0 ¼
E 10L K 0L N 0 T 1n G 00
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o 2 1 À E 10L
ð
Þ
2
20 T 1n G 00
ð
Þ
6 2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o
2
4
3
5
À E 10L :
ð4:49Þ
The equation for the frequency amendment takes the following form:
0 ¼ À2πν 0F
K 0L N 0
Q 12 Q 0F
∂
2
∂E
E 10L 2π ν À ν 0F
ð
ÞT 1n
½
1 þ T 1n G 00 E
2
0n
À
Á 2 þ 2π ν À ν 0F
ð
ÞT 1n
½
2
(
)
:
ð4:50Þ
Now we simplify these equations substituting in them the approximate expression
for the nonlinearity in the form of the third power polynomial. In this case, the
equation under the analysis takes a form:
0 ¼ E 10L
6N 0 T 1n G 00 j4π ν À ν 0
ð
ÞT 1n
½
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o 2
À E 10L
ð
Þ
3 20N 0 T 1n G 00
ð
Þ
2 j6π ν À ν 0
ð
ÞT 1n
½
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o 3
:
ð4:51Þ
For values close to ν ¼ ν 0 , we obtain for the amplitude and the frequency
amendment the following formulas:
E 10L
ð
Þ
2 ¼
1
3:3 T 1n G 00
ð
Þ
1 À
1
K 0L N 0 T 1n G 00
!
,
ð4:52Þ
ν À ν 0n
ð
Þ¼
1
2πT 1n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
5 Á E 10L
ð
Þ
2 T 1n G 00
ð
ÞÀ1
q
ð4:53Þ
These formulas coincide with the calculated functions obtained at computer
modeling. From Eq. (4.52) we deduce the inequality: K 0L N 0 T 1n G 00 > 1. Equation
for the frequency amendment has the real solution for: E 10L
ð
Þ
2 >
1
5Á T 1n G 00
ð
Þ . This
condition is satisfied in the steady-state point at
dS E 10n
ð
Þ
dE 10n
< 0. For the linear-hyperbolic
nonlinearity of the form S E n
ð Þ ¼
E 10n
1þT 1n G 00 ÁE
2
10n
, the derivative value S
0 (E n ) should
satisfy to the inequality: S
0 E n
ð Þ ¼
1ÀT 1n G 00 E
2
10n
1þT 1n G 00 ÁE
2
10n
ð
Þ
2 < 0. From last it follows that in the
steady-state point E
2
10n >
1
T 1n G 00
.
Our analysis of the linear-hyperbolic function of the nonlinear laser model allows
performing of its approximation with the help of polynomial of third and fifth power
taking into account the inertial properties (or the lifetime of carriers). The transfer to
approximate expressions for nonlinearity essentially simplifies the deduction of
168
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .
E 10L K 0L N 0 T 1n G 00
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o 2 1 À E 10L
ð
Þ
2
20 T 1n G 00
ð
Þ
6 2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o
2
4
3
5
À E 10L :
ð4:49Þ
The equation for the frequency amendment takes the following form:
0 ¼ À2πν 0F
K 0L N 0
Q 12 Q 0F
∂
2
∂E
E 10L 2π ν À ν 0F
ð
ÞT 1n
½
1 þ T 1n G 00 E
2
0n
À
Á 2 þ 2π ν À ν 0F
ð
ÞT 1n
½
2
(
)
:
ð4:50Þ
Now we simplify these equations substituting in them the approximate expression
for the nonlinearity in the form of the third power polynomial. In this case, the
equation under the analysis takes a form:
0 ¼ E 10L
6N 0 T 1n G 00 j4π ν À ν 0
ð
ÞT 1n
½
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o 2
À E 10L
ð
Þ
3 20N 0 T 1n G 00
ð
Þ
2 j6π ν À ν 0
ð
ÞT 1n
½
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o 3
:
ð4:51Þ
For values close to ν ¼ ν 0 , we obtain for the amplitude and the frequency
amendment the following formulas:
E 10L
ð
Þ
2 ¼
1
3:3 T 1n G 00
ð
Þ
1 À
1
K 0L N 0 T 1n G 00
!
,
ð4:52Þ
ν À ν 0n
ð
Þ¼
1
2πT 1n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
5 Á E 10L
ð
Þ
2 T 1n G 00
ð
ÞÀ1
q
ð4:53Þ
These formulas coincide with the calculated functions obtained at computer
modeling. From Eq. (4.52) we deduce the inequality: K 0L N 0 T 1n G 00 > 1. Equation
for the frequency amendment has the real solution for: E 10L
ð
Þ
2 >
1
5Á T 1n G 00
ð
Þ . This
condition is satisfied in the steady-state point at
dS E 10n
ð
Þ
dE 10n
< 0. For the linear-hyperbolic
nonlinearity of the form S E n
ð Þ ¼
E 10n
1þT 1n G 00 ÁE
2
10n
, the derivative value S
0 (E n ) should
satisfy to the inequality: S
0 E n
ð Þ ¼
1ÀT 1n G 00 E
2
10n
1þT 1n G 00 ÁE
2
10n
ð
Þ
2 < 0. From last it follows that in the
steady-state point E
2
10n >
1
T 1n G 00
.
Our analysis of the linear-hyperbolic function of the nonlinear laser model allows
performing of its approximation with the help of polynomial of third and fifth power
taking into account the inertial properties (or the lifetime of carriers). The transfer to
approximate expressions for nonlinearity essentially simplifies the deduction of
168
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .