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98 Fiber Optic Communications

where
k T
B
= . (3.17)
c
ℏ
Typically for an optical source, > . Therefore, spontaneous emission dominates stimulated emission.
c
As an example, consider an optical source at temperature T = 300 K:
k T 1.38 × 10 −23 × 300
B
13
= = = 3.92 × 10 rad∕s. (3.18)
c
ℏ 1.054 × 10 −34
13
For > 3.92 × 10 rad∕s, radiation would mostly be due to spontaneous emission. If the operating wave-
15
length of the optical source is 700 nm, = 2.69 × 10 rad/s and
R ( 15 )
spont 2.69 × 10 29
= exp − 1 ≃ 6.34 × 10 . (3.19)
R stim 3.92 × 10 13
The above equation indicates that, on average, one out of 6.34 × 10 29 emissions is a stimulated emission.
Thus, at optical frequencies, the emission is mostly due to spontaneous emission and hence the light from the
usual light sources is not coherent. From Eqs. (3.2) and (3.3), we see that
R N
stim 2
= . (3.20)
R abs N 1
Therefore, the stimulated emission rate exceeds the absorption rate only when N > N . This condition is
2 1
called population inversion. For systems in thermal equilibrium, from Eq. (3.10), we find that N is always
2
less than N and population inversion can never be achieved. Therefore, all lasers should operate away from
1
thermal equilibrium. To achieve population inversion, atoms should be pumped to the excited state by means
of an external energy source known as a pump. A flash pump could act as an optical pump and atoms are
excited into higher-energy states through absorption of the pump energy. Alternatively, an electrical pump
can be used to achieve population inversion as discussed in Section 3.8.
The photons generated by stimulated emission have the same frequency, phase, direction, and polariza-
tion as the incident light. In contrast, the spontaneous emission occurs randomly in all directions and both
polarizations, and often acts as noise. In lasers, we like to maximize the stimulated emission by achieving
population inversion.
The Einstein coefficient A is related to the spontaneous emission lifetime associated with state 2 to state 1
transition. Let us consider a system in which stimulated emission is negligible and atoms in the excited state
spontaneously emit photons and return to the ground state. Considering only spontaneous emission, the decay
rate of the excited level is given by Eq. (3.4),
dN 2
=−A N . (3.21)
21 2
dt
The solution of Eq. (3.21) is
N (t)= N (0) exp (−t∕t ), (3.22)
2 2 sp
where
1
t = . (3.23)
sp
A 21
−1
At t = t , N (t)= N (0)e . Thus, the population density of level 2 reduces by e over a time t which is
sp 2 2 sp
known as the spontaneous lifetime associated with 2 → 1 transition.
So far we have assumed that the energy levels are sharp, but in reality these levels consist of several
sublevels or bands. The spectrum of the electromagnetic waves due to spontaneous or stimulated emission