Rami Arieli: "The Laser Adventure" Chapter 2.10 page 1
2.10 Amplification

We saw in the rate equation that:

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:
dUadt = Kn(t)[N1(t)-N2(t)]*hn

Question 2.6:

Compare this equation to the rate equation. What is the difference? explain why.

The incoming energy is:

Usignal(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:

dUsignal(t)/dt = -K*n(t)*[N1(t)-N2(t)]*hn = -K*[N1(t)-N2(t)]* Usignal(t)

The same equation can be written for the photon density:

dn(t)/dt = -K* [N1(t)-N2(t)]*n(t)