Page 295 - Fiber Optic Communications Fund

276 Fiber Optic Communications

If a light wave of energy E crosses the area A over the time interval Δt, the optical intensity is
p
E p
 = . (6.163)
p
AΔt
Since the energy E = n ℏ , where is the frequency of the pump wave, we find
p p p p
 p
= . (6.164)
p
ℏ p
Using Eqs. (6.161) and (6.164) in Eq. (6.160), we find

R abs = N , (6.165)
13 1 p
where
ℏ B
p 13
13 = (6.166)

is known as the absorption cross-section associated with the transition from level 1 to level 3. The physical
meaning of is as follows. The optical power absorbed by an erbium ion is proportional to the optical
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intensity  of the light wave incident,
p
P = k , (6.167)
abs p
where k is a constant of proportionality that depends on the medium. Since P abs ∕ℏ is the number of photons
p
absorbed per unit time by an erbium ion (photon absorption rate) and  ∕ℏ is the photon flux density
p
p
(Eq. (6.164)), dividing Eq. (6.167) by ℏ , we find
p
P
abs
= k . (6.168)
p
ℏ p
If there are N erbium ions per unit volume in the ground state, the total absorption rate is
1
R abs = kN , (6.169)
1 p
which is the same as Eq. (6.165) if k = . Thus, the absorption cross-section can be imagined as an effective
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area that “captures” a fraction of the incident photons [1]. Similarly, the stimulated emission rate from level
3tolevel 1isgiven by
R =− N , (6.170)
stim 31 3 p
where = B ℏ ∕ is the cross-section associated with the transition from j to k and is the energy
jk
jk
jk
jk
difference between the levels j and k. Since the transition from level 3 to level 2 is mostly non-radiative,
absorption and stimulated emission between level 3 and level 2 can be ignored. Using Eqs. (6.165) and (6.170)
in Eq. (6.159), we find
dN 3 N 3
=( N − N ) − . (6.171)
31 3
p
13 1
dt 32
In the case of the two-level atomic system discussed in Chapter 3, Section 3.2, we found that B 12 = B , which
21
implies that the emission and absorption cross-section are equal. However, in general, they could be different