228 CHAPTER 6. OPTICAL AMPLIFIERS
Figure 6.1: Lorentzian gain profileg(ω)and the corresponding amplifier-gain spectrumG(ω)
for a two-level gain medium.
whereP(z)is the optical power at a distancezfrom the input end. A straightforward
integration with the initial conditionP( 0 )=Pinshows that the signal power grows
exponentially as
P(z)=Pinexp(gz). (6.1.6)By noting thatP(L)=Poutand using Eq. (6.1.4), the amplification factor for an ampli-
fier of lengthLis given by
G(ω)=exp[g(ω)L], (6.1.7)where the frequency dependence of bothGandgis shown explicitly. Both the amplifier
gainG(ω)and the gain coefficientg(ω)are maximum whenω=ω 0 and decrease with
the signal detuningω−ω 0. However,G(ω)decreases much faster thang(ω). The
amplifier bandwidth∆νAis defined as the FWHM ofG(ω)and is related to the gain
bandwidth∆νgas
∆νA=∆νg[
ln 2
ln(G 0 / 2 )] 1 / 2
, (6.1.8)
whereG 0 =exp(g 0 L). Figure 6.1 shows the gain profileg(ω)and the amplification
factorG(ω)by plottingg/g 0 andG/G 0 as a function of(ω−ω 0 )T 2. The amplifier
bandwidth is smaller than the gain bandwidth, and the difference depends on the am-
plifier gain itself.