Page 1 of 10

Chapter Ten

WAVE OPTICS

Introduction

Wave optics deals with the wave

behavior of light. Plane wavefront : - wavefront at
Christian Huygens proposed wave large distances from a point source.
theory of light.
According to wave theory, a luminous
body is a source of disturbance and the
disturbance propagated in the form of
waves and energy is distributed equally
in all directions. Huygen’s Principle
Interference, diffraction, polarization,
According to Huygens principle, each
etc can be explained by wave optics.
point of the wavefront is the source of a
Thomas Young confirmed the wave nature
secondary disturbance and the wavelets
of light through double-slit experiment.
(secondary wavelets) emanating from
Maxwell proposed electromagnetic
these points spread out in all directions
theory of light.
with the speed of the wave.
Transverse nature of light is
By drawing a common tangent to all
established by polarization.
these spheres, we obtain the new
Wavefront position of the wavefront at a later time.

A wavefront is locus of all points in a

medium which are at the same phase
of vibration.
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 • Huygens argued that the amplitude of
direction perpendicular to the wavefront. the secondary wavelets is maximum in
the forward direction and zero in the
Types of wavefront backward direction
Spherical wavefront – wavefront from Refraction of a plane wave
a point source

Cylindrical wavefront- wavefront from

a linear source.
 Page 2 of 10

• Let τ be the time taken by the • The angle of refraction will be

wavefront to travel the distance BC. greater than angle of incidence.
• Thus • Thus, if i = ic then sin r = 1 and r = 90°.
• From the triangle ABC we get

• Therefore
• Also from triangle AEC
• The angle ic is known as the critical
angle and for all angles of incidence
greater than the critical angle the wave
• Thus
will undergo total internal reflection.

Reflection of a plane wave by a plane surface

• If c represents the speed of light

in vacuum, then,

• Therefore

• If v represents the speed of the wave in

• This is the Snell’s law of refraction. the medium and if τ represents the time
• If λ1 and λ 2 denote the wavelengths taken by the wavefront to advance from
of light in medium 1 and medium 2, the point B to C then
respectively, then
• Also
• The triangles EAC and BAC are congruent
• Therefore the angles i and r would be
• That is equal. This is the law of reflection.

A plane wave passing through a thin prism.

• This implies that when a wave gets

refracted into a denser medium, the
wavelength and the speed of
propagation decrease but the frequency
ν (=v/λ) remains the same.

Refraction at a rarer medium

A plane wave incident on a thin convex lens
 Page 3 of 10

A plane wave is incident on a concave mirror the vector sum of displacements due
to individual waves.
Intensity of a wave is proportional to
the square of its amplitude.

Coherent Sources of light

The coherent sources which emits light waves of :
Same wavelength or frequency
Nearly equal amplitude
The Doppler effect
Are in phase or having a constant
• The apparent change in frequency of light
phase difference
seen by an observer ,whenever there is a
Eg: light from a double slit
relative motion between source and
Interference
observer is called Doppler Effect.
The modification in the distribution of
• When the source is moving towards
light energy when waves from more
the observer with a velocity v, then
than one coherent sources superpose
the apparent frequency of light
v each other.
' (1 ) Constructive interference
c When crests of two waves or two
Where ν- actual frequency, v – velocity troughs meet together the amplitude of
Therefore the fractional change in the resultant wave becomes maximum.
frequency This is called constructive interference.
v The intensity of the wave is given by
c c
• If the source is moving away from the
observer, the apparent frequency Condition for constructive interference
v For constructive interference, the path
' (1 ) difference of the waves reaching at point
c
• Hence the fractional change in is the integral multiple of the wavelength.
frequency is
v
The resultant intensity of constructive
c c interference is 4I0., where I0 is the
Red shift intensity of the wave before interference.
When the source moves away from the
observer, there is an apparent decrease
in the frequency of light. This is called
red shift.
Blue shift
When the source moves towards the
observer, there is an apparent increase in
the frequency of observed light. This is Destructive interference
called blue shift. When crest of one wave meet with
INTERFERENCE OF LIGHT trough of the other the amplitude of the
Superposition Principle resultant wave becomes minimum. This
When more than one wave is passed is called destructive interference.
through the same medium at the same The intensity of the wave is given by
instant, then the resultant displacement is
 Page 4 of 10

• The dark and bright bands appear on

Condition for Destructive Interference the screen are called fringes.
The path difference is given by • The distance between two consecutive
bright fringes or two consecutive dark
fringes is called the fringe width.
The resultant intensity of destructive • From the triangle S1AP
interference will be zero.
• That is

• From triangle S2BP

Relation between Path Difference and • That is

Phase Difference
Path difference of λ corresponds to
a phase difference of 2π. • Subtracting equation 1 from 2, we get
If Δx is the path difference, then the
phase difference

YOUNG’S DOUBLE- SLIT EXPERIMENT

• Thomas Young designed an double slit • Thus

arrangement to study interference.
• Young derived two coherent sources
of light using a double slit. • Or

• If the point P is near to O, then

• Therefore
• When light from two coherent sources S1
and S2 superimpose, alternate dark and • Or
bright bands are formed on the screen.

Expression for band width ( Fringe width)

• Thus
xnd
Path Difference =
D
• For the point P to be bright, the
path difference =nλ, thus
x d
n n
D
th
• Therefore the distance to n band is,
 Page 5 of 10

Some observations
• If one of the slits is covered with black
•
th
Thus distance to (n+1) band is paper – no interference pattern.
• If the source is moved towards the
slits, the fringe width do not change
but intensity increases.
• The band width is given by • If white light is used, then a white band
at the centre and colored bands on
either side are formed.
• If the system is immersed in a medium
• Thus of refractive index n, then the new
fringe width β’, is given by
'
n
• This is the combined width of a dark • The color of thin films of soap solution, or
band and a bright band. oil or petrol spread over water is due to
• The dark and bright bands are interference.
equally spaced. DIFFRACTION
• If P is dark, then • It is the bending of the light at the
sharp corners of obstacles.
• Diffraction of light occurs when the size
of obstacle is comparable to the
wavelength of light.
Fringe width can be increased by • All types of waves namely , light
waves, sound waves, matter waves,
• Increasing the wavelength of light (λ) waves on water etc shows diffraction.
• Increasing the distance between • The finite resolution of our eye or of
the sources and screen (D) optical instruments such as telescopes
• By decreasing the distance between or microscopes is limited due to the
the two coherent sources (d). phenomenon of diffraction.
Conditions for getting sustainable interference The single slit Diffraction
pattern • When the double slit in Young’s
experiment is replaced by a single
• The two sources must be coherent narrow slit (illuminated by a
• The coherent sources must be narrow monochromatic source), a broad pattern
and very close to each other. with a central bright region is seen.
• The screen must be at large distance • On both sides, there are alternate dark
from the sources. and bright regions, the intensity
The intensity distribution of light on the becoming weaker away from the centre.
screen in Young’s Double Slit Experiment
 Page 6 of 10

• The path difference NP – LP between

the two edges of the slit is

Some observations
• To find the intensity at any point P on the
• Diffraction is more when the slit width
screen we divide the slit into much
is decreased.
smaller parts, and add their contributions
• When the wavelength of the source
at P with the proper phase differences.
is increased the angular deviation
Central maximum also increases.
• At ‘C’, the path difference between • Only few band are observable
the rays coming from LM and MN is in diffraction.
zero. Hence constructive interference
Comparison between interference
takes place. This point is called central
and diffraction
maximum or principal maximum.
• Since the light rays from different points
Interference Diffraction
of the slit interfere constructively, the
It is the superposition It is the superposition
point C is maximum bright.
of secondary waves of secondary waves
Positions of secondary minima from two different from different parts of
• The secondary minima occurs at wave fronts. the same wave front.
Fringes may or may not Fringes are never of
be of equal width. equal width.
All bright fringes have Intensity of bright
• Where n= 1, 2, 3,……
same intensity. fringes decreases as we
Positions of secondary maxima move from the central
• The secondary minima occur at bright fringe.
The regions of The regions of
minimum intensity are minimum intensity are
• Where n= 1, 2, 3,…… perfectly dark. not perfectly dark.
The conditions for minima and maxima of
diffraction at a single slit experiment Energy conservation in interferncce
• For minima and diffraction

• Energy is conserved in both

interference and diffraction
• For maxima • The total energy is redistributed
in interference and diffraction
Resolving power of optical instruments

• The ability to resolve two neighboring

Intensity distribution of diffraction pattern objects which are very close to each
other is the resolving power.
• Human eye can resolve two objects if they
subtend an angle of one minute at the eye
• The resolving power is measured as the
reciprocal of the angle subtended by
the object.
 Page 7 of 10

• The resolving power of optical POLARISATION

instruments is limited by diffraction.
• The phenomenon by is called polarization.
Resolving power of a Telescope • When ordinary light passes through
certain crystals like tourmaline crystal,
the vibrations of electric field vector are
restricted. This phenomenon is called
polarization.
• Polarization shows that light is
a transverse wave.
• Sound waves cannot
polarize. Unpolarised light

•The ordinary light which contains the

vibrations of electric field vector in every
plane perpendicular to the direction of
propagation is called unpolarised light.
Representation of unpolarised light

Plane polarized light

• The polarized light in which the electric
field vibrations of light are confined to a
single plane are called plane
• This implies that the telescope will have polarised light.
better resolving power if a is large. Representation of plane polarised light
• It is for this reason that for better
resolution, a telescope must have a
large diameter objective.

Resolving power of a microscope

Plane of vibration
• It is the plane in which the vibrations
of the polarized light take place.
Plane of polarization
• It is the plane perpendicular to the plane
of vibration of the plane polarized light.
Polarizer
• The crystal which produces polarized
light is called a polarizer.
• The product n sinβ is called the Analyzer
numerical aperture • The crystal which is used to check
• The resolving power can be increased whether the light is polarized or not
is called analyzer or detector.
by choosing a medium of higher
An experiment to study polarization of light
refractive index. Such an arrangement • When unpolarized light passes through
is called an ‘oil immersion objective’. polarizer the light coming out of it is plane
 Page 8 of 10

Methods of polarization

• Polarization by scattering
• Polarisation by reflection
Polarization by scattering

• When sunlight is incident on the

• If the polarizer and analyser are parallel gas molecules in the atmosphere, it
the intensity of light coming through gets scattered.
the analyser will be maximum. • The scattered light seen in a direction
0
• If the analyser is rotated through 90 the perpendicular to the direction of
intensity of light coming out of it incidence is found to be plane polarised.
becomes zero. This phenomenon is called polarisation
Polaroids by scattering.
• Polaroid is an artificially made polarising • When this polarised light is viewed
through a polaroid which is rotated, then
material that produce intense beam of
the intensity changes with rotation.
polarised light by selective absorption.
• Polaroids are in sunglasses, • The scattering of light by molecules was
windowpanes, photographic cameras, intensively investigated by C.V. Raman
3D movie cameras etc. and his collaborators .Raman was
awarded the Nobel Prize for Physics in
Malus’ law for this work.

• Malus’s law states that when a beam of Polarisation by reflection

plane polarised light is incident on the
analyser, then the intensity of the
emergent light is directly proportional to
square of the cosine of the angle
between the polariser and analyser.

• Where θ is the angle between the axes

of polarizer and analyzer.
• When ordinary light falls on a surface
separating two transparent media, a
part of the light is reflected and the
other part is transmitted (refracted).
• When reflected wave is perpendicular
to the refracted wave, the reflected
wave is a totally polarised wave.
 Page 9 of 10

Brewster’s angle (polarizing angle)

• The angle of incidence at which the

reflected ray is totally polarized is called
Brewster’s angle and is denoted by iB.

Brewster’s law

• Brewster’s law states that the tangent

of the Brewster’s angle is equal to the
refractive index of the medium.
tan iB
Proof

• From Snell’s law

Distinguishing a polarized light and

unpolarized light

• When we observe unpolarised light

(ordinary light) through a Nicol prism
(tourmaline crystal), the intensity of
the light coming out of the prism does
not change if the crystal is rotated.
• But when we observe polarized light
through a Nicol prism, the intensity of
the light coming out of the prism
changes if the crystal is rotated.
PROBLEMS
Page 10 of 10

********