Page 129 - Fiber Optic Communications Fund

110 Fiber Optic Communications

The energy and momentum of a free non-relativistic particle are related by
2 2
p 2 ℏ k ℏ 2 ( 2 ) 2
E = = = , (3.66)
2m 0 2m 0 2m 0
where m is the mass of a particle. Thus, an electron with high energy has a large momentum or in other words,
0
has a short wavelength (see Eq. (3.65)). Typically, the highest achievable resolution of an optical microscope is
of the order of the wavelength of the light used (∼0.4 − 2 μm) and, therefore, it is hard to study the structure
of a nanoparticle using optical microscopes. If we increase the energy of an electron, the corresponding
De Broglie wavelength can be made quite small. This property is the principle behind electron microscopy.
The energy can be measured in several units, such as Joules (J) and electron volts (eV). Voltage is defined
as the potential energy E per unit charge,
E
V = . (3.67)
q
The energy required to carry an electron of charge 1.602 × 10 −19 C over a potential barrier of 1 V is 1 eV.
With q = 1.602 × 10 −19 C and V = 1 volt, from Eq. (3.67) we have
E = qV

= 1.602 × 10 −19 × 1J
= 1 eV. (3.68)
Note that the electron volt is not a unit of voltage, but of energy.

Example 3.5
The energy difference between the two states of an ammonia maser is 10 −4 eV. Calculate the frequency of
the electromagnetic wave emitted by stimulated emission.
Solution:
E = 10 −4 eV = 10 −4 × 1.602 × 10 −19 J,
E 10 −4 × 1.602 × 10 −19
f = = = 24 GHz.
2ℏ 2 × 1.054 × 10 −34

3.6 Laser Rate Equations
In this section, we consider the gain rate and loss rate of photons and population densities in states 1 and 2 due
to stimulated emission, spontaneous emission, and various loss mechanisms. This is similar to the population
growth rate of a country. The population of a country increases due to new births and immigration, while it
decreases due to death and migration to other countries. Suppose N(t) is the population at t, the net rate of
population growth may be modeled as
dN
= R born + R immigration + R death + R migration . (3.69)
dt
To model lasers, we follow a similar approach. Let us consider the atomic system with two levels. The popula-
tion density of the excited state decreases due to stimulated emission, spontaneous emission, and non-radiative