Page 14 - Photonics Essentials an introduction with experiments

Electrons and Photons

8 Introductory Concepts

electrons or photons. However, two fundamental physical laws always
apply: conservation of energy and conservation of momentum. The be-
havior of electrons and photons can be tracked by their respective en-
ergies and momenta. The band structure is a particularly useful tool
for this task.

2.2 The Fundamental Relationships
There are two simple principles that support almost all the science of
photonic devices. One is the Boltzmann relationship and the other is
Planck’s equation relating the energy of a photon to the frequency of
the light wave associated with the photon.

Ludwig Boltzmann
Boltzmann studied gases and the motion of molecules in gases. In a
dense gas, Boltzmann said, the velocities of the molecules are statisti-
cally distributed about the average velocity v 0 = 0. Since the Law of
Large Numbers in statistics says that all distributions tend toward a
Gaussian or normal distribution, Boltzmann started from this point,
too.
The probability of finding a particular velocity v 1 is given by a
Gaussian distribution:

–(v 1 – v 0 ) 2

Pr(v = v 1 ) = A·e v 2 (2.1)
2
where v means the average velocity = 0, and v means the average
0
2
of the square of the velocity. Even though v = 0, v is definitely not
0
equal to zero. This is the “spread” of the distribution.
Remember that:
1
E kinetic = – mv 2
2
1
2
––m(v 1 )
2

1
2
Pr(v = v 1 ) = A·e –m v 2
1 – m v = spread in the energy = E
2
2

Pr(v = v 1 ) = A·e –(E/E) (2.2)
From Brownian motion studies more than a century earlier, as well
as mechanical equivalent of heat studies, energy is proportional to
temperature. That is, E = constant · T and
Pr(v = v 1 ) = Pr(E = E 1 ) = A·e –(E/constant · T)

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