6.1. BASIC CONCEPTS 227
gain coefficientof such a medium can be written as [1]
g(ω)=
g 0
1 +(ω−ω 0 )^2 T 22 +P/Ps, (6.1.1)
whereg 0 is the peak value of the gain,ωis the optical frequency of the incident signal,
ω 0 is the atomic transition frequency, andPis the optical power of the signal being
amplified. The saturation powerPsdepends on gain-medium parameters such as the
fluorescence timeT 1 and the transition cross section; its expression for different kinds
of amplifiers is given in the following sections. The parameterT 2 in Eq. (6.1.1), known
as thedipole relaxation time, is typically quite small (<1 ps). The fluorescence timeT 1 ,
also called thepopulation relaxation time, varies in the range 100 ps–10 ms, depending
on the gain medium. Equation (6.1.1) can be used to discuss important characteristics
of optical amplifiers, such as the gain bandwidth, amplification factor, and output satu-
ration power.
6.1.1 Gain Spectrum and Bandwidth.................
Consider the unsaturated regime in whichP/Ps1 throughout the amplifier. By ne-
glecting the termP/Psin Eq. (6.1.1), the gain coefficient becomes
g(ω)=g 0
1 +(ω−ω 0 )^2 T 22. (6.1.2)
This equation shows that the gain is maximum when the incident frequencyωcoincides
with the atomic transition frequencyω 0. The gain reduction forω=ω 0 is governed
by a Lorentzian profile that is a characteristic of homogeneously broadened two-level
systems [1]. As discussed later, the gain spectrum of actual amplifiers can deviate con-
siderably from the Lorentzian profile. The gain bandwidth is defined as the full width
at half maximum (FWHM) of the gain spectrumg(ω). For the Lorentzian spectrum,
the gain bandwidth is given by∆ωg= 2 /T 2 ,orby
∆νg=∆ωg
2 π=
1
πT 2. (6.1.3)
As an example,∆νg∼5 THz for semiconductor optical amplifiers for whichT 2 ∼60 fs.
Amplifiers with a relatively large bandwidth are preferred for optical communication
systems because the gain is then nearly constant over the entire bandwidth of even a
multichannel signal.
The concept ofamplifier bandwidthis commonly used in place of the gain band-
width. The difference becomes clear when one considers the amplifier gainG, known
as theamplification factorand defined as
G=Pout/Pin, (6.1.4)wherePinandPoutare the input and output powers of the continuous-wave (CW) signal
being amplified. We can obtain an expression forGby using
dP
dz=gP, (6.1.5)