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Stimulated Raman scattering in a field of ultrashort light
pulses
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1982 Sov. J. Quantum Electron. 12 98
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S t i m u l a t e d R a m a n s c a t t e r i n g in a f i e l d
of u l t r a sh o r t l i g h tp u l s e sV. A. Gorbunov(Submitted
February 16, 1981)Kvantovaya Elektron. (Moscow)9,152-155(January
1982)Numerical methods are used to study the evolution of
ultrashort light pulses of arbitrary intensity in the caseof
concurrent stimulated Raman scattering in a dispersion-free medium.
It is shown that a characteristicfeature of transient stimulated
Raman scattering is a strong amplitude and phase modulation of the
pumppulse resulting from a periodic reversal of the direction of
energy transfer between the pump and Stokeswaves.This lowers the
efficiency of conversion to the Stokescomponent.PACS numbers:
42.65.Cq
1. A theory of transient stimulated Raman scatter inghas been
developed in detail for the linear case, whenth e pump field is
assumed to be constant and thechanges in the level populat ions of
a substance are ig-nored. This theory deals with transient effects
as-sociated with the inertia of the nonlinear polarizationof the
medium1 and with the difference between thegroup velocit ies of the
pump and Stokes pulses. 2 Thepossibility of self- induced t
ransparen cy in stimulatedRaman scatt erin g (STRS) has been
investigated in thenonlinear case. Steady-state scatt ering regimes
havebeen predicted theoretically for high- intensity fields
ofultrashort pulses altering greatly the level populations(STRS
solitons).3 However, further studies are neededof the possibility
of formation of such steady- statepulses in real system s.
T h e present paper gives a numerical analysis of thesolution of
the equations describing the STRS process.We wish to draw attent
ion to a charact eristic effect ofa n amplitude and phase
modulation of ultrashort pumppulses in the process of scat tering
when an allowanceis made for the reaction of the Stokes on the
pumpwave. This modulation affects pulses of moderateintensity12 as
well as high- power pulses3 if their shapediffers greatly from that
of a soliton corresponding to agiven dispersion of the medium. This
modulation isdu e to the influence on the scat ter ing pr ocess of
thedifference between the phases of the pump, Stokes, andcoherent
polarization waves, which change as a resultof changes in the
amplitudes of these waves. Similarbehavior of the phase difference
between the waves isknown to play an important ro le in resonant
four-waveparamet ric interactions in steady- state theory
ofSTRS.4"6 The modulation appears most clearly in thecase discussed
below when the scattering is concurrentan d there is no group delay
of the pump and Stokeswaves. This situation is normal in the case
of STRSin gases. The results of numerical calculations forth e
countercu rren t STRS (see Fig. 3 in Ref. 7) can alsobe regarded as
a manifestation of such modulation.
2. We shall assume that the scatt erin g occurs onon e pair of
levels of scattering molecules and we shallconsider the one-
dimensional problem. The evolutionof the pump and Stokes pulses is
described by a systemof equations consisting of the reduced Maxwell
equa-tions for the pulse envelopes and the equations of mo-tion for
the slow amplitude of a nondiagonal element
of the density mat rix and for the population difference . In
the resonance approximation when - , = 21+ ( , are the frequencies
of the pump and Stokesfields, respectively; 21 is the frequency of
the in-vestigated transit ion; is the detuning from resonance)
,this system is of the form (see, for example, Ref. 8)
das
cuts s sin ( );
= 5 sin ( ) ;
~ p - - j - s = Con paj sin ( ) ;
- + ' / = Coapc, s sin ( - ).
(1)(2)(3)(4)(5)(6)
T he equations are given in the real form.
Here,at,,M>i)=Attt{ ,t)/AP(/ni are the dimensionless
realamplitudes of the pump and Stokes waves; = - $ isth e
difference between their phases; A
pmaiis the max-
imum amplitude of the pump pulse at the entry to theinvestigated
medium; t,=z/lt is the dimensionlesslength; 1~ = 2 ( 3/ ,) ; = 3/ (
, ); nps are therefractive indices at the frequencies ,\ is the
den-sity of the scattering molecules; is the matrix e le-ment of
the scatter ing process; s and are the realamplitude and phase [a=s
( )]; 0 is the equi-librium value of the population difference ; =
(t - z/c)/T2 is the dimensionless time; 0 = 2 ; = 1/ 2;T l and 2
are the longitudinal and tr ansverse relaxationtimes of the medium;
C o= ^ n^ V* As mentionedear lier , we shall consider the
concurrent scatteringan d ignore the group delay of the waves.
T h e essence of the effect can be explained by the fol-lowing
qualitative considerat ions. Let us assume thatth e pump and Stokes
pulses at the entry (at = 0) arefree of phase modulation [(3 0/ 9)
{.o = O] and that theirduration is ,
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th e pulse power but the one- dimensional approximationc o n d i
t i o n s must be satisfied over distances of a fewlengths L t.Th e
growth of regula r modulation of the pump pulsemay be disturbed by
the transfer of energy to the an ti-Stokes and higher Stokes
components which are ignored
in our analysis. Special calculations allowing for thepresence
of the pump and Stokes components and oneo t h e r component (for
example, anti- Stokes) showedt h a t parametric energy transfer to
higher componentsis slight if = / 1 l,whereltm=2ir/Ak is the
lengthof the linear phase matching of the waves; A = 2kp - k t- k s
is the difference between the wave vectors. Wecan thus see that the
above analysis is valid if lt/ lpm * 1(in the case of pulses of
different initial area whenL l/ lpm^ 1). In the example of hydrogen
we have elO"2;th e value of can be increased by employing a
mixtureof hydrogen with a buffer gas, as has been done inRef. 6.
Then, the value of le is governed by the partialpr essur e of the
Raman - active hydrogen and lm isgoverned by the total pressure in
the mixture.
Th e auth or is grateful to G. N. Vinokurov and S. B.P a p e r n
y i for valuable discussions.
*S . A. Akhmanov, . . D rabovich, A. P. Sukhorukov, andA. S. Ch
i rk i n , Zh. Eksp. T e o r . Fiz. 59, 485 (1970) [Sov.Phys . J E
T P 3 2 , 266 (1971)].2S . A. Akhmanov, . . D rabovich, A. P.
Sukhorukov, andA. K. Shchednova, Zh. Ek s p . Te o r . Fiz. 62, 525
(1972)[Sov. Phys . J E T P 3 5 , 279(1972)] .3 T . M. M akhviladze
and . . Sary chev , Zh. Ek s p . Te o r .F i z . 71, 896 (1976)
[Sov. Phys . J E T P 44, 471 (1976)].4V. S. Butylkin, G. V. Venkin,
V. P. P ro t a s o v , P. S. F i s h e r ,Yu. G. Khronopulo , and
M. F . Shalyaev, Zh. Ek s p . Te o r .Fiz . 70, 829 (1976) [Sov.
Phys . J E T P 4 3 ,4 3 0 (1976)].5V. S. Butylkin, G. V. Venkin, L.
L. Kulyuk, D. I. M aleev ,Yu. G. Khronopulo , and M. F . Shalyaev,
Kvantovaya Elek-t r o n . (M oscow) 4, 1537 (1977) [Sov. J. Quantum
E l e c t r o n . 7,867 (1977)].6V. A. G orbunov, K. Sh. Must aev ,
S. B. Papernyi , and V. A.Sereb ry akov , P i s ' m a Zh. Te k h .
F iz. 5, 1244 (1979) [Sov.Tech. Phys . Le t t . 5, 522 (1979)].7S .
F. Mo ro z o v , L. V. Piskunova, . . Sushchik, and G. 1.F r e i d
m a n , Kvantovaya Ele k t ro n . (Moscow) 5, 1005 (1978)[Sov. J .
Quantum E l e c t r o n . 8, 576 (1978)].8 I . A. Poluektov, Yu. M.
Popov, and V. S. Roi tberg , Kvanto -vaya Ele k t ro n . (M oscow)
4, 651 (1977) [Sov. J. QuantumEle c t ro n . 7, 362 (1977)].
Tra n s la t e d by A. Tybulewicz
N e e d to a l l ow for slow n o n l i n e a r i t y in m e a s
u r e m e n t s of n.R. A. Garaev, D. V. Vlasov, and V. V.
KorobkinP. N. Lebedev Physics Institute, Academy of Sciences of the
USSR , Moscow( S u b m i t t e d Ma rch 17, 1981Kvantovaya
Elektron. (Moscow) 9, 155- 157 (Jan uary 1982)T he componen ts of
the no nlinear susceptibility tensor ^ S , were determ ined by a
method eliminating theco n t r i b u t i o n of the slow part of
the nonlinearity. A comparison with an interferometric method made
itpossible to demo nstra te a considerable contribut ion of the
slow part of the non linearity in the measurementsof j under
conditions of har d focusing of radiation in a sample. This
comparison also showed t h a t allowancefor the slow part of the
nonlinearity is essential in calorimetric measurements of 2 in the
case of pulsed u r a t i o n s typical of - switching
conditions.PACS num bers: 42.55.Bi
The exact knowledge of the nonlinear refractive indexw2 of an
active medium is essential in the developmentof high- power laser s
for scientific r esearch and pract i-cal applica tions. Measurem
ent s of this refract ive in-dex have been made on many occasions,
1"3 but to thebest of our knowledge the problem of separate deter-m
i n a t i o n of the fast and slow parts of n2 has not
beendiscussed in the lit erat ur e. It should be pointed outt h a t
the fast (electronic and electron - nuc lear) and slow( t h e r m a
l and strict ion) pa rt s may generally have dif-ferent signs and
their contribution to n2 depends on thea c t u a l case. The time
to establish the slow part of then o n l i n e a r i t y depends on
the structure and dimensionsof a beam.
Th e characteristic time for the establishment of theslow part ,
= l/ v is governed by the characteristic
size / of the tr an sverse struc tu re of the beam and byth e
velocity of sound v. We shall consider the specificcase of the time
needed for the establishment of theslow pa rt when radiation is
focused in a sample in aspot of ~100 size. If = 5 5 cm/ sec, we
obtain , 20 nsec. This simple estima te shows that even inth e case
of relatively shor t laser pulses the influenceof the slow
nonlinearity may be considerable.
We shall consider not only the results of determina-t ion of the
component ^ by eliminating the contribu-t ion of the slow part ,
butfor the sake of comparisonwe shall show oscillogram s obtained
in the course ofth e determination of xJJ^ by an interferometric
method,which demonstrate clearly the importance of the con-t r i b
u t i o n of slow nonlinearity.
The values of an d were measured using the10 0 Sov. J. Quantum
Electron. 12(1), Jan. 1982 0049-1748/82/010100-03$04.10 1982
American Institute of Physics 10 0
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