1
ε 0
2d
2
e
3 Á 2πh
p
2 N Á E n ¼
1
ε 0
2d
2
e N 00
3 Á 2πh
p
2 N
N 00
Á E n ¼ K α p
2 N
N 00
Á E n
¼ K α p
2 S L N Á E n
ð
Þ:
ð4:30Þ
At values N 00 ¼ 10
+16
– 10
+17 m
À3 and d e ¼ 1.2 Á 10
À30 [C Á m], the gain, which
is defined by the difference of N 0 and the dipole moment d e , is approximately equal:
K ¼
2Á2d
2
e
ε 0 2πh N 00 ¼
2Á2Á 1:2Á10
À30
ð
Þ
2 Á10
þ16 À10
þ17 Á10
þ17
8:85Á10
À12 Á6Á6:6Á10
À34
% 1 À 10 . Now we obtain for N the
formal notation at p ¼ 0. We call the extracted nonlinear function S L (E n ) ¼ N Á E n
as the linear-hyperbolic function of the nonlinear element of the laser model. For it,
the following formula is true:
S L E n
ð Þ ¼ N Á E n ¼
T 1 α N0 E n
1 þ T 1 G 00 Á Re E n E
Ã
n
Â
ü
T 1 α N0 E n
1 þ T 1 G 00 Á E 0n
ð Þ
2
:
ð4:31Þ
Figures 4.11 and 4.12 show the functions of population difference N versus E n , as
well as its derivative for different values of pumping K.
Presented functions of the amplitude E 0n (a), the product N Á E 0n and the
derivative (N Á E n )
0 permit the determination of oscillation amplitude in the steadystate mode.
Figure 4.12 shows the functions of the product N Á E n in the laser versus the
amplitude E n for different values of pumping K.
From Figs. 4.11 and 4.12, we see that steady-state points of the generator (the
laser model) correspond to interception points of plots E n and N Á E n . At weak
pumping, i.e., when the threshold inversed population does not achieve, these
interception points are absent. At pumping growth, on the branch, where the slope
of the nonlinear function N Á E n is less than zero, functions have the interception
point. From the plot, it is clear that one of conditions of laser generation existence
25
5
4
3
2
1 2.4
27.2
K = 43.2
S'(E n )
-1
2
4
6
8
1 0
20
15
10
5
2
4
6
a)
b)
8
1 0
E n
E n
N · E n
N, N · E n
N
Fig. 4.11 Functions of population difference N and the product N Á E n versus amplitude E n as the
function of amplitude E n (a), and the derivative (N Á E n )
0 (the slope of the nonlinear characteristic)
for different values of pumping (b)
156
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .
ε 0
2d
2
e
3 Á 2πh
p
2 N Á E n ¼
1
ε 0
2d
2
e N 00
3 Á 2πh
p
2 N
N 00
Á E n ¼ K α p
2 N
N 00
Á E n
¼ K α p
2 S L N Á E n
ð
Þ:
ð4:30Þ
At values N 00 ¼ 10
+16
– 10
+17 m
À3 and d e ¼ 1.2 Á 10
À30 [C Á m], the gain, which
is defined by the difference of N 0 and the dipole moment d e , is approximately equal:
K ¼
2Á2d
2
e
ε 0 2πh N 00 ¼
2Á2Á 1:2Á10
À30
ð
Þ
2 Á10
þ16 À10
þ17 Á10
þ17
8:85Á10
À12 Á6Á6:6Á10
À34
% 1 À 10 . Now we obtain for N the
formal notation at p ¼ 0. We call the extracted nonlinear function S L (E n ) ¼ N Á E n
as the linear-hyperbolic function of the nonlinear element of the laser model. For it,
the following formula is true:
S L E n
ð Þ ¼ N Á E n ¼
T 1 α N0 E n
1 þ T 1 G 00 Á Re E n E
Ã
n
Â
ü
T 1 α N0 E n
1 þ T 1 G 00 Á E 0n
ð Þ
2
:
ð4:31Þ
Figures 4.11 and 4.12 show the functions of population difference N versus E n , as
well as its derivative for different values of pumping K.
Presented functions of the amplitude E 0n (a), the product N Á E 0n and the
derivative (N Á E n )
0 permit the determination of oscillation amplitude in the steadystate mode.
Figure 4.12 shows the functions of the product N Á E n in the laser versus the
amplitude E n for different values of pumping K.
From Figs. 4.11 and 4.12, we see that steady-state points of the generator (the
laser model) correspond to interception points of plots E n and N Á E n . At weak
pumping, i.e., when the threshold inversed population does not achieve, these
interception points are absent. At pumping growth, on the branch, where the slope
of the nonlinear function N Á E n is less than zero, functions have the interception
point. From the plot, it is clear that one of conditions of laser generation existence
25
5
4
3
2
1 2.4
27.2
K = 43.2
S'(E n )
-1
2
4
6
8
1 0
20
15
10
5
2
4
6
a)
b)
8
1 0
E n
E n
N · E n
N, N · E n
N
Fig. 4.11 Functions of population difference N and the product N Á E n versus amplitude E n as the
function of amplitude E n (a), and the derivative (N Á E n )
0 (the slope of the nonlinear characteristic)
for different values of pumping (b)
156
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .