Page 131 - Fiber Optic Communications Fund

112 Fiber Optic Communications

Since u ∝ N , the time rate of change of photon density is
ph
dN ph N ph
= GN − , (3.78)
ph
dt
ph
where
1
= (3.79)
ph

cav
is the photon lifetime. In the absence of gain (G = 0), Eq. (3.78) can be solved to yield
N (t)= N (0) exp (−t∕ ). (3.80)
ph
ph
ph
−1
At t = t , N (t)= N (0)e . Thus, the photon density reduces by e over a time t . In Eq. (3.78), G repre-
ph ph ph ph
sents the net gain coefficient due to stimulated emission and absorption and, therefore, the first term on the
right-hand side of Eq. (3.78) can be identified as
R stim + R abs = GN , (3.81)
ph
or
BuN − BuN = GN . (3.82)
ph
2
1
Since u = N ℏ, from Eq. (3.82) we find
ph
G = B(N − N )ℏ. (3.83)
2 1
In Eq. (3.78), the second term represents the loss, rate due to scattering, mirror loss, and other possible loss
mechanisms,
N ph
R =− . (3.84)
loss

ph
Eq. (3.78) does not include the photon gain rate due to spontaneous emission. Using Eqs. (3.81), (3.84), and
(3.4) in Eq. (3.74), we find
dN N
ph ph
= GN + AN − . (3.85)
ph
2
dt ph
Note that when N > N , population inversion is achieved, G > 0 (see Eq. (3.83)) and amplification of photons
2
1
takes place. In other words, the energy of the atomic system is transferred to the electromagnetic wave. When
N < N , the electromagnetic wave is attenuated and the energy of the wave is transferred to the atomic system.
1
2
Using Eq. (3.82), Eqs. (3.72) and (3.73) can be rewritten as
dN 2 N 2
= R pump − GN − , (3.86)
ph
dt 21
dN N
1 = GN + 2 , (3.87)
dt ph
21
where
1
21 = (3.88)
A + C
is the lifetime associated with spontaneous emission and non-radiative decay from the excited state to the
ground state. Eqs. (3.86) and (3.87) can be simplified further under the assumption that the population density