Page 1
255
Wave Optics
Chapter Ten
WAVE OPTICS
10.1 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 – it is this wave model of light that we will discuss in
this chapter. As we will see, 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.
The wave theory was not readily accepted primarily because of
Newton’s authority and also because light could travel through vacuum
Rationalised 2023-24
Page 2
255
Wave Optics
Chapter Ten
WAVE OPTICS
10.1 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 – it is this wave model of light that we will discuss in
this chapter. As we will see, 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.
The wave theory was not readily accepted primarily because of
Newton’s authority and also because light could travel through vacuum
Rationalised 2023-24
Physics
256
and it was felt that a wave would always require a medium to propagate
from one point to the other. However, when Thomas Young performed
his famous interference experiment in 1801, it was firmly established
that light is indeed a wave phenomenon. The wavelength of visible
light was measured and found to be extremely small; for example, the
wavelength of yellow light is about 0.6 mm. Because of the smallness
of the wavelength of visible light (in comparison to the dimensions of
typical mirrors and lenses), light can be assumed to approximately
travel in straight lines. This is the field of geometrical optics, which we
had discussed in the previous chapter. Indeed, the branch of optics in
which one completely neglects the finiteness of the wavelength is called
geometrical optics and a ray is defined as the path of energy
propagation in the limit of wavelength tending to zero.
After the interference experiment of Young in 1801, for the next 40
years or so, many experiments were carried out involving the
interference and diffraction of lightwaves; these experiments could only
be satisfactorily explained by assuming a wave model of light. Thus,
around the middle of the nineteenth century, the wave theory seemed
to be very well established. The only major difficulty was that since it
was thought that a wave required a medium for its propagation, how
could light waves propagate through vacuum. This was explained
when Maxwell put forward his famous electromagnetic theory of light.
Maxwell had developed a set of equations describing the laws of
electricity and magnetism and using these equations he derived what
is known as the wave equation from which he predicted the existence
of electromagnetic waves*. From the wave equation, Maxwell could
calculate the speed of electromagnetic waves in free space and he found
that the theoretical value was very close to the measured value of speed
of light. From this, he propounded that light must be an
electromagnetic wave. Thus, according to Maxwell, light waves are
associated with changing electric and magnetic fields; changing electric
field produces a time and space varying magnetic field and a changing
magnetic field produces a time and space varying electric field. The
changing electric and magnetic fields result in the propagation of
electromagnetic waves (or light waves) even in vacuum.
In this chapter we will first discuss the original formulation of the
Huygens principle and derive the laws of reflection and refraction. In
Sections 10.4 and 10.5, we will discuss the phenomenon of interference
which is based on the principle of superposition. In Section 10.6 we
will discuss the phenomenon of diffraction which is based on Huygens-
Fresnel principle. Finally in Section 10.7 we will discuss the
phenomenon of polarisation which is based on the fact that the light
waves are transverse electromagnetic waves.
* Maxwell had predicted the existence of electromagnetic waves around 1855; it
was much later (around 1890) that Heinrich Hertz produced radiowaves in the
laboratory. J.C. Bose and G. Marconi made practical applications of the Hertzian
waves
Rationalised 2023-24
Page 3
255
Wave Optics
Chapter Ten
WAVE OPTICS
10.1 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 – it is this wave model of light that we will discuss in
this chapter. As we will see, 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.
The wave theory was not readily accepted primarily because of
Newton’s authority and also because light could travel through vacuum
Rationalised 2023-24
Physics
256
and it was felt that a wave would always require a medium to propagate
from one point to the other. However, when Thomas Young performed
his famous interference experiment in 1801, it was firmly established
that light is indeed a wave phenomenon. The wavelength of visible
light was measured and found to be extremely small; for example, the
wavelength of yellow light is about 0.6 mm. Because of the smallness
of the wavelength of visible light (in comparison to the dimensions of
typical mirrors and lenses), light can be assumed to approximately
travel in straight lines. This is the field of geometrical optics, which we
had discussed in the previous chapter. Indeed, the branch of optics in
which one completely neglects the finiteness of the wavelength is called
geometrical optics and a ray is defined as the path of energy
propagation in the limit of wavelength tending to zero.
After the interference experiment of Young in 1801, for the next 40
years or so, many experiments were carried out involving the
interference and diffraction of lightwaves; these experiments could only
be satisfactorily explained by assuming a wave model of light. Thus,
around the middle of the nineteenth century, the wave theory seemed
to be very well established. The only major difficulty was that since it
was thought that a wave required a medium for its propagation, how
could light waves propagate through vacuum. This was explained
when Maxwell put forward his famous electromagnetic theory of light.
Maxwell had developed a set of equations describing the laws of
electricity and magnetism and using these equations he derived what
is known as the wave equation from which he predicted the existence
of electromagnetic waves*. From the wave equation, Maxwell could
calculate the speed of electromagnetic waves in free space and he found
that the theoretical value was very close to the measured value of speed
of light. From this, he propounded that light must be an
electromagnetic wave. Thus, according to Maxwell, light waves are
associated with changing electric and magnetic fields; changing electric
field produces a time and space varying magnetic field and a changing
magnetic field produces a time and space varying electric field. The
changing electric and magnetic fields result in the propagation of
electromagnetic waves (or light waves) even in vacuum.
In this chapter we will first discuss the original formulation of the
Huygens principle and derive the laws of reflection and refraction. In
Sections 10.4 and 10.5, we will discuss the phenomenon of interference
which is based on the principle of superposition. In Section 10.6 we
will discuss the phenomenon of diffraction which is based on Huygens-
Fresnel principle. Finally in Section 10.7 we will discuss the
phenomenon of polarisation which is based on the fact that the light
waves are transverse electromagnetic waves.
* Maxwell had predicted the existence of electromagnetic waves around 1855; it
was much later (around 1890) that Heinrich Hertz produced radiowaves in the
laboratory. J.C. Bose and G. Marconi made practical applications of the Hertzian
waves
Rationalised 2023-24
257
Wave Optics
10.2 HUYGENS PRINCIPLE
We would first define a wavefront: when we drop a small stone on a
calm pool of water, waves spread out from the point of impact. Every
point on the surface starts oscillating with time. At any instant, a
photograph of the surface would show circular rings on which the
disturbance is maximum. Clearly, all points on such a circle are
oscillating in phase because they are at the same distance from the
source. Such a 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.
If we have a point source emitting waves uniformly in all directions,
then 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
wave as shown in Fig. 10.1(a). 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 wave [Fig. 10.1(b)].
Now, if we know the shape of the wavefront at t = 0, then Huygens
principle allows us to determine the shape of the wavefront at a later
time t. Thus, Huygens principle is essentially a geometrical construction,
which given the shape of the wafefront at any time allows us to determine
the shape of the wavefront at a later time. Let us consider a diverging
wave and let F
1
F
2
represent a portion of the spherical wavefront at t = 0
(Fig. 10.2). Now, according to Huygens 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 and if we draw a common tangent
to all these spheres, we obtain the new position of the wavefront at a
later time.
FIGURE 10.1 (a) A
diverging spherical
wave emanating from
a point source. The
wavefronts are
spherical.
FIGURE 10.2 F
1
F
2
represents the spherical wavefront (with O as
centre) at t = 0. The envelope of the secondary wavelets
emanating from F
1
F
2
produces the forward moving wavefront G
1
G
2
.
The backwave D
1
D
2
does not exist.
FIGURE 10.1 (b) At a
large distance from
the source, a small
portion of the
spherical wave can
be approximated by a
plane wave.
Rationalised 2023-24
Page 4
255
Wave Optics
Chapter Ten
WAVE OPTICS
10.1 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 – it is this wave model of light that we will discuss in
this chapter. As we will see, 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.
The wave theory was not readily accepted primarily because of
Newton’s authority and also because light could travel through vacuum
Rationalised 2023-24
Physics
256
and it was felt that a wave would always require a medium to propagate
from one point to the other. However, when Thomas Young performed
his famous interference experiment in 1801, it was firmly established
that light is indeed a wave phenomenon. The wavelength of visible
light was measured and found to be extremely small; for example, the
wavelength of yellow light is about 0.6 mm. Because of the smallness
of the wavelength of visible light (in comparison to the dimensions of
typical mirrors and lenses), light can be assumed to approximately
travel in straight lines. This is the field of geometrical optics, which we
had discussed in the previous chapter. Indeed, the branch of optics in
which one completely neglects the finiteness of the wavelength is called
geometrical optics and a ray is defined as the path of energy
propagation in the limit of wavelength tending to zero.
After the interference experiment of Young in 1801, for the next 40
years or so, many experiments were carried out involving the
interference and diffraction of lightwaves; these experiments could only
be satisfactorily explained by assuming a wave model of light. Thus,
around the middle of the nineteenth century, the wave theory seemed
to be very well established. The only major difficulty was that since it
was thought that a wave required a medium for its propagation, how
could light waves propagate through vacuum. This was explained
when Maxwell put forward his famous electromagnetic theory of light.
Maxwell had developed a set of equations describing the laws of
electricity and magnetism and using these equations he derived what
is known as the wave equation from which he predicted the existence
of electromagnetic waves*. From the wave equation, Maxwell could
calculate the speed of electromagnetic waves in free space and he found
that the theoretical value was very close to the measured value of speed
of light. From this, he propounded that light must be an
electromagnetic wave. Thus, according to Maxwell, light waves are
associated with changing electric and magnetic fields; changing electric
field produces a time and space varying magnetic field and a changing
magnetic field produces a time and space varying electric field. The
changing electric and magnetic fields result in the propagation of
electromagnetic waves (or light waves) even in vacuum.
In this chapter we will first discuss the original formulation of the
Huygens principle and derive the laws of reflection and refraction. In
Sections 10.4 and 10.5, we will discuss the phenomenon of interference
which is based on the principle of superposition. In Section 10.6 we
will discuss the phenomenon of diffraction which is based on Huygens-
Fresnel principle. Finally in Section 10.7 we will discuss the
phenomenon of polarisation which is based on the fact that the light
waves are transverse electromagnetic waves.
* Maxwell had predicted the existence of electromagnetic waves around 1855; it
was much later (around 1890) that Heinrich Hertz produced radiowaves in the
laboratory. J.C. Bose and G. Marconi made practical applications of the Hertzian
waves
Rationalised 2023-24
257
Wave Optics
10.2 HUYGENS PRINCIPLE
We would first define a wavefront: when we drop a small stone on a
calm pool of water, waves spread out from the point of impact. Every
point on the surface starts oscillating with time. At any instant, a
photograph of the surface would show circular rings on which the
disturbance is maximum. Clearly, all points on such a circle are
oscillating in phase because they are at the same distance from the
source. Such a 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.
If we have a point source emitting waves uniformly in all directions,
then 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
wave as shown in Fig. 10.1(a). 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 wave [Fig. 10.1(b)].
Now, if we know the shape of the wavefront at t = 0, then Huygens
principle allows us to determine the shape of the wavefront at a later
time t. Thus, Huygens principle is essentially a geometrical construction,
which given the shape of the wafefront at any time allows us to determine
the shape of the wavefront at a later time. Let us consider a diverging
wave and let F
1
F
2
represent a portion of the spherical wavefront at t = 0
(Fig. 10.2). Now, according to Huygens 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 and if we draw a common tangent
to all these spheres, we obtain the new position of the wavefront at a
later time.
FIGURE 10.1 (a) A
diverging spherical
wave emanating from
a point source. The
wavefronts are
spherical.
FIGURE 10.2 F
1
F
2
represents the spherical wavefront (with O as
centre) at t = 0. The envelope of the secondary wavelets
emanating from F
1
F
2
produces the forward moving wavefront G
1
G
2
.
The backwave D
1
D
2
does not exist.
FIGURE 10.1 (b) At a
large distance from
the source, a small
portion of the
spherical wave can
be approximated by a
plane wave.
Rationalised 2023-24
Physics
258
Thus, if we wish to determine the shape of the wavefront at t = t, we
draw spheres of radius vt from each point on the spherical wavefront
where v represents the speed of the waves in the medium. If we now draw
a common tangent to all these spheres, we obtain the new position of the
wavefront at t = t. The new wavefront shown as G
1
G
2
in Fig. 10.2 is again
spherical with point O as the centre.
The above model has one shortcoming: we also have a backwave which
is shown as D
1
D
2
in Fig. 10.2. 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, Huygens could
explain the absence of the backwave. However, this adhoc assumption is
not satisfactory and the absence of the backwave is really justified from
more rigorous wave theory.
In a similar manner, we can use Huygens principle to determine the
shape of the wavefront for a plane wave propagating through a medium
(Fig. 10.3).
10.3 REFRACTION AND REFLECTION OF
PLANE WAVES USING HUYGENS PRINCIPLE
10.3.1 Refraction of a plane wave
We will now use Huygens principle to derive the laws of refraction. Let PP¢
represent the surface separating medium 1 and medium 2, as shown in
Fig. 10.4. Let v
1
and v
2
represent the speed of light in medium 1 and
medium 2, respectively. We assume a plane wavefront AB propagating in
the direction A¢A incident on the interface at an angle i as shown in the
figure. Let t be the time taken by the wavefront to travel the distance BC.
Thus,
BC = v
1
t
FIGURE 10.3
Huygens geometrical
construction for a
plane wave
propagating to the
right. F
1
F
2
is the
plane wavefront at
t = 0 and G
1
G
2
is the
wavefront at a later
time t. The lines A
1
A
2
,
B
1
B
2
… etc., are
normal to both F
1
F
2
and G
1
G
2
and
represent rays.
FIGURE 10.4 A plane wave AB is incident at an angle i on the surface
PP¢ separating medium 1 and medium 2. The plane wave undergoes
refraction and CE represents the refracted wavefront. The figure
corresponds to v
2
< v
1
so that the refracted waves bends towards the
normal.
Rationalised 2023-24
Page 5
255
Wave Optics
Chapter Ten
WAVE OPTICS
10.1 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 – it is this wave model of light that we will discuss in
this chapter. As we will see, 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.
The wave theory was not readily accepted primarily because of
Newton’s authority and also because light could travel through vacuum
Rationalised 2023-24
Physics
256
and it was felt that a wave would always require a medium to propagate
from one point to the other. However, when Thomas Young performed
his famous interference experiment in 1801, it was firmly established
that light is indeed a wave phenomenon. The wavelength of visible
light was measured and found to be extremely small; for example, the
wavelength of yellow light is about 0.6 mm. Because of the smallness
of the wavelength of visible light (in comparison to the dimensions of
typical mirrors and lenses), light can be assumed to approximately
travel in straight lines. This is the field of geometrical optics, which we
had discussed in the previous chapter. Indeed, the branch of optics in
which one completely neglects the finiteness of the wavelength is called
geometrical optics and a ray is defined as the path of energy
propagation in the limit of wavelength tending to zero.
After the interference experiment of Young in 1801, for the next 40
years or so, many experiments were carried out involving the
interference and diffraction of lightwaves; these experiments could only
be satisfactorily explained by assuming a wave model of light. Thus,
around the middle of the nineteenth century, the wave theory seemed
to be very well established. The only major difficulty was that since it
was thought that a wave required a medium for its propagation, how
could light waves propagate through vacuum. This was explained
when Maxwell put forward his famous electromagnetic theory of light.
Maxwell had developed a set of equations describing the laws of
electricity and magnetism and using these equations he derived what
is known as the wave equation from which he predicted the existence
of electromagnetic waves*. From the wave equation, Maxwell could
calculate the speed of electromagnetic waves in free space and he found
that the theoretical value was very close to the measured value of speed
of light. From this, he propounded that light must be an
electromagnetic wave. Thus, according to Maxwell, light waves are
associated with changing electric and magnetic fields; changing electric
field produces a time and space varying magnetic field and a changing
magnetic field produces a time and space varying electric field. The
changing electric and magnetic fields result in the propagation of
electromagnetic waves (or light waves) even in vacuum.
In this chapter we will first discuss the original formulation of the
Huygens principle and derive the laws of reflection and refraction. In
Sections 10.4 and 10.5, we will discuss the phenomenon of interference
which is based on the principle of superposition. In Section 10.6 we
will discuss the phenomenon of diffraction which is based on Huygens-
Fresnel principle. Finally in Section 10.7 we will discuss the
phenomenon of polarisation which is based on the fact that the light
waves are transverse electromagnetic waves.
* Maxwell had predicted the existence of electromagnetic waves around 1855; it
was much later (around 1890) that Heinrich Hertz produced radiowaves in the
laboratory. J.C. Bose and G. Marconi made practical applications of the Hertzian
waves
Rationalised 2023-24
257
Wave Optics
10.2 HUYGENS PRINCIPLE
We would first define a wavefront: when we drop a small stone on a
calm pool of water, waves spread out from the point of impact. Every
point on the surface starts oscillating with time. At any instant, a
photograph of the surface would show circular rings on which the
disturbance is maximum. Clearly, all points on such a circle are
oscillating in phase because they are at the same distance from the
source. Such a 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.
If we have a point source emitting waves uniformly in all directions,
then 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
wave as shown in Fig. 10.1(a). 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 wave [Fig. 10.1(b)].
Now, if we know the shape of the wavefront at t = 0, then Huygens
principle allows us to determine the shape of the wavefront at a later
time t. Thus, Huygens principle is essentially a geometrical construction,
which given the shape of the wafefront at any time allows us to determine
the shape of the wavefront at a later time. Let us consider a diverging
wave and let F
1
F
2
represent a portion of the spherical wavefront at t = 0
(Fig. 10.2). Now, according to Huygens 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 and if we draw a common tangent
to all these spheres, we obtain the new position of the wavefront at a
later time.
FIGURE 10.1 (a) A
diverging spherical
wave emanating from
a point source. The
wavefronts are
spherical.
FIGURE 10.2 F
1
F
2
represents the spherical wavefront (with O as
centre) at t = 0. The envelope of the secondary wavelets
emanating from F
1
F
2
produces the forward moving wavefront G
1
G
2
.
The backwave D
1
D
2
does not exist.
FIGURE 10.1 (b) At a
large distance from
the source, a small
portion of the
spherical wave can
be approximated by a
plane wave.
Rationalised 2023-24
Physics
258
Thus, if we wish to determine the shape of the wavefront at t = t, we
draw spheres of radius vt from each point on the spherical wavefront
where v represents the speed of the waves in the medium. If we now draw
a common tangent to all these spheres, we obtain the new position of the
wavefront at t = t. The new wavefront shown as G
1
G
2
in Fig. 10.2 is again
spherical with point O as the centre.
The above model has one shortcoming: we also have a backwave which
is shown as D
1
D
2
in Fig. 10.2. 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, Huygens could
explain the absence of the backwave. However, this adhoc assumption is
not satisfactory and the absence of the backwave is really justified from
more rigorous wave theory.
In a similar manner, we can use Huygens principle to determine the
shape of the wavefront for a plane wave propagating through a medium
(Fig. 10.3).
10.3 REFRACTION AND REFLECTION OF
PLANE WAVES USING HUYGENS PRINCIPLE
10.3.1 Refraction of a plane wave
We will now use Huygens principle to derive the laws of refraction. Let PP¢
represent the surface separating medium 1 and medium 2, as shown in
Fig. 10.4. Let v
1
and v
2
represent the speed of light in medium 1 and
medium 2, respectively. We assume a plane wavefront AB propagating in
the direction A¢A incident on the interface at an angle i as shown in the
figure. Let t be the time taken by the wavefront to travel the distance BC.
Thus,
BC = v
1
t
FIGURE 10.3
Huygens geometrical
construction for a
plane wave
propagating to the
right. F
1
F
2
is the
plane wavefront at
t = 0 and G
1
G
2
is the
wavefront at a later
time t. The lines A
1
A
2
,
B
1
B
2
… etc., are
normal to both F
1
F
2
and G
1
G
2
and
represent rays.
FIGURE 10.4 A plane wave AB is incident at an angle i on the surface
PP¢ separating medium 1 and medium 2. The plane wave undergoes
refraction and CE represents the refracted wavefront. The figure
corresponds to v
2
< v
1
so that the refracted waves bends towards the
normal.
Rationalised 2023-24
259
Wave Optics
In order to determine the shape of the refracted
wavefront, we draw a sphere of radius v
2
t from the point
A in the second medium (the speed of the wave in the
second medium is v
2
). Let CE represent a tangent plane
drawn from the point C on to the sphere. Then, AE = v
2
t
and CE would represent the refracted wavefront. If we
now consider the triangles ABC and AEC, we readily
obtain
sin i =
1
BC
AC AC
v t
=
(10.1)
and
sin r =
2
AE
AC AC
v t
=
(10.2)
where i and r are the angles of incidence and refraction,
respectively. Thus we obtain
1
2
sin
sin
i v
r v
=
(10.3)
From the above equation, we get the important result
that if r < i (i.e., if the ray bends toward the normal), the
speed of the light wave in the second medium (v
2
) will be
less then the speed of the light wave in the first medium
(v
1
). This prediction is opposite to the prediction from
the corpuscular model of light and as later experiments
showed, the prediction of the wave theory is correct. Now,
if c represents the speed of light in vacuum, then,
1
1
c
n
v
=
(10.4)
and
n
2
=
2
c
v
(10.5)
are known as the refractive indices of medium 1 and
medium 2, respectively. In terms of the refractive indices, Eq. (10.3) can
be written as
n
1
sin i = n
2
sin r (10.6)
This is the Snell’s law of refraction. Further, if l
1
and l
2
denote the
wavelengths of light in medium 1 and medium 2, respectively and if the
distance BC is equal to l
1
then the distance AE will be equal to l
2
( because
if the crest from B has reached C in time t, then the crest from A should
have also reached E in time t ); thus,
1 1
2 2
BC
AE
v
v
?
?
= =
or
1 2
1 2
v v
? ?
=
(10.7)
CHRISTIAAN HUYGENS (1629 – 1695)
Christiaan Huygens
(1629 – 1695) Dutch
physicist, astronomer,
mathematician and the
founder of the wave
theory of light. His book,
Treatise on light, makes
fascinating reading even
today. He brilliantly
explained the double
refraction shown by the
mineral calcite in this
work in addition to
reflection and refraction.
He was the first to
analyse circular and
simple harmonic motion
and designed and built
improved clocks and
telescopes. He discovered
the true geometry of
Saturn’s rings.
Rationalised 2023-24
Read More