Chemical Physics 173 (1993) 291-303 North-Holland

Time dependence of the refractive index of SF6 and SF6-Ar mixtures during a TEA CO2 laser pulse. A focusing effect preceding thermal lensing

M. Lenzi a*b, E. Molinari ‘, G. Piciacchia b and M.L. Terranova ’ ’ Istituto di Metodologie Avanzate Inorganiche, CNR, P.O. Box 10, Monterotondo Scala 00016, Rome, Italy b Area della Ricerca di Roma, CNR, Servizio Laser, P.O. Box IO, Monterotondo Scala 00016, Rome, Italy ’ Dipartimento di Scienze e Tecnologie Chimiche, Vniversitb di Roma “Tor Vergata’: Via 0. Raimondo, Rome, Italy

Received 14 July 1992; in final form 17 February 1993

The time dependence of the refractive index in SF6 and SF&r mixtures during the absorption of the infrared radiation from a TEA CO2 laser beam is monitored by a cw He-Ne probe laser. A focusing signal which occurs before the onset of the thermal lensing is mostly attributed to an increase of the local SF, gas density. The extent of this effect is evaluated in terms of the formation of a symmetric positive Gaussian gas lens in the absorption cell.

1. Introdnctlon

The kinetics of the infrared multiple-photon ab- sorption by SF6 during a TEA COZ laser pulse has re- cently been investigated in this laboratory by a light attenuation technique. Results obtained on both neat SF6 [ 1 ] at pressures of 0.2-l Torr and in mixtures [2] of SF6 at pressures less than 0.2 Torr with He, Ne, Ar, Kr and Xe (200-760 Torr), lead to an eval- uation of the average number of photons (n) ab- sorbed per molecule as a function of time during the laser pulse. (n) (t) values thus obtained are not in agreement with those expected on the basis of ab- sorption cross sections given in the literature [ 31. The analysis of the temporal evolution of ( n) and, where possible, of the absorption cross section, indicates that a radial concentration profile of vibrationally excited SF, molecules should form during the irradiation leading to a higher axial gas density. This effect has been described theoretically in the literature [ 41 un- der the name “laser induced osmosis”. A tentative experimental evidence has been provided by the same author [ 5 1.

It is worth stressing the relevance of this effect in the interpretation of some kinetic experiments based on the evaluation of the initial vibrational energy

content of molecules such as in sensitised photolysis or laser stimulated chemical reactions where the pos- sibility of an increase of the concentration of the ab- sorber along the axis of the exciting laser beam has never been taken into consideration. It appears therefore of interest to verify this hypothesis by some more “direct” observation of the existence of such a radial profile of the gas density. The formation of a nearly Gaussian profile inside the CO2 laser beam should give rise to a focusing gas lens coaxial to the beam and therefore should be detectable with an ex- perimental approach similar to that used in thermal lensing studies. Accordingly experiments have been carried out using a He-Ne laser output as a probe beam. The results clearly indicate the presence of a time dependence of the refractive index. An ob- served initial focusing signal can be attributed to the onset of a Gaussian profile either in polarisability or in density while the following defocusing effect is due as expected to thermal lensing. A quantitative evalu- ation of the overall refractivity variation in the irra- diated volume has been made using a simplified op- tical model based on the formation of a positive Gaussian lens in both neat SF, and SFs-Ar mixtures. An evaluation of the contribution due to the polaris- ability increase of the excited SF, molecule is made

0301-0104/93/S 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.

292 h4. Lenzi et al, / Chemical Physics 173 (1993) 291-303

on the basis of available literature data [ 61. This ap- pears to be minor in most cases and the observed fo- cusing is thus tentatively attributed for the larger part to the onset of a radial distribution of the gas density. This is in qualitative agreement with previous exper- iments [ 1,2].

2. Experimental

The experimental set up (fig. 1) is of the type adopted in some cases for thermal lensing studies. It is based on the detection of density fluctuations of the gas leading to modulations of the refractive in- dex. These are revealed by the intensity changes of the light passing through a pinhole centered on the axis of a TEM, He-Ne probe laser and detected by a photodiode.

SF6 or SF6-Ar mixtures are irradiated by a CO2 TEA laser equipped with a low pressure discharge (4 Tot-r) smoothing tube and with an intracavity aper- ture. The resulting single mode output, tuned to the P( 20) line of the 10.6 pm band, has a power of 60 mJ per pulse and is focused by a two ZnSe lens tele- scope to an incident effective radius of 0.3 1 cm. The corresponding fluence is 0.20 J cm-’ at the entrance of the irradiation cell. The gas mixture in the laser cavity is the same as previously employed [ 1 ] and

l-l L2

I I

Phd SPh W C W

-Ad1 DS I Ll

Fig. I. Schematic experimental setup. Ll: TEA CO2 laser; L2: He-Ne laser; DS: digital scope; C: cell; Phd: photodiode; PD: photon drag; M 1 and M2: gold coated mirrors; M3 and M4: alu- minium coated mirrors; BS: beam splitter, BC: beam combiner; Ph: pinhole; W: windows; S: quartz flat; T: timer.

the time profile of the CO2 laser pulse consists of a main peak with 300 ns fwhm followed by a tail of about 4.5 us containing about 80% of the total en- ergy. The average effective radius into the cell is about 0.3 1 cm in vacuum and the autofocusing effect in presence of SFs produces only a very limited reduc- tion of this value as shown in table 1. Part of the CO2 laser output is diverted to a photon drag by a ZnSe 30% beam splitter placed before the absorption cell. The transmitted part of the COz laser beam after ab- sorption is stopped by a pinhole of 0.5 mm radius and by a quartz flat set in front of a photodiode.

A cw He-Ne laser is used as a probe. Properties of this beam are: 10 mW power, 0.44 mm waist, 0.5 x 1 0e3 rad divergence half angle 0, and 1.6 X 10’ mm confocal parameter b. Vertical polarisation was adopted in order to increase the performance of the two aluminium mirrors used in the iterative align- ment procedure. The probe beam is directed through the cell coaxially to the CO2 laser beam using an Ar- Ar coated Ge flat as a beam combiner.

The absorption cell, 5 cm long and 4.2 cm in di- ameter, is equipped with two 2.2 cm diameter NaCl windows. The aperture is sufficient to avoid Fresnel diffraction and the diameter gives echo times out of the range of our interest. The cell is placed at a 195 cm distance S from the waist of the incident probe beam which occurs at the laser output mirror.

The intensity of the He-Ne laser passing through the 0.5 mm radius pinhole was monitored by a pho- todiode (EG&G model SGD-444) before (lo) and after (I) the onset of the CO2 laser pulse. The device has an active area of 1 cm’ and is wired in the pho- toconductive mode to have a rise time of 10 ns. It is followed by well screened x 10 impedance matching amplifier. The response of this device is linear over many decades and the amplifier introduces a 7 ns de- lay with respect to the photon drag output. The out- puts of the photon drag and of the amplifier of the photodiode are fed in the two input channels of a 300 MHz, 500 MS/s digital scope (Tektronix model 2440) giving a nominal time resolution of 2 ns. The reported results are averages of 256 curves and it should be noticed that the disturbances induced by the laser triggering device vanish before the onset of the laser signal. The measurements are carried out at a repetition rate of 10 Hz which allows a thermal equilibration of the irradiated volume between pulses

M. kmzi et al. / Chemical Physics 173 (1993) 291-303 293

Table 1 F: mean peak fluence and r& mean effective radius in the cell. t: time of the maximum of the signal. An,,,/ ( P+- 1): the overall fractional increaseoftherefractionduetoSF,(f?omeq. (12)).AN/N:defocusingbyAr(fromeq. (l),scetext). (n)otheGaussianaverageof the mean number of photons absorbed per SF6 molecule per pulse. (Jo-JI)/Jo: fraction of enezgy absorbed. The incident energy Jo is 0.06 per pulse, the effective incident CO2 laser radius is r& = 0.3 1 cm and the corresponding incident IR laser flue.nce is FO= 0.20 J cm’

SF, Ar F kf Focusing Thermal lensing Absorption (mbar) (mbar) (J cm-‘) (mm)

t hJ(~o- 1) t ANIN (n>G (Jo-J,)IJo (IQ) (ms) (photons molecule- ’ )

4 0.14 3.1 5.5 0.11 10.5 0.48 6 0.12 3.0 4.2 0.28 10.9 0.70 8 0.08 3.0 3.8 0.33 9.8 0.84

10 0.07 2.9 3.6 0.26 9.1 0.91

2 400 0.13 3.0 4.3 1.25 6.04 -0.20 29.8 0.64 5 120 0.09 3.0 4.2 1.15 1.65 -0.49 15.1 0.81 5 200 0.07 2.9 4.1 1.34 2.74 -0.14 17.9 0.90 5 400 0.05 2.9 3.4 1.12 6.04 -0.28 18.9 0.95 5 600 0.02 2.9 3.4 1.24 7.14 -0.22 19.7 0.99

3. Results

Fig. 2. Thermal lensing signals at 10 Hz: cooling between pulses in a 5 mbar SF,, 600 mbar Ar mixture.

(fig. 2). The distance L between the cell and the pin- hole is 86 cm and the unperturbed 1 /e* radius of the probe laser at this distance is 1.6 mm. Both the ab- sorption and the detection steps are thus carried out in the far field of the probe beam.

The determination of the fraction of energy ab- sorbed per pulse and of the extent of the autofocusing effect in the CO? beam were performed by a digital pyroelectric energy meter. Effective l/e radii were estimated from the ratio between the total energy and that detected behind a 0.250 mm diameter pinhole as in a previous study [ 11.

Results obtained in a few representative runs on SF, and SF,-Ar mixtures are shown in figs. 3, 5 and 6. Pure SF6 (fig. 3) shows a sharp focusing signal starting very early during the CO* laser pulse and reaching the peak value at a time which is very close to the end of the pulse of the exciting laser. Excepting a small defocusing backshoot signal which occasion- ally appears after the tail of these features, no other

t (PSI Fig. 3. Focusing signal at 10 mbar SF, pressure. The baseline is obtained with the CO2 laser on but intercep@ the beam before entering the cell. The time profile of the COz laser pulse is also shown.

294 M. Lend et al. / Chemical Physics I73 (1993) 291-303

effect is detectable in these conditions. In fig. 4 the focusing signal has been measured at different times and by setting the pinhole at various distances from the axis common to the lasers. The fractional incre- ments of the He-Ne beam intensity are reported as seen through the 1 mm diameter aperture. This ap erture size was selected to obtain signal to noise ra- tios workable by our averaging procedure and it can be shown that it introduces a decrease of the peak in- tensity < 14%. Anyway it will be seen in the next sec- tion that only intensity ratios are used in the calcula- tions. To the positive increments in the near-to-axis values of the He-Ne intensity, negative variations must correspond in the periphery. The magnitude of these, for a given cross section, are much smaller than their positive counterpart and are not detectable in this system.

The response of SF6-Ar mixtures is more complex (fig. 5). In the short time range (microseconds) a focusing peak is detectable with a time history simi- lar to that of pure SFs. This is followed by a defocus-

s A I \ I , I I I I I

I I

I I

r Imm)

Fig. 4. Fractional variations of the peak He-Ne laser intensity as a function afthe distance from the beams axis at d&rent times. (0) 2.9ps, (0) 4.5~~~ (+) ll.Ops, (x) 18.0ps.InitialSF, pressure: 10 mbar.

:: 0 4 8 12 16 9) L t(e)

8

P

I a 1 , I , I I

0 20 40 60 f

t(ms)

3

Fi8.5. (a) Full lines: focusing and defocusing signal in the early stages of the experiment. Broken lines: the thermal lensing over- shoot as calculated according to ref. [ 111. (b) overall thermal lensing signal in SF,-Ar mixtures. SF, pressure: 5 mbar; Ar pres- sure(I)6OOmbar,(2)4OOmbar,(3)2OOmbar.In(a)thetime profile of the CO2 laser pulse is also shown.

ing signal which can be easily attributed to the over- shoot of the thermal lensing effect. On a longer time scale (milliseconds), after a few echoes not shown in the figure, the full evolution of the thermal signal is seen until the system reaches a new equilibrium. It appears that, while the thermal signal shows a depen- dence on the Ar and SF, pressures (see also fig. 6), the initial focusing effect seems to be more or less in- dependent of the gas pressures in the experimental conditions adopted.

Absorption measurements were also performed at the end of the CO2 laser pulse in order to comple- ment the optical data with boundary conditions on the incident Jo and transmitted J, energies, with an approximate evaluation of the vibrational energy content per molecule ( n) o, and with the mean effec-

iU. Lenzi et al. / Chemical Physics 173 (1993) 291-303 295

Fig. 6. Focusing and defocusing signals in SFs-Ar mixtures. Ar pressure: 400 mbar, SF6 pressure ( 1) 5 mbar, (2) 2 mbar. Also shown are the time profile of the CO2 laser pulse (3) and the baseline of the photodiode.

tive radii in the cell. All these values are calculated according to the procedures described in a preceding paper [ 1 ] and are listed in table 1.

The present results are discussed in the next sec- tion in separate subsections. The first deals with the negative lens signal that is easily identified with a thermal lensing effect and the second deals with the positive lens signal which is mostly attributed to the laser induced osmosis effect with minor contribu- tions due to the enhanced polarisability of the ab- sorbing medium. A quantitative evaluation of the re- sults obtained is also given in a third subsection.

4. Discussion

4.1. Thermal lensing

The defocusing signal, present in the experiments on SF&r mixtures, shows a first maximum in the short time range at about 7-8 us and another in the long time range ( z l-7 ms) . This behaviour is easily attributable to thermal lensing. This type of signal is not detectable in pure SF6 at pressures used in this study.

The thermal lensing effect has been extensively de- scribed by Bailey [ 7 ] and Barker [ 8 1. The model is character&d by four time constants: the acoustic time rP, the pseudo first order energy transfer time ‘I~, the

diffusion time rn and the time of thermal R.kWttiOIi

of the system rK. Definition of these constants and indicative values calculated at 298 K for the present system are collected in table 2. Depending on the rel- ative values of these time constants, different limit- ing cases are possible. When the diffusion and the thermal relaxation times are longer than rn and 7,

two cases can be considered: 7p e TV, i.e. the V-T energy transfer is slower than the pressure wave crossing of the irradiated volume. In this case the gas density change, responsible of the lens effect, is at all times proportional to the temperature increment. When 7p xa 7vVT the behaviour of the system is deter- mined by a pressure wave propagation and a charac- teristic overshoot appears in the signal at a time close to TV. For this particular case a very useful treatment has been given in ref. [ 111 and simple solutions are possible if the laser pulse is assumed to be instanta- neous with a Gaussian radial profile and if 7J

T~=OO. An optically thin sample is also required. These are clearly conditions different from those

encountered in the present study where in many cases the CO2 laser pulse duration t,, rvr, and 7,, are of the same order of magnitude. This notwithstanding we have tried to retrace the signal using the relation of Bailey [ 111

(~-Ldl~o

In this expression s is a dimensionless time given as t/7,; F(s) reduces, along the axis, to Dawson’s integral:

s

exp( -s2) I

exp(x2) dx, 0

and Qo/C2 is the density change when all the ab- sorbed energy is released as heat at constant pressure. This value has been calculated from the data given in table 1 according to

&_ (JO-JIM c2 - l&o cyc2 ’

where 1, is the cell length and C is the speed of the sound at the proper temperature. A is a calibration factor determined at high s values where

296 M. L.enzi et al. / Chemical Physics I73 (I 993) 291-303

Table 2 Timeconstants((rsat298K).Forreferencesee:r~[9,10];7~[11,12];7,7,[7];7.[8].~/Efractionoftheenergycontentre1eased per molecule per V-T collision, SF6-SF, 1.4x lo-’ [9], SF,-Ar 4.0~ IO-’ [lo]; C,: heat capacity at constant pressure (Cal mol-’ deg-' ), for SF, is 23.234 at 298 K [ 131; p: Pressure (N m-l); rd: mean effective radius of the CO1 laser beam in the cell; R: gas constant (caldcg-‘mal-‘):K:thermal~nductivitycodficient (Wm-‘K-r), 17.3x10-“forArand 14.2x10-3forSF~ [12];D=D,,p,,/p;D,: diffusion coeflicient, 0.341 for Ar and 0.0396 for SF, (cm2 s-t) at po= 1 atm and 293 K, C= [ (C&“)/(C,M)]1/2: velocity of sound (cm 5-I) at 300 K, 32.198X 1O’for Arand 13.648X lo3 for SF,; r,: the radius ofthe cell, 2.1 (cm); t,: the time length ofthe laser pulse

Process Constant

V-T energy transfer rvr= (ZNAE/E)-’ thermal relaxation r~=pC,,r&j4RTK diffusion 7r,=&4D pressure wave 7&&&/C echo r,=2r,fC laser emission t1

SF, (mbar) SFJAr (mbar)

4 10 S/120 S/600

26.7 10.7 2.7 0.60 2.5x lo3 6.5x lo3 14x 103 70x 103 2.4x lo3 6.0x lo3 8.7 x 10’ 42x lo3

22.7 9.6 308 130

-4.5

A= (l-Az/Z&l’2-1

2Qo/C2 .

The fractional decreases of the mixture gas density due to the thermal effect, reported in table 1, are ob- tained from the relation

&f’J_& Qo N -7=c2p-

The results of the reconstruction of the signal accord- ing to Bailey’s relation are given as broken lines in fig. 5a using the maximum signal in fig. 5b at the proper time as calibration point. The correspond- ence with the experiment is satisfactory as far as the peak time and amplitude are concerned. Deviations must probably be related to the non-instantaneous character of the laser pulse and to other characteris- tics of the system such as rW and the optical thick- ness which are far from those of the ideal system de- scribed by Bailey.

Also the thermal recovery times appear to be shorter than those calculated in table 2. Apart from these deviations the presence of a positive lens signal is clearly not a part of the thermal lensing effect and requires therefore some additional considerations. It should be noticed that previous similar studies on SF,-inert gas systems gave no evidence of an initial focusing signal [ 14,15 1.

In pure SF6 the thermal lensing is nearly non- observable in the present experimental conditions. This is a feature common to other low pressure ex-

periments [ 8 1. The energy transfer time becomes similar to that of the acoustic wave thus decreasing the overshoot profile and the increase of the thermal conductivity and of the diffusion leads to a fast dis- sipation of the gas lens.

4.2. The focusing signal

Focusing signals in experiments dealing with ther- mal lensing have been found in a number of cases. They have been reported for CH3F [ 161, CD4, SO2 [ 17 1, and CO2 [ 18 ] and attributed, under the name of kinetic cooling, to two different mechanisms. The original interpretation postulates the existence, in the levels ladder of the molecule, of a possible relatively small endothermic step. The resulting fast endother- mic T-V energy transfer leads to an equilibrium at the expense of the translation energy. In the case of CO2 it has been shown that the operating mechanism is more general. Kinetic cooling is the result of the excitation of the molecule from a level above the ground state in a given vibrational manifold to a level in a different manifold. The first manifold is thus vi- brationally cooled and endothermic V-T processes tend to restore equilibrium with the bulk. Given a fa- vourable balance between this V-T transfer rate and that of the corresponding exothermic processes of the newly excited manifold, kinetic cooling is observed. No positive lens signal has been reported [ 141 for SF6 when irradiated by a TEA CO2 laser with a pre- sumable pulse duration of a few hundred nanosec-

M. L.enzi et al. / Chemical Physics 173 (1993) 291-303 297

ond. The kinetics of the IR absorption by SF6 is very different from that occurring with the above men- tioned species. The SF6 case is a multiple-photon process in the presence of collisions. At the pressures adopted in this work the number of SF,-SF, colli- sions is of the order of lo-100 us-’ molecule-‘. It has been calculated by Oref [ 19 ] that in a 15 oscilla- tor system such as SF6 only a few collisions are suffi- cient to obtain a quasi-Boltzmann equilibrium of the vibrational energy among the absorbing molecules. This distribution corresponds to a temperature higher than that of the system before irradiation and de- scribing the rotational and translational modes dis- tribution. In this case no net V-T endothermic pro- cess is possible as already experimentally evidenced in ref. [ 141.

The effect described in this study seems to be of a different nature. It reaches its maximum at the end of the laser pulse which lasts about 4.5 us then van- ishes. The rise and fall times are of the order of a few microseconds and not very responsive to the pres- sure of the gas in contrast to what is shown for the kinetic cooling of CO*. Two different mechanisms can at this point be considered to rationalise the forma- tion of a positive gas lens. The radial profile in refrac- tivity can be related to the presence of gradients either in polarisability or in gas density or both.

The increase of the electronic polarisability due to vibrational excitation has been observed in SF6 [ 6 1. In a vibrationally excited gas optical properties can be different from those of the unexcited molecules and the difference is mainly due to the perturbation of the mean polarisability. As vibrational excitation grown in the quasi-continuum all modes are excited and be- cause of the anharmonicity of the motion the mean spacing between the nuclei increases. The change in molecular dimensions leads to an increase of the po- larisability. Measures of the variation of the refrac- tive index under pulsed IR irradiation were per- formed by an interferometric technique where the experimental conditions “guaranteed the constancy of the energy density of the exciting radiation both along and across the observation zone” [ 6 1. The changes in transmittance thus detected were only due to a homogeneous increase of the refractive index in SF6 and no gas lens effect was present. The increase of the refractive index due to vibrational excitation by the P( 20) line of the 10.6 urn band was found to

be linear with the CO* laser fluence up to 1 J cm-’ and was given as An/ (n- 1) = 10 -’ per vibrational energy content of one photon per molecule. This re- lation makes possible a quantitative evaluation of the role played by polarisability in the observed increase of refractivity once an estimation of the vibrational content of the molecules is made.

The focusing signal can be related also to the sec- ond mechanism above mentioned. It involves the formation of a number density profile resulting in a positive gas lens. This would bc in agreement with a previous observation in a kinetic study of the multi- ple-photon IR absorption [ 1,2]. Light attenuation experiments were performed for SF6 and SF6-rare gas mixtures with a pulsed TEA COZ laser. Results were compared with model calculations based on the small signal absorption cross sections available in the lit- erature [ 3 ] and obtained from measurements with a weak cw CO1 laser in monophotonic absorption con- ditions [ 20,2 11. In order to rationalise the results the assumption was made of an increase, during the laser pulse, of the density of the absorber along the axis of the cell. This effect, under the name of “laser induced osmosis”, had previously been theoretically de- scribed by de Hemptinne [ 41 and some tentative ex- perimental evidence was also presented [ 5 1. Accord- ing to this author when a gas is irradiated by a laser and there is resonance between the field and the mol- ecules, the system is described as an “open” one. The population of the resonators in the irradiated zone can change either by the well known relaxation pro- cesses overcoming the rotational hole burning or by a flow of matter thus creating a density ,gradient. Should the equilibrium be reached, the gradients of temperature, matter and polarisation amplitude are in balance.

The time resolved light attenuation technique util- ised in our previous work led to the evaluation of the average number of IR photons ( n) absorbed per SF, molecule as a function of time and of laser fluence during the laser pulse The temporal profile of the COZ laser pulse was identical with that reported in the present work but the intensity was higher due to the smaller effective radius adopted. The kinetic of IR photon absorption is quite different in neat SF6 or in its mixtures with the noble gases where a large frac- tion of the absorbed photons is transferred by V-T processes. The total number of photons absorbed per

298 h4. L.enzi ei al. /Chemical Physics 173 (1993) 291-303

molecule(n),canbewrittenas (n)=(n,>+(n~>, where (&) gives the number of photons per mole- cule stored as vibrational energy and ( nT) the num- ber of photons transferred per molecule. While in pure SF6, at pressure investigated, (n) z (&) and ( nT) =O, in the presence of the buffer gas (n,) < ( nT) x ( n) . From these measurements one derives the following quantities as a function of time and of fluence: (n), (II,), (n,>, the differential absorp- tion cross section u and ( AE), the average energy transferred per collision. The results of these deter- minations are compared with model calculations. The conclusion, for both neat SF, and for its mixtures with the noble gases, was that the comparison was favour- able provided the models included the possibility of the formation of time dependent concentration pro- files within the Gaussian laser beam which signifi- cantly perturb the absorption kinetics.

In refs. [ 1,2 ] gas density variations along the axis of the irradiated volume were given in terms of a pa- rameter q defined as

4=4142 = W”IN) (NtINo) ,

where &, N,, and NV are the number densities of SF6 respectively at the beginning of the experiment, at any time during and after the CO2 pulse, and of the mol- ecules actually resonating with the CO2 laser fteld. In the present experimental condition, after about 1 ps it can safely be assumed that all molecules are above the vibrational bottleneck of the y ladder so that qr x 1 and q= q2.

4.3. Evaluation of the focusing eflect

The positive gas lens due to one or both of the above mechanisms, that is formation of gradients in pola- risability or density, can be described in terms of the resulting refractive index distribution across the CO2 beam. The maximum value An, of this radial profile can be evaluated as follows.

The decrease of the 1 /e2 radius r2 of the TEhl, probe laser at the pinhole is seen by the photodiode as an increase in the light intensity Z through the pin- hole. Its relative value for a Gaussian beam is

Ar2/r2 = 1 - (IO/Z) ‘I2 .

The aperture is placed in the far field of the He-Ne beam at a distance L from the cell. The fractional de-

crease of the far field half-angle of divergence 8 is

A9/9=Ar2/8L.

This value can be related to the focal length f of the gas lens using the magnification m given in the stan- dard treatment for a thin lens in geometrical optics:

where S and S’ are the object and image distances, and 8 and 8’ the corresponding half-angles of diver- gence. S is the distance between the probe beam 1 /e2 waist at the front mirror of the laser and the cell. Hence the focal length is

f=SS/A6J.

The use of a more precise magnification relation, given for Gaussian beams [ 221 in cases when b2 GZ (S-f )2 does not apply, gives the same results for the present experimental values.

The focal length can be given as a function of the height An,,, of the refractive index profile in the irra- diated gas volume. In the aberrationless, paraxial ray approximation the focus of a thin symmetric spheri- cal lens, given as a function of the radius of curvature R, is

f= J!- 2(n- 1) *

In this case a radial distribution in refractivity is as- sumed which replicates the Gaussian form of the CO2 laser beam with the same rl/, effective radius. Hence the spherical curvature must be substituted by the corresponding Gaussian value. For a solid plano- Gaussian lens with a maximum thickness Al,,, the curvature is d21/dr2=2A1Jr:,e so that the focal length of a symmetric Gaussian lens is

f= rCe

4AZ,,,(n-1). (2)

It is known that a gas lens can be treated approxi- mately as a thin lens even if it is of sizable thickness [ 23 1. The refractive index of a gas with unexcited in- ternal modes depends on its density [ 23 1,

n=l+(k-lhdp,, (3)

nR is the index at some reference value m( of the gas density. For SF6, the literature [ 121 gives nR= 1.000783 at 589 nm (sodium line), 0°C and 1 atm.

M. Lenzi et al. / Chemical Physics 173 (1993) 291-303 299

This value is adopted because the wavelength of the probe laser (632.8 nm) is sufftciently close to that of the sodium line. From (3), using as a reference value the initial gas density p. in the cell:

An,l(n,-1)=&,/p,. (4)

For a symmetric Gaussian gas lens A&,, can be de- rived in terms of the maximum value An,,, of the re- fractivity distribution considering an equivalent gas lens of plano-Gaussian profile having the original re: fraction index no. The actual lens is due to the for- mation, in a cell of length lo of a radial distribution of the index having a Gaussian profile:

An(r)=An,exp[ -(r/r,n)2] .

According to (4) this is optically equivalent to a ra- dial distribution of the gas density:

~(r)=~~exp[-(r/r,,)*].

The distribution generates a gas lens and the number of molecules herein contained is

10 Qrn a&i . (5)

This lens is also equivalent to a symmetric Gaussian gas lens, at the uniform gas density po, having a Gaussian profile of thickness 1 of the type I(r) = 2A1, exp [ - (r/r&)*]. The number of molecules contributing to the lens effect is now

2p,,Al,,,n&. (6)

Equating (5 ) and (6):

AL, = &,4d2~o . (7)

Substituting (7) into (2) and making use of (4) one obtains

rL f= - 21, An,,,

for the focal length, and

_&r!& rfle no - 1 210(%- 1 If’

The final expression used to calculate the ratio An,/ (no- 1) evidenced by the focusing effect is

(8)

It should be noticed that An, indicates the value of the height of the Gaussian function at the maximum that is the “thickness” of the gas lens on the axis.’

The relative contributions of polarisability and density are estimated as follows. The vibrational con- tent (n,) of the irradiated molecules is evaluated by integration of the rate relation [ 2 ]

d(nv) - = aw- (ZNAE),, - (ZNAE),, , dt (9)

where the rate of vibrational excitation is expressed as the result of a competition between the rate of the absorption process, d(n) /dt=aW, and the rate of loss by V-T energy transfer to the surrounding mol- ecules, d ( nT) /dt. Z is the collision number (cm3 molecules-’ s-r). In the initial part of the pulse the rate of absorption prevails and (n,) increases up to a maximum value. From a certain time on, the deac- tivation processes take over and (n,) diminishes. In (9) u is the absorption cross section for the P(20) line of the CO2 laser and is taken from the literature [ 3,20,2 1 ] as discussed in refs. [ 1,2], W( t ) is the laser intensity (photons s-r cm-*) and is known from ex- periment. Al& and AEAr are reported in the note to table 2. Values of ( n,) , as a function of time, are ob- tained by numerical integration and their maximum under different conditions is reported in table 3. Once values of ( n,) are available the increment in refrac- tivity An, due to the increased polarisability of the SF6 molecules can be derived according to ref. [ 6 ] :

An,/(nO-l)=lO-*(n,). (10)

Curve 3 of fig. 7 shows the outcome of this calcula- tion for two typical runs.

The refractive index is the function of both polaris- ability and density, and the overall increment An must be given in terms of the two contributions. From (3) and (10)

n=l+(n,-l)(l+lO-*(n,))N(t)/N,

and

An m=l+(nO-l)(l+lO-*(n,))N(t)/NO-no.

‘_( 11)

Let AnNl ( no- 1) =AN/N, be the fractional incre- ment ofthe:r&activeindex due to density vari&ions only. Substitution in ( 11.) gives

300 hf. Lenzi et al. / Chemical Physics I73 (1993) 291-303

Table 3 (n,.): approximate calculated maximum value of the mean vibrational energy content of SF, molecules (944 cm-’ photons). &a/ ( PI,,- 1): overall fractional increase of refractivity (from eq. ( 12 ) ) and fractional increase of refractivity due to the increase of the polarisability evabutted from ref. [ 61. U/N,,: gas density e&t due to laser induced osmosis (LIO ). q2 = NJN, (see text )

SF6 (mbar)

Ar

(mbar) <ny>llwm W(no-1)

overall PO1

AN/NO LIO

42

4 10.5 0.11 0.11 0.00 1.00 6 9.6 0.28 0.10 0.17 1.17 8 8.2 0.33 0.08 0.23 1.23

10 7.8 0.26 0.08 0.17 1.17

2 400 8.3 1.25 0.08 1.18 2.18 5 120 6.3 1.15 0.06 1.06 2.06 5 200 5.2 1.34 0.05 1.24 2.24 5 400 4.2 1.12 0.04 1.08 2.08 5 600 2.4 1.24 0.02 1.21 2.21

0.

0.

0

E \ 4

0

r

.3 - 0 0

Fig. 7. Fractional increase of the refractive in&x(a) in pure SF, at 8 mbar and (b).in a mixture SFJAr at 5/200 mbar. ( 1) The overall fractional increase, (2) and (3): the components due to density and polarisability respectively.

Relation (8) can now be rewritten to obtain A+/

(no- 1) from the intensity measurements: with nP = no + An,. The above relation can be used to evaluate the gas density increment introducing a

M. Lenzi et al. / Chemical Physics 173 (1993) 291-303 301

proper value for the radius of the gas lens. This, to a first approximation, should be close to the effective radius of the CO* laser beam.

An estimate of the expected rising time of the fo- cusing effect can also be attempted. The action of the gradient of the laser beam electric field on the excited SF6 molecules should be felt by the gas as an addi- tional internal pressure. This leads to a squeeze of the Gaussian distribution of the molecules contributing to the optical effect. An increase of the axial value by a quantity AN corresponds to a decrease of the effec- tive radius by a quantity

Art/c= r,/,(O)--r*,,(l) ,

with rl,=( 0) the effective radius of the COZ beam in cell. During this process the number of excited mol- ecules remains constant:

4l&(O)& = &&,(W, *

This relation gives a new definition of the quantity 92:

qz(t)=r:,,(O)lr:,,(t) 7

which, coupled with a slightly modified eq. ( 13 ) ,

no-1 r:,,r2[l-(z0/z)1’2] 92(t)= ---y +

nP- 2L8SI,( np - 1) ’

provides a system to obtain corresponding values of q2 and rI ,e at various times.

The rate of decrease of r&t) with time thus ob- tained can be compared with the velocity of sound C( 7’) at the proper temperature in the same gas mix- ture. This is estimated from the relations [ 21

and

T(t)=298+(nT)hv/CV,

with x: mole fractions, C: velocity of sound, hv: 2.7x 10’ cal mol-‘, Cy molar heat capacity at con- stant volume and ( nT) = (n) - (n,) is the number of photons transferred per SF6 molecule to the bath from eq. (9).

The results of the above treatment of data are pre- sented in tables 1 and 3 and in figs. 7, 8 and 9. The peak values of the fractional increments of refractive index for the overall process are reported in tables 1 and 3. The time dependence of this quantity is shown,

10

I-

5

t(P*)

Fig. 8. The decrease of the effective radius oftbe Gaussian distri- bution of the excited SF6 molecules versus time (broken lines) and same quantity as calculated ftom the velocity of sound (Ml liaes). (a) SFs/Ar mixture at 5/200 mbar, (b) pum SF6 at 8 mbar.

for two selected runs, in fig. 7 along with the separate contributions by density and polarisability. Correc- tions of the overall values due to polarisability are smaller for mixtures. This is due to stronger focusing signals by molecules in mixtures with a vibrational energy content of the same order of magnitude as those of pure SF+

The characteristic time of the focusing effect is shown for two cases in fig. 8 and compared with sound velocity. It seems reasonable to ascribe the focusing signal to a compression of the vibrationally excited SF, molecule and it appears that the related rate has, as an upper limit, the speed of sound in the various gaseous media at the appropriate temperature. No

M. L.enzi et al. / Chemical Physics 173 (1993) 291-303 302

5

r

t (ts)

Fig. 9. q,=N(t)/& versus time. Full lines from refs. [ 1,2]: (1) SFs/Ar, 0.27/1000 mbar, 0.97 J cm-2fluence; (2) and (3) SF6, 1.33 mbar, 0.9 and 0.25 J cm-’ fluence respectively. Broken lines (this work): (1) SF,/Ar at 5/200 mbar and (2) SF, at 8 mbar.

signal is detectable during the time interval which in- cludes the sharp initial peak of the CO;! laser pulse. This induction period could be the reason why focus- ing signals have not been detected with laser pulses shorter than about 500 ns [,14,15 1.

In fig. 9 one reports the experimental q2(t) func- tions for Ar/SF, and for neat SF,. Included for com- parison are the time profiles derived in ref. [ 1 ] for neat SF6 and in ref. [ 2 ] for mixturesValues of qnux

compare satisfactorily in both cases; differences in the slopes might be attributed, in part, to the lower laser fluences utilised in the present work. Table 3 collects, for. various experimental conditions, the maximum calculated values of (n,) , the corresponding values of: &/ (no 11) attributable to polarisability changes only, and the maximum overall values of this qua,n- tityl ,The’Jast column contains the values of (12 calcu- lated, as discussed, from the signals after correction for polarisability changes.

5. Conclusions ~

It can be cancluded thatthe results reported in figs. 8 ‘and .9 indicate that a cause common to both the

focusing signals of the present work and the “anom- alies” reported in refs. [ 1,2] for the kinetics of the multiple-photon absorption could be identified with the onset of radial density profiles of excited SF6 molecules inside the region irradiated by the COP. laser and provide therefore additional evidence of the ex- istence of the effect named “laser induoed osmosis” and discussed in refs. [ 4,5 1,

As to the dynamics of generation of these density gradients, there is, to our knowledge, no indication in the literature. It should in any case involve some kind of suction of the induced dipoles towards the beam axis under the action of the radial gradients of the COz laser field. The density pro&es develop in a time of the order of microseconds under the present conditions and proceed at a maximum rate close to the sound velocity in either neat SF6 or in its mix- tures with Ar at the appropriate gas temperature.

Descriptions of chemical processes, induced by pulsed IR lasers, should therefore pay due attention to the possibility of formation of these gradients which alter both the concentration and the excitation of the species involved in the processes by a factor which is of the order of 1.2 to 2.3 in the cases here presented and which certainly depends on the experimental conditions. Further and more detailed work is nec- essary to reach a better definition of this focusing ef- fect and of its possible consequences in ‘laser induced processes. The main open question remains the defi- nition, at a microscopic level, of the dynamics of for- mation of the density gradients.

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