Page 155 - Photonics Essentials an introduction with experiments

Lasers

Lasers 149

absorbing transitions equals the number of emitting transitions. We
can summarize the discussion so far in a set of simple equations:

Emission rate = W 21 = B 21 ( ) + A 21
Absorption rate = W 12 = B 12 ( )
(7.1)
N 1 B 12 ( ) = N 2 B 21 ( ) + N 2 A 21
This allows us to solve for the photon density at the energy of the
transition:

A 21

N 2 A 21 B 21
( ) = = (7.2)
N 1 B 12 – N 2 B 21 N 1 B 12
– 1
N 2 B 21
Now we will compare this expression for ( ) to another one based
on the Planck radiation law. We discussed Planck’s experiments in
Chapter 1. The result of his work was to derive an expression for the
energy density of photons. We recall that Planck discovered that the
energy density depends on the temperature and on the color, or ener-
gy, of an individual photon. Planck’s radiation law states

16 1
2
( ) = (7.3)
3 e /k B T –1
In comparing Eqs. 7.2. and 7.3, we can see some similarities. For
example, we know from Boltzmann statistics that N 2 /N 1 = e E/k B T .
Therefore, it follows that N 1 /N 2 = e E/k B T = e /k B T . We can see that
the two equations are identical when

B 12 = B 21
and
3
16 8 n hf 3
2
A 21
= = (7.4)
3 c 3
B 21
The two expressions in Eq. 7.4 are called the Einstein relations, in
which c is the speed of light and n is the index of refraction of the
medium involved. For semiconductors like GaAs or InP, n is about
3.4.
The ratio of the spontaneous emission rate to the stimulated emis-
sion rate is:

A 21 /k B T
R = = e –1 (7.5)
( )B 21

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies. All rights reserved.
Any use is subject to the Terms of Use as given at the website.