6.4. ERBIUM-DOPED FIBER AMPLIFIERS 253
length because of pump power variations. The total gain is obtained by integrating over
the amplifier length. This feature can be used to realize EDFAs that provide amplifica-
tion in the L band covering the spectral region 1570–1610 nm. The wavelength range
over which an EDFA can provide nearly constant gain is of primary interest for WDM
systems. This issue is discussed later in this section.
6.4.3 Simple Theory
The gain of an EDFA depends on a large number of device parameters such as erbium-
ion concentration, amplifier length, core radius, and pump power [64]–[68]. A three-
level rate-equation model commonly used for lasers [1] can be adapted for EDFAs. It
is sometimes necessary to add a fourth level to include theexcited-state absorption.In
general, the resulting equations must be solved numerically.
Considerable insight can be gained by using a simple two-level model that is valid
when ASE and excited-state absorption are negligible. The model assumes that the
top level of the three-level system remains nearly empty because of a rapid transfer
of the pumped population to the excited state. It is, however, important to take into
account the different emission and absorption cross sections for the pump and signal
fields. The population densities of the two states,N 1 andN 2 , satisfy the following two
rate equations [55]:
∂N 2
∂t=(σapN 1 −σpeN 2 )φp+(σsaN 1 −σseN 2 )φs−N 2
T 1
, (6.4.2)
∂N 1
∂t=(σepN 2 −σapN 1 )φp+(σseN 2 −σsaN 1 )φs+N 2
T 1
, (6.4.3)
whereσajandσejare the absorption and emission cross sections at the frequencyωj
withj=p,s. Further,T 1 is the spontaneous lifetime of the excited state (about 10 ms
for EDFAs). The quantitiesφpandφsrepresent the photon flux for the pump and
signal waves, defined such thatφj=Pj/(ajhνj), wherePjis the optical power,σjis
the transition cross section at the frequencyνj, andajis the cross-sectional area of the
fiber mode forj=p,s.
The pump and signal powers vary along the amplifier length because of absorption,
stimulated emission, and spontaneous emission. If the contribution of spontaneous
emission is neglected,PsandPpsatisfy the simple equations
∂Ps
∂z=Γs(σseN 2 −σsaN 1 )Ps−αPs, (6.4.4)s∂Pp
∂z=Γp(σepN 2 −σapN 1 )Pp−α′Pp, (6.4.5)whereαandα′take into account fiber losses at the signal and pump wavelengths,
respectively. These losses can be neglected for typical amplifier lengths of 10–20 m.
However, they must be included in the case of distributed amplification discussed later.
The confinement factorsΓsandΓpaccount for the fact that the doped region within the
core provides the gain for the entire fiber mode. The parameters=±1 in Eq. (6.4.5)