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Influence of the population change on the stimulated Raman scattering by short light pulses
Volume 43A, number 2
PHYSICS LETTERS
26 February 1973
INFLUENCE OF THE POPULATION CHANGE ON THE STIMULATED RAMAN SCAUERING BY SHORT LIGHT PULSES J.HERRMANN Dept. of Physics, University of Jena, Jena, German DemocraticRepublic Received 26 October 1972 The theory of the stimulated Raman scattering has been extended to effects in the non-steady-state by population change of the molecular levels.
Recently the development of methods for producing ultrashort light pulses in the picosecond region excited interest in the theory of transient stimulated Raman scattering (SRS) [1, 2]. The time intervals of such ultrashort pulses are comparable to or shorter than the transversal relaxation time ~‘2~Transient effects in the SRS may appear in the case of nanosecond pulses too, caused by the population change of the molecular levels. In the present paper we shall investigate the theory of such processes. We suppose that the typical times of the phase and amplitude change (‘r0, TA) of the Stokes and laser cornponent satisfy the condition r0, TA >> T2, whereas we do not make any supposition for T1. From the density matrix formalism and components the Maxwell of equations for and the quasimonochromatic the Stokes
~ exp (—~/T1),devide the second equation of (1) by N5(N0— Ns) exp (—~/T1)and differentiate with respect After\ some we get 2 I tofl. Ns / i calculations ~N a 0~1 a Ns =
~—~—-
\
In
—
iv °
—N~, ~T1
—i--— J~— ~ln
—
vqiiv
a~ ~ ~+
~ = (~ ~o) —
1
aNSIa~=ANLNS& aNL/a~—aNS/a~,
(1)
with the definitions NS,L =(nI2hws,L)IEs,LI2v’~IV~,
+
(2)
.
1
After integration with respect to E~eq. (2) leads to
(
—
an
=
) r/
°
I(
N0
L\
—~
—
2ANONS + AN°~ I + ~ —
—
a~N0
T11 -1
N0—N5 (3)
,
T j
with the abbreviation ~i
=
~
The functionf(~)
Ng~j/No
to
x(l—x)
n = (2AN~)~jS N
5/N0
2)
+
1
iz0w~w5T2(Ia12I 2
A— =
2
t — nzfc
p~
=
is calculated from the initial condition for ~ = 0. In the special case fo rectangular light pulses for NLO andN~ eq. (3) can be resolved. We get for n r with a =
~~LNS,
=
s
—
aN~ A N0~0 —2AN0
1 laser waves Es and EL we get the following system of differential equations [1, 3]:
‘~
°
‘
~=
n
(A difference of the population, a12 polarizabiity).
Eqs.(1) lead to the Manley-Rowe-relationNL + Ns = N0(n); therefore we can eliminate NL. We substitute
2AN~T
ln
1—~--_
exp(—AN
~-
1~
a
°
~1)l
.
(4)
j
1 L The function Ns = N = 0) calculated from the initial condition n = 05(~, of nthe eq.is(1): = N~exp (AN0~1) [1 + a exp (AN0A1)] 1. An analytical integration of(4) is not possible. Therefore we calculated the integral (4) for some interesting
133
Volume 43A, number 2
PHYSICS LETTERS
26 February 1973
For large longitudinal relaxation times
T
1 >>i~, TA 4s) the integration in (formay 112 or we have ~ 10 in a good approxi(4) be 02 carried out T1 analytically mation by neglecting the logarithmic term in (4)
tb’s-lao 85
(l/2AN 2T 0 1 <<1). In the case of c<< I we get r~= (2AN~)
—
ltNs-Ns N+ln [(NoNs)Ns NsNs (N0N5)Af~]) _
~
.
(5)
This special case has also been considered by Wilhelmi in [3] but the result of this paper is only true for small signal amplification.
8-5
,
0,2
2
6
8
10
12
lti’
~
Fig. 1. Temporal behavior ofthe Stokes radiation for different values of the parameters B = 2A N~T1 and S (Nso/No) exp
The author is grateful to Prof. G. Weber and Dr. B. Wilhelmi for very useful discussions.
(,4N0A0t).
References values of the parameters numerically. Fig. 1 clearly shows in which manner the Stokes signal decreases and tends to a steady-state value in the case of an intensive laser pulse. Such a decrease results from a population change. The steady-state value N5 is defined by the singularity of the integrand in (4).
134
[1] S.A. Akhmanov, K.N. Drabovich, A.P. Suchonikov and A.C. Chirkin, Zh. Eksp. i Theor. Fiz 59 (1970) 485. [2] R.L. Carman, F. Shiinizu, CS. Wang, N. Bloembergen, Phys. Rev. 2 (1970) 59. [3) B. Withelmi, Wissensch. Zeitschr. d. Univ. Jena, to be published.
PHYSICS LETTERS
26 February 1973
INFLUENCE OF THE POPULATION CHANGE ON THE STIMULATED RAMAN SCAUERING BY SHORT LIGHT PULSES J.HERRMANN Dept. of Physics, University of Jena, Jena, German DemocraticRepublic Received 26 October 1972 The theory of the stimulated Raman scattering has been extended to effects in the non-steady-state by population change of the molecular levels.
Recently the development of methods for producing ultrashort light pulses in the picosecond region excited interest in the theory of transient stimulated Raman scattering (SRS) [1, 2]. The time intervals of such ultrashort pulses are comparable to or shorter than the transversal relaxation time ~‘2~Transient effects in the SRS may appear in the case of nanosecond pulses too, caused by the population change of the molecular levels. In the present paper we shall investigate the theory of such processes. We suppose that the typical times of the phase and amplitude change (‘r0, TA) of the Stokes and laser cornponent satisfy the condition r0, TA >> T2, whereas we do not make any supposition for T1. From the density matrix formalism and components the Maxwell of equations for and the quasimonochromatic the Stokes
~ exp (—~/T1),devide the second equation of (1) by N5(N0— Ns) exp (—~/T1)and differentiate with respect After\ some we get 2 I tofl. Ns / i calculations ~N a 0~1 a Ns =
~—~—-
\
In
—
iv °
—N~, ~T1
—i--— J~— ~ln
—
vqiiv
a~ ~ ~+
~ = (~ ~o) —
1
aNSIa~=ANLNS& aNL/a~—aNS/a~,
(1)
with the definitions NS,L =(nI2hws,L)IEs,LI2v’~IV~,
+
(2)
.
1
After integration with respect to E~eq. (2) leads to
(
—
an
=
) r/
°
I(
N0
L\
—~
—
2ANONS + AN°~ I + ~ —
—
a~N0
T11 -1
N0—N5 (3)
,
T j
with the abbreviation ~i
=
~
The functionf(~)
Ng~j/No
to
x(l—x)
n = (2AN~)~jS N
5/N0
2)
+
1
iz0w~w5T2(Ia12I 2
A— =
2
t — nzfc
p~
=
is calculated from the initial condition for ~ = 0. In the special case fo rectangular light pulses for NLO andN~ eq. (3) can be resolved. We get for n r with a =
~~LNS,
=
s
—
aN~ A N0~0 —2AN0
1 laser waves Es and EL we get the following system of differential equations [1, 3]:
‘~
°
‘
~=
n
(A difference of the population, a12 polarizabiity).
Eqs.(1) lead to the Manley-Rowe-relationNL + Ns = N0(n); therefore we can eliminate NL. We substitute
2AN~T
ln
1—~--_
exp(—AN
~-
1~
a
°
~1)l
.
(4)
j
1 L The function Ns = N = 0) calculated from the initial condition n = 05(~, of nthe eq.is(1): = N~exp (AN0~1) [1 + a exp (AN0A1)] 1. An analytical integration of(4) is not possible. Therefore we calculated the integral (4) for some interesting
133
Volume 43A, number 2
PHYSICS LETTERS
26 February 1973
For large longitudinal relaxation times
T
1 >>i~, TA 4s) the integration in (formay 112 or we have ~ 10 in a good approxi(4) be 02 carried out T1 analytically mation by neglecting the logarithmic term in (4)
tb’s-lao 85
(l/2AN 2T 0 1 <<1). In the case of c<< I we get r~= (2AN~)
—
ltNs-Ns N+ln [(NoNs)Ns NsNs (N0N5)Af~]) _
~
.
(5)
This special case has also been considered by Wilhelmi in [3] but the result of this paper is only true for small signal amplification.
8-5
,
0,2
2
6
8
10
12
lti’
~
Fig. 1. Temporal behavior ofthe Stokes radiation for different values of the parameters B = 2A N~T1 and S (Nso/No) exp
The author is grateful to Prof. G. Weber and Dr. B. Wilhelmi for very useful discussions.
(,4N0A0t).
References values of the parameters numerically. Fig. 1 clearly shows in which manner the Stokes signal decreases and tends to a steady-state value in the case of an intensive laser pulse. Such a decrease results from a population change. The steady-state value N5 is defined by the singularity of the integrand in (4).
134
[1] S.A. Akhmanov, K.N. Drabovich, A.P. Suchonikov and A.C. Chirkin, Zh. Eksp. i Theor. Fiz 59 (1970) 485. [2] R.L. Carman, F. Shiinizu, CS. Wang, N. Bloembergen, Phys. Rev. 2 (1970) 59. [3) B. Withelmi, Wissensch. Zeitschr. d. Univ. Jena, to be published.