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# Birefringent Materials Optic Axis

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What is meant by the optic axis of a uniaxial birefringent crystal? Could you please describe and explain how birefringent materials may be used to produce linearly and circularly polarized light from unpolarized light? Thank you.

https://brainmass.com/physics/velocity/birefringent-materials-optic-axis-169309

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My answer to the question is in the attached word document.

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

And thickness of a half wave plate is given by the equation

Note that nE greater that nO for quartz.

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