S L ¼ N 0
E 10L
1 þ T 1n G 00 E
2
10n
À
Á 2 þ 2π ν À ν 0
ð
ÞT 1n
½
2
(
)
Á 1 þ T 1n G 00 E
2
0n
À
Á À j 2π ν À ν 0
ð
ÞT 1n
½
È
É :
ð4:41Þ
Now we obtain the following expression:
S L ¼ N 0
E 10L
1 þ T 1n G 00 E
2
10n
À
Á 2 þ 2π νÀ ν 0
ð
ÞT 1n
½
2
n
o 1=2 exp ÀjArctan
2π νÀ ν 0
ð
ÞT 1n
1 þ T 1n G 00 E
2
0n
!
:
ð4:42Þ
With the aim to ensure notation compactness for S L , we introduce the module S L0 :
S L0 ¼ N 0
E 10L
1þT 1n G 00 E
2
10n
ð
Þ
2 þ 2π νÀν 0
ð
Þ T 1n
½
2
: Then, the formula for S L takes a form:
S L ¼ S L0 Á exp ÀjArctan
2π ν À ν 0
ð
ÞT 1n
1 þ T 1n G 00 E
2
0n
!
:
ð4:43Þ
Having approximated S L by the fifth-order polynomial and taking sequentially the
first and second derivative on E 10L , we obtain the expression for the nonlinearity
slope on the stationary mode for the laser model in the dipole approximation.
For the second derivative (on E 10L ) we obtain after some transformations:
S
00
L ¼ ÀE 10L
6N 0 T 1n G 00
2π ν À ν 0
ð
ÞT 1n
ð
Þ
2 þ 1
h
i 2 1 À 2j2π ν À ν 0
ð
ÞT 1n
½
À E 10L
ð
Þ
3
20N 0 T 1n G 00
ð
Þ
2
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o 3 1 À 3j Á 2π ν À ν 0
ð
ÞT 1n
½
:
ð4:44Þ
The last expression can be presented (using the Euler formula) in the form. Then,
in the compact notation of the expression S
00
L ¼ α 0L E n À β 0L E n Re E n Á E
Ã
n
Â
Ã
, coefficients of the first α 0L and third β 0L powers of the S
00
L polynomial, relatively, are equal:
α 0L ¼
6N 0 T 1n G 00
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o 2
ffiffi ffi
5
p
exp ÀjArctan 4π ν À ν 0
ð
ÞT 1n
½
f
g
ð4:45Þ
and
β 0L ¼
20N 0 T 1n G 00
ð
Þ
2
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o 3
ffiffiffiffiffi
10
p
exp ÀjArctan 6π ν À ν 0
ð
ÞT 1n
½
f
g
ð4:46Þ
In last expressions, arguments of exponential functions represent phase shifts due
to population inertial properties or the carriers’ lifetime on the upper energy level.
164
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .
E 10L
1 þ T 1n G 00 E
2
10n
À
Á 2 þ 2π ν À ν 0
ð
ÞT 1n
½
2
(
)
Á 1 þ T 1n G 00 E
2
0n
À
Á À j 2π ν À ν 0
ð
ÞT 1n
½
È
É :
ð4:41Þ
Now we obtain the following expression:
S L ¼ N 0
E 10L
1 þ T 1n G 00 E
2
10n
À
Á 2 þ 2π νÀ ν 0
ð
ÞT 1n
½
2
n
o 1=2 exp ÀjArctan
2π νÀ ν 0
ð
ÞT 1n
1 þ T 1n G 00 E
2
0n
!
:
ð4:42Þ
With the aim to ensure notation compactness for S L , we introduce the module S L0 :
S L0 ¼ N 0
E 10L
1þT 1n G 00 E
2
10n
ð
Þ
2 þ 2π νÀν 0
ð
Þ T 1n
½
2
: Then, the formula for S L takes a form:
S L ¼ S L0 Á exp ÀjArctan
2π ν À ν 0
ð
ÞT 1n
1 þ T 1n G 00 E
2
0n
!
:
ð4:43Þ
Having approximated S L by the fifth-order polynomial and taking sequentially the
first and second derivative on E 10L , we obtain the expression for the nonlinearity
slope on the stationary mode for the laser model in the dipole approximation.
For the second derivative (on E 10L ) we obtain after some transformations:
S
00
L ¼ ÀE 10L
6N 0 T 1n G 00
2π ν À ν 0
ð
ÞT 1n
ð
Þ
2 þ 1
h
i 2 1 À 2j2π ν À ν 0
ð
ÞT 1n
½
À E 10L
ð
Þ
3
20N 0 T 1n G 00
ð
Þ
2
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o 3 1 À 3j Á 2π ν À ν 0
ð
ÞT 1n
½
:
ð4:44Þ
The last expression can be presented (using the Euler formula) in the form. Then,
in the compact notation of the expression S
00
L ¼ α 0L E n À β 0L E n Re E n Á E
Ã
n
Â
Ã
, coefficients of the first α 0L and third β 0L powers of the S
00
L polynomial, relatively, are equal:
α 0L ¼
6N 0 T 1n G 00
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o 2
ffiffi ffi
5
p
exp ÀjArctan 4π ν À ν 0
ð
ÞT 1n
½
f
g
ð4:45Þ
and
β 0L ¼
20N 0 T 1n G 00
ð
Þ
2
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o 3
ffiffiffiffiffi
10
p
exp ÀjArctan 6π ν À ν 0
ð
ÞT 1n
½
f
g
ð4:46Þ
In last expressions, arguments of exponential functions represent phase shifts due
to population inertial properties or the carriers’ lifetime on the upper energy level.
164
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .