Page 158 - Photonics Essentials an introduction with experiments

Lasers

152 Photonic Devices

To continue our discussion of absorption, consider what happens to
the number of photons N, per unit volume, or the photon density, as a
function of time. The photon density will decrease as the number of
electron transitions from level 1 to level 2 increases. The density will
increase when the number of transitions from state 2 to state 1 in-
creases:

d
N = –N 1 (hf)B 12 + N 2 (hf)B 21
dt
(7.7)
= (N 2 – N 1 ) (hf)B 21
The photon density is closely related to the energy density: (hf) = N·
hf. Similarly, the intensity is related to the energy density:

c hfc
I = (hf)· = N
n n
In Eq. 7.6 we derived a relationship between the intensity and the
distance. Because of the relationship between the intensity and the
photon density, we can write another expression for the gradient:
d hfc d hfc d dt
I(x) = N = N· (7.8)
dx n dx n dt dx
For the case of light, dx/dt = c/n. Since this is a simple constant, the
inverse expression that we would like to substitute in Eq. 7.8 is the
arithmetic inverse; that is: dt/dx = n/c.
Using these results we can determine the condition for generating
optical gain:

d 1 d 1 c
N = I = I(x)·(– ) = – (hf)
dt hf dx hf hf n
Using Eq. 7.7,
d c
N = (N 2 – N 1 ) (hf)B 21 = – (hf)
dt hf n
nhf
= (N 1 – N 2 )B 21 (7.9)
c
So is positive, and absorption occurs when N 1 > N 2 . On the other
hand, is negative and amplification occurs when N 2 > N 1 . This sim-
ple condition is called population inversion. You may notice that al-
though simple, it appears to violate the requirements of Boltzmann
statistics. The art of making a laser is understanding how this condi-
tion can be achieved in real materials.

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