Rami Arieli: "The
Laser Adventure" Chapter 2.10 page 1
2.10 Amplification
We saw in the rate equation that:

Two processes decrease the population number of the excited level: Spontaneous
emission and stimulated emission.

One process increases the population number of the excited state  absorption.
Since the same amount of energy (hn)
is involved in every transition up or down, than the rate at which energy
is absorbed in a unit volume of atoms is given by the transition rate times
the unit of energy:
dU_{a}dt = Kn(t)[N_{1}(t)N_{2}(t)]*hn
U_{a} = energy density in stimulated transitions.
Question 2.6:
Compare this equation to the rate equation. What is the difference?
explain why.
The incoming energy
is:
U_{signal}(t) = n(t)*hn
The absorbed energy is taken out of the incoming signal, so the rate
of loss of energy from the incoming signal is:
dU_{signal}(t)/dt = K*n(t)*[N_{1}(t)N_{2}(t)]*hn
= K*[N_{1}(t)N_{2}(t)]* U_{signal}(t)
The same equation can be written for the photon
density:
dn(t)/dt = K* [N_{1}(t)N_{2}(t)]*n(t)