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.
.
In this case
and
,so that the resulting wave is given by
.
(14.5)
Notice that the scalar amplitude, E0,
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 right circularly polarized. In a similar
way, if 
,
then the resulting wave is
.
(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 EE = Ex + Ey, this can be solved to yield
.Squaring both sides and rearranging yields
.
(14.10)
This is the equation of an ellipse making an
angle 
with the (Ex,Ey) coordinate system
such that
.
(14.11)
If we rotate the (Ex,Ey)
coordinate system through an angle ,
this takes on the more familiar form,
.
(14.12)
To see that both circularly polarized and plane
polarized light are special cases of equation (14.10), consider the following.
If 
,
and Ex,0 = Ey,0 = E0,
then (14.10) becomes
(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 E0 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.
,
is near the resonant frequency of the atom, then a significant amount of
energy can be transferred. In particular, when the field frequency equals
the resonance, the atom will transition from the ground state into one
of its excited states. In this case, all of the energy in the photon in
absorbed by the atom.
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 I0, I1, I2,
and I3. Then the operational definition of Stokes parameters
are
(14.17)
where S0 is the incident irradiance,
and S1, S2 and S3
specify the state of polarization. Physically, we can interpret the last
three as follows. S1 is a statement of the amount that
the polarization resembles either a horizontal Pstate (so that S1
> 0) or a vertical P-state (where S1 < 0). When S1
= 0, the beam is either elliptical at 45,
circularly polarized, or unpolarized. In a similar way, S2 represents
the likelihood that the light is in a P-state oriented at +45
(when S2 > 0), -45,
(when S2 < 0), or neither (S2 =
0). Finally, S3 is related to the probability that the
beam is in a R-state (S3 > 0), L-state (S3
< 0), or neither (S3 = 0).
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 = Ex + Ey. From the definition of irradiance, Stokes parameters become
,
(14.20)
where 
and we've dropped the constant 
.
If the beam is unpolarized, then 
.
In this case 
,
and S1 = S2 = S3
= 0. It is convenient to normalize the Stokes parameters by dividing each
one by S0. This implies that the incident light has an
irradiance of unity. The set of parameters (S0, S1,
S2, S3) for unpolarized light is then
(1, 0, 0, 0). If the light is horizontally polarized, the normalized parameters
are (1, 1, 0, 0). Similar results can be found for other orientations.
Notice that for completely polarized light,
,
(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, 
x
and y,
enables us to also handle coherent waves. The sum of two beams is formed
by the sum of the corresponding components. Thus, when E0,x
= E0,y = E0 and x
= y
= ,
E is given by
.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 Eh and Ev
form an orthonormal set. In other words,
and
.In a similar manner, EL and ER are orthonormal. Thus, any polarization state can be described as a linear combination of either one of the orthonormal sets. This approach has great importance in quantum mechanics, where we are dealing with orthonormal wave functions.
Last updated: July 24, 1997
Comments to: D-Suson@tamuk.edu