Page 319 - Electrical Properties of Materials

Resonators and conditions of oscillation 301

12.4 Absorption and amplification
Let us look now at energy levels 2 and 3 and consider the induced transition
rate between them. It is

3
c ρ(ν)
W 32 = B 32 ρ(ν)= 3 3 , (12.28)
8πn hν t spont
where eqns (12.11) and (12.13) have been used. The transition rate will of
course depend on the lineshape function, so we need to multiply eqn (12.28)
–2
by g(ν). We shall also introduce the power density (measured in W m ) instead
–3
of the radiation density (measured in J m ) with the relation,
c
P d = ρ, (12.29)
n
leading to the form,

2
c P d g(ν)
W 32 = . (12.30)
3
2
8πn hν t spont
Now the number of induced transition per second is N 3 W 32 per unit volume,
and the corresponding energy density per second is N 3 W 32 hν. For upward
transitions, we obtain similarly N 2 W 32 hν, and hence the power lost in a dz
thickness of the material is (N 3 – N 2 )W 32 hν dz. Denoting the change in power
density across the dz element by dρ d , we obtain the differential equation,

2
dρ d c g(ν)
= γ (ν)ρ d , γ (ν)=(N 3 – N 2 ) , (12.31)
2 2
dz 8πn ν t spont
which has the solution,

P d (Z)= P d (0) exp γ (ν)z. (12.32)

Under thermal equilibrium conditions N 3 < N 2 , and consequently, the input
light suffers absorption. However, when N 3 > N 2 , that is there is a population
inversion, the input light is amplified.

12.5 Resonators and conditions of oscillation
As we have said before, the energy levels are not infinitely narrow, hence emis-
sion occurs in a finite frequency band. For single-frequency emission (by single
frequency, we mean here a single narrow frequency range) all the excited states
should decay in unison. But how would an atom in one corner of the mater-
ial know when its mate in the other corner decides to take the plunge? They
need some kind of coordinating agent or—in the parlance of the electronic
engineer—a feedback mechanism. What could give the required feedback?
The photons themselves. They stimulate the emission of further photons as
discussed in the previous section and also ensure that the emissions occur
at the right time. If we want to form a somewhat better physical picture of