Page 296 - Fiber Optic Communications Fund

Optical Amplifiers 277

if the two levels are combinations of sublevels that are populated to various extents depending on the thermal
distribution [1]. It is straightforward to write the rate equations for population densities N and N as
2
1
dN 2 N 3 N 2
= −(N − N ) − , (6.172)
2 21
1 12
s
dt 32 21
dN N
1 =(N − N ) +(N − N ) + 2 , (6.173)
dt 2 21 1 12 s 3 31 1 13 p
21
where is the photon flux density of the signal. Consider the rate equation (6.171). As soon as the pump
s
photons cause transition from level 1 to level 3, erbium ions relax to level 2 by the non-radiative processes
involving interaction with phonons of the glass matrix. Therefore, the second term N in Eq. (6.171)
3 31 p
that accounts for the stimulated emission can be ignored. Under steady-state conditions, dN ∕dt = 0 and from
3
Eq. (6.171), we obtain
N 3
=(N − N ) ≈ N . (6.174)
1 13
p
1 13 p
3 31

32
Substituting Eq. (6.174) into Eqs. (6.172) and (6.173) and ignoring N , we obtain [1]
3 31 p
dN N
2 2
= N − −(N − N ) , (6.175)
1 12
s
1 13 p
2 21
dt 21
dN 1 N 2
=(N − N ) − N + . (6.176)
1 13 p
2 21
s
1 12
dt 21
Note that Eq. (6.175) is similar to Eq. (3.89), corresponding to the two-level system with R pump = N .
1 13 p
The erbium ions are excited to level 2 from level 1 by an alternate route, i.e., first they make an upward
transition to level 3 from level 1 by absorbing pump photons and they relax to level 2 by means of non-radiative
processes. If the population inversion is achieved ( N > N ), the energy of the pump is transferred to
12 1
21 2
the signal.
Adding Eqs. (6.175) and (6.176), we find
d(N + N )
1 2 = 0or N + N = N (a constant). (6.177)
dt 1 2 T
Here, N denotes the erbium ion density. The steady-state solution of Eqs. (6.175) and (6.176) can be obtained
T
by setting
dN 1 dN 2
= = 0. (6.178)
dt dt
Using Eq. (6.177) in Eq. (6.175), we obtain under the steady-state condition
N [ + ] 21
12 s
13 p
T
N = . (6.179)
2
1 + +( + )
s 21
13 21 p
21
12
The photon flux density and optical power P are related by Eq. (6.164),
p
p
 P p
= = , (6.180)
p
ℏ p A ℏ p
eff
where A is the cross-section of the erbium ion distribution. Similarly, we have
eff
P s
= . (6.181)
s
A ℏ
eff s