N j2π ν À ν 0
ð
Þ
½
¼
1
1 þ T 1 Á j2π ν À ν 0
ð
Þ
Á
N 0
1 þ
T 1 ÁG 00 ÁE
2
0n
1þT 1 Áj2π νÀν 0
ð
Þ
:
ð4:33Þ
We introduce the coefficient K N [j2π(ν À ν 0 )], which we define as:
K N j2π ν À ν 0
ð
Þ
½
¼
N j2π νÀν 0
ð
Þ
½
N 0
. The last representation allows the solution of the
task on transformation of external pumping block in the laser analog model (Fig. 4.7)
into the laser analog models shown in Fig. 4.13.
The laser analog model in the dipole approximation in the quasi-stationary mode
is presented in Fig. 4.13c. Its peculiarity consists in the fact that the pimping block is
located into a loop (a circuit) of the positive feedback. Restrictions of such a “singleloop” (or single-circuit) laser model are discussed later. To find out solutions of the
symbolic differential equations, we perform the normalization of the laser
symbolic DE.
In the laser analog model presented in Fig. 4.13a, the closed loop of the positive
FB with the active medium (OA) is shown. The optical oscillations of the strength
E L. in act in the OA input. Amplified strength oscillations E L. out act in the optical
output of OA. The laser analog model in the quasi-stationary mode is presented in
Fig. 4.13b with the help of the control operator transfer function K L ( p 00 ) ¼ E L. out /
E L. in and the nonlinear element S(E n ).
4.3.4 Normalized Differential Equations of the Laser
We introduce the operator p 00 ¼ p/(2πν 0n ) and normalized variables: G 0n ¼
G 0
2πν 0n T 1 N 00
, τ ¼ t Á 2πν 0n , T 1n ¼ 2πν 0n T 1 , N n ¼
N
N 00
, α nN0 ¼
α N0
N 00 2πν 0n
and use the
approximation of N n in the vicinity of optical frequencies 2π(ν À ν 0n ):
N n ¼
N 0
1 þ j ν=ν 0F
ð
ÞÀ1
½
Þ T 1n þ T 1n G 00n Re E n Á E
Ã
n
Â
Ã:
ð4:34Þ
Now we introduce the designation of the operator function:
Y p 00
ð Þ¼
1
p 00
2
 p
4
00 þ
1
Q 0F
þ
1
Q 02
p
3
00 þ 2 þ
1
Q 0F
Á
1
Q 02
p
2
00 þ
1
Q 0F
þ
1
Q 02
p 00 þ 1
!
:
ð4:35Þ
After performing of the normalization at β ¼ [(ν 0n À ν 12 )/ν 12 ]
2
% 0.001, the last
expression can be written in the compact form:
4.3 Laser Kinetic Equations and the Pumping System
159
ð
Þ
½
¼
1
1 þ T 1 Á j2π ν À ν 0
ð
Þ
Á
N 0
1 þ
T 1 ÁG 00 ÁE
2
0n
1þT 1 Áj2π νÀν 0
ð
Þ
:
ð4:33Þ
We introduce the coefficient K N [j2π(ν À ν 0 )], which we define as:
K N j2π ν À ν 0
ð
Þ
½
¼
N j2π νÀν 0
ð
Þ
½
N 0
. The last representation allows the solution of the
task on transformation of external pumping block in the laser analog model (Fig. 4.7)
into the laser analog models shown in Fig. 4.13.
The laser analog model in the dipole approximation in the quasi-stationary mode
is presented in Fig. 4.13c. Its peculiarity consists in the fact that the pimping block is
located into a loop (a circuit) of the positive feedback. Restrictions of such a “singleloop” (or single-circuit) laser model are discussed later. To find out solutions of the
symbolic differential equations, we perform the normalization of the laser
symbolic DE.
In the laser analog model presented in Fig. 4.13a, the closed loop of the positive
FB with the active medium (OA) is shown. The optical oscillations of the strength
E L. in act in the OA input. Amplified strength oscillations E L. out act in the optical
output of OA. The laser analog model in the quasi-stationary mode is presented in
Fig. 4.13b with the help of the control operator transfer function K L ( p 00 ) ¼ E L. out /
E L. in and the nonlinear element S(E n ).
4.3.4 Normalized Differential Equations of the Laser
We introduce the operator p 00 ¼ p/(2πν 0n ) and normalized variables: G 0n ¼
G 0
2πν 0n T 1 N 00
, τ ¼ t Á 2πν 0n , T 1n ¼ 2πν 0n T 1 , N n ¼
N
N 00
, α nN0 ¼
α N0
N 00 2πν 0n
and use the
approximation of N n in the vicinity of optical frequencies 2π(ν À ν 0n ):
N n ¼
N 0
1 þ j ν=ν 0F
ð
ÞÀ1
½
Þ T 1n þ T 1n G 00n Re E n Á E
Ã
n
Â
Ã:
ð4:34Þ
Now we introduce the designation of the operator function:
Y p 00
ð Þ¼
1
p 00
2
 p
4
00 þ
1
Q 0F
þ
1
Q 02
p
3
00 þ 2 þ
1
Q 0F
Á
1
Q 02
p
2
00 þ
1
Q 0F
þ
1
Q 02
p 00 þ 1
!
:
ð4:35Þ
After performing of the normalization at β ¼ [(ν 0n À ν 12 )/ν 12 ]
2
% 0.001, the last
expression can be written in the compact form:
4.3 Laser Kinetic Equations and the Pumping System
159