WAVE-OPTICS
Wave frontAccording to wave theory of light, a source of light send out disturbance in all directions and if
the medium is homogeneous, the disturbance reaches all those particles of the medium in same
phase, which are located at same distance from the source of light. All such particles at equal
distance from the source of light are thus vibrating in same phase.
The locus of all the particles of the medium which at any instant are vibrating in the same
phase is called the wavefront. The shape of the wavefront depends on the source of light.
Wavefront can be 3 types:
[a] Spherical Wavefront: If we have a point source of light then the locus of all the points
which are vibrating in same phase lies on a sphere and the wave front formed is called spherical
wavefront.
[b]Cylindrical Wavefront: When the source of light is linear in shape then the locus of all
the point which are equidistant from the linear source lies on a cylinder and the wavefront
formed is called cylindrical wavefront.
[c] Plane Wave front: It is a small part of spherical or cylindrical wavefront when the source
of light lies at large distance from the wavefront.
Ray of light: An arrow drawn normal to the wavefront and pointing in the direction of
propagation of disturbance is called ray of light and as it is always normal to the wavefront is it
is sometimes called wave normal.
Huygens’ Principle:
Huygens’s principle is geometrical construction which is used to
determine the new position of the wavefront at a later time from
its position at any instant. According to Huygens’s principle
[a] Light travels in the form of wave fronts and each point on
the wave front acts as a new source of light from which fresh
light waves called secondary wavelets travel in all directions
with the same speed.
[b] The envelope of all the particles vibrating in the same
phase gives the new position of wave front [called secondary
wave front] at an instant.
Consider a point source of light and AB be a section of primary spherical wave-front at any time
t. In order to find the new position of wavefront at time (t + Δt), consider points a, b, c… on the
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Wave Optics by Umesh Tyagi
primary wave front (AB). These points acts as source of secondary wavelets. In the time Δt the
secondary wavelets travels a distance cΔt. Taking points a, b, c… as centers, draw spheres of
radius cΔt. The tangents on these spheres represents the position of secondary wave front at
time (t + Δt).
Note- Backward wave front doesn’t exist at all because energy can’t flow in opposite direction.
Laws of reflection on the Basis of Wave Theory:
Let a wave front AP strikes a reflecting surface XY obliquely. First of all the primary wave front
reaches point ‘A’ and the incident rays makes an angle i with the normal AN. The disturbance
from P reaches point P’ at later time t w.r.t. A. Therefore PP '  c  t . Where c is the speed of light
in vacuum.
In the same time t, the secondary reflected
wavelet from point A reaches at point A’.
Therefore AA'  c  t .
To draw the reflected wavefront [after
reflection from face XY] we draw a sphere of
radius AA’ from A, Now the tangent on this
sphere from point P’ represents the reflected wave front i.e. P’A’ represents the reflected
wavefront.
To prove laws of reflection we use the principle that time taken by the ray to move
from incident to reflected wavefront should be same for all rays.
( )
( )
From equation (1) & (2)
This is the law of reflection. Further incident ray reflected and normal all three lie in the plane
perpendicular to the plane of reflection, therefore laws of reflection are verified.
Laws of Refraction on the Basis of Huygens Wave TheorySuppose a plane wave front AB strike an
interface XX’ at A. The ray from point B
will strike the interface at A’ after time t
with respect to A.
Therefore
Where c is the speed of light in
vacuum.
B
First Medium
Velocity = c
i
X
i
A’
A
r
r
B’
X’
Second Medium
Velocity = v
In the same time period the secondary
wavelets from point A will cover a
distance
in second medium, where v is the velocity of light in second medium. Draw the
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Wave Optics by Umesh Tyagi
sphere of radius
from A, as radius and the tangent on this sphere from A’ gives the
position of refracted wave front A’B’ after time t.
Therefore
( )
------------(2)
Divide equation (1) by (2)⁄
⁄
(
)
This is the Snell’s law and μ is the refractive index if second medium w. r. t. first medium.
Principle of Superposition:
According to this principle whenever a number of waves travel through a same medium
then the resultant displacement of a particle is equal to the vector sum of individual displacement
produced by the waves. i.e.




Amplitude
Amplitude
y  y1  y 2  . . . . . .  y n
t
t
Waves in Same Phase
t
Waves in Opposite Phase
Interference of Light:
The phenomenon of the redistribution of light energy in a medium due to the superposition of two
light waves having same frequency and constant or zero phase difference, is called as Interference
of light.
It should be noted that in interference there is only transfer of energy from one region to another, the
energy missing in one region reappears at another region. The regions with maximum intensity are
termed as constructive interference while the regions with minimum intensity are termed as
destructive interference.
Coherent Sources:
The sources of light emitting the waves having same frequency and zero or constant phase
difference are called as coherent sources.
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Wave Optics by Umesh Tyagi
Two independent sources are never coherent-To understand this concept let us go back to
the origin of any light source. Whenever energy is supplied to any object (for e.g. filament of bulb)
through some means, then the atoms of the substance gets excited with electron jumping from ground
state to outer energy orbits. These excited states of atom are unstable and electron jumps back to
ground state directly or through the intermediate states by emitting the radiation. If the radiation
frequency lies in the visible region of e.m. spectra then light is emitted. For any source of light
millions of such transitions takes place per second. In maintain constant or zero phase difference two
sources should have exactly identical transitions which is never possible. Thus, two independent
sources never be coherent. Therefore, we have to generate sources from single parent source.
Methods to generate the coherent sources- There are two methods(a) By the division of wave front
(b) By the division of amplitude
Young's double slit experiment.
This experiment was performed by Thomas Young to demonstrate the phenomenon of
interference. In this experiment, a source of monochromatic light (e.g., a sodium vapor lamp)
illuminates a rectangular narrow slit S, as shown in Fig. S 1 and S2 are the two parallel narrow
slits which are arranged symmetrically and parallel to the slit S at the same distance.
An observation screen is placed at
some distance (about =2 m) from the
two slits. Alternate bright and dark
bands appear on the observation
screen which are called interference
as fringes.
As the light waves coming out from S1
and S2 are derived from the same
parent source S, so they are always in
same phase i.e., S1 and S2 act as
coherent sources. Light wave from S1
and S2 travel in the form of crest and
trough which are shown by dotted line
and continuous line. At the lines leading to O, P2 and P’2 the crest of one wave falls over the
crest of other wave or the trough of one wave falls over it the trough of other wave, the
amplitudes of the two waves get added up and hence the intensity becomes maximum. This is
called constructive interference. At the lines leading to P1 and P’1, the crest of one wave falls
over the trough of other or the amplitudes of the two waves subtracted and hence the intensity
becomes minimum. This is called destructive interference.
Condition for Interference:
Let two waves of amplitudes a1 and a2 same frequency (ω) and the constant phase
difference ϕ. Their displacement equations at time ‘t’ are given as-
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Wave Optics by Umesh Tyagi
y1
 a1 sin t                    1
y2  a2 sin ( t   )                  2 
Resultant displacement, y  y1  y2
y  a1 sin t  a2 sin (t   )
y  a1sint  (a2 sint cos  a2cost sin )
y  sin t ( a1 + a2 cos )  a2 cost sin
( a1 + a2 cos )  Acos                 (3)
Let
a2 sin  A sin 
               ( 4)
∴
y  A sint cos  A sin cost
Or
y  A sin ( t   )                 5 
This is the equation of same type as the equations (1) & (2).
To find the amplitude of resultant wave, squaring and adding the equations (3) and (4), we get
( A cos ) 2  ( A sin ) 2  ( a1  a2 cos ) 2  ( a2 sin ) 2
A2  ( a12  a22 cos 2   2 a1a2 cos  )  a22 sin 2 
A = a12 + a 22 + 2a1a 2cos            (6)
We know that,
⟹
(
)
I  I1  I 2  2 I1 I 2 cos            (7)
Condition for intensity to be maximum (Constructive Interference)⟹
Or
Phase Difference,   2 nπ where n  0,1, 2, ..
Thus, for constructive interference, the phase difference should be 2n or phase difference
should be n . Where n=0, 1, 2, 3…….
∴ Maximum intensity, I max 
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
I1  I 2

2
and Maximum Amplitude, Amax  a1  a2
Wave Optics by Umesh Tyagi
Condition for intensity to be minimum (Destructive Interference)cos  min  1
Phase Difference,    2 n  1 
Path difference x 
    ,3 ,5 ........
Where n= 0, 1, 2, 3…



 
 (2n  1)  (2n  1)
2
2
2
Thus, for destructive interference, the phase difference should be (2n  1) or path
difference should be (2n  1) / 2 .Where n=1, 2, 3 …
∴ Minimum Intensity, I min 

I1  I 2

2
and Minimum Amplitude, Amin  ( a1  a2 ) .
Special Cases- (i) If a1  a 2  a  let  i.e. the waves have same amplitudes, then equations (6)
and (7) reduces to
A2  4a 2 cos 2

2
I  4 I 0 cos 2
and

2
,
Where I0 is the intensity of each source
(2) If w1 and w2 are the widths of slits then
(3)
(√
√ )
(√
√ )
(4) Intensity distribution curve-
-3λ
- λ -λ
λ
Path difference
λ
3λ
Theory of fringes and fringe width:
Let the light from a single source be divided into two parts using slits separated by
distance d. The distance between source and screen is D. These two slits acts as two
independent source of light.
Consider any point P on screen, path difference between
the waves reaching at P from S1 and S2 is given as,
Path Difference  x  (S 2 P  S1 P )        (1)
In S2PB,
d

S2 P  D   y  
2

In S1AP,
d

S1 P 2  D 2   y  
2

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2
2
2
2
Wave Optics by Umesh Tyagi
2
d 
d

S 2 P  S1 P   y     y  
2 
2

2
2
2
( S 2 P  S1 P ) ( S 2 P  S1 P )  4 y
( S 2 P  S1 P ) 
d
2
2 yd
-----------------(2)
( S 2 P  S1 P )
From equation (1) and (2),
Path difference, x 
2 yd
( S 2 P  S1 P )
If P is very close to O the S2 P  D & S1 P  D
∴Path difference, x 
2 yd
2D
x
yd
        (3)
D
But the path difference for nth bright fringe is n
∴ Position of nth bright fringe on screen from central bright fringe becomes
yn 
n D
d
For nth dark fringes the path difference is (2 n  1)

2
.
∴Position of nth dark fringe on screen from central bright fringe is given asy n  (2 n  1)
D
2d
Fringe Width: The distance between two consecutive bright fringes or dark fringes is called fringe width.
Width of dark fringe, 1  yn  yn 1 
n D ( n  1) D  D


       (4)
d
d
d
Width of bright fringe,  2  yn  yn 1 
(2n  1) D [2( n  1)  1) D  D


     (5)
d
d
d
Thus, from equation (4) and (5), it is clear that β1 = β 2 i.e. the bright fringes as well as dark fringes
have the same width.
Fringe width β = λD
∴
d
Angular Fringe Width   
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
D


d
Wave Optics by Umesh Tyagi
Factors affecting the width of fringe- (i ) B  
(ii ) B  D
and
B
1
d
Note- If the YDSE arrangement is completely immersed in a liquid of refractive index 𝝁 then
'


  '


Conditions for Sustained Interference:
In sustained interference pattern the positions of maximum and minimum intensity do not change
with time. To obtain sustained interference pattern the following conditions should be satisfied:
(1)
(2)
(3)
(4)
(5)
The two sources should continuously emit waves of same frequency and zero or constant
phase difference.
For good interference pattern the amplitude of two waves should be preferably equal
otherwise total darkness will not be produced at any point.
Sources should be monochromatic.
The two sources should lie close to each other to have well defined and sharp interference
pattern.
The sources should preferably be point source because a broad source of light acts as
number of point sources placed together. Interference can place between two such point
sources which are part of broad sources resulting in large number of interference patterns.
FRINGE SHIFTWhen a thin transparent plate of thickness t and refractive index μ is introduced in the path of one of the
interfering waves then the effective path in air is increased by (μ – 1) t due to the introduction of the
plate
Effective path difference in air = S 2 P   S1 P  (   1)t 
= S 2 P  S1 P  (   1)t
=
For maxima, path difference
yd
 (   1)t
D
λ w
3
Therefore the position of nth maxima is given by;
yn d
 (   1)t  n
D
Or
yn 
D
 n  (   1)t           (1)
d
In the absence of the plate (i.e. t=0), the position of the nth maxima is
n D
            (2)
d
Wave Optics by Umesh Tyagi
yn 
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t
∴ Displacement of fringe- y 
y 
D
n D
 n  (   1)t  
d
d
D

(   1)t     1 t
d

Δy =
β
 μ - 1 t
λ
Thus we find that with the introduction of the transparent plate in the path of one of the slits, the entire
pattern is displaced through a distance of D (   1)t or     1 t towards the side on which the
d

plate is introduced and there is no other change in the pattern.
Diffraction of Light:
The phenomenon of bending of light around the corners of an obstacle or
an aperture into their geometrical shadow is called diffraction of light
Types of diffraction-
(1)
F s
l s Diffraction: In this source and slit are at finite distance from the obstacle.
(2)
Fraunhoffer Diffraction:
In this category source and slit are at infinite distance from obstacle.
Note- Condition for the diffraction of light the dimensions of the obstacle should be comparable to the
wavelength of light.
Diffraction at A Single Slit:
Consider a narrow slit AB illuminated by a beam of light coming
from infinity. After passing through slit beam is focused by means of convex lens. Rays going straight
comes to focus at O. Thus, O should be a bright spot. Since all the rays have been brought to focus at O,
there should have been complete darkness on either side of O. On the contrary alternate dark and bright
fringes are observed on both sides of O.
This can be explained on the basis of diffraction. Consider any point P at a distance y from O. A beam of
light will reach at point P only if it is bent through an angle θ. Draw AN as perpendicular to the ray
diffracted from B then ‘AN will be a diffracted wave front. The intensity at P will depend upon the path
difference between the secondary waves emitted from corresponding points of the wave front. Path
difference between the waves coming from A and B to P is
BN  AB sin   d sin 
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Wave Optics by Umesh Tyagi
If this path difference is λ, then P will be point of minimum intensity. To explain this, the whole wave
front can be divided into two equal halves AC and CB. The path difference between A and C will be λ/2.
Also for every point in AC there will be a point in CB for which the path difference between secondary
waves is λ /2. Thus destructive interference takes place and P is point of secondary minimum. If path
difference s λ, then wavefront is divided into equal four parts and second secondary minimum occurs.
Thus, for secondary minima path difference should be n λ w
3
d sin  n  n
∴
If θ is very small then sin  n   n . Hence angular position of nth minima is given as θ n =
nλ
d
If the point P is such that, d s θ= 3λ/ , then wave front can be divided into 3 equal parts, destructive
interference takes place between first two parts. The secondary waves from third part however remain
unused and therefore reinforce each other and first secondary maxima is there.
Thus, for nth secondary maxima, the path difference should be (
(
∴
If θ is very small then
given as
(
) / w
3
) /
sin  n   n . Hence angular position of nth secondary minima on the screen is
)
Linear Width of Secondary Maxima- In any diffraction pattern
Direction of nth minimum
n 
Direction of (n+1)th minimum
 n 1 
n
d
 n  1 
d
Angular width of secondary maximum =  n 1   n 
Hence
 n  1 
d

n


d
d
Linear Width  Angular Width  D
 
D
d
Angular Width of Central Maxima-
It is the
separation between the directions of the first minima on the
two sides of central maximum.
The direction of 1st maxima on either side of central
maxima is given by


θ
x
θ
d
Angular width of central maxima  2  
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d
2
d
Wave Optics by Umesh Tyagi
D
Linear Width of Central Maxima  x   2   D 
2 D
d
Thus, width of central maxima  2  width of secondary maxima
Factors on which width of central maxima depends(1)
(2)
width of central maxima ∝ λ
Width of central maxima ∝ 1/ width of slit
If white light is used, then, central fringe is white while other fringes are colored.
Intensity distribution curve-
Intensity
(I)
This shows that as we go away from central
maxima, the intensity of secondary maxima
decreases rapidly.
-4
-3
-2

-
2
3
4
Path Difference
DIFFERENCE BETWEEN INTERFERENCE AND DIFFRACTIONThe importance differences between interference and diffraction of light are listed below
S. No.
Interference
Diffraction
1
Interference occurs due to the
superposition of coming from two
coherent sources
Diffraction occurs due to the superposition of
light waves coming from different parts of the
same wave front
2
All bright fringes are of the same
intensity.
The intensity of bright fringes decreases with
increasing distance from the central maximum.
3
For a monochromatic light, the fringe
independent of the order of fringe i.e.
it is constant
Fringe width is not a constant quantity
4
There is better contrast between bright
and dark fringes
The contrast between bright and dark fringes is
poor
Rayleigh’s Criterion for ResolutionWhen a point source of light is imaged by an optical system with a circular aperture, the image is an Airy disc.
If two points are very close, their Airy discs will overlap and we may not be able to resolve them, i.e.,
distinguish separate images. The following figure shows the images of two points that are : a) resolved (b)
barely resolved and (c) unresolved, along clearly with their intensity patterns.
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Wave Optics by Umesh Tyagi
As a quantitative measure of the resolution of two points, Lord Rayleigh proposed the following criterion,
called the Rayleigh criterion, which states that:
Two points are barely resolved when the centre of the one’s Airy disc is at the edge of the other’s i.e.,
one’s Airy disc falls on the first minimum of the other’s.
Rayleigh Criterion in terms of Intensity PatternsTwo points are just resolved by an optical system when the central maximum of the diffraction pattern due
to one falls on the first minimum of the diffraction pattern of the other.
  1.22
Thus the limit of resolution,

d
RESOLVING POWER OF A TELESCOPE
The resolving power of a telescope is defined as the reciprocal of the smallest angular separation ( θ)
between two distant objects whose images are just seen as separate through the telescope.
According to Rayleigh’s criterion, limit of angular resolution is given as
  1.22

d
Where λ is the wavelength of light used and ‘d’ is the diameter of telescope objective.
Thus resolving power of a telescope,
R.P. 
1
d

  1.22 
NOTE- (1) Maximum useful magnification of a telescope, M  2.5  10 4
d

(2) Relation between magnifying power of a telescope and the aperture of the eye-
M 
Aperture of the Objective
Aperture of the eye
(3) Followings re the factors affecting the resolving power of telescopei.
ii.
Diameter of objective of telescope i.e. R.P.∝ diameter
Wavelength of light i.e. R.P. ∝1/𝛌
Resolving Power of a MicroscopeThe resolving power of a microscope represents its ability to form distinctly separate images of two objects
lying very close together.
The minimum distance (d) between two point objects, whose images in the microscope are just seen as
separate is called limit of resolution. The limit of resolution is given as-
d=
Where 𝝁 = refractive index,
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
2  sin 
= wavelength of light used and = half of the cone angle formed by objective
Wave Optics by Umesh Tyagi
The reciprocal of the limit of resolution is called resolving power of the microscope.
Resolving Power of Microscope = R.P. =
1 2μsinθ
=
d
λ
The R. P. can be increase by decreasing the wavelength so the object is seen in violet light as it has the least
wavelength. 2 sin𝜃 is called as numerical aperture of objective.
Polarization of Light Waves
Transverse Nature of Light: Huygen's initially assumed light waves to be longitudinal in nature and
proved many properties of light using longitudinal behaviour of light waves. But in 1690, when Huygen's
himself discovered polarization he could not explain it using longitudinal nature of light.
The transverse nature of light can be experimentally verified using apparatus in which two slits S 1 and
S2 are cut in two card board pieces. A string is passed through S1 and S2 and transverse vibrations parallel to
length of slit S1 are given to the end of the string near S1. If the length of slit S2 is parallel to S1 then the
vibrations will pass through S2 also. But if S2 is rotated in its plane, the amplitude of vibrations passing
through S2 goes on decreasing and when the length of S2 becomes parallel to S1 then vibrations do not come
out of S2 i.e. amplitude of emergent vibrations becomes zero. But if longitudinal waves are passed through
slits then there is no effect of relative orientation of S1 and S2 on amplitude.
We thus conclude that if a wave produced at first end of the string passes through S 1 and S2, and on
rotating the amplitude of vibration varies, then waves passing through S1 are transverse.
C1
C2
Maximum Light
Intensity
Ordinary Light
(Axes are parallel to each other)
C1
C2
Ordinary Light
No light
(Axes are perpendicular to each
A similar behaviour was observed in light waves by using two tourmaline crystals which are cut that
their axis lies in their plane.
To demonstrate the transverse nature of light, the ordinary light is incident normally on the pair of two Nichol
Prism or tourmaline crystal cut parallel to crystallographic axis, the intensity of emergent beam is maximum
when the axis of two crystals is parallel to each other and becomes minimum when axis are perpendicular to
other. It confirms that the light is transverse in nature.
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Polarization of light- The phenomenon due to which the vibrations of light are restricted in
a particular plane is called as polarization of light.
Un-polarized and plane polarized lights & their pictorial representation
A light wave, in which vibrations are present in all possible direction in a plane
perpendicular to the direction of propagation is said to be unpolarised light.
If the vibrations of light wave are present just in one direction in a plane perpendicular to
the direction of propagation, then the light is said to be plane polarized.
&
&
PLANE POLARISED LIGHT
ORDINARY LIGHT OR UNPOLARISED
Plane of vibration and plane of polarizationThe plane containing the direction of vibration and the direction of wave propagation is
called the plane of vibration. In the following figure, ABCD is a plane of vibration.
A plane perpendicular to the plane of vibration is called the plane of polarization. In figure,
PQRS is the plane of polarization.
P
Q
B
A
UNPOLARIZED
C
D
S
PLANE OF POLARISATION
R
PLANE OF VIBRATION
Polarization of light by reflectionWhen an ordinary beam of light is incident on the
surface of a transparent medium then the reflected
light becomes partially plane polarized. The degree of
polarization depends upon the angle of incidence. The
angle of incidence at which the reflected ray becomes
completely plane polarized is called the angle of
polarization.
Explanation: - When the light is incident at the
polarizing angle the vibration perpendicular to plane of
paper or parallel to the reflecting surface are reflected
along BC, while the other vibrations are transmitted and are not reflected. The reflected
light is therefore, completely polarized in the plane of incidence.
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Bwester’s lawAccording to this law the refractive index of the medium is equal to the tangent of
angle of polarization.
If ip is the angle of polarization and is  the refractive index of the medium then
𝝁
From Snell’s law
Thus
=
s (
)
Thus, Brewster law can also be stated as that " if angle of incidence is equal to the polarizing angle
then reflected and refracted ray are perpendicular to each other."
Malus law- It states that when a complete plane
The intensity of the transmitted light is given by
Ι α Cοs 2 θ
 I  I o Cos 2
Where I0 is the maximum intensity of transmitted
light.
Note: - If the incident light is unpolarised then the
intensity of transmitted light is given by
I = I0 / 2
I=I0 Cos2
Intensity (I)
polarized light is incident on an analyzer, the intensity
of the emergent light varies as the square of the cosine
of the angle between the planes of transmission of
analyzer and the polarizer.
90º

180º
270º
360º
Graph between intensity of transmitted
light and the angle between polarizer
and analyzer.
here I0 is the maximum intensity
What is the Polaroid. How are these constructed? Give their important uses.
A Polaroid is a device, which is used as a polarizer as well as analyzer for the polarized
light. A large number of (small needle shaped) crystals of iodosulphate of quinine called
Herpathite, are spread over a transparent nitrocellulose film so that their axis remain
parallel to each other. Now this film is sandwiched between the two transparent glass
sheets. The sheet so obtained is called as Polaroid.
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P2
P1
Un polarized Light
P1
Polarized Light
Parallel
POLARISER
P2
Crossed
ANALYSER
Uses of Polaroid: Some important uses of Polaroid’s are given as –
1.
2.
3.
Polaroids are used in sunglasses to protect the eyes from the glare.
The windshield of an automobile is made of Polaroid, which protects the eyes of the
driver from the dazzling light of approaching vehicles.
The pictures taken by a stereoscopic camera, when seen with the help of Polaroid
spectacles, create the 3-D effect.
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