POLARISATION : OBJECTIVES Notes | EduRev

: POLARISATION : OBJECTIVES Notes | EduRev

 Page 1


84
L6 POLARISATION
 OBJECTIVES
Aims
Once you have studied this chapter you should understand the concepts of transverse waves, plane
polarisation, circular polarisation and elliptical polarisation.  You should be able to relate this
understanding to a knowledge of methods for producing the different types of polarised light in
sufficient detail so that you can explain the basic principles of those methods.
Minimum learning goals
1. Explain, interpret and use the terms:
polarised light, unpolarised light, randomly polarised light, linear polarisation (plane
polarisation), partially polarised light, polarising axis, polariser, ideal polariser, analyser,
crossed polarisers, Malus's law, circular polarisation, elliptical polarisation, birefringence
(double refraction), dichroic material, dichroism, optical activity, quarter-wave plate,
polarising angle (Brewster angle).
2. Describe how plane polarised light can be produced by dichroic materials, by birefringent
materials, by reflection and by scattering. 
3. State and apply Malus's law.
4. Explain how circularly or elliptically polarised light can be regarded as a superposition of
plane polarisations.
5. Describe how circularly polarised light can be produced from unpolarised or plane polarised
light.
6. Describe the phenomenon of optical activity and describe one example of its application.
Extra Goals
7. Describe and discuss various applications of polarised light and explain how they work.
 TEXT
6-1 PLANE OR LINEAR POLARISATION
In light and all other kinds of electromagnetic waves, the oscillating electric and magnetic fields are
always directed at right angles to each other and to the direction of propagation of the wave.  In other
words the fields are transverse, and light is described as a transverse wave. (By contrast sound
waves are said to be longitudinal, because the oscillations of the particles are parallel to the direction
of propagation.)   Since both the directions and the magnitudes of the electric and magnetic fields in
a light wave are related in a fixed manner, it is sufficient to talk about only one of them, the usual
choice being the electric field.  Now although the electric field at any point in space must be
perpendicular to the wave velocity, it can still have many different directions; it can point in any
direction in the plane perpendicular to the wave's direction of travel.
Any beam of light can be thought of as a huge collection of elementary waves with a range of
different frequencies.  Each elementary wave has its own unique orientation of its electric field; it is
polarised (figure 6.1).  If the polarisations of all the elementary waves in a complex beam can be
made to have the same orientation all the time then the light beam is also said to be polarised.  Since
there is then a unique plane containing all the electric field directions as well as the direction of the
light ray, this kind of polarisation is also called plane polarisation.  It is also known as linear
polarisation.  However, the usual situation is that the directions of the electric fields of the
component wavelets are randomly distributed; in that case the resultant wave is said to be randomly
polarised or unpolarised.
Page 2


84
L6 POLARISATION
 OBJECTIVES
Aims
Once you have studied this chapter you should understand the concepts of transverse waves, plane
polarisation, circular polarisation and elliptical polarisation.  You should be able to relate this
understanding to a knowledge of methods for producing the different types of polarised light in
sufficient detail so that you can explain the basic principles of those methods.
Minimum learning goals
1. Explain, interpret and use the terms:
polarised light, unpolarised light, randomly polarised light, linear polarisation (plane
polarisation), partially polarised light, polarising axis, polariser, ideal polariser, analyser,
crossed polarisers, Malus's law, circular polarisation, elliptical polarisation, birefringence
(double refraction), dichroic material, dichroism, optical activity, quarter-wave plate,
polarising angle (Brewster angle).
2. Describe how plane polarised light can be produced by dichroic materials, by birefringent
materials, by reflection and by scattering. 
3. State and apply Malus's law.
4. Explain how circularly or elliptically polarised light can be regarded as a superposition of
plane polarisations.
5. Describe how circularly polarised light can be produced from unpolarised or plane polarised
light.
6. Describe the phenomenon of optical activity and describe one example of its application.
Extra Goals
7. Describe and discuss various applications of polarised light and explain how they work.
 TEXT
6-1 PLANE OR LINEAR POLARISATION
In light and all other kinds of electromagnetic waves, the oscillating electric and magnetic fields are
always directed at right angles to each other and to the direction of propagation of the wave.  In other
words the fields are transverse, and light is described as a transverse wave. (By contrast sound
waves are said to be longitudinal, because the oscillations of the particles are parallel to the direction
of propagation.)   Since both the directions and the magnitudes of the electric and magnetic fields in
a light wave are related in a fixed manner, it is sufficient to talk about only one of them, the usual
choice being the electric field.  Now although the electric field at any point in space must be
perpendicular to the wave velocity, it can still have many different directions; it can point in any
direction in the plane perpendicular to the wave's direction of travel.
Any beam of light can be thought of as a huge collection of elementary waves with a range of
different frequencies.  Each elementary wave has its own unique orientation of its electric field; it is
polarised (figure 6.1).  If the polarisations of all the elementary waves in a complex beam can be
made to have the same orientation all the time then the light beam is also said to be polarised.  Since
there is then a unique plane containing all the electric field directions as well as the direction of the
light ray, this kind of polarisation is also called plane polarisation.  It is also known as linear
polarisation.  However, the usual situation is that the directions of the electric fields of the
component wavelets are randomly distributed; in that case the resultant wave is said to be randomly
polarised or unpolarised.
L6:   Polarisation 85
Figure 6.1.  A polarised elementary wave
The picture shows a perspective plot of the instantaneous electric field vectors which all lie in the same plane
(shaded).  Every such elementary harmonic wave is plane polarised.
It is quite common to find partially polarised light which is a mixture of unpolarised
(completely random polarisations) and plane polarised waves, in which a significant fraction of  the
elementary waves have their electric fields oriented the same way.
Components of polarisation
Since electric field is a vector quantity it can be described in terms of components referred to a set of
coordinate directions.  In the case of polarised waves we can take any two perpendicular directions in
a plane perpendicular to the wave's direction of travel.   An electric field E which makes an angle a
with one of these directions can then be described completely as two components with values Ecosa
and Esina .  We can think of these components as two independent electric fields, each with its own
magnitude and direction, which are together equivalent in every respect to the original field.  So any
elementary wave can be regarded as a superposition of two elementary waves with perpendicular
polarisations.
x
y
E
E
is
equivalent
to
y
x
E
a
Figure 6.2.  Components of the instantaneous electric field
In just the same way, any plane polarisation can be described in terms of two mutually
perpendicular component polarisations.  In the schematic diagrams that we use to represent
polarisation such a line can be drawn as a double-headed arrow, representing the two opposite
directions that a plane polarised wave can have at any point.  The instantaneous value of an electric
field (which has a unique direction at any instant of time) will be shown as a single-headed arrow.
Page 3


84
L6 POLARISATION
 OBJECTIVES
Aims
Once you have studied this chapter you should understand the concepts of transverse waves, plane
polarisation, circular polarisation and elliptical polarisation.  You should be able to relate this
understanding to a knowledge of methods for producing the different types of polarised light in
sufficient detail so that you can explain the basic principles of those methods.
Minimum learning goals
1. Explain, interpret and use the terms:
polarised light, unpolarised light, randomly polarised light, linear polarisation (plane
polarisation), partially polarised light, polarising axis, polariser, ideal polariser, analyser,
crossed polarisers, Malus's law, circular polarisation, elliptical polarisation, birefringence
(double refraction), dichroic material, dichroism, optical activity, quarter-wave plate,
polarising angle (Brewster angle).
2. Describe how plane polarised light can be produced by dichroic materials, by birefringent
materials, by reflection and by scattering. 
3. State and apply Malus's law.
4. Explain how circularly or elliptically polarised light can be regarded as a superposition of
plane polarisations.
5. Describe how circularly polarised light can be produced from unpolarised or plane polarised
light.
6. Describe the phenomenon of optical activity and describe one example of its application.
Extra Goals
7. Describe and discuss various applications of polarised light and explain how they work.
 TEXT
6-1 PLANE OR LINEAR POLARISATION
In light and all other kinds of electromagnetic waves, the oscillating electric and magnetic fields are
always directed at right angles to each other and to the direction of propagation of the wave.  In other
words the fields are transverse, and light is described as a transverse wave. (By contrast sound
waves are said to be longitudinal, because the oscillations of the particles are parallel to the direction
of propagation.)   Since both the directions and the magnitudes of the electric and magnetic fields in
a light wave are related in a fixed manner, it is sufficient to talk about only one of them, the usual
choice being the electric field.  Now although the electric field at any point in space must be
perpendicular to the wave velocity, it can still have many different directions; it can point in any
direction in the plane perpendicular to the wave's direction of travel.
Any beam of light can be thought of as a huge collection of elementary waves with a range of
different frequencies.  Each elementary wave has its own unique orientation of its electric field; it is
polarised (figure 6.1).  If the polarisations of all the elementary waves in a complex beam can be
made to have the same orientation all the time then the light beam is also said to be polarised.  Since
there is then a unique plane containing all the electric field directions as well as the direction of the
light ray, this kind of polarisation is also called plane polarisation.  It is also known as linear
polarisation.  However, the usual situation is that the directions of the electric fields of the
component wavelets are randomly distributed; in that case the resultant wave is said to be randomly
polarised or unpolarised.
L6:   Polarisation 85
Figure 6.1.  A polarised elementary wave
The picture shows a perspective plot of the instantaneous electric field vectors which all lie in the same plane
(shaded).  Every such elementary harmonic wave is plane polarised.
It is quite common to find partially polarised light which is a mixture of unpolarised
(completely random polarisations) and plane polarised waves, in which a significant fraction of  the
elementary waves have their electric fields oriented the same way.
Components of polarisation
Since electric field is a vector quantity it can be described in terms of components referred to a set of
coordinate directions.  In the case of polarised waves we can take any two perpendicular directions in
a plane perpendicular to the wave's direction of travel.   An electric field E which makes an angle a
with one of these directions can then be described completely as two components with values Ecosa
and Esina .  We can think of these components as two independent electric fields, each with its own
magnitude and direction, which are together equivalent in every respect to the original field.  So any
elementary wave can be regarded as a superposition of two elementary waves with perpendicular
polarisations.
x
y
E
E
is
equivalent
to
y
x
E
a
Figure 6.2.  Components of the instantaneous electric field
In just the same way, any plane polarisation can be described in terms of two mutually
perpendicular component polarisations.  In the schematic diagrams that we use to represent
polarisation such a line can be drawn as a double-headed arrow, representing the two opposite
directions that a plane polarised wave can have at any point.  The instantaneous value of an electric
field (which has a unique direction at any instant of time) will be shown as a single-headed arrow.
L6:   Polarisation 86
y
x
is
equivalent
to
Polarisation
Component
polarisations
a
Figure 6.3  Components of the polarisation
Polarisers and Malus's law
A ideal polariser, or polarising filter, turns unpolarised light into completely plane polarised light.
Its action can be described in terms of its effect on elementary waves with different polarisations;
waves whose polarisation is parallel to an axis in the polariser, called its polarising axis, are
transmitted without any absorption but waves whose polarisation is perpendicular to the polarising
axis are completely absorbed.  An elementary wave whose polarisation is at some other angle to the
polarising axis is partly transmitted and partly absorbed but it emerges from the other side of the
polariser with a new polarisation, which is parallel to the polariser's axis.  This can be described in
terms of components of the original wave.  Since we can use any reference directions for taking
components we choose one direction parallel to the polariser's axis and the other one perpendicular
to it.  If the angle between the original polarisation and the polariser's axis is ?, then the component
parallel the the polariser's axis, which gets through, has an amplitude of E
0
 cos?.  Since the other
component is absorbed, the wave which emerges has a new amplitude E
0
 cos? and a new
polarisation.   Since the irradiance or "intensity" of light is proportional to the square of the electric
field's amplitude,
I
out 
= I
in
 cos
2
?. ...(6.1)
This result is known as Malus's law.
?
Incident linearly
polarised light
Polarising axis
Emerging light is
polarised parallel to
the axis of the polariser
Polariser
Figure 6.4.  Effect of a polariser on plane polarised light
Many practical polarisers do not obey Malus's law exactly, firstly because they absorb some
of the component with polarisation parallel to the polarising axis and secondly because some of the
component polarised perpendicular to the axis is not completely absorbed.
Malus's law also describes the action of an ideal polariser on unpolarised light.  Unpolarised
light is really a vast collection of polarised elementary waves whose polarisations are randomly
Page 4


84
L6 POLARISATION
 OBJECTIVES
Aims
Once you have studied this chapter you should understand the concepts of transverse waves, plane
polarisation, circular polarisation and elliptical polarisation.  You should be able to relate this
understanding to a knowledge of methods for producing the different types of polarised light in
sufficient detail so that you can explain the basic principles of those methods.
Minimum learning goals
1. Explain, interpret and use the terms:
polarised light, unpolarised light, randomly polarised light, linear polarisation (plane
polarisation), partially polarised light, polarising axis, polariser, ideal polariser, analyser,
crossed polarisers, Malus's law, circular polarisation, elliptical polarisation, birefringence
(double refraction), dichroic material, dichroism, optical activity, quarter-wave plate,
polarising angle (Brewster angle).
2. Describe how plane polarised light can be produced by dichroic materials, by birefringent
materials, by reflection and by scattering. 
3. State and apply Malus's law.
4. Explain how circularly or elliptically polarised light can be regarded as a superposition of
plane polarisations.
5. Describe how circularly polarised light can be produced from unpolarised or plane polarised
light.
6. Describe the phenomenon of optical activity and describe one example of its application.
Extra Goals
7. Describe and discuss various applications of polarised light and explain how they work.
 TEXT
6-1 PLANE OR LINEAR POLARISATION
In light and all other kinds of electromagnetic waves, the oscillating electric and magnetic fields are
always directed at right angles to each other and to the direction of propagation of the wave.  In other
words the fields are transverse, and light is described as a transverse wave. (By contrast sound
waves are said to be longitudinal, because the oscillations of the particles are parallel to the direction
of propagation.)   Since both the directions and the magnitudes of the electric and magnetic fields in
a light wave are related in a fixed manner, it is sufficient to talk about only one of them, the usual
choice being the electric field.  Now although the electric field at any point in space must be
perpendicular to the wave velocity, it can still have many different directions; it can point in any
direction in the plane perpendicular to the wave's direction of travel.
Any beam of light can be thought of as a huge collection of elementary waves with a range of
different frequencies.  Each elementary wave has its own unique orientation of its electric field; it is
polarised (figure 6.1).  If the polarisations of all the elementary waves in a complex beam can be
made to have the same orientation all the time then the light beam is also said to be polarised.  Since
there is then a unique plane containing all the electric field directions as well as the direction of the
light ray, this kind of polarisation is also called plane polarisation.  It is also known as linear
polarisation.  However, the usual situation is that the directions of the electric fields of the
component wavelets are randomly distributed; in that case the resultant wave is said to be randomly
polarised or unpolarised.
L6:   Polarisation 85
Figure 6.1.  A polarised elementary wave
The picture shows a perspective plot of the instantaneous electric field vectors which all lie in the same plane
(shaded).  Every such elementary harmonic wave is plane polarised.
It is quite common to find partially polarised light which is a mixture of unpolarised
(completely random polarisations) and plane polarised waves, in which a significant fraction of  the
elementary waves have their electric fields oriented the same way.
Components of polarisation
Since electric field is a vector quantity it can be described in terms of components referred to a set of
coordinate directions.  In the case of polarised waves we can take any two perpendicular directions in
a plane perpendicular to the wave's direction of travel.   An electric field E which makes an angle a
with one of these directions can then be described completely as two components with values Ecosa
and Esina .  We can think of these components as two independent electric fields, each with its own
magnitude and direction, which are together equivalent in every respect to the original field.  So any
elementary wave can be regarded as a superposition of two elementary waves with perpendicular
polarisations.
x
y
E
E
is
equivalent
to
y
x
E
a
Figure 6.2.  Components of the instantaneous electric field
In just the same way, any plane polarisation can be described in terms of two mutually
perpendicular component polarisations.  In the schematic diagrams that we use to represent
polarisation such a line can be drawn as a double-headed arrow, representing the two opposite
directions that a plane polarised wave can have at any point.  The instantaneous value of an electric
field (which has a unique direction at any instant of time) will be shown as a single-headed arrow.
L6:   Polarisation 86
y
x
is
equivalent
to
Polarisation
Component
polarisations
a
Figure 6.3  Components of the polarisation
Polarisers and Malus's law
A ideal polariser, or polarising filter, turns unpolarised light into completely plane polarised light.
Its action can be described in terms of its effect on elementary waves with different polarisations;
waves whose polarisation is parallel to an axis in the polariser, called its polarising axis, are
transmitted without any absorption but waves whose polarisation is perpendicular to the polarising
axis are completely absorbed.  An elementary wave whose polarisation is at some other angle to the
polarising axis is partly transmitted and partly absorbed but it emerges from the other side of the
polariser with a new polarisation, which is parallel to the polariser's axis.  This can be described in
terms of components of the original wave.  Since we can use any reference directions for taking
components we choose one direction parallel to the polariser's axis and the other one perpendicular
to it.  If the angle between the original polarisation and the polariser's axis is ?, then the component
parallel the the polariser's axis, which gets through, has an amplitude of E
0
 cos?.  Since the other
component is absorbed, the wave which emerges has a new amplitude E
0
 cos? and a new
polarisation.   Since the irradiance or "intensity" of light is proportional to the square of the electric
field's amplitude,
I
out 
= I
in
 cos
2
?. ...(6.1)
This result is known as Malus's law.
?
Incident linearly
polarised light
Polarising axis
Emerging light is
polarised parallel to
the axis of the polariser
Polariser
Figure 6.4.  Effect of a polariser on plane polarised light
Many practical polarisers do not obey Malus's law exactly, firstly because they absorb some
of the component with polarisation parallel to the polarising axis and secondly because some of the
component polarised perpendicular to the axis is not completely absorbed.
Malus's law also describes the action of an ideal polariser on unpolarised light.  Unpolarised
light is really a vast collection of polarised elementary waves whose polarisations are randomly
L6:   Polarisation 87
spread over all directions perpendicular to the wave velocity.  Since these elementary waves are not
coherent, their intensities, rather than their amplitudes, can be added, so Malus's law works for each
elementary wave.  To work out the effect of the polariser on the whole beam of unpolarised light we
take the average value of I
in
 cos
2
?  over all possible angles, which gives
I
out 
= 
1
2
  I
in
 .
Polarising axis
Emerging light is
polarised parallel to
the axis of the polariser
Polariser
Incident unpolarised light
Figure 6.5.  Effect of an ideal polariser on unpolarised light
If we send initially unpolarised light through two successive polarisers, the irradiance
(intensity) of the light which comes out depends on the angle between the axes of the two polarisers.
If one polariser is kept fixed and the axis of the other is rotated, the irradiance of the transmitted light
will vary.  Maximum transmission occurs when the two polarising axes are parallel.  When the
polarising axes are at right angles to each other the polarisers are said to be crossed and the
transmitted intensity is a minimum.   A pair of crossed ideal polarisers will completely absorb any
light which is directed through them (figure 6.6).  Note that the polarisation of the light which comes
out is always parallel to the polarising axis of the last polariser.
Figure 6.6.  Crossed polarisers
Each polariser on its own transmits half the incident irradiance of the unpolarised light.
So far we have considered a polariser as something which produces polarised light.  It can
also be considered as a device for detecting polarised light.  When it is used that way it may be
called an analyser.  For example, in the case of crossed polarising filters above, you can think of the
first filter as the polariser, which makes the polarised light, and the second filter as the analyser
which reveals the existence of the polarised light as it is rotated.
Page 5


84
L6 POLARISATION
 OBJECTIVES
Aims
Once you have studied this chapter you should understand the concepts of transverse waves, plane
polarisation, circular polarisation and elliptical polarisation.  You should be able to relate this
understanding to a knowledge of methods for producing the different types of polarised light in
sufficient detail so that you can explain the basic principles of those methods.
Minimum learning goals
1. Explain, interpret and use the terms:
polarised light, unpolarised light, randomly polarised light, linear polarisation (plane
polarisation), partially polarised light, polarising axis, polariser, ideal polariser, analyser,
crossed polarisers, Malus's law, circular polarisation, elliptical polarisation, birefringence
(double refraction), dichroic material, dichroism, optical activity, quarter-wave plate,
polarising angle (Brewster angle).
2. Describe how plane polarised light can be produced by dichroic materials, by birefringent
materials, by reflection and by scattering. 
3. State and apply Malus's law.
4. Explain how circularly or elliptically polarised light can be regarded as a superposition of
plane polarisations.
5. Describe how circularly polarised light can be produced from unpolarised or plane polarised
light.
6. Describe the phenomenon of optical activity and describe one example of its application.
Extra Goals
7. Describe and discuss various applications of polarised light and explain how they work.
 TEXT
6-1 PLANE OR LINEAR POLARISATION
In light and all other kinds of electromagnetic waves, the oscillating electric and magnetic fields are
always directed at right angles to each other and to the direction of propagation of the wave.  In other
words the fields are transverse, and light is described as a transverse wave. (By contrast sound
waves are said to be longitudinal, because the oscillations of the particles are parallel to the direction
of propagation.)   Since both the directions and the magnitudes of the electric and magnetic fields in
a light wave are related in a fixed manner, it is sufficient to talk about only one of them, the usual
choice being the electric field.  Now although the electric field at any point in space must be
perpendicular to the wave velocity, it can still have many different directions; it can point in any
direction in the plane perpendicular to the wave's direction of travel.
Any beam of light can be thought of as a huge collection of elementary waves with a range of
different frequencies.  Each elementary wave has its own unique orientation of its electric field; it is
polarised (figure 6.1).  If the polarisations of all the elementary waves in a complex beam can be
made to have the same orientation all the time then the light beam is also said to be polarised.  Since
there is then a unique plane containing all the electric field directions as well as the direction of the
light ray, this kind of polarisation is also called plane polarisation.  It is also known as linear
polarisation.  However, the usual situation is that the directions of the electric fields of the
component wavelets are randomly distributed; in that case the resultant wave is said to be randomly
polarised or unpolarised.
L6:   Polarisation 85
Figure 6.1.  A polarised elementary wave
The picture shows a perspective plot of the instantaneous electric field vectors which all lie in the same plane
(shaded).  Every such elementary harmonic wave is plane polarised.
It is quite common to find partially polarised light which is a mixture of unpolarised
(completely random polarisations) and plane polarised waves, in which a significant fraction of  the
elementary waves have their electric fields oriented the same way.
Components of polarisation
Since electric field is a vector quantity it can be described in terms of components referred to a set of
coordinate directions.  In the case of polarised waves we can take any two perpendicular directions in
a plane perpendicular to the wave's direction of travel.   An electric field E which makes an angle a
with one of these directions can then be described completely as two components with values Ecosa
and Esina .  We can think of these components as two independent electric fields, each with its own
magnitude and direction, which are together equivalent in every respect to the original field.  So any
elementary wave can be regarded as a superposition of two elementary waves with perpendicular
polarisations.
x
y
E
E
is
equivalent
to
y
x
E
a
Figure 6.2.  Components of the instantaneous electric field
In just the same way, any plane polarisation can be described in terms of two mutually
perpendicular component polarisations.  In the schematic diagrams that we use to represent
polarisation such a line can be drawn as a double-headed arrow, representing the two opposite
directions that a plane polarised wave can have at any point.  The instantaneous value of an electric
field (which has a unique direction at any instant of time) will be shown as a single-headed arrow.
L6:   Polarisation 86
y
x
is
equivalent
to
Polarisation
Component
polarisations
a
Figure 6.3  Components of the polarisation
Polarisers and Malus's law
A ideal polariser, or polarising filter, turns unpolarised light into completely plane polarised light.
Its action can be described in terms of its effect on elementary waves with different polarisations;
waves whose polarisation is parallel to an axis in the polariser, called its polarising axis, are
transmitted without any absorption but waves whose polarisation is perpendicular to the polarising
axis are completely absorbed.  An elementary wave whose polarisation is at some other angle to the
polarising axis is partly transmitted and partly absorbed but it emerges from the other side of the
polariser with a new polarisation, which is parallel to the polariser's axis.  This can be described in
terms of components of the original wave.  Since we can use any reference directions for taking
components we choose one direction parallel to the polariser's axis and the other one perpendicular
to it.  If the angle between the original polarisation and the polariser's axis is ?, then the component
parallel the the polariser's axis, which gets through, has an amplitude of E
0
 cos?.  Since the other
component is absorbed, the wave which emerges has a new amplitude E
0
 cos? and a new
polarisation.   Since the irradiance or "intensity" of light is proportional to the square of the electric
field's amplitude,
I
out 
= I
in
 cos
2
?. ...(6.1)
This result is known as Malus's law.
?
Incident linearly
polarised light
Polarising axis
Emerging light is
polarised parallel to
the axis of the polariser
Polariser
Figure 6.4.  Effect of a polariser on plane polarised light
Many practical polarisers do not obey Malus's law exactly, firstly because they absorb some
of the component with polarisation parallel to the polarising axis and secondly because some of the
component polarised perpendicular to the axis is not completely absorbed.
Malus's law also describes the action of an ideal polariser on unpolarised light.  Unpolarised
light is really a vast collection of polarised elementary waves whose polarisations are randomly
L6:   Polarisation 87
spread over all directions perpendicular to the wave velocity.  Since these elementary waves are not
coherent, their intensities, rather than their amplitudes, can be added, so Malus's law works for each
elementary wave.  To work out the effect of the polariser on the whole beam of unpolarised light we
take the average value of I
in
 cos
2
?  over all possible angles, which gives
I
out 
= 
1
2
  I
in
 .
Polarising axis
Emerging light is
polarised parallel to
the axis of the polariser
Polariser
Incident unpolarised light
Figure 6.5.  Effect of an ideal polariser on unpolarised light
If we send initially unpolarised light through two successive polarisers, the irradiance
(intensity) of the light which comes out depends on the angle between the axes of the two polarisers.
If one polariser is kept fixed and the axis of the other is rotated, the irradiance of the transmitted light
will vary.  Maximum transmission occurs when the two polarising axes are parallel.  When the
polarising axes are at right angles to each other the polarisers are said to be crossed and the
transmitted intensity is a minimum.   A pair of crossed ideal polarisers will completely absorb any
light which is directed through them (figure 6.6).  Note that the polarisation of the light which comes
out is always parallel to the polarising axis of the last polariser.
Figure 6.6.  Crossed polarisers
Each polariser on its own transmits half the incident irradiance of the unpolarised light.
So far we have considered a polariser as something which produces polarised light.  It can
also be considered as a device for detecting polarised light.  When it is used that way it may be
called an analyser.  For example, in the case of crossed polarising filters above, you can think of the
first filter as the polariser, which makes the polarised light, and the second filter as the analyser
which reveals the existence of the polarised light as it is rotated.
L6:   Polarisation 88
6-2 CIRCULAR POLARISATION
Plane polarisation is not the only way that a transverse wave can be polarised.  In circular
polarisation the electric field vector at a point in space rotates in the plane perpendicular to the
direction of propagation, instead of oscillating in a fixed orientation, and the magnitude of the electric
field vector remains constant.
Looking into the oncoming wave the electric field vector can rotate in one of two ways.  If it
rotates clockwise the wave is said to be right-circularly polarised and if it rotates anticlockwise the
light is left-circularly polarised.
0 T/4 T/8 T/2 3T/4 T 3T/8 5T/8 7T/8
Right
polarised
Left
polarised
Time
Figure 6.7.  Circularly polarised waves
The diagrams show the electric field vector of an elementary wave at successive time intervals of 1/8  of a wave
period, as the wave comes towards you.
Actually circular polarisation is not anything new.  A circularly polarised elementary wave can
be described as the superposition of two plane polarised waves with the same amplitude which are
out of phase by a quarter of a cycle (p/2) or three quarters of a cycle (3p/2).  Figure 6.8 shows how.
 + + + + +
= = = = = 
 t =3T/8 t = 0 t = T/8 t = T/4
 t = T/2
Figure 6.8  Circular polarisation as the superposition of two linear polarisations
The illustrations show the two linearly polarised electric fields with the same amplitude plotted
at intervals of one eighth of a wave period.  When these are combined the resultant electric field
vector always has the same magnitude, but its direction rotates.  Note that the amplitude of the
circularly polarised wave is equal to the amplitude of each of its linearly polarised components. Its
period and frequency are also identical with those of the component waves.
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