Page 146 - Fiber Optic Communications Fund

Lasers 127

than the surrounding layers and, therefore, it acts as a waveguide. The tails of an optical mode extend well
into the surrounding regions, but they do not contribute to the photon density in the active region. Since the
electron–hole recombination by photon emission depends on the photon density in the active region, we
introduce a confinement factor Γ,
G =Γg, (3.109)
where Γ is the ratio of optical power in the active region to total optical power carried by the mode.
Let us consider the growth of photons due to stimulated emission alone. Eqs. (3.107) and (3.108) become
dN
e
=−GN , (3.110)
ph
dt
dN
ph
= GN . (3.111)
ph
dt
Adding Eqs. (3.110) and (3.111), we find
d(N + N )
e ph
= 0 (3.112)
dt
or
N + N = Const. (3.113)
e ph
This implies that the total number of electrons and photons is conserved under these conditions. In other
words, if you lose 10 electrons per unit volume per unit time by recombination, you gain 10 photons per unit
volume per unit time.
Now, let us find an expression for R . The electrons and holes are consumed by stimulated emission.
pump
Therefore, the external power supply should inject electrons continuously. The current is
n q
e
I = , (3.114)
T
where n is the number of electrons, q is the electron charge = 1.602 × 10 −19 C, and T is the time interval.
e
The number of electrons crossing the active region per unit time is
n
e I
= . (3.115)
T q
The above equation gives the electron pumping rate. We divide it by the volume of the active region to obtain
the electron pumping rate per unit volume,
n e I
R pump = = , (3.116)
TV qdL
where d, , and L are thickness, width, and length of the active layer, respectively, as shown in Fig. 3.35(b).
Using Eq. (3.116) in Eqs. (3.107) and (3.108), we find

dN N
e I e
= − GN − , (3.117)
ph
dt qV e
dN ph N ph
= GN + R − . (3.118)
sp
ph
dt ph
In the case of an atomic system, we have derived an expression for G (see Eq. (3.91)). But in the case of a
semiconductor laser, it is hard to find an exact analytical expression for the gain coefficient g. Instead, we use