Unit 1 optics jntu anantapur

GLOBAL COLLEGE OF ENGINEERING &TECHNOLOGY: YSR DIST.
Unit –I OPTICS - Interference Engineering Physics
1. Introduction
In the 17th century, the properties of light were explained by Sir Isaac Newton
and Christian Huygens. Sir Isaac Newton was explained the properties of light by introducing
corpuscular theory in 1675. It explains reflection, refraction and dispersion properties of light.
It fails to explain interference, diffraction, polarization, photo electric effect and double
refraction.
In 1679, Christian Huygens proposed the wave theory of light. According to Huygen’s
wave theory, each point on the wave front is to be considered as a source of secondary
wavelets. It explains reflection, refraction, dispersion, double refraction, diffraction,
interference and polarization properties of light. It fails to explain, photo electric effect, black

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body radiation etc,.

2. Coherence

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Coherence is a property of waves. Two produce interference pattern coherent sources
are required. Coherent light sources cannot be obtained from the separate light sources, but
they are obtained from the single light source.

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Two waves are said to be coherent, the waves must have
• Same wavelength
• Same amplitude and

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• Constant phase difference s.w
Definition:-

A predictable correlation of the amplitude and phase at any one point with other point is
called coherence.
sic

There are two types of coherence

i) Temporal coherence (or longitudinal coherence)
hy

ii) Spatial coherence (or transverse coherence)

Temporal coherence (or longitudinal coherence):-
gp

It is possible to predict the amplitude the amplitude and phase at a one point on the
in

wave w.r.t another point on the same wave, then the wave is in temporal coherence.
To understand this, let us consider two points P1 and P2 on the same wave as in shown
er

in figure. Suppose the phase and amplitude at any one point is known, then we can easily
calculate the amplitude and phase for any other point on the same wave by using the wave
e

equation
2ߨ
‫ ݕ‬ൌ ܽ sin ቆ ሺܿ ‫ ݐ‬െ ‫ݔ‬ሻቇ
gin

ߣ
Where ‘a’ is the amplitude of the wave and ‘x’ is the displacement of the wave at any instant of
time‘t’.
en

P1
P2
Spatial coherence (or transverse coherence)
It is possible to predict the amplitude the amplitude and phase at a one point on the
wave w.r.t another point on the second wave, then the wave is in spatial coherence.

P.Sreenivasula Reddy M.Sc, (PhD) Website: www.engineeringphysics.weebly.com Page 1

Independent sources are not coherent
Two independent sources of light with same power can never be treated as coherent
sources; even it emits same wavelength and same amplitude waves. Generally the light is
emitted from the source through spontaneous emission process. But the waves emitted
trough spontaneous emission process is in out of phase. So two waves generated by two light
sources are always in out of coherence. Even the waves are emitted from the different parts of
the lamp, no longer they are coherent.

3. Principle of superposition of waves
Definition:-
Principle of superposition of waves states that, the resultant displacement of two or
more waves traveling simultaneously in a medium at any point is equal to the algebraic sum of
the displacements due to individual waves.

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To understand this concept, Let us consider two waves traveling simultaneously in the

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medium. Let ܻଵ and ܻଶ are the displacements of first and second waves at any point in the
absence of other.
According to principle of superposition, the resultant displacement ሺܻሻ of two waves

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can be written as
ܻ ൌ ܻଵ േ ܻଶ

ee
Wave 1
+
=
s.w
Wave 2

Let two waves are in phase, their resultant amplitude is
ܻ ൌ ܻଵ ൅ ܻଶ
sic

Two waves are in opposite phase, their resultant amplitude is
ܻ ൌ ܻଵ ~ ܻଶ
hy

Wave 1
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+ =
Wave 2
in
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4. Young’s double slit experiment

The first experimental evidence of interference pattern was given by Thomas Young In
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1801. He explained the interference pattern based on the wave nature of light.
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Experimental setup:
His experimental arrangement is shown in figure. He passed the sun light through a
narrow pin hole S on a opaque surface and then through closely spaced pin holes S1 and S2
in another opaque surface. A screen XY was arranged in front of the pin holes S1 and S2. He
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observed few coloured bright and dark bands on the screen. Now a day, the pin holes are
replaced by slits and sun light is replaced by monochromatic light.

X

S1

S2
S Screen

Y

Explanation of wave theory
When sun light is passes through pin hole S, spherical waves are spread out.
According to Huygen’s wave theory, each point on the wave front is to be considered as a
source of secondary wavelets. Hence spherical waves also spread out from pin holes S1 and
S2. The radii of the wave fronts increase as they move away from S1 and S2 and hence
superimpose on each other.
In the above figure, continuous circular arcs represent the wave crests while dotted
circular arcs represent the troughs of each wave. A crest fall on the other crest constructive
interference will occurs and their resultant amplitude is equal to the sum of the amplitudes
individual waves.
ܻ ൌ ܻଵ ൅ ܻଶ

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A crest fall on the other trough destructive interference will occurs and their resultant

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amplitude is equal to the difference of the amplitudes of individual waves.
ܻ ൌ ܻଵ െ ܻଶ
Thus a number of dark and bright bands having equal width is observed on the screen.

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5. Interference of light

The best evidence for the wave nature of light is interference phenomenon. This was

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experimentally demonstrated by Thomas Young in 180, through double slit experiment. Due
to interference, we will observe many observations in our day today life, such as multiple
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colours on soap bubbles as well as on oil film when viewed under sun light. Interference
concept is explained on the basis of superposition of wave’s concept.
When two light waves superimpose, then the resultant amplitude or intensity in
sic

the region of superposition is different than the amplitude of individual waves.

Definition:-
hy

The modification in the distribution of intensity in the region of superposition is known
as interference.
gp

In case of interference pattern we observe two cases
• Constructive interference
in

• Destructive interference
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Constructive interference
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The waves are reaching at a point are in phase constructive interference occurs
In constructive interference, the resultant amplitude is always equal to the sum of the
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amplitudes of two individual waves.
Condition
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The path difference between the two waves is equal to the integral multiple of wave
length ሺߣሻ the constructive interference occurs.
‫ ݁ܿ݊݁ݎ݂݁݅݀ ݄ݐܽ݌‬ൌ ݊ߣ Where n = 0, 1, 2, 3, 4 …

Wave 1
+
=
Wave 2


Destructive interference
The waves are reaching at a point are in out of phase desstructive interference occurs
In Destructive interference, the resultant amplitude is always equal to the difference of the
amplitudes of two individual waves.
Condition
The path difference between the two waves is equal to the odd integral multiple of λ/2
destructive interference occurs
ሺ2݊ െ 1ሻߣ
‫ ݁ܿ݊݁ݎ݂݁݅݀ ݄ݐܽ݌‬ൌ Where n ൌ 1, 2, 3, 4 … ‫ݎ݋‬
2
ሺଶ௡ାଵሻఒ
‫ ݁ܿ݊݁ݎ݂݁݅݀ ݄ݐܽ݌‬ൌ Where n ൌ 0, 1, 2, 3, 4 …
ଶ

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Wave 1
+

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=
Wave 2

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6. Types of interference:-
For the formation of interference pattern, two coherent light sources are required. To

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get two coherent sources form a single light source, two techniques are used. They are
1. Division of wave front
2. Division of amplitude
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Division of wave front
The wave front from a single light source is divided into two parts using the
phenomenon of reflection, refraction or diffraction. Young’s double slit experiment is belongs
sic

to this class of interference.
Division of amplitude
The amplitude of a single light beam is divided into two parts by parallel reflection or
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refraction. Newton’s ring experiment, Michelson’s interferometer is belongs to this class of
interference.
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7. Conditions for interference
in

1) Two light sources of emitting light waves should be coherent.
2) Two sources must emit continuous light waves of same wavelengths or frequency.
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3) The separation between the two sources should be small.
4) The distance between the two sources and the screen should be large.
5) To view interference fringes, the background should be dark.
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6) The amplitude of light waves should be equal or nearly equal.
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7) The sources should be narrow.
8) The sources should be monochromatic.
8. Interference in thin films by reflection
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Principle:-
The formation of colours in thin films can explained as due to the phenomenon of
interference. In this example, the formation of interference pattern is by the division of
amplitude.
. Consider a thin film of uniform thickness ’‫ ’ݐ‬and refractive index ‘µ’. Let a
monochromatic light ray AB is incident is on the upper surface of the film with an angle ‘i’.
This ray is partially reflected along BC and the remaining part is refracted along BD with an
angle ‘r’. The refracted ray BD is reflected into DE from the lower surface of the film and it
emerges through the film in the form of light ray EF. The two light rays BC and EF are parallel
and superimpose and produce interference. The intensity of interference fringe depends up
on the path difference between the ray 1 and ray 2.

Ray 1

Ray 2

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To find out the path difference between the light rays (1) and (2), draw one

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normal DG to BE and from point E to BC. From point E and H onwards the two light
rays 1 and 2 travels equal distances. By the time the light ray 1 travels from B to H, the
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transmitted ray has to travel from B to C and D to E.
The path difference between the light rays (1) and (2) is
‫ ݁ܿ݊݁ݎ݂݂݁݅݀ ݄ݐܽ݌‬ൌ ߤሺ‫ ܦܤ‬൅ ‫ܧܦ‬ሻ ݅݊ ݂݈݅݉ െ ‫ݎ݅ܽ ݊݅ ܪܤ‬ (1)
sic

஽ீ ௧
From ∆‫ܩܦܤ‬ cos ‫ ݎ‬ൌ ൌ
஻஽ ஻஽
‫ݐ‬
hy

‫ ܦܤ‬ൌ
cos ‫ݎ‬
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஽ீ ௧
Similarly from ∆‫ܩܧܦ‬ cos ‫ ݎ‬ൌ ൌ
஽ா ஽ா
‫ݐ‬
in

‫ ܧܦ‬ൌ
cos ‫ݎ‬
er

௧
‫ ܦܤ ׵‬ൌ ‫ ܧܦ‬ൌ (2)
௖௢௦ ௥
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஻ு ஻ு
∆‫ܪܧܤ‬ sin ݅ ൌ ൌ ஻ீାீா
gin

From
஻ா

‫ ܪܤ ׵‬ൌ ሺ‫ ܩܤ‬൅ ‫ܧܩ‬ሻ. sin ݅
From ∆‫ܩܧܦ∆ ݀݊ܽ ܩܦܤ‬ ‫ ܩܤ‬ൌ ‫ ܧܩ‬ൌ ‫ ݐ‬tan ‫ݎ‬
en

‫ ܪܤ‬ൌ ሺ2‫ ݐ‬tan ‫ݎ‬ሻ. sin ݅
From Snell’s law at point B
sin ݅ ൌ ߤ sin ‫ݎ‬
‫ ܪܤ ׵‬ൌ 2ߤ‫ ݐ‬tan ‫ . ݎ‬sin ‫ݎ‬ (3)
Substituting the equations (2) and (3) in equation (1), we get
ଶఓ௧
ܲܽ‫ ݁ܿ݊݁ݎ݂݂݁݅݀ ݄ݐ‬ൌ ୡ୭ୱ ௥ െ 2ߤ‫ ݐ‬tan ‫ . ݎ‬sin ‫ݎ‬


2ߤ‫ݐ‬ sin ଶ ‫ݎ‬
ൌ െ 2ߤ‫.ݐ‬
cos ‫ݎ‬ cos ‫ݎ‬
2ߤ‫ݐ‬
ൌ ሺ1 െ sin ଶ ‫ݎ‬ሻ
cos ‫ݎ‬
2ߤ‫ݐ‬
ൌ ܿ‫ ݏ݋‬ଶ ‫ݎ‬
cos ‫ݎ‬
=2ߤ‫ ݐ‬cos ‫ݎ‬
At point B the light ray (1) is reflected at the surface of thin film (denser medium). So the
light ray (1) undergoes a phase change π or an additional path difference λ/2.
ఒ
Total path difference ൌ 2ߤ‫ ݐ‬cos ‫ ݎ‬െ ଶ
Constructive interference (or Bright fridge)

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General condition; ‫ ݁ܿ݊݁ݎ݂݂݄݁݅݀ݐܽ݌‬ൌ ݊ߣ
ߣ
2ߤ‫ ݐ‬cos ‫ ݎ‬െ ൌ ݊ߣ

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2
ߣ
2ߤ‫ ݐ‬cos ‫ ݎ‬ൌ ݊ߣ ൅
2

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ሺ2݊ ൅ 1ሻߣ
2ߤ‫ ݐ‬cos ‫ ݎ‬ൌ
2

ee
Destructive interference (or Dark fridge)
ఒ
General condition: ‫ ݁ܿ݊݁ݎ݂݂݄݁݅݀ݐܽ݌‬ൌ ሺ2݊ ൅ 1ሻ ଶ
s.w
ߣ ሺ2݊ െ 1ሻߣ
2ߤ‫ ݐ‬cos ‫ ݎ‬െ ൌ
2 2
2ߤ‫ ݐ‬cos ‫ ݎ‬ൌ ݊ߣ
sic

9. Newton’s rings
hy

Principle:-
The formation of Newton’s rings can explained as due to the phenomenon of
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interference. In this example, the formation of interference pattern is obtained by the division
of amplitude.
in

Experimental arrangement
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The experimental arrangement of
Newton’s rings is shown in figure.
e

The Plano -convex lens (L) of large
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radius of curvature is placed with its
M
convex surface on the glass plate (P).
The lens touches the glass plate at
en

O. A monochromatic light is allowed
to fall normally on the lens with the
help of glass plate M kept at 450 to
the incident monochromatic light.

A part of light is reflected by
the curved surface of the lens ‘L’ and
a part of light is transmitted is partly O
reflected back by the upper surface of
the plane glass plate P as shown in figure. These reflected rays interfere and give rise to an
interference pattern in the form of circular fringes. These rings are seen through microscope.

Explanation of formation of Newton’s rings

Newton’s rings are formed due to the interference between the light rays reflected from
the lower surface of the lens and the upper surface of the glass plate (or top and bottom
surfaces of the air film).
Let R be the radius of curvature of the lens and ‫ ݐ‬be the thickness of the air film at point
P. At point P a part of the incident monochromatic light is reflected from the bottom surface of
the glass plate without phase change. The other part of the light is refracted from the upper
surface of glass plate with additional phase change of π or path difference λ/2.
The path difference between the two rays is
ߣ

m
2ߤ‫ ݐ‬cos ‫ ݎ‬൅
2
For air film ߤ ൌ 1 and for normal incidence ‫ ݎ‬ൌ 0, so

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ఒ
The path difference ൌ 2‫ ݐ‬൅
ଶ

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Constructive interference (or Bright fridge)
General condition: ‫ ݁ܿ݊݁ݎ݂݂݄݁݅݀ݐܽ݌‬ൌ ݊ߣ
ߣ
2‫ ݐ‬൅ ൌ ݊ߣ
2

ee
ߣ
2‫ ݐ‬ൌ ሺ2݊ െ 1ሻ
2
Where ݊ ൌ 0, 1, 2, … … … ..
s.w
Destructive interference (or Dark fridge)
ఒ
General condition: ‫ ݁ܿ݊݁ݎ݂݂݄݁݅݀ݐܽ݌‬ൌ ሺ2݊ ൅ 1ሻ ଶ
sic

ߣ ߣ
2‫ ݐ‬൅ ൌ ሺ2݊ ൅ 1ሻ
2 2
hy

2‫ ݐ‬ൌ ݊ ߣ
Where ݊ ൌ 0, 1, 2, … … … ..
gp

ఒ
At the point of contact ‫ ݐ‬ൌ 0, path difference is ଶ
i.e., the reflected and incidence light are out
in

of phase and destructive interference occur. So the center fringe is always dark.
Theory of Newton’s rings
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To find the diameters of a dark and bright fringes construct a circle with the radius of
curvature R of a lens L and let us choose a point P at a distance ‘r’ from the center of lens and
e

‫ ݐ‬be the thickness of air film at point ܲ.
gin

From the property of a circle ܰܲ. ܰ‫ ܤ‬ൌ ܱܰ. ܰ‫ܦ‬
‫ ݎ .ݎ‬ൌ ܶ. ሺ2‫ ݎ‬െ ‫ݐ‬ሻ
‫ ݎ‬ଶ ൌ 2ܴ‫ ݐ‬െ ‫ ݐ‬ଶ
en

If t is small ‫ ݐ‬ଶ is negligible.
‫ ݎ‬ଶ ൌ 2ܴ‫ݐ‬
‫ݎ‬ଶ r r
‫ݐ‬ൌ
2ܴ
Bright rings
ఒ
For bright ring, the condition is 2‫ ݐ‬ൌ ሺ2݊ െ 1ሻ ଶ
2 ‫ݎ‬ଶ ߣ
ൌ ሺ2݊ െ 1ሻ
2ܴ 2
ሺ2݊ െ 1ሻߣܴ
‫ݎ‬ଶ ൌ
2

By replacing r by D/2 the diameter of the bright ring is
‫ܦ‬ଶ ሺ2݊ െ 1ሻߣܴ
ൌ
4 2
‫ܦ‬ଶ ൌ 2ሺ2݊ െ 1ሻߣܴ
‫ ܦ‬ൌ ඥ2ሺ2݊ െ 1ሻߣܴ

‫ ܦ‬ൌ ඥሺ2݊ െ 1ሻ√2ߣܴ

‫ ן ܦ‬ඥሺ2݊ െ 1ሻ
‫ݎܾ݁݉ݑ݊ ݈ܽݎݑݐܽ݊ ݀݀݋√ ן ܦ‬
Dark rings

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For dark rings, the condition is
2‫ ݐ‬ൌ ݊ ߣ

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2 ‫ݎ‬ଶ
ൌ݊ߣ
2ܴ

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‫ ݎ‬ଶ ൌ ݊ ߣܴ
By replacing r by D/2 the diameter of the nth dark ring is

ee
‫ܦ‬ଶ
ൌ ݊ ߣܴ
4
s.w
‫ ܦ‬ൌ √4݊ ߣܴ
‫ ܦ‬ൌ 2√݊ ߣܴ
‫݊√ ן ܦ‬
sic

‫ݎܾ݁݉ݑ݊ ݈ܽݎݑݐܽ݊√ ן ܦ‬
hy

Note: suppose a liquid is taken in between the lens and glass plate having refractive index ߤ,
then the diameter of the dark nth dark ring can be written as
gp

√4݊ ߣܴ
‫ܦ‬ൌ
ߤ
in

10. Determination of wave length of sodium light using Newton’s rings
er

By forming Newton’s rings and measuring the radii of the rings formed, we can
e

calculate the wavelength of the light used if the radius of curvature of the lens is known. Let R
be the radius of curvature of the lens and ߣ is the wavelength of the light used.
gin

So the diameter of the nth dark ring can be written as
‫ܦ‬௡ ଶ ൌ 4 ݊ ߣ ܴ (1)
en

Similarly the diameter of the (n+m)th dark ring is
‫ܦ‬௡ା௠ ଶ ൌ 4 ሺ݊ ൅ ݉ሻ ߣ ܴ (2)
Subtracting equation (1) from (2) we get
‫ܦ‬௡ା௠ ଶ െ ‫ܦ‬௡ ଶ ൌ 4 ሺ݊ ൅ ݉ሻߣ ܴ െ 4 ݊ ߣ ܴ
‫ܦ‬௡ା௠ ଶ െ ‫ܦ‬௡ ଶ ൌ 4 ݉ ߣ ܴ
‫ܦ‬௡ା௠ ଶ െ ‫ܦ‬௡ ଶ
ߣൌ
4ܴ݉
Using the above relation wavelength can be calculated

11. Determination of refractive index of a liquid using Newton’s rings

By forming Newton’s rings and measuring the diameter of the rings formed, we can
calculate the refractive index of the liquid.
In air film, the diameter of the nth and mth dark rings are ‫ܦ‬௡ and ‫ܦ‬௠ are measured
with the help of travelling microscope.
The diameter of the nth dark ring is
‫ܦ‬௡ ଶ ൌ 4 ݊ ߣ ܴ

m
(1)
The diameter of the mth dark ring is

.co
‫ܦ‬௠ ଶ ൌ 4 ݉ߣ ܴ (2)
So ‫ܦ‬௠ ଶ െ ‫ܦ‬௡ ଶ ൌ 4 ݉ߣ ܴ െ 4 ݊ ߣ ܴ

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‫ܦ‬௠ ଶ െ ‫ܦ‬௡ ଶ ൌ 4 ሺ݉ െ ݊ሻ ߣ ܴ (3)
The Newton’s rings setup is taken in a liquid. Now the air film is replaced by liquid film.
In liquid film, the diameters of the same nth and mth dark rings are ‫ ′ܦ‬௡ and ‫ ′ܦ‬௠ are measured

ee
with the help of travelling microscope.
ଶ ସ௡ఒோ
‫ ′ܦ‬௡ ൌ
s.w
and
ఓ

ଶ 4݉ߣܴ
‫′ܦ‬௠ ൌ
ߤ
sic

ଶ ଶ ସ ሺ௠ି௡ሻ ఒ ோ
So ‫ ′ܦ‬௠ െ ‫ ′ܦ‬௡ ൌ (4)
ఓ
hy

Dividing equation (3) by (4)
‫ܦ‬௠ ଶ െ ‫ܦ‬௡ ଶ 4 ሺ݉ െ ݊ሻ ߣ ܴ
ൌ
ଶ ଶ 4 ሺ݉ െ ݊ሻ ߣ ܴ
gp

‫′ܦ‬௠ െ ‫ ′ܦ‬௡
ߤ
‫ܦ‬௠ ଶ െ ‫ܦ‬௡ ଶ
ൌߤ
in

ଶ ଶ
‫ ′ܦ‬௠ െ ‫ ′ܦ‬௡
er

Using the above relation ߤ can be calculated.
e
gin
en


en
gin
ee
rin
gph
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Unit –I OPTICS - Diffraction Engineering Physics
Introduction
The wave nature of light is first confirmed by the phenomenon of interference.
Further it is confirmed by the phenomenon of diffraction. The word ‘diffraction’ is derived
from the Latin word diffractus which means break to piece. When the light waves encounter
an obstacle, they bend round the edges of the obstacle. The bending is predominant when
the size of the obstacle is comparable with the wavelength of light. The bending of light

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waves around the edges of an obstacle is diffraction. It was first observed by Gremaldy.

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1. Diffraction
Definition:-
When the light falls on the obstacle whose size is comparable with the wavelength of

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light then the light bends around the obstacle and enters in the geometrical shadow. This
bending of light is called diffraction.

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When the light is incident on an obstacle AB, their corresponding shadow is
completely dark on the screen. Suppose the width of the slit is comparable to the

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wavelength of light, then the shadow consists of bright and dark fringes. These fringes are
formed due to the superposition of bended waves around the corners of an obstacle. The
amount of bending always depends on the size of the obstacle and wavelength of light used.
ics
ys
g ph
rin

2. Types of diffraction
The diffraction phenomena are classified into two ways
ee

I. Fresnel diffraction
II. Fraunhofer diffraction.
gin

Fresnel diffraction:-
In this diffraction the source and screen are separated at finite distance. To study this
en

diffraction lenses are not used because the source and screen separated at finite distance.
This diffraction can be studied in the direction of propagation of light. In this diffraction the
incidence wave front must be spherical of cylindrical.
Fraunhofer diffraction:-

In this diffraction the source and screen are separated at infinite distance. To study
this diffraction lenses are used because the source and screen separated at infinite
distance. This diffraction can be studied in any direction. In this diffraction the incidence
wave front must be plane.

3. Fraunhofer single slit diffraction:-
Let us consider a slit AB of width ‘e’. Let a plane wavefront of monochromatic
light of wavelength λ is incident on the slit AB. According to Huygens principle, every point
on the wavefront is a source of secondary wavelets. Those secondary wavelets travelling
normal to the slit are brought to focus at point P0 on the screen by using the lens. The
secondary wavelets traveling at an angle θ with the normal are focused at point P1.

m
.co
bly
Intensity at point P1 depends up on the path difference between the wavelets A and B

ee
reaching to point P1.

.w
2

2 sin
ics
Let the width of the slit is divided into ‘n’ equal parts and the amplitude of the wave front
each part is ‘a’. Then the phase difference between any two successive waves from these
ys

parts would be
1 1 2 sin
ph

Using the vector addition method, the resultant amplitude R is
sin
g

2
rin

sin
2
1 2 sin
. sin
ee

sin sin
2
1 2 sin sin
sin
sin
gin

2
sin sin
sin
en

sin
sin

sin

sin

Therefore resultant intensity

Principal maximum:-
The resultant amplitude R can be written as

3! 5! 7!

1
3! 5! 7!

m
1
3! 5! 7!
In the above expression for 0 values the resultant amplitude is maximum

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Then
sin
0

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sin 0
0
For 0 value the resultant intensity is maximum at P0 and is known as principal

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maximum.

Minimum intensity positions

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Will be minimum when sin 0
1,2,3,4,5 … … ..
sin
ics
sin
ys

So we obtain the minimum intensity positions on either side of the principal maxima for all
values.
ph

Secondary maximum
In between these minima secondary maxima positions are located. This can be
obtained by differentiating the expression of w.r.t and equation to zero
g

sin
0
rin

2sin sin
0
ee

2sin cos sin
· 0
gin

In the above expression can never equal to zero, so
Either sin 0 or cos sin 0
sin 0 gives the positions of minima.
en

The condition for getting the secondary maxima is
cos sin 0
cos sin
tan
The values of satisfying the above equation are obtained graphically by plotting the curves
and tan on the same graph. The plots of and tan is shown
in figure.


m
From the graph intersecting points are 0, , , …………

.co
From the above concepts the intensity distribution curve verses is shown in figure.

bly
ee
.w
ics
ys
ph

4. Fraunhofer double slit diffraction
Let S1and S2 be the two slits equal width e and separated by a distance d. The
g

distance between the two slits is (e + d). A monochromatic light of wavelength λ is incident
on the two slits. The diffracted light from these slits is focused on the screen by using a
rin

lens. The diffraction of a two slits is a combination of diffraction and interference. The
wavelets travelling normal to the slit is focused at P0. The wavelets travelling at an angle θ
ee

with the incident light are focused at point P1.
From Fraunhofer single slit experiment, the resultant amplitude is
gin

So the amplitude of each secondary wavelet travelling with an angle θ can be taken
sin
as . These two wavelets interference and meet at point P1 on the screen.
en

To find out the path difference between the two wavelets, let us draw a normal s1k to
the wavelet S2.
S
S S
From ∆S S k sin
S S
S sin
2 2
sin

m
By using vector addition method, we can calculate the resultant amplitude at point P1
2 cos

.co
sin sin sin sin
2 cos
sin sin
2 2 cos

bly
sin
2 1 cos

ee
sin
2 1 2 cos 1
sin
2 2 cos

4
sin
.w cos
ics
sin
4 cos
sin
ys

sin
ph

4 cos

The resultant intensity 4 cos
g

From the above equation, it is clear that the resultant intensity is a product of two factors i.e.,
rin

1. represents the diffraction pattern due to a single slit.
2. cos represents the interference pattern due to wavelets from double slit.
ee

Diffraction effect
In diffraction pattern the variation of w.r.t as shown in figure.
gin
en


The diffraction pattern consists of central principal maxima for 0 value.
The secondary maxima of decreasing intensity are present on either side of the central
maxima for , , values.
Between the secondary maxima the minima values are present for , 2 ,
values.

Interference effect
In interference pattern the variation of cos w.r.t as shown in figure.

m
.co
bly
ee
cos represents the interference pattern.
Interference maximum will occur for cos 1

.w
0, 1, 2, 3, 4 … … …
, 2 , 3 , 4 ………
ics
sin

sin
ys

Interference minima will occur for cos 0
ph

2 1
2
sin
2 1 2
g

sin 2 1
rin

2
The resultant intensity variation due both interference and diffraction patterns is shown in
figure. Due to interference the resultant minimum in not exactly equal to zero.
ee
gin
en

θ
Construction of plane diffraction grating
Or
Construction of grating

A set of large number of parallel slits of same width and separated by opaque spaces
is known as diffraction grating. Fraunhofer used the first grating consisting of a large
number of parallel wires placed side by side very closely at regular separation. Now the

m
gratings are constructed by ruling the equidistance parallel lines on a transparent material

.co
such as glass with fine diamond point. The ruled lines are opaque to light while the space
between the two lines is transparent to light and act as a slit.

bly
Let ‘e’ be the width of line and‘d’ be the width of the slit. Then (e + d) is known as
grating element. If N is the number of lines per inch on the grating then

ee
1 2.54
.

.w
ics
ys
g ph
rin
ee

Plane Diffraction grating ( or diffraction at N - slits)
Let a plane wavefront of monochromatic light of wavelength λ is incident on the
gin

slits. Let ‘e’ be the width of line and ‘ ’ be the width of the slit. Then (e + d) is known as
grating element. According to Huygens principle, every point on the wavefront is a source
of secondary wavelets. The wavelets spread out to the right in all directions. Those
en

secondary wavelets travelling normal to the slit are brought to focus at point P0 on the
screen by using the lens. . The wavelets travelling at an angle θ with the incident light are
focused at point P1.


sin
The resultant amplitude due to diffraction of a single slit is , where
sin .
Let us assume that there are N slits in the grating. The path difference between the
corresponding points is
sin 2 .
The resultant of amplitude R of the N amplitudes of is
sin sin
1
sin

m
And intensity is

.co
sin sin
. 2
sin

bly
In the above equation shows the intensity distribution due to a single slit

diffraction and shows the combined intensity distribution due to interference at all

ee
the N slits.
Principal Maxima: from equation 2, we know that the intensity would be maximum

.w
when sin 0, where 0, 2 , … . . and n= 0,1,2,3…
Indeterminate
ics
To find out its value, L- Hospitals rule is applied

sin sin cos
ys

lim lim lim
sin sin cos
ph

Hence the resultant intensity is can be written as
sin
g
rin

So the maxima are obtained for
sin or
ee

sin
Minima:
From equation 2 the resultant intensity is minimum for sin 0 but sin 0
gin

sin
en

sin
can be take integral values except 0, N, 2N,………..nN. For these values we bet maxima
Secondary maxima:
Between two adjacent principal maxima there are 1 minima and 2 other maxima
called secondary maxima.b to get the position so secondary maxima, differentiate equation
2 w.r.t and then equate to zero.

sin sin cos sin sin cos
2 . 0
sin 2

cos sin sin cos 0 or
tan tan
The roots of the above equation (other than , this corresponds to principal maxima)
give the secondary maxima.

m
The variation of intensity due to the interference factor is shown in figure

.co
bly
ee
.w
The variation of intensity due to the diffraction factor of is shown in figure
ics
ys
g ph
rin

The variation of resultant intensity is shown in figure.
ee
gin
en


5. Grating spectrum
When white light is fall on the grating, the light gets diffracted through each slit. As a
result of both interference and diffraction, diffraction pattern will occur. This pattern is known
as grating spectrum or diffraction spectrum.

m
.co
bly
ee
.w
ics
ys
ph

The condition for the principal maximum in grating spectrum is
sin
g

Where is the grating element and is the order of spectrum.
From the grating equation, we observe that
rin

1. For a particular wavelength λ the angle of diffraction θ is different for principal maxima
and for different orders.
ee

2. For white light and for a particular order , the light of different wavelengths will be
diffracted in different directions.
3. Angle of diffraction increases with wavelength.
gin

4. In the spectrum violet is in the innermost position and red at the outermost position.
5. Most of the energy is distributed for zero order and the rest is distributed among the
other orders.
en

6. Spectrum different orders are situated symmetrically on both sides of zero order.
7. Sharpness of maxima increases with increasing of number of lines in the grating.
8. The colour of central maxima is same as the incident light.
9. The condition for finding the maximum number of orders formed by the grating is

Unit –I OPTICS - polarisation Engineering Physics
Introduction

From the previous chapter’s interference and diffraction, we can understand that the light is
exhibiting wave nature. But these phenomenon don’t explain the character of these wave motion i.e.,
whether it is longitudinal or transverse. In longitudinal waves, the particles of the medium move to
and fro in the direction of propagation of the wave as in case of sound waves. In transverse waves,
the particles of the medium vibrate up and down perpendicular to the direction of propagation of wave

m
as ripples on a water surface.

.co
1. Polarization
Light is electromagnetic in nature, it consists of oscillating electric and magnetic fields of
vectors. They are perpendicular to each other and perpendicular to the propagation of light.
In light, the electric field direction is always changes in random ways. So the unpolarized or

bly
ordinary light is represented as shown in figure.

ee
If the direction of electric field vector is confined to one fixed direction, that light beam is

.w
called linearly polarized or plane polarized light. So the plane polarized light is represented as shown
in figure
ics
ys

Definition:
ph

The process of restricting the vibration of electric field vector is called polarization.

2. Plane of polarization and plane of vibration
g

Plane of vibration: -
rin

The plane which contains the vibrations of polarized is known as plan of vibration.
Plane of vibration:-
ee

The plane perpendicular to the plane of vibration is known as plane of polarization.
When an unpolarized light is incident on the polarizer, the emergent light beam always has
vibrations parallel to the optic axis.
gin
en

In the figure the plane parallel to the optic axis of the polarizer and containing the
vibrations of polarized light and hence it is known as plane of vibration the plane is
perpendicular to the plane of vibration and hence it is known as plane of polarization.

P.Sreenivasula Reddy M.Sc, (PhD) Page 1 of 9

3. Definitions:
Blunt corners:- If any corner contain all obtuse angles (is an angle between 90 and 180 degrees),
which is called blunt corner.
Optic axis: -
A line joining the two blunt corners is known as optic axis of the crystal. An optic axis is not
an axis, but it is a direction of a crystal. Any line parallel to the optic is also known as optic axis.
Uniaxial crystal: -
If any crystal having only one optic axis is known as uniaxial crystal.
Examples: - calcite, quartz.

m
Biaxial crystal: -

.co
If any crystal having two optic axes is known as biaxial crystal.
Examples: -borax, mica.

bly
Principal section: -
A plane containing the optic axis and perpendicular to two opposite faces of the crystal is
known as the principal section of the crystal.

ee
Principal planes: -
The plane having the optic axis and ordinary ray of the crystal is known as principal plane of

.w
the ordinary ray. The plane having the optic axis and extraordinary ray of the crystal is known as
principal plane of the extraordinary ray.
ics
4. Production of polarized light
For the production of plane polarized light different methods are followed, they are
ys

a) Polarization by reflection
b) Polarization by transmission
c) Polarization by absorption
ph

d) Polarization by double refraction
g

a) Polarization by reflection
rin

When the unpolarized light is fall on the smooth surface such as glass or water surface, the
reflected light is partially plane polarized. The degree of polarization changes with the angle
ee

of incidence. For a particular angle of incidence the reflected light was completely polarized
transmitted light ray is partially plane polarized. The vibration of the reflected light is
gin

perpendicular to the plane of incidence. The polarizing angle for glass is 57.50
At which angle of incidence the reflected ray is completely plane polarized, that
angle is called polarizing angle or angle of polarization.
en

b) Polarization by transmission
When the unpolarized light incident on smooth surface at polarizing angle the reflected light is
plane polarized and the transmitted light ray is partially plane polarized.
When the unpolarized light incident on the pile of plates at polarizing angle, the transmitted
light ray is purely plane polarized. The polarizing angle for glass is 57.50. The vibration of the
transmitted light is parallel to the plane of incidence.

m
.co
bly
ee
c) Polarization by absorption
When un polarized light is passing through the tourmaline ( , , , it is split into
horizontal and vertical components. The horizontal components are completely absorbed by the

.w
crystal and the vertical component comes out. The light coming from the crystal is linearly polarized.
ics
ys
ph

d) Double refraction: -
When an unpolarized light is passes through a calcite crystal, it is split up into two plane
g

polarized lights. This phenomenon is known as double refraction.
rin
ee
gin

In which, one ray obeys the laws of refraction and has vibrations perpendicular to the principal
section is known as ordinary ray. The other ray does not obey the laws of refraction and has
en

vibrations parallel to the principal section is known as extraordinary ray. The refractive index of
ordinary and extraordinary rays for calcite crystal is
sin sin

sin sin
In calcite crystal the velocity light for O- ray inside the crystal is less than the E- ray. The
refractive index of O-ray is same for all angle of incidence and hence it is represented by spherical
wave front. The refractive index of E-ray is always varies with the angle of incidence and hence it is
represented by ellipsoidal wave front. But both rays are travel with same velocity along the optic axis
of the calcite crystal.


5. Polarization of light waves
Let us consider an unpolarized light is passed through the tourmaline crystal as shown in
figure. Even when we are rotating the tourmaline crystal by taking the incident ray as an axis, there is
no intensity variation in the emergent beam. The tourmaline crystal only allows the vibrations of light
that are parallel to the axis of the crystal, and it does not allow the other vibrations. So the emergent
beam from the tourmaline crystal is a polarized light. So the first tourmaline crystal is called
polarizer.

m
.co
bly
If the second tourmaline crystal is rotated by taking the incident ray as an axis, the intensity of
emergent beam from the second tourmaline crystal has two times minimum intensity and two times
maximum intensity is observed within one complete rotation. When the second tourmaline crystal is

ee
parallel to the first one the emergent beam from the second crystal is maximum. When the second
tourmaline crystal is perpendicular to the first one the emergent beam from the second crystal is
minimum. Therefore the second crystal is called analyzer.

.w
So this property of light of rays confirms the transverse nature of light rays.
6. Nicol prism
ics
Nicol prism is an optical device made from calcite and is used to produce and analyze plane
polarized light. It was invented by Nicol in 1828 and is known as Nicol prism.
ys

Principle:-
Nicol prism is worked on the principle of double refraction exhibited by calcite crystal.
Construction:-
ph

Nicol prism is constructed by using calcite crystal. Its length and breadth dimensions are in
the ratio 3:1. Consider a calcite crystal, in which is the principal section and B and D are blunt
g

corners. The angles of the principal section are 710 and 1090. The angle of the principal section is
rin

reduced from 710 to 680 by grinding the faces of AB and CD. After grinding the new principal section
is
The crystal is cut into two pieces perpendicular to the principal section along . The two
ee

cut surfaces are highly polished into optical flat and joined together by means of Canada balsam. The
refractive index of Nicol prism for sodium wave length 5893 and the refractive indices of
gin

ordinary ray, extraordinary ray and Canada balsam are 1.658, 1.46 and 1.55 respectively. The two
faces of the Nicol prism is kept open, while the two sides are blackened to absorb the totally internally
reflected rays.
en

B
C

Canada balsam
cement

68 71
A
D

Working of Nicol prism
When an unpolarized light is passes through a calcite crystal, it is split up into two beams, namely,
ordinary and extraordinary rays.

m
E ray

.co
When the ordinary ray passing through the calcite, the Canada balsam layer acts as a rarer medium.

bly
Therefore the ordinary ray is travelling from denser (calcite) to rarer (Canada balsam) medium. For
ordinary ray, the angle of incidence at the interface of calcite and Canada balsam is greater than the
critical angle, it undergoes total internal reflection.

ee
The critical angle for the ordinary ray is
1.55
sin 69.2

.w
1.658
The angle of the principal section is greater than the critical angle, so the upper and lower surfaces of
the Nicol prism are grounded from 710 to 680. For the extraordinary ray the Canada balsam is acts as
ics
a dense medium. Hence it travels in a straight line and emerges out from the Nicol prism. The
emergent beam is plane polarized light.
ys

7. Plane polarized light by Nicol prism.
A Nicol prism is used to produce and analyze plane polarized light. When an unpolarized
ph

light is passed through a Nicol prism, it is split up into ordinary and extraordinary rays. The ordinary
ray is cut of when it is passing from calcite to Canada balsam layer. The extraordinary ray is passed
g

through the Nicol prism.
rin
ee
gin
en

If the second Nicol prism is rotated by taking the incident ray as an axis, the intensity of
emergent beam from the second Nicol prism has two times minimum intensity and two times
maximum intensity is observed within one complete rotation. The first Nicol prism is known as
polarizer and the second Nicol prism is known as analyzer. Thus a Nicol prism can be used as a
polarizer as well as analyzer.


8. Limitations of Nicol prism
1. The Nicol prism acts as a polarizer only for the light rays having slightly convergent and slightly
divergent beam.
2. The Nicol prism does not acts as a polarizer for the light rays having highly convergent and highly
divergent beam
3. If the incident divergent beam is making larger angle with the front face, the extraordinary ray
will become parallel to the optics axis of the crystal. Extraordinary ray also totally internally
reflected by the Canada balsam layer. Therefore no light emerges our from the Nicol prism.
4. If the incident convergent beam is making small angle with the front face, the ordinary makes less
critical angle at the calcite and Canada balsam interface. Then the ordinary ray also emerges out

m
from the Nicol prism.
9. Quarter wave plate

.co
A wave plate is an optical device, is used to produce path difference or phase difference
between the ordinary and extraordinary rays. It is made from the doubly refracting uniaxial crystals.
Its faces are cut parallel to the direction of the optic axis.

bly
When the polarized light is incident on the calcite crystal, the light splits up into O-ray and E-
ray. The thickness of the device is cut in such way it can produce the path difference 4 or phase
difference 2 between the O-ray and E-ray is called quarter wave plate.

ee
Let be the thickness of the crystal. Let and be the refractive indices of E-ray and O-
ray. Then the path difference between E-ray and O-ray is

For quarter wave plate path difference = 4
.w
ics
4

4
ys

For negative crystals
For positive crystals
ph

10. Half wav e plate
A wave plate is an optical device, is used to produce path difference or phase difference
g

between the ordinary and extraordinary rays. It is made from the doubly refracting uniaxial crystals.
rin

Its faces are cut parallel to the direction of the optic axis.
When the plane polarized light is incident on the calcite crystal, the light splits up into O-ray and E-
ee

ray. When the refracting faces are cut parallel to the optic axis, O-ray and E-ray travels with different
velocities along the same path.
The thickness of the device is cut in such way it can produce the path difference 2 or path
gin

difference between the O-ray and E-ray is called half wave plate.
Let be the thickness of the crystal. Let and be the refractive indices of E-ray and O-ray.
Then the path difference between E-ray and O-ray is
en

For half wave plate path difference = 2
2

2
For negative crystals

For positive crystals


11. Theory of circular and elliptical polarized lights
A beam of plane polarized light is obtained by passing the unpolarized light through the Nicol
prism. Let be the maximum amplitude of the incident light which makes an angle with the
optic axis. When the plane polarized light is passes through a wave plate, it is split up into two
ordinary and extraordinary rays. The amplitudes of ordinary and extraordinary rays are sin   and
cos respectively. A sin

m
Q

.co
P A cos

bly
The ordinary and extraordinary rays travel with different velocities in the same direction, and

ee
hence a phase difference is introduced between these two rays. Let be the phase difference between
the ordinary and extraordinary rays. The emergent beam from the wave plate can be represented as

.w
cos sin (1)
sin sin (2)
ics
Taking cos and sin , the above equations can be written as
sin (3)
ys

sin (4)
Equations (3) and (4) can be written as
ph

sin (5)
g

sin (6)
rin

Equation (5) can be written as
sin cos cos sin (7)
ee

Substituting the value of sin and cos from equation (6) in equation (7)
gin

cos 1   sin (8)

Squaring equation (8), we get
en

cos 2 cos 1   sin

cos 2 cos   sin   sin

cos   sin 2 cos   sin

2 cos   sin (9)


Equation (9) is a general expression for an ellipse.
when 0,   sin 0 and cos 1
Equation (9) becomes
2 0

0

m
The above expression represents a straight line. Therefore the light is plane polarized.
when ,   sin 1 and cos 0

.co
Equation (9) becomes

1

bly
This represents the equation of a ellipse. Thus the emergent light in this case will be elliptically
polarized light.
when ,   sin 1, cos 0 and

ee
1

.w
This represents the equation of a circle. Thus the emergent light in this case will be circularly
ics
polarized light

12. Differences between polarized light and unpolarized light
ys

Unpolarized light Polarized light
ph

When an unpolarized light is passed through the When a polarized light is passed through the
rotating tourmaline crystal, no variation intensity rotating tourmaline crystal, variation in the
of light is observed. intensity of light is observed.
g

The unpolarized light is symmetrical about the The polarized light is lack symmetry about the
direction of propagation of light. direction of propagation light.
rin

In unpolarized light the vibrations are confined to In polarized light the vibrations are confined to
any direction. one particular direction.
ee

The unpolarized light is specified as The polarized light is specified as
gin
en

Differences between O-ray and E-ray
O-ray E-ray
The O-ray obeys the laws of refraction The E-ray does not obeys the laws of refraction
The O-ray has vibrations perpendicular to the The E-ray has vibrations parallel to the principal
principal section. section
In negative crystal the velocity light for O- ray is In positive crystal the velocity light for E- ray is
less than the E- ray less than the O- ray
The refractive index of O-ray is same for all The refractive index of E-ray is always varies
angle of incidence with the angle of incidence

GLOB
BAL COLLE
EGE OF EN
NGINEERI
ING &TEC
CHNOLOGY YSR DIS
Y: ST.
Unit –I
– OPTICS - polarisat
tion Engineer
E ring Phys
sics

13. A
Analysis or detection of unknown p
d f polarization
n

m
.co
bly
ee
.w
ics
ys
g ph
rin
ee
gin
en

P.Sr
reenivasu Reddy
ula M.Sc, (PhD) Page 9 of 9
e

Unit 1 optics jntu anantapur

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