Page 314 - Electrical Properties of Materials

296 Lasers

E

E
3

Fig. 12.1
Number of atoms in a natural
two-state system as a function of E
1
energy. The dotted line shows the
Boltzmann function, decaying
N N log N(E)
exponentially with increasing energy. 3 1 e

population of the energy levels is, therefore, accurately described by the tem-
perature, T, of the system and its appropriate statistics, which we shall take as
Boltzmann statistics.
The two levels we are considering are labelled E 1 and E 3 in Fig. 12.1. Later
on we shall see what happens in a three-level system, with the third level called
E 2 ; but for the moment do not be put off by this notation; we are still talking
of only two levels. The numbers of electrons N 1 , N 3 in the levels E 1 , E 3 are
related by the Boltzmann function, so that they will be of the general form

N = N 0 exp(–E/k B T). (12.1)

N 0 is a constant. Therefore,

E 3 – E 1
N 3 = N 1 exp – . (12.2)
k B T
As I said above, the atoms are in dynamic equilibrium, which means that the
number of atoms descending from E 3 to E 1 is the same as the number leaping
from E 1 to E 3 .Anatomat E 3 can lose the energy E 3 – E 1 either by radiative or
by non-radiative processes. I shall consider only the former case here. When a
radiative transition between E 3 and E 1 occurs during the thermal equilibrium
process, it is called spontaneous emission for the ‘down’ process and photon
We shall follow here the custom absorption for the ‘up’ process. In each case, the photon energy is given by
adopted in laser theory of using
the frequency, ν, instead of the hν 31 = E 3 – E 1 . (12.3)
angular frequency, ω.
What do we mean by talking about photons being present? It is a very basic
law of physics that every body having a finite temperature will radiate thermal
or ‘black body’ radiation. This radiation comes from the sort of internal trans-
itions that I have just mentioned. As we saw in Chapter 2, the whole business of
quantum theory historically started at this point. In order to derive a radiation
law that agreed with experiments, Planck found it necessary to say that atomic
ρ(ν) is the radiation density em- radiation was quantized. This famous radiation equation is
anating from a body at temperat-
3
ure, T, in a band of the frequency 8πn hν 3 dν
spectrum of width, dν, and at a ρ(ν)dν = c 3 exp(hν/k B T)–1 . (12.4)
frequency, ν.
The derivation can be found in many textbooks.