254 CHAPTER 6. OPTICAL AMPLIFIERS
Figure 6.16: Small-signal gain as a function of (a) pump power and (b) amplifier length for
an EDFA assumed to be pumped at 1.48μm. (After Ref. [64];©c1991 IEEE; reprinted with
permission.)
depending on the direction of pump propagation;s=−1 in the case of a backward-
propagating pump.
Equations (6.4.2)–(6.4.5) can be solved analytically, in spite of their complexity,
after some justifiable approximations [65]. For lumped amplifiers, the fiber length is
short enough that bothαandα′can be set to zero. Noting thatN 1 +N 2 =NtwhereNtis
the total ion density, only one equation, say Eq. (6.4.2) forN 2 , need be solved. Noting
again that the absorption and stimulated-emission terms in the field and population
equations are related, the steady-state solution of Eq. (6.4.2), obtained by setting the
time derivative to zero, can be written as
N 2 (z)=−T 1
adhνs∂Ps
∂z−
sT 1
adhνp∂Pp
∂z, (6.4.6)
wheread=Γsas=Γpapis the cross-sectional area of the doped portion of the fiber
core. Substituting this solution into Eqs. (6.4.4) and (6.4.5) and integrating them over
the fiber length, the powersPsandPpat the fiber output can be obtained in an analyt-
ical form. This model has been extended to include the ASE propagation in both the
forward and backward directions [68].
The total amplifier gainGfor an EDFA of lengthLis obtained using
G=Γsexp[∫L
0(σseN 2 −σsaN 1 )dz]
, (6.4.7)
whereN 1 =Nt−N 2 andN 2 is given by Eq. (6.4.6). Figure 6.16 shows the small-signal
gain at 1.55μm as a function of the pump power and the amplifier length by using
typical parameter values. For a given amplifier lengthL, the amplifier gain initially
increases exponentially with the pump power, but the increase becomes much smaller
when the pump power exceeds a certain value [corresponding to the “knee” in Fig.
6.16(a)]. For a given pump power, the amplifier gain becomes maximum at an optimum
value ofLand drops sharply whenLexceeds this optimum value. The reason is that
the latter portion of the amplifier remains unpumped and absorbs the amplified signal.