Chapter XIX Polarization of light
Interference and diffraction of light are arguments for wave
charactiristics of light.
We know that there are two types of wave processes: transverse &
longitutional waves →to what are light waves belong ?
Study of polarization of light makes clear that
This conclusion is in according to the concept that light waves are
electromagnetic waves with a definte band of frequencies. Recall that
electromagnetic waves are transervse waves in which e-vectors &
m-vectors oscillate in such follwing directions...
GENERAL PHYSICS III
Optics
&
Quantum Physics
Chapter XIX
Polarization of light
§1. Natural and polarized light. Malus’s law
§2. Polarization in reflection and refraction
§3. Polarization by birefringent
§4. Rotation of polarizing plane
Interference and diffraction of light are arguments for wave
charactiristics of light.
We know that there are two types of wave processes: transverse &
longitutional waves → to what are light waves belong ?
Study of polarization of light makes clear that
light waves is transverse waves
This conclusion is in according to the concept that light waves are
electromagnetic waves with a definte band of frequencies. Recall that
electromagnetic waves are transervse waves in which e-vectors &
m-vectors oscillate in such follwing directions
x
z
y
§1. Natural and polarized light:
1.1 Definitions:
• Light waves emmitted from natural
sources are from excited molecules
or atoms.
• The light waves emitted by any
individual molecule can be linearly Natural (unpolarized) light ray
polarized (definite in direction),
but any natural light source consists of a very large number of molecules
with random orientation → natural light is a random mixture of linearly
polarized waves with all possible transvere directions.
• Definitons:
Polarized light → light waves that have oscillation direction oriented
in any definite way
Linearly (or planely) polarized light → oscillation of light vector ( E )
is in one plane only
Oscillation Light ray
Oscillation plane → the plane in which plane
the light vector E oscillates
Polarization
Polarization plane → the plane that plane
is perpendicular to osc. plane
(it is used rarely)
1.2 Polarizer:
How can make a polarized light beam from natural light ?
→ using a polarizer
There are some crystal materials that have dichroism, that is, a
selective absorption: one of components of the light vector is
absorbed much more strongly than the other
Polarizing axis Polarizing axis
a) b)
Therefore if a polarizer (polaroid) is illuminated by natural light →
after polarizer there will be a linearly polarized light.
The direction of E vector of transmitted light is parallel to the
direction called polarizing axis of polarizer.
A||=A.cos
1.3 Malus’s law:
A
Now we are interested in the
light intensity after a polarizer, Polarizer
if the incident light is natural:
• Before P → I ~ A2
• After P → I’ = < I cos2> = I/2 (the average over all equi-probability
direction)
( : the angle between the incident and transmitted light vectors)
If we rotate the polarizer around the propagation direction → we
don’t observe any variation in light intensity.
What happens when the I
linearly polarized light I0
emerging from a polarizer
passes through a second
polaroid ? Analyzer
Then I = I0 cos2
Polarizer
This is Malus’s law
Intensity of the the angle between
:
transmitted light two polarizing axises
Intensity of the incident
linearly polarized light of polarizer and analyzer
• Rotate the second polaroid → varies → variation of light intensity I
is observed.
• The second polaroid helps us know about is the incident light either
natural or polarized → it is called analyzer.
1.4 Partially polarized light:
We have known about linearly polarized light in which the light vector
E oscillates in one definite direction.
There is the case of polarized light in which the light vector
E oscillates in different directions (that are all, of course,
perpendicular to the propagation direction), but one direction
predominates over all other → it is called partially polarized light.
It can be considered as a mixing of natural and linearly polarized light.
When a partially polarozed lighi transmits through an analyzer → the
light intensity varies from Imax to Imin . One cycle of analyzer rotation
the intensity has twice got maximum anf twice got minimum.
It is useful to introduce the degree of polarization P :
Imax - Imin
P=
Imax + Imin
The limit cases: Natural light → Imax = Imin → P = 0;
Linear polarized light → Imin = 0 → P = 1
1.5 Cicular and elliptical polarization:
Consider two coherent linearly polarized waves, and their oscillation
directions are perpendicular each to other: the first is along x-axis,
the second – y-axis (xOy plane the ray)
We have
The composition of two such waves is the
vector E whose arrowhead draws an ellipse.
→ then we have a polarized light wave in
which the oscilation plane rotates around
the propagation direction, and the head end
of the light vector moves describes an
ellipse. This light is called elliptically polarized.
Some limit cases:
• When phase difference equals m.→ the ellipse collapses
to a straight line, we have a linearly polarized light
• When = (2m + 1) ( and the amplitudes of two waves are equal
/2)
→ the ellipse becomes a circle, the light is circularly polarized
§2. Polarization in reflection and refraction:
Unpolarized light can be polarized, either partially or totally,
by reflection.
2.1 Polarization by reflection: Plane of incidence
Suppose that unpolarized natural
light is incident on a reflecting
surface between two transparent
materials. Then by experiment one
discovered that:
• Waves for which E the plane of
incidence are reflected more strongly
than waves for which E lies in this plane
• As a result, the reflected light is
partially polarized in the direction to
the plane of incidence.
• The level of polarization depends on the angle of incidence.
• Application: sunglasses that are made from polaroids can defend from
the light reflected from water or road surfaces.
2.2 Polarizing angle of incident (Brewster angle):
• For every reflecting surface there exists one definite angle of incidence,
at which a critical case takes place: the light waves E lies in the plane of
incidence is not reflected at all and the incident beam is completely
refracted. This special angle is called the polarizing angle .
p
• At the polarizing angle of incident :
p
The reflected light is linearly polarized p
and E the plane of incidence na
The refracted light is partially polarized
nb
and E lies in the plane of incidence
David Brewster discovered experimentally ’
that the reflected ray refraction ray
(this property can also be derived by
Maxwell’s theory of electromagnetic waves)
• This property gives us the way to determine :
p
na sin = nb sin(90o - ) = nb cos → tg = nb/na
p p p p Brewster’s law
§3. Polarization by birefringence:
3.1 Ordinary and extraordinary rays:
In some crystals there exists the birefringent
property, that is, when a light ray transmits
through crystal, it is separated into two parts:
One part satisfies the ordinary refraction law
(“o” ray), and the other does not (“e” ray).
Properties of the “e” ray:
• It does not lie in the plane of incidence
• For it the expression n1sini1/n2sini2 doesn’t remain constant
with variation of the angle of incidence
• Even for the case of the incident angle 90 o → the “e” ray deviates from
the incident direction (see picture).
The birefringent phenomenon appears in allmost any tranferent crystal,
besides of the crystals that are belong to the cubic system.
In some crystals there exists one direction the “o” and “e” rays are not
separated and have the same propagation velocity. This direction is called
optical axis of the crystal, and this type of crystals is called uniaxial
crystals. Any line parallel to this direction is also a optical axis.
Any plane passes the optical axis is called the principal section of
crystal. Usually, one is interested in the
principal section that passes the light ray.
3.2 Polarization of “o” and “e” rays: Optical
axis
Both the “o” and “e” rays are linearly polarized
The oscillation direction of E in the “o” ray is perpendicular to the
principal section, and oscillation of E in the “e” ray is in this section.
These properties of polarization are applied to make polarizers:
• Turmalin plate:
The turmalin crystal has the dichroism property, that is, the different
absorption for the “o” ray and “e” ray. The “o” ray is completely absorbed
in a path length ~ 1mm, and the “e” ray remains only after turmalin plate
→ a turmalin plate is used as a polarizer.
• Nicol prism:
William Nicol (1768-18570)
discovered a way to make
a polarizer, based on the
birefringence. Optical axis
• Two prisms are cut from a calcite crystal and polished to the form as
shown in picture. The two prisms then are glued together with a cement
called Canada balsam.
• The specific values of angles and the choice of the cement (the choice
of the index of refraction) are determined by the difference
between critical angle (for the complet reflection) of the “o” and
the “e” rays.
As shown in the picture, after Nicol prism one has a liearly polarized
light, it is the remained extraodinary ray.
§4.Birefringence in electric field – Kerr effect:
4.1 Birefringer in electric field:
The birefringence expresses the optically anisotropic property of
crystals. But the befringence appears also in isotropic materials in e-field.
John Kerr (1824-19070) observed that optically isotropic materials
exhibited uniaxial birefringent behavior when placed in an electric field.
• The central part, called Kerr cell, is a cuvette with liquid and a capacitor
• In the absence of electric field, the liquid is optically isotropic. If the
optical axis of the analyzer is cross with respect to that of the
polarizer → the light does not transmit through the system
• When an uniform electric field is applied between the capacitor plates,
the liquid behaves like an uniaxial crystal with the optical axis directed
along the electric field → the birefringence appears
• After the capacitor the light is elliptically polarized: It is the
composition of o and e rays - two waves that have perpendicular
oscillation directions.
By experimental measurement one obtains the following equations:
• The difference in the index of refraction of o & e rays:
• The path difference of the two rays:
and the corresponding phase difference:
• The last equation can be written in the form:
Kerr constant
The Kerr constant B is characterictic of a material,
B depends also on temperature and the wave length
4.2 Rotation of polarizing plane:
When a linear polarized light transmits through some materials, one
observed the rotation of the oscillation plane of E . Materials that have
such ability are called optically active.
Examples of such materials: some solid crystals (quartz), liquid
(turpentine, nicotin), solutions (sugar, wine acid),…
This effect is popular in sugar industry to measure the cyrup
concentration (sugar-meter).
• When the optical axis of the analyzer P’ is cross with respect to that
of the polarizer P → it is dark after P’.
• If the cylindrical vessel is filled with sugar solution → it becomes bright
after the analyzer
• Rotate the analyzer by an angle one makes dark again.
• By experiment one obtains the following law:
where l is the path length of light in the solution, c – the concentration
[ – coefficient that is called specific rotation constant. It is
]
characteristic of material, and depends on wavelength
• With this formula, one can determine the concentration of the solution
by measuring the angle by that the polarization plane was rotated.
Resume
Light is a transverse electromagnetic wave, and the oscillation direction
of the vector E determines polarization property of light:
• E oscillates in a plane (passing through the ray) → light is called
linearly (or planely, or completely) polarized
• E oscillates in many different directions, but one direction
predominates over all other → light is called partially polarized
• The arrowhead of E rotates and draws an ellipse (circular) →
light is called elliptically (circularly) polarized
Light reflected from a surface is mostly partially polarized. The critical
case when the angle of incidence is the polarizing angle , light is
p
completely linearly polarized. The polarizing angle is determined by
the formula tg = n /n
p b a
In birefrengent crystals, both the “o” and the “e” rays are linearly
polarized, for the “o” ray E the principal plane and for the “e” ray
E lies in this plane.
The birefringence expresses the optically anisotropic property of
material.
In isotropic materials, the birefringence can appear when an electric
field is applied.
In some materials that are called optically active, the polarizing plane
rotates. For solutions, the rotation angle depends on the concentration
and this effect can be applied as a way to determine the concentration
of solutions.
