Rami Arieli: "The
Laser Adventure" Chapter 2.8 page 1
2.8 Rate Equations for spontaneous
(Advanced level only, need to know first order
For simplicity we shall assume:
1. The material is composed of many identical atoms.
The rate at which the excited atom population N2(t) decays
from the higher energy level (E2) to the lower energy
level (E1), by stimulated emission is given by the decay
coefficient g21 multiplied by the instantaneous
population number of this level N2(t) as seen by the
first order differential equation:
d N2(t)/dt = - g21 N2(t)
= - N2(t)/ t2
2. Each atom has only two energy levels: E1 and E2.
3. The only decay mechanism of energy level E2 is
4. At time t N1 atoms are in energy level E1,
and N2 atoms in energy level E2.
This equation defines the lifetime
t2 of energy level E2:
t2 = 1/
(For a reminder about the solution for a differential
equation of this kind, Click
The solution to the rate (differential) equation is:
N2(t) = N2(0) exp(-g21t)
= N2(0) exp(-t/ t2)
check by substitution that the solution is valid.