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Optical Amplifiers 275

component amplified. This corresponds to the very narrow (impulse) gain spectrum. However, level 2 is a
collection of sublevels. Broadening of the energy levels occurs when the erbium ions are incorporated into
the glass of optical fibers and, thus, the gain spectrum is also broadened. This broadening is both homogeneous
(all erbium ions exhibit the identical broadened spectrum) and inhomogeneous (different ions in different glass
locations exhibit different spectra). Homogeneous broadening is due to the interactions with photons of the
glass, whereas inhomogeneous broadening is caused by differences in the glass sites where different ions
are hosted.

6.7.2 Rate Equations ∗
Consider a three-level system as shown in Fig. 6.16. An optical pump beam of frequency causes upward
p
transitions from level 1 to level 3. Let the population densities of the level j be N , j = 1, 2, 3. The erbium
j
ions excited to level 3 relax to level 2 by spontaneous emission and non-radiative processes. In practice, it
is mostly non-radiative. The stimulated emission occurring between level 2 and level 1 is responsible for
the signal amplification. Let the lifetime associated with spontaneous emission and non-radiative processes
between any levels j and k be . First consider the gain and loss rates for level 3. The population density of
jk
level 3 increases because of the net absorption of pump photons and it decreases because of the non-radiative
emission,
dN
3
= R abs + R stim + R + R . (6.159)
sp
nr
dt
Consider the pump absorption. From the Einstein relation (see Eqs. (3.2) and (3.30)), we have
R = B N u , (6.160)
abs 13 1 p
where u is the energy density of the pump. We will write Eq. (6.160) in a slightly different form. The pump
p
intensity  and energy density are related by (Eq. (3.50))
p
 p
u = , (6.161)
p

where is the speed of light in the medium. The photon flux density is defined as the mean number of photons
per unit area per unit time. In other words, if n photons cross the area A over the time interval Δt, the photon
p
flux density is
n p
= . (6.162)
p
AΔt
Level 3

Non-radiative transition

Level 2

ħω p ħω s

Ground level 1

Figure 6.16 A three-level system. Signal amplification in erbium-doped fiber.