Page 276 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Electron-Photon
is called the inversion and we have added a phe-
N –
where D =
N
2
1
γ
nomenological damping term with damping constant the inverse of the
so called dipole decay time. For the inversion we obtain an equation of
motion of the following form
2 D – D 0
˙
(
D = ----- α∗ t()V 21 – α t()V ) – ----------------- (7.86)
12
i— T
where we also have introduced a phenomenological decay time and an
T
equilibrium inversion D which is given by the temperature distribution
0
of the carriers.
We formally integrate (7.85) and make use of the rotating wave approxi-
mation to give
t
— ∫ 0 21
(
γ
d
t'
α t() = --- i exp ( – ( iω + ) t – t'))V ()Dt'() t' (7.87)
– ∞
We now assume that the dipole decay time 1 γ⁄ is much smaller than
D
times in which the inversion or the electric field amplitude E change.
0
We therefore put these quantities outside the integration and obtain
µE 0 1
α t() = – i-----------------------------------------D (7.88)
2— γ + i ω –( ω)
0
Insert this in (7.86) and we have
D – D 0 γ 2 I
˙
D = – ----------------- – ---------------------------------------D (7.89)
2
T γ + ( ω – ω) 2 I s
0
2 2 – 1
⁄
where I = cε E 0 is the light field intensity and I = cε — µ T is
0
0
s
the saturation intensity.
2
We have to find an equation for the laser field intensity, i.e., the E 0 that
is generated by the inversion. This can be achieved by inserting the polar-
ization generated from the inversion density into the amplitude equation

Semiconductors for Micro and Nanosystem Technology 273