Page 316 - Electrical Properties of Materials

298 Lasers

What is the physical significance of A 31 ? It is a measure of the spontaneous
depopulation of state 3. Assuming, as usual, an exponential decay, the rate of
change of population is
dN 3
– = A 31 N 3 , (12.12)
dt
which leads to a decay time constant, called spontaneous lifetime, by defining
1
t spont = . (12.13)
A 31
We should, by now, have quite a good picture of what happens when light
of frequency v 31 shines on the two-state system. In the presence of an input
light spontaneous decay is usually negligible, and although the probabilities of
upward and downward transitions are exactly equal, there will be more trans-
itions from E 1 to E 3 because there are many more atoms in the lower state.
In other words, the result is a net absorption of photons. This we often see in
nature. For example, many crystalline copper salts have two energy bands, sep-
arated by photon energy corresponding to yellow light. Thus, when viewed in
white light, the yellow part is absorbed, and the crystal transmits and reflects
the complementary colour, blue. Ruby (chromium ions in crystalline alumina)
has an absorption band in the green by this mechanism, and hence looks red in
white light.
E When light is absorbed, the population of the upper level is increased.
Excited population Normally this perturbation from the equilibrium condition is small. But if we
Slope defining have an increasingly intense ‘pump’ light source, the number in level 3 will go
E
3 temperature on increasing, by the same amount as those in level 1 decrease. Fairly obvi-
E
2 ously, there is a limit, when the levels are equally populated, and the pump is
Population
removed infinitely strong. This is illustrated in Fig. 12.2. For the case of intense pump-
E ing, the non-equilibrium level populations (denoted by an asterisk) become
1
almost equal,
N N N log N(E)
3 2 1 e
N 1 + N 3
∗ ∗
N N . (12.14)
3
1
Fig. 12.2 2
The three-level system. The strong Now let us consider a three-level system, with the third level E 2 between
E 1 and E 3 , also shown in Fig. 12.2. The pumping will have no effect on its
‘pump’ signal has equalized levels E 1
and E 3 ,sothat E 3 now has a greater
population, which is the equilibrium value N 2 . So with the three-level system
population than E 2 . The dotted line
strongly pumped, the number of electrons in the three states are N , N 2 , and
∗
shows how population is changing 1
N . Suppose that some photons come along with energy
∗
with energy, as in Fig. 12.1, but now it 3
has a positive slope. hν 32 = E 3 – E 2 . (12.15)
They will clearly interact with the system, causing stimulated emission by
transitions from E 3 to E 2 and absorption by transitions from E 2 to E 3 .But
now we have an unnatural occurrence: there are more electrons in the upper
state (E 3 ) than in the lower (E 2 ). So instead of there being a net absorp-
tion of photons of energy, hv 32 , there will be a net emission. The three-level
system will amplify a photon of frequency, v 32 , which is called the signal fre-
quency. The whole thing is called a laser, which stands for light amplification
by stimulated emission of radiation.
When there are more atoms in an upper than a lower level, as in the case
of E 3 and E 2 in Fig. 12.2, it is justifiable jargon to speak of an ‘inverted