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Nature of Light and Huygens’ Wave Theory
Paper: Optics
Lesson: Nature of Light and Huygens’ Wave Theory
Author: Dr. D. V. Chopra
College/Department: Associate Professor (Retired),
Department of Physics and Electronics, Rajdhani College,
University of Delhi
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Nature of Light and Huygens’ Wave Theory
Objective: After studying this chapter you should:
1. Be able to know about light and its nature
2. Be able to study various theories regarding nature of light
3. Be able to understand Huygens ‘principle of wave propagation of light and its
application to derive laws of reflection and refraction of light
4. Be able to derive thin lens formula on the basis of Huygens’ wave theory
5. Be able to relate Huygens’ wave theory to Fermat’s principle
6. Be able to solve problems involving nature of light and Huygens’ wave theory
1. What is light and its nature ?
So far, we have considered the theorems which may be deduced from the axioms
formulated for rays of light and the optical instruments based on them. Such a study is
made in Geometrical Optics which is developed without going into detail with regard to the
nature of light. We shall see that the knowledge of the nature of light has an important
bearing on the design of optical instruments, e.g., resolving power of a telescope using the
phenomenon of diffraction which is based on the wave nature of light. In geometrical
optics, it is assumed that light travels in straight lines and we assume particle nature of
light. We are going to discuss theories of light; that is, answers to the question: What is
light; What is nature of light ? The answers to the question and the experimental evidence
bearing on it are called Physical Optics to distinguish it from the preceding geometrical
optics. In physical optics, we deal with wave nature of light.
There are only two possible types of theory, for there are only two possible ways of
transmitting energy through empty space. One is by a stream of moving material particles,
when the energy is transmitted as the kinetic energy of the particles, and other is by means
of waves, when the energy is transmitted as the energy of the waves without any motion of
medium transmitting them. And so the following theories of light arise:
(i)
(ii)
(iii)
(iv)
Newton’s corpuscular theory of light (1665)
Huygens’ wave theory of light (1678)
Maxwell’s electromagnetic theory of light ( or radiation) (1873)
Planck’s quantum theory of radiation (1900)
2. Theories regarding Nature of Light
(i)
Newton’s Corpuscular theory : In 1665, Newton gave the corpuscular theory of
light. It states that light consists of a large number of minute material corpuscles (i.e.
particles) emitted by a luminous body which produces the sensation of sight when these
corpuscles strike the eye. According to this theory, light travels with a tremendous speed
(3 x 108 ms1) in straight lines is a homogeneous medium. Different colours of light were
ascribed to different – sized corpuscles. The corpuscles travel with tremendous speed in
straight lines, because of their insignificant masses pull of gravity on them is negligible.
This theory was able to explain the principle of rectilinear propagation of light, laws
of reflection as well as the law of refraction.
Failures of this theory: Explanation of laws of refraction leads to the result that light
travels with a higher speed in denser medium. Again, the experiment of Faucault on the
direct determinal of the velocity of light in different media gives an opposite result. The
phenomenon of diffraction, viz, a small bending of the rays round a small object shows that
the propagation of light is not strictly rectilinear but approximately so. Thus a correct
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Nature of Light and Huygens’ Wave Theory
theoty should explain the approximate rectilinear propagation of light but Newton’s
corpuscular theory failed to explain this.
(ii)
Huygen’s wave theory:
This wave theory was given by Huygens in 1678.
According to this theory, a source of light propagates spherical waves in hypothetical
medium called the ether which pervades the whole of universe, in much the same way that
a stone sets up circular ripples when it falls into a pond. But as light can travel even in
vacuum, Huygens had to assume the existence of this medium ‘ether’ in much the similar
way as sound water waves require medium for its propagation. On the basis of this theory,
these light waves produce the sensation of sight when they enter the eyes. It explains that
light travels more slowly in water than air, the precise opposite of the corpuscular theory.
Failures:
Huygens in 1678 considered these light waves as longitudinal and the
phenomenon of reflection, refraction, interference and diffraction were successfully
explained. Difficulty aread to explain the phenomenon of polarization of light, but that
difficulty was surmounted by Fresnel, who assumed the light waves to be transverse.
Fresnel explained the approximate rectilinear propagation of light on the basis of wave
theory and Fresnel half-period zones vis-à-vis diffraction. Though the Huygens’ wave
theory, as modified by Fresnel, successfully supplented Newton’s corpuscular theory, yet it
had many drawbacks. It necessitated the adoption of a hypothetical medium called ether
possessing an extraordinary property of elastic solid. The velocity of transverse wave in a
solid medium is given by
is the modulus rigidity and , the density of the medium. Hence, to account high
velocity of light, ether must possess high rigidity and low density – the elasticity of ether
must be many times, greater than that of steel and its density many times less than that of
the best vacuum we can produce ! card of etis asumeher medium.
Discard of ‘either’ medium:
On the basis of Huygens’ wave theory, light waves are transverse in nature and a
longitudinal light wave has not been detected so far. Since the ether is assumed to have
property of an elastic solid and elastic solid is capable of transmitting both transverse and
longitudinal waves, this difficulty is removed by supposing that ether is incompressible. For
incompressible fluid, bulk modulus (k) is infinite. The velocity of longitudinal wave
This makes the velocity of longitudinal wave infinitely small, but can a medium be
stable under such conditions ! Michlson – Morely experiment, conducted in 1887, attempted
to measure the velocity of the earth through the ether. This experiment proved that there
is no relative velocity between the earth and the ether. Nor only this, all experiments
designed to detect the presence of ether have, so far failed and hence its contradictory
properties cannot be tested. Hence, ether concept was abandoned.
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Nature of Light and Huygens’ Wave Theory
It must be emphasized that Fresnel (1788-1827) gave a satisfactory interpretation of
many of the diffraction phenomena, assuming Huygens’ wave theory to which he added one
or two assumptions.
(iii)
Maxwell’s electromagnetic theory of light (1873):
Before the Michelron – Morley experiment was performed in 1887, it was assumed
by Faraday that a medium like ether was an essential feature of the electrical theory. It
was however not a mechanical medium in which material displacements took place but
rather an electromagnetic phenomenon in which displacement currents and magnetic fields
occur. The periodic disturbances which are supposed to constitute these waves, were
called displacement currents by Maxwell.
In 1873, Clark Maxwell working on the hypothesis of displacement current in
dielectrics arrived at the conclusion that an electromagnetic wave travelling in space has a
velocity given by
where
= permeability of the medium,
= permittivity of the medium.
In vaccum,
equals velocity of light c in free space
This led Maxwell to put forward his famous Electro-magnetic Theory of light.
Maxwell therefore concluded theoretically that light is an electromagnetic wave.
In 1873
In 1888, fifteen years later, Hertz experimentally proved the existence of
electromagnetic
waves and thus verified Maxwell’s electromagnetic theory.
These
electromagnetic waves are transverse in nature and differ from light waves in wavelength.
In Maxwel’s electromagnetic theory, the vibrations in wave theory is replaced by the
oscillations of electric and magnetic fields which are mutually perpendicular to each other
and also perpendicular to direction of waves propagation. In fact, it is the electric wave that
really constitutes light, while the magnetic wave, though no less real, is less important.
Further confirmation of the electromagnetic theory of light is obtained from (a)
Faraday effect, (b) Kerr effect, (c) Zeeman effect, (d) Electrostatic double refraction (Kerr),
and (e) Stark-effect. But it failed to explain Photoelectric effect, Compton effect.
(iv)
Plancks Quantum theory of light (1900):
In 1900, Planck gave quantum theory of radiation according to which exchanges of
energy between ether and matter, instead taking place continuously, can occur only in
discrete steps i.e., in multiples of some small unit, called the quantum. His assumption is
that an oscillator or a vibrating electron absorbs or emits radiations having integral
multiples of energy quanta and energy of one quantum is
where
is Planck’s
constant and is the frequency of radiation. This idea of Planck was verified by experiment.
In 1904, Einstein gave quantum theory of light; and thus, there was again the revival of
Newton’s corpuscular theory of light. These corpuscles corparticles are considered, quanta
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Nature of Light and Huygens’ Wave Theory
called photons. According to this Einstein’s theory of light, every radiation is to consist of
indivisible ‘radiation quanta’ of energy
. Einstein (1904) used the concept of black body
radiation put forth by Planck (1900). Quantum theory of light was able to explain the
photoelectric effect, the atomic structure, the Compton effect and Raman effect, but it failed
to explain the phenomena of interference, diffraction, etc. The latter phenomena were
explainable on the basis of wave theory of light. Thus for one phenomenon we regard light
as quanta (or particles) while for another phenomenon we regard light as waves. A
satisfactory theory must therefore combine these two conceptions into one general coordinating principle. This led to dual nature of light.
Co-relation between wave theory and quantum theory of light (or Dual nature of
light)
In 1926, de Broglie worked out mathematically that material particle may be
supposed to be consisting of certain wave frequencies. Soon after Schrödinger extended
Broglie’s theory and formulated a system of mechanics known as wave mechanics. This is
how the wave and particle aspect of matter are co-related. Mathematically it is given by
where m is mass of particle and
its velocity where as
is the wavelength of the wave
associated with the moving particles called de Broglie wave and h, the Planck’s constant.
Conclusion:
Various phenomena of light propagation are explainable on the basis of
electromagnetic wave theory of light, while the interaction of light with matter in the
processes of emission and absorption is explainable on the basis of quantum theory of light.
The former theory of light has wave concept while the latter theory has particle concept.
This is known as dual nature of light which is valid even to-day.
Similarly, electron appears to have a dual nature; it is a wave as well as a particle.
An electon in motion behaves as a wave, but when it lands anywhere it lands as a particle.
In 1926, Schrödinger described matter waves by formulating the equation called
Schrödinger equation. The de Broglie’s theoretical prediction of the existence of matter
waves was first verified experimentally in 1927 by C.J. Davisson and L.H. Germer and by
G.P. Thomson.
Value Addition:
Ether: As sound needs a medium to propagate, so light must also need a medium to
propagate. Ether was suggested a medium for light to propagate but ether is now a
discarded hypothetical medium once thought to fill all space and to be responsible for
carrying light waves and other electromagnetic waves. Such a medium of ether was
postulated with mechanical properties adjusted to provide a consistent theory. For
propagation of electromagnetic radiation, it was assumed to pervade all space and matter,
to be extremely elastic yet extremely light, to transmit transverse waves with the speed of
light to have a greater density in matter than in free space.
A search for “ether” medium was made by Michelson and Morley in their famous Michelson –
Morley experiment to detect the presence of ether. But, they failed to detect its existence.
On this result obtained by Michelson and Morley in 1887, Albert Einstein built up his
revolutionary Relativity and Quantum theories. Further, Einstein concluded that light rays do
not travel in straight lines but they curve under the influence of a gravitational field.
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Nature of Light and Huygens’ Wave Theory
3. Definition of Wave-Front and Ray
If a stone is dropped into a pond, circular ripples spread out from the point O where
the stone entered the water, as shown in Fig 1. They are produced by the vibration of the
water due to the entry of the stone and the ripples are circles with O as centre. Each one of
these circles is called a wave front and the direction OA, in which any very small portion of a
wave front travels, is normal to the wave front and is called a ray. If a graph of the
displacement (y) of the particles, which normally lie on the line OA, against their distance
(x) from O is drawn, it will resemble the wave form shown in Fig. 2 and the wave form
travels through the medium without any
bodily motion of the medium itself. But
we are dealing with waves travelling in two
A
dimensions, and hence the concept of
wave front comes in.
O
A wave front at any instant of time
is defined as the locus of all the
neighboring particles in the medium which
are being just disturbed at that instant of
time and are consequently in the same
Fig. 1 Wave-front
phase of vibration. The direction in which
the disturbance is propagated in a
Y
homogeneous medium is called a ray. It is
always normal to the wave front. A wave
x
O
front is the locus of a particular phase of
the vibration of the particles of the
B
medium. The phase most usually chosen
is the position of maximum positive
Fig. 2 Wave-form
displacement and so, the locus is the line
joining the crests.
The wave front is
clearly a circle in the case of the waves
produced by dropping a stone in a pond. If a plank BC
O
A
floating on the surface of a pond is moved up and down, it
will produce waves roughly of the form shown in Fig. 3, in
which the wave fronts are straight lines, a ray being straight
line, OA, normal to the wave fronts.
C
Fig. 3 Plane Wave Fronts
In the case of waves in three dimensional, a point source
will produce spherical waves, the rays being radii of the
spheres, while a plane source will produce plane waves in which the rays are lines normal to
the plane.
4.
Huygens’ principle of wave propagation of light :
We know from simple experience that a circular wave front grows into a circle of ever
increasing radius; a linear wave front moves as a line parallel to itself; a spherical wave
front spreads as a sphere of ever increasing radius; a plane wave front moves as a plane
parallel to itself. There is no propagation of waves (or energy) in the backward direction.
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Nature of Light and Huygens’ Wave Theory
The theory of predicting the future course of wave fronts was put forward by Huygens and is
known as Huygens’ principle.
Explanation of Huygens’ theory of wave propagation:
According to Huygens’s wave theory of light, each point on a source of light is a
centre of periodic disturbance (=wave) from which transverse waves spread out in all
direction, the velocity of propagation of the disturbance varying with the nature of the
medium. In a homogeneous medium the waves travel with the same velocity in all
directions. For a point source of light, in a homogeneous medium, the wave front is a
sphere. At a very large distance from the point source the radius of he sphere is very large
so that a small position of its is sensibly plane. The wave front in such a case is called a
plane wave front.
Hugens’ principle is a geometrical
method of ding the position of the new
wave front after a certain time.
It
states that the position of a wave front t
seconds after its present position is
found by considering each point of the
given wave front as the source of
secondary wavelets. With each point on
the given wave front as centre a sphere
of radius vt is drawn, where v is the
velocity of the waves in the medium.
The required position of the wave front is
the envelope of the spherical wave lets in
the forward direction. We shall illustrate
its principle by applying it to the
propagation of spherical and plane wave
fronts.
vt
Ray
O
B
Fig. 4 Construction of New Spherical wave front
A vt
In Fig 4 let O be a point source of light,
placed in a homogeneous isotropic
medium. Let v be the velocity of the light
in the medium.
A point source emits
spherical waves in all directions, with O as
centre
Ray
AB be the portion of the spherical wave
front at any instant in order to obtain the
wave front after a time t.
B
Consider a number of points on AB.
Fig. 5 Construction of New Plane Wave
Each point is a source of secondary
Front
wavelets. With each point on the given
wave front as centre a sphere of radius vt is drawn. The sections of these secondary waves
of a plane will be circles as shown in the figure. The required position of the new wave front
is the envelope or common tangent of the spherical wavelets in the forward direction. This
is shown by A/B/ which is the front envelope of these circles. By this type of construction
we can construct one wave-front from another.
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Nature of Light and Huygens’ Wave Theory
The rear envelope of these wavelets is shown by
which cannot
represent the next wave-front after the time t. This can be explained using Stokes law.
According to this law, the amplitude at each point of the secondary wave is proportional to
(1+cos) where  is the angle between the wave normal (OA or OB) and the direction of
propagation which is the line, joining the point of the secondary wave to its centre. For
example, for the first wavelet, A is its centre,
is the direction of the normal OA and for
point
the direction of propagation is along
, therefore angle  is . For the point of
the secondary which is directly behind the wave, the value of = and hence (1+cos) is
zero. Thus, the intensity of the secondary waves in the rear is zero and hence the rear
envelope of the circles cannot represent the section of the wave front. Hence, in Fig 4, the
rear envelope
cannot represent the section of the wavefront after time t.
Similarly, we can construct new plane wave front
from a plane wave
front AB at any time t, as shown in Fig 5, by drawing spherical wavelets of radius vt from a
number of points on AB. The rear envelope
cannot be new plane wave front after time
t.
The direction in which a small part of the wave front advances in a
medium is called a ray. In a homogeneous isotropic medium, the rays are normal to the
wave front, as shown by arrows in Fig. 4 and 5.
Applications of Huygens’ Principle of wave theory of light:
Huygens’ principle of wave propagation of light may be used to account
for the phenomena of reflection and refraction from the plane and curved surfaces. One can
prove
(i) Laws of reflection
(ii)
Laws of refraction (and hence Snell’s law)
(iii)
The mirror formula in terms of u, v and f
(iv)
The lens formula in terms of u, v and f.
In order to explain the above mentioned phenomena, we shall apply
Huygens’ postulates. These postulates are as follows:
1. Each point on a wave-front acts as the centre of a new disturbance
called secondary wavelet. These wavelets travel out with the same
speed as the orginal wave so long as the medium remains the same.
2. The envelope, namely, the tangitial plane to these secondary
wavelets constitute the wave-front.
3. Only forward wavefront exists and no backward wave-front exixts.
Explanation of Absence of a back wave:
It is explained on the basis of Stokes’ law. According to this law, the
amplitude propagated in any direction due to secondary wave is proportional to
where
is the angle between the wave normal and the diretrion of wave propagation.
For the backward wave-front,
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Nature of Light and Huygens’ Wave Theory
Hence no effect of secondary wavelets is observed in the backward direction because
amplitude is zero.
5. Laws of Reflection derived from the Huygens’ wave theory:
We shall consider refletion of a plane wave front from a plane reflecting
surface (or a plane mirror). Consider the incidence of a plane wave front AB upon a plane
reflecting surface XY. Assume rays PM, QN etc.(which are normal to AB) to be in the plane
of paper. Hence the sections AB and XY will be perpendicular to the plane of paper. Let v
be the velocity of light and the wave front travelling along AC, strikes the reflecting surface
at A. The angle of incidence, i, between the normal and the incident ray is equal to
which the wave front AB makes with the reflecting surface XY. Let t be the time taken by
the disturbance from B to . Then, during this time interval, the spherical wavelet from A
will have spread out in a sphere of radius vt which is equal to AC, and the plane wave front
AB will have occupied its parallel position
if the reflecting surface were not present.
According to the Huygens’ principle, each point on the wave front AB is a source of
disturbance, and gives rise to a secondary wavelet. Owing to the presence of the reflecting
surface, the disturbances from the various points on AB, after the striking surface XY spread
out in spheres and travel in the upper portion, the envelope of which gives rise to a new
wave front called the reflected
wave front.
To find the position of
the reflected wave front, with A
as centre and radius =
= AC
= vt, describe a sphere. From
draw a tangent plane to the
sphere touching the latter at .
Then
is the trace of the
reflected wave front . Let
make an angle r with the
reflecting surface XY , where r
is the angle of reflection.
B’
B
i
X
vt
vt
Q
S
Vt2
R
P vt
1
i
A
r
r
M
Y
N
E
D
To prove that
is the
C
common reflecltled wave-front,
we are required to show that
the disturbances from each
point on AB, such as P,Q, etc
Fig. 6 Laws of Reflection from the Huygens’
after being reflected from the
wave theory
points M, N, etc. respectively of
the surface XY, reach the wavefront
exactly at the same time. Let t 1 be the time taken by the disturbance from P to
reach reflecting surface XY at M and also let t2 be the time taken by the disturbance from M
to reach
.
Then PM = vt1, MS = vt2
With M as centre draw a sphere so that it touches the tangent plane
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at S. Join MS
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Nature of Light and Huygens’ Wave Theory
Let AM = x, and
In 
,
and PM is parallel to
. Hence s APM and
are
similar
Similary
are similar
Adding Eq.5-1 and 5-2, t1+ t2 = t
Hence, the wavelet from P, after reflection at M, reaches the point S on the wave
front
exactly at the same time as the disturbance from A reaches . Similarly it can be
proved that disturbances from other points, such as Q etc. reach
at the same time t, or
is the trace of the common reflected wave front.
Hence, a plane wave front is reflected back from a plane reflecting surface as a
plane wave front.
[The reader will recall that Huygens’ principle cannot predict the way the wave front will
travel]
In
. We have
is common
These triangles are congruent
Hence
This is the second law of reflection.(Proved)
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Nature of Light and Huygens’ Wave Theory
Also AB,
and
are all section of the plane perpendicular to the plane of paper; hence
normal to the planes are in the same plane. But normal to AB and
are incident and
reflected rays, hence the incident and reflected rays and normal to the surface at the point
of incidence lie in the same plane. This is the first law of reflection (Proved).
6. Laws of Refraction on the basis of the Huygens’ wave theory:
Fig 7 shows the trace XY of a plane surface perpendicular to the plane of paper,
separating the two media of refractive indices 1 and 2, such that 2>1; and v1 and v2
being the velocities of light in the upper and lower media respectively in such a way that v 2
< v1.
Let AB be the trace of a plane wave front from a monochromatic source of light,
perpendicular to the plane of paper, incident at an angle i with surface XY, and travelling in
the direction AE from less dense to
more dense medium (Fig. 7).
1, v1
According
to
Huygens’
principle, each point on the wave
front AB is a source of secondary
(Rarer)
B
wavelet. Assume the wave front AB
Q
at time t=0. Let t be the time taken
v1t i
P
i
by the disturbance from B to reach
v
1t1
A
i
X
Y
the surface XY at C. Then during
L
r
C
this time t, the spherical wavelet
v2t2
r
L
from A will have spread out in a
v2t
M
sphere of radius v1t and the plane
N
2 , v2
E
wave front will have occupied its
D
parallel position CE if the refracting
(Denser)
surface XY were not present. But
owing to the presence of the
Fig.7 Laws of Refraction from Huygens’ wave theory
refracting surface XY, during this
time t, the wavelet from A which
started earlier, instead of spreading into the lower medium 2 as a sphere of radius AE =
BC, moves into it with a velocity v2 and spreads out as a sphere of radius AD = v2t.
An envelope (or a tangent plane) perpendicular to the plane of paper drawn from C
to the sphere, i.e., the tangent plane CD would represent the refracted wave front.
To prove that CD is the common refracted wave front for all disturbances which
originate from points such as P, Q etc. on AB, one has to show that during the time the
wavelet from A travels to D or from B to C, the wavelet from P reaches the wave front CD.
Let t1 be the time taken by the wavelet from P to reach XY at L and also let t 2 be the time
taken by the disturbance from L to reach CD at M.
Then
Let
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Nature of Light and Huygens’ Wave Theory
have PL parallel to BC and
.
Since these triangles are
similar,
Similarly, since
are similar,
This gives ,
Adding Eqs. (6-1) and (6-2)
t1+ t2 = t
Hence, the wavelet from P after refraction at L, reaches the point M on the wave front CD
exactly at the same time as the wavelet from A reaches D. Similarly, it can be proved that
wavelets from other points, such as Q etc., reach the same wave front at the same time or
CD is the trace of common rerfracted wave front.
Let r be the angle if refraction corresponding to angle of incidence, i.
From
, in which
From
, in which
These equations give
We have, since lower medium is denser than upper medium i.e.
Therefore
.
is greater than one.  is refractive index of second medium with respect to
first medium.
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Nature of Light and Huygens’ Wave Theory
This proves second law of refraction. This is Snell’s law of refraction, i.e.
Since
This means that the velocity of light in a denser medium is less than that in a rarer medium.
This has been verified experimentally.
Also AB, CE and CD aer all traces of the planes perpendicular to the plane of paper;
hence normals to these planes are in the same plane. Since normals AB and CD are the
incident and refracted rays, hence the incident ray, refracted ray and normal to the
refracting surface at the point of incidence lie in the same plane. This proves the first law of
refraction.
Total internal reflection on the basis of Huygens’ wave theory:
From Eq,(6-4), we have
Putting the value of BC from Eq. (6-3), we get,
When the lower medium (see Fig.7) is denser, 2>1 and because maximum value of
sin i is one then AD is always less than AC. Here AD is the radius of refracted wavelet.
The tangent CD can always be drawn. Since CD is refracted wave front, there is always
refracted ray in the lower medium.
From Eq.(6-5), (or Snell’s law),
Since
Suppose the lower medium is rarer, then 2<1 From Eq.(7-1) it follows that the
following three cases arise, depending upon whether AD is less than, equal
to or greater than AC.
(i) Case I: AD<AC, (Fig. 8)
X
A
(Dens
(Denser)
er)
B
i
C
r
Y
D
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Fig.8. AD<AC
(Rarer)
Nature of Light and Huygens’ Wave Theory
It means that the radius of the refracted
spherical wavelet AD is less than AC. Therefore,
the point C lies outside the sphere. Hence a real
tangent CD which is the refracted wave front can
be obtained. That is to say, refraction is possible.
(ii). Case II AD = AC, (Fig.9)
Refracted wave front AD lies on the surface
XY which separates the two media and point D
coincides with C. The tangent CE drawn from C to
the wavelet sphere will be normal to the surface
XY. Since this tangent is also refracted wave front,
therefore the acted rays will be parallel to the
surface, along AC as shown in Fig 9. From Eq.7-1,
putting AD = AC, replacing i by ic, we have,
(Denser)
B
ic
X
A
C
r=900
=AD
where  is refractive index of denser medium with respect to
rarer medium. Here µ is always greater than 1. This angle i c
of incidence at which r is 900, is called critical angle.
Fig.9.AD=AC
Y
E
(Rarer)
(iii). Case III AD>AC
This is shown in Fig. 10. Point C will lie inside the spherical wavelet drawn from A in the
lower medium. In such a case, no real tangent from C to the wavelet is possible. Hence no
refracted wave front i.e. refracted ray is possible. Since reflection is possible, the reflected
wavelet from A propagates in the first medium and the radius of this spherical wavelet in
AD’, as shown in Fig.10.
The radius AD’ of this wavelet is less than AC;
therefore a real tangent CD’ may be drawn from C to
the reflected wavelet. In this case, no refracted
wave front is possible but only reflected wavefront
CD’ is possible when the incident wave front travels
from denser medium to rarer medium. From Eq. (71) and (7-2), it follows that, since AD>AC, we have i
> ic . This is the case of total internal reflection. The
conditions of total reflection are :
(Denser)
i
X
A
B
Y
D C
(i) The incident ray must proceed from denser
medium to rarer medium.
(ii) The angle of incidence (i) must exceed the
critical angle (ic) between the two media.
(Rarer)
Fig.10. AD>AC
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Nature of Light and Huygens’ Wave Theory
8. Reflection of a spherical wave front a spherical surface (concave mirror)
Consider reflection of a spherical wave from a concave mirror MPM’, as shown in Fig.
11. Let R be the radius of curvature of the mirror, C centre of curvature and P its pole. Let
A be the point source, emitting spherical wave fronts. Let XLY be a spherical wave front at
any instant. This wave front touches the concave mirror at point M and M’. By the time
wavelet from L touches P, wavelet from M and M’ reach E and D respectively.
LP=ME=M’D
During this time, the secondary wavelets from
all other points on MLM’ reach the surface EPD
to produce reflected wave front of incident
wave front MLM’. The reflected wave front
EPD converges towards point B. Thus B is the
image of A.
X
M
E
Let AP = u
PB = v
PC = R
Assume small aperture as compared to its
radius of curvature. Then we can write
A
C
B
G H
L
P
From the geometry of the figure for spherical
surface
, we have
D
Y
Similarly, for the surface
Fig.11 Reflection from a concave
Mirror MP
Similarly for the surface EPD,
To the first approximation,
Substitute Eq. 8-1, 8-2, 8-3 in Eq 8-4.
According to sign convention, v, u and R are all negative,
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Nature of Light and Huygens’ Wave Theory
This is the required expression.
N.B. Reflection of a spherical wave front from a convex surface can be studied in the same
way. In this case the incident spherical wave front XLY would meet pole P of convex mirror.
9.
Refraction of a spherical wave front at a spherical source (using Huygens’
wave theory)
Refraction at a convex surface.
K
air
We are to obtain the relation
D

We assume small aperture of the convex
surface.
Consider a convex surface KPL where pole
is P and is refractive index of the medium
having convex surface. Let A be a point
object at a distance u from P. Since A is a
point object, it emits spherical wave front.
GPH is a spherical wave front touching the
convex surface at P. Obviously the radius
of curvature of the spherical wave front will
be u. Each point on GPH is a source of
secondary wavelets, as follows from
Huygens’ wave theory. By the time wavelet
from point D in air reaches B on the convex
distance PQ in the medium where B Q S is the
J
G
I
A
B

y
E P L
Q
P
u
R

v
H

C
S
L
Fig.12. Refraction at a convex surface
(using Huygens’ wave theory)
surface, wavelet from P will have covered a
refracted wave front.
DB = .P Q
The refracted spherical wave front BQS appears to come from point I where I is the centre
of this spherical wave front. Thus I is point image of point object A, BJ is refracted ray and
the distance IQ = v. In other words, the radius of curvature of the refracted wave front is
v. Now, draw perpendicular DE and BL on the principal axis
Let DE=BL=y, say
DB=EL=.P Q……………………………(9-1)
EL=EP+PL ……………………………….(9-2)
PQ=PL+LQ ……………………………….(9-3)
EP+PL=(PL+LQ) ;
[Putting Eq.8-2 and 8-3 in Eq.8-1]
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EP+PL= .PL+.LQ
EP.LQ=.PL.PL…………………………………(9-4)
From Fig. 13, using the property of circle,
BL.LS=PL.LT
=PL(PTPL)
neglecting small quantity PL2.
Hence R is radius of curvature of convex surface.
Since BL=LS=y and
 y2 = PL 2R
B
Similarly, we can write
P
L
C
T
S
Putting above values in Eq. 9-4,
Fig. 13
According to sign convention, u and v are negative and R is positive
This is the required relation between u, v and R. If object is in medium of
refractive index
and the convex surface has medium of refractive index , then
the above relation becomes
where
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Nature of Light and Huygens’ Wave Theory
Refraction at a concave surface:
A is a point object: Spherical wave front from A is
G D H which reaches concave surface kPL of a
medium of refractive index . Then A is centre of
spherical wave front GDL whose radius is u
G
Air
K D

y
Each point on wave front GDH is a source of
secondary wavelets. By the time, wavelet from
point D reaches P, the wavelet from K and L
reaches D and E in the medium . Distance KD in
the medium is equivalent to a distance .KD in air
A
y
M D N P
I
v
u
Th refracted wave front is DPE which appears to
originate from point I (I is centre of spherical
(reracted) wave front DPE).
PI = v
L
E
H
Fig.14. Refraction at concave surface
where I is virtual image of point object A and DD’
is refracted ray Draw perpendiculars KM and DN from K and D on the principle
axis
Using property of circle,
K
C
M P
R
L
(Neglecting small quantity PM2 in comparison to
)
Fig.15
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Nature of Light and Huygens’ Wave Theory
Similarly, we can prove that
Substituting the above values in Eq. 8-5
According to sign convention, u, v and R are all negative,
This is the required expression.
10. Thin Convex Lens Formula on the basis of Huygens’ wave theory of light:
In Fig 16, if A is a point object on the axis of a converging lens, it will
produce a spherical wave represented by K1P1, touching the surface of the lens at
P1. Let v1 and v2 be the velocities of light wave in air (medium refractive index 1)
and medium of the lens (of refractive index
2) respectively. Since the light
travels more slowly in the material of lens than in the surrounding medium, the
portion of the wave front round and about P1 will travel more slowly than the outer
parts and the wave front will turn inside out to become spherical and concave in
the direction of motion. It is represented by K 2P2 with centre B, which is therefore
the point image of the point object A. According to Huygens’ principle, each point
on K1P1 is source of secondary wavelet. Since the wave diverges from A and
converges to B again, the time taken for the light from A to reach B is the same by
all paths; otherwise no image will be formed. Hence, the time taken by light to
travel the path AP1LP2B is the same as that taken to travel the pathAK 1MK2B. The
refractive index of the material of the lens is
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Nature of Light and Huygens’ Wave Theory
1
v1
K2
p
Let AM = p, MB = q; AL = u, LB = v, P1L
= a, LP2 = b, ML=y
C2
1
M
K1
q
y
P1
A
v1
L
P2
C1
B
a
b
As stated above regarding the time taken
being equal along the two paths; one has
u
v
2
Fig. 16 Thin convex lens formula
From right angled triangle AML
[By binomial theorem
is small]
Similarly, from right angled triangle MLB,
Adding Eq (10-2) and (10-3),
From Eq.(10-2) and (10-4),
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Let C1 be the centre of curvature of the surface of the lens at which the light enters
and C2 that at which it leaves, as shown in Fig. 16. The
corresponding radii of curvature of the two surfaces of the
lens are
M
y
P1C1 = R1
P1
P2C2 = R2
From the property of circle in geometry, one can write, as
shown in Fig. 17,
P1LLQ = MLL
a L C1
y
2R1a
Q
M
Fig.17 Property of circle
a(2R1a) = y2
2R1aa2 = y2
For a thin lens, a is small; therefore a2 is negligibly small compared to 2R1a.
Similarly,
Substituting these values in Eq. (10-4), we have
This, on simplification, gives
Using sign conventions of optical distances, u and R 2 are negative whereas v and
R1 are positive. Rewriting the above equation,
This is the general formula for lenses. For object lying at infinity u = ,
(second) focal length of the lens.
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Nature of Light and Huygens’ Wave Theory
From Eq.(10-7) and (10-8), we have
N.B. (i) A similar formula for refraction by a concave lens, on the basis of Huygens’
wave theory, can be derived
(ii) The reader will recall that the above procedure uses the same
principle(regarding the time taken along the two paths) which was used in section
2.17 in proving the lens formula by Fermat’s principle and so the above proof from
Huygens’ principle follows just the same lines.
11. Thin concave lens formula on the basis of Huygens’ wave theory of light.
Fig. 18 shows a thin concave lens
where A is a point object being on the
principal axis of the lens.
is the
portion of the wave front coming from A,
which touches the lens surface at
. The wavelet from K1 reaches K2
in a time interval equal to that time taken
by the wavelet from D to E. The wavelet
from D partly travels through air and
partly through the medium of the lens.
Assume small thickness in the middle of
the lens. Velocity v1 of light in air is
greater than velocity v2 of light inside the
lens. Hence the refracted wave front will
take the form
being convex
outside. It will appear to come from B,
(see Fig. 18).
K1
q
p
D
B
A
C1
K2
G
L1 L2 E
C2
H
Fig. 18 Thin Concave Lens formula
Hence B is the virtual point image of the point object A.
Draw K1D and K2E perpendicular to the principal axis. Suppose
Time taken by wavelet to travel from K1 to K2 in the lens is
Time taken by light to travel from D to G partly in air and partly in lens is
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Since t1 = t2
Now,
Multiply throughout by v1,
Cancelling
Since ,
This gives
Let
R1= radius of curvature of curved surface
curvature C1.
having centre of
R2= radius of curvature of curved surface
curvature C2.
having centre of
AH = u
BH = v
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Nature of Light and Huygens’ Wave Theory
Substituting the above values,
Using sign convention, u, v, and R1 are negative
A
O
P
For a point object at infinity
B
This gives the required expression, namely,
12.
Fig.19 Ray OP is perpendicular
to wave front AB
Relation between Huygens’ wave theory and Fermat’s principle:
Consider a plane wave front AB at any instant of time. We are to find the effect of
AB at any point P on a screen. From P draw perpendicular PO and AB. Each point on AB is
a source of secondary wavelets which are superimposed on P. It can be proved that the
resultant effect at P due to all the secondary waves from the wave front AB will be due to a
small element around O whereas the effect due to the others parts of the wave front will
cancel each others effect and its resultant effect is nil. This is due to their mutual
cancellation by the phenomenon called interference of light. Thus, the light travels along
the path OP which is the shortest (or the least) path between the wave front AB and the
point P. In other words, disturbance travels along the minimum path. It shows that it
follows Fermat’s principle which is also the principle of extreme path. Thus Fermat’s
principle follows from the concept of the wave theory of light. The straight line OP along
which the disturbance travels is called a ray. Thus, a ray is a straight line which is
perpendicular to wave front.
13. Solved Examples:
Examples 1:
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Nature of Light and Huygens’ Wave Theory
(a) Calculate the optical path for two wavelengths 1 and 2 of light propagating
in a medium of refractive index .
(b) Calculate also for two lights of frequencies 1 and 2 .
Solution:
In air, all the waves (or wave-fronts) of different wavelets travel with the
same speed of light. Therefore, the frequencies of light are inversely proportional
to the wavelengths. If v1 and v2 are the velocities of waves in a medium, and 1
and 2 are their wavelengths,
If v1 is velocity of light in air and v2 in a medium of refractive index ,
Optical path is the distance traversed by light in a medium of refractive index  in
time t.
Let d be the distance traversed.
Where v is velocity of light in the medium of refractive index .
Distance travelled by light in the same time t in air is
This distance is the equivalent distance in vacuum. It is called optical path.
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Nature of Light and Huygens’ Wave Theory
If d1 and d2 are distances of light travelled in medium ‘1’ and ‘2’ ,
(c) Velocity of light for each wavelength is different, but the frequency for each
light remains the same
Example 2:
Calculate the total optical path when light travels d1, d2 and d3 in three different
media with refractive indices 1, 2 and 3. How much distance in the third medium
3 will have the same optical path ?
Solution:
Total optical path is
This is the answer for initial part of the question. Let l be the distance in the third
medium 3 with the same optical path d.
Example 3.
A plane wave front in a medium of refractive index
1 is incident on a convex refracting suface of
refractive index 2. If R is the radius of curvature
of convex surface, show that the incident plane
wave front will turn into a spherical refracted wave
front and its curvature will be
A
A
A0
1
D
Medium 1
N
P M
C
F
Solution:
Let A0B0 be a plane wave front moving towards
convex surface
. At any instant A0B0 touches
E
B0
B
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Page 26
Fig.20, Ex.3
Nature of Light and Huygens’ Wave Theory
the convex surface. Let it be AB . As rays are perpendicular to plane wave front,
and
are the rays meeting convex surface on
and . Let 1 and v1 be the
refractive index and velocity of light in incident medium and the corresponding
quantities be 2 and v2 in refracting medium such that v1 > v2 and  1<2. If the
convex surface were absent, plane wave front AB would have occupied the position
after a time t. Due to the presence of refracting medium in which v2 < v1 the
wavelet from D and E reach
and
while the wavelet from P reaches M instead of
N.
From Fig.20, it follows
Because
Thus the plane wave front takes the shape
whose radius of curvature is
larger than R.
is radius of curvature of
, then from the property of circle.
From Eq.(1) and (2)
Thus, a plane wave front becomes a spherical wave front whose curvature is less
than that of the convex refracting surface (2 > 1) and is of the same sign as that
of the convex surface.
Summary:
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Nature of Light and Huygens’ Wave Theory
Light is a form of energy which can be transferred/transported in vacuum as well as in medium.
In Geometrical Optics, it is assumed that light travels in straight lines and we assume particle
nature of light. In Physical Optics, we deal with wave nature of light. In order to study the nature
of light, following theories of light were developed, viz,




Newton’s corpuscular theory of light (in year 1665)
Huygens’s wave theory of light (in year 1678)
Maxwell’s electromagnetic theory of light (or radiation) (in year 1873)
Planck’s quantum theory of radiation (in year 1900)
Newton’s Corpuscular theory of light: According to this theory, light consists of a large
number of minute material corpuscles (or particles) emitted by a luminous body and it travels
with a tremendous speed in straight lines in a homogeneous medium. This theory of light could
explain rectilinear propagation of light, laws of reflection and refraction. It led to wrong result
that light travels with a higher speed in denser medium.
Huygens’s wave theory states that a source of light propagates spherical waves in hypothetical
medium called ether, which pervades the whole of universe. It led to correct result that light
travels more slowly in water than air. Michelson- Morley experiment conducted in 1887, failed
to detect the presence of ether. Hence ether concept was abandoned.
Maxwell’s electromagnetic wave theory of light concluded theoretically that an
electromagnetic wave travelling in space has a velocity given by
where  = permeability of the medium
= permittivity of the medium
In vacuum, v equals velocity of light c in free space i.e. c = v = 3108 ms-1
This led Maxwell to put forward his famous electromagnetic theory of light in 1873.
According to this theory, light is an electromagnetic wave. In 1888, fifteen years later, Hertz
experimentally proved the existence of electromagnetic waves and thus verified Maxwell’s
electromagnetic theory that electromagnetic waves are transverse in nature. Maxwell’s
electromagnetic theory failed to explain phenomena like photoelectric effect and Compton
Effect.
In 1900, Planck gave quantum theory of radiation. According to this theory, exchanges of energy
between ether and matter can occur only in discrete steps, i.e. in multiples of some small unit,
called the quantum and energy of one quantum is
where h is Planck’s constant and is
the frequency of radiation.
This idea of Planck’s constant was verified by experiment.
In 1904, Einstein gave quantum theory of light. He used the concept of black body radiation put
forth by Planck in year 1900. Quantum theory of light was able to explain the photoelectric
effect, the atomic structure, the Compton Effect and Raman Effect, but it failed to explain the
phenomena of interference, diffraction etc. The latter phenomena were explainable on the basis
of wave theory of light. In other words, for one phenomenon we regard light as waves while for
another phenomenon we regard light as quanta (particles). This led to dual nature of light.
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Nature of Light and Huygens’ Wave Theory
In 1924, de Broglie worked out mathematically to correlate dual nature of light. Soon after, in
year 1926, Schrödinger extended Broglie’s theory and formulated a system of mechanics known
as wave mechanics. This is how the wave and particle aspect of matter are co-related.
Mathematically it is given by,

where m is mass of particle and is its velocity, whereas  is the wavelength of the wave
associated with moving particle called the de Broglie wavelength and h, the Planck’s constant.
De Broglie’s theoretical prediction of the existence of matter waves was first verified
experimentally in 1927 by C.J. Davisson and L.H. Germer and by G.P. Thomson.
A wavefront at any instant of time is defined as the locus of all the neighboring particles in the
medium which are being just disturbed at that instant of time and are consequently in the same
phase of vibration. The direction in which the disturbance is propagated in a homogeneous
medium is called a ray. A ray is always normal to the wave front. In the case of waves in three
dimensional, a point source will produce spherical waves, the rays being radii of the spheres,
while a plane source will produce plane waves in which the rays are lines normal to the plane.
Huygens’s principle of wave propagation of light is a geometrical method of finding the position
of the new wave front after a certain time. It states that the position of a wave front t seconds
after its present position is found by regarding each point of the given wave front as the source of
secondary wavelets. With each point on the given wave front as centre a sphere of radius vt is
drawn, where v is the velocity of the waves in the medium. The required position of the wave
front is the envelope of the spherical wavelets in the forward direction. Only forward wave front
exists and no backward wavefront exists. Application of Huygens’ principle of wave theory of
light:
One can prove
(i) Laws of reflection
(ii) Laws of refraction (and hence Snell’s law)
(iii) The mirror formula in terms of u, and f, i.e.
(iv) Thin lens formula viz.

Also
(v) Fermat’s principle which is also the principle of extreme path
Exercise:
1.
2.
3.
4.
5.
State four theories regarding nature of light.
Briefly outline the Huygens’ wave theory of light. What is a wave-front ? Ray ? How
is wave front produced and propagated ?
State Huygens’ theory of secondary wavelets. Explain clearly Huygens’ principle for
the propagation of light.
Sate Hugen’s principle and show how it be used to explain the laws of reflection of
light.
(a) Explain Huygens’ principle for propagation of light.
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Nature of Light and Huygens’ Wave Theory
(b) State the laws of reflection and refraction of light and obtain them form the wave
theory of light.
6.
Explain the laws of refraction of light on wave theory of light. What is meant by
refractive index and how is it related to the velocities of light in different media.
Derive Snell’s law on the basis of wave theory of light.
8. How is refraction of a plane wave explained on wave theory of light. When does total
internal reflection occure ?
9. Explain the phenomenon of total internal reflection on the basis of Huygens’ principle and
obtain the value of critical angle.
10. Derive a formula for refraction of a spherical wave at a spherical surface according to
the wave theory of light ?
11. Derive a formula for reflection of a spherical wave at a spherical surface according to
the wave theory of light.
12. Account for the formation of images by refraction through lenses on the wave theory
and prove the relation
where the symbols have their usual significance
13. What is a wave-front and what is the mode of its propagation in a medium ? Using the
wave theory establish the thin lens formula
14. Using the concept of wave theory prove the thin lens formula
15. State Huygens’s principle for propagation of light. Using the same, deduce the formula
connecting object and image distances with the constants of a thin lens.
16.
Derive the refraction formula for a thin concave lens on the basis of Huygens’ wave
theory of light.
1 State and explain Huygens’ principle of secondary waves. Apply this principle for
explaining the simultaneous reflection and refraction of a plane light wave from a plane
surface of separation of two optical media.
18. Explain how Huygens’ wave theory of light is related to Fermat’s principle.
19. write a short essay on the nature of light.
20. Give an elementary account of the wave theory of light, and prove by its means that the
focal length of a concave mirror is equal to half its radius of curvature.
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21. Explain, on the basis of wave theory of light, how a lens forms an image of a point on
its axis. Use the ideas of the wave theory to deduce an expression for the focal length
of the lens, pointing out very clearly the assumptions made. Point out which of these
assumptions are not true is practice, and in what way an actual lens may differ form
the simple theoretical lens as a result.
22. (a) Apply the principles of the wave theory to find the relation between the focal length
of a thin lens, the radii of curvature of the surfaces, and the refractive index of the
material.
(b) If a convex lens of focal length 15 cm, made of glass of refractive index 1.52 is totally
immersed in a liquid of refractive index 1.35, how will its focal length be affected ?
[Ans: Increased to 61.9 cm.]
23. Apply the principles of the wave theory to establish the formula
for a thin lens.
A ray of light is incident in a direction parallel to the axis on one face of a thin double
convex lens of refractive index 1.5, and after two internal reflections, emerges from the
second face. Show that it will cut the axis at a point at a distance from the centre of
the lens.
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