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Threshold of stimulated Raman scattering in liquids and self-focusing of laser beams
Volume
21,
number
THRESHOLD
5
OF
.,..
..
PHYSICS
LETTERS
STIMULATED SELF-FOCUSING
RAMAN OF
15 June 1966
SCATTERING LASER BEAMS
IN
LIQUIDS
AND
M. MAIER and W. KAISER Physics-Department
der
Technischen
Received
Hochschule
7 May
power for beam trapping: The factor n2 is connected with the Kerr constant by the relation n2 = 2hB/3, no is the linear index of refraction, c the velocity of light in vacuum, X the wave length of the laser light, a is the beam radius, andfis a characteristic number of the individual laser beam [ 51. Eq. (1) can be rewritten in the form P = /3/B(2) where p = 3c/2h(l.22h/16 +noa2/4fl)2. P represents the threshold for self-focusing. In the theory of stimulated Raman scattering the threshold value is determined by the balance of the gain factor g and the losses of the system
P cr PC,
is the critical
= (1.22N2c/256n2.
German>
1966
The threshold power of stimulated Raman scattering in pure tional to the reciprocal Kerr constant. This fact is expected liquids,
It has been noticed for some time that for many liquids the gain for stimulated Raman scattering deviates strongly from the values calculated from the spontaneous Raman data [l]. Recently a substantial lowering of the Raman threshold was observed when certain liquids were diluted bv a second component [2,3]. This result was contrary to the expected increase in threshold values due to the decrease of molar concentration. It was suggested by a number of authors [2-51 that the selffocusing of a laser beam is of major importance for the onset of stimulated Raman scattering. We wish to present data of Raman threshold measurements in pure and mixed liquids which can be interpreted quantitatively in terms of the self-focusing properties of laser beams. The self-focusing effect [6,7] is essentially determined by the Kerr constant of the substances investigated here. At a definite threshold power the laser beam is focused within a certain length, called the self-focusing length I, which was calculated by Kelley [7] and modified by Wang [5].
Miinchen,
and mixed liquids was found to be proporfrom the self-focusing properties of the
CONCENTRATION
OF Cs, (ML%)
E E 0
20
COtKENTRATK)N
LO
60 Ct= TOLCEK
IO)
m IKlL%I
$ km
esE2
1
I b)
Fig. la. Ramnn threshold power as a function of concentration. Curve I represents the measured and curve Ia the calculated threshold power for the CS2 vibration (x) in mixtures of CS2-n-heptnne. Curve II and Iln show the measured and calculated threshold power of the toluene vibration (e) in toluene-nitrobenzene mixtures. Fig. lb. Raman threshold power as a function of the reciprocal Kerr constant B. CS2 (*). vibrational frequency 65.5 cm-l: nitrobenzene (0). 1345 cm-l: bromobenzene (a). 1000 cm-l: chlorobenzene (A). 1002 cm-l: m-xylene (A). 2916 cm-l: toluene (0). 1004 cm-l: benzene (a). 992 cm-l. CS2 (,x) in mixtures of CS2-n-heptnne: toluene (a) and nitrobenzene (0) in mixtures of toluenenitrobenzene.
The gain factor depends on the classical Raman data, the spontaneous Raman cross section and the line width. Raman threshold measurements were made with a giant pulse ruby laser, the properties of which were described previously [2]. In all measurements reported here, liquid cells of 20 cm length were used. The Raman threshold was ob[8].
529
Volume
21,
number
Raman threshold the Kerr constants 12) (all values
5
PHYSICS
Table 1 power measured and calculated and from spontaneous Raman normalized to carbon disulfide).
experimental -
from data
Threshold power calculated from Kerr spontaneous constants Raman data
cs2
1.0
1 .o
I.0
nitrobenzene
1.3
1.2
6.1
toluene
4.9
5.0
9 .3
benzene
5.2
3 .7
3.6
tained by analysing the rapid rise of stimulated Stokes emission over several orders of magnitude. The incident laser power was attenuated by glass plates and dielectric mirrors. Fig. lb shows the Raman thresholds, pth, of seven pure liquids as a function of the reciprocal Kerr constant B. There is a good proportionality between the measured values of Pth and the magnitude of l/B. (The experimental error for Pth is approximately 10 per cent; the values of the Kerr constants * were taken from ref. 9. ) This result is expected from eq. (2) when the occurence of self-focusing leads simultaneously to the appearance of stimulated Raman action. The data in table 1 demonstrate that the threshold for stimulated Raman action is not determined by the spontaneous Raman values. The measured threshold values of stimulated Raman scattering (normalized to CS2 = 1 .O) are compared with threshold data calculated from the spontaneous Raman effect [ 121. The large disagreement between the calculated and measured threshold values is quite obvious. We have extended the threshold measurements to mixed liquids (see fig. 1). In the system CS2-n-heptane the threshold of the CS2 vibration is found to increase monotonically with the heptane concentration (curve I of fig. la). Curve Ia represents the threshold values as a result of the dilution of CS2 by n-heptane calculated from spontaneous Raman data [ll]. There is substantial disagreement between the caiculated and measured data. Similarly curve II and IIa are the measured and calculated threshold values of toluene in the system toluene-nitrobenzene. * There
are variations of the values of the Kerr constant in the literature [lo] of the order of 20 percent. These numbers are still in reasonahle approximity of the straight line in fig. 1. There are, however. Kerr constants of benzene in the literature which cannot be reconsiled with our results in fig. 1.
530
LETTERS
15 June
1966
The curves differ in a striking manner. From spontaneous Raman data an increase in threshold is expected while actually a lowering of the Raman threshold is observed. It should be noted that the threshold values for the toluene vibration go through a minimum and increase again for a 40 per cent nitrobenzene and 60 per cent toluene mixture (see below). In order to explain the experimental observations of fig. la the Pth Values are replotted as a function of the reciprocal Kerr constant of the individual mixture ** (see fig. lb). There is again good proportionality between the Raman threshold and l/B. It should be emphasized that there is a common slope for all the pure and mixed liquids. From the threshold measurements reported here, we wish to conclude that - for the liquids investigated - the threshold of stimulated Raman scattering is essentially determined by the Kerr constant. When the threshold for self-focusing is exceeded, laser light is trapped in filaments of high intensity. If the Raman gain in the filaments is sufficiently high then the self-focusing threshold and the Raman threshold are reached at the same input intensity. If the Raman gain within the filaments is insufficient then laser power exceeding the threshold value for self-focusing is required. Such a case seems to occur at the mixture of 40 per cent nitrobenzene and 60 per cent toluene. The strong dilution requires a high threshold intensity for the toluene vibration. The threshold value of self-focusing is lower than the measured threshold of the stimulated Raman action (see fig. lb). 1. D. Weiner.
‘2 . 3.
4.
5
6. 7. 8. 9. 10. 11. 12.
S.E.Schwarz and F..J.McCiung, J.Appl. Phys. 36 (1965) 2395; G. Bret and G. Mayer in Proc. Intern. Conf. on Physics of quantum electronics. Puerto Rico. 1965. bl.Maier, Physics letters 20 (1966) 388. P. Lallemand and N. Bloemhergen, Phys. Rev. Let ters I.5 (196.5) 1010. G. Hauchecornc and G. Mayer ,Compt. Rend.261 (l!)(i3) 4014: Y, R . Shen and Y. J. Shaham, Phys. Rev. Ixttcrs 1.j (1965) 1008. C. C. Wang, Phys.Rev. I.ettcrs 16 (1966) 344. G.A. Askar’yan, Soviet Phys.JETP 1.5 (1962) 1088; R. Y. Chiao. E. Garmire and C. H. Townes 1 Phys. Rev. Letters 13 (1964) 479. P.L.Kelley, Phys.Rev.Letters 15 (1966) 1005. R.W.Hellwarth, Phys.Rrv.130 (1963) 18.50. Y.R.Shen, Physics Letters 20 (1966) 378, Landolt-Bornstein Vol.11, part 8. J.A.Koningstein. thesis. IJnivcrsity of Amsterdam. E.Garmire, thesis, Massachusetts Institute of Technology (1965) unpuhlishetl.
** Kerr constants of the mixed system CS2-n-hcptnnc wcrc taken from ref. 10. The Kerr constants of the tolucne-nitrobenzene mixtures were obtained by linear interpolation of the two pure components.
21,
number
THRESHOLD
5
OF
.,..
..
PHYSICS
LETTERS
STIMULATED SELF-FOCUSING
RAMAN OF
15 June 1966
SCATTERING LASER BEAMS
IN
LIQUIDS
AND
M. MAIER and W. KAISER Physics-Department
der
Technischen
Received
Hochschule
7 May
power for beam trapping: The factor n2 is connected with the Kerr constant by the relation n2 = 2hB/3, no is the linear index of refraction, c the velocity of light in vacuum, X the wave length of the laser light, a is the beam radius, andfis a characteristic number of the individual laser beam [ 51. Eq. (1) can be rewritten in the form P = /3/B(2) where p = 3c/2h(l.22h/16 +noa2/4fl)2. P represents the threshold for self-focusing. In the theory of stimulated Raman scattering the threshold value is determined by the balance of the gain factor g and the losses of the system
P cr PC,
is the critical
= (1.22N2c/256n2.
German>
1966
The threshold power of stimulated Raman scattering in pure tional to the reciprocal Kerr constant. This fact is expected liquids,
It has been noticed for some time that for many liquids the gain for stimulated Raman scattering deviates strongly from the values calculated from the spontaneous Raman data [l]. Recently a substantial lowering of the Raman threshold was observed when certain liquids were diluted bv a second component [2,3]. This result was contrary to the expected increase in threshold values due to the decrease of molar concentration. It was suggested by a number of authors [2-51 that the selffocusing of a laser beam is of major importance for the onset of stimulated Raman scattering. We wish to present data of Raman threshold measurements in pure and mixed liquids which can be interpreted quantitatively in terms of the self-focusing properties of laser beams. The self-focusing effect [6,7] is essentially determined by the Kerr constant of the substances investigated here. At a definite threshold power the laser beam is focused within a certain length, called the self-focusing length I, which was calculated by Kelley [7] and modified by Wang [5].
Miinchen,
and mixed liquids was found to be proporfrom the self-focusing properties of the
CONCENTRATION
OF Cs, (ML%)
E E 0
20
COtKENTRATK)N
LO
60 Ct= TOLCEK
IO)
m IKlL%I
$ km
esE2
1
I b)
Fig. la. Ramnn threshold power as a function of concentration. Curve I represents the measured and curve Ia the calculated threshold power for the CS2 vibration (x) in mixtures of CS2-n-heptnne. Curve II and Iln show the measured and calculated threshold power of the toluene vibration (e) in toluene-nitrobenzene mixtures. Fig. lb. Raman threshold power as a function of the reciprocal Kerr constant B. CS2 (*). vibrational frequency 65.5 cm-l: nitrobenzene (0). 1345 cm-l: bromobenzene (a). 1000 cm-l: chlorobenzene (A). 1002 cm-l: m-xylene (A). 2916 cm-l: toluene (0). 1004 cm-l: benzene (a). 992 cm-l. CS2 (,x) in mixtures of CS2-n-heptnne: toluene (a) and nitrobenzene (0) in mixtures of toluenenitrobenzene.
The gain factor depends on the classical Raman data, the spontaneous Raman cross section and the line width. Raman threshold measurements were made with a giant pulse ruby laser, the properties of which were described previously [2]. In all measurements reported here, liquid cells of 20 cm length were used. The Raman threshold was ob[8].
529
Volume
21,
number
Raman threshold the Kerr constants 12) (all values
5
PHYSICS
Table 1 power measured and calculated and from spontaneous Raman normalized to carbon disulfide).
experimental -
from data
Threshold power calculated from Kerr spontaneous constants Raman data
cs2
1.0
1 .o
I.0
nitrobenzene
1.3
1.2
6.1
toluene
4.9
5.0
9 .3
benzene
5.2
3 .7
3.6
tained by analysing the rapid rise of stimulated Stokes emission over several orders of magnitude. The incident laser power was attenuated by glass plates and dielectric mirrors. Fig. lb shows the Raman thresholds, pth, of seven pure liquids as a function of the reciprocal Kerr constant B. There is a good proportionality between the measured values of Pth and the magnitude of l/B. (The experimental error for Pth is approximately 10 per cent; the values of the Kerr constants * were taken from ref. 9. ) This result is expected from eq. (2) when the occurence of self-focusing leads simultaneously to the appearance of stimulated Raman action. The data in table 1 demonstrate that the threshold for stimulated Raman action is not determined by the spontaneous Raman values. The measured threshold values of stimulated Raman scattering (normalized to CS2 = 1 .O) are compared with threshold data calculated from the spontaneous Raman effect [ 121. The large disagreement between the calculated and measured threshold values is quite obvious. We have extended the threshold measurements to mixed liquids (see fig. 1). In the system CS2-n-heptane the threshold of the CS2 vibration is found to increase monotonically with the heptane concentration (curve I of fig. la). Curve Ia represents the threshold values as a result of the dilution of CS2 by n-heptane calculated from spontaneous Raman data [ll]. There is substantial disagreement between the caiculated and measured data. Similarly curve II and IIa are the measured and calculated threshold values of toluene in the system toluene-nitrobenzene. * There
are variations of the values of the Kerr constant in the literature [lo] of the order of 20 percent. These numbers are still in reasonahle approximity of the straight line in fig. 1. There are, however. Kerr constants of benzene in the literature which cannot be reconsiled with our results in fig. 1.
530
LETTERS
15 June
1966
The curves differ in a striking manner. From spontaneous Raman data an increase in threshold is expected while actually a lowering of the Raman threshold is observed. It should be noted that the threshold values for the toluene vibration go through a minimum and increase again for a 40 per cent nitrobenzene and 60 per cent toluene mixture (see below). In order to explain the experimental observations of fig. la the Pth Values are replotted as a function of the reciprocal Kerr constant of the individual mixture ** (see fig. lb). There is again good proportionality between the Raman threshold and l/B. It should be emphasized that there is a common slope for all the pure and mixed liquids. From the threshold measurements reported here, we wish to conclude that - for the liquids investigated - the threshold of stimulated Raman scattering is essentially determined by the Kerr constant. When the threshold for self-focusing is exceeded, laser light is trapped in filaments of high intensity. If the Raman gain in the filaments is sufficiently high then the self-focusing threshold and the Raman threshold are reached at the same input intensity. If the Raman gain within the filaments is insufficient then laser power exceeding the threshold value for self-focusing is required. Such a case seems to occur at the mixture of 40 per cent nitrobenzene and 60 per cent toluene. The strong dilution requires a high threshold intensity for the toluene vibration. The threshold value of self-focusing is lower than the measured threshold of the stimulated Raman action (see fig. lb). 1. D. Weiner.
‘2 . 3.
4.
5
6. 7. 8. 9. 10. 11. 12.
S.E.Schwarz and F..J.McCiung, J.Appl. Phys. 36 (1965) 2395; G. Bret and G. Mayer in Proc. Intern. Conf. on Physics of quantum electronics. Puerto Rico. 1965. bl.Maier, Physics letters 20 (1966) 388. P. Lallemand and N. Bloemhergen, Phys. Rev. Let ters I.5 (196.5) 1010. G. Hauchecornc and G. Mayer ,Compt. Rend.261 (l!)(i3) 4014: Y, R . Shen and Y. J. Shaham, Phys. Rev. Ixttcrs 1.j (1965) 1008. C. C. Wang, Phys.Rev. I.ettcrs 16 (1966) 344. G.A. Askar’yan, Soviet Phys.JETP 1.5 (1962) 1088; R. Y. Chiao. E. Garmire and C. H. Townes 1 Phys. Rev. Letters 13 (1964) 479. P.L.Kelley, Phys.Rev.Letters 15 (1966) 1005. R.W.Hellwarth, Phys.Rrv.130 (1963) 18.50. Y.R.Shen, Physics Letters 20 (1966) 378, Landolt-Bornstein Vol.11, part 8. J.A.Koningstein. thesis. IJnivcrsity of Amsterdam. E.Garmire, thesis, Massachusetts Institute of Technology (1965) unpuhlishetl.
** Kerr constants of the mixed system CS2-n-hcptnnc wcrc taken from ref. 10. The Kerr constants of the tolucne-nitrobenzene mixtures were obtained by linear interpolation of the two pure components.