Frequency-angular spectra of the stimulated Raman scattering parametric components under self-action
It’s theoretically proved that in self-focusing mode of exciting radiation, selfphase modulation, and the relaxation oscillations of the nonlinear polarization amplitude, the frequency-angular structure of the parametric components of the stimulated Raman scattering has special features. Maximum...
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Ivanisik, А.І. Isaenko, O.Iu. Korotkov, P.A. 2017-05-31T05:24:52Z 2017-05-31T05:24:52Z 2012 Frequency-angular spectra of the stimulated Raman scattering parametric components under self-action / А.І. Ivanisik, O.Iu. Isaenko and P.A. Korotkov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 4. — С. 370-375. — Бібліогр.: 9 назв. — англ. 1560-8034 PACS 42.65.Dr https://nasplib.isofts.kiev.ua/handle/123456789/118729 It’s theoretically proved that in self-focusing mode of exciting radiation, selfphase modulation, and the relaxation oscillations of the nonlinear polarization amplitude, the frequency-angular structure of the parametric components of the stimulated Raman scattering has special features. Maximum of radiation energy shifts relative to Raman frequencies, and this shift depends on the scattering angle. The largest Stokes shift (about -10 cm⁻¹ ) of the maximum energy in the spectrum of parametric components corresponds to axial radiation and is determined by the size of the focal area, time of polarization existence as well as mismatch of the polarization wave vectors and parametric component. The obtained estimations of the self-phase modulation influence on radiation in the case of relaxation oscillations show that this influence is insignificant. The effect has been interpreted in terms of short-term phase matching. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Frequency-angular spectra of the stimulated Raman scattering parametric components under self-action Article published earlier |
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Frequency-angular spectra of the stimulated Raman scattering parametric components under self-action |
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Frequency-angular spectra of the stimulated Raman scattering parametric components under self-action Ivanisik, А.І. Isaenko, O.Iu. Korotkov, P.A. |
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Frequency-angular spectra of the stimulated Raman scattering parametric components under self-action |
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Frequency-angular spectra of the stimulated Raman scattering parametric components under self-action |
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Frequency-angular spectra of the stimulated Raman scattering parametric components under self-action |
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Frequency-angular spectra of the stimulated Raman scattering parametric components under self-action |
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frequency-angular spectra of the stimulated raman scattering parametric components under self-action |
| author |
Ivanisik, А.І. Isaenko, O.Iu. Korotkov, P.A. |
| author_facet |
Ivanisik, А.І. Isaenko, O.Iu. Korotkov, P.A. |
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2012 |
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English |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Article |
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It’s theoretically proved that in self-focusing mode of exciting radiation, selfphase
modulation, and the relaxation oscillations of the nonlinear polarization amplitude,
the frequency-angular structure of the parametric components of the stimulated Raman
scattering has special features. Maximum of radiation energy shifts relative to Raman
frequencies, and this shift depends on the scattering angle. The largest Stokes shift (about
-10 cm⁻¹
) of the maximum energy in the spectrum of parametric components
corresponds to axial radiation and is determined by the size of the focal area, time of
polarization existence as well as mismatch of the polarization wave vectors and
parametric component. The obtained estimations of the self-phase modulation influence
on radiation in the case of relaxation oscillations show that this influence is insignificant.
The effect has been interpreted in terms of short-term phase matching.
|
| issn |
1560-8034 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118729 |
| citation_txt |
Frequency-angular spectra of the stimulated Raman scattering parametric components under self-action / А.І. Ivanisik, O.Iu. Isaenko and P.A. Korotkov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 4. — С. 370-375. — Бібліогр.: 9 назв. — англ. |
| work_keys_str_mv |
AT ivanisikaí frequencyangularspectraofthestimulatedramanscatteringparametriccomponentsunderselfaction AT isaenkooiu frequencyangularspectraofthestimulatedramanscatteringparametriccomponentsunderselfaction AT korotkovpa frequencyangularspectraofthestimulatedramanscatteringparametriccomponentsunderselfaction |
| first_indexed |
2025-11-25T10:04:19Z |
| last_indexed |
2025-11-25T10:04:19Z |
| _version_ |
1850509637518884864 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 4. P. 370-375.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
370
PACS 42.65.Dr
Frequency-angular spectra of the stimulated Raman scattering
parametric components under self-action
А.І. Ivanisik1, O.Iu. Isaenko and P.A. Korotkov
Taras Shevchenko Kyiv National University,
4g, Academician Glushkov Avenue, 03127 Kyiv, Ukraine
1Phone: +38 (044)-526-0570; e-mail: aivan@univ.kiev.ua
Abstract. It’s theoretically proved that in self-focusing mode of exciting radiation, self-
phase modulation, and the relaxation oscillations of the nonlinear polarization amplitude,
the frequency-angular structure of the parametric components of the stimulated Raman
scattering has special features. Maximum of radiation energy shifts relative to Raman
frequencies, and this shift depends on the scattering angle. The largest Stokes shift (about
10 cm1) of the maximum energy in the spectrum of parametric components
corresponds to axial radiation and is determined by the size of the focal area, time of
polarization existence as well as mismatch of the polarization wave vectors and
parametric component. The obtained estimations of the self-phase modulation influence
on radiation in the case of relaxation oscillations show that this influence is insignificant.
The effect has been interpreted in terms of short-term phase matching.
Keywords: stimulated Raman scattering, self-focusing, self-phase modulation, relaxation
oscillations.
Manuscript received 27.08.12; revised version received 25.09.12; accepted for
publication 17.10.12; published online 12.12.12.
1. Introduction
Stimulated Raman scattering (SRS) is widely used to
control the shape of optical pulses and their spectra, the
statistics of laser radiation and its spatial distribution, to
create devices conjugating the wave front [1], and to
amplify weak signals in optical fibers [2]. On the other
hand, the parameters of the scattered radiation,
especially spectrum, spatial-temporal structure, and
coherence are also an important source of information
about the nature of the scattering process itself and
properties of the scattering medium.
However, many features of SRS remain unclear. In
particular, in self-focusing liquids, even in the
nanosecond duration range of excitation laser pulse, the
spectral lines broadening of SRS parametric components
significantly exceeds the line width of spontaneous
Raman scattering. The proposed list of mechanisms,
classification of such broadenings [3-6] is not conclusive
and needs further details and additions. In particular, this
paper examines the action of self-phase modulation and
relaxation oscillations of the nonlinear polarization
amplitude under self-focusing conditions.
2. Theoretical consideration
In our studies, we assumed that the experimental
threshold of SRS in self-focusing liquids is mainly
determined by the threshold of exciting radiation self-
focusing. If self-focusing threshold is exceeded 2-3
times, then, in the case of light pulse duration close to
10 ns and cell length close to 10 cm, the focal area
appears first at the output side of the medium. The
growth of instantaneous power at the front of the laser
pulse is accompanied by the movement of the focal area
in the direction of input side of the medium, i.e. towards
the laser source. When the power reaches its maximum
value, the focus stops at the minimum distance from the
input border (“turning point”), and at the drop of the
laser pulse the focus moves in the opposite direction, to
the output side [7]. The pass from the input border of the
medium to the center of focal area is considered as the
“input side” of the medium, while pass from the center
of focal area to the output border is taken into account as
the “output side”.
The amplitude of induced nonlinear polarization,
which is the source of parametric components, reaches
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 4. P. 370-375.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
371
the maximum value at the focal area [8]. Due to the focal
area movement the maximum polarization amplitude
area moves as well. As shown in [6], this itself causes
the frequency shift (a few 1cm ) of parametric
components against the Raman frequencies. The shift
occurs in the direction of the frequency, which satisfies
the condition of amplitude-phase matching: the
condition of synchronous movement of focal area and
the area of polarization phase synchronism and the
generated wave.
However, most of the time at the top of the laser
pulse power, when the focal area is near the turning
point, it has a velocity close to zero. In this case, the
frequency shift under the mentioned mechanism [6] is
insignificant as compared to the frequency shift under
other mechanisms. In our opinion, the shift can be
caused by the mechanism associated with the relaxation
oscillations of the nonlinear polarization amplitude.
Under relaxation oscillations, the SRS mode in the
focal area is developed not only in exciting laser beam
direction but also in the opposite direction. If the focal
area is moving slowly or is stationary (as at the turning
point), then, in addition to the Stokes wave of the SRS in
the opposite direction, the Stokes wave of the stimulated
Mandelstam-Brillouin scattering can be generated,
which requires a longer developing time than SRS. The
radiation scattered in the opposite direction meets the
exciting laser beam and amplifies itself, contributing to
the effective exhaustion of exciting radiation. As a result
of exciting radiation exhaustion, the self-focusing
disappears, and its disappearance stops the scattering.
Over the time, during which the earlier scattered
radiation leaves the medium, the threshold of self-
focusing is reached again and the process repeats.
Oscillation period T of the Stokes component
intensity variance is determined by the time needed for
the light to pass the distance from the input side of the
medium to the focal area and backward [9]:
cnzT f2 , where fz is the self-focusing distance; c
– speed of light in vacuum; n – refractive index of light
in the medium. For zf = 10 cm and n = 1.5, the
oscillation period is T 1 ns.
The amplitude of nonlinear polarization induced in
the focal area, which is the source of SRS parametric
components, oscillates simultaneously with the Stokes
components intensity at the period of T . In each
oscillation the increase of the polarization amplitude
occurs during the SRS development time g , which is
1 ps [7]. The decline in the amplitude of polarization
occurs during the time d of light passing through the
focal area, but is not less than the duration of the
longitudinal relaxation of Raman-active transition, for
the focal area length b = 0.1 cm and n = 1.5,
d cbn / = 5 ps. Thus, we can expect that the time
of polarization existence in each of the oscillations lies
within = 1…5 ps by the order of magnitude for typical
Raman-active self-focusing liquids, and it is much
smaller than the oscillations period T . This allows
limiting the review to the calculation of frequency-
angular structure of the parametric SRS components
spectra for one sub-pulse of exciting self-focused
radiation. Oscillation periodicity leads only to more sub-
structures in the spectra in scale of T/1 /1 , which is
difficult to observe experimentally, as
)2/(1 cT 1cm0.01 .
3. Calculation methodology
To simplify calculations, it’s assumed that all waves are
polarized identically and nonlinear polarization NLP ,
which is the source of some parametric component in the
focal area, can be represented as:
),(exp)()(, 0 rr tizzFrFtFPtP pzt
NLNL
f ,
(1)
where NLP0 is the maximum polarization amplitude in
the focal area; )(tFt is the function that describes the
temporal change in the amplitude due to polarization
relaxation oscillations; )( rF is the lateral distribution
function of the polarization amplitude, which value is
determined by the distance r from the axis z;
)( fzzFz is the longitudinal distribution function of
the polarization amplitude, which argument is the
distance )( fzz from the focal area center; ),( rtp is
the phase of the polarization; t is time, and r is the
radius vector.
There is a connection between each Fourier
component of the polarization
tdtitPP NLNL exp,ω r , (2)
and generated field
tdtitEE exp,r (3)
in the far field area. This connection in the case of small
scattering angles can be represented as [6]:
V
NL rdiPikR
Rc
E 3
2
2
exp,exp, krrR , (4)
where R is the vector that connects any point within the
area V of polarization and an observation point;
kk Rcn // R – wave vector of parametric
component field with length k , unidirectional with R .
The formula for frequency-angular energy density can
be obtained as:
2
3
32
4
ωθ exp,
8
V
NL titdtPrd
c
n
W krr , (5)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 4. P. 370-375.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
372
which after substitution of ),( rtPNL , specification of the
parametric process and the functions )(tFt , )( rF ,
)( fzzFz in (1) allows to calculate the frequency-
angular spectrum of the selected parametric component
of SRS under self-focusing mode of exciting radiation
and the relaxation oscillation of the nonlinear
polarization amplitude.
4. Calculation results
To obtain the expression of energy density W , it’s
necessary to determine the phase of nonlinear
polarization, i.e. find the phase delay of radiation that
includes delays at the input and output sides of the
medium. For simplicity, let us consider only axial
radiation, allowing to assume that x = 0, y = 0, resulting
in 1)( rF .
The wave vector length k at the focal area
increases up to some value Γ. The maximum of this
value Γ = Γf is gained at z = 0, where the center of the
focal area is located (zf = 0), and t = 0, when light
intensity in the focal area is maximal. In this case Γ can
be determined as:
22 /exp/exp, tbzzt f . (6)
A fragment of laser pulse, which is located in a
point z at the moment of time t, was located at some
point (on the pass from input border to the point z) at
gztt / , where g is the group velocity of
laser radiation. According to the specified considerations
phase of laser changes to:
dztbzt
z
L
gfL
2/
22 /)/)((exp/exp, ,
(7)
where L is the length of the medium with the center at
z = 0. The medium border in our model is located at a
considerable distance from the focal area, i.e. L/2 >> b,
which is equivalent to replacing the lower limit of
integral (7) with . Since we are interested in the
maximum amplitude of polarization, the phase L
should be determined for coordinate and time z = 0,
t = 0:
.)/(1/
2
)/(/exp0,0
2
0
22
gf
gfL
bb
db
(8)
The same initial conditions allow to calculate
analytically the following derivatives of the phase:
.
/)(1
1
0,0
,
/)(1
1
0,0
2
2
f
g
f
L
g
gf
L
bz
bt
(9)
Fig. 1 shows the dependences of the phase and its
derivatives in time and coordinate under the conditions
z = 0 and )2/();2/( gg LLt . For better
clearness, the values on the ordinates axis here and
below are given in arbitrary units with normalized scale.
Now, let us consider the phase delay at the output
side of the medium. In this case, the distribution of
increase of the wave vector length remains
unchanged. However, a wave front that came out of the
point z at time t will get to the point (on the pass from
z to the output border) at
phph
z
t
z
tt ,
where υph is the phase velocity of scattered radiation.
Expressions for t and t are identical and differ
only by the values of speed, thus:
d
zt
bzt
L
z
f
2/ 2
ph2
ω
/)(
exp/exp, .
(10)
As L/2 >> b, the upper limit of integral (10) can be
replaced with . Let us calculate ω for conditions
specified earlier z = 0, t = 0:
.
)/(1
1
2
)/(/exp0,0
2
ph
0
2
ph
2
ω
b
b
db
f
f
(11)
-1·10-11-2·10-11 1·10-11 2·10-11
t, s
a
b
c
a.
u.
1.0
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
0
0
-0.8
Fig. 1. Phase delay of the laser radiation at the input side of the
medium (a) and its derivatives on time (b) and coordinate (c) in
arbitrary units.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 4. P. 370-375.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
373
Similarly, we calculate the derivatives on time and
coordinate at the output side of the medium:
.
/)(1
1
0,0
,
/)(1
1
0,0
2
ph
ω
2
ph
ph
ω
ff
f
bz
bt
(12)
The phase and its derivatives, obtained using (10),
are shown in Fig. 2 under the same conditions as for the
input (laser) radiation.
The total phase delay and its derivatives could be
found as a sum of specified values at the input and
output sides of the medium:
2
ph
2
1111
2
0,0
bb
b
g
f ,
(13)
22
ph
ph
11
0,0
bb
t
g
g
ff , (14)
22
ph 1
1
1
1
0,0
bb
z
g
ff . (15)
Resulting dependences are shown in Fig. 3 under
the earlier accepted values z and t .
Taking into account the above considerations, the
analytical calculation of polarization phase can be
simplified by using the results (14) and (15) for
derivatives and linear interpolation:
-1·10-11-2·10-11 1·10-11 2·10-11
t, s
a
b
c
a.
u.
1.0
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
0
0
-0.8
Fig. 2. The phase delay of the scattered radiation at the output
side of the medium (a) and its derivatives on time (b) and
coordinate (c) in arbitrary units.
-1·10-11-2·10-11 1·10-11 2·10-11
t, s
a
b
c
a.
u.
1.0
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
0
0
-0.8
Fig. 3. The total phase delay of the scattered radiation (a) and
its derivatives on time (b) and coordinate (c) in arbitrary units.
ω0 Lpp zkt =
,
/)(1
1
/)(1
1
/)(1
1
/)(1
1
2
ph
2
2
ph
ph20
z
bb
k
t
bb
f
g
fp
f
g
gf
(16)
where p is the phase of polarization, kp – initial wave
vector length of polarization without self-phase
modulation and 0 – initial frequency.
The expression (16) allows estimating the influence
of self-phase modulation on the resulting wave vector
length and frequency.
In case of toluene, the group velocity of ruby laser
light is υg = 0.65c, refractive index nL = 1.49, i.e. υph =
c/nL = 0.67c, relaxation time, as stated earlier, is τ = 5 ps
and, finally, length of the focal area b = 0.1 cm. Under
these parameters the following values are obtained:
6
2
ph
ph
2
1052
/)(1/)(1
c
bbg
g ,
01.0
/)(1
1
/)(1
1
2
ph
2
bbg
,
meaning that in the first approximation the self-phase
modulation of polarization can be neglected, i.e. (16) can
be written as zkt pp 0 .
Functions )(tFt , )( rF , )( fzzFz in (1) can be
specified as Gaussian functions with according
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 4. P. 370-375.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
374
halfwidths , fa , b at the level 1e ; also 0fz (the
condition of stationary focal area).
Using (1), (5) and taking into consideration
zkt pp 0 , we find that the frequency-angular
density of radiation energy for parametric components is
described by the expression:
W =
3
42
8c
n 2422
0 baP f
NL
2222 2
||
25.0exp bkak f , (17)
where ||k = pkk cos and k = sink are
wave vector difference projections onto axis z and axis
perpendicular to it, i.e. the longitudinal and lateral
mismatch of wave vector k of the parametric
components radiation and wave vector pk of
polarization; is the scattering angle determined by the
angle between vector k and axis z, along which the
polarization wave vector is oriented.
The most optimal conditions to generate radiation
are obtained, when the sum
222222
|| bkakF f
in (6) gains its minimum value. The search for these
conditions can be done through finding an extremum
0ddF . In the approximation of linear dispersion
and small , the energy density maximum W is
shifted relative to the Raman frequency 0 as
2222
22
002
0
2
00
11
211
bba
bakk
k
gf
f
g
, (18)
θ,
a
ng
. m
in
.
-5-10 5 10
140
70
0
-70
-140
0
/ 2 c , cm–1
Fig. 4. Frequency-angular spectrum of the first anti-Stokes
component in toluene.
1.0
-5 -2.5/ 2 c , cm–1
a
b
c
W , a.u.
0.8
0.6
0.4
0.2
Fig. 5. Comparison of axial radiation energy density, when
self-phase modulation is considered (a) and neglected: (b) for
b = 0.1 cm, (c) for b = 0.3 cm.
where 0k = pkk )( 00 is the difference of wave
vector length of radiation ( 0k ) at Raman frequency 0
and wave vector length of polarization ( pk ); 0 =
00 /2 kk is the scattering angle, for which
)( 0|| k =0.
Additional simplification under conditions
fa << b , 0k << 0k provides:
2
2
0
0
1
1
b
k
g
g
. (19)
Using (17) we can calculate a normalized spectrum
of the first anti-Stokes component, when pk = 2 Lk sk ,
0 = sL 2 , where s and sk are the Stokes
Raman frequency and wave vector length, L and Lk
are laser radiation frequency and wave vector length,
respectively. In toluene with SRS excitation by ruby
laser, the following values are used: frequency of
exciting radiation cL 2/ = 1cm14403 ; frequency of
Raman-active vibration = 1cm1004 ; focal area
length b = 0.1 cm; focal area radius fa =5 μm; existence
duration of nonlinear polarization = 2.5 ps.
The resulting frequency-angular spectrum structure
is shown in Fig. 4.
Finally, let us estimate in more detail, how self-
phase modulation affects the spectrum characteristics,
for example, of the axial radiation. The energy density
that accounts self-phase modulation can be calculated
using (5) and (16), which was done numerically. After
that, the calculated spectrum that accounts self-phase
modulation and the one obtained from (17) are compared
under varying values of b, as one of the focal area
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 4. P. 370-375.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
375
characteristics. Comparison of the normalized spectra
for axial radiation is shown in Fig. 5. As can be seen, the
larger is b, the closer match results obtained between
spectrum that accounts self-phase modulation and that
neglecting it.
5. Conclusions
The obtained results indicate that self-phase modulation
in the self-focusing and relaxation oscillations mode has
a weak effect on the radiation spectra of SRS.
Nevertheless, parametric components have the
frequency-angular structure, shaped as parabolic
branches. The shift of the parametric component peak
energy in spectrum, obtained in media with normal
dispersion at transparency area, is determined by the size
of the focal area, its existence duration, and mismatch
0k between polarization and wave vectors field of the
corresponding parametric component at the frequency of
exact Raman resonance.
The important point is that the generation of
radiation at shifted frequencies occurs under short-term
phase matching, and it is a non-stationary process. The
principal difference between amplitude-phase matching
[6] and short-term phase matching is that in the first case
the area of the maximum polarization amplitude moves,
and matching of the polarization phases and the field,
which is generated at a shifted frequency, remains
constant in the case of steady motion. In the case of
short-term phase, matching the area of the polarization
amplitude is stationary, and therefore matching of phases
is rapidly disturbed.
Influence of the short-term phase matching
conditions ( 0k , 0 ) can cause a frequency
shift of radiation in other non-stationary processes, not
related to SRS. In particular, this relates to the frequency
doubling of picosecond focused light pulses in samples,
which length is much shorter than that of the pulse, in
powdery nonlinear optical media, or on the surface of
absorbing media by “slide reflection” scheme, etc.
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multiwave pumped fiber Raman amplifiers //
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3. C.A. Sacchi, C.H. Townes, J.R. Lifshitz, Anti-
Stokes generation in trapped filaments of light //
Phys. Rev. 174(2), p. 439-448 (1968).
4. N.E. Kornienko, V.I. Maliy, G.V. Ponezha,
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scattering in liquids // Izvestiia Akademii Nauk
UkrSSR, Ser. A, 4, p. 65-68 (1983), in Russian.
5. A.I. Ivanisik, V.I. Maliy, G.V. Ponezha,
Cherenkov-type radiation under conditions of
Raman light scattering in self-focusing liquids //
Optics and Spectroscopy, 82(3), p. 410-416 (1997).
6. A.I. Ivanisik, V.I. Maliy, G.V. Ponezha, Spectrum
of anti-Stokes stimulated Raman scattering from
the moving focal regions of self-focusing // Optics
and Spectroscopy, 90(4), p. 625-629 (2001).
7. Y.R. Shen, The Principles of Nonlinear Optics.
New York, Wiley, 1984.
8. A.I. Ivanisik, V.I. Maliy, G.V. Ponezha, Effect of
self-focusing on the angular spectra of stimulated
Raman scattering // Optics and Spectroscopy 85(1),
p. 78-84 (1998).
9. R.V. Johnson, J.H. Marburger, Relaxation
oscillation in stimulated Raman and Brillouin
scattering // Phys. Rev. A, 4(3), p. 1175-1182 (1971).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 4. P. 370-375.
PACS 42.65.Dr
Frequency-angular spectra of the stimulated Raman scattering parametric components under self-action
А.І. Ivanisik1, O.Iu. Isaenko and P.A. Korotkov
Taras Shevchenko Kyiv National University,
4g, Academician Glushkov Avenue, 03127 Kyiv, Ukraine
1Phone: +38 (044)-526-0570; e-mail: aivan@univ.kiev.ua
Abstract. It’s theoretically proved that in self-focusing mode of exciting radiation, self-phase modulation, and the relaxation oscillations of the nonlinear polarization amplitude, the frequency-angular structure of the parametric components of the stimulated Raman scattering has special features. Maximum of radiation energy shifts relative to Raman frequencies, and this shift depends on the scattering angle. The largest Stokes shift (about ( 10 cm(1) of the maximum energy in the spectrum of parametric components corresponds to axial radiation and is determined by the size of the focal area, time of polarization existence as well as mismatch of the polarization wave vectors and parametric component. The obtained estimations of the self-phase modulation influence on radiation in the case of relaxation oscillations show that this influence is insignificant. The effect has been interpreted in terms of short-term phase matching.
Keywords: stimulated Raman scattering, self-focusing, self-phase modulation, relaxation oscillations.
Manuscript received 27.08.12; revised version received 25.09.12; accepted for publication 17.10.12; published online 12.12.12.
1. Introduction
Stimulated Raman scattering (SRS) is widely used to control the shape of optical pulses and their spectra, the statistics of laser radiation and its spatial distribution, to create devices conjugating the wave front [1], and to amplify weak signals in optical fibers [2]. On the other hand, the parameters of the scattered radiation, especially spectrum, spatial-temporal structure, and coherence are also an important source of information about the nature of the scattering process itself and properties of the scattering medium.
However, many features of SRS remain unclear. In particular, in self-focusing liquids, even in the nanosecond duration range of excitation laser pulse, the spectral lines broadening of SRS parametric components significantly exceeds the line width of spontaneous Raman scattering. The proposed list of mechanisms, classification of such broadenings [3-6] is not conclusive and needs further details and additions. In particular, this paper examines the action of self-phase modulation and relaxation oscillations of the nonlinear polarization amplitude under self-focusing conditions.
2. Theoretical consideration
In our studies, we assumed that the experimental threshold of SRS in self-focusing liquids is mainly determined by the threshold of exciting radiation self-focusing. If self-focusing threshold is exceeded 2-3 times, then, in the case of light pulse duration close to 10 ns and cell length close to 10 cm, the focal area appears first at the output side of the medium. The growth of instantaneous power at the front of the laser pulse is accompanied by the movement of the focal area in the direction of input side of the medium, i.e. towards the laser source. When the power reaches its maximum value, the focus stops at the minimum distance from the input border (“turning point”), and at the drop of the laser pulse the focus moves in the opposite direction, to the output side [7]. The pass from the input border of the medium to the center of focal area is considered as the “input side” of the medium, while pass from the center of focal area to the output border is taken into account as the “output side”.
The amplitude of induced nonlinear polarization, which is the source of parametric components, reaches the maximum value at the focal area [8]. Due to the focal area movement the maximum polarization amplitude area moves as well. As shown in [6], this itself causes the frequency shift (a few
1
cm
-
) of parametric components against the Raman frequencies. The shift occurs in the direction of the frequency, which satisfies the condition of amplitude-phase matching: the condition of synchronous movement of focal area and the area of polarization phase synchronism and the generated wave.
However, most of the time at the top of the laser pulse power, when the focal area is near the turning point, it has a velocity close to zero. In this case, the frequency shift under the mentioned mechanism [6] is insignificant as compared to the frequency shift under other mechanisms. In our opinion, the shift can be caused by the mechanism associated with the relaxation oscillations of the nonlinear polarization amplitude.
Under relaxation oscillations, the SRS mode in the focal area is developed not only in exciting laser beam direction but also in the opposite direction. If the focal area is moving slowly or is stationary (as at the turning point), then, in addition to the Stokes wave of the SRS in the opposite direction, the Stokes wave of the stimulated Mandelstam-Brillouin scattering can be generated, which requires a longer developing time than SRS. The radiation scattered in the opposite direction meets the exciting laser beam and amplifies itself, contributing to the effective exhaustion of exciting radiation. As a result of exciting radiation exhaustion, the self-focusing disappears, and its disappearance stops the scattering. Over the time, during which the earlier scattered radiation leaves the medium, the threshold of self-focusing is reached again and the process repeats.
Oscillation period
T
of the Stokes component intensity variance is determined by the time needed for the light to pass the distance from the input side of the medium to the focal area and backward [9]:
c
n
z
T
f
2
»
, where
f
z
is the self-focusing distance;
c
– speed of light in vacuum; n – refractive index of light in the medium. For zf = 10 cm and
n
= 1.5, the oscillation period is
T
( 1 ns.
The amplitude of nonlinear polarization induced in the focal area, which is the source of SRS parametric components, oscillates simultaneously with the Stokes components intensity at the period of
T
. In each oscillation the increase of the polarization amplitude occurs during the SRS development time
g
t
, which is (1 ps [7]. The decline in the amplitude of polarization occurs during the time
d
t
of light passing through the focal area, but is not less than the duration of the longitudinal relaxation of Raman-active transition, for the focal area length
b
= 0.1 cm and
n
= 1.5,
d
t
(
c
bn
/
= 5 ps. Thus, we can expect that the time
t
of polarization existence in each of the oscillations lies within
t
= 1…5 ps by the order of magnitude for typical Raman-active self-focusing liquids, and it is much smaller than the oscillations period
T
. This allows limiting the review to the calculation of frequency-angular structure of the parametric SRS components spectra for one sub-pulse of exciting self-focused radiation. Oscillation periodicity leads only to more sub-structures in the spectra in scale of
T
/
1
(
t
/
1
, which is difficult to observe experimentally, as
)
2
/(
1
cT
p
(
1
cm
0.01
-
.
3. Calculation methodology
To simplify calculations, it’s assumed that all waves are polarized identically and nonlinear polarization
NL
P
, which is the source of some parametric component in the focal area, can be represented as:
(
)
(
)
[
]
)
,
(
exp
)
(
)
(
,
0
r
r
t
i
z
z
F
r
F
t
F
P
t
P
p
z
t
NL
NL
f
j
-
-
=
^
^
,
(1)
where
NL
P
0
is the maximum polarization amplitude in the focal area;
)
(
t
F
t
is the function that describes the temporal change in the amplitude due to polarization relaxation oscillations;
)
(
^
^
r
F
is the lateral distribution function of the polarization amplitude, which value is determined by the distance
^
r
from the axis z;
)
(
f
z
z
F
z
-
is the longitudinal distribution function of the polarization amplitude, which argument is the distance
)
(
f
z
z
-
from the focal area center;
)
,
(
r
t
p
j
is the phase of the polarization;
t
is time, and
r
is the radius vector.
There is a connection between each Fourier component of the polarization
(
)
(
)
ò
¥
¥
-
w
=
t
d
t
i
t
P
P
NL
NL
exp
,
ω
r
,
(2)
and generated field
(
)
(
)
ò
¥
¥
-
w
w
=
t
d
t
i
t
E
E
exp
,
r
(3)
in the far field area. This connection in the case of small scattering angles
q
can be represented as [6]:
(
)
[
]
(
)
[
]
ò
w
w
=
w
w
w
V
NL
r
d
i
P
ikR
R
c
E
3
2
2
exp
,
exp
,
kr
r
R
,
(4)
where
R
is the vector that connects any point within the area
V
of polarization and an observation point;
(
)
=
w
=
k
k
(
)
R
c
n
/
/
R
w
– wave vector of parametric component field with length
k
, unidirectional with
R
. The formula for frequency-angular energy density can be obtained as:
(
)
(
)
[
]
2
3
3
2
4
ωθ
exp
,
8
ò
ò
¥
+
¥
-
-
w
p
w
=
V
NL
t
i
t
dtP
r
d
c
n
W
kr
r
,
(5)
which after substitution of
)
,
(
r
t
P
NL
, specification of the parametric process and the functions
)
(
t
F
t
,
)
(
^
^
r
F
,
)
(
f
z
z
F
z
-
in (1) allows to calculate the frequency-angular spectrum of the selected parametric component of SRS under self-focusing mode of exciting radiation and the relaxation oscillation of the nonlinear polarization amplitude.
4. Calculation results
To obtain the expression of energy density
wq
W
, it’s necessary to determine the phase of nonlinear polarization, i.e. find the phase delay of radiation that includes delays at the input and output sides of the medium. For simplicity, let us consider only axial radiation, allowing to assume that x = 0, y = 0, resulting in
1
)
(
=
^
^
r
F
.
The wave vector length
k
at the focal area increases up to some value Γ. The maximum of this value Γ = Γf is gained at z = 0, where the center of the focal area is located (zf = 0), and t = 0, when light intensity in the focal area is maximal. In this case Γ can be determined as:
(
)
(
)
[
]
(
)
[
]
2
2
/
exp
/
exp
,
t
-
-
G
=
G
t
b
z
z
t
f
.
(6)
A fragment of laser pulse, which is located in a point z at the moment of time t, was located at some point
x
(on the pass from input border to the point z) at
(
)
g
z
t
t
u
x
-
-
=
¢
/
, where
g
u
is the group velocity of laser radiation. According to the specified considerations phase of laser changes to:
(
)
(
)
[
]
(
)
[
]
x
t
u
x
-
-
-
x
-
G
=
dj
ò
-
d
z
t
b
z
t
z
L
g
f
L
2
/
2
2
/
)
/
)
(
(
exp
/
exp
,
,
(7)
where L is the length of the medium with the center at z = 0. The medium border in our model is located at a considerable distance from the focal area, i.e. L/2 >> b, which is equivalent to replacing the lower limit of integral (7) with
-¥
. Since we are interested in the maximum amplitude of polarization, the phase ((L should be determined for coordinate and time z = 0, t = 0:
(
)
(
)
(
)
[
]
(
)
.
)
/(
1
/
2
)
/(
/
exp
0
,
0
2
0
2
2
t
u
+
G
p
=
=
x
t
u
x
-
x
-
G
=
dj
ò
¥
-
g
f
g
f
L
b
b
d
b
(8)
The same initial conditions allow to calculate analytically the following derivatives of the phase:
(
)
(
)
(
)
(
)
.
/
)
(
1
1
0
,
0
,
/
)
(
1
1
0
,
0
2
2
f
g
f
L
g
g
f
L
b
z
b
t
G
+
t
u
+
G
-
=
¶
dj
t
u
+
u
G
=
¶
dj
(9)
Fig. 1 shows the dependences of the phase and its derivatives in time and coordinate under the conditions z = 0 and
{
}
)
2
/(
);
2
/(
g
g
L
L
t
u
u
-
=
. For better clearness, the values on the ordinates axis here and below are given in arbitrary units with normalized scale.
Now, let us consider the phase delay at the output side of the medium. In this case, the distribution of increase of the wave vector length
G
remains unchanged. However, a wave front that came out of the point z at time t will get to the point
x
(on the pass from z to the output border) at
ph
ph
u
x
-
-
=
u
-
x
+
=
¢
¢
z
t
z
t
t
, where υph is the phase velocity of scattered radiation.
Expressions for
t
¢
and
t
¢
¢
are identical and differ only by the values of speed, thus:
(
)
(
)
[
]
x
ú
ú
û
ù
ê
ê
ë
é
÷
÷
ø
ö
ç
ç
è
æ
t
u
x
-
-
-
x
-
G
=
dj
ò
d
z
t
b
z
t
L
z
f
2
/
2
ph
2
ω
/
)
(
exp
/
exp
,
.
(10)
As L/2 >> b, the upper limit of integral (10) can be replaced with
+¥
. Let us calculate
ω
dj
for conditions specified earlier z = 0, t = 0:
(
)
(
)
(
)
[
]
(
)
.
)
/(
1
1
2
)
/(
/
exp
0
,
0
2
ph
0
2
ph
2
ω
tu
+
G
p
=
=
x
tu
x
-
x
-
G
=
dj
ò
¥
b
b
d
b
f
f
(11)
-
1·10
-
11
-
2·10
-
11
1·10
-
11
2·10
-
11
t
, s
a
b
c
a.u.
1
.0
0.8
0.6
0.4
0.2
-
0.2
-
0.4
-
0.6
0
0
-
0.8
Fig. 1. Phase delay of the laser radiation at the input side of the medium (a) and its derivatives on time (b) and coordinate (c) in arbitrary units.
Similarly, we calculate the derivatives on time and coordinate at the output side of the medium:
(
)
(
)
(
)
(
)
.
/
)
(
1
1
0
,
0
,
/
)
(
1
1
0
,
0
2
ph
ω
2
ph
ph
ω
f
f
f
b
z
b
t
G
-
t
u
+
G
=
¶
j
¶
t
u
+
u
G
-
=
¶
j
¶
(12)
The phase and its derivatives, obtained using (10), are shown in Fig. 2 under the same conditions as for the input (laser) radiation.
The total phase delay and its derivatives could be found as a sum of specified values at the input and output sides of the medium:
(
)
÷
÷
÷
ø
ö
ç
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
tu
+
+
÷
÷
ø
ö
ç
ç
è
æ
tu
+
G
p
=
dj
2
ph
2
1
1
1
1
2
0
,
0
b
b
b
g
f
,
(13)
(
)
2
2
ph
ph
1
1
0
,
0
÷
÷
ø
ö
ç
ç
è
æ
t
u
+
u
G
+
÷
÷
ø
ö
ç
ç
è
æ
t
u
+
u
G
-
=
¶
dj
¶
b
b
t
g
g
f
f
,
(14)
(
)
2
2
ph
1
1
1
1
0
,
0
÷
÷
ø
ö
ç
ç
è
æ
t
u
+
G
-
÷
÷
ø
ö
ç
ç
è
æ
t
u
+
G
=
¶
dj
¶
b
b
z
g
f
f
.
(15)
Resulting dependences are shown in Fig. 3 under the earlier accepted values
z
and
t
.
Taking into account the above considerations, the analytical calculation of polarization phase can be simplified by using the results (14) and (15) for derivatives and linear interpolation:
-
1·10
-
11
-
2·10
-
11
1·10
-
11
2·10
-
11
t
, s
a
b
c
a.u.
1
.0
0.8
0.6
0.4
0.2
-
0.2
-
0.4
-
0.6
0
0
-
0.8
Fig. 2. The phase delay of the scattered radiation at the output side of the medium (a) and its derivatives on time (b) and coordinate (c) in arbitrary units.
-
1·10
-
11
-
2·10
-
11
1·10
-
11
2·10
-
11
t
, s
a
b
c
a.u.
1
.0
0.8
0.6
0.4
0.2
-
0.2
-
0.4
-
0.6
0
0
-
0.8
Fig. 3. The total phase delay of the scattered radiation (a) and its derivatives on time (b) and coordinate (c) in arbitrary units.
ω
0
dj
-
dj
-
-
w
=
j
L
p
p
z
k
t
=
(
)
(
)
(
)
(
)
,
/
)
(
1
1
/
)
(
1
1
/
)
(
1
1
/
)
(
1
1
2
ph
2
2
ph
ph
2
0
z
b
b
k
t
b
b
f
g
f
p
f
g
g
f
÷
÷
ø
ö
ç
ç
è
æ
t
u
+
G
+
t
u
+
G
-
-
-
÷
÷
ø
ö
ç
ç
è
æ
t
u
+
u
G
+
t
u
+
u
G
-
w
=
(16)
where
p
j
is the phase of polarization, kp – initial wave vector length of polarization without self-phase modulation and
0
w
– initial frequency.
The expression (16) allows estimating the influence of self-phase modulation on the resulting wave vector length and frequency.
In case of toluene, the group velocity of ruby laser light is υg = 0.65c, refractive index nL = 1.49, i.e. υph = c/nL = 0.67c, relaxation time, as stated earlier, is τ = 5 ps and, finally, length of the focal area b = 0.1 cm. Under these parameters the following values are obtained:
(
)
(
)
6
2
ph
ph
2
10
5
2
/
)
(
1
/
)
(
1
-
×
»
p
÷
÷
ø
ö
ç
ç
è
æ
t
u
+
u
-
t
u
+
u
c
b
b
g
g
,
(
)
(
)
01
.
0
/
)
(
1
1
/
)
(
1
1
2
ph
2
»
t
u
+
-
t
u
+
b
b
g
,
meaning that in the first approximation the self-phase modulation of polarization can be neglected, i.e. (16) can be written as
z
k
t
p
p
-
w
=
j
0
.
Functions
)
(
t
F
t
,
)
(
^
^
r
F
,
)
(
f
z
z
F
z
-
in (1) can be specified as Gaussian functions with according halfwidths
t
,
f
a
,
b
at the level
1
-
e
; also
0
=
f
z
(the condition of stationary focal area).
Using (1), (5) and taking into consideration
z
k
t
p
p
-
w
=
j
0
, we find that the frequency-angular density of radiation energy for parametric components is described by the expression:
wq
W
=
3
4
2
8
c
n
w
p
EMBED Equation.2 (
)
´
t
2
4
2
2
0
b
a
P
f
NL
(
)
[
]
2
2
2
2
2
||
2
5
.
0
exp
t
w
D
+
D
+
D
-
´
^
b
k
a
k
f
,
(17)
where
||
k
D
=
(
)
p
k
k
-
q
w
cos
and
^
D
k
=
(
)
q
w
sin
k
are wave vector difference projections onto axis z and axis perpendicular to it, i.e. the longitudinal and lateral mismatch of wave vector
(
)
w
k
of the parametric components radiation and wave vector
p
k
of polarization;
q
is the scattering angle determined by the angle between vector
(
)
w
k
and axis z, along which the polarization wave vector is oriented.
The most optimal conditions to generate radiation are obtained, when the sum
2
2
2
2
2
2
||
t
w
D
+
D
+
D
=
^
b
k
a
k
F
f
in (6) gains its minimum value. The search for these conditions can be done through finding an extremum
0
=
w
d
dF
. In the approximation of linear dispersion and small
q
, the energy density maximum
is shifted relative to the Raman frequency
as
(
)
(
)
(
)
2
2
2
2
2
2
0
0
2
0
2
0
0
1
1
2
1
1
b
b
a
b
a
k
k
k
g
f
f
g
tu
+
+
-
q
-
D
+
q
q
-
u
D
-
=
w
-
w
=
w
D
,
(18)
θ, ang. min.
-5
-10
5
10
140
70
0
-70
-140
0
/2c
, cm
–1
Fig. 4. Frequency-angular spectrum of the first anti-Stokes component in toluene.
1.0
-5
-2.5
/2c
, cm
–1
a
b
c
W
, a.u.
0.8
0.6
0.4
0.2
Fig. 5. Comparison of axial radiation energy density, when self-phase modulation is considered (a) and neglected: (b) for b = 0.1 cm, (c) for b = 0.3 cm.
where
0
k
D
=
p
k
k
-
w
)
(
0
0
is the difference of wave vector length of radiation (
0
k
) at Raman frequency
0
w
and wave vector length of polarization (
p
k
);
0
q
=
0
0
/
2
k
k
D
is the scattering angle, for which
)
(
0
||
w
D
k
=0.
Additional simplification under conditions
f
a
<<
b
,
0
k
D
<<
0
k
provides:
(
)
(
)
2
2
0
0
1
1
b
k
g
g
tu
-
q
q
-
u
D
-
=
w
D
.
(19)
Using (17) we can calculate a normalized spectrum of the first anti-Stokes component, when
= 2
(
,
=
, where
and
are the Stokes Raman frequency and wave vector length,
and
are laser radiation frequency and wave vector length, respectively. In toluene with SRS excitation by ruby laser, the following values are used: frequency of exciting radiation
(
)
c
L
p
w
2
/
=
1
cm
14403
-
; frequency of Raman-active vibration
W
=
1
cm
1004
-
; focal area length b = 0.1 cm; focal area radius
f
a
=5 μm; existence duration of nonlinear polarization
t
= 2.5 ps.
The resulting frequency-angular spectrum structure is shown in Fig. 4.
Finally, let us estimate in more detail, how self-phase modulation affects the spectrum characteristics, for example, of the axial radiation. The energy density that accounts self-phase modulation can be calculated using (5) and (16), which was done numerically. After that, the calculated spectrum that accounts self-phase modulation and the one obtained from (17) are compared under varying values of b, as one of the focal area characteristics. Comparison of the normalized spectra for axial radiation is shown in Fig. 5. As can be seen, the larger is b, the closer match results obtained between spectrum that accounts self-phase modulation and that neglecting it.
5. Conclusions
The obtained results indicate that self-phase modulation in the self-focusing and relaxation oscillations mode has a weak effect on the radiation spectra of SRS. Nevertheless, parametric components have the frequency-angular structure, shaped as parabolic ((( branches. The shift of the parametric component peak energy in spectrum, obtained in media with normal dispersion at transparency area, is determined by the size of the focal area, its existence duration, and mismatch
between polarization and wave vectors field of the corresponding parametric component at the frequency of exact Raman resonance.
The important point is that the generation of radiation at shifted frequencies occurs under short-term phase matching, and it is a non-stationary process. The principal difference between amplitude-phase matching [6] and short-term phase matching is that in the first case the area of the maximum polarization amplitude moves, and matching of the polarization phases and the field, which is generated at a shifted frequency, remains constant in the case of steady motion. In the case of short-term phase, matching the area of the polarization amplitude is stationary, and therefore matching of phases is rapidly disturbed.
Influence of the short-term phase matching conditions (
0
=
D
k
,
0
¹
w
D
) can cause a frequency shift of radiation in other non-stationary processes, not related to SRS. In particular, this relates to the frequency doubling of picosecond focused light pulses in samples,
which length is much shorter than that of the pulse, in powdery nonlinear optical media, or on the surface of absorbing media by “slide reflection” scheme, etc.
References
1. V.I. Bespalov, G.A. Pasmanik, Nonlinear Optics and Adaptive Laser Systems. Moscow, Nauka, 1985 (in Russian).
2. G.S. Felinskiy, P.A. Korotkov, Simulation of multiwave pumped fiber Raman amplifiers // Semiconductor Physics, Quantum Electronics & Optoelectronics, 9(3), p. 35-39 (2006).
3. C.A. Sacchi, C.H. Townes, J.R. Lifshitz, Anti-Stokes generation in trapped filaments of light // Phys. Rev. 174(2), p. 439-448 (1968).
4. N.E. Kornienko, V.I. Maliy, G.V. Ponezha, E.A. Ponezha, The nature of the frequency-angular structure of radiation in stimulated Raman scattering in liquids // Izvestiia Akademii Nauk UkrSSR, Ser. A, 4, p. 65-68 (1983), in Russian.
5. A.I. Ivanisik, V.I. Maliy, G.V. Ponezha, Cherenkov-type radiation under conditions of Raman light scattering in self-focusing liquids // Optics and Spectroscopy, 82(3), p. 410-416 (1997).
6. A.I. Ivanisik, V.I. Maliy, G.V. Ponezha, Spectrum of anti-Stokes stimulated Raman scattering from the moving focal regions of self-focusing // Optics and Spectroscopy, 90(4), p. 625-629 (2001).
7. Y.R. Shen, The Principles of Nonlinear Optics. New York, Wiley, 1984.
8. A.I. Ivanisik, V.I. Maliy, G.V. Ponezha, Effect of self-focusing on the angular spectra of stimulated Raman scattering // Optics and Spectroscopy 85(1), p. 78-84 (1998).
9. R.V. Johnson, J.H. Marburger, Relaxation oscillation in stimulated Raman and Brillouin scattering // Phys. Rev. A, 4(3), p. 1175-1182 (1971).
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
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