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Lasers 95

where R stim is the stimulated emission rate that is equal to the number of atoms undergoing stimulated emission
per unit time per unit volume, and B 21 is a constant.
(c) An atom in the excited state of energy E can also make a spontaneous emission and return to the ground
2
state whether or not the radiation at frequency is present, as illustrated in Fig. 3.4. This occurs randomly and
is called spontaneous emission. The rate of spontaneous emission is independent of the intensity of incident
radiation. Therefore,
( )
dN 2
R spont ≡ − = A N , (3.4)
21 2
dt
spont
where A 21 is a constant and R spont is the spontaneous emission rate.
At thermal equilibrium between the atomic system and the radiation field, the number of upward transitions
must be equal to the number of downward transitions. If not, for example if downward transition occurs more
frequently than upward transition, this would result in an increase in radiation with time, which implies it is
not in equilibrium. At thermal equilibrium, we have

R = R down , (3.5)
up
R abs = R stim + R spont , (3.6)
B u ()N = B u ()N + A N , (3.7)
12 s
2
21 s
1
21 2
A 21
u ()= . (3.8)
s
(N ∕N )B − B
1 2 12 21
According to Boltzmann’s law, the ratio of populations of level 1 and level 2 at equilibrium is
N
2
= exp (−ΔE∕k T), (3.9)
B
N 1
where ΔE = E − E is the energy difference, k = 1.38 × 10 −23 J/K is Boltzmann’s constant, and T is the
2
B
1
absolute temperature in Kelvin. Since the energy difference ΔE = ℏ, Eq. (3.9) can be written as
N = N exp (−ℏ∕k T). (3.10)
1
B
2
Eq. (3.9) is valid for any systems having different energy levels. For example, the number of air molecules
decreases as we go to higher altitudes. If N and N are the number of molecules near the ground and at
2
1
height h, respectively, at thermal equilibrium, their ratio is
( ) ( )
N mgh
2 ΔE
= exp − = exp − , (3.11)
N k T k T
1 B B
E 2 N 2 E 2 N 2
E 1 N 1 E 1 N 1
(a) (b)
Figure 3.4 Two-level atomic system emitting a photon due to spontaneous emission (a) before spontaneous emission
and (b) after spontaneous emission.