Explain the change in intensity as polarized light passes
through a polarizing filter
Calculate the effect of polarization by reflection and
Brewster’s angle
Describe the effect of polarization by scattering
Explain the use of polarizing materials in devices such as
LCDs
Polarizing sunglasses are familiar to most of us. They have a
special ability to cut the glare of light reflected from water or
glass (Figure \(\PageIndex{1}\)). They have this ability because of
a wave characteristic of light called polarization. What is
polarization? How is it produced? What are some of its uses? The
answers to these questions are related to the wave character of
light.
Figure \(\PageIndex{1}\): These two photographs of a
river show the effect of a polarizing filter in reducing glare in
light reflected from the surface of water. Part (b) of this figure
was taken with a polarizing filter and part (a) was not. As a
result, the reflection of clouds and sky observed in part (a) is
not observed in part (b). Polarizing sunglasses are particularly
useful on snow and water. (credit a and credit b: modifications of
work by “Amithshs”/Wikimedia Commons)
Malus’s Law
Light is one type of
electromagnetic (EM) wave. EM waves are transverse
waves consisting of varying electric and magnetic fields that
oscillate perpendicular to the direction of propagation (Figure
\(\PageIndex{2}\)). However, in general, there are no specific
directions for the oscillations of the electric and magnetic
fields; they vibrate in any randomly oriented plane perpendicular
to the direction of propagation. Polarization is the attribute that
a wave’s oscillations do have a definite direction relative to the
direction of propagation of the wave. (This is not the same type of
polarization as that discussed for the separation of charges.)
Waves having such a direction are said to be polarized. For an EM
wave, we define the direction of polarization to be the direction
parallel to the electric field. Thus, we can think of the electric
field arrows as showing the direction of polarization, as in Figure
\(\PageIndex{2}\).
Figure \(\PageIndex{2}\): An EM wave, such as light, is
a transverse wave. The electric \(\overrightarrow{E}\) and magnetic
\(\overrightarrow{B}\) fields are perpendicular to the direction of
propagation. The direction of polarization of the wave is the
direction of the electric field.
To examine this further, consider the transverse waves in the
ropes shown in Figure \(\PageIndex{3}\). The oscillations in one
rope are in a vertical plane and are said to be vertically
polarized. Those in the other rope are in a horizontal plane and
are horizontally polarized. If a vertical slit is placed on the
first rope, the waves pass through. However, a vertical slit blocks
the horizontally polarized waves. For EM waves, the direction of
the electric field is analogous to the disturbances on the
ropes.
Figure \(\PageIndex{3}\): The transverse oscillations
in one rope (a) are in a vertical plane, and those in the other
rope (b) are in a horizontal plane. The first is said to be
vertically polarized, and the other is said to be horizontally
polarized. Vertical slits pass vertically polarized waves and block
horizontally polarized waves.
The Sun and many other light sources produce waves that have the
electric fields in random directions (Figure \(\PageIndex{1a}\)).
Such light is said to be unpolarized, because it is composed of
many waves with all possible directions of polarization. Polaroid
materials—which were invented by the founder of the Polaroid
Corporation, Edwin Land—act as a polarizing slit for light,
allowing only polarization in one direction to pass through.
Polarizing filters are composed of long molecules aligned in one
direction. If we think of the molecules as many slits, analogous to
those for the oscillating ropes, we can understand why only light
with a specific polarization can get through. The axis of a
polarizing filter is the direction along which the filter passes
the electric field of an EM wave.
Figure \(\PageIndex{4}\): The slender arrow represents
a ray of unpolarized light. The bold arrows represent the direction
of polarization of the individual waves composing the ray. (a) If
the light is unpolarized, the arrows point in all directions. (b) A
polarizing filter has a polarization axis that acts as a slit
passing through electric fields parallel to its direction. The
direction of polarization of an EM wave is defined to be the
direction of its electric field.
Figure \(\PageIndex{5}\) shows the effect of two polarizing
filters on originally unpolarized light. The first filter polarizes
the light along its axis. When the axes of the first and second
filters are aligned (parallel), then all of the polarized light
passed by the first filter is also passed by the second filter. If
the second polarizing filter is rotated, only the component of the
light parallel to the second filter’s axis is passed. When the axes
are perpendicular, no light is passed by the second filter.
Figure \(\PageIndex{5}\): The effect of rotating two
polarizing filters, where the first polarizes the light. (a) All of
the polarized light is passed by the second polarizing filter,
because its axis is parallel to the first. (b) As the second filter
is rotated, only part of the light is passed. (c) When the second
filter is perpendicular to the first, no light is passed. (d) In
this photograph, a polarizing filter is placed above two others.
Its axis is perpendicular to the filter on the right (dark area)
and parallel to the filter on the left (lighter area). (credit d:
modification of work by P.P. Urone)
Only the component of the EM wave parallel to the axis of a
filter is passed. Let us call the angle between the direction of
polarization and the axis of a filter θ. If the electric field has
an amplitude E, then the transmitted part of the wave has
an amplitude \(E\cos θ \) (Figure \(\PageIndex{6}\)). Since the
intensity of a wave is proportional to its amplitude squared, the
intensity I of the transmitted wave is related to the
incident wave by
\[I=I_0 \cos^2θ \label{Malus's
Law} \nonumber \]
where \(I_0\) is the intensity of the polarized wave before
passing through the filter. This equation is known as Malus’s
law.
Figure \(\PageIndex{6}\): A polarizing filter transmits
only the component of the wave parallel to its axis, reducing the
intensity of any light not polarized parallel to its
axis.
This Open
Source Physics animation helps you visualize the electric field
vectors as light encounters a polarizing filter. You can rotate the
filter—note that the angle displayed is in radians. You can also
rotate the animation for 3D visualization.
Example
\(\PageIndex{1}\): Calculating Intensity Reduction by a Polarizing
Filter
What angle is
needed between the direction of polarized light and the axis of a
polarizing filter to reduce its intensity by
90.0%?
Strategy
When the intensity is reduced by 90.0%, it is
10.0% or 0.100 times its original value. That is,
I=0.100I0. Using this
information, the equation
I=I0cos2θ can be
used to solve for the needed angle.
Solution
Solving Malus's law (Equation \ref{Malus's Law}) for \(\cos θ\)
and substituting with the relationship between I and
I0 gives
A fairly large angle between the direction of polarization and
the filter axis is needed to reduce the intensity to 10.0% of its
original value. This seems reasonable based on experimenting with
polarizing films. It is interesting that at an angle of 45°, the
intensity is reduced to 50% of its original value. Note that 71.6°
is 18.4° from reducing the intensity to zero, and that at an angle
of 18.4°, the intensity is reduced to 90.0% of its original value,
giving evidence of symmetry.
Exercise \(\PageIndex{1}\)
Although we
did not specify the direction in Example \(\PageIndex{1}\), let’s
say the polarizing filter was rotated clockwise by 71.6° to reduce
the light intensity by 90.0%. What would be the intensity reduction
if the polarizing filter were rotated counterclockwise by
71.6°?
Answer
also 90.0%
Polarization by Reflection
By now, you can probably guess that polarizing sunglasses cut
the glare in reflected light, because that light is polarized. You
can check this for yourself by holding polarizing sunglasses in
front of you and rotating them while looking at light reflected
from water or glass. As you rotate the sunglasses, you will notice
the light gets bright and dim, but not completely black. This
implies the reflected light is partially polarized and cannot be
completely blocked by a polarizing filter.
Figure \(\PageIndex{7}\) illustrates what happens when
unpolarized light is reflected from a surface. Vertically polarized
light is preferentially refracted at the surface, so the reflected
light is left more horizontally polarized. The reasons for this
phenomenon are beyond the scope of this text, but a convenient
mnemonic for remembering this is to imagine the polarization
direction to be like an arrow. Vertical polarization is like an
arrow perpendicular to the surface and is more likely to stick and
not be reflected. Horizontal polarization is like an arrow bouncing
on its side and is more likely to be reflected. Sunglasses with
vertical axes thus block more reflected light than unpolarized
light from other sources.
Figure \(\PageIndex{7}\): Polarization by reflection.
Unpolarized light has equal amounts of vertical and horizontal
polarization. After interaction with a surface, the vertical
components are preferentially absorbed or refracted, leaving the
reflected light more horizontally polarized. This is akin to arrows
striking on their sides and bouncing off, whereas arrows striking
on their tips go into the surface.
Since the part of the light that is not reflected is refracted,
the amount of polarization depends on the indices of refraction of
the media involved. It can be shown that reflected light is
completely polarized at an angle of reflection θb given
by
\[tan \, θ_b=\frac{n_2}{n_1} \nonumber \]
where n1 is the medium in which the incident and
reflected light travel and n2 is the index of refraction
of the medium that forms the interface that reflects the light.
This equation is known as Brewster’s law and θb is known
as Brewster’s angle, named after the nineteenth-century Scottish
physicist who discovered them.
This Open
Source Physics animation shows incident, reflected, and
refracted light as rays and EM waves. Try rotating the animation
for 3D visualization and also change the angle of incidence. Near
Brewster’s angle, the reflected light becomes highly polarized.
Example
\(\PageIndex{2}\): Calculating Polarization by Reflection
(a) At what
angle will light traveling in air be completely polarized
horizontally when reflected from water? (b) From glass?
Strategy
All we need to solve these problems are the indices of
refraction. Air has n1=1.00, water has
n2=1.333, and crown glass has n′2=1.520. The
equation \(tan \, θ_b=\frac{n_2}{n_1}\) can be directly applied to
find θb in each case.
Light reflected at these angles could be completely blocked by a
good polarizing filter held with its axis vertical. Brewster’s
angle for water and air are similar to those for glass and air, so
that sunglasses are equally effective for light reflected from
either water or glass under similar circumstances. Light that is
not reflected is refracted into these media. Therefore, at an
incident angle equal to Brewster’s angle, the refracted light is
slightly polarized vertically. It is not completely polarized
vertically, because only a small fraction of the incident light is
reflected, so a significant amount of horizontally polarized light
is refracted.
Exercise \(\PageIndex{2}\)
What happens
at Brewster’s angle if the original incident light is already 100%
vertically polarized?
Answer
There will be only refraction but no reflection.
Atomic Explanation of Polarizing Filters
Polarizing filters have a polarization axis that acts as a slit.
This slit passes EM waves (often visible light) that have an
electric field parallel to the axis. This is accomplished with long
molecules aligned perpendicular to the axis, as shown in Figure
\(\PageIndex{8}\).
Figure \(\PageIndex{8}\): Long molecules are aligned
perpendicular to the axis of a polarizing filter. In an EM wave,
the component of the electric field perpendicular to these
molecules passes through the filter, whereas the component parallel
to the molecules is absorbed.
Figure \(\PageIndex{9}\) illustrates how the component of the
electric field parallel to the long molecules is absorbed. An EM
wave is composed of oscillating electric and magnetic fields. The
electric field is strong compared with the magnetic field and is
more effective in exerting force on charges in the molecules. The
most affected charged particles are the electrons, since electron
masses are small. If an electron is forced to oscillate, it can
absorb energy from the EM wave. This reduces the field in the wave
and, hence, reduces its intensity. In long molecules, electrons can
more easily oscillate parallel to the molecule than in the
perpendicular direction. The electrons are bound to the molecule
and are more restricted in their movement perpendicular to the
molecule. Thus, the electrons can absorb EM waves that have a
component of their electric field parallel to the molecule. The
electrons are much less responsive to electric fields perpendicular
to the molecule and allow these fields to pass. Thus, the axis of
the polarizing filter is perpendicular to the length of the
molecule.
Figure \(\PageIndex{9}\): Diagram of an electron in a
long molecule oscillating parallel to the molecule. The oscillation
of the electron absorbs energy and reduces the intensity of the
component of the EM wave that is parallel to the
molecule.
Polarization by Scattering
If you hold your polarizing sunglasses in front of you and
rotate them while looking at blue sky, you will see the sky get
bright and dim. This is a clear indication that light scattered by
air is partially polarized. Figure \(\PageIndex{10}\) helps
illustrate how this happens. Since light is a transverse EM wave,
it vibrates the electrons of air molecules perpendicular to the
direction that it is traveling. The electrons then radiate like
small antennae. Since they are oscillating perpendicular to the
direction of the light ray, they produce EM radiation that is
polarized perpendicular to the direction of the ray. When viewing
the light along a line perpendicular to the original ray, as in the
figure, there can be no polarization in the scattered light
parallel to the original ray, because that would require the
original ray to be a longitudinal wave. Along other directions, a
component of the other polarization can be projected along the line
of sight, and the scattered light is only partially polarized.
Furthermore, multiple scattering can bring light to your eyes from
other directions and can contain different polarizations.
Figure \(\PageIndex{10}\): Polarization by scattering.
Unpolarized light scattering from air molecules shakes their
electrons perpendicular to the direction of the original ray. The
scattered light therefore has a polarization perpendicular to the
original direction and none parallel to the original
direction.
Photographs of the sky can be darkened by polarizing filters, a
trick used by many photographers to make clouds brighter by
contrast. Scattering from other particles, such as smoke or dust,
can also polarize light. Detecting polarization in scattered EM
waves can be a useful analytical tool in determining the scattering
source.
A range of optical effects are used in sunglasses. Besides being
polarizing, sunglasses may have colored pigments embedded in them,
whereas others use either a nonreflective or reflective coating. A
recent development is photochromic lenses, which darken in the
sunlight and become clear indoors. Photochromic lenses are embedded
with organic microcrystalline molecules that change their
properties when exposed to UV in sunlight, but become clear in
artificial lighting with no UV.
Liquid Crystals and Other Polarization Effects in
Materials
Although you are undoubtedly aware of liquid crystal displays
(LCDs) found in watches, calculators, computer screens, cellphones,
flat screen televisions, and many other places, you may not be
aware that they are based on polarization. Liquid crystals are so
named because their molecules can be aligned even though they are
in a liquid. Liquid crystals have the property that they can rotate
the polarization of light passing through them by 90°. Furthermore,
this property can be turned off by the application of a voltage, as
illustrated in Figure \(\PageIndex{11}\). It is possible to
manipulate this characteristic quickly and in small, well-defined
regions to create the contrast patterns we see in so many LCD
devices.
In flat screen LCD televisions, a large light is generated at
the back of the TV. The light travels to the front screen through
millions of tiny units called pixels (picture elements). One of
these is shown in Figure \(\PageIndex{11}\). Each unit has three
cells, with red, blue, or green filters, each controlled
independently. When the voltage across a liquid crystal is switched
off, the liquid crystal passes the light through the particular
filter. We can vary the picture contrast by varying the strength of
the voltage applied to the liquid crystal.
Figure \(\PageIndex{11}\): (a) Polarized light is
rotated 90° by a liquid crystal and then passed by a polarizing
filter that has its axis perpendicular to the direction of the
original polarization. (b) When a voltage is applied to the liquid
crystal, the polarized light is not rotated and is blocked by the
filter, making the region dark in comparison with its surroundings.
(c) LCDs can be made color specific, small, and fast enough to use
in laptop computers and TVs.
Many crystals and solutions rotate the plane of polarization of
light passing through them. Such substances are said to be
optically active. Examples include sugar water, insulin, and
collagen (Figure \(\PageIndex{11}\)). In addition to depending on
the type of substance, the amount and direction of rotation depend
on several other factors. Among these is the concentration of the
substance, the distance the light travels through it, and the
wavelength of light. Optical activity is due to the asymmetrical
shape of molecules in the substance, such as being helical.
Measurements of the rotation of polarized light passing through
substances can thus be used to measure concentrations, a standard
technique for sugars. It can also give information on the shapes of
molecules, such as proteins, and factors that affect their shapes,
such as temperature and pH.
Figure \(\PageIndex{11}\). Optical activity is the
ability of some substances to rotate the plane of polarization of
light passing through them. The rotation is detected with a
polarizing filter or analyzer.
Glass and plastic become optically active when stressed: the
greater the stress, the greater the effect. Optical stress analysis
on complicated shapes can be performed by making plastic models of
them and observing them through crossed filters, as seen in Figure
\(\PageIndex{12}\). It is apparent that the effect depends on
wavelength as well as stress. The wavelength dependence is
sometimes also used for artistic purposes.
Figure \(\PageIndex{13}\): Optical stress analysis of a
plastic lens placed between crossed polarizers. (credit:
“Infopro”/Wikimedia Commons)
Another interesting phenomenon associated with polarized light
is the ability of some crystals to split an unpolarized beam of
light into two polarized beams. This occurs because the crystal has
one value for the index of refraction of polarized light but a
different value for the index of refraction of light polarized in
the perpendicular direction, so that each component has its own
angle of refraction. Such crystals are said to be birefringent,
and, when aligned properly, two perpendicularly polarized beams
will emerge from the crystal (Figure \(\PageIndex{14}\)).
Birefringent crystals can be used to produce polarized beams from
unpolarized light. Some birefringent materials preferentially
absorb one of the polarizations. These materials are called
dichroic and can produce polarization by this preferential
absorption. This is fundamentally how polarizing filters and other
polarizers work.
Figure \(\PageIndex{14}\): Birefringent materials, such
as the common mineral calcite, split unpolarized beams of light
into two with two different values of index of
refraction.