XII - PHYSICS Prof.

Prakash Patil (9821271237)

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7. Wave Optics - Interference of light 3

❖ Principle of superposition of waves

When two or more waves of same type, travelling through a medium, arrive at a point
simultaneously, each wave produces its own displacement at that point which does not depend
upon displacements produced by other waves. The resultant displacement at that point is
equal to the vector sum of the displacements due to all waves separately.
❖ Interference of light
Phenomenon of enhancement (addition) or cancellation (subtraction) of displacement produced
due to superposition of waves is called interference.
OR
The physical effect observed due to superposition of two waves in which there takes place
redistribution of energy is called interference.
In case of light waves, depending upon the resultant intensity which is maximum or minimum,
the interference of light is of two types.
1. Constructive Interference
2. Destructive interference

❖ Constructive interference
This type of interference occurs when,
Two light waves of same frequency (coherent) travelling through the same medium, in nearly
same directions are arriving at a point of the medium in phase.
The displacements produced at the point of arrival are in phase, that is crest of one wave falls
on the crest of another wave and trough of one wave falls on the trough of the other wave.
Then resultant displacement at that point is maximum.
(If waves are of different frequency(non-coherent), then resultant displacement varies with time)
Then resultant amplitude of oscillation at that point is maximum.
(It the arriving waves are of same amplitude, there is duplication of amplitude)
(It the arriving waves are of different amplitude, there is addition of amplitude)
Since intensity of light is directly proportional to the square of amplitude of wave, intensity at that
point becomes maximum(brightness).
The point appears bright.

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So, the point of arrival appears bright if the phase difference between two interfering waves is,
integral multiple of 2 radian.
So, the point of arrival appears bright if the path difference between two interfering waves is,
integral multiple of wavelength.
Phase difference = 0, 2π, 4π, − − − − − − n(2π)
Where, n = 0, 1, 2, 3 − − − − − −
Since phase difference of 2 corresponds to a path difference of .
Path difference = 0, λ, 2λ, − − − − − − n(λ)
Where, n = 0, 1, 2, 3 − − − − − −
❖ Destructive interference
This type of interference occurs when,
Two light waves of same frequency (coherent) travelling through the same medium, in nearly
same directions are arriving at a point of the medium 1800 out of phase.
The displacements produced at the point of arrival are 1800 out phase, that is crest of one wave
falls on the trough of another wave and trough of one wave falls on the crest of the other wave.
Then resultant displacement at that point is minimum.
(If waves are of different frequency(non-coherent), then resultant displacement varies with time)
Then resultant amplitude of oscillation at that point is minimum.
(It the arriving waves are of same amplitude, there is cancellation of amplitude)
(It the arriving waves are of different amplitude, there is subtraction of amplitude)
Since intensity of light is directly proportional to the square of amplitude of wave, intensity at that
point becomes minimum(darkness).
The point appears dark.
So, the point of arrival appears dark if the phase difference between two interfering waves is,
odd integral multiple of  radian.
So, the point of arrival appears dark if the path difference between two interfering waves is, odd
integral multiple of half wavelength.
Phase difference = π, 3π, 5π, − − − − − − (2n − 1)(π)
Where, n = 1, 2, 3 − − − − − −
Since phase difference of 2 corresponds to a path difference of .
λ λ λ λ
Path difference = , 3 (2) , 5 (2) , − − − − − − (2n − 1) (2)
2

Where, n = 1, , 2, 3, − − − − − −
❖ Interference pattern on water surface (surface interference)
Consider that, tuning fork of fixed frequency is held over the water surface, such that its
Prong is parallel to water surface. Two needles are pasted by wax to lower end of prong.
When tuning fork is strike gently, needles touched the water surface at points S1 and S2
Water waves are produced in the form of concentric ripples. In diagram, let continue circles
represent crest and dotted circles represent trough. On superposition of waves at different points
on water surface, interference pattern obtained is shown in figure.

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Consider that, tuning fork of fixed frequency is held over the water surface, such that its Prong
is parallel to water surface. Two needles are pasted by wax to lower end of prong.
When tuning fork is strike gently, needles touched the water surface at points S 1 and S2
Water waves are produced in the form of concentric ripples. In diagram, let continue circles
represent crest and dotted circles represent trough. On superposition of waves at
different points on water surface, interference pattern obtained is shown in figure.
The points where crest of one wave falls on the crest of another wave (trough of one wave falls
on the trough of other wave), constructive interference takes place and point vibrate with
maximum amplitude.
The points where crest of one wave falls on the trough of another wave or vice-versa, destructive
interference takes place and point vibrate with minimum (zero) amplitude.
❖ Conditions to obtain steady interference pattern.
The pattern containing alternate dark and bright bands is called steady or stationary interference
pattern.
• The two sources of light must be coherent.
• The two sources of light must be monochromatic.
• The two sources must be equally bright.
• The sources should be narrow.
• The two sources should be close to each other.
• The distances between the screen and the sources should be large.
• The sources must be in a same state of polarization.
• The interfering waves must be propagating in nearly same direction.
❖ Coherent Sources of light
Two sources emitting the light waves (sound waves) of same frequency and constant phase
difference (does not change with time and space), are called coherent sources.
It is found that so long as two different sources of light are used, waves emitted by the sources
undergo rapid and irregular changes in phase. Therefore, due to change in phase difference
between emitted waves, steady interference pattern cannot be obtained.

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1. In Young’s double experiment, a single slit is illuminated by a monochromatic light.
Cylindrical wave front emitted by this slit is allow to incident on two equally narrow and
parallel slits. These slits are considered as two coherent sources.
2. In Fresnel’s biprism experiment, an optical device, called biprism is used. Biprism
produces two virtual and equally bright images of original source in its plane. These virtual and
monochromatic images are considered as two coherent sources.
3. In Lloyd’s mirror experiment, an optical device, called Lloyd’s mirror is used. This mirror
On reflection gives a virtual image of original source in its own plane. The source and its
image both are considered as two coherent sources.
❖ Monochromatic Sources of light
For a given source, colour of light is determined by the wavelength emitted. The original
source from which the two coherent sources are derived, must emit light of one single
wavelength. If this is not so, then the light of each colour, on interference will form its own
interference pattern and the interference pattern so obtained is diffused.
❖ Sources must be equally bright
Brightness or intensity of light is directly proportional to square of amplitude. If amplitudes of two
sources are not same then,
(i) During the constructive interference resultant amplitude is obtained by addition of individual
amplitudes. Point appears bright.
(ii) During the destructive interference resultant amplitude is obtained by subtraction of individual
amplitudes. Due non zero subtraction, point appears less bright. Which is not
expected (expected cancellation of amplitudes and point should appear dark)
Hence two sources should be equally bright.
❖ Sources must be narrow
The coherent sources must be narrow, because a broad source is equivalent to a number of
narrow sources placed side by side. Each of these narrow sources form their own interference
pattern. This produces general illumination on a screen.
❖ Distance between two sources should be very small
This experiment was performed to determine wavelength of monochromatic light.
Fringe width is given by,
λD
β=
d
1
β ∝
d
Where,
𝑑 - distance between two sources
𝐷 - distance between sources and screen
 − distance between two successive bright or dark bands (fringe width)
So, two get well defined interference pattern, d should small, so that  is large.
❖ Distance between sources and screen should large
This experiment was performed to determine wavelength of monochromatic light.
Fringe width is given by,

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𝜆𝐷
𝛽=
𝑑
𝛽 ∝ 𝐷
Where,
𝑑 - distance between two sources
𝐷 - distance between sources and screen
 − distance between two successive bright or dark bands (fringe width)
So, two get well defined interference pattern, D should large, so that  is large.
❖ The source must be in a same state of polarization.
This condition is applicable, only when polarized light is used to study interference experiment.
Light waves having their plane of vibration perpendicular to each other do
Produce interference.
❖ The interfering waves must be propagating in nearly same direction.
According to principle of superposition of waves, resultant displacement and hence resultant
amplitude is vector sum of individual displacements and amplitudes respectively. It is calculated
by law of parallelogram of two vectors. If two interfering are travelling in nearly same direction,
resultant can be found by simple algebraic addition.
❖ How coherent sources are obtained
1. Young’s double slit experiment.

Source of monochromatic light is kept behind the partition S1 on thin slit a is provided.
Light emerging from slit a is allow to fall on two more equidistance slits b and c, provided
On next partition S2. Cylindrical wavefront emitted from line source a is incident on partition S2.
Slits b and c are equidistance from slit a, so according Huygens’s principle Slits b and c acts
secondary sources of light as well as coherent sources of light.Secondary waves emitted from
them on superposition gives line or band interference pattern on the screen.
2. Fresnel’s Biprism experiment.
Fresnel used a biprism to obtain the coherent sources. Biprism produces the two virtual images
of the slit by refraction. A biprism consists of two identical prisms of very small vertex angles
(from 30’ to 10) with their bases connected together. Actually, it is a single prism of refracting
angle nearly equal to1800.
A narrow and vertical slit S is illuminated by a monochromatic light. The biprism is arranged in
front of the slit with its refracting edge parallel to the slit. The light from S passes through the
two halves of the biprism and the refracted rays appear to come from the two virtual images S1
and S2 of the slits. These two virtual images are in the plane of the slit S and serve as two
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coherent monochromatic sources. Virtual images are sufficiently closed to each other, due to
small refracting angles of prisms. Thus they give rise to clear, well-spaced, interference
fringes on screen or in the focal plane of an eye-piece, in the region where waves coming from
S1 and S2 overlap.

3. Lloyd’s mirror experiment.

A light from the source S1 is allow to fall obliquely on the Lloyd’s mirror. Reflected ray meet the
screen at point P. The ray of light from source S1 meet the screen at point P. This ray of light
appears to come from the point at S2, in the plane of source S1. Thus, the real source S1 and
virtual source S2 acts as two coherent sources of light. The waves emitted from them give
interference pattern on the screen.
❖ Young’s double slit experiment.

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Above diagram shows the Young’s double slit experiment.
A linear source S of monochromatic light is kept at the principal focus of double convex lens. A
double convex lens converts a cylindrical wave front incident on it into plane wave front on
refraction. A plane wave front is allowed to incident on opaque screen AB, on which two slits, S1
and S2 are provided. These slits are narrow and close to each other. Plane incident wave front
make them secondary and coherent sources of light. Light waves emitted from the slits
propagate in the form of cylindrical wave fronts and their superposition gives interference pattern
on the screen. Interference pattern consists alternate dark and bright bands, with bright band at
the Centre. This band is called central bright or zero order bright band.
[ Above diagram is two-dimensional look of experiment. In diagram continuous arc and dotted
arc represent crest and trough respectively. During the propagation points where Dotted arcs or
continuous arcs coincide, appear bright, due to constructive interference. During the propagation
points where dotted arc of one and continuous arc the other coincide, appear dark, due to
destructive interference. If propagation is terminated on the screen, interference pattern is seen
in the form of alternate dark and bright bands.
➢ Importance of Young’s experiment
(i) This experiment confirms the wave nature of light.
(ii) This experiment, was the first demonstration on interference of light.
(iii) This experiment was performed to determine wavelength of monochromatic light.
❖ Expression for path difference and fringe width
(Analytical treatment)

Consider S1 and S2 are two coherent, monochromatic and equally bright sources of light
(slits) separated by a distance d. Let a screen is placed at a distance D from the coherent sources
and parallel to slits as shown in Figure. Draw perpendicular from midpoint of slit separation at
point O on the screen. Draw S1A and S2B. Cylindrical wave fronts emitted from sources S1 and
S2 propagate towards screen and on superposition gives interference pattern on screen.
Interference pattern consists alternate dark and bright bands, with bright band at the centre. This
band is called central bright or zero order bright band.
For point O on the screen, light waves arrived in phase.

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Optical path difference is,
(a) S2 O − S1 O = 0
So, point O appears bright.
Consider the point P on the screen, at a distance y, from the centre of interference pattern.
Nature of illumination at point p depends upon the phase difference between arriving waves and
path difference (S2 P − S1 P)
From right angled triangles S1AP and S2BP
S1 P 2 = S1 A2 + AP 2
d
S1 P 2 = D2 + (y − 2)2 -----(i)

S2 P 2 = S2 B2 + BP 2
d
S2 P 2 = D2 + (y + 2)2 -----(ii)

Eq.(ii) – Eq.(i) gives,

S2 P 2 − S1 P 2 = 2yd -----(iii)
(S2 P − S1 P)(S2 P + S1 P) = 2yd
In practice, d << D
(S2 P ≈ S1 P ≈ D) and (S2 P − S1 P ≠ 0)
(S2 P − S1 P)(2D) = 2yd
yd
(S2 P − S1 P) = -----(iv)
D

This is the expression optical path difference.

(b) Point P appears bright, if optical path difference is integral multiple of wavelength.

yd
(S2 P − S1 P) = = nλ
D

n = 0, 1, 2, --------------
nλD
ynB =
d

This is the expression for distance of nth order bright band from the central bright.
(c) Point P appears dark, if optical path difference is odd multiple of half wavelength.
yd λ
(S2 P − S1 P) = = (2n − 1) 2
D

n = 1, 2, 3, --------------
(2n − 1)λD
ynD =
2d
This is the expression for distance of nth order dark band from the central bright.

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❖ Fringe width or Band width
Distance between two successive bright bands or two successive dark bands, interference pattern
is called as band width or fringe width.
A. Considering two successive bright bands
Distance of nth order bright fringe from central bright fringe is,
nλD
ynB = -------(a)
d
Distance of (n+1) th order bright fringe from central bright fringe is,
(n+1)λD
y(n+1)B = -------(b)
d
Fringe width is,
β = y(n+1)B − y(n)B
(n + 1)λD (n)λD
β= −
d d
λD
β= -------(c)
d
B. Considering two successive dark bands
Distance of nth order dark fringe from central bright fringe is,
(2n−1)λD
ynD = -------(a)
2d
Distance of (n+1) th order dark fringe from central bright fringe is,
(2n+1)λD
y(n+1)D = -------(b)
2d
Fringe width is,
β = y(n+1)D − y(n)D
(2n + 1)λD (2n − 1)λD
β= −
2d 2d
λD λD
β= +
2d 2d
λD
β= -------(c)
d

Equation (c) represents expression for band width.

❖ Conclusions
1. Band width is directly proportional to the wavelength of monochromatic light used.
2. Band width is directly proportional to the distance between slits and screen.
3. Band width is inversely proportional to separation between slits (two coherent sources).
βmedium λmedium m μa
4. = = μ =
βair λair a μm

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❖ Intensity distribution on superposition of light waves

• Light waves are electromagnetic waves, which consists oscillating and transverse electric field
→ →
vector 𝐸 and magnetic field vector 𝐵 . Optical properties of light are governed by oscillating
electric field.
• Consider two monochromatic and equally bright sources of light situated at point S1 and S2. Light
waves emitted by them reach at point P.
• Displacement of point P due their individual arrivals are,
E1 = E0 sin(ωt) ---(i)
E2 = E0 sin(ωt + ϕ) ---(ii)
Where E – electric field vector
E0 – amplitude of electric field vector
If waves arriving at point P travelled in nearly same direction (llel), resultant electric field is given
by,
E = E1 + E2
E = E0 sin(ωt) + E0 sin(ωt + ϕ)
C+D C−D
SinC + sin D = 2 sin ( ) cos ( )
2 2
ωt+ωt+ϕ ωt−ωt−ϕ
E = 2E0 sin ( ) cos ( )
2 2

ϕ ϕ
E = 2E0 sin (ωt + ) cos (− )
2 2
cos(−θ) = cos θ
ϕ ϕ
E = 2E0 cos ( 2 ) sin (ωt + 2 ) -------(iii)

This is the expression for resultant electric field at point P.

Here,
ϕ
2E0 cos ( 2 ) = ER is amplitude value of resultant electric intensity
ϕ
(2) = δ initial phase of resultant intensity

E = ER sin(ωt + δ) ------(iv)
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❖ As intensity of light, I directly proportional to square of amplitude,
ϕ
I = ER2 = 4E02 cos 2 ( 2 )
ϕ
I = 4I0 cos2 ( 2 ) ----(v)

❖ Special cases
Case I
For maximum intensity (brightness or constructive interference)
I𝑚𝑎𝑥.
ϕ
cos2 ( 2 ) = ±1
ϕ
( 2 ) = 0, π, 2π, − − − − − − −

ϕ = 0,2π, 4π, − − − − − − −n(2π) Where n =0,1,2,3, -------

As,
2π
phase difference = × path difference
λ
λ
path difference = 2π × phase difference
λ
x = S2 P − S1 P = 2π × [ 0,2π, 4π, − − − − − − n(2π)]

x = S2 P − S1 P = 0, λ, 2λ, 3λ, − − − − − − nλ Where n =0,1,2,3, -------

Thus, for point P to be bright or for constructive interference, phase difference between
Arriving waves is even multiple of  radian and path difference is even multiple of 
Case II
For minimum intensity (darkness or destructive interference)
Imin.
ϕ
cos2 ( 2 ) = 0
ϕ π 3π 5π
(2) = 2 , 2
,
2
,− − − − − − −
φ = π, 3π, 5π − − − − − − − (2n − 1)(π) Where n =1,2,3, -------
As,
2π
phase difference = × path difference
λ
λ
path difference = 2π × phase difference
λ
x = S2 P − S1 P = 2π × [ π, 3π, − − − − − − (2n − 1)(π)]
λ 3λ 5λ λ
x = S2 P − S1 P = , , , − − − − − − (2n − 1) Where n =1,2,3, -------
2 2 2 2
Thus, for point P to be dark or for destructive interference, phase difference between
Arriving waves is odd multiple of  radian and path difference is odd multiple of 
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