Waves and Optics

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Chap. 5 Polarization
5.1 Nature of polarized light
5.1.1 Linear polarization
Fig. 5.1(a) shows an electromagnetic wave with
its electric field oscillating parallel to the
vertical y axis. The plane containing the
vectors is called the plane of oscillation or
vibration. We can represent the waves
polarization by showing the extent of the
electric field oscillations in a head-on view
of the plane of oscillation, as in Fig. 5.1 (b).
Fig. 5.1 (a) The plane of oscillation of a
polarized electromagnetic wave. (b) To represent
the ploarization, we view the plane of
oscillation head-on and indicate the amplitude
of the oscillating electric field.
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Consider two orthogonal optical disturbances
and
where is the relative phase difference
between the waves, both of which are traveling in
the z-direction.
Fig. 5.2 Linear light.
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The resultant optical disturbance is the vector
sum of these two perpendicular waves
If or is an integral multiple of
, the waves are said to be in phase. In that
particular case Eq. 5.3 becomes
Obviously, the resultant wave is also linearly
polarized, as shown in Fig. 5.2. A single
resultant electric-field oscillates, along a
tilted line, consinusoidally in time Fig. 5.2
(b). This process of addition can be carried out
equally well in reverse that is, we resolve any
plane-polarized wave into two orthogonal
components.
5.1.2 Circular polarization
Now we consider another particular case. That is,
, in addition,
where
Accordingly,
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and
The resultant wave is given by
as shown in Fig. 5.3.
Fig 5.3 Right-circular light.
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The resultant electric field vector is
rotating clockwise at an angular frequency of
. Such a wave is said to be right-circularly
polarized. In comparison, if
where then

The amplitude is unaffected, but now
rotates counter-clockwise, and the wave is
referred to as left-circularly polarized. A
linearly polarized wave can be synthesized from
two oppositely polarized circular waves of equal
amplitude. In particular, if we add Eq. 5.7 to
Eq. 5.8, we get a linearly polarized wave,
5.1.3 Elliptical polarization
Now we consider a general case. Recall that
and
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Expand the expression for into
and combine it with
to yield
It follows from Eq. 5.10 that
So Eq. 5.12 leads to
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Finally, on rearranging terms, we have
This is the equation of an ellipse making an
angle with the (Ex, Ey)-coordinate system
(Fig. 5.4) such that
Fig. 5.4 Elliptical light.
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If or equivalently
we have
familiar form
Furthermore, if
this can be reduced to
Clearly, it is a circle. If is an even
multiple of , Eq. 5.13 results in
And similarly for odd multiples of ,
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These are both straight lines having slopes of
in other words, we have linear
light. So, both linear and circular light may be
considered to be special cases of elliptically
polarized light. Fig. 5.5 gives various
polarization configurations. This very important
diagram is labeled across the bottom
leads by
where these are the positive values of
to be used in Eq. 5.2.
Fig. 5.5 (a) Various polarization
configurations. (b) . leads by
, or alternatively , leads
by .
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5.1.4 Natural light
The electromagnetic waves emitted from any
common source of light are polarized randomly or
unpolarized. That is, the electric field changes
directions randomly. We can use the mess like
that in Fig. 5.6 (a) to represent the unpolarized
light. In principle, we can simplify the mess by
resolving each electric field in Fig. 5.6 (a)
into y and z components and then finding the net
fields along the two directions. In doing
Fig. 5.6 (a) and (b) Two different drawings to
represent natural light.
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so, we mathematically change unpolarized light
into the superposition of two polarized waves
whose planes of oscillation are perpendicular to
each other. The result is the double-arrow
representation of Fig. 5.6 (b), which simplifies
drawing of unpolarized light. Actually, light is
generally neither completely polarized nor
completely unpolarized. More often, the electric
field vector varies in a way that is neither
totally regular nor totally irregular, and one
refers to such an optical disturbance as being
partially polarized. For this situation, we can
draw one of the arrows of the double-arrow
representation longer than the other arrow.

5.2 Polarizers
An optical device whose input is natural
light and whose output is some form of polarized
light is a polarizer.
Fig. 5.7 A linear polarizer.
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Fig. 5.7 shows a linear polarizer. Depending on
the form of the output, we could also have
circular or elliptical polarizers. Polarizing
direction of a linear polarizer An electric
filed component parallel to the polarizing
direction is passed (transmitted) by a polarizer
a component perpendicular to it is
absorbed. Intensity of transmitted light through
a linear polarizer When unpolarized light
reaches a polarizer, the intensity of the
emerging polarized light is then
Now we suppose that the light reaching a
polarizer is already polarized. Fig. 5.8 shows a
linear polarizer and the electric-field of
such a polarized light traveling toward the
polarizer. We can resolve into two
components relative to the polarizing direction
of the polarizer parallel component is
transmitted by the polarizer, and the
perpendicular component is absorbed. Since
is the angle between and the polarizing
direction, the transmitted parallel component is
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Recall that the intensity (irradiance) of an
electromagnetic wave is proportional to the
square of the electric-fields magnitude. In our
present case then, the intensity of the
emerging wave is proportional to and the
intensity of the original wave is
proportional to . Hence, from Eq. 5.20 we can
write
This is knows as Maluss law. We also call this
the cosine-sequared rule.
Fig. 5.8 Polarized light approaching a linear
polarizer.
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Fig. 5.9 shows an arrangement in which
initially unpolarized light is sent through two
polarizers and . (often, the first
polarizer is called the polarizer, and the second
the analyzer.) If their polarizing directions are
parallel, all the light passed by the first
polarizer is passed by the second polarizer. If
those directions are perpendicular (the
polarizers are said to be crossed), no light is
passed by the second polarizer.
Finally, if the two polarizing directions
make an angle between and , some of the
light transmitted by will be transmitted by
. The intensity of that light is determined
by Eq. 5.21. We can use the setup of Fig.
5.9 along with Maluss law to determine whether a
particular device is linear polarizer.
Fig. 5.9 A linear polarizer and analyzer.
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Fig. 5.10 Polarizing sunglasses consist of sheets
whose polarizing directions are vertical when the
sunglasses are worn. (a) Overlapping sunglasses
transmit light fairly well when their polarizing
directions have the same orientation, but (b)
they block most of the light when ther are
crossed.
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5.3 Birefringence 5.3.1 birefringence
The refractive index n is described in Eq.
(3.44), where are the
angular frequency of the driving force and the
natural angular frequency of the electric dipole,
respectively. In a isotropic material, is a
constant along all directions. A material can
have an anisotropy in the bound force as shown in
Fig 5.11. It leads to different and
therefore different refractive index n along
different directions. If the stiffness of the
spring in x direction has one value while has
another value along any direction perpendicular
to x-direction, the material has two different
refractive indices and is said to be
birefringent. The x-direction is called the
optic axis. Any plane contains the optic axis is
called the principal section. The birefringent
property is determined by the materials atomic
structure.
x
Fig 5.11 mechanical model depicting a negative
charged shell bound to a positive nucleus by
pairs of springs having different stiffness
Fig 5.12 shows what we can see through a
calcite crystal (CaCO3) two images of an object.
One of the image is formed by the ordinary rays
(o-rays) with linear polarization perpendicular
to the principal section (fig 5.13). Another
image is formed by the extraordinary rays
(e-rays) with linear polarization parallel to the
principal section (Fig 5.14).
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Fig 5.12 Double image formed by a calcite crystal
Fig 5.13 An incident plane wave polarized
perpendicular to the principal section (o-ray)
Fig 5.15 wavelets within calcite for e-rays
Fig 5.14 An incident plane wave polarized
parallel to the principal section (e-ray)
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In the case of o-ray (Fig 5.13), the E-field and
the wavelets is everywhere normal to the optic
axis, they will expand into the crystal in all
directions with a speed as they would in an
isotropic medium. In contrast, in the case of
e-ray (Fig. 5.14), the E-field has a component
normal to the optic axis, as well as a component
parallel to it. Since the medium is
birefringent, light polarized parallel to the
optic axis propagates with a speed
in the case of calcite. Therefore the
wavelet will elongate in all directions normal to
the optic axis, forming wave fronts of ellipsoids
of revolution about the optic axis as shown in
Fig 5.15. The envelope of all the ellipsoidal
wavelets is still a plane wave parallel to the
incident wave (Fig 5.14). However, the beam
moves in a direction parallel to the lines
connecting the origin of each wavelet and the
point of tangency with the planar envelope.
Therefore the e-ray travels a different way in
the crystal, forming double image. Crystals
belonging to the hexagonal, tetragonal and
trigonal systems have their atoms arranged so
that light propagating in some direction will
encounter an asymmetric structure. Such
substances are optically anisotropic and
birefringent. The optic axis corresponds to a
direction
about which the atoms are arranged
symmetrically. Crystals like these, for which
there is only one such direction, are known as
uniaxial. o-ray is everywhere normal to the
optic axis, so it moves at a speed in all
directions, corresponding to .
e-ray has a speed only in the direction of
the optic axis and has a speed normal to the
optic axis, corresponding a refractive index
. Table 5.1 shows some uniaxial
crystals. Crystals with
Table 5.1 Refractive indices of some uniaxial
birefringent crystals ( 589.3 nm)
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Fig. 5.16 Wavelets in a negative uniaxial crysal
Fig. 5.17 Wavelets in a positive uniaxial crysal
are negative uniaxials, such as Calcite. The
ellipsoidal e-ray wavelets are out the circular
o-ray wavelets as shown in Fig. 5.16. Crystals
with are positive
uniaxials, such as quarts. The ellipsoidal e-ray
wavelets are enclosed within the circular o-ray
wavelets as shown in Fig 5.17. 5.3.2
Birefringent polarizers The Glan-Foucault
polarizer is composed of two pieces of calcite
triangles with a thin air gap in between as shown
in Fig 5.18. The optic axis is perpendicular to
the paper plane as being noted with dots. The
incoming ray strikes the surface normally, and E
can be resold into components that are either
completely parallel or perpendicular to the optic
axis. The two rays traverse the first calcite
section without any deviation . Arrange the
incident angle, , on the
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middle calcite-air boundary to be
. The
o-ray will be internally reflected, leaving the
e-ray out of the crystal. Therefore the
polarization direction of the light is parallel
to the optic axis. The Wollaston prism
(Fig 5.19) is a polarizing beam-splitter. It is
composed of two pieces of calcite or quartz.
crystals with their optic axes perpendicular to
each other. The o-ray and
Fig 5.19 The Wollaston prism
Fig 5.18 The Glan-Foucault prism
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e-ray interchange after the diagonal interface.
There, the e-ray becoms an o-ray, changing its
index accordingly. In calcite neltno, and the
emerging o-ray us bent towards the normal.
Similarly, the o-ray becomes an e-ray and is bent
away from the normal. Out of the prism, the
o-ray and e-ray are spited. 5.3.3 Retarders
Cut the calcite such that its optic axis is
parallel to front and back surfaces as shown in
Fig 5.20. If the E-field of an incident
monochromatic plane wave has components parallel
and perpendicular to the optic axis, two separate
plane waves will propagate through the crystal.
Since , and the e-wave will
move across the plate more rapidly than the
o-wave. After traversing a plate of thickness d
the resultant electromagnetic waves is the
superposition of the e- and o-waves, which now
have a relative phase difference of .
where is the wavelength in vacuum. The
polarization of the emergent light depends on the
amplitudes of the incoming orthogonal field
components and of course on .
Fig 5.20 A calcite plate cut parallel to the
optic axis
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• The half-wave plate

A retardation plate that introduces a
relative phase difference of 180o between the o-
and e-waves is known as the half-wave plate.
Suppose that the plane of vibration of an
incoming beam of linear light makes an angle
with the optical axis. As show in Fig. 5.21. In
a negative material the e-wave will have a higher
speed and a longer wavelength than the o-wave.
When the waves emerge from the plate there will
be a relative phase shift of 180o, with the
effect that E will have rotated 2 . Going
back to Fig. 5.5, it should be evident that a
half-wave plate will similarly flip elliptical
light. In addition, it will invert the
handedness of circular or elliptical light,
changing right to left and vice versa.
Evidently if the thickness of the material is
such that ,
where m is an integer, it will function as a
half-wave plate.
Fig. 5.21 A half-wave plate
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(B) The quarter-wave plate
The quarter-wave plate is an optic element that
introduces a relative phase shift of
between the o- and e-wave. From Fig. 5.5, a
quarter-wave plate will convert linear to
elliptical light and vice versa. When linear
light at 45o to the optical axis is incident on a
quarter-wave plate, its o- and e-components have
equal amplitudes. The out-going light will be a
circular light. Similarly, an incoming circular
beam will emerge linearly polarized. It should
be apparent that linear light incident parallel
or perpendicular to the optic axis will be
unaffected by the retardation plate. You cant
have a relative phase difference without having
two components. If the thickness of the material
is such that
, where m is an integer, it will function as a
quarter-wave plate.
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5.4 Faraday effect 5.4.1 The effect
In 1845, Michael Faraday found that the
plane of vibration of linearly polarized light
incident on a piece of glass rotated when a
strong magnetic field was applied in the
(d)
Fig. 5.22 A Faraday effect modulator..
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propagation direction. This is known as Faraday
effect. Fig. 5.22 shows the experimental setup of
Faraday effect.
The angle through which the plane of
vibration rotates is given by a empirically
determined expression
where is the static magnetic flux density
(usually in Gauss), is the length of medium
traversed (in cm), and is a factor of
proportionality known as the Verdet constant. The
Verdet constant for a particular medium varies
with both frequency and temperature. Table 5.1
lists Verdet constants of some materials.
Table 5.1 Verdet constants for some substances.
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5.4.2 The explanation
Fig. 5.23 The superposition of an R- and an
L-state
A incident linear wave can be represented as
a superposition of R- and L-states as shown in
Fig 5.23. An elastically bound electron in the
material will take on steady-state circular orbit
driven by the rotating E-field of the wave (while
the effect of the waves B-field is negligible).
The introduction of a large constant applied
magnetic field perpendicular to the plane of the
orbit will result in a radial force FM on the
electron. That force can point either toward or
away from the circles center, depending on the
handedness of the light and the direction of the
constant B-field. The total radial force (FM plus
the elastic restoring force) can therefore have
two different values and so too can the radius of
the orbit. Consequently, for a given magnetic
field there will be two possible values of the
electric dipole moment, the polarization , and
the permittivity, as well as two values of the
index of refraction, nR and nL. Suppose the
linear polarization is along the positive x-axis
at the entrance of the material
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After traveling through the sample with thickness
d, the R-state rotates an angle
clockwise and the L-state rotates an angle
anti-clockwise. The resultant linear
polarization direction rotates an angle
as shown in Fig. 5.24. Since

The difference nR-nL is proportional to the
applied B-filed. Therefore Eq. (5.24) is the
same as Eq. (5.23).
Fig. 5.24 The superposition of an R- and an
L-state after traveling a sample of thickness d.