Download A transparent material like glass allows light to pass

A transparent material like glass allows light to pass through it. The velocity of
light inside any transparent material like glass is decided by the refractive index of the
material. The more the absolute refractive index, the less is the velocity of light in the
medium. But in a material like glass, the refractive index doesn’t depend on the direction
of passage of light inside the material, or on the plane of polarization of the light with
respect to the material. The refractive index is the same in all directions. Or the material
is isotropic.
But this is not the situation in many materials, like calcite and quartz. These
materials are crystalline in nature. The refractive index of these materials is not a constant
one. The refractive index can vary with the direction light make with certain internal
symmetry directions of the crystal. In other words the material is anisotropic.
Let us take the example of a calcite crystal. A calcite crystal has the shape of
rhombohedra. Each of the six faces of this crystal is a parallelogram with angles 78o 5’
and 101o 55’. When light falls on the face of a calcite crystal, it doesn’t get transmitted
like a single beam as in glass. Instead the incident beam of light gets split up into two
beams and these two beams travels in different directions. One of these two beams has a
constant velocity in all directions, or this ray of light has an ordinary behavior. This ray is
called as the Ordinary ray (O-ray). The other ray of light behaves extra-ordinarily. It has
different refractive indices in different directions and therefore travels with different
velocities. The velocity of the ordinary ray is determined by the crystal symmetry.
In a uniaxial crystal there is only one optic axis. Optic axis is the direction inside
the crystal along which both the rays (O-ray and E-ray) have equal velocities and
refractive indices. In a calcite crystal the optic axis is in the direction of the line joining
the two blunt corners of the crystal. Blunt corner is the corner at which three
parallelograms meet with obtuse angles. There will be two blunt corners. In a uniaxial
crystal there is only one optic axis. There some other crystals with more than one optic
axis and they may be called biaxial crystals. It should be remembered that, optic axis is
not a single line. It is a direction inside the crystal. All lines drawn parallel to the line
joining the blunt corners are optic axes.
Normally when a ray of light falls on any face of a crystal, it gets split up into two
beams. These two beams are plane polarized in mutually perpendicular planes. Even
along the direction of the optic axis rays get split up, and they are polarized in mutually
perpendicular planes, but travels with equal velocities.
A unique thing happens in the direction perpendicular to the optic axis. Along this
direction both rays of light travels with different velocities, even though they are
traveling in the same direction. They are plane polarized in mutually perpendicular planes
as usual.
It is easy to produce linearly (or plane polarized) polarized light from a
birefringent material. By some technique if one of the rays can be eliminated from
coming out, the remaining beam of light will be linearly polarized. This is what exactly
done in a Nicol Prism. A Nicol Prism is a calcite crystal cut in to two halves, and again
joined by using a transparent paste called Canada Balsam. The ordinary ray in a calcite
crystal has a refractive index 1.658 and refractive index of extra ordinary ray is 1.486
(this is not a constant, and can vary with the design of the Nicol Prism since refractive
index of E ray depends on direction of light). The Canada balsam layer can act as rarer
medium for O-ray and a denser medium for E-ray. So by adjusting angle of the Canada
balsam layer makes with the O-ray, it is possible to make this ray to undergo total
internal reflection and not transmitted. The transmitted ray is in now plane polarized. The
formation of O-ray and E-ray is base on the direction of the light makes with the optic
axis.
We have already seen that in a direction perpendicular to the optic axis of the
crystal both ray travels along the same direction but with different velocities. Also both
rays are linearly polarized. This situation can be exploited to make some highly useful
optical devices like half wave plate and quarter plate. Consider that a calcite crystal is cut
in the form of a thin plate of uniform thickness so that the optic axis is lying on the plane
of the face of the plate. A direction perpendicular to the face of this plate is also a
direction perpendicular to the optic axis. A beam of light falls normally on this plate is
transmitted without deviation. But they get split up into two beams internally. These two
beams are linearly polarized in mutually perpendicular planes. They travel with unequal
velocities. So when emerging out of the opposite face of the plate there will be a phase
difference between these two rays. They combine to form a single beam at the point of
emergence. But due to the phase difference introduced between two components the
properties of light is different from that of the incident beam.
Now consider the thickness of the plate is so adjusted that the two rays make a
path difference exactly equal to one fourth of the wavelength of light used. (Here we
assume that light is monochromatic). We call this a quarter wave plate. One fourth of
wavelength is equal to a phase difference of π/2. So by using a quarter wave-plate, on the
point of emergence we combine two plane waves which are in perpendicular planes and
with a phase difference of π/2. Such a combination will give an elliptically polarized light
if the amplitudes of the two components are unequal. And if the amplitudes of the two
plane waves are equal we can produce circularly polarized light. How can we make the
amplitudes equal? We need to make a plane polarized light to fall on one of the faces of
the quarter wave plate. And plane of polarization should make an angle 45o with the optic
axis. Remember that the optic axis of the quarter wave plate is lying on the plane of the
faces. In this way it is possible to make a circularly polarized beam of light from the
linearly polarize beam using a quarter wave plate. If a linearly polarized beam of light is
made to fall on the quarter wave plate, normally we get an elliptically polarized light and
circularly polarized light is produced only if the plane of polarization of the linearly
polarized light makes equal components of O-ray and E-ray.
A half wave plate also works on the same manner. But the path difference
produced is one half of a wavelength. Such a device can rotate the plane of polarization
of the linearly polarized light through 180o.
It may be noted that both calcite and quartz exhibits the phenomenon of
birefringence. But in calcite crystal the E-ray is faster than O-ray. For this matter calcite
crystal is called a negative crystal. The O-ray in quartz is faster than the E-ray and quartz
is known to be positive crystal.
Usually quartz is used to make half wave and quarter wave plate. For quartz the
difference in refractive index is smaller than that in calcite. So the plates made of quartz
are thicker than that of calcite and therefore it is easier to cut and polish quartz wave
plates.
In the direction perpendicular to the optic axis let no and nE are the refractive
indices corresponding to the O-ray and E-ray of a quartz crystal, then the thickness of the
quarter wave plate is given by the relation
tq 

4(nE  nO )
And thickness of a half wave plate is given by the equation
th 

2(n E  nO )
Note that nE greater that nO for quartz.