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Angular distribution of stimulated Brillouin scattering gain
Volume 3, number
ANGULAR
July 1971
OPTICS COMMUNICATIONS
5
DISTRIBUTION
OF
STIMULATED
BRILLOUIN
SCATTERING
GAIN*
F. BAROCCHI Istituto
di Ricerca
sulle
Onde Elettromagnetiche
de1 C.N.R.,
Florence,
Italy
and
M. ZOPPI Istituto
di Fisica
Superiore
dell’Universit&
Received
We present an beams interacting An expression for the cases in which
29 April
di Firenze,
Florence,
Italy
1971
extension of the stimulated Brillouin scattering steady-state amplification theory for at angles tl # 8. the steady-state amplified power as a function of a generalized gain factor is given for the volume of interaction of the two beams is completely contained in the amplifier cell.
Recently stimulated Brillouin scattering (S.B.S.) has been extensively studied theoretically [l-3] and experimentally [3-lo] by a number of authors. Kaiser and coworkers [3,6,8,10] performed measurements of the S.B.S. steady-state gain and of the phonon lifetime of several liquids and solids with good precision, while Litovitz and Goldblatt [5], measured the phonon velocity of several liquids with the precision of a few O/oo. As a result S.B.S. becomes a technique as useful as the normal Brillouin scattering for studying the phonon propagation properties in condensed matter and is particularly useful for liquids which have the phonon dispersion in the GHz region of the phonon spectrum [ll]. At the present time only one important difference still remains in the application of the two techniques, i.e. the possibility of measuring the phonon propagation parameters as functions of the phonon frequency. Whereas experiments and theories of S.B.S. have been worked only in the case of the backscattering angle, the normal Brillouin scattering allows measurements of the phonon velocity and absorption as function of the phonon frequency simply by varying the angle between the incident and scattered radiation. Clearly, as tunable laser sources useful for S.B.S. do not exist at the present level of the art of lasers, it is interesting to consider also for S.B.S. the possiblity of varying the angle between the incident and the scattered radiation. In this paper we shall generalize the theory of S.B.S. steady-state amplification and deduce the expression of the steady-state gain factor as a function of the angle between the incident and the amplified radiation in the case of beams with gaussian wave-front distributions. Let us consider an isotropic system and start from the coupled wave equations for the longitudinal elastic displacement u and the electric field E = EL + ES, where EL and Es are the laser and the S.B.S. field components respectively.
at2
L372+2$
1 2po
(1)
“=57(E2),
vx+~~+(~,"~]E=~~[~V."~E],
(2)
where v p, F, y, po, c, (Y, n are the phonon velocity and line width, the elasto-optic coupling coefficient, the density of the medium, the velocity, absorption and refractive index of the electromagnetic wave respectively and we assumed the last two equal for both the waves. The two electric fields are polarized in the same direction e and propagate forming an angle 0. * This work was partially supported by the Laboratorio
di Elettronica
Quantistica
de1 C.N.R.,
Florence.
Italy.
335
Volume 3, number 5
OPTICS COMMUNICATIONS
In the slowly varying the form: EL
= $ {EL(r,t)
ES =
plane wave approximation
exp[i(wLt
-kL.
$ {.??s(r,t) exp[i(wSt+ks.r)]
U = ${G( r,t) exp[i(wpt-kpar)]
the electric
fields
July 1971 and the displacement
r)] + c.c.> e , ,
+ c.c.}e + c.c.} kp/kp
(34
(3b) (3c)
.
If we consider the following approximations: (1) that in a small-signal pumping field EL changes negligibly during the interaction, i.e.
where 1 is the interaction iength between the two beams (2) that the amplitudes of the fields vary slowly, i.e.
(3) that the spatial length of the electromagnetic (2) reduce to:
EL Eg exp[i(Awwt - Ak. r)]
theory the amplitude
is much larger
- Ak. r-)]}
than I; then eqs. (1) and
(44
,
(4b)
,
where AW =WL-WS-Wpmd Ak= kL - ks - kp. Firstly, we integrate eq. (4b) in a reference frame where the z and kp directions that the radial electric field distributions are gaussian i.e. exp
[-$(Ef],
Es(r,t) = E=S(r,t) exp[-+(gf]
where R, is a measure of the spot size, that the relative respect to the phonon absorption ap = lY/up, i.e.
=--
~- pp-exp[i(Awt 4poz/p [r - iAw sin &Q]
where the Ak is supposed
336
to be caused by a frequency
Assuming
(5)
are very small with
u [12]:
- Aker)]
mismatch
coincide.
,
spatial field variations
<< oP and that R, >> ljcvp one finds for the amplitude
u(r,tj
of the
in the kL direction;
wave packets
Re{%?LEiu*exp[i(Awt
EL(r,t) = E,(t)
can be set in
,
of the pumping fields
(6) so that
Volume 3. number 5
OPTICS COMMUNICATIONS
July 1971
Fig. 1. TWOgaussian beams of spot size R interacting at the angle ellm in a cell of length I,. The equation for the Stokes intensity iS = $eo nc l&12 by means of eqs. (4a) and (6) neglecting electromagnetic wave absorption (Yassumes the following form: kS. V ~ fS( r, t) = A sin+0 I2+Am2 kS
r I= t f
sin2 $% L( ) S(
r,t)exp
[ -ckLRo “““T,“],
the
(7)
where: Y2,i
A=
2nPoVpC3
*
We now integrate
eq. (7) in a reference (l?/sin$%) ( r/sin$6)2
+ Au2
frame
:” Z1
where the z and kS directions
ew[
-(j!g)2]dr/
,
coincide
(fig. 1) obtaining:
(8)
where zl and 22 indicate the two limits of the amplifier cell. To give a convenient form to eq. (8) we must perform the integration in the curly brackets. If the stimulated interaction is supposed to take place over the main parts of the two gaussian cross sections and is completely contained in the amplifier of length I,, one can extend the integral from -m to +m with a small error. Let us designate by R the size of the cross sections useful for the stimulated amplification, which is taken equal for both the two beams. In this case the interaction volume is completely contained in the amplifier cell (fig. 1) if 7r- ‘lim a%‘%lim
where %lim = 2 arctg (2R/Z,)
and the error which one does extending the integral (8) from -Q) to +m depends directly on the value of R,/R. For example it is easily seen that R,/R = 1 corresponds to an error less than lo/r. When x - ‘lim a%a%lim eq. (8) can be writteh in the form:
1TL
7, = ii exp A
(r/sin (r/sin+Q2
$ %)
(9)
+ aw
where x is the coordinate in the direction perpendicular and the amplified Stokes intensities respectively.
to the plane of incidence,
fi andTa are the signal
337
Volume
3. number
5
OPTICS COMMUNICATIONS
0
8’
,
I
1lAW
I I
41.60"
77.32O
22.62O
Fig. 2. The normalized
gain multiplied
I 900102!6W
61.92"
July 1971
I
I I
136.400 116.08"
I
168.: 157.360
by the effective length as a function of 0 (continuous tion of Olim (-. - curves).
,
00
curves),
and as a func-
Let us now calculate the amplified Stokes power as a function of the signal and laser powers and of the parameters of the material inside the amplifier cell. We expand the exp[-(x/Ro)2] of eq. (9) in a power series then we integrate over the Stokes cross section, obtaining:
(10) where: y2,2 g(Aw,@)
=
L
(r/sin
Rod?
&0)
2nc3 po2ip (lY/sin+B)2 + Au2
’
l (j,Q ) =
(j+l)1’2jsin
8
In eq. (10) the Stokes power is expressed by means of the power of the generalized gain factorg(Aw,Q) and of the effective interaction length l(j, Q). The number of the terms of eq. (10) that one must retain to compare the theory with the experiments, depends on the precision one wishes to have. In the case in which the requestion precision is less than 10% as in the experiments which were performed up to date [3,6,8,10] it is sufficient to consider only the first two terms. In this case one has:
338
Volume 3. number 5
OPTICS COMMUNICATIONS
July 1971
which is a generalization of the usual expression of Pa( 0 = a) when Q f IT. We note here that Pa(O) for 0 + 71does not coincide with Pa( 0 = rr) because range 7-r - @lima’8 2e lim
where elim = 2 arctg (2R/Z,).
In particular
eq. (11) is valid only in the
in eq. (10) the quantity Z(j, 0) is directly
connected with the limitation on 0 and it is not comparable with I, as it goes to 00 when f3+ pi while g(Aw,e)-g(Aw,a). Fig. 2 shows the behaviour of the functionf( Q), which is the product g(0, 0) I(1 ,B ) normalized to the back-scattering value g(O,a)Z,, for various values of R,/I, as a function of 8. To stress the importance of the limitation which we imposed on 0 we also reported in fig. 2 the function f(0 lim) for the values of R,/R = 1 considering Slim a function of R,. Clearly for a certain value of R/R0 the simple theory we performed gives consistent results only for values of f( .9) contained inside of the space limited by f(elim) andf(n-Qim).
ACKNOWLEDGEMENTS We like to thank Professor G. Toraldo di Francis under whose direction and Dr. R. Vallauri for helpful and stimulating discussions.
this work was carried
out,
REFERENCES [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12]
Y. R. Shen and N. Bloembergen, Phys. Rev. 137 (1965) A1784. C. L.Tang, J. Appl. Phys. 37 (1966) 2945. D. Pohl and W. Kaiser, Phys. Rev. Bl (1970) 31. A. S. Pine, Phys. Rev. 149 (1966) 113. N.R. Goldblatt and T. A. Litovitz. J. Acoust. Sot. Am. 41 (1967) 1301. M. Maier, Phys. Rev. 166 (1968) 113. J. L. Emmett and A. L. Schawlow, Phys. Rev. 170 (1968) 358. D. Pohl, M. Maier and W. Kaiser, Phys. Rev. Letters 20 (1968) 366. M. Denariez and G. Bret, Phys. Rev. 171 (1968) 160. D. Pohl, I. Reinhold and W. Kaiser, Phys. Rev. Letters 20 (1968) 1141. F. Barocchi, J. Appl. Phys. 40 (1969) 2867. M. Zoppi, Thesis, Univ. of Florence (1970)) unpublished.
339
ANGULAR
July 1971
OPTICS COMMUNICATIONS
5
DISTRIBUTION
OF
STIMULATED
BRILLOUIN
SCATTERING
GAIN*
F. BAROCCHI Istituto
di Ricerca
sulle
Onde Elettromagnetiche
de1 C.N.R.,
Florence,
Italy
and
M. ZOPPI Istituto
di Fisica
Superiore
dell’Universit&
Received
We present an beams interacting An expression for the cases in which
29 April
di Firenze,
Florence,
Italy
1971
extension of the stimulated Brillouin scattering steady-state amplification theory for at angles tl # 8. the steady-state amplified power as a function of a generalized gain factor is given for the volume of interaction of the two beams is completely contained in the amplifier cell.
Recently stimulated Brillouin scattering (S.B.S.) has been extensively studied theoretically [l-3] and experimentally [3-lo] by a number of authors. Kaiser and coworkers [3,6,8,10] performed measurements of the S.B.S. steady-state gain and of the phonon lifetime of several liquids and solids with good precision, while Litovitz and Goldblatt [5], measured the phonon velocity of several liquids with the precision of a few O/oo. As a result S.B.S. becomes a technique as useful as the normal Brillouin scattering for studying the phonon propagation properties in condensed matter and is particularly useful for liquids which have the phonon dispersion in the GHz region of the phonon spectrum [ll]. At the present time only one important difference still remains in the application of the two techniques, i.e. the possibility of measuring the phonon propagation parameters as functions of the phonon frequency. Whereas experiments and theories of S.B.S. have been worked only in the case of the backscattering angle, the normal Brillouin scattering allows measurements of the phonon velocity and absorption as function of the phonon frequency simply by varying the angle between the incident and scattered radiation. Clearly, as tunable laser sources useful for S.B.S. do not exist at the present level of the art of lasers, it is interesting to consider also for S.B.S. the possiblity of varying the angle between the incident and the scattered radiation. In this paper we shall generalize the theory of S.B.S. steady-state amplification and deduce the expression of the steady-state gain factor as a function of the angle between the incident and the amplified radiation in the case of beams with gaussian wave-front distributions. Let us consider an isotropic system and start from the coupled wave equations for the longitudinal elastic displacement u and the electric field E = EL + ES, where EL and Es are the laser and the S.B.S. field components respectively.
at2
L372+2$
1 2po
(1)
“=57(E2),
vx+~~+(~,"~]E=~~[~V."~E],
(2)
where v p, F, y, po, c, (Y, n are the phonon velocity and line width, the elasto-optic coupling coefficient, the density of the medium, the velocity, absorption and refractive index of the electromagnetic wave respectively and we assumed the last two equal for both the waves. The two electric fields are polarized in the same direction e and propagate forming an angle 0. * This work was partially supported by the Laboratorio
di Elettronica
Quantistica
de1 C.N.R.,
Florence.
Italy.
335
Volume 3, number 5
OPTICS COMMUNICATIONS
In the slowly varying the form: EL
= $ {EL(r,t)
ES =
plane wave approximation
exp[i(wLt
-kL.
$ {.??s(r,t) exp[i(wSt+ks.r)]
U = ${G( r,t) exp[i(wpt-kpar)]
the electric
fields
July 1971 and the displacement
r)] + c.c.> e , ,
+ c.c.}e + c.c.} kp/kp
(34
(3b) (3c)
.
If we consider the following approximations: (1) that in a small-signal pumping field EL changes negligibly during the interaction, i.e.
where 1 is the interaction iength between the two beams (2) that the amplitudes of the fields vary slowly, i.e.
(3) that the spatial length of the electromagnetic (2) reduce to:
EL Eg exp[i(Awwt - Ak. r)]
theory the amplitude
is much larger
- Ak. r-)]}
than I; then eqs. (1) and
(44
,
(4b)
,
where AW =WL-WS-Wpmd Ak= kL - ks - kp. Firstly, we integrate eq. (4b) in a reference frame where the z and kp directions that the radial electric field distributions are gaussian i.e. exp
[-$(Ef],
Es(r,t) = E=S(r,t) exp[-+(gf]
where R, is a measure of the spot size, that the relative respect to the phonon absorption ap = lY/up, i.e.
=--
~- pp-exp[i(Awt 4poz/p [r - iAw sin &Q]
where the Ak is supposed
336
to be caused by a frequency
Assuming
(5)
are very small with
u [12]:
- Aker)]
mismatch
coincide.
,
spatial field variations
<< oP and that R, >> ljcvp one finds for the amplitude
u(r,tj
of the
in the kL direction;
wave packets
Re{%?LEiu*exp[i(Awt
EL(r,t) = E,(t)
can be set in
,
of the pumping fields
(6) so that
Volume 3. number 5
OPTICS COMMUNICATIONS
July 1971
Fig. 1. TWOgaussian beams of spot size R interacting at the angle ellm in a cell of length I,. The equation for the Stokes intensity iS = $eo nc l&12 by means of eqs. (4a) and (6) neglecting electromagnetic wave absorption (Yassumes the following form: kS. V ~ fS( r, t) = A sin+0 I2+Am2 kS
r I= t f
sin2 $% L( ) S(
r,t)exp
[ -ckLRo “““T,“],
the
(7)
where: Y2,i
A=
2nPoVpC3
*
We now integrate
eq. (7) in a reference (l?/sin$%) ( r/sin$6)2
+ Au2
frame
:” Z1
where the z and kS directions
ew[
-(j!g)2]dr/
,
coincide
(fig. 1) obtaining:
(8)
where zl and 22 indicate the two limits of the amplifier cell. To give a convenient form to eq. (8) we must perform the integration in the curly brackets. If the stimulated interaction is supposed to take place over the main parts of the two gaussian cross sections and is completely contained in the amplifier of length I,, one can extend the integral from -m to +m with a small error. Let us designate by R the size of the cross sections useful for the stimulated amplification, which is taken equal for both the two beams. In this case the interaction volume is completely contained in the amplifier cell (fig. 1) if 7r- ‘lim a%‘%lim
where %lim = 2 arctg (2R/Z,)
and the error which one does extending the integral (8) from -Q) to +m depends directly on the value of R,/R. For example it is easily seen that R,/R = 1 corresponds to an error less than lo/r. When x - ‘lim a%a%lim eq. (8) can be writteh in the form:
1TL
7, = ii exp A
(r/sin (r/sin+Q2
$ %)
(9)
+ aw
where x is the coordinate in the direction perpendicular and the amplified Stokes intensities respectively.
to the plane of incidence,
fi andTa are the signal
337
Volume
3. number
5
OPTICS COMMUNICATIONS
0
8’
,
I
1lAW
I I
41.60"
77.32O
22.62O
Fig. 2. The normalized
gain multiplied
I 900102!6W
61.92"
July 1971
I
I I
136.400 116.08"
I
168.: 157.360
by the effective length as a function of 0 (continuous tion of Olim (-. - curves).
,
00
curves),
and as a func-
Let us now calculate the amplified Stokes power as a function of the signal and laser powers and of the parameters of the material inside the amplifier cell. We expand the exp[-(x/Ro)2] of eq. (9) in a power series then we integrate over the Stokes cross section, obtaining:
(10) where: y2,2 g(Aw,@)
=
L
(r/sin
Rod?
&0)
2nc3 po2ip (lY/sin+B)2 + Au2
’
l (j,Q ) =
(j+l)1’2jsin
8
In eq. (10) the Stokes power is expressed by means of the power of the generalized gain factorg(Aw,Q) and of the effective interaction length l(j, Q). The number of the terms of eq. (10) that one must retain to compare the theory with the experiments, depends on the precision one wishes to have. In the case in which the requestion precision is less than 10% as in the experiments which were performed up to date [3,6,8,10] it is sufficient to consider only the first two terms. In this case one has:
338
Volume 3. number 5
OPTICS COMMUNICATIONS
July 1971
which is a generalization of the usual expression of Pa( 0 = a) when Q f IT. We note here that Pa(O) for 0 + 71does not coincide with Pa( 0 = rr) because range 7-r - @lima’8 2e lim
where elim = 2 arctg (2R/Z,).
In particular
eq. (11) is valid only in the
in eq. (10) the quantity Z(j, 0) is directly
connected with the limitation on 0 and it is not comparable with I, as it goes to 00 when f3+ pi while g(Aw,e)-g(Aw,a). Fig. 2 shows the behaviour of the functionf( Q), which is the product g(0, 0) I(1 ,B ) normalized to the back-scattering value g(O,a)Z,, for various values of R,/I, as a function of 8. To stress the importance of the limitation which we imposed on 0 we also reported in fig. 2 the function f(0 lim) for the values of R,/R = 1 considering Slim a function of R,. Clearly for a certain value of R/R0 the simple theory we performed gives consistent results only for values of f( .9) contained inside of the space limited by f(elim) andf(n-Qim).
ACKNOWLEDGEMENTS We like to thank Professor G. Toraldo di Francis under whose direction and Dr. R. Vallauri for helpful and stimulating discussions.
this work was carried
out,
REFERENCES [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12]
Y. R. Shen and N. Bloembergen, Phys. Rev. 137 (1965) A1784. C. L.Tang, J. Appl. Phys. 37 (1966) 2945. D. Pohl and W. Kaiser, Phys. Rev. Bl (1970) 31. A. S. Pine, Phys. Rev. 149 (1966) 113. N.R. Goldblatt and T. A. Litovitz. J. Acoust. Sot. Am. 41 (1967) 1301. M. Maier, Phys. Rev. 166 (1968) 113. J. L. Emmett and A. L. Schawlow, Phys. Rev. 170 (1968) 358. D. Pohl, M. Maier and W. Kaiser, Phys. Rev. Letters 20 (1968) 366. M. Denariez and G. Bret, Phys. Rev. 171 (1968) 160. D. Pohl, I. Reinhold and W. Kaiser, Phys. Rev. Letters 20 (1968) 1141. F. Barocchi, J. Appl. Phys. 40 (1969) 2867. M. Zoppi, Thesis, Univ. of Florence (1970)) unpublished.
339