Page 128 - Fiber Optic Communications Fund

Lasers 109

where is the angular frequency. In the wave picture, the optical field of a plane wave can be written as

= A exp [−i(t − k x − k y − k z)]. (3.57)
x y z
It has four degrees of freedom, , k , k , and k . If we imagine light as a particle, it has four degrees of freedom
x
z
y
too. They are energy E and momenta p , p , and p in the x-, y- and z-directions, respectively. Energy (particle
x
y
z
picture) and frequency (wave picture) are related by Eq. (3.56). Similarly, the wave vector components are
related to the momenta by
p = ℏk , p = ℏk , p = ℏk , (3.58)
x
y
z
z
x
y
or
p = ℏk, (3.59)
where
p = p x + p y + p z, (3.60)
z
y
x
k = k x + k y + k z. (3.61)
x y z
From Eq. (3.59), we see that the photon carries a momentum in the direction of propagation. The magnitude
of the momentum is
p = |p| = ℏ|k| = ℏk. (3.62)
If k = k x + k y + k z,
x
y
z
√
2
2
2
p = ℏ k + k + k . (3.63)
x y z
Light is a wave in free space, but it sometimes acts like a particle when it interacts with matter. In the early
1920s, De Broglie proposed that every particle (atom, electron, photon, etc.) has a wave nature associated
with it. If a particle has energy E and momentum p, the angular frequency associated with its wave part is
E∕ℏ and the wave vector is k = p∕ℏ or the wavenumber is
2 p
k = = , (3.64)
ℏ
or
2ℏ
= . (3.65)
p
This wavelength is called the De Broglie wavelength. The wave nature of particles has been confirmed by
several experiments. The first electron-diffraction experiment was done by Davisson and Germer in 1927 [8].
In this experiment, the incident beam was obtained by the acceleration of electrons in an electrical potential
and diffraction of the electron beam by a single crystal is studied. This experiment showed that electrons
behave as waves, exhibiting the features of diffraction and interference. From the electron interference pattern,
it is possible to deduce the experimental value of the electron wavelength which is in good agreement with
De Broglie’s formula, Eq. (3.65). However, it turns out that these are not real waves, but probabilistic waves.
If electrons were matter waves, we would expect that the intensity of the interference pattern should reduce
as the intensity of the incident beam decreases, but the interference pattern should not become discontinuous;
electron diffraction experiments contradict the above property of a matter wave. If the intensity of the incident
electron beam in these experiments is reduced to a very low value, we would observe a single impact either on
the central spot or on one of the diffraction rings, which shows the particle nature of electrons. The simplest
interpretation we could give of wave–particle duality is a statistical interpretation: the intensity of the wave
at each point on the diffraction pattern gives the probability of occurence of an impact at that point [9].