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Nonlinear refractive index of CS2 in small scale filaments
Volume
3, number
NONLINEAR
5
OPTICS COMMUNICATIONS
REFRACTIVE
Laboratorio
INDEX
OF
CS2
IN
July 1971
SMALL
R. CUBEDDU and F. ZARAGA di Fisica de1 Plasma ed Elettronica Quantistica Istituto di Fisica de1 Politecnico, Milano, Italy Received
SCALE
FILAMENTS*
de1 C.N.R.,
3 May 1971
From measurements on spectral broadening of light filaments in CS3, under picosecond excitation, we calculate the corresponding time behaviour of the nonlinear refractive index. Our results seem to be in rough agreement with the moving focus picture.
A large number of experimental and theoretical works have been devoted in the last few years to the phenomenon of small scale filaments of light in liquids ]1,2]. While for nanosecond excitation the moving focus picture seems to be satisfactory [3-61, for picosecond excitation neither the self trapping [7,8] nor the moving focus [6,9] picture have so far been able to account for all the experimental data [lo]. For the latter case, therefore, further experimental data which may shed some light on the phenomena occurring in the filaments seem to be appropriate. In this letter we present a calculation of the time behaviour of the nonlinear refractive index 6n in CS2, as deduced from the experimental spectra of filaments under picosecond excitation. Our calculations rely only on the method of stationary phase [ll] and on the use of the parabolic equations [12]; we do not need any particular model for trapping. The idea of measuring 6n is as follows. Let us write the electric field of the light pulse in the filament as E(t,z ,Y) = {$A exp [ i(w t -kz - ks)] + c.c.} where cylindrical coordinates with the z axis along the propagation axis have been assumed. In the usual parabolic approximation the equation of the phase of the field is [12]: 2 a24 18A 2 as as as 26n 1 --+2ai+ 72 ( > =-;+rA v at ( -#+rz > ,
(1)
where u is group velocity in the medium and n is the refractive index. In a previous paper [lo P we have shown that the experimental data are well fitted by the following assumptions:
* Work supported
310
by Consiglio
Nazionale
delle Ricerche.
A(z,r.t)
A(r) =A,
=A(t-z/v)A(r)
exp [-(r/ro)2]
;
.
Then eq. (1) becomes:
with the new coordinate r = t-z/v. It was shown in ref. [ll] that, at a given z (e.g., the exit face of the trapping cell), the phase ks can be deduced from the spectral broadening of the filament. Therefore, if a simultaneous measurement of the spectral broadening of a given filament at two slightly different z positions is made, the quantity a.s/az can be deduced. From eq. (3) one then gets
6n(t, Y). To measure 8s as, the unfocused beam of a mode-locked ruby laser [13] giving pulses 5 to 10 psec wide was sent into a trapping cell in which two mylar beam-splitters (= 5n thick) were inserted at N 10 cm from the entrance face of the cell. The two beam splitters were separated by = 6 mm and both placed at an angle of 45O to the laser beam. The two light beams reflected from the mylar sheets were imaged on the entrance slit of the spectrograph. A typical spectrum obtained in this way is shown in fig. 1. In every picture we were able, generally, to correlate the filaments reflected by the two beam splitters, as is also apparent from fig. 1. A numerical example of the result for a typical filament is shown in fig. 2. The curve gives the time behaviour of 6n at Y = 0 (i.e.on the axis of the filament). The radial behaviour of 6n at a fixed r is obviously parabolic. as one can immediately see from eq. (2). With reference to fig. 2,
Volume
3, number
5
July 1971
OPTICS COMMUNICATIONS
2.4
2.2
21) c(
Fig. 1. Typical spectra showing the difference in broadening for the same filament at two points along the propagation direction, which are separated by a distance of N 6 mm.
a few comments are appropriate here: (i) although fig. 2 has been obtained for a particular filament, other measurements on different filaments have given a similar behaviour for bn (the difference of the various curves being = 40%); (ii) the peak value of 6n is in agreement with that calculated elsewhere [ll, 14,151; (iii) the time behaviour of 6n has been calculated up to * 2 psec from its maximum (solid part of the curve of fig. 2), which is the limit of our measuring method. Outside of this region, however, we do know that the time behaviour must proceed with a slope smaller than that at points A and B of fig. 2. Just as a representative example, a possible extrapolation of the curve is also indicated in fig. 2 (dashed part of the curve). Therefore, at a given ?‘(e.g. ‘Y = 0) the nonlinear refractive index &z does not drop to zero outside the region in which the field is phase modulated. In conclusion we can say that our results, and in particular the one discussed under point (iii) above, seem to be only explainable with the idea of a pulse propagating into an already existing waveguide, which is produced by the moving focus [6]. The later analytical development of this idea by Shen and Loy [9] does not seem to be consistent, however, with our results or with those of a previous paper [lo]. The detailed mechanism by which the waveguide is formed and, in particular, the value of the waveguide diameter still remain to be explained.
A
B/
A
‘\
‘.
. .
r, I
-4
I
-3
I
-2
I
-1
I
0
I
1
I
2
I
I
3
.!
t(psec) Fi .2. Temporal behaviour of the nonlinear refractive in 8 ex an( t, 0) , as derived in the text from experimental data (solid curve). The dashed curve is a representative extrapolation outside the calculated region.
REFERENCES [l] R. Y. Chiao, E. Garmire and C. H. Townes, [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13]
The authors would like to thank Professor 0. Svelto for many helpful discussions and for reading the manuscript.
/‘f /’
[14] [15]
Phya. Rev. Letters 13 (1964) 479. R. G. Brewer, J. R. Lifshitz, E. Garmire, R. Y. Chiao and C. H. Townes, Phys. Rev. 166 (1968) 326. A. L. Dyshko, V. N. Lugovoi and A.M. Prokhorov, JETP Letters 6 (1967) 146. V. N. Lugovoi and A.M. Prokhorov, JETP Letters ‘7 (1968) 117. M. M. T. Loy and Y. R. Shen, Phys. Rev. Letters 22 (1969) 994. M. M. T. Loy and Y. R. Shen, Phys. Rev. Letters 25 (1970) 1333. R. G. Brewer and C. H. Lee, Phys. Rev. Letters 21 (1968) 267. R. Polloni, C. A. Sacchi and 0. Svelto, Phys. Rev. Letters 23 (1969) 690. Y. R. Shen and M. M. T. Loy, to be published. R. Cubeddu, R. Polloni, C. A. Sacchi, 0. Svelto and F. Zaraga, Opt. Commun. 3 (1971) 9. R. Cubeddu, R. Polloni, C. A. Sacchi and 0. Svelto, Phys. Rev. A2 (1970) 1955. S. A. Akhmanov, A. P. Sukhorukov and R. V. Khoklov, JETP 24 (1967) 198. R. Cubeddu, R. Polloni, C. A. Sacchi and 0. Svelto, IEEE J. Quantum Electron. BE-5 (1969) 470. M. M. Denariez-Roberge and J.-P.E. Taran, Appl. Phys. Letters 14 (1969) 205. F. Shimizu and B. P. Stoicheff, IEEE J. Quantum Electron. QE-5 (1969) 544.
311
3, number
NONLINEAR
5
OPTICS COMMUNICATIONS
REFRACTIVE
Laboratorio
INDEX
OF
CS2
IN
July 1971
SMALL
R. CUBEDDU and F. ZARAGA di Fisica de1 Plasma ed Elettronica Quantistica Istituto di Fisica de1 Politecnico, Milano, Italy Received
SCALE
FILAMENTS*
de1 C.N.R.,
3 May 1971
From measurements on spectral broadening of light filaments in CS3, under picosecond excitation, we calculate the corresponding time behaviour of the nonlinear refractive index. Our results seem to be in rough agreement with the moving focus picture.
A large number of experimental and theoretical works have been devoted in the last few years to the phenomenon of small scale filaments of light in liquids ]1,2]. While for nanosecond excitation the moving focus picture seems to be satisfactory [3-61, for picosecond excitation neither the self trapping [7,8] nor the moving focus [6,9] picture have so far been able to account for all the experimental data [lo]. For the latter case, therefore, further experimental data which may shed some light on the phenomena occurring in the filaments seem to be appropriate. In this letter we present a calculation of the time behaviour of the nonlinear refractive index 6n in CS2, as deduced from the experimental spectra of filaments under picosecond excitation. Our calculations rely only on the method of stationary phase [ll] and on the use of the parabolic equations [12]; we do not need any particular model for trapping. The idea of measuring 6n is as follows. Let us write the electric field of the light pulse in the filament as E(t,z ,Y) = {$A exp [ i(w t -kz - ks)] + c.c.} where cylindrical coordinates with the z axis along the propagation axis have been assumed. In the usual parabolic approximation the equation of the phase of the field is [12]: 2 a24 18A 2 as as as 26n 1 --+2ai+ 72 ( > =-;+rA v at ( -#+rz > ,
(1)
where u is group velocity in the medium and n is the refractive index. In a previous paper [lo P we have shown that the experimental data are well fitted by the following assumptions:
* Work supported
310
by Consiglio
Nazionale
delle Ricerche.
A(z,r.t)
A(r) =A,
=A(t-z/v)A(r)
exp [-(r/ro)2]
;
.
Then eq. (1) becomes:
with the new coordinate r = t-z/v. It was shown in ref. [ll] that, at a given z (e.g., the exit face of the trapping cell), the phase ks can be deduced from the spectral broadening of the filament. Therefore, if a simultaneous measurement of the spectral broadening of a given filament at two slightly different z positions is made, the quantity a.s/az can be deduced. From eq. (3) one then gets
6n(t, Y). To measure 8s as, the unfocused beam of a mode-locked ruby laser [13] giving pulses 5 to 10 psec wide was sent into a trapping cell in which two mylar beam-splitters (= 5n thick) were inserted at N 10 cm from the entrance face of the cell. The two beam splitters were separated by = 6 mm and both placed at an angle of 45O to the laser beam. The two light beams reflected from the mylar sheets were imaged on the entrance slit of the spectrograph. A typical spectrum obtained in this way is shown in fig. 1. In every picture we were able, generally, to correlate the filaments reflected by the two beam splitters, as is also apparent from fig. 1. A numerical example of the result for a typical filament is shown in fig. 2. The curve gives the time behaviour of 6n at Y = 0 (i.e.on the axis of the filament). The radial behaviour of 6n at a fixed r is obviously parabolic. as one can immediately see from eq. (2). With reference to fig. 2,
Volume
3, number
5
July 1971
OPTICS COMMUNICATIONS
2.4
2.2
21) c(
Fig. 1. Typical spectra showing the difference in broadening for the same filament at two points along the propagation direction, which are separated by a distance of N 6 mm.
a few comments are appropriate here: (i) although fig. 2 has been obtained for a particular filament, other measurements on different filaments have given a similar behaviour for bn (the difference of the various curves being = 40%); (ii) the peak value of 6n is in agreement with that calculated elsewhere [ll, 14,151; (iii) the time behaviour of 6n has been calculated up to * 2 psec from its maximum (solid part of the curve of fig. 2), which is the limit of our measuring method. Outside of this region, however, we do know that the time behaviour must proceed with a slope smaller than that at points A and B of fig. 2. Just as a representative example, a possible extrapolation of the curve is also indicated in fig. 2 (dashed part of the curve). Therefore, at a given ?‘(e.g. ‘Y = 0) the nonlinear refractive index &z does not drop to zero outside the region in which the field is phase modulated. In conclusion we can say that our results, and in particular the one discussed under point (iii) above, seem to be only explainable with the idea of a pulse propagating into an already existing waveguide, which is produced by the moving focus [6]. The later analytical development of this idea by Shen and Loy [9] does not seem to be consistent, however, with our results or with those of a previous paper [lo]. The detailed mechanism by which the waveguide is formed and, in particular, the value of the waveguide diameter still remain to be explained.
A
B/
A
‘\
‘.
. .
r, I
-4
I
-3
I
-2
I
-1
I
0
I
1
I
2
I
I
3
.!
t(psec) Fi .2. Temporal behaviour of the nonlinear refractive in 8 ex an( t, 0) , as derived in the text from experimental data (solid curve). The dashed curve is a representative extrapolation outside the calculated region.
REFERENCES [l] R. Y. Chiao, E. Garmire and C. H. Townes, [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13]
The authors would like to thank Professor 0. Svelto for many helpful discussions and for reading the manuscript.
/‘f /’
[14] [15]
Phya. Rev. Letters 13 (1964) 479. R. G. Brewer, J. R. Lifshitz, E. Garmire, R. Y. Chiao and C. H. Townes, Phys. Rev. 166 (1968) 326. A. L. Dyshko, V. N. Lugovoi and A.M. Prokhorov, JETP Letters 6 (1967) 146. V. N. Lugovoi and A.M. Prokhorov, JETP Letters ‘7 (1968) 117. M. M. T. Loy and Y. R. Shen, Phys. Rev. Letters 22 (1969) 994. M. M. T. Loy and Y. R. Shen, Phys. Rev. Letters 25 (1970) 1333. R. G. Brewer and C. H. Lee, Phys. Rev. Letters 21 (1968) 267. R. Polloni, C. A. Sacchi and 0. Svelto, Phys. Rev. Letters 23 (1969) 690. Y. R. Shen and M. M. T. Loy, to be published. R. Cubeddu, R. Polloni, C. A. Sacchi, 0. Svelto and F. Zaraga, Opt. Commun. 3 (1971) 9. R. Cubeddu, R. Polloni, C. A. Sacchi and 0. Svelto, Phys. Rev. A2 (1970) 1955. S. A. Akhmanov, A. P. Sukhorukov and R. V. Khoklov, JETP 24 (1967) 198. R. Cubeddu, R. Polloni, C. A. Sacchi and 0. Svelto, IEEE J. Quantum Electron. BE-5 (1969) 470. M. M. Denariez-Roberge and J.-P.E. Taran, Appl. Phys. Letters 14 (1969) 205. F. Shimizu and B. P. Stoicheff, IEEE J. Quantum Electron. QE-5 (1969) 544.
311