- Linear Polarization
- Circular Polarization
- Elliptical Polarization
- Angular Momentum
- Malus's Law
- Dichroism
- Birefringence
- Scattering
- Stokes Parameters
- Jones Vectors

When there are two or more waves superimposed
in a region the situation becomes more complicated. For simplicity, we
can restrict our attention to two waves. When the electric field vectors
are collinear, the resulting field will simply combine to form another
linearly polarized wave. However, if the two waves are not collinear, then
the second wave can be split into collinear and perpendicular components.
The collinear component will form a linearly polarized wave again. The
resultant wave formed by the perpendicular components can have various
**states of polarization**.

(14.1)

and

(14.2)

where
is the relative phase difference between the waves, both of which are traveling
in the *z* direction. The resulting superposition of these waves is

. (14.3)

If
the waves are said to be **in phase**. In this case,

. (14.4)

This wave has a fixed amplitude equal to ,
which shows that this wave is also linearly polarized. Similarly, if ,
the resultant wave is again linearly polarized, but now the original waves
are said to be **out of phase**.

and

so that the resulting wave is given by

. (14.5)

Notice that the scalar amplitude, *E _{0}*,
is a constant, but the direction of the amplitude varies with time. This
implies that the amplitude is not restricted to a single plane as before,
but instead rotates so that the axis of rotation (as given by the right
hand rule) is opposite the direction of motion. The angular frequency of
the rotation is .
This wave is said to be

. (14.6)

In this case the wave rotates with the axis of
rotation in the same direction as the motion, and the wave is said to be
**left circularly polarized**.

As an interesting aside, a linearly polarized wave can be constructed from two oppositely polarized waves,

(14.7)

This wave has an amplitude of twice the original
wave and is linearly polarized in the *x* plane.

. (14.8)

From (14.1), we have that

so that (14.8) becomes

(14.9)

Since *E _{E}* =

Squaring both sides and rearranging yields

. (14.10)

This is the equation of an ellipse making an
angle
with the (*E _{x}*,

. (14.11)

If we rotate the (*E _{x}*,

. (14.12)

To see that both circularly polarized and plane
polarized light are special cases of equation (14.10), consider the following.
If ,
and *E _{x,0}* =

(14.13)

which is the equation of a circle. Similarly, if , equation (14.10) is

(14.14)

This is the equation of a straight line with a slope of .

Using our results so far, we can now describe
a particular wave of light in terms of its specific state of polarization.
If the light is linearly, or plane, polarized, then we say it is in the
**Pstate**. Light that is in right (left) circularly polarized
is in the **R-** (**L-**) **state**. Finally, elliptically polarized
light is referred to as being in the **E-state**.

(14.15)

and the **intrinsic angular momentum**, or
**spin**, of a photon is either -
or +,
where the sign indicates right- or left-handedness, respectively. It is
important to note that **the angular momentum of a photon is completely
independent of its energy**. Whenever a charged particle emits or absorbs
electromagnetic radiation, along with changes to its energy and linear
momentum, it will undergo a change of
in its angular momentum.

The energy transferred to a target by an incident monochromatic wave can be viewed as being a stream of identical photons. Thus, a purely leftcircularly polarized plane wave will impart angular momentum to the target as if all the constituent photons in the beam had their spins aligned in the direction of propagation. Similarly, right-circularly polarized wave have their spin opposite the direction of propagation.

Now suppose that a second polarizer is inserted
into the light path. This second polarizer is known as the **analyzer**.
If the analyzer is placed at an angle
with respect to the polarizer, then the amplitude of the electric field
that exits the analyzer is *E _{0}* cos.
From this, we see that the irradiance behind the analyzer is

. (14.16)

When = 0, a maximum amount of irradiance is transmitted, so (14.16) can be rewritten as

, (14.17)

where .
Equation (14.17) is known as **Malus's law**. By using Malus's law,
we can determine experimentally if a material is a linear polarizer or
not.

The simplest form of a dichroic polarizer is
a grid of parallel conducting wires. There are also certain materials which
are inherently dichroic due to the anisotropy which exists in their crystal
structure. One of the best known naturally occurring examples is the mineral
tourmaline. Certainly the most famous dichroic material is the **Polaroid
sheet**. The initial Polaroid was known as the *Jsheet*. It
was made from the mineral *herapathite*. These crystals were ground
into submicroscopic crystals, which we aligned by using magnetic or electric
fields. They could also be mechanically aligned using a viscous colloidal
suspension. By drawing this suspension through a long, narrow slit, the
resulting sheet acts effectively like a large flat dichroic crystal. Unfortunately,
the individual submicroscopic crystals still scattered light so that the
*J-sheet* resulted in images which were somewhat hazy. The more common
type of Polaroid is the *Hsheet*. It does not use dichroic crystals,
but instead consists of a molecular analogue of the wire grid. Each separate
dichroic entity is known as a dichromiphore. In the Hsheet the dichromiphores
are molecular sized, so scattering is reduced to minimal levels.

The usefulness of this model can be seen by considering
how light propagates through a transparent medium in reality. We have seen
that the light excites the electrons within the medium. The electrons are
driven by the electric field; they re-radiate secondary wavelets, which
then recombine and the resultant wave continues on. The speed of the wave,
and thus the index of refraction, is determined by the difference between
the frequency of the electric field and the natural, or characteristic,
frequency of the electrons. **An anisotropy in the binding force will
therefore be manifest in an anisotropy in the index of refraction**.
A material of this sort, which displays two different indices of refraction,
is said to be **birefringent**.

An atom in the excited state is not stable, and
it will eventually return to its ground state. If the atom is in a dense
media, the excess energy will usually be dissipated thermally. On the other
hand, when the atom is in a rarefied gas, the atom will usually release
the energy through the emission of a new photon. This process is known
as **resonance radiation**. The emitted photon will have no knowledge
of its previous direction and will be sent with equal probability in any
direction.

When the frequency is away from resonance, the
electrons vibrate with respect to the nucleus. The atom can then be viewed
as a set of oscillating dipoles. This causes the atom to radiate electromagnetic
energy at a frequency equal to that of the incident field. This nonresonant
emission propagates out in the standard dipole radiation pattern described
earlier. The absorption of energy out of an incident wave by the atom,
and the subsequent reemission of some of that energy is known as **scattering**.
It is the underlying physical mechanism responsible for reflection, refraction
and diffraction.

In addition to electron oscillations in an atom, which generally result in absorption and emission of ultraviolet light, there are also atomic oscillators. In this case the constituent atoms in a molecule vibrate about the center of mass of the molecule. These oscillations usually have resonances in the infrared region of the spectrum due to the larger masses associated with atoms.

The amplitude of an oscillator, and thus the amount of energy removed from the incident wave, increases as the frequency of the wave approaches one of the natural frequencies of the atom. Assuming that the energy is not lost via thermal interactions, we find that the scattered wave carries of more and more energy as the incident frequency approaches a resonance.

Lord Rayleigh was the first to work out the dependence
of the scattered flux density on frequency. He found that **the scattered
flux density is directly proportional to the fourth power of the incident
frequency**. This result is in agreement with the radiation pattern associated
with an oscillating dipole. The scattering of light by objects that are
small in comparison with the wavelength of the incoming wave is known as
**Rayleigh scattering**.

The modern formulation of polarized light has
its origins in the work of G. G. Stokes in 1852. He introduced four quantities
which are functions only of observables of the electromagnetic wave. These
quantities are now know as the **Stokes parameters**. In order to understand
the parameters, image that we have a set of four filters, each of which,
under natural illumination, will transmit half of the incident light, discarding
the other half. The choice of filters is not a unique one, and a number
of equivalent possibilities exist. For our purposes, suppose that the first
filter is an isotropic one, passing all polarization states equally. Let
the second and third filters be linear polarizers whose transmission axes
are horizontal and at +45,
respectively. Choose the last filter so that it is a circular polarizer
which only transmits R-states. If each filter is placed singly in the path
of the light beam, then we can assume that the transmitted irradiances
are measured to be *I _{0}*,

(14.17)

where *S _{0}* is the incident irradiance,
and

We can connect the Stokes parameters with the electromagnetic field directly by using the expressions for quasimonochromatic waves,

(14.18)

and

(14.19)

where ** E** =

, (14.20)

where and we've dropped the constant .

If the beam is unpolarized, then .
In this case ,
and *S _{1}* =

, (14.21)

while for partially polarized light

, (14.22)

where *V* is defined as the **degree of
polarization**.

. (14.23)

Such a vector is called a **Jones vector**.
Notice that a knowledge of ** E** allows us to know everything
about the polarization state, while the phase information,

Frequently we do not need to know the exact amplitude and phase of the polarized components. In this case we can normalize the irradiance to unity to get

. (14.24)

Notice that *E _{h}* and

and

In a similar manner, *E _{L}* and

Last updated: July 24, 1997

Comments to: D-Suson@tamuk.edu