Page 120 - Fiber Optic Communications Fund

Lasers 101

(c) From Eq. (3.20), we have
R stim N 2
= .
R N
abs 1
According to Boltzmann’s law,

N = N e −ℏ∕k B T ,
2 1
( )
R −19
stim −ℏ∕k B T −1.26 × 10
= e = exp
R abs 1.38 × 10 −23 × 300
= 5.29 × 10 −14 .
(d) The population density of the excited level is
N = N e −ℏ∕k B T
2 1
−3
5
= 5.29 × 10 cm .

3.3 Conditions for Laser Oscillations
Consider a lossless gain medium as shown in Fig. 3.8, in which the incident light wave is amplified by stim-
ulated emission. The optical intensity at z can be phenomenologically described as

(z)= (0) exp (gz), (3.31)

where g is the gain coefficient associated with stimulated emission. For the atomic system with two levels, an
expression for g can be obtained in terms of the population densities N , N and the Einstein coefficient B (see
2
1
Section 3.6 for details). By differentiating (z) with respect to z, Eq. (3.31) can be rewritten in differential
form as
d
= g(0) exp (gz)
dz
= g. (3.32)
The optical field is attenuated in the gain medium due to scattering and other possible loss mechanisms
similar to attenuation in optical fibers. The effect of loss is modeled as
(z)= (0) exp (− z), (3.33)
int

I(0) I(L) = I(0)exp(gL)
Gain medium

L

Figure 3.8 Light amplification in a gain medium.