Prakash Gupta Polarization, B. Sc. 2nd Year

Polarization

1 Q. Explain Malus Law. Explain polarization by reﬂection

and scattering.

When unpolarized light is incident on a polarizer, the transmitted light is linearly

polarized. If this light further passes through an analyser, the intensity varies with

the angle between the transmission axes of the polarizer and analyzer.

According to Malus, the intensity of polarized light tranmitted through the analyzer

id proportional to cosine square of the angle between the plane of transmission of the

analyzer and the plane of tranmission of the polarizer. This is Malus’s Law.

If unpolarized light of intensity Iois incident on a polarizer, plane polarized light

of intensity I1is transmitted by it. This plane polarized light passes through the

analyzer. Let abe the amplitude of vibration and θbe the angle that this vibration

makes with the axis of analyzer.

acan be resolved into two components; ax=asinθ perpendicular to plane of analyzer

and ay=acosθ parallel to plane of analyzer.

Only the component ayis transmitted by the analyzer.

∴Intensity corresponding to this component is

I=a2cos2θ

=I1cos2θ[∴I1=a2]

where I1is the intensity of polarized light.

If θ= 0oaxes is parallel, I=I1

θ= 90oaxes is perpendicular, I= 0

θ= 180oaxes is parallel, I=I1

θ= 270oaxes is perpendicular, I= 0

Prakash Gupta Polarization, B. Sc. 2nd Year

Thus, we obtain two positions of maximum intensity and minimum (zero) intensity

when we rotate the axis of analyzer with respect to polarizer.

2 Polarization by reﬂection

When an unpolarized light is incident on a plane surface of denser medium like glass

or water at a certian angle, the ray reﬂected from the surface is plane polarized.

This is because of the reason that every vibration can be resolved into two perpendic-

ular components, one perpendicular to the plane of incidence and the other lying in

the plane of incidence. The two components are reﬂected to diﬀerent extents.

At a particular angle of incidence θp, the reﬂected ray contain only perpendicular com-

ponent but not any parallel component. Hence, it is totally linearly polarized. The

angle θpis called polarizing angle.

3 Polarization of Scattering

If a narrow beam of natural (unpolarized) light is incident on aa transparent medium

containing a suspension of ultramicroscopic particles, the light scattered is partially

polarized. The degree of polarization depends on the angle of scattering. The beam

scattered at 90owith respect to the incident direction is linearly polarized.

Prakash Gupta Polarization, B. Sc. 2nd Year

The direction of vibration in the scattered light will be perpendicular to the plane de-

ﬁned by the direction of propagation. For example, sunlight scattered by air molecules

is polarized.

4 Polarization by Refraction

A beam of ordinary light is incident at the polarizing angle on the piles of plates.

Light vibrating parallel to the plane of incidence is transmitted 100% whereas the

perpendicular vibrations is only 85% transmitted & 15% reﬂected.

When the light is passed through a series of plates, some perpendicular vibrations are

reﬂected, and sone is transmitted at each plate. But the parallel vibration is com-

pletely transmitted. After few plates, the transmitted light is completely free from the

perpendicular vibrations and have vibrations only in the plane of incidence. Thus, we

get plane-polarized light by reﬂection with the help of a pile of plate.

Prakash Gupta Polarization, B. Sc. 2nd Year

The pile of plates consistes of number of glass plates (microscope cover slips) and are

supported in a tube of suitable size and are inclined at an angle of 32.5oto the axis of

the tube.

5 Polarization by Double Refraction

When a beam of unpolarized light is allowed to pass through a quartz crystal, it splits

up into two refracted beams instead of one. This phenomenon is called double refrac-

tion.

If a ray of light from a point source is incident on a calcite crystal making an angle of

incidence iit is refracted along two paths AB and AC making angle of refraction r1

and r2respectively. These rays emerge out along BO and CE parallel to each other

as shown in ﬁgure.

Prakash Gupta Polarization, B. Sc. 2nd Year

If the ray BO obeys the oridinary laws of refraction, it is called ordinary ray and the

other ray that does not obey Snell’s law is called extraordinary ray.

For ordinary ray,

the refractive index µo=sini

sinr1= constant

and for extraordinary ray

the refractive index µe=sini

sinr2is not constant

but varies with the angle of incidence i.

Both the rays are plane polarized, but their plane of polarization are at right angle to

each other. The vibration of o-ray are normal to the plane of paper while those e-ray

are in the plane of paper as shown in ﬁgure.

6 What is Nicol’s Prism? Explain how it is used to produce

and analyze plane polarized light.

Nicol prism is an optical device used for producing and analyzing plane polarized light,

When a beam of light is transmitted through a calcite crystal, it breaks into two rays,

the ordinary ray having vibration perpendicular to princcipal section of the crystal

and the extraordinary ray which has vibration parallel to principal section.

The Nicol prism is designed in such a way that it can eliminate one of the two rays by

total internal reﬂection. It is generally found that the ordinary ray is eliminated and

only the extraordinary ray is transmitted through the prism.

Let us consider a section of crystal ACFC. The section is cut into parts and joined

together by Canada balsam whose refractive index lies between refractive indices for

ordinary and extraordinary rays for the crystal.

Canada balsam acts as a rarer medium for an ordinary ray and it acts as denser

medium for extraordinary ray. When ordinary ray passes through the crystal and

incident on the layer of Canada balsam, the ordinary and extraordinary ray undergo

following phenomenon:

Prakash Gupta Polarization, B. Sc. 2nd Year

i) If the angle of incidence is more than the critical angle for ordinary light, it is

totally internally reﬂected and only the extraordinary ray passes through the nicol

prism. Therefore a ray of unpolarized light on passing through the nicol prism in this

position becomes plane-polarized.

ii) If the angle of incidence is less than the critical angle for the ordinary ray, it is not

reﬂected and is transmitted through the prism. In this position both the ordinary and

extraordinary rays are transmitted through the prism.

iii) If the angle of incidence is more than both the critical angle for the ordinary and

extraordinary ray, both the rays are totally internally reﬂected at the Canada balsam

layer.

The sides of the Nicol prism are coated with black paint to absorb the ordinary ray

that are reﬂected towards the sides by the Canada balsam layer.

6.1 Nicol Prism as an analyzer

When two Nicol prism P1and P2are placed adjacent to each other, on of them acts

as a polarizer and the other acts as an analyzer. The position of two parallel Nicols

are such that only the extraordinary ray passes through both the prisms.

If the prism A is gradually rotated, the intensity of the extraordinary ray decreases

in accordance with Malus Law and when the two prisms are at right angles at each

other, then no light comes out of the prism A. It means that light coming out of prism

P is plane polarized. Thus, the prism P produces plane-polarized light and prism A

detects it.

Hence, Prism P and Prism A are called the polarizer and the analyzer respectively.

7 Q. Give an account of Huygen’s theory of double refraction

in a uniaxial crystal.

Huygen’s explain the phenomenon of double refraction by extending his theory of sec-

ondary wavelets. According to his theory,

Prakash Gupta Polarization, B. Sc. 2nd Year

i) Every point on the crystal surface is distributed by incident wavefront and becomes

the origin of two wavelets; ordinary and extraordinary.

ii) The ordinary ray obeys both the laws of refraction. The velocity of ordinary ray is

same in all direction. So, refractive index for ordinary ray, µo=sini

sinr =c

vo= constant.

So, the wavefront of ordinary ray is spherical.

iii) The extraordinary ray doesnot obey the laws of refraction. The ratio sini

sinr =c

varies with direction of incidence. The velocity varies with angle of incidence. So, the

wavefront is an ellipsoid in nature.

iv) The properties of uniaxial crystal are perfectly symmetrical about the optic axis.

v) The velocity of extraordinary ray is equal to ordinary ray along optic axis. So,

uniaxial crystal do not show double refraction along the optic axis.

Thus, according to Huygen’s, the point source in a double refracting medium is the

origin of two wavefronts. For the ordinary rays, the velocity is constant and so the

wavefront is spherical. For the extraordinary ray, the velocity varies with the direc-

tion and the wavefront is an ellipsoid of revolution. The velocities of ordinary and

extraordinary rays are same along the optic axis.

Optic axis

Negative Crystal

(Calcite)

Optic axis

Positive Crystal

(Overty)

OE E O

Negative uniaxial crystal

If the ordinary wave surface lies in the extraordinary wave surface, such crystal are

known as Negative crystal. Here, µe< µo.

The velocity of extraordianry ray is least & equal to velocity of the ordinary ray along

the optic axis but it is maximum at right angles to the direction of the optic axis.

Positive uniaxial crystal

Prakash Gupta Polarization, B. Sc. 2nd Year

If the extraordinary wave surface lies in the ordinary wave surface, such crystal are

known as Positive crystal. Here, µe> µo.

The velocity of extraordinary ray is least in a direction at right angle to the optic axis.

It is maximum and is equal to the velocity of the ordinary ray along the optic axis.

Hence, from Huygen’s theory, the wavefront or surface in uniaxial crystals are a sphere

and an ellipsoid and there are two points where these wavefronts touch each other.

The direction of line joining these two points is the optic axis.

8 Q. Sketch the formation of ordinary and extraordinary rays

for oblique/normal incidence when the optic axis is in the

plane of incidence and parallel/perpendicular to the crystal

surface.

A) Optic Axis in the plane of incidence and parallel to the crystal surface.

Oblique Incidence

Let AB be the incident plane wavefront of the rays falling obliquely on the surface

MN of the negative crystal. The optic axis is in the plane of incidence and parallel to

the crystal surface. The spherical wavefront and the ellipsoidal wavefront originating

from point A touch each other along the line MN.

optic axis

air

crystal

Here, BC

=AP

=AQ

(1)

where, vais velocity of light in air along BC.

vois velocity of ordinary light along AP.

veis velocity of extraordinary light along AQ.

From 1st two,

AP =BC

va×vo=BC

µo

(2)

Prakash Gupta Polarization, B. Sc. 2nd Year

and from 1st and last,

AQ =BC

va×ve=BC

µe

(3)

where, µoand µeare the refractive indices of ordinary and extraordinary ray along

AP and AQ respectively. The ordinary and extraordinary rays travel with diﬀerent

velocities along diﬀerent directions.

Here, µo> µe

Normal Incidence

AB is the incident plane wavefront of the rays falling normally on the surface MN

of the negative crystal. PR is the refracted wavefront for the extraordinary ray. The

wavefront PR and QS are parallel. The ordinary & the extraordinary rays travel along

the same direction but with diﬀerent velocities in the crystal. The extraordinary ray

travels faster than the ordinary ray.

optic axis

A B

E,O E,O

P R

air

crystal

B) Optic Axis in the plane of incidence and perpendicular to the crystal surface.

Oblique Incidence

Let AB be the incident plane wavefront of the rays falling obliquely on the surface

MN of the negative crystal. The optic axis is in the plane of incidence and normal to

the crystal surface. The spherical wavefront and the ellipsoidal wavefront originating

from point A touch each other in the direction of the optic axis.

Prakash Gupta Polarization, B. Sc. 2nd Year

M N

optic axis

air

crystal

Here, BC

=AP

=AQ

(4)

where, vais velocity of light in air along BC.

vois velocity of ordinary light along AP.

veis velocity of extraordinary light along AQ.

From 1st two,

AP =BC

va×vo=BC

µo

(5)

and from 1st and last,

AQ =BC

va×ve=BC

µe

(6)

where, µoand µeare the refractive indices of ordinary and extraordinary ray along

AP and AQ respectively. The ordinary and extraordinary rays travel with diﬀerent

velocities along diﬀerent directions.

Normal Incidence AB is the incident plane wavefront of the rays falling normally

on the surface MN of the negative crystal.

M N

optic axis

A B

O,E O,E

air

crystal

Prakash Gupta Polarization, B. Sc. 2nd Year

The spherical and ellipsoidal wavelets originating from point A touch each other at P.

There is no separation of ordinary and extraordinary rays. Both travel with the same

velocity along the optic axis. Here, µo=µealong AQ.

9 Q. Explain with necessary theory about plane, elliptical

and circularly polarized light?

Let a monochromatic light be incident on a Nicol prism, it splits into E-ray and O-ray.

The ordinary ray is totally internally reﬂected. The E-ray from Nicol prism is incident

normally on a calcite crystal with its face cut parallel to the optic axis. The plane

polarized light again gets double refracted into O-component and E-component. They

move along the same direction with diﬀerent velocity and introduce a path diﬀerence

or phase diﬀerence between them.

Plane

polarized

Light

Nicol Prism Calcite

Plane

polarized

Light Optic

axis

Let OR = A be the amplitude of the incident plane polarized light on the crystal and

θbe the angle between the direction of incident vibration and optic axis. Thus,

The two vibration components are: -

Acosθ along OP (E-ray)

Asinθ along OQ (O-ray)

Asinθ

Acosθ

E-ray and O-ray can be expressed as

x=Acosθsin(ωt +δ) = asin(ωt +δ)

y=Asinθsin(ωt) = bsin(ωt)

Prakash Gupta Polarization, B. Sc. 2nd Year

So,

sinωt =y

band cosωt =1−y2

b21/2

∴x=asin(ωt +δ)

or, x=a(sinωt.cosδ +cosωt.sinδ)

or, x

a=y

bcosδ +1−y2

b21/2

sinδ

or, x

a−y

bcosδ =1−y2

b21/2

sinδ

Squaring on both sides, we get,

a2−2xy

ab cosδ +y2

b2cos2δ=1−y2

b2sin2δ

or, x2

a2−2xy

ab cosδ +y2

b2=sin2δ

(7)

This is general equation of ellipse.

CASE I: When δ= 0, cosδ= 1 and sinδ= 0

Equation (1) becomes,

a2−2xy

ab +y2

b2= 0

or, x

a−y

b2= 0

or, y=b

This is the equation of Straight line passing through origin. Thus, the resultant light

is a plane polarized light.

CASE II: When δ=π

2, cosδ= 0 and sinδ= 1

Equation (1) becomes,

a2+y2

b2= 1

This is the equation of an ellipse and hence the resultant light is called elliptically

polarized light, if a 6= b

Case III: When δ=π

2and θ= 45o

a=Acosθ =1

√2A

b=Asinθ =1

√2A

∴a=b

Prakash Gupta Polarization, B. Sc. 2nd Year

Equation (1) becomes,

x2+y2=a2(orb2)

This is the equation of a circle having radius a. Thus, the resulting light is a circulat-

ing polarized light. Here, the vibrations of the incident plane polarized light on the

crystalm make an angle of 45owith the direction of the optic axis.

10 Polarization:

Polarization of light is the process of conﬁning the vibrations of the electric vector of

light waves to any particular directions.

In polarized light, the electric ﬁeld vibrates in all direction perpendicular to the direc-

tion of propagation.

If the vibration is conﬁned to one direction perpendicular to the direction of propaga-

tion, it is plane-polarized light.

When two plane polarized light having diﬀerent phase overlap each other, the resul-

tant is partially polarized light.

If the two overlaping waves have same amplitude and a constant phase diﬀerence of

2, the resultant is circularly polarized light.

If the two overlaping wave have diﬀerent amplitude and a constant phase diﬀerence of

2, the resultant is elliptically polarized light.

11 Q. What is quarter wave plate? How is it used to produce

and detect elliptically and circularly polarized light?

A quarter wave plate is a retardation plate which produces a phase diﬀerence of 90o

or a path diﬀerence of λ

4between the ordinary ray and extra ordinary ray from it.

It is a thin plate of birefrigerant crystal having the optic axis cut parallel to its refract-

ing faces. Its thickness is adjusted in such a way that it introduces a quarter wave λ

path diﬀerence between the e-ray and the o-ray propagating through it.

Prakash Gupta Polarization, B. Sc. 2nd Year

When a plane polarized light wave is incident on a quarter wave plate, the wave splits

into e-ray and o-ray. The two waves travel along the same direction but with diﬀerent

velocities. As a result, an optical path diﬀerence of λ

4would be developed between

them, when they emerge from the rear face of the crystal.

If t = thickness of the quarter wave plate.

µo= refractive index for ordinary light.

µe= refractive index for extraordinary light.

Then, path diﬀerence = µet−µot( for positive crystal)

∴

4= (µe−µo)t

or, t=λ

4(µe−µo)

Similarly, for negative crystal

t=λ

4(µo−µe)

For a path diﬀerence x = λ

φ=2πx

λ=2π

λ.λ

4=π

i.e., a quarter wave plate introduces a phase angle of π

Production of ELliptically and Circularly polarized light and detection.

A quarter wave plate is used in producing elliptically & circularly polarized light. It

converts plane polarized light into elliptically and circularly polarized light depending

upon the angle that the incident light makes with the optic axis of the quarter wave

plate.

Elliptically Polarized light

For elliptically polarized light, two vibrations which are at right angle to each other

should have unequal amplitudes with path diﬀerence of λ

4or a phase diﬀerence of π

between them.

The following arrangement can be used for this purpose. A parallel beam of

monochromatic light is allowed to fall on the nicol prism N1. The prism N1and

Prakash Gupta Polarization, B. Sc. 2nd Year

N2are cross and the ﬁeld of view is dark. A quarter wave plate is introduced between

N1and N2. The plane polarized light from the Nicol prism N1falls normally of the

quarter wave plate. The ﬁeld of view is illuminated and the light coming from the

quarter wave plate is elliptically polarize (angle of vibration of incident light with op-

tic axis of quarter wave plate is not equal to 45o). The nicol prism N2is rotated, if

intensity of illumination of the ﬁeld of view varies between a maximum and minimum,

the light will be elliptical polarized light.

Circularly Polarized Light

For circularly polarized light, two vibrations which are at right angles to each other

should have equal amplitude with path diﬀerence of λ

4or a phase diﬀerence of π

2be-

tween them.

The following arrangement can be used for this purpose. A parallel beam of monochro-

matic light is allowed to fall on the nicol prism N1. The prism N1and N2are cross

and the ﬁeld of view is dark. A quarter wave plate is introduced between N1and N2.

The plane polarized light from the Nicol prism N1falls normally on the quartz wave

plate. The wave plate is rotated such that the angle of vibration of incident light with

the optic axis of wave plate is 45o. Now the light coming from the quarter wave plate

is circularly polarized light.

The intensity of illumination of the ﬁeld of view does not varies with the rotation of

N2. But, this case also applied for ordinary light.

So, to identify between ordinary and circularly polarized light, it is again passed

through 2nd quarter wave plate that produces a further change of path diﬀerence of

4. The resulting light is plane polarized. When viewed through rotating nicol prism.

The light coming from 1st quarter wave plate is circularly polarized.

12 Half Wave Plate

It is a thin plate of birefrigent crystal having optic axis cut parallel to the refracting

faces. Its thickness is adjusted that it introduces a half wave λ

2path diﬀerence between

the e-ray and o-ray.

When a plane polarized light is incident on a half wave plate, the wave splits into e-ray

Prakash Gupta Polarization, B. Sc. 2nd Year

and o-ray. The two waves travel along the same direction but with diﬀerent velocities.

As a result, an optical path diﬀerences of λ

2would be developed between them, when

they emerge from the rear face of the crystal.

Then, path diﬀerence = µet−µot( for positive crystal)

∴

2= (µe−µo)t

or, t=λ

2(µe−µo)

Similarly, for negative crystal

t=λ

2(µo−µe)

For a path diﬀerence x = λ

φ=2πx

λ=2π

λ.λ

2=π

i.e., a half wave plate introduces a phase diﬀerence of π

13 Q. What is Optical Activity? Outline the principle of

half shade polarimeter. Explain with necessary theory to

determine the optical activity of a solution.

There are certain substances that can rotate the phase of vibration of plane polarized

light when passed through it eg. Sugar solution, sodium chloride etc. This phe-

nomenon of rotating the plane of vibration by certain substance is known as optical

activity and the substance is said to be an optically active substance.

If the substance rotates the plane of vibration in the clockwise direction, it is called

dextrorotatory (right-handed). If the substance rotates the plane of vibration in the

anticlockwise direction, it is said to be laevo-rotatory (left-handed).

Biot studied the phenomenon of optical rotation and proposed the following laws:-

i) Optical rotation is directly proportional to the length of optically active substance.

i.e., θ∝l

ii) Optical rotation is directly proportional to the concentration of the substance.

i.e., θ∝c

iii) Optical rotation is inversely proportional to square of wavelength of light passing

through it

i.e., θ∝1

λ2

iv) The optical Activity depends upon the temperature

i.e., θ=F(T)

Prakash Gupta Polarization, B. Sc. 2nd Year

The speciﬁc rotation for a given wavelength of light at a particular temperature is

deﬁned as the rotation of plane of polarization in degrees produced by a path of one

decimeter length of the solution of concentration 1.0 gm/cc.

Mathematocally,

λ=θ

Polarimeter

It is a optical device used to measure the optical rotation of the plane vibration of the

polarized light by the optical. One of the type of polarimeter is Laurent’s half shade

polarimeter.

It consists of Nicol prisms N1and N2and a half shade device. One half portion of the

half shade is made of quartz and other half is made up of glass plate which absorbs or

emits the same quantity of light as is done by quartz half. A vernier scale is used to

measure the rotation.

A monochromatic light from source S incident on N1after passing through it becomes

plane polarized. Half of the polarized light passes through quartz and other half

through the glass. The light which passes through the quartz splits into E and O

components. On emergence, a phase diﬀerence of πis introduced between them. The

light which passes through glass has no such process.

The vibration of the emerging out of quartz will be along CD and that of glass will

be along AB. If the analyzer N2has its principle section parallel to YY‘ bisecting the

angle AOC, the amplitude of light incident of N2from half will be equal. So, the ﬁeld

of view will be equally bright as shown in the ﬁgure.

θ θ

x‘

Glass x

Quartz

y‘

Prakash Gupta Polarization, B. Sc. 2nd Year

If N2is rotated to the left of YY‘, then left half will be brighter than right half. If N2

is rotated to the right of YY‘, then the right half will be brighter than the left is.

If the principle section of the analyzer is perpendicular to YY‘, the ﬁeld of view is

equally dark.

Determination of Speciﬁc Rotation

Sugar solution can be used as an optically active substance. At ﬁrst, the analyzer is set into the position

of equal brightness of the ﬁeld of view without sugar solution in tube T. The tube T is ﬁlled with pure

water, and the analyzer is again set to equally bright ﬁeld of view. The position of analyzer is noted.

Now, the tube is ﬁlled with sugar solution of known concentration. The sugar solution rotates the

plane of vibration in the clockwise direction. The analyzer is rotated in the clockwise direction to

obtain equally bright ﬁeld of view. This position is again noted.

The diﬀerence between the two positions gives the angle of rotation θ.θis measured for diﬀerent

concentration of sugar solution. A graph is plotted between concentration and angle of rotation θ. The

graph is a straight line passing through the origin.

The speciﬁc rotation of sugar solution is then calculated using the relation.

λ=10 ×θ

where,

l= length of sugar solution in cm.

c= concentration of sugar solution in gm/cc.

θ= angle of rotation in degree.

rotation

concentration