Polarization

of light waves
Wavelength 𝜆
Figure: Polarization of light
 Unpolarized light

Polarized light

Polarizing filter

Direction of
propagation of light
Figure: Polarization of light
Polarization:
Light is an electromagnetic wave of transverse nature. In an
ordinary light (known as unpolarized light) both electric vector
𝐸 and magnetic field vector 𝐵 oscillate in all possible directions
in a plane perpendicular to its direction of propagation.
When this unpolarized light is passed through a special device
(called polarizer) the electric field vectors are restricted to a
single plane by filtration and the light is said to be polarized
with respect to the direction of propagation and all waves
vibrate in the same plane.
This light having vibrations in only a particular direction is
called polarized light and the process is called polarization. The
polarized light has half the intensity of normal light.
Representation of Unpolarized and Polarized Light:

All the symmetrically distributed electric vibrations of

unpolarized light can be resolved into two mutually
perpendicular electric vibrations of equal amplitude, one
along the plane of the paper and the other perpendicular
to the plane of the paper.

The polarized light is the one in which only one

direction of vibration is allowed, the rest is removed
from the unpolarized light. So, either it is oscillating
along the plane of the paper or perpendicular to the plane
of the paper.
Symbols for Unpolarized light
Symbols for Polarized light
Types of Polarization:
The polarization of light wave describes the shape and
locus of the tip of the 𝐄 vector at a given point in space as
a function of time. Depending upon the locus of the tip of
the 𝐄 vector, light may exhibit three different states of
polarization.
1) Plane or linear polarization.
2) Circular polarization.
3) Elliptical polarization.
An unaided human eye cannot identify the state of
polarization of light. Some insects and animals possess
polarization sensitive vision.
1) Plane or linear polarization:
If the electric field oscillates in fixed orientation in a single
plane, the light is said to be linearly polarized. If linearly
polarized light contains an additional component of natural
unpolarized light, it is called partially polarized light. Partially
polarized light could come from a perfectly polarizing filter
with some holes in it or from a low-quality filter.
2) Circular polarization:
In circular polarization, the electric vector no longer oscillates
in a plane (as in linear polarization). Instead the vector is
constant in magnitude and proceeds in the form of a helix,
around (rather than through) the axis of propagation.
Within one wavelength, the E vector completes one
revolution. If the tip of the vector, when seen looking
towards the light source, rotates clockwise, the light is
said to be right circularly polarized. If the tip of the
vector rotates anticlockwise, the light is said to be left
circularly polarized.
3) Elliptical polarization:
If the magnitude of the rotating E vector changes with
time and proceeds in the form of a flattened helix, then
the polarization is called elliptical polarization. It is the
most general type of polarization; linear and circular are
the two extremes of elliptical polarization,
Methods of Polarizing Unpolarized light:
Plane polarized light can be obtained from an unpolarized light
by the five optical phenomena:
(a) Reflection, (b) Refraction, (c) Double refraction or
Birefringence, (d) Dichroism and (e) Scattering.
1) Polarization by reflection:
If an unpolarized light is incident at an angle ∠𝑖 on a dielectric
like glass or water, it will be reflected; as well as refracted. It
has been observed that the reflected ray is partially plane
polarized and that only at a certain definite angle (about 57°
for ordinary glass) it is plane polarized. It was Brewster who
first discovered that at the polarizing angle ∠𝒊𝒑 , the reflected
and refracted rays are just 90° apart. The angle ∠𝒊𝒑 is also
called Brewster’s angle.
Referring to Fig-7 one can write,
∠𝐴𝑂𝐵 = ∠𝑖 = ∠𝑖𝑝 = ∠𝐵𝑂𝐶;
∠𝐵𝑂𝐸 = ∠𝐵𝑂𝐶 + ∠𝐶𝑂𝐸 = 90°
Again ∠𝐶𝑂𝐸 + ∠𝐸𝑂𝐷 = 90°;
𝟗𝟎° − ∠𝒊
∴ ∠𝐸𝑂𝐷 = ∠𝐵𝑂𝐶 = ∠𝑖.
Now, ∠𝐹𝑂𝐷= ∠𝑟
∠𝐹𝑂𝐷 + ∠𝐸𝑂𝐷 = 90°;
𝑜𝑟, ∠𝐹𝑂𝐷 = 90° − ∠𝐸𝑂𝐷
∠𝒓
∴ ∠𝑟 = 90° − ∠𝑖.
Now we have
𝝁𝟐 sin 𝑖 sin 𝑖 sin 𝑖
= = =
𝝁𝟏 sin 𝑟 sin(90° − 𝑖) cos 𝑖
= tan 𝑖.
 𝜇2
At ∠𝑖 = ∠𝑖𝑝 ; = tan 𝑖𝑝
𝜇1
−𝟏 𝝁𝟐
and ∠𝒊𝒑 = 𝐭𝐚𝐧
𝝁𝟏

This is Brewster’s law, which shows that the angle of

incidence for maximum polarization depends only on the
refractive indices of the two medium. It varies somewhat
with wavelength of light, but for ordinary glass the
dispersion is such that the polarizing angle ∠𝑖𝑝 does not
change much over the whole visible spectrum.

For μ1 = 1 , ∠ 𝒊𝒑 = 𝐭𝐚𝐧−𝟏 𝝁𝟐
If μ2 = 1.55 for glass
and μ1 = 1 for air, ∠𝒊𝒑 = 𝟓𝟕. 𝟏𝟕°.
Applications of Brewster’s law:

1. Brewster's law can be used to determine the

refractive indices of opaque materials.

2. It is used to calculate the polarizing angle for total

polarization of reflected light, if refractive index of
the material is known.

3. Brewster's angle can be utilized for transmitting a

light beam in into or out of an optical fiber without
reflections losses.
2) Polarization by refraction:
If a beam of ordinary light is incident at the polarizing angle
(𝜃𝐵 ) on a pile of glass plates, some of the vibrations
perpendicular to the plane of incidence are reflected at each
surface and all those parallel to it are refracted.
The net result is that the reflected beams are all plane
polarized in the same plane, and the refracted beam, having
lost more and more of its perpendicular vibrations is partially
plane polarized. Reflections from successive surfaces of each
glass plate filter the perpendicular component from the
transmitted ray.
The larger the number of glass surfaces (glass plates), the
more nearly plane polarized is this transmitted beam. The
transmitted ray consists of parallel components.
Fig-8: Polarization by refraction
The degree of polarization 𝑃 of the transmitted light can be
calculated by summing the intensities of the parallel and
perpendicular components.
If these intensities are Ip and Is, respectively, then
𝐼𝑝 − 𝐼𝑠 𝑚
𝑃= = 2
𝐼𝑝 + 𝐼𝑠 2𝑛
𝑚+
1 − 𝑛2
Here, 𝑚 is the number of glass plates (𝑖. 𝑒. 2𝑚 surfaces) and
𝑛 their refractive index. This equation shows that by the use
of enough plates the degree of polarization can be made to
approach unity, i.e. 100%.
The drawback of this method is that a good portion of light
is lost in reflections.
3) Polarization by double refraction or Birefringence:
When a ray of unpolarized light is incident on a calcite or
quartz crystal, there will be, in addition to the reflected
ray, two refracted rays (in place of the usual single one)
are observed. This phenomenon observed in crystals of
calcite (CaCO3), quartz (SiO2) etc. is called double
refraction or birefringence.

Double refraction
Upon measuring the angles of refraction 𝑟 for different
angles of incidence ∠𝑖, one finds that the Snell’s law of
sin 𝑖
refraction, = 𝜇, holds for only one of the rays known
sin 𝑟
as ordinary ray or O-ray, but not for the other, known as
extraordinary ray or E-ray.
O-rays obey the laws of refraction and maintains a constant
refractive index µO. E-rays do not obey the laws of
refraction and have variable refractive index in between µO
and µE.
In double refraction two images of a single object are
formed. Both the calcite and quartz crystals are found in
nature to be transparent to visible light as well as ultra-violet
light.
When an ink dot on a sheet of
paper is viewed through a calcite
crystal, two images of the dot will
be seen. On rotating the crystal,
one image remains stationary,
while the other rotates around the
first. The stationary image is the
ordinary image (O), the other is
extra-ordinary image (E).

Since the two opposite faces of a calcite crystal are always

parallel, the two refracted rays emerge parallel to each other.
In O-ray the vibration is perpendicular to the plane of
incidence and in E-ray the vibration is parallel to the plane of
incidence.
Double refraction
If the incident light is normal to the surface, the E-ray will be
refracted at some angle that is not zero and will come out
parallel to, but dispersed from the incident beam. The O-ray
will pass straight through without deviation.
Inside a double refracting crystal the O-ray travels with same
velocity in all directions of incident angle ∠𝑖 but the E-ray
travels with different velocities along different directions of the
incident angle ∠𝑖 . The difference between the refractive
indices between E-ray and O-ray is known as the amount of
double refraction or birefringence Δ𝜇 = 𝜇𝐸 − 𝜇𝑂 .
A point source inside a refracting crystal produces spherical
wave front corresponding to O-ray and elliptical wave front
corresponding to E-ray. In some cases the O-ray is not
completely polarized but partially polarized. E-ray is the one
which is completely polarized.
Positive Crystal and Negative Crystal:
Positive crystals are defined as the crystals in which the
refractive index for E-ray is greater than refractive index for
O-ray 𝑖. 𝑒. µ𝐸 > µ𝑂 . In positive crystals E-ray travels slower
than O-ray in all directions of incident angle ∠𝑖 , except along
the optic axis 𝑖. 𝑒. 𝑉𝐸 < 𝑉𝑂 .
Example: Quartz and ice.
Negative crystals are defined as the crystals in which the
refractive index for O-ray is greater than reflective index for
E-ray 𝑖. 𝑒. µ𝑂 > µ𝐸 . In negative crystals O-ray travels slower
than E- ray in all directions of incident angle ∠𝑖 except along
the optic axis 𝑖. 𝑒. 𝑉𝑂 < 𝑉𝐸 .
Example: Calcite and tourmaline.
E-ray

O-ray
Optic axis:
In calcite and quartz crystals there is a single direction, which is
an axis of symmetry with respect to both the crystal form and
arrangement of atoms, called the optic axis.

The double refraction in uniaxial crystals disappears when the

light is made to enter the crystal in a direction so that it travels
along the optic axis. That is, there is no separation of the O-rays
and E-rays. This is also true in directions at right angles to the
axis but here O-rays and E-rays differ in speed.

The optic axis is not a particular line through the crystals but a
direction only. For any given point in the crystal an optic axis
may be drawn which will be parallel to that for any other point.
 E-ray

O-ray

Thus, the optic axis of a crystal is the direction in which the

transmitted light suffers no birefringence (double refraction)
𝑖. 𝑒. through which E-ray and O-ray travel with the same
speed. Non-crystalline materials have no double refraction
and thus, no optic axis. Some solid materials under specific
conditions can demonstrate double refractions and optic axes.
Polarization by double refraction using Nicol prism:
A Nicol prism is a type of polarizer, an optical device used to
produce a polarized beam of light from an unpolarized beam.
It is made in such a way that it eliminates the O-ray by total
internal reflection and only the extraordinary ray is
transmitted through the prism.

O-ray

E-ray

Nicol prism
It was the first type of polarizing prism to be invented, in
1828 by Scottish physicist and geologist William Nicol
(1770–1851) of Edinburg, the capital city of Scotland, UK.

It consists of a rhombohedral crystal of Iceland spar (a

variety of calcite) that has been cut at an angle of 68° with
respect to the crystal axis. The crystal is then cut apart
again diagonally. The two cut surfaces are ground and
polished optically flat and then cemented together with
Canada balsam as glue. Canada balsam is used because it
is clear transparent substance with an index of refraction
about midway between the refractive index of the O and E
rays.
Iceland Spar, Calcite (Calcium Carbonate)
Iceland Spar (Optical Calcite)
Unpolarized light ray enters through the left face of the crystal,
as shown in the diagram, and is split into two orthogonally
polarized, differently directed rays by the birefringence
property of the calcite.
O-ray, experiences a refractive index of 𝜇𝑜 = 1.658 in the
calcite and undergoes total internal reflection at the calcite–
glue interface because its angle of incidence at the glue layer
(refractive index 𝜇 = 1.55 ) exceeds the critical angle for the
interface. It passes out from the top/bottom side of the
upper/lower half of the prism with some refraction.
E-ray, experiences a lower refractive index (𝜇𝐸 = 1.486) in
the calcite and is not totally reflected at the interface because it
strikes the interface at a sub-critical angle.
The E-ray merely undergoes a slight refraction, or bending,
as it passes through the interface of the prism. It finally
leaves the prism as a ray of plane-polarized light, undergoing
refraction, as it exits the far right side of the prism.

The two exiting rays have polarizations perpendicular to each

other, but the E-ray is more commonly used for further
experimentation because it travels in the original horizontal
direction. The direction of the O-ray is quite different from
its original direction because it alone suffers total internal
reflection at the glue interface, as well as a final refraction on
exit from the upper/lower side of the prism.