4.3.1 Laser Pumping System
The population difference N(t) ¼ N 2 À N 1 (where N 2 and N 1 are populations of the
second and first levels, relatively) is determined from the third DE (4.17) for particles
being under impact of pumping α N0 , it follows that at pumping variation α N0 (t) in the
form of the stepped pulse:
α N0 t
ð Þ ¼
0,
t < 0
N 0P =T 1 , t > 0
&
,
ð4:25Þ
before the moment of generation beginning (i.e., stimulated laser emission E n ¼ 0)
the population difference N increases obeying
N t
ð Þ ¼ N 0 1 À exp Àt=T 1
ð
Þ
½
,
ð4:26Þ
where lifetime on the upper level T 1 defines the growth constant, and 1/T 1 is a
probability of stimulated transition. At achievement of the threshold value, in the
quasi-stationary mode, the population depends on the amplitude square E
2
0n , i.e.,
E
2
0n ¼ Re E n Á E
Ã
n
Â
Ã
, where “Ô is the operation of conjugation. In stationary
(or quasi-stationary) mode
dE
2
10n
dτ ¼ 0 ,
dN
dτ ¼ 0 and from Eq. (4.24), it follows that
N ¼
1
G 00 Q 0F
and 0 ¼ α N0 À
N
T 1
À G 00 Á N Â E
2
10n .
At achievement of the threshold, the laser transfers in the generation state. Variables values in the steady-state mode are E
2
10n ¼ E
2
0 and N ¼ N 0 :
N ¼
N 0P
1 þ T 1 G 00 E
2
10n
Â
à :
ð4:27Þ
Figure 4.8 shows the transient scenario in the laser at stepped variation of the
normalized pumping level α N0 ¼ J 0 ¼ 20. The steady-state point is the interception
point (Fig. 4.8d) of the plot 1 N ¼
1
G 00 Q 0F
and the plot 2 calculated on Eq. (4.27).
4.3.1.1 Laser Modulation by Sine Oscillation
At laser modulation of the harmonic oscillation, the population difference N(t)
delays relative the pumping pulse α N0 (t). This delay is determined by the lifetime
T 1 on the operating level. At pumping variation on the harmonic law (the case of DM
in OEO) with some radio frequency f o and the modulation index m 0 in the form:
α N0 t
ð Þ ¼
0, t < 0
N 0 =T 1
ð
ÞÁ 1 þ m 0 cos 2πf o t
ð
Þ
½
, t > 0
&
:
4.3 Laser Kinetic Equations and the Pumping System
151
The population difference N(t) ¼ N 2 À N 1 (where N 2 and N 1 are populations of the
second and first levels, relatively) is determined from the third DE (4.17) for particles
being under impact of pumping α N0 , it follows that at pumping variation α N0 (t) in the
form of the stepped pulse:
α N0 t
ð Þ ¼
0,
t < 0
N 0P =T 1 , t > 0
&
,
ð4:25Þ
before the moment of generation beginning (i.e., stimulated laser emission E n ¼ 0)
the population difference N increases obeying
N t
ð Þ ¼ N 0 1 À exp Àt=T 1
ð
Þ
½
,
ð4:26Þ
where lifetime on the upper level T 1 defines the growth constant, and 1/T 1 is a
probability of stimulated transition. At achievement of the threshold value, in the
quasi-stationary mode, the population depends on the amplitude square E
2
0n , i.e.,
E
2
0n ¼ Re E n Á E
Ã
n
Â
Ã
, where “Ô is the operation of conjugation. In stationary
(or quasi-stationary) mode
dE
2
10n
dτ ¼ 0 ,
dN
dτ ¼ 0 and from Eq. (4.24), it follows that
N ¼
1
G 00 Q 0F
and 0 ¼ α N0 À
N
T 1
À G 00 Á N Â E
2
10n .
At achievement of the threshold, the laser transfers in the generation state. Variables values in the steady-state mode are E
2
10n ¼ E
2
0 and N ¼ N 0 :
N ¼
N 0P
1 þ T 1 G 00 E
2
10n
Â
à :
ð4:27Þ
Figure 4.8 shows the transient scenario in the laser at stepped variation of the
normalized pumping level α N0 ¼ J 0 ¼ 20. The steady-state point is the interception
point (Fig. 4.8d) of the plot 1 N ¼
1
G 00 Q 0F
and the plot 2 calculated on Eq. (4.27).
4.3.1.1 Laser Modulation by Sine Oscillation
At laser modulation of the harmonic oscillation, the population difference N(t)
delays relative the pumping pulse α N0 (t). This delay is determined by the lifetime
T 1 on the operating level. At pumping variation on the harmonic law (the case of DM
in OEO) with some radio frequency f o and the modulation index m 0 in the form:
α N0 t
ð Þ ¼
0, t < 0
N 0 =T 1
ð
ÞÁ 1 þ m 0 cos 2πf o t
ð
Þ
½
, t > 0
&
:
4.3 Laser Kinetic Equations and the Pumping System
151