E n ¼
1
ε 0
p
2
p 2 þ
1
T 0F
p þ 2πν 0n
ð
Þ
2
P n
P n ¼ K D Á
1
p 2 þ 1=T 2
ð
Þp þ 2πν 12
ð
Þ
2
N Â E n
pN ¼ α N0 À
N
T 1
À G 00 Á N Â Re E n Á E
Ã
n
Â
Ã
8
> > > > > > > > <
> > > > > > > > :
:
ð4:18Þ
We introduce the operator transfer functions (OTF) K P ( p) и K E ( p):
K E0 p
ð Þ ¼
1
ε 0
p
2
p 2 þ 1=T 2OF
ð
Þp þ 2πν 0n
ð
Þ
2
,
ð4:19Þ
K P0 p
ð Þ ¼ K D Á
1
p 2 þ 1=T 2
ð
Þp þ 2πν 12
ð
Þ
2
:
ð4:20Þ
The total OTF K E Á K P is:
K E Á K P ¼
K D
ε 0
p
2
p 2 þ 1=T 2OF
ð
Þp þ 2πν 0n
ð
Þ
2
Á
1
p 2 þ 1=T 2
ð
Þp þ 2πν 12
ð
Þ
2
, ð4:21Þ
where K D0 ¼ K 00 ¼
2d
2
e
3Áε 0 2πh .
Table 4.1 Variables and parameters
Variable
Essence
Formula
E n
Dimensionless normalized EMF strength of the main
mode E n ¼ E L
E
2
L ¼
c
n0
εε0
2 E
2 % 3 Á 10
À3 E
2
, E [V/m]
T 0F
Time constant of the optical resonator
T OF ¼ 10
À12 – 10
À8 s
ν 0n ¼ ν 0OF Fundamental optical frequency of the resonator
ν 0n ¼ 231 Á 10
12 [1/s]
P n
Polarization of the active material of the main mode
T 2
Time constant of the transverse relaxation
(polarization)
For QWLD T 2 ¼ 10
À12 s
ν 12
The optical frequency of transition
ν 12 % 231 Á 10
12 [1/s]
p e ¼ d e
Dipole moment
d e ¼ p e ¼ 10
À30 [C Á m]
N
Population difference between excited and
non-excited levels
N ¼ 10
16 – 10
18
[1/м
3
]
T 1
Lifetime of excited particles on upper energy level
T 1 ¼ 10
À9 s
α N0
α N0 ¼ N 00 /T 1 , where N 00 is the average population
difference produces by the pumping
ħ ¼ 2πh
ε 0
The Planck constant
Permittivity
ħ ¼ 6.6 Á 10
À34 J Á s
ε 0 ¼ 8.85 Á 10
À12 C
2
/
(N Á m
2
)
144
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .