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128 Fiber Optic Communications

Metal contact Current, I
Dielectric layer Current direction
Active region
d
p-type
w
n-type
L

Active region Cleaved facet

(a) (b)

Figure 3.35 (a) Forward-biased heterojunction laser. (b) Active region.

the following approximation [16]:
g = (N − N ), (3.119)
g e e0
where and N e0 are parameters that depend on the specific design. is called the gain cross-section and
g
g
N is the value of the carrier density at which the gain coefficient becomes zero. Using Eq. (3.109), we find
e0
G =Γg = G (N − N ) (3.120)
e
e0
0
where
G =Γ . (3.121)
g
0
3.8.4 Steady-State Solutions of Rate Equations
Eqs. (3.117) and (3.118) describe the evolution of electron density and photon density in the active region,
respectively. In general, they have to be solved numerically on a computer. However, the steady-state solution
can be found analytically under some approximations. First, we ignore the spontaneous emission rate since it
is much smaller than the stimulated emission rate for a laser. Second, we use Eq. (3.120) for the gain, which
is an approximation to the calculated/measured gain. Now, Eqs. (3.117) and (3.118) become
dN ph N ph
= GN − , (3.122)
ph
dt ph
dN N
e e I
=−GN − + . (3.123)
ph
dt e qV
We assume that the current I is constant. Under steady-state conditions, the loss of photons due to cavity loss
is balanced by the gain of photons due to stimulated emission. As a result, the photon density does not change
as a function of time. Similarly, the loss of electrons due to radiative and non-radiative transitions is balanced
by electron injection from the battery. So, the electron density does not change with time too. Therefore, under
steady-state conditions, the time derivatives in Eqs. (3.122) and (3.123) can be set to zero,
dN
ph dN e
= = 0. (3.124)
dt dt