Page 549 - Fiber Optic Communications Fund

530 Fiber Optic Communications

Next we consider the photon net gain rate due to absorption and spontaneous/stimulated emission. In this
analysis, we ignore the loss of photons due to scattering and other possible mechanisms. From Eqs. (3.74)
and (3.81), we have
dN
ph
= R stim + R abs + R spont
dt
= ℏBN (N − N )+ AN . (A.20)
2
ph
2
1
3
Let n be the number of photons in the volume L ,
ph
3
n ph = N L . (A.21)
ph
Using Eqs. (A.21) and (A.19), Eq. (A.20) may be rewritten as
dn
ph
= ℏBn (N − N )+ ℏBN . (A.22)
2
2
ph
1
dt
Using = dz∕dt and simplifying Eq. (A.22), we obtain
dn ph ℏBdz
= . (A.23)
(N − N )n + N
2 1 ph 2
Eq. (A.23) can be rewritten as
dn
ph ℏB(N − N )dz
2
1
= , (A.24)
n + n
ph sp
where
N 2
n = (A.25)
sp
N − N
2 1
is known as the spontaneous emission factor or population-inversion factor. For an amplifier, N > N and,
1
2
therefore, n ≥ 1. Integrating Eq. (A.24) from 0 to L, we find
sp
ln [n (L)+ n ]− ln [n (0)+ n ]= gL, (A.26)
sp
sp
ph
ph
ℏB(N − N )
1
2
g = (A.27)

or
n (L)= n (0) exp(gL)+ n [exp (gL)− 1]. (A.28)
ph ph sp
Since G = exp (gL), Eq. (A.28) can be written as
n (L)= n (0)G + n (G − 1). (A.29)
sp
ph
ph
Eq. (A.29) is of fundamental significance. The first and second terms on the right hand side represent the
photon gain due to stimulated emission and spontaneous emission, respectively.
Next, let us consider the average noise power due to spontaneous emission. A photon of energy ℏ is
0
assumed to occupy a length L or equivalently time L∕[1, 2]. The noise power of a photon is
ℏ 0
P = . (A.30)
0
L∕