PADMA SESHADRI BALA BHAVAN SR. SEC.

SCHOOL
SUBJECT: PHYSICS
STD XII WAVE OPTICS NOTES

INTRODUCTION
• In 1637 Descartes gave the corpuscular model of light and derived Snell’s
law. It explained the laws of reflection and refraction of light at an
interface.
• The corpuscular model predicted that if the ray of light (on refraction)
bends towards the normal then the speed of light would be greater in the
second medium. This corpuscular model of light was further developed by
Isaac Newton in his famous book entitled OPTICKS and because of the
tremendous popularity of this book, the corpuscular model is very often
attributed to Newton.

• In 1678, the Dutch physicist Christiaan Huygens put forward the wave
theory of light. The wave model could satisfactorily explain the phenomena
of reflection and refraction; however, it predicted that on refraction if the
wave bends towards the normal then the speed of light would be less in the
second medium. This is in contradiction to the prediction made by using the
corpuscular model of light. It was much later confirmed by experiments
where it was shown that the speed of light in water is less than the speed in
air confirming the prediction of the wave model; Foucault carried out this
experiment in 1850.

WAVEFRONT
The locus of points, which oscillate in phase is called a wavefront. Thus a
wavefront is defined as a surface of constant phase. The speed with which the
wavefront moves outwards from the source is called the speed of the wave. The
energy of the wave travels in a direction perpendicular to the wavefront.
 Spherical wavefront
For a point source emitting waves uniformly in all directions, the locus of points
which have the same amplitude and vibrate in the same phase are spheres and we
have what is known as a spherical wavefront.
 Cylindrical wavefront
A line source can be considered to be made up of a number of point sources. For
each point source, the locus of all points vibrating in phase is a circle.

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For a line source, the locus of all points that receive the disturbance at the same
time and oscillate in phase is thus a cylinder and hence the wavefront is a
cylindrical wavefront.
 Plane wavefront
At a large distance from the source, a small portion of the sphere can be
considered as a plane and we have what is known as a plane wavefront.

Note:
 Whatever be the shape of the wavefront, the disturbance travels outward
along straight lines i.e. energy of a wave travels in a direction perpendicular
to the wavefront.
 A ray of light represents the path along which the light travels. Rays
are perpendicular to the wavefront.
 A spherical wavefront can be a converging spherical wavefront or a
diverging spherical wavefront.

HUYGEN’S PRINCIPLE
 Each point of the wavefront is the source of a secondary disturbance and
the wavelets emanating from these points spread out in all directions with
the speed of the wave. These wavelets emanating from the wavefront are
usually referred to as secondary wavelets.
 A common tangent drawn to all these spheres in the forward direction gives
the position of the new wavefront at a later time.

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 1) A, B, C, D and E are points on the spherical wavefront that is the source of
secondary wavelets.
2) The forward envelop or surface tangent at A’, B’, C’, D’ and E’ gives the new
wavefront.
What about the backward envelop?
Huygens argued that the amplitude of the secondary wavelets is maximum in the
forward direction and zero in the backward direction; by making this adhoc
assumption he stated that there is no backward envelop.

At large distances, the spherical wavefront becomes a plane wavefront as shown

LAWS OF REFLECTION
 Let PQ be a plane mirror and A1B1 a plane wavefront incident at an angle i.
 At t=0, let the secondary wavelet from A1 reach the mirror at A as shown.

 From Huygen’s principle, every point on the wavefront is a source of

secondary wavelets
 Let t be the time for the secondary wavelet from B to reach the mirror at C.
BC = ct.
 During this time, the secondary wavelet from A spreads into a sphere of
radius ct.
 With A as centre draw an arc of radius ct.
 The surface tangent from C to the arc gives the reflected wavefront CD.

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  The angle the incident wavefront AB makes with the mirror is angle of
incidence and the angle the reflected wavefront makes with the mirror is
angle of reflection.
 Rays are perpendicular to the wavefront and hence ∠𝐀𝐁𝐂 = ∠𝐀𝐃𝐂 = 900.
Consider 𝚫𝐀𝐁𝐂 and 𝚫𝐀𝐃𝐂
∠ABC = ∠ADC = 900
AC = AC – Common
AD = BC = ct
𝚫𝐀𝐁𝐂 and 𝚫𝐀𝐃𝐂 are congruent (RHS)
Thus ∠𝐢 = ∠𝐫 proving the law of reflection.

LAWS OF REFRACTION
 Let PQ be the interface and AB a plane wavefront incident at an angle i.
 At t=0, let the secondary wavelet has reached interface as shown.

 From Huygen’s principle, every point on the wavefront is a source of

secondary wavelets.
 Let t be the time for the secondary wavelet from B to reach the interface
at B’. BB’ = 𝐯𝟏 t.
 During this time, the secondary wavelet from A spreads into a sphere of
radius 𝐯𝟐 t in the denser medium
 With A as centre draw an arc of radius 𝐯𝟐 t.
 The surface tangent from B’ to the arc gives the refracted wavefront A’B’.
 The angle the incident wavefront AB makes with the interface is angle of
incidence and the angle the refracted wavefront makes with the interface is
angle of refraction.
 Rays are perpendicular to the wavefront and hence ∠𝐀𝐁𝐁′=∠𝐀𝐀′ 𝐁′ = 900.
𝐁𝐁 ′ 𝐯𝟏 𝐭
In ∆ABB′ sini = = ---- (1)
𝐀𝐁 ′ 𝐀𝐁 ′
𝐀𝐀′ 𝐯𝟐 𝐭
In ∆AA′B′ sinr = = ---- (2)
𝐀𝐁 ′ 𝐀𝐁 ′
Dividing (1) and (2) we get
𝐬𝐢𝐧𝐢 𝐯𝟏 𝐭 𝐯𝟏 𝛍𝟐
= = =
𝐬𝐢𝐧𝐫 𝐯𝟐 𝐭 𝐯𝟐 𝛍𝟏
This is Snell’s law of refraction.
If λ1 and λ2 are the wavelength of light in the first and second medium then
𝐬𝐢𝐧𝐢 𝐯𝟏 𝛌𝟏 𝛎 𝛌𝟏
= = = (Frequency remains the same in both the media)
𝐬𝐢𝐧𝐫 𝐯𝟐 𝛌𝟐 𝛎 𝛌𝟐

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For refraction of a plane wave at a rarer medium, i.e., v2 > v1. Proceeding in an
exactly similar manner we can construct a refracted wavefront as shown in the
figure. The angle of refraction will now be greater than angle of incidence and
𝐬𝐢𝐧𝐢 𝐯𝟏 𝛍𝟐 𝛌𝟏 𝛎 𝛌𝟏
= = = =
𝐬𝐢𝐧𝐫 𝐯𝟐 𝛍𝟏 𝛌𝟐 𝛎 𝛌𝟐

PLANE WAVEFRONT INCIDENT ON

A) CONVEX LENS
AC is a plane wavefront incident on a convex lens. The secondary wavelet from the
central part B has to travel through the greatest thickness of the lens and is
therefore slowed down the most. The secondary wavelets from A and C have to
travel through the least thickness of the lens and hence they are slowed down the
least. Hence the emerging wavefront is a converging spherical wavefront and
converges to focus F.

B) CONCAVE MIRROR
AC is a plane wavefront incident on a concave mirror. The secondary wavelet from
the central part B has to travel through the greatest distance before getting
reflected. The secondary wavelets from A and C have to travel the least distance
to get reflected. Hence the emerging wavefront is a converging spherical
wavefront and converges to focus F.

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C) PRISM
Let a plane wavefront be incident on a prism. Since the speed of light in glass is
less than that in air the secondary wavelets from the lower part C of the
wavefront which travels the greatest thickness of the glass prism is slowed down
the most. The secondary wavelet from the upper part of the wavefront travels the
least thickness of the glass prism and is slowed down the least. Hence the
emerging wavefront is a plane wavefront tilted towards the base of the prism.

NOTE:
When a plane wavefront is incident on a convex mirror or a concave lens, the
emerging wavefront is a diverging spherical wavefront.

INTEREFERENCE OF LIGHT
SUPERPOSITION PRINCIPLE: When two or more waves travelling through a
medium superpose, the resultant displacement at any point at a given instant is
equal to the vector sum of the displacements due to the individual waves at that
point.
⃗ = ⃗⃗⃗⃗
𝒚 𝒚𝟏 + ⃗⃗⃗⃗
𝒚𝟐 + ⃗⃗⃗⃗
𝒚𝟑 …….
INTERFERENCE OF LIGHT: When two light waves of the same frequency
travelling in the same direction and having same phase or a constant phase
difference superpose, the intensity of light in the region of superposition gets
redistributed. This phenomenon is called interference of light.
COHERENT SOURCES: Two sources of light which continuously emit light waves
of the same frequency (or wavelength) with zero or constant phase difference
between them are called coherent sources.

CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE

Let y1 and y2 be the displacement of light waves from two coherent sources S1 and
S2.
y1 = asinωt
y2 = bsin(ωt + φ)
Here a and b are the amplitudes of the two waves and φ is the constant phase
difference between them.
By superposition principle

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y = y1 + y2 = asinωt + bsin(ωt + φ)
y = asinωt + bsinωtcosφ + bcosωtsinφ
y = sinωt(a + bcosφ) + cosωt(bsinφ) ------ (1)
Let a + bcosφ = Acosθ ----- (2)
bsinφ = Asinθ ------ (3)
Substituting (2) and (3) in (1) we get
y = Asinωtcosθ + Acosωtsinθ
y = Asin(ωt + θ)
The resultant wave is also a harmonic wave of amplitude A and it leads the
first harmonic wave by phase angle θ.
Squaring and adding (2) and (3) we get
2 2 2 2
(a + bcosφ) + (bsinφ) = (Asinθ) + (Acosθ)
a2 + b2cos2φ + 2abcosφ + b2sin2φ = A2sin2θ + A2cos2θ
A2 = a2 + b2 + 2abcosφ
A = √a2 + b2 + 2abcosφ
Intensity α (amplitude)2
I = kA2 = A2 = a2 + b2 + 2abcosφ
Intensity of the first wave I1 = a2
Intensity of the second wave I2 = b2
Thus I = I1 + I2 + 2√𝐈𝟏 𝐈𝟐 cosφ

CONDITION FOR CONSTRUCTIVE INTERFERENCE:

The resultant intensity at a point will be maximum when cosφ = +1
Or φ = 0, 2π, 4π, 6π ..... = 2nπ
Maximum Amplitude Amax = √a2 + b2 + 2ab = (a + b)
Maximum Intensity Imax = (a + b)2 = I1 + I2 + 2√𝐈𝟏 𝐈𝟐
For a phase difference of 2π, path difference is λ
Thus for path difference 0, λ, 2λ, 3λ ….. = nλ, constructive interference
takes place.
If a = b, Imax = 4a2

CONDITION FOR DESTRUCTIVE INTERFERENCE:

The resultant intensity at a point will be minimum when cosφ = -1
Or φ = π, 3π, 5π..... = (2n-1)π
Minimum Amplitude Amin = √a2 + b2 - 2ab = (a - b)
Minimum Intensity Imin = (a - b)2 = I1 + I2 - 2√𝐈𝟏 𝐈𝟐
For a phase difference of 2π, path difference is λ
Thus for path difference λ/2, 3λ/2, 5λ/2 ….. = (2n-1)λ/2, destructive
interference takes place.
If a = b, Imin = zero

YOUNG’S DOUBLE SLIT EXPERIMENT

In Young’s double slit experiment, sunlight illuminates a slit S. Two slits S1 and S2
are equidistant from S.
Spherical wavefronts emerge out of S. S1 and S2 are two points of the wavefront
from S and hence act as sources of secondary wavelets.
Interference takes place between the waves from S1 and S2. Alternate bright and
dark regions are obtained on the screen and are called interference bands or
fringes.

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 • In the direction where the crest of one wave falls on the crest of the other
or trough of one wave falls on the trough of the other, amplitude of the two
waves get added and hence intensity is maximum. This is called constructive
interference.
• In the direction where the trough of one wave falls on the crest of the
other, amplitude of the two waves get subtracted and hence intensity is
minimum. This is called destructive interference.

THEORY OF INTERFERENCE FRINGES

S is a narrow slit illuminated by a monochromatic source of light of wavelength λ.
S1 and S2 are two narrow slits equidistant from S. S1 and S2 act as coherent
sources. Let d be the separation between S1 and S2 and D the distance between
the slits and the screen.

The centre O on the screen is equidistant from S1 and S2. Thus path difference is
zero, phase difference is zero, constructive interference takes place and hence O
is maxima or position of central maxima.
Let P be a point on the screen at a distance x from the central maxima. The nature
of interference at P depends on the path difference S2P – S1P.
S2P2 = D2 + (x + d/2)2 ----- (1)
S1P2 = D2 + (x - d/2)2 ----- (2)
Subtracting (1) and (2) we get
S2P2 – S1P2 = 2xd ------ (3)
S2P2 – S1P2 = (S2P – S1P) (S2P + S1P) ------ (4)
Since D is of the order of few metre and d in millimetre,
S2P ≈ S1P = D
Equation (4) is thus
S2P2 – S1P2 = (S2P – S1P) (2D)
Substituting in (3) we get

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 𝟐𝐱𝐝 𝐱𝐝
(S2P – S1P) (2D) = 2xd S2P – S1P = =
𝟐𝐃 𝐃
𝐱𝐝
Path difference S2P – S1P =
𝐃

Position of bright fringes (Constructive interference)

𝐱𝐝
= nλ
𝐃
𝐧𝛌𝐃
x =
𝐝
• For n = 0, x0 = 0 ( Central maxima)
𝛌𝐃
• For n = 1, x1 = (first bright fringe)
𝐝
𝟐𝛌𝐃
• For n = 2, x2 = (second bright fringe)
𝐝
…………………………
𝐧𝛌𝐃
Thus nth bright fringe is at xn =
𝐝

Position of dark fringes (Destructive interference)

𝐱𝐝 𝛌
= (2n-1)𝟐
𝐃
𝛌𝐃
x =(2n-1)𝟐𝐝
𝛌𝐃
• For n = 1, x1 = ( first dark fringe)
𝟐𝐝
𝟑𝛌𝐃
• For n = 2, x2 = (second dark fringe)
𝟐𝐝
𝟓𝛌𝐃
• For n = 3, x3 = (third dark fringe)
𝟐𝐝
…………………………
𝛌𝐃
Thus nth dark fringe is at xn = (2n-1)
𝟐𝐝

FRINGE WIDTH
It is the separation between two successive bright or dark fringes.
 Width of a dark fringe is the separation between two successive bright
fringe.
𝐧𝛌𝐃 (𝐧−𝟏)𝛌𝐃 𝛌𝐃
βdark = xn – xn-1 = - =
𝐝 𝐝 𝐝

 Width of bright fringe is the separation between two successive dark

fringes.
(𝟐𝐧−𝟏)𝛌𝐃 [𝟐(𝐧−𝟏)−𝟏]𝛌𝐃 𝛌𝐃
βbright = xn – xn-1 = - =
𝟐𝐝 𝟐𝐝 𝐝
Both the bright and dark fringes have the same width in the interference
𝛌𝐃
pattern and hence fringe width β =
𝐝

INTENSITY PATTERN FOR INTERFERENCE

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YOUNG’S DOUBLE SLIT EXPERIMENT (YDSE)
In YDSE, since a single monochromatic light illuminates two slits S1 and S2
y1 = acosωt
y2 = acos(ωt + φ)
y = y1 + y2 = acosωt + acos(ωt + φ)
(C+D) (C−D)
Using cosC + cosD = 2cos cos we get
2 2
𝛗 𝛗
y = 2acos 𝟐 cos(ωt + 𝟐 )
The amplitude of the resultant wave is
𝛗
A = 2acos 𝟐
𝛗
Intensity I = A2 = 4a2cos2( 𝟐 )
If φ = 0, ± 2π, ± 4π…… we will have constructive interference leading to
maximum intensity. On the other hand, if φ = ± π, ± 3π, ± 5π … we will have
destructive interference

Conditions for sustained interference pattern

The interference pattern in which the positions of maxima and minima on the
screen do not change with time is called a sustained interference pattern.
For obtaining a sustained pattern
 The two sources must be coherent
 The two sources must continuously emit light waves of the same frequency
or wavelength i.e. the sources must be monochromatic.
 For a better contrast between maxima and minima, the amplitude of the
interfering waves must be equal.
 The distance between the two coherent sources must be small and the
distance between the sources and screen must be large.

Conservation of energy in interference

In interference pattern
Imax = (a + b)2 and Imin = (a – b)2
(𝐚 + 𝐛 )𝟐 + (𝐚 − 𝐛)𝟐
Iaverage = = a2 + b2
𝟐
If there was no interference,
I1 = a2
I2 = b2
Intensity at every point is I = I1 + I2 = a2 + b2 = Iaverage
Thus interference of light is redistribution of energy due to superposition of
waves from two coherent sources.

Why cannot two different sources of light be coherent?

Each source consists of millions of atoms which when excited comes to the ground
state randomly emitting light waves. Since these waves are emitted randomly the
phase difference between the waves emitted by the two sources changes
rapidly and randomly. Hence two different sources are incoherent.
When two sources are incoherent, they do not maintain a constant phase
difference. The interference pattern will also change with time and, if the phase
difference changes very rapidly with time, the positions of maxima and minima will
also vary rapidly with time and we will see a “time-averaged” intensity distribution.
When this happens, we will observe an average intensity that will be given by
φ 0+1 1
= )
𝛗
Iaverage = <4a2cos2( 𝟐 )> = 2a2 (As average of cos2( ) =
2 2 2
2
Our eye sees a uniform illumination of 2a .

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What happens if monochromatic light is replaced by white light?
The interference pattern will have a central white fringe followed by overlapping
coloured fringes.

The first bright fringe closest to the central white fringe is blue.

What is diffraction?
The phenomenon of bending of light around the corners of an obstacle or an
aperture into the region of geometrical shadow is called diffraction of light.

The diffraction of light is more pronounced when the dimensions of the slit or
obstacle is comparable to the wavelength of light.

TYPES OF DIFFRACTION
The two types of diffraction are
a) Fresnel Diffraction: In this type of diffraction, the source and screen are
both at finite distance from the obstacle or the aperture causing
diffraction.
b) Fraunhofer Diffraction: In Fraunhofer diffraction, the source and screen
are at infinite distance from the obstacle or aperture causing diffraction.

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DIFFRACTION OF LIGHT DUE TO SINGLE SLIT
S is a monochromatic source placed at the focus of a convex lens L1. The emerging
plane wavefront is incident on a rectangular slit AB of width ‘a’. The resulting
diffraction pattern is obtained on a screen placed at the focus of lens L2.

 For point O on the screen, the secondary wavelets travelling in the direction
of incident light have path difference zero and hence constructive
interference takes place.
 For a point P on the screen it will be maxima or minima depending on the path
difference.
 Path difference BN = asinθ

CENTRAL MAXIMA:
 For point O, θ = 0, hence path difference and phase difference is zero and
constructive interference takes place. O is the position of central maxima

First minima:
For asinθ = λ, we can imagine the slit AB to be divided into two equal parts, AC
and CB. For every point on the first part of the wavefront AC, there is a point on
CB that give out secondary wavelets with a path difference of λ/2. Thus
destructive interference takes place and P will be a point of minima.

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 𝝀
Hence for first minima sinθ =
𝒂

Second minima:
For asinθ = 2λ, we can imagine the slit AB to be divided into four equal parts. For
every point on the first part of the wavefront, there is a point on the second part
that give out secondary wavelets with a path difference of λ/2. Similarly the
secondary wavelets from the third and fourth parts interfere destructively. Thus
destructive interference takes place and P will be a point of minima.
𝟐𝛌
Hence for first minima sinθ =
𝐚

Thus the condition for minima is asinθ = nλ.

Secondary maxima
𝟑𝛌
For asinθ = , we can imagine the slit AB to be divided into three equal parts.
𝟐
For every point on the first part of the wavefront, there is a point on the second
part that give out secondary wavelets with a path difference of λ/2 and hence
destructive interference takes place. The wavelets from the third part of the
wavefront contributes to intensity on the screen and hence P will be the position of
maxima.

𝟓𝛌
Similarly for asinθ = , we can imagine the slit to be divided into five equal parts.
𝟐
Wavelets from four parts interfere destructively and the resultant intensity on
the screen will be due to the fifth part of the wavefront.

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 (𝟐𝐧+𝟏)𝛌
Hence the condition for secondary maxima is asinθ =
𝟐

Intensity of secondary maxima decreases with order of maxima:

Intensity of the central maxima is due to secondary wavelets from all parts of the
wavefront. But the first secondary maxima is due to wavelets from one-third part
of the slit as the first two parts of the slits give out secondary wavelets which
interfere destructively.
The second secondary maxima is due to secondary wavelets from one-fifth part of
the wavefront as the secondary wavelets from the first four parts interfere
destructively.
Hence intensity of secondary maxima decreases with order of maxima.

DIFFRACTION PATTERN DUE TO A SINGLE SLIT

LINEAR AND ANGULAR WIDTH OF MAXIMA AND MINIMA

Angular width of central maxima is defined as the angle between the directions
of the first minima on either side of the central maxima.
Thus angular width of central maxima is 2θ.
The direction of the first minima on either side of the central maxima is at
𝛌
θ = ---- (1)
𝐚
𝟐𝛌
Therefore angular width of central maxima is 2θ =
𝐚
If D is the distance between the slit and the screen, the linear distance of the
first minima from the central maxima is x.
𝐱 𝛌
tanθ ≈ θ = = [From (1)]
𝐃 𝐚
𝛌𝐃
Thus x = θD =
𝐚
𝟐𝛌𝐃
Thus linear width of central maxima is 2x =
𝐚
Linear width of central maxima is the linear distance between the first minima on
either side of the central maxima.

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Angular width of nth secondary maxima is the angular separation between the nth
and (n+1)th minima.
𝐧𝛌
From asinθ = nλ ; sinθ ≈ θn =
𝐚
(𝐧+𝟏)𝛌
For (n+ 1)th minima, asinθ = (n+1)λ ; sinθ ≈ θn+1 =
𝐚
(𝐧+𝟏)𝛌 𝐧𝛌 𝛌
Angular width of nth maxima is θn+1 – θn = - =
𝐚 𝐚 𝐚

Angular width of nth secondary minima is the angular separation between the nth
and (n+1)th maxima.
(𝟐𝐧+𝟏) (𝟐𝐧+𝟏)
From asinθ = λ ; sinθ ≈ θn = λ
𝟐 𝟐𝐚
(𝟐(𝐧+𝟏)+𝟏)
For (n+ 1)th maxima, asinθ = ( )λ
𝟐
(𝟐(𝐧+𝟏)+𝟏)
sinθ ≈ θn+1 = ( )λ
𝟐𝐚
(𝟐(𝐧+𝟏)+𝟏) (𝟐𝐧+𝟏) 𝛌
Angular width of nth minima is θn+1 – θn = ( )λ - λ =
𝟐𝐚 𝟐𝐚 𝐚
𝛌𝐃
Linear width of secondary maxima and minima is
𝐚
Linear and angular width of central maxima is twice that of the secondary
maxima and minima.

COMPARISION BETWEEN INTERFERENCE AND DIFFRACTION

When the width of each slit is comparable to the wavelength of the wave in
YDSE, the resultant interference pattern is modulated by the diffraction
pattern.

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IMPORTANT NOTE:
In order to determine the shape of the interference pattern on the screen we
note that a particular fringe would correspond to the locus of points with a
constant value of S2P – S1P. Whenever this constant is an integral multiple of λ ,
the fringe will be bright and whenever it is an odd integral multiple of λ/2 it will be
a dark fringe. Now, the locus of the point P lying in the x-y plane such that S2P –
S1P is a constant, is a hyperbola. Thus the fringe pattern will strictly be a
hyperbola; however, if the distance D is very large compared to the fringe width,
the fringes will be very nearly straight lines as shown.

***************************************************

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