Polarization of Light: Notes with Examples - IIT JAM PDF Download

Plane of Incidence

Incident ray, reflected ray, refracted ray and normal to incidence forms a plane that plane

is plane of incidence.

Plane of Polarization

The plane passing through the direction of propagation and containing no vibration is

called the “Plane of Polarization”.

Production of Plane Polarized Light

Different methods of production of polarized light

• Polarization by reflection
• Polarization by refraction
• Polarization by selective absorption
• Polarization by double refraction
• Polarization by scattering

Polarization by Reflection

If a linearly polarized wave (Electric vector associated with the incident wave lies in the

plane of incidence) is incident on the interface of two dielectrics with the angle of

incidence equal toθ . If the angle of incidence θ is such that

then the reflection coefficient is zero.

Thus if an unpolarized beam is incident with an angle of incidence equal to p θ , the

reflected beam is plane polarized whose electric vector is perpendicular to the plane of

incidence.

Above equation is known as Brewster’s law. The angle θ_p is known as the polarizing

angle or the Brewster angle. At this angle, the reflected and the refracted rays are at right

angle to each other i.e.

θ_{p + r =  π/2  ⇒ n =} = ⇒ tanθ_p = n

For the air-glass interface, n₁ =1 and n₂ ≈1.5 giving θ_p ≈ 57⁰

Polarization by Refraction

If an unpolarized beam is incident with an angle of incidence equal to p θ , the reflected

beam is plane polarized whose electric vector is perpendicular to the plane of incidence.

The transmitted beam is partially polarized and if this beam is made to undergo several

reflections, then the emergent beam is almost plane polarized with its electric vector in

the plane of incidence.

If I_p and I_s be the intensity of the parallel and perpendicular component in refracted

light, then the degree of polarization is given by 

where m is the number of plate and n is the refractive index.

Polarization by selective absorption

A simple method for eliminating one of the beams is through selective absorption; this

property of selective absorption is known as dichroism. A crystal such as tourmaline has

different coefficients of absorption for the two linearly polarized beams into which the

incident beam splits up. Consequently, one of the beams gets absorbed quickly, and the

other component passes through without much attenuation. Thus, if an unpolarized beam

is passed through a tourmaline crystal, the emergent beam will be linearly polarized

Polarization by Double Refraction

When a ray of unpolarized light passed through doubly refracting crystal (calcite or

quartz), it split up into two refracted rays. One of the ray obeys the ordinary laws of

refraction i.e. ( n remains constant) and it is called ‘ordinary ray’ (o -ray). The other

behaves in an extra ordinary way (i.e. n varies) and called ‘extraordinary ray’ ( e -ray).

It is found that both the ordinary and extraordinary rays are plane-polarized having

vibration perpendicular to each other.

If one can sandwich a layer of a material whose refractive index lies between the two,

then for one of the beams, the incidence will be at a rarer medium and for the other it will

be at a denser medium. This 

principle is used in a Nicol prism

which consists of a calcite crystal

cut in such a way that for the

beam, for which the sandwiched

material is a rarer medium, the angle of incidence is greater than the critical angle. Thus

this particular beam will be eliminated by total internal reflection. Following figure

shows a properly cut calcite crystal in which a layer of Canada balsam has been

introduced so that the ordinary ray undergoes total internal reflection. The extraordinary

component passes through, and the beam emerging from the crystal is linearly polarized.

Polarization by Scattering

When an unpolarized beam moving in z -axis, strike a gas atom than the electric field in

unpolarized beam sets the electric charges in the gas atom in vibration. Since the E -field

is in xy plane, therefore, the vibration will take place in xy plane only. These vibration

can resolve in x and y direction and can said that two dipoles oscillating one in x and

other in y -direction with the frequency of incident light. Since an oscillating dipole does

not radiate in the direction of its own length.

Therefore, observer along x -axis will receive light component vibrating only in y -

direction and along y -axis the component will vibrate only x axis. Hence observer will

receive Plane Polarized light.

Blue colour of sky

Lord Rayleigh Formula: Intensity of light scattered from firie particles (dimension ≤ λ )

varies inversely as the fourth power of λ . i.e. Scattered intensity of blue light is more than red blue red ∵ λ_blue < λ_red .

Why blue scattered more: Natural frequency of bound electron in gas lies in ultra-violet region. Blue light closer to natural frequency than red light, therefore, blue is more effective in causing the electron to oscillate and thus it get more effectively scattered than

red.

Red colour of sunrise and sunset

The path of the light through the atmosphere at sunrise and sunset is greatest. Since the

violet & blue light is largely scattered and get removed and what we see is rest having red

component.

Malus’ law

An unpolarized light beam gets polarized after passing through the Polaroid 1 P which has

a pass axis parallel to the x axis. When this x -polarized light beam incident on the

second Polaroid 2 P whose pass axis makes an angle θ with the x axis, than the intensity

of the emerging beam will vary as

I = I₀ cos²θ

where I₀ represents the intensity of the emergent beam when the pass axis of P₂ is also

along the x axis (i.e., when θ = 0 ), above equation known as Malus’ law.

Thus, if a linearly polarized beam is incident on a Polaroid and if the Polaroid is rotated

about the z axis, then the intensity of the emergent wave will vary according to the

above law.

Superposition of Two Disturbances and Production of Polarized Wave 

Superposition of Two Waves with Parallel Electric Field

Let us consider the propagation of two linearly polarized electromagnetic waves (both

propagating along the z axis) with their electric vectors oscillating along the x axis. The

electric fields associated with the waves can be written in the form

E₁ = xˆa₁ cos (kz −ω t +θ₁)                                             (1)

E₂ = xˆa₂ cos (kz −ω t +θ₂)                                             (2)

where a₁ and a₂ represent the amplitudes of the waves, ˆx represents the unit vector

along the x axis, and θ₁ and θ₂ are phase constants. The resultant of these two waves is

given by

E = E₁ + E₂                                                                     (3)

which can always be written in the form

E = xˆa cos (kz −ω t +θ )                                                  (4)

where                              (5)

represents the amplitude of the wave. Equation (4) tells us that the resultant is also a

linearly polarized wave with its electric vector oscillating along the same axis.

Superposition of Two Waves with Mutually Perpendicular Electric field

We next consider the superposition of two linearly polarized electromagnetic waves (both

propagating along the z axis) but with their electric vectors oscillating along two

mutually perpendicular directions. Thus, we may have

E₁ = xˆa₁ cos kz −ωt                                                      (6)

E₂ = yˆa₂ cos kz −ωt +θ                                                 (7)

For θ = nπ , the resultant will also be a linearly polarized wave with its electric vector

oscillating along a direction making a certain angle with the x axis; this angle will

depend on the relative values of a₁ and a₂ .