PH 201

OPTICS & LASERS

Lecture_Polarization_3
 Production of Polarized Light
1. Wire Grid Polarizer & Polaroid
2. Polarization by Reflection
3. Polarization by Double Refraction
4. Polarization by Scattering
 Wire Grid Polarizer & Polaroid
 It consists of a large number of thin copper wires placed parallel to
one another. When an unpolarized em wave is incident on it, then
component of electric vector along length of wire is absorbed.

 This is so because electric field does work on electrons inside thin

wires, & energy associated with electric field is lost in Joule heating
of wires. On the other hand, since wires are assumed to be very
thin, component of electric vector along x axis passes through
without much attenuation.

 Emergent wave is linearly polarized with electric vector along x axis.

 However, for system to be effective (for Ey component to be almost

completely attenuated) spacing between wires should be <~ λ.

 Fabrication of such a polarizer for a 3 cm microwave is relatively

easy because spacing has to be <~ 3 cm.
Wire grid polarizer
 On the other hand, since light waves are associated with a very
small wavelength (~ 5 ×10-5 cm) fabrication of a polarizer in which
wires are placed at distances ~ 5 ×10-5 cm is extremely difficult.

 Bird & Parrish [1950] did succeed in putting about 30,000 wires in
about 1".

 It is extremely difficult to fabricate a wire grid polarizer which would

be effective for visible light.

 However, instead of long, thin wires, one may employ long chain
polymer molecules that contain atoms (such as iodine) which
provide high conductivity along length of chain.

 Long chain molecules are aligned so that they are almost parallel to
one another.

 Because of high conductivity provided by iodine atoms, electric field

parallel to molecules gets absorbed.
 A sheet containing such long chain polymer molecules (which are
aligned parallel to one another) is known as a Polaroid.

 When a light beam is incident on such a Polaroid, molecules

(aligned parallel to one another) absorb component of electric field
which is parallel to direction of alignment because of high
conductivity provided by iodine atoms; component perpendicular to
it passes through.

 Aligned conducting molecules act similar to wires in wire grid

polarizer, & since spacing between two adjacent long chain
molecules is small compared to optical wavelength, Polaroid is
usually very effective in producing linearly polarized light.

 Aligning of long chain conducting molecules is not very difficult.

 Polarization by Reflection
 Consider incidence of a plane wave on a dielectric assuming that
electric vector associated with incident wave lies in plane of
incidence as shown.

If a linearly polarized wave (with its E in plane of incidence) is

incident on interface of two dielectrics with angle of incidence equal
to θp [= tan-1 (n2/n1)], then reflection coefficient is zero.
 If angle of incidence θ is such that

θ = θp = tan-1(n2/n1)

then reflection coefficient is zero.

 If an unpolarized beam is incident at this angle, then reflected beam

will be linearly polarized with its electric vector perpendicular to
plane of incidence.

 Above Eq. is called Brewster’s law & at this angle of incidence,

reflected & transmitted rays are at right angles to one another;
angle θp is known as polarizing angle or Brewster angle.

 For air-glass interface, n1 = 1 & n2 ≈ 1.5, giving θp ≈ 57º. The

transmitted beam is partially polarized, & if one uses a large
number of reflecting surfaces, one obtains an almost plane
polarized transmitted beam.
If an unpolarized beam is incident with an angle of incidence θp, reflected
beam is plane polarized whose electric vector is perpendicular to plane of
incidence. Transmitted beam is partially polarized, & if this beam is made
to undergo several reflections, then emergent beam is almost plane
polarized with its electric vector in plane of incidence.
 For air-water interface,

n1 ≈ 1 & n2 ≈ 1.33 & polarizing angle θp ≈ 53º.

 If sunlight is incident on sea at an angle close to polarization angle,

then reflected light will be almost polarized.

 If we now view through a rotating Polaroid, sea will appear more

transparent when Polaroid blocks the reflected light.

 If Polaroid allows the (almost polarized) reflected beam to pass

through, we see the glare from water surface; the glare can be
blocked by using a vertical polarizer, & one can see inside of water.
If sunlight is incident on water surface at an angle close to polarizing
angle, then reflected light will be almost polarized.
(a) If Polaroid allows the (almost polarized) reflected beam to pass
through, we see glare from water surface.
(b) Glare can be blocked by using a vertical polarizer, & one can see
inside the water.
 Polarization by Double Refraction
 Double Refraction: When an unpolarized beam enters an anisotropic
crystal, it splits up into two beams, each being characterized by a
certain state of polarization.

 If, by some method, we could eliminate one of the beams, then we

would obtain a linearly polarized beam.

 Eliminating one of the beams is through selective absorption, this

property is known as dichroism.

 A crystal, tourmaline has different coefficients of absorption for the two

linearly polarized beams into which the incident beam splits up.

 Consequently, one of the beams gets absorbed quickly, & the other
component passes through without much attenuation.

 If an unpolarized beam is passed through a tourmaline crystal,

emergent beam will be linearly polarized.
When an unpolarized beam enters a dichroic crystal (tourmaline), it
splits up into two linearly polarized components. One of the
components gets absorbed quickly, & other component passes
through without much attenuation.
 Another method for eliminating one of the polarized beams is
through total internal reflection.

 The two beams have different velocities, & as such corresponding

refractive indices will be different.

 If one can sandwich a layer of a material whose refractive index lies

between the two, then for one of the beams, the incidence will be at
a rarer medium & for the other it will be at a denser medium.

 This principle is used in a Nicol prism which consists of a calcite

crystal cut in such a way that for the beam, for which the
sandwiched material is a rarer medium, the angle of incidence is
greater than the critical angle. Thus this particular beam will be
eliminated by total internal reflection.

 Ordinary ray undergoes total internal reflection & extraordinary

component passes through, & beam emerging from the crystal is
linearly polarized.
Nicol prism. Dashed outline corresponds to natural crystal which is
cut in such a way that ordinary ray undergoes total internal
reflection at Canada balsam layer.
 Double Refraction

 When an unpolarized light beam is incident normally on a calcite crystal, it

would in general, split up into two linearly polarized beams.

 Beam which travels undeviated is known as ordinary ray (o-ray) & obeys
Snell’s laws of refraction.

 2nd beam, which in general does not obey Snell’s laws, is known as
extraordinary ray (e-ray).

 Appearance of two beams is due to the phenomenon of double refraction,

& a crystal such as calcite is usually referred to as a double-refracting
crystal.
(a) When an unpolarized light beam is incident normally on a calcite
crystal, it would in general, split up into two linearly polarized beams.
(b) If we rotate crystal about NN‘, then e-ray will rotate about NN‘.
 If we put a Polaroid PP' behind calcite crystal & rotate Polaroid (about NN'),
then for two positions of Polaroid (when pass axis is perpendicular to plane
of paper) the e-ray will be completely blocked & only o-ray will pass
through.

 When pass axis of Polaroid is in plane of paper (i.e., along line PP'), then
o-ray will be completely blocked & only e-ray will pass through.

 If we rotate the crystal about NN', then e-ray will rotate about the axis.

A calcite crystal
showing double
refraction.
 Velocity of o-ray is same in all directions & velocity of e-ray is
different in different directions.

 Anisotropic substances: Calcite, Quartz

 Along a particular direction (fixed in the crystal), the two velocities

are equal; this direction is known as the optic axis of crystal.

 Uniaxial crystals: two rays have same speed only along one
direction (which is optic axis).

 Biaxial crystals: there may be two directions along which the two
rays have same speed.
 Velocities of ordinary & extraordinary rays are given by following equations:

c
vro  ordinary ray
n0
1 sin 2  cos 2 
  extraordin ary ray
vre (c / ne ) (c / no ) 2
2 2

where no & ne are constants of crystal & θ is the angle that the ray makes
with optic axis; we have assumed the optic axis to be parallel to z axis.

 Thus, c/no & c/ne are velocities of extraordinary ray when it propagates
parallel & perpendicular to optic axis.
 Equation of an ellipse (in zx plane) is given by:

z 2 x2
2
 2 1
a b
 If (ρ,θ) represents the polar coordinates, then z = ρcosθ & x = ρsinθ, &
equation of ellipse can be written in the form:

1 cos 2  sin 2 
  2
 2
a 2
b
 In three dimensions this Eq. will represent an ellipsoid of revolution with the
optic axis as the axis of revolution.

 Thus if we plot vre as a function of θ, we obtain an ellipsoid of revolution; on

the other hand, since vro is independent of θ, if we plot vro (as a function of
θ), we obtain a sphere. Along the optic axis, θ = 0 &
c
vro  vre 
n0
 Consider the value of vre perpendicular to optic axis (i.e., for θ = π/2).

 For a negative crystal, ne < no &

  c
vre      vro
 2  ne
 Minor axis will now be along optic axis, & ellipsoid of revolution will lie
outside the sphere.

(a) In a negative crystal, ellipsoid of revolution (which corresponds to

the extra ordinary ray) lies outside sphere; the sphere corresponds
to ordinary ray. (b) In a positive crystal, ellipsoid of revolution
(which corresponds to extraordinary ray) lies inside sphere.
 For a positive crystal, ne > no &

  c
vre      vro
 2  ne

 Major axis will now be along optic axis, & ellipsoid of revolution will lie
inside the sphere.

 Ellipsoid of revolution & sphere are known as ray velocity surfaces.

 Polarization by Scattering
 If an unpolarized beam is allowed to fall on a gas, then beam
scattered at 90° to incident beam is linearly polarized.

 This follows from the fact that the waves propagating in y direction
are produced by x component of dipole oscillations & y component
of dipole oscillations will produce no field in y direction.

 Clearly, if incident beam is linearly polarized with its electric vector

along x direction, then there will be no scattered light along x axis.

 Blue color of sky is due to Rayleigh scattering of sunlight by

molecules in our atmosphere.

 When sun is about to set, if we look vertically upward, light will have
a high degree of polarization; this is so because angle of scattering
will be very close to 90°.

 If we view blue sky (which is vertically above us) with a rotating

Polaroid, we will observe considerable variation of intensity.
If electromagnetic wave is propagating along z direction, then
scattered wave along any direction that is perpendicular to z axis
will be linearly polarized.
If a linearly polarized wave (with its E oscillating along x direction) is
incident on a dipole, then there will be no scattered wave in x
direction.
 Malus’ Law
 Consider a polarizer P1 which has a pass axis parallel to x axis; i.e., if
an unpolarized beam propagating in z direction is incident on
polarizer, then electric vector associated with the emergent wave will
oscillate along x axis.

 If polarizer is a Polaroid, then for pass axis to be along the x

direction, the long chain molecules must be aligned along the y axis.

 Consider incidence of the x-polarized beam on the Polaroid P2

whose pass axis makes an angle θ with the x axis.

 If amplitude of incident electric field is E0, then the amplitude of the

wave emerging from the Polaroid P2 will be E0cosθ , & thus intensity
of emerging beam will be
I = I0cos2θ

where I0 represents intensity of emergent beam when pass axis of P2

is along x axis (i.e., when θ = 0). This equation is called Malus’ law.
An unpolarized light beam gets x-polarized after passing through the
polaroid P1, the pass axis of the second polaroid P2 makes an angle
θ with the x axis. Intensity of emerging beam will vary as cos2θ.
 If a linearly polarized beam is incident on a Polaroid & if Polaroid is
rotated about z axis, then intensity of the emergent wave will vary
according to Malus’ law.

 If Polaroid P2 is rotated in clockwise direction, then intensity will

increase until pass axis is parallel to x axis; a further rotation will
result in a decrease in intensity until the pass axis is parallel to the
y axis, where intensity will be almost zero.

If we further rotate it, it will pass through a maximum & again a

minimum before it reaches its original position.