];.k (jPk,OSj,kc2), VT v-r = iA-‘[p, H]3; -(Ku f Ts, fK3.1 h
-(l/T2)jlcPjk dp,,ldr + (Kz
+ TwINa+ K~:Px,
where the nonzero elements H are:
(3) of the hamiltonian (4)
Ha, = E,, H,-,_,=H
(2)
,,.” -1=cr “.,, -iAsinot.
(5)
Eqs. (l)-(5) include the effects of coherent pumping (ih-‘[p, HI), dephasing processes (l/T?) and V-T vibration-transIation energy transfer (K”). The Bloch equations have been solved in the rotating wave approximation in order to render the Hn--l.n coefficients independent of time (~y,.,-i are the transition dipole moments, A and o are the intensity and the frequency of the laser). The second dynamic region is governed by a set of rate equations for the total population Ni of vibrational levels:
present calculations will be given in a forthcoming paper [14]. In our caiculation throughout the second dynamic region (q-c_) fifty energy levels have been included while a simple anharmonic mode1 was used to obtain the vibrational energies c,, of SF6 U; mode [12, 151. Only one rotational pathway has been considered in the present calculation, since the introduction of all rotational pathways (PQR, PQQ, PPR, and so on) should require a long computational time which does not appear justified at this stage of approximation. Present calculations have been performed by using eqs. (l)-(6) with the first laser field (wl) switched on for a time comparable with the pulse length, followed by the second laser field (ol) at different delay times. This theoretical approach was also used in calculations to be compared with experiments performed at zero delay time. In the last case the delay was considered equal to the largest experimental uncertainty (50 ns) in synchronization, so that the interference of the laser fields had not to be included in eqs. (l)-(6).
dhiijdr=(?;-l.j+K,~,.j)Ni-l +(Tf+l.i+K>:.i)Nj+l -(~;;-1+~;i;l+K,~1
4. Results +K,yl
+kj)Nj
(6)
(k; are specific reaction rate constants). The transition rates T appearing in eq. (6) as well as in eq. (3) take account of the coupling with the other vibrational modes through the dephasing rates l/T? (see ref. [12]), which depend on the number of quanta transferred to the heat bath. Both V-T rates and dissociation rates have been taken from ref. [13], while the dephasing rates have been calculated according to the rules given by Horsley and co-workers [12]. These rates depend on the anharmonic coupling between the pumped mode and the other SF6 modes, (Hanh)as well as on the density of states. A value of Han,,= 0.04 cm-’ has been used, while the density of states has been calculated according to the Whitten-Rabinovitch formula (see ref. [12j for details). A complete discussion of the influence of these parameters on the
Fig. 2a shows the quantity )I ,? - n 1 as a function of the fluence &I of the first exciting laser (oi = 948 cm-‘) for two different frequencies of the probe laser, shifted respectively to the blue (02 = 976 cm-‘) and to the red (wz = 929 cm-‘) of the main transition. The quantities rzr~ and ni represent respectively the absorbed quanta per molecule in the presence of both lasers and of the first exciting laser only. Thus ni2- tzi results to be proportional to the absorption cross section of the second laser in the presence of the first one [6]_ The following features are apparent in fig. 2a: (i) the quantity IZiz - n r, after an initial sharp increase varies slowly with (5i, [ii) the change in the slope takes place at a different value of ~$i depending on the frequency wz of the second laser, (iii) the absorbed number of photons nlz--nl is larger when the frequency of the second laser is shifted to the red. The theoretical results reported in fig. 2b
E. Borselia et ni. / Mtdciphoton excitation of SF6
Fig. 2. Sumber of absorbed qilanrafrom the second laser in the presence of the first one (psFa = 0.8 Torr, LI)~= 937.7 cm-‘. & = 0.7 J/cm’) versus the resonant laser Ruence bl: (0) w2 = 929 cm-’ left scale, (A! wz = 976 cm-’ right scale: (a) experimental results, tb) theoretical predictions. reproduce qualitatively well two features of the experimental curves: (i) the presence of two different slopes at both the considered frequencies wl, and (ii) the larger number of the absorbed quanta at 929 cm-’ with respect to those measured at 976 cm-‘. Quantitatively, however, some discrepancies between theory and experiments are still retained: in particular the theory underestimates by a factor of 2 the absorbed quanta at 929 cm-l, while overestimates by a factor of 3 those measured at 976 cm-‘. In order to understand the behaviour of n12 -nl versus 41 it is interesting to investigate the distribution of the vibrational states before and after the onset of the second laser. Significant exampIes of these calcuIared distributions are reported in fig. 3 for o2 = 929 cm-‘. It can be seen that the distribution of population created by the first laser is shifted towards higher levels as 41 increases; thus the energy of second laser field, red-shifted
with respect to the fundamental transition, can be more and more absorbed since the energy spacing between the adjacent upper levels tends to decrease as the vibrational quantum number L’ grows up. The distribution of population originated after the absorption of the first laser is driven by the second one to higher vibrational states thus determining the steep increase of the absorbed quanta n 12- nl. When states of sufficiently high energy are considerably populated by the first laser field a matching can be easily found between the red shifted second laser frequency and the energy spacing among the states. In this condition the transfer of the vibrational population towards the higher energy states, or the quasicontinuum, approaches its maximum rate. When this is the case there are only slight variations of nlr - n 1 with &. Moreover, the anharmonicity favours the absorption of energy from the second laser field at frequencies shifted to the red, while it is less effective in the case of blue-shifted frequencies: this could justify the lower absorption of radiation at wa = 976 cm-’ in comparison with the case of o2 = 929 cm-‘. Fig. 4a shows tll? as a function of the frequency of the second laser measured at different delay times, the frequency of the first laser being fixed at 948 cm-‘. It can be noticed that there is a strong red-shift as compared with the one frequency multiphoton absorption spectrum. The response curve at zero delay time shows the presence of two maxima which merge in only one peak centered at 935 cm-’ in a time of the order of 15 ps. A similar behaviour has been found by other authors [8,9], by using a cw probe laser of low intensity. Bagratashvili et al. [S] proposed also a qualitative explanation of this effect: they considered the system after having been pumped by the first laser as formed by two ensembles of particles: one vibrationally hot, the other one cold. As a consequence of the collision processes between the two subsystems a common vibrational temperature should be reached. In the present case (0.8 Torr of SF6, & =0.4 J/cm’) the features shown by the spectrum at zero delay time, which are related to the non-equilibrium energy distribution, disap-
a
b
.-_ /’ I
i
oi I _i -i -i _i i qi)I
i.
..!I -Ii _ii ii
i i i
j
i i
i i
\P i ! $
i
i
\
1’
I I
ii ii!
1 ii:
Fig. 3. Calculated distribution of population in the energy levels versus the quantum number (u) [ps~* = 0.8 Torr, w1 = Ia) $I =O.l J/cm’, lb) 6, = 0.7 J/cm’. 947.7 cm-‘. &,=O.SJ/cm’): c---j w1 only; C-.-b) O,+OJ~ with wz=929cm-‘. (cl *, = 1.1 J/cm’.
pear in about 15 ~LScorresponding to the establishment of the equilibrium temperature. Our results give a further 193 evidence of the nontermal vibrational energy distribution produced by absorption of intense laser radiation. Fig. 4b shows the two-frequency theoretical spectrum calculated as a function of 02 at zero delay time. A qualitative agreement with the experimental spectrum is found. We observe in fact an appreciable red-shift of the theoretical response curve, even if the strong maximum at 945 cm-‘, which can be theoreticalIy ascribed to a twophoton transition, is not evident in the experimental spectrum. Moreover, the theoretical and experimental response curves appear to be frequency shifted. At this stage of approximation, the theoretical curve calculated with T = 15 ps cannot reproduce the experimental data, because only V-T relaxation processes are
taken into account while V-V exchanges, which are effective in vibrationa energy redistribution, are neglected. The experimental red-shift in the absorption spectrum at : = 0 is more evident when the quantity n ,Z - (n 1+ nl) is reported as a function of the frequency of the second Iaser (see fig. 5a). Once again the theoretica results, repcrted in fig. 5b, qualitativeIy follow the experimental trend. Finally, it is worthwhile to stress that the twofrequency method allows for measuring the lifetime of the highly excited states by varying the time delay between the two lasers. As an example measurements of the enhancemen in the two-laser multiphoton absorption A 12(A,+ AZ) versus the time interval are reported in fig. 6a. As the delay increases the molecules excited by the first laser undergo a progressive
Fig. 4. Absorbed quanta n12 versus WI (psF,, = 0.8 Torr, & = 0.4 J/cm’. & = 0.8 J,‘cm’, o1 = 947.7 cm-‘): (0) only one laser; two lasers: (0) zero delay time; (A) r =7 p.s, (x) : = 15 p.s. (a) experimental results. (b) theoretical predictions.
collisional deactivation and the absorption of the second laser radiation is reduced. When all excited molecules are deactivated in the interval between the hvo pulses, the quantity Al*(AlsAl) is equal to zero since the cooperative effect of the lasers vanishes. Since V-T relaxation is the deactivation process with the longest decay time r when compared with the V-V (1.6 &sTorr) and R-R (36 nsTorr) ones [16] our experimental curves, at least for a delay ~20 ps, are essentially affected by V-T relaxation. Interpolation of data (see fig. 6a) for the above specified range of delay gives: ~v-T== 25 ps Torr in agreement with results reported in ref. [17]. Calculated values nl,-((nl+nz) of the enhancement in the absorption measured when both lasers are present as a function of the delay time between the two laser pulses are reported in fig. 6b. It must be emphasized that only V-T relaxation is included at present in the used model. The theoretical results (reported in fig. 6b), which have been obtained by increasing the collisional rates of ref. [13] by a factor of 5, are in satisfactory agreement with the experimental values especially after lo-20 ps.
-I 12 c? T -=
Ta -n z
Fig. 5. Enhancement in the absorbed quanta versus u2 as measured
reported
in fig. 4.
(a)
and
calculated
(b)
under the same conditions as
E. Borsella er al. / Mulriphoron
excirarion of SF6
225
(3) Inclusion of appropriate energy exchange processes (in particular the V-V and V-R ones). (.a) Evaluation of hot bands contributions. AI1 these refinements, which are at present under investigation, would probably increase the agreement between theory and experiment. It is worthwhile to notice that the incoherent master equation approach, widely used to interpret the one-laser absorption measurements (see for example ref. [13]) fails to reproduce the twofrequency absorption measurements. In fact, values of nlz- (nl + nZ) equal to zero have been calculated on the basis of this model for ail the second laser frequencies. Thus, despite the limitation above mentioned, we suggest that the heat bath feedback model can be considered as a satisfactory first approach for understanding the two-laser absorption measurements.
I SO
I
Acknowledgement
100
Fig. 6. Enhancement A,-(A, -iA,) (arbitrary units) in the absorbed quanta versus delay time (wI =947.7 cn~‘), oz= 925 cm-‘, Q, = 0.1 J/cm’, & = 0.5 Jf cm’, psFD= 0.8 Torr): (a) experimental results, (b) theoretical predictions [nlz-(n,+&)].
5. Conclusions
The results reported here show that the heat bath feedback model can be utilized with a fair amount of confidence for the interpretation of the two-frequency multiphoton absorption measurements. However, in order to obtain a quantitative agreement it is required to introduce further improvements in the theory, which can be summarized as follows: (1) Inclusion of all the rotational pathways in the structure of the first few levels treated by the Bloch equations. (2) Better characterization of the vibrational energies of the first levels by removal of the degeneracy of the v3 pumped mode.
We wish to thank Professor A. GiardiniGuidoni and Professor J. Reuss for many valuable discussions and suggestions, Dr. Cacciatore for help in the calculations and Dr. M. Bernardini for cooperation in developing the opto-acoustic cell. We also acknowledge the technicians of the Molecular Spectroscopy Laboratory of CNEN. (R. Belardinelli, P. Cardoni, I. Cenciarelli, M. Nardelli, S. Riioeuo, G. Schina) for their technical assistance in the course of the experiments.
References r11 C.D. Cantrell, H.W. Galbraith
and J.R. Ackerhalr, in: ?ilultiphoton processes, Proceedings of the International Conference at the University of Rochester, June 6-9, 1977. New York, ed. J. Eberly. 121 M. Janossy and S. Van0 eds, Proceedings of the 2nd ICOMP, Budapest, April 1980 (Budapest, 1981). excitation and dis 131 CD. Cantrell ed., Multiple-photon sociation of polyatomic molecules (Springer, Serlin, 1981).
E.
‘26
Borsch
er al. / Mdriphoron
excirarion
[4] V.S. Letokhov,
in: Multiphoton processes, Proceedings of the International Conference at the University of
Rochester, June 6-9, 1977, New York, ed. J. Eberly, pp. 331-347, and references therein. [5 j N.V. Karlov, in: Proceedings of the 2nd ICQMP. Budapest, April 1980, eds. M. Yanossy and S. Varro (Budapest, 1981) pp. 149-191. and references rherein. [6] G.P. Quiglsy, Opt. Letters 4 (1979) 84. 173 W. Fuss and J. Hartmann, J. Chem. Phys. 70 (19791 5468. [8] V.N. Bagratashvili. VS. Dolzhikov and VS. Letokhov. Soviet Phys. JETP 49 (1978) 8. [91 J.L. Lyman, L.J. Radzicnski and A.C. Nilsson, IEEE JQE. QE-16 (1980~ 1174. [lo] J. Stone and M.F. Goodman, J. Chem. Phys. 71 (1979) 408. [ill (a) M. Bcmardini. M. Sardi al:d G. Sanna. to be published;
[lZ] j13]
[IS] [15] [16]
[17]
of SF,
(b) $1. Bernardini. E. Borsella, i&4.Nardi and G. Sanna, to be published. J.A. Horsley, J. Stone, MF. Goodman and D.A. Dow, Chem. Phys.. Letters 66 (1979) 461. J.L. Lyman, J. Chem. Phys., 67 (1977) 1868; _v. Bernardini. M. Cacciatore, M. Capitelli and G. Sanna, Nuovo Cimento 63 B (1981) 36; M. Dilonardo, Nuovo Cimento 63 B (1981) 131. M. Diionardo et al., to be published. J.R. Ackerhalt and H.W. Galbraith, J. Chem. Phys. 69 (1976) 1200. J.L. Lyman, G.P. Quigley and 0-P. Judd, in: Multiplephoton excitation and dissociation of polyatomic molecules. ed. CD. Cantrell (Springer, Berlin, 1981); LA-UR 79-2605. S.A. Akmanov, V.N. Gordienko, A.V. Nikheenko and V. Ya. Panchenko, JETP Letters 26 (1979) 454.
STUDY OF MULTIPHOTON EXCITATION IN TWO STRONG PR LASER FLELDS E. BORSELLA,
R. FANTONI,
OF SFs MOLECULES
G. PETROCELLI,
G. SANNA
CNEN, Centro di Frascati, 00044 Fmscati, Roma, Itaiy
and M. CAPITELLI
and M. DELONARDO
Centro di Studio per la Chimica dei Plasmi de1 CNR and Istitrtto di Chbnica Generale de!l*Utlicersirri, 701% Bnri. Iraly Received
22 June
1981
Experiments of multiphoton excitation of SF6 molecules in two strong IR laser fields have been performed as a function of significant physical parameters (fluence and wavelength of the two lasers, time delay between laser pulses). The measured average number of absorbed photons has been compared wirh calculations based on the heat bath feedback model extended IO the case of two-frequency excitation.
I. Introduction Among the processes involving the interaction of laser light with matter, multiphoton absorption and dissociation of polyatomic molecules in a strong IR field is a very interesting one [l-3]. A systematic approach to an understanding of these processes requires correct models for the molecular energy levels and the dynamics of interaction. It is important to distinguish between the initial excitation of the molecules up to few discrete energy levels and the subsequent excitation to the so caIled quasicontinuum region. The two-frequency technique, where the molecules are irradiated by light at two different wavelengths helps to obtain information separately on the two regions [4-73. In these experiments the first laser, tuned on the linear absorption frequency of the studied molecule is responsible of its vibrational excitation up to the discrete levels. The second laser, red or blue detuned with respect to the frequency of the first one, brings the excited molecules through the quasicontinuum. The experiments performed in two 0301-0104/81/0000-0000/$02.75
strong fields require an appropriate theoretical approach which must account for the effects of both lasers. A detailed theoretical analysis has not been performed up to now since much work is, 91 concerned the case where the second laser had such a low intensity to act only as a probe of the effect of the first one. In this paper we present a set of multiphoton absorption measure-. ments performed on SF6 together with a theoretical analysis. The theory is essentially based on the heat bath feed back model recently proposed and successfully appIied by Stone and Goodman [lo] to explain the absorption of SF6 under the action of one IR laser. Here the model has been extended in order to include the effect of the second laser. At this stage of development we cannot expect that the adopted approach will quantitatively reproduce all the experimental results either for the still retained approximations or for the influence of collisions in our measurements which are performed in a cell. However, it comes out that the adopted approach reproduces qualitatively the experimental features, thus allowing for further under-
@ 1981 North-Holland
standing of the process of multiple-photon tation of polyatomic molecules.
exci-
2. Experimentai We have performed the two-frequency multiphoton absorption experiments on SF6 molecules in a reaction cell 30 cm long, kept at room temperature. The experimental apparatus is sketched in fig. 1. Two pulsed TEA CO2 lasers (Lumonics mod. 102 and 2031 have been used as radiation sources. Using a typical CO1 gas mixture, pulse shapes consisted of a ~100 ns peak followed by a tail of about 1 ps as monitored by two fast pyroelectric detectors (Molectron P3-001. i\‘?-lean gas mixtures were used to shorten the pulses by suppressing the tail. Both lasers ii-ere equipped with low jitter units to reduce the jitter to less than 20 ns, thus allowing for a synchronization within 50 ns. When required a variable time delay (O-200 KS) was inserted between the two laser pulses. The laser beams were collimated and completely superimposed in the central region of the cell for 10 cm along the cell optical axis, as monitored on burned patterns of photosensible paper. This geometr) was obtained by means of twin optic lines in which iong focal (f= 1.5 m) spherical mirrors were inserted to correct the divergence of the two counter-propagating laser
beams. The spherical mirrors were rotated of a small angie (2”30’) in order to avoid the laser exit windows to be damaged. Two pyroelectric detectors (20D-Lumonics) were employed to measure the incident pulse energy, after calibration against a disc calorimeter (Scienteck mod 360 2033. Attenuation of the incident energy was accomplished by inserting in the optical lines either an absorption cell filled with propylen at a variable pressure or a couple of NaCl plates mounted at the Brewster angie which could rotate about the optical axis. The stainless steel cell sealed by two ZnSe windows, was equipped by a homemade capacitance microphone for low absorption measurements. The device is able to detect the acoustic pulse originated after V-T relaxation of the vibrationally excited molecules. In ref. [lla] it was shown that the maximum of the opto-acoustic signal is proportional to the average vibrational energy deposited in the molecules. A calibration of the microphone response is accomplished by means of conventional measurements of the incident and transmitted laser intensity in the case of appreciably high absorption. After processing [lib], the maximum of the optoacoustic signal is stored and averaged over a selected number of laser shots together with the signals coming from the two pyroelectrics.
3. Theoretical
analysis
The heat bath feed-back model proposed by Stone and Goodman [lo] for explaining the single laser absorption measurements has been extended in this work to two frequency absorption measurements. According to the model the first few vibrational levels t‘~; of the pumped ~a mode (uva< 3) are treated by Bloch equations (the notation is the same as used in ref. [12]): dpi//dt =ih-‘[p,
H];i-iKz1
+K,~I
)pij
+Ki\::.jpj+l.j+l +KjT,iP
j-l.j-tr
(05zis2)
(1)
dpJdt
= ih-‘[ p, H];.k (jPk,OSj,kc2), VT v-r = iA-‘[p, H]3; -(Ku f Ts, fK3.1 h
-(l/T2)jlcPjk dp,,ldr + (Kz
+ TwINa+ K~:Px,
where the nonzero elements H are:
(3) of the hamiltonian (4)
Ha, = E,, H,-,_,=H
(2)
,,.” -1=cr “.,, -iAsinot.
(5)
Eqs. (l)-(5) include the effects of coherent pumping (ih-‘[p, HI), dephasing processes (l/T?) and V-T vibration-transIation energy transfer (K”). The Bloch equations have been solved in the rotating wave approximation in order to render the Hn--l.n coefficients independent of time (~y,.,-i are the transition dipole moments, A and o are the intensity and the frequency of the laser). The second dynamic region is governed by a set of rate equations for the total population Ni of vibrational levels:
present calculations will be given in a forthcoming paper [14]. In our caiculation throughout the second dynamic region (q-c_) fifty energy levels have been included while a simple anharmonic mode1 was used to obtain the vibrational energies c,, of SF6 U; mode [12, 151. Only one rotational pathway has been considered in the present calculation, since the introduction of all rotational pathways (PQR, PQQ, PPR, and so on) should require a long computational time which does not appear justified at this stage of approximation. Present calculations have been performed by using eqs. (l)-(6) with the first laser field (wl) switched on for a time comparable with the pulse length, followed by the second laser field (ol) at different delay times. This theoretical approach was also used in calculations to be compared with experiments performed at zero delay time. In the last case the delay was considered equal to the largest experimental uncertainty (50 ns) in synchronization, so that the interference of the laser fields had not to be included in eqs. (l)-(6).
dhiijdr=(?;-l.j+K,~,.j)Ni-l +(Tf+l.i+K>:.i)Nj+l -(~;;-1+~;i;l+K,~1
4. Results +K,yl
+kj)Nj
(6)
(k; are specific reaction rate constants). The transition rates T appearing in eq. (6) as well as in eq. (3) take account of the coupling with the other vibrational modes through the dephasing rates l/T? (see ref. [12]), which depend on the number of quanta transferred to the heat bath. Both V-T rates and dissociation rates have been taken from ref. [13], while the dephasing rates have been calculated according to the rules given by Horsley and co-workers [12]. These rates depend on the anharmonic coupling between the pumped mode and the other SF6 modes, (Hanh)as well as on the density of states. A value of Han,,= 0.04 cm-’ has been used, while the density of states has been calculated according to the Whitten-Rabinovitch formula (see ref. [12j for details). A complete discussion of the influence of these parameters on the
Fig. 2a shows the quantity )I ,? - n 1 as a function of the fluence &I of the first exciting laser (oi = 948 cm-‘) for two different frequencies of the probe laser, shifted respectively to the blue (02 = 976 cm-‘) and to the red (wz = 929 cm-‘) of the main transition. The quantities rzr~ and ni represent respectively the absorbed quanta per molecule in the presence of both lasers and of the first exciting laser only. Thus ni2- tzi results to be proportional to the absorption cross section of the second laser in the presence of the first one [6]_ The following features are apparent in fig. 2a: (i) the quantity IZiz - n r, after an initial sharp increase varies slowly with (5i, [ii) the change in the slope takes place at a different value of ~$i depending on the frequency wz of the second laser, (iii) the absorbed number of photons nlz--nl is larger when the frequency of the second laser is shifted to the red. The theoretical results reported in fig. 2b
E. Borselia et ni. / Mtdciphoton excitation of SF6
Fig. 2. Sumber of absorbed qilanrafrom the second laser in the presence of the first one (psFa = 0.8 Torr, LI)~= 937.7 cm-‘. & = 0.7 J/cm’) versus the resonant laser Ruence bl: (0) w2 = 929 cm-’ left scale, (A! wz = 976 cm-’ right scale: (a) experimental results, tb) theoretical predictions. reproduce qualitatively well two features of the experimental curves: (i) the presence of two different slopes at both the considered frequencies wl, and (ii) the larger number of the absorbed quanta at 929 cm-’ with respect to those measured at 976 cm-‘. Quantitatively, however, some discrepancies between theory and experiments are still retained: in particular the theory underestimates by a factor of 2 the absorbed quanta at 929 cm-l, while overestimates by a factor of 3 those measured at 976 cm-‘. In order to understand the behaviour of n12 -nl versus 41 it is interesting to investigate the distribution of the vibrational states before and after the onset of the second laser. Significant exampIes of these calcuIared distributions are reported in fig. 3 for o2 = 929 cm-‘. It can be seen that the distribution of population created by the first laser is shifted towards higher levels as 41 increases; thus the energy of second laser field, red-shifted
with respect to the fundamental transition, can be more and more absorbed since the energy spacing between the adjacent upper levels tends to decrease as the vibrational quantum number L’ grows up. The distribution of population originated after the absorption of the first laser is driven by the second one to higher vibrational states thus determining the steep increase of the absorbed quanta n 12- nl. When states of sufficiently high energy are considerably populated by the first laser field a matching can be easily found between the red shifted second laser frequency and the energy spacing among the states. In this condition the transfer of the vibrational population towards the higher energy states, or the quasicontinuum, approaches its maximum rate. When this is the case there are only slight variations of nlr - n 1 with &. Moreover, the anharmonicity favours the absorption of energy from the second laser field at frequencies shifted to the red, while it is less effective in the case of blue-shifted frequencies: this could justify the lower absorption of radiation at wa = 976 cm-’ in comparison with the case of o2 = 929 cm-‘. Fig. 4a shows tll? as a function of the frequency of the second laser measured at different delay times, the frequency of the first laser being fixed at 948 cm-‘. It can be noticed that there is a strong red-shift as compared with the one frequency multiphoton absorption spectrum. The response curve at zero delay time shows the presence of two maxima which merge in only one peak centered at 935 cm-’ in a time of the order of 15 ps. A similar behaviour has been found by other authors [8,9], by using a cw probe laser of low intensity. Bagratashvili et al. [S] proposed also a qualitative explanation of this effect: they considered the system after having been pumped by the first laser as formed by two ensembles of particles: one vibrationally hot, the other one cold. As a consequence of the collision processes between the two subsystems a common vibrational temperature should be reached. In the present case (0.8 Torr of SF6, & =0.4 J/cm’) the features shown by the spectrum at zero delay time, which are related to the non-equilibrium energy distribution, disap-
a
b
.-_ /’ I
i
oi I _i -i -i _i i qi)I
i.
..!I -Ii _ii ii
i i i
j
i i
i i
\P i ! $
i
i
\
1’
I I
ii ii!
1 ii:
Fig. 3. Calculated distribution of population in the energy levels versus the quantum number (u) [ps~* = 0.8 Torr, w1 = Ia) $I =O.l J/cm’, lb) 6, = 0.7 J/cm’. 947.7 cm-‘. &,=O.SJ/cm’): c---j w1 only; C-.-b) O,+OJ~ with wz=929cm-‘. (cl *, = 1.1 J/cm’.
pear in about 15 ~LScorresponding to the establishment of the equilibrium temperature. Our results give a further 193 evidence of the nontermal vibrational energy distribution produced by absorption of intense laser radiation. Fig. 4b shows the two-frequency theoretical spectrum calculated as a function of 02 at zero delay time. A qualitative agreement with the experimental spectrum is found. We observe in fact an appreciable red-shift of the theoretical response curve, even if the strong maximum at 945 cm-‘, which can be theoreticalIy ascribed to a twophoton transition, is not evident in the experimental spectrum. Moreover, the theoretical and experimental response curves appear to be frequency shifted. At this stage of approximation, the theoretical curve calculated with T = 15 ps cannot reproduce the experimental data, because only V-T relaxation processes are
taken into account while V-V exchanges, which are effective in vibrationa energy redistribution, are neglected. The experimental red-shift in the absorption spectrum at : = 0 is more evident when the quantity n ,Z - (n 1+ nl) is reported as a function of the frequency of the second Iaser (see fig. 5a). Once again the theoretica results, repcrted in fig. 5b, qualitativeIy follow the experimental trend. Finally, it is worthwhile to stress that the twofrequency method allows for measuring the lifetime of the highly excited states by varying the time delay between the two lasers. As an example measurements of the enhancemen in the two-laser multiphoton absorption A 12(A,+ AZ) versus the time interval are reported in fig. 6a. As the delay increases the molecules excited by the first laser undergo a progressive
Fig. 4. Absorbed quanta n12 versus WI (psF,, = 0.8 Torr, & = 0.4 J/cm’. & = 0.8 J,‘cm’, o1 = 947.7 cm-‘): (0) only one laser; two lasers: (0) zero delay time; (A) r =7 p.s, (x) : = 15 p.s. (a) experimental results. (b) theoretical predictions.
collisional deactivation and the absorption of the second laser radiation is reduced. When all excited molecules are deactivated in the interval between the hvo pulses, the quantity Al*(AlsAl) is equal to zero since the cooperative effect of the lasers vanishes. Since V-T relaxation is the deactivation process with the longest decay time r when compared with the V-V (1.6 &sTorr) and R-R (36 nsTorr) ones [16] our experimental curves, at least for a delay ~20 ps, are essentially affected by V-T relaxation. Interpolation of data (see fig. 6a) for the above specified range of delay gives: ~v-T== 25 ps Torr in agreement with results reported in ref. [17]. Calculated values nl,-((nl+nz) of the enhancement in the absorption measured when both lasers are present as a function of the delay time between the two laser pulses are reported in fig. 6b. It must be emphasized that only V-T relaxation is included at present in the used model. The theoretical results (reported in fig. 6b), which have been obtained by increasing the collisional rates of ref. [13] by a factor of 5, are in satisfactory agreement with the experimental values especially after lo-20 ps.
-I 12 c? T -=
Ta -n z
Fig. 5. Enhancement in the absorbed quanta versus u2 as measured
reported
in fig. 4.
(a)
and
calculated
(b)
under the same conditions as
E. Borsella er al. / Mulriphoron
excirarion of SF6
225
(3) Inclusion of appropriate energy exchange processes (in particular the V-V and V-R ones). (.a) Evaluation of hot bands contributions. AI1 these refinements, which are at present under investigation, would probably increase the agreement between theory and experiment. It is worthwhile to notice that the incoherent master equation approach, widely used to interpret the one-laser absorption measurements (see for example ref. [13]) fails to reproduce the twofrequency absorption measurements. In fact, values of nlz- (nl + nZ) equal to zero have been calculated on the basis of this model for ail the second laser frequencies. Thus, despite the limitation above mentioned, we suggest that the heat bath feedback model can be considered as a satisfactory first approach for understanding the two-laser absorption measurements.
I SO
I
Acknowledgement
100
Fig. 6. Enhancement A,-(A, -iA,) (arbitrary units) in the absorbed quanta versus delay time (wI =947.7 cn~‘), oz= 925 cm-‘, Q, = 0.1 J/cm’, & = 0.5 Jf cm’, psFD= 0.8 Torr): (a) experimental results, (b) theoretical predictions [nlz-(n,+&)].
5. Conclusions
The results reported here show that the heat bath feedback model can be utilized with a fair amount of confidence for the interpretation of the two-frequency multiphoton absorption measurements. However, in order to obtain a quantitative agreement it is required to introduce further improvements in the theory, which can be summarized as follows: (1) Inclusion of all the rotational pathways in the structure of the first few levels treated by the Bloch equations. (2) Better characterization of the vibrational energies of the first levels by removal of the degeneracy of the v3 pumped mode.
We wish to thank Professor A. GiardiniGuidoni and Professor J. Reuss for many valuable discussions and suggestions, Dr. Cacciatore for help in the calculations and Dr. M. Bernardini for cooperation in developing the opto-acoustic cell. We also acknowledge the technicians of the Molecular Spectroscopy Laboratory of CNEN. (R. Belardinelli, P. Cardoni, I. Cenciarelli, M. Nardelli, S. Riioeuo, G. Schina) for their technical assistance in the course of the experiments.
References r11 C.D. Cantrell, H.W. Galbraith
and J.R. Ackerhalr, in: ?ilultiphoton processes, Proceedings of the International Conference at the University of Rochester, June 6-9, 1977. New York, ed. J. Eberly. 121 M. Janossy and S. Van0 eds, Proceedings of the 2nd ICOMP, Budapest, April 1980 (Budapest, 1981). excitation and dis 131 CD. Cantrell ed., Multiple-photon sociation of polyatomic molecules (Springer, Serlin, 1981).
E.
‘26
Borsch
er al. / Mdriphoron
excirarion
[4] V.S. Letokhov,
in: Multiphoton processes, Proceedings of the International Conference at the University of
Rochester, June 6-9, 1977, New York, ed. J. Eberly, pp. 331-347, and references therein. [5 j N.V. Karlov, in: Proceedings of the 2nd ICQMP. Budapest, April 1980, eds. M. Yanossy and S. Varro (Budapest, 1981) pp. 149-191. and references rherein. [6] G.P. Quiglsy, Opt. Letters 4 (1979) 84. 173 W. Fuss and J. Hartmann, J. Chem. Phys. 70 (19791 5468. [8] V.N. Bagratashvili. VS. Dolzhikov and VS. Letokhov. Soviet Phys. JETP 49 (1978) 8. [91 J.L. Lyman, L.J. Radzicnski and A.C. Nilsson, IEEE JQE. QE-16 (1980~ 1174. [lo] J. Stone and M.F. Goodman, J. Chem. Phys. 71 (1979) 408. [ill (a) M. Bcmardini. M. Sardi al:d G. Sanna. to be published;
[lZ] j13]
[IS] [15] [16]
[17]
of SF,
(b) $1. Bernardini. E. Borsella, i&4.Nardi and G. Sanna, to be published. J.A. Horsley, J. Stone, MF. Goodman and D.A. Dow, Chem. Phys.. Letters 66 (1979) 461. J.L. Lyman, J. Chem. Phys., 67 (1977) 1868; _v. Bernardini. M. Cacciatore, M. Capitelli and G. Sanna, Nuovo Cimento 63 B (1981) 36; M. Dilonardo, Nuovo Cimento 63 B (1981) 131. M. Diionardo et al., to be published. J.R. Ackerhalt and H.W. Galbraith, J. Chem. Phys. 69 (1976) 1200. J.L. Lyman, G.P. Quigley and 0-P. Judd, in: Multiplephoton excitation and dissociation of polyatomic molecules. ed. CD. Cantrell (Springer, Berlin, 1981); LA-UR 79-2605. S.A. Akmanov, V.N. Gordienko, A.V. Nikheenko and V. Ya. Panchenko, JETP Letters 26 (1979) 454.