Page | 1

Name of Faculty; Dr.Jitendra kumar

Department of Physisc

Unit-1

Interference of light

Contents

1. Newtons ring

2. Application of Newtons ring

3. Michelson Interferometer

4. Application of Michelson Interferometer

5. Anti-reflection coating

6. Interference filter

Page | 2

Newtons Ring

Formation of Newtons rings When a Plano-convex lens of large radius of curvature is

placed with its convex surface in contact with a plane glass plate, an air-film of gradually

increasing thickness from the point of contact is formed between the upper surface of the plate

and the lower surface of the lens. If monochromatic light is allowed to fall normally on this film,

then alternate bright and dark concentric rings with their centre dark are formed. These rings are

known as Newtons rings. The fringes are circular because the air film is symmetrical about the

point of contact of the Plano-convex lens with the plane glass plate.

Newtons rings are formed because of the interference (by division of amplitude) between the

waves reflected from the top and bottom surfaces of an air-film formed between the plano-

convex lens and the plate.

Newtons rings in reflected light We know that when monochromatic light falls normally on

a plano-convex lens resting on a plane glass plate, alternate bright and dark concentric rings

with dark centre are formed due to waves reflected from the top and bottom surfaces of an air-

film or any other medium of refractive index between the plano-convex lens and plane glass

plate.

For reflected system, the effective path difference is given by

2 cos 22 2

t r t

(since light is falling normally, cos 1r ) 1

Page | 3

At the point of contact 0t , therefore, effective path difference 2

This is the condition for minimum intensity. Hence, the centre of Newtons rings is dark.

For constructive interference (bright fringes/maxima)

22

t n

n = 0, 1, 2, 3..

Or 2 2 12

t n

2

For destructive interference (dark fringes/minima)

2 2 12 2

t n

n = 0, 1, 2, 3. .

Or 2 t n 3

Diameters of Bright and dark Rings

Page | 4

Therefore, 2 2 2ON PN OP

Or 2 2 2

n nR t r R

Or 2 2 2 22n n nR t Rt r R

Or

2

2 nnr

tR

neglecting 2nt , since nt is small 4

Determination of refractive index of a liquid - Newtons ring experiment can be used to

determine the refractive index of a liquid. The liquid whose refractive index is to be determined

is placed between the plano-convex lens L and the glass plate P of the Newtons ring set-up. In

case liquid is rarer than glass, a phase change of will occur at reflection from the lower surface

of the liquid, but if the liquid is denser than glass, phase change will occur at reflection from the

upper surface of the film. Hence, in both the cases, path difference will be equal to /2.

therefore, effective path difference 2 cos2

t r

for normal incidence 0,cos 1r r

Or effective path difference 22

t

and from eq4 2

2 nnr

tR

Page | 5

for nth bright fringe, 22

t n

Or 2 1

22

nt

Or 2 2 1

2

nnr

R

Or 2 2 1

2n

nr R

5

If nd is diameter of the thn ring, then 2n nd r

therefore, 2 2 2 1

n

nd R

6

If n pd is the diameter of th

n p ring,

then,

22 2 1

n p

n pd R

7

Or 2 24p

n p nliquid

Rd d

8

Since, 1 , for air

Page | 6

therefore, 2 2 4n p nair

d d p R 9

Or dividing eq9 by eq8

2 2

n p n

2 2

n p n

d d

d d

air

liquid

10

By measuring diameters of thn and

thn p rings for medium as air and liquid and substituting

the values in the eq10, refractive index of the liquid can be determined.

since, liquid > 1, n liquidd < n aird

therefore, when liquid is introduced between the lens and the plate, the diameters of the rings

decrease, that is, rings are contracted.

1

filmair in ring same theofdiameter

film liquidin ring a ofdiameter

Michelsons interferometer

Construction and working -

M.i is a device that can be used to measure lengths or changes in

length with great accuracy by means of interference fringes. The basic principle of this

instrument was given by A.A. Michelson in 1881 according to which when a parallel beam of

monochromatic light coming from an extended source is incident on a half silvered glass plate

(also called as beam splitter), it is divided into two parts. One part is reflected wave and the

other part is a refracted wave and both are coherent. In this experiment, coherent waves are

Page | 7

produced by the method of division of amplitude. These waves proceed in the perpendicular

directions and are incident normally on the two mirrors. After reflections from these mirrors, they

superpose and produce interference fringes, which are observed with the help of a telescope as

shown in the figure.

Michelsons interferometer consists of two highly polished mirrors M1 and M2 and two plane

glass plates A and B parallel to each other but inclined at 45. The glass plate A is half silvered

so that half of the intensity of the incident wave is reflected and rest is transmitted. Mirrors M1

and M2 are perpendicular to each other. Mirror M2 is kept fixed while mirror M1 is movable. Light

from a monochromatic source S after passing through the lens becomes parallel and falls on the

plate A. The half silvered plate A reflects one half of the energy towards the mirror M1 and the

other half is transmitted towards the mirror M2. These two beams (reflected and transmitted)

travel along two mutually perpendicular paths and are reflected back by the mirrors M1 and M2.

The beam going towards the mirror M1 and reflected back passes twice through the glass

plate A whereas the beam going towards the mirror M2 and reflected back does not pass even

once through the glass plate A. Therefore, to compensate for the path, the plane glass plate B

of same thickness and refractive index as A is inserted between the mirror M2 and A. That is

why this plate B is also called as compensating plate. Thus, the light beam going towards the

mirror M2 and reflected back towards A also passes twice through the plate B. Therefore, the

paths of the two waves in glass are the same. If t is thickness of the glass plates A and B and

the refractive index, extra optical path 2 1 t is introduced in 1. Hence for observing

achromatic fringes with white light, it is essential to compensate for this optical path 2 1 t

for all wavelengths. Thus, the introduction of plate B nullifies the effect of extra path difference

created. The mirror M1 is movable and the distance moved by it can be read on the scale

attached. Mirror M2 is fixed. Since 1 and 2 are derived from the same source, hence they are

coherent and when superpose, interference pattern can be observed through the telescope.

Page | 8

When the mirrors are orthogonal to the optical axis and the half silvered glass plate A (beam

splitter) is at an angle of 45, the interference may be considered due to hypothetical air film

between enclosed between mirror M1 and image M2 of mirror M2. In this case the air film has

constant thickness and since the source is broad, circular fringes of constant inclination are

formed. If mirrors are not orthogonal, localized (straight) fringes are formed.

Type of fringes

In Michelsons interferometer, the form of fringes depends on the separation d between M1 and

M2 and the shape of hypothetical air film formed between M1 and M2

, which is virtual image of

M2.

Circular fringes are produced when the mirrors M1 and M2 are perpendicular to each other

and thickness of air film between M1 and M2 is not equal to zero, that is 0d . If 0d then,

the whole pattern becomes dark.

Appearance of fringes in the Michelsons interferometer as the mirrors are moved away from each

other. Arrows on the far right figure indicate motion of the fringes.

s

Page | 9

If thickness of air film is d , the light waves reflected from the mirror M1 and M2 and reaching

towards the telescope will coming parallel from M1 and M2 and will be equal to 2d . If these

parallel waves make an angle with the normal, the path difference between them will be

2 cosd .

We know that when a wave is reflected from a denser medium and another wave are reflected

from a rarer medium, path difference of 2

is created between them.

Hence, effective path difference between these waves will be 2 cos2

d

.

If 2 cos2

d n

n = 1, 2, 3

Or 2 cos 2 12

d n

1

Then a bright fringe will form due to constructive interference. Same condition will be at all

points on the circle of inclination and bright fringe will appear circular.

If the effective path difference 2 cos 2 12

d n

n = 0, 1, 2

Or 2 cosd n 2

Then a dark fringe will form due to destructive interference. Same condition will be at all points

on the circle of inclination and dark fringe will appear circular. Hence, alternate bright and dark

circular fringes are observed.

Localized fringes are formed when mirrors are not orthogonal, that is, M1 and M2 are not

exactly parallel. A wedge shaped air film is formed between them giving rise to fringes of equal

thickness. The path of the two waves reflected from mirrors M1 and M2 and originating from a

single wave, are no more parallel but intersect near M1 as shown in the figure below and so

fringes are localized near M1. The shapes of these fringes are curved with convex side towards

thin edge of the wedge. As mirror M2 is moved gradually the air film wedge varies successively

and fringes change the shape and when mirrors M1 and M2 intersect each other, fringes

become straight as shown in the figure.

Page | 10

s

Radii of circular fringes- It is clear that in the fringe system of Michelsons interferometer, for

given d , as n increases, cos increases and hence decreases, that is, order of fringes

increases towards center and decreases as we move away from it. For central fringe 0 and order is n , then order of the successive fringes from the central fringe are

1 , 2 , 3 ...n n n and so on.

Then from eq2 2d n 3

If 1st, 2nd, 3rd mth circular fringes subtend semi-angles 1 2 3, , ... m respectively from the

telescope, then

1

2

3

2 cos 1

2 cos 2

2 cos 3

2 cos m

d n

d n

d n

d n m

4

Page | 11

Thus, if fringes are counted from the central fringe (assuming its order zero), then subtracting

eq4 from eq3, we get

2 1 cos md m m = 1, 2, 3 5

Or cos 12

m

m

d

6

If radius of mth fringe is rm and final image of circular fringes is observed at a distance D (least

distance of distinct vision), then

2 2cos 1

2m

m

D m

dr D

Or

12 2

1 12

m

mr D

d

7

If angle m is very small, or 2m d , then using binomial approximation we get

mm

r Dd

8

That is, near the central fringe, radius of fringes is directly proportional to square root of natural

numbers.

Application of Michelson Interferometer;

1. Determination of wavelength of monochromatic light; For this, monochromatic light from source is allowed to fall on half silvered plate A and Michelsons interferometer is adjusted

for circular fringes. Then, mirror M1 is moved such that AM1 = BM2. The mirror M1 and M2 are

Page | 12

made perfectly perpendicular to each other. Thus, concentric circular fringes are observed

through telescope.

Let the separation between real mirror M1 and virtual mirror M2 is such that bright fringe of thn

order is formed at the center of the field of view and let reading of micrometer screw is say 1x .

Then, path difference,

02 cos0d n

Or 2d n 1

Where d is separation between M1 and M2.

Adding on both sides of the eq1, we get

2 1d n

Or 2 12

d n

2

From the above eq2, it is observed that when d becomes2

d

, the thn fringe at the

center is replaced by 1th

n fringe. We can also say that if M1 is moved by distance2

, one

fringe is displaced in the telescope. Now the mirror M1 is gradually moved and number of fringes

displaced is counted and reading of micrometer screw is say 2x . If M1 is moved through distance

2 1x x x and the number of fringes displaced is N . That is, by moving the mirror by2

, the

number of fringes displaced is one.

Page | 13

Therefore, on moving the mirror by distance 2 1x x x , the number of fringes displaced will

be

2x

N

Or wavelength 2x

N 3

Hence, by knowing the values of x and N experimentally, wavelength of monochromatic light used can be calculated.

2.Determination of separation between two close wavelengths - For this, light is

allowed to fall on half silvered plate A and Michelsons interferometer is adjusted for circular

fringes. Let two wavelengths 1 and 2 are very close to each other. The two wavelengths form

their separate fringe patterns, but because of very small difference in wavelengths, the two

patterns overlap. As the mirror M1 is moved slowly, the two patterns separate out slowly and

when the path difference is such that the dark fringe due to 1 falls on the bright fringe due to 2 ,

the result is maximum indistinctness. When the path difference is such that, bright fringe due to

1 falls on the bright fringe due to 2 , or vice-versa, the result is maximum distinctness.

Let the mirror M1 is moved through a distance 2 1x x x between two positions 1x and 2x of

successive distinctness. In this position thn fringe due to 1 must coincide with 1

thn fringe

due to 2 . Therefore,

21 1

2 2

nnx

Or 1

2xn

1

Page | 14

And 2

21

xn

2

Subtracting eq...1 from eq2, we get

2 1

1 11 2x

Or 1 2

1 2

1 2x

Or 1 21 2

2x

Or 2

1 22x

3

Where 1 2 is geometric mean of the two wavelengths. Thus, by measuring the distance x moved by the mirror M1, the difference between two close wavelengths can be

determined.

Compare the rings formed by Michelsons interferometer and Newtons rings.

1. The fundamental difference between the two is that in Michelsons

interferometer rings originate as locus of equal inclination (also called as

Page | 15

Haidingers fringes) whereas the Newtons rings are locus of the air film of

equal thickness (also called as Fizeau fringes).

2. In Michelsons interferometer rings are located at infinity and are therefore

viewed by a telescope whereas Newtons rings are located in the plane of the

film and hence viewed by traveling microscope.

3. The air film in Michelsons interferometer is imaginary (hypothetical) whereas

in Newtons rings experiment it is real.

4. Center of circular rings in Michelsons interferometer can be dark or bright

whereas in Newtons rings, in case of reflected light it is dark and in case of

transmitted light it is bright.

5. In Michelsons interferometer, order of the rings decrease when one moves

outwards from the center whereas in Newtons rings order of the rings

increase when one moves away from the center.

6. In both, Michelsons interferometer and Newtons rings, the thickness of the

rings decreases as radius of the rings increases, which is a common feature.

Anti-reflection coating - Whenever a ray of light moves from one medium to another, for

example, when light enters a sheet of a glass after traveling through air, some portion of the

light is reflected from the surface (known as interface) between the two media. The strength of

the reflection depends on the refractive indices of the two media as well as the angle of the

surface to the beam of light. When the light meets the interface at normal incidence

(perpendicularly to the surface), the intensity of light reflected is given by the reflection

coefficient or reflectanceR .

If 1 and 2 are refractive indices of the two media, then reflectance, R is given by

Page | 16

2

2 1

2 1

R

1

It is clear from the above eq1 that reflection will not occur if 1 2

One of the practical applications of the interference phenomenon is the anti-reflection coating on

the glass. The reflection from a lens or a prism can be decreased to a minimum by coating a

thin transparent film of proper refractive index and proper thickness.

The idea behind anti-reflection coatings is that the creation of a double interface by means of a thin transparent film gives two reflected waves. If these waves are of nearly equal amplitude and out of phase, they partially or totally cancel. If the coating is of quarter wavelength thickness and has refractive index less than that of the glass then the two reflections are 180 degrees out of phase and complete destructive interference occurs and no reflected waves will emerge from the film.

The thickness of coating and refractive index is chosen in such a way that light waves reflected from the two layers have the same amplitude and out of phase so as to cancel one another.

If refractive index of coating be c , that of glass be g and that of air be 0 , then the

amplitude of reflected wave from the first surface (air to coating) is given by

2

0

1

0

c

c

R

2

and the amplitude of reflected wave from the second surface (coating to glass) is given by

Page | 17

2

2

g c

g c

R

3

The condition of equality of amplitude, that is, 1 2R R , at two reflections yield,

22

0

0

g cc

c g c

4

Or 0

0

g cc

c g c

Or 2 2

0 0 0 0c g c g c c g c g c

Or 2

02 2c g

Or 0c g 5

Or c g since, 0 1 (for air) 6

That is, refractive index of coating should be equal approximately to the geometric mean of

refractive indices of media on either side.

The -phase condition gives the thickness of the coating film to be

2 2 12

ct n

7

Page | 18

For minimum thickness, 1n

Or 22

ct

Or 4 c

t

8

That is, optical thickness of the coating must be equal to one-quarter of a wavelength. Thus

transparent coating satisfying eq6 and eq8 eliminates reflection completely. The best

material known for this is MgF2 for which refractive index 1.38 . To have suitability at multi-

wavelengths, for example, white light, multi-layer coating is used. Each layer is optically quarter

wave thick.

Interference filters - are multilayer thin-film devices. They can be designed to function as an

edge filter or band pass filter. In either case, wavelength selection is based on the property of

destructive light interference. This is the same principle underlying the operation of a Fabry-

Perot interferometer.

Incident light is passed through two coated reflecting surfaces. The distance between the

reflective coatings determines which wavelengths destructively interfere and which wavelengths

are in phase and will ultimately pass through the coatings. If the reflected beams are in phase,

the light is passed through two reflective surfaces. If, on the other hand, the multiple reflections

are not in phase, destructive interference reduces the transmission of these wavelengths

through the device to near zero. This principle strongly attenuates the transmitted intensity of

light at wavelengths that are higher or lower than the wavelength of interest.

In many spectroscopic studies, it is required to have a narrow frequency band of light of width

about 100 or less, centered on a chosen wavelength of visible light. It can be obtained by an

interference filter. In an interference filter, a thin transparent dielectric spacer like magnesium

fluoride (MgF2) or cryolite is sandwiched between glass plates. Reflecting surfaces are coated

by extremely thin semi-transparent layers of a good reflecting material like silver, deposited by

vacuum evaporation method or a dielectric of desired characteristics. When a beam of light is

incident normally on the filter, multiple reflections take place within the film.

Page | 19

Path difference between successive pair of emergent parallel rays is 2 t , for normal incidence.

With white light, the transmitted beam will be maximum for only those wavelengths which

satisfies the condition

2 t n n = 1, 2, 3

Or 2

nt

Where, is refractive index of the dielectric and t is its thickness.

If the effective thickness of the spacer is integral multiple of half of the desired wavelength, then

other wavelengths will be attenuated by destructive interference and wavelength ,2 ... will be

transmitted through the filter. If for a particular thickness there are two maxima in the visible

region, one of them can be eliminated by using colored glass filter. However, if the angle of

incidence is and angle of refraction in the spacer is , then, the wavelength of light passing

through the filter can be obtained from

2 cost n

Page | 20

2

2

sin2 1t n

using

sin

sin

and 2 2sin cos 1

For 1n , 2

2

sin2 1t

Or

2

0 2

sin1

where 0 2 t

Unit-11(Polarization of light)

Un-polarized light Ordinary light is a collection of wave trains emitted by atoms or group of

atoms with coherent time no longer than 10-8 second. Each wave train has different orientation

and phase of the electric field. Consequently, it is a mixture of light polarized in different ways to

different degrees, that is, it is randomly polarized or generally referred to as un-polarized light.

Plane polarized light If electric field vector of light oscillates in some definite orientation, that is, it oscillates in a fixed plane as light progresses, the light is said to be plane polarized or

linearly polarized light.

Page | 21

Partially polarized light If a plane or linearly polarized light contains an additional

component of natural or randomly polarized light then light is said to be partially linearly

polarized light.

Elliptically polarized light If electric filed vector rotates on a flattened helix as the wave

progresses, that is, electric field vector rotates on the circumference of an ellipse when

observed from fixed point in space, the light is said to elliptically polarized light. Elliptically

polarized light may be left handed or right handed depending on the rotation of the electric field

vector in anti-clockwise or clockwise direction when seen towards the source.

Left handed elliptical polarization

Right handed elliptical polarizationss

Page | 22

Circularly polarized light If the tip of electric field vector of constant magnitude rotates on the

helix as the wave progresses, that is, when seen from fixed point in space, the electric field

vector appears to rotate on the circumference of a circle, the light is said to be circularly

polarized light. If the tip of electric vector when seen towards the wave (source) rotates

clockwise, the light is said to be right circularly polarized. On the other hand, if it rotates anti-

clockwise, the light is said to be left circularly polarized light.

Left handed circular polarization

Right handed circular polarization

Plane of vibration The plane containing the direction of vibrations of electric field vectors and

the direction of propagation of light in plane polarized light is called as plane of vibration.

Page | 23

Plane of polarization The plane perpendicular to the plane of vibration in which components

of electric field vectors are zero but contains the direction of propagation of light is called as

plane of polarization.

Thus, the plane of vibration and the plane of polarization are perpendicular to each other.

Brewsters law states that the tangent of the angle of polarization pi for a given reflecting

medium is numerically equal to the refractive index of the medium relative to incident medium.

That is,

2

1

tan pi

1

Where, 2 and 1 are refractive indices of reflecting and incident medium respectively.

Consider incidence of monochromatic light at polarizing angle pi .

According to Snells law of refraction 1 2sin sinpi r 2

According to Brewsters law 2

1

sintan

cos

p

p

p

ii

i

Or 1 2sin cosp pi i 3

Comparing eq2 and eq3 cos sin cos2

pi r r

Page | 24

Or 2

pi r

That is, reflected and refracted rays are mutually perpendicular at polarizing angle of incidence.

Also, as a consequence, we get reflected light as completely plane polarized and refracted light

as partially polarized.

Law of Malus- According to Malus, when a completely plane polarized light is incident on an

analyzer; the intensity I of the emergent light is directly proportional to the square of the cosine

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

That is, I 2cos

Or 2

0 cosI I 1

Where 0I is the maximum intensity of the plane polarized light incident on the analyzer.

To prove this law, let 0E be the amplitude of the plane polarized light transmitted by the

polarizer and incident on the analyzer and be the angle between the plane of transmission of

the polarizer and that of the analyzer. The amplitude 0E may be resolved into two components

0 cosE parallel to the plane of transmission of the analyzer and 0 sinE normal

(perpendicular) to it. Because of polarizing ability of the analyzer, only parallel component

0 cosE will be transmitted through the analyzer. Since, intensity is directly proportional to the

square of the amplitude, therefore,

Page | 25

I 2

0 cosE

Or 2 20 cosI kE

If 1k , then 2 20 cosI E

Or 2

0 cosI I , where 2

0 0I E 2

If 0 , then 2cos 1 , hence. 0I I . Thus, when planes of transmission of polarizer and

analyzer are parallel to each other, maximum light is transmitted through the analyzer.

If2

, then 2cos 0 , hence 0I . Thus, when planes of transmission of polarizer and

analyzer are perpendicular to each other, no light is transmitted through the analyzer.

If un-polarized light is incident on the polarizer, the intensity of light transmitted by the polarizer

is one-half the intensity of un-polarized light incident on it.

Page | 26

To prove this, let be the angle between the plane of vibration of the electric field vector of the

incident light and the plane of transmission of the polarization. According to law of Malus, the

intensity of light transmitted by the polarizer will be

' 20 cosuI I 3

Since, incident light is un-polarized; its electric field vector vibrates randomly in all directions

from 0 to 2 . Therefore, the intensity of light transmitted through the polarizer will be

the average of eq3. Hence,

0I '

0I = uI 2cos

24

Since, 2cos

1

2 therefore

0

2

uII 4

Thus, the intensity of light transmitted through the polarizer and the analyzer is given by

2cos

2

uII 5

Where uI is the intensity of incident un-polarized light

Page | 27

Double refraction There are certain transparent substances such as calcite (crystalline CaCO3 or Icelandspar), topaz, quartz, mica, argonite etc. which forms two images of an object,

that is, incident ray splits into two refracted rays. This phenomenon is called as double refraction

or birefringence and substances showing this phenomenon are called double refracting crystals.

For example, when a calcite crystal is placed on a white paper having a small ink dot, two

images of ink dot are seen when viewed through the upper surface of the crystal. If the calcite

crystal is rotated about the direction of incident ray; out of two images, one image remains

stationary while the other image rotates around the stationary image. The first type of image

(stationary) is formed by a ray which obeys ordinary laws of refraction (Snells laws) and is,

therefore, called as ordinary ray (o-ray). It always remains in the same plane of incidence and

its velocity in crystal is found same in all directions. The second type of image (rotating) is

formed by a ray which does not obey ordinary laws of refraction and whose refractive index is a

function of direction, that is, its velocity in crystal is found different in different directions, and is,

therefore called as extra-ordinary ray (e-ray).

When a tourmaline crystal is placed on the calcite crystal and rotated about the direction of incident ray, if intensity of ordinary image decreases, then intensity of extra-ordinary image

increases. At one position of the tourmaline crystal, when intensity of ordinary image becomes

zero, the intensity of extra-ordinary image becomes maximum. If the crystal is further rotated by

90 degree, then the intensity of ordinary image becomes maximum and that of extra-ordinary

image becomes zero. Thus, it is clear that both o-ray and e-ray is plane polarized light and

planes of vibration of these rays are perpendicular to each other.

Huygens explanation of double refraction Double refraction or birefringence is analyzed in

light of Huygens theory which is generalization of the Huygens construction for isotropic media.

The basic postulates of this theory of wave front construction are:

Page | 28

1. When a beam of light strikes the surface of double refracting crystal, each

point on the surface becomes a source of secondary emission. It originates

two types of disturbances: one each corresponding to ordinary ray (o-ray) and

extra-ordinary ray (e-ray).

2. For o-ray, the velocity of light is same in all the directions and, therefore, it

Spreads on the surface of a sphere, that is, o-wave front is spherical.

3. The crystal is anisotropic for e-ray. Its velocity is same as that of o-ray along

The optic axis and differ maximum in the direction perpendicular to optic axis.

Thus, e-wave front spreads over the ellipsoid of revolution along optic axis.

4. Since, along the optic axis, e ov v and for negative crystal , ,e o e ov v

the ellipsoid encloses the sphere and touches along optic axis. For positive

crystals, the otherwise is true, the sphere encloses the ellipsoid and the two

Page | 29

touch along the optic axis. The vibrations of e-ray lie in the principal section

but that of o-ray lie perpendicular to it.

Phase retardation plates When a light ray travels perpendicular to optic axis, the o-wave and

e-wave travel in the same direction but with different velocities. Thus, after traveling certain

length of the crystal, one wave shall lead in phase to the other wave. Such plates are called

phase retardation plates.

Consider a monochromatic plane polarized light of wavelength incident normally on a double

refracting crystal like calcite, with optic axis parallel to the surface. It splits into o-ray and e-ray

which travels in the same direction but with different velocities and the phase of one wave shall

retard as compared to the other. In calcite crystal, the velocity ev of e-ray is greater than

velocity ov of o-ray. So the difference in time T taken by the two waves to cross the plate of

thickness t will be

o e

t tT

v v

Therefore, path difference between the e-ray and o-ray will be

o ex t

Where o and e are refractive indices of calcite for o-ray and e-ray respectively.

Hence, the phase difference between the e-ray and o-ray will be

2

x

Or 2

o e

t

1

Eq1 represents the relative phase retardation between o-ray and e-ray.

Two main types of phase retardation plates are: Quarter wave plate and half wave

Page | 30

Quarter wave plate (QWP) A plate of double refracting uniaxial crystal of suitable thickness t cut with its optic axis parallel to its refracting faces and capable of producing a path

difference of 4

or a phase difference of

2

between two mutually perpendicular o-waves and e-

waves is called as quarter wave plate (QWP).

For a quarter wave plate

5 9 1

, , ... 22 2 2 2

n

where 0,1,2,3...n 2

Comparing above eq2 with eq1, we get

4 1 2

2o e

n t

Or

4 1

4 o e

nt

3

Quarter wave plate is used for producing elliptically or circularly polarized light.

When a plane polarized light (PPL) is incident normally on a quarter wave plate and its plane of

vibration does not makes an angle of 45 with the optic axis, then the emergent light is

elliptically polarized light (EPL).

Page | 31

When a plane

polarized light is incident normally on a quarter wave plate and its plane of vibration makes an

angle of 45 with the optic axis, then the emergent light is circularly polarized light (CPL).

Half wave plate (HWP) - A plate of double refracting uniaxial crystal of suitable thickness t cut with its optic axis parallel to its refracting faces and capable of producing a path difference of

2

or a phase difference of between two mutually perpendicular o-waves and e-waves is

called as half wave plate (HWP).

For a half wave plate

,3 ,5 ... 2 1n where 0,1,2,3...n 4

Comparing above eq4 with eq1, we get

2

2 1 o et

n

Or

2 1

2 o e

nt

5

Half wave plate is used for changing the direction of plane of vibration of the plane polarized

light.

When a plane polarized light is incident normally on a half wave plate (HWP), the emergent ray

remains plane polarized but the plane of vibration of the emergent ray rotates through an angle

2 from the plane of vibration of incident ray, where is the angle between the plane of vibration of the incident light and the optic axis of a half wave plate. Half wave plate changes left

Page | 32

handed elliptically or circularly polarized light into respective right handed elliptically or circularly

polarized light and vice-versa.

Production of elliptically or circularly polarized light Elliptically or circularly polarized light may be produced using an experimental arrangement shown in the figure below.

Light from a monochromatic source, say sodium lamp S1 falls on Nicol prism N1 which produces

plane polarized light. Nicol N2 is adjusted in crossed position so that the field of view is dark.

Now a quarter wave plate (QWP) mounted on a tube T1, which is free to rotate about the outer

tube T2, is placed between the crossed Nicol prisms. Due to introduction of quarter wave plate

between N1 and N2 some light is observed through N2.

Page | 33

The quarter wave plate is rotated until the field of view is again dark. It means that incident

vibrations are making angle 0 or 2 with optic axis. Keeping quarter wave plate fixed, T1 is

rotated so that the mark S on plate coincides with zero mark on T1. Now by rotating the quarter

wave plate, if mark S is made to coincide with 45 or 135 marks on T1, then the light coming out

of quarter wave plate will be circularly polarized. At these positions, the amplitudes of ordinary

ray and extra-ordinary become equal.

For rotation of quarter wave plate by an angle other than 45 will produce elliptically polarized

light, that is, amplitudes of ordinary ray and extra-ordinary ray become un-equal.

N2 shall analyze light coming out from quarter wave plate. If on rotating N2, intensity varies

between maximum and minimum, that is, not equal to zero, then light coming out from quarter

wave plate is elliptically polarized light. If it shows no variation in intensity, then light coming out

from quarter wave plate is circularly polarized light

Page | 34

Analyze the light of unknown polarization by quarter wave plate (QWP) and rotating

Nicol?

The following can be used to analyze the light of unknown polarization by quarter wave plate

and rotating Nicol.

How shall you distinguish quarter wave plate (QWP), plane glass plate and half wave

plate (HWP)?

The following table can be used to distinguish quarter wave plate, plane glass plate and half

wave plate.

Page | 35

Laurents half-shade polarimeter Laurents half shade polarimeter is suitable only for one

wavelength for which the path difference between o-ray and e-ray is2

. It is usually constructed

for sodium source.

When position is adjusted for equally dark halves, it gives fairly accurate observation as slight

rotation of the analyzer changes the intensity of two halves.

Bi-quartz polarimeter In bi-quartz polarimeter, transition from red to blue is very rapid,

hence, zero position can be obtained very accurately.

Page | 36

In this polarimeter, white light can be used, that is, mercury lamp can be used.

It is highly sensitive device for measuring optical rotation.

This instrument does not give accurate results for colorless optically active substances due to

rotatory dispersion produced by the substance itself.

It is not possible for color blind person to use this instrument.