Polarization

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Polarization
Linear, circular, and elliptical
polarization Mathematics of polarization Uniax
ial crystals Birefringence Polarizers
Prof. Rick Trebino Georgia Tech www.physics.gatech
.edu/frog/lectures
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45 Polarization
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Arbitrary-Angle Linear Polarization
Here, the y-component is in phase with the
x-component, but has different magnitude.
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The Mathematics of Polarization
Define the polarization state of a field as a 2D
vector Jones vectorcontaining the two complex
amplitudes For many
purposes, we only care about the relative
values (alternatively normalize this
vector to unity magnitude) Specifically
0 linear (x) polarization Ey /Ex 0
90 linear (y) polarization Ey /Ex
45 linear polarization Ey /Ex 1
Arbitrary linear polarization






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Circular (or Helical) Polarization
Or, more generally,
Here, the complex amplitude of the y-component is
-i times the complex amplitude of the
x-component. So the components are always 90
out of phase.
The resulting E-field rotates counterclockwise
around the k-vector (looking along k).
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Right vs. Left Circular (or Helical) Polarization
Or, more generally,
Here, the complex amplitude of the y-component is
i times the complex amplitude of the
x-component. So the components are always 90
out of phase, but in the other direction.
The resulting E-field rotates clockwise around
the k-vector (looking along k).
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Unequal arbitrary-relative-phase components yield
elliptical polarization
Or, more generally,
The resulting E-field can rotate clockwise or
counter-clockwise around the k-vector (looking
along k).
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The mathematics of circular andelliptical
polarization
Circular polarization has an imaginary Jones
vector y-component Right circular
polarization Left circular polarization Elli
ptical polarization has both real and imaginary
components We can calculate the
eccentricity and tilt of the ellipse if we feel
like it.
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When the phases of the x- and y-polarizations
fluctuate, we say the light is unpolarized.
where qx(t) and qy(t) are functions that vary on
a time scale slower than 1/w, but faster than you
can measure. The polarization state (Jones
vector) will be As long as the time-varying
relative phase, qx(t) - qy(t), fluctuates, the
light will not remain in a single polarization
state and hence is unpolarized.
In practice, the amplitudes vary, too!
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Light with very complex polarizationvs. position
is also unpolarized.

• Light that has passed through cruddy stuff is
  often unpolarized for this reason. Well see how
  this happens later.

The polarization vs. position must be
unresolvable, or else, we should refer to this
light as locally polarized.
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An intentionally spatially varying polarization
can be interesting.
Imagine a donut-shaped beam with radial
polarization.
Focus
Longitudinal electric field at the focus!
Lens
This type of beam can also focus to a smaller
spot than a beam with a spatially uniform
polarization!
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A complex polarization spatial dependence
An optical vortex
y
x
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Birefringence
The molecular "spring constant" can be different
for different directions.
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Birefringence
The x- and y-polarizations can see different
refractive index curves.
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Uniaxial crystals have an optic axis
Uniaxial crystals have one refractive index for
light polarized along the optic axis (ne) and
another for light polarized in either of the two
directions perpendicular to it (no). Light
polarized along the optic axis is called the
extraordinary ray, and light polarized
perpendicular to it is called the ordinary ray.
These polarization directions are the crystal
principal axes.
Light with any other polarization must be broken
down into its ordinary and extraordinary
components, considered individually, and
recombined afterward.
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Birefringence can separate the twopolarizations
into separate beams
Due to Snell's Law, light of different
polarizations will bend by different amounts at
an interface.
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Calcite
Calcite is one of the most birefringent materials
known.
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Birefringent Materials
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How to make a polarizer
Well use birefringence, and well take advantage
of the Fresnel equations we derived last time.
When ni gt nt, theres a steep variation in
reflectivity with incidence angle. This
corresponds to a steep variation with refractive
index.
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Polarizers take advantage of birefringence,
Brewster's angle, and total internal reflection.
The Glan-Thompson polarizer uses two prisms of
calcite (with parallel optical axes), glued
together with Canada balsam cement (n 1.55).
Alternatively, the optical axis points out of
the screen, and the polarizations are also
rotated.
Snells Law separates the beams at the entrance.
The perpendicular polarization then goes from
high index (1.66) to low (1.55) and undergoes
total internal reflection, while the parallel
polarization is transmitted near Brewster's
angle. If the gap is air, its called the
Glan-Taylor polarizer.
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Even better, rotate the second prism The Nicol
polarizer.
Combine two prisms of calcite, rotated so that
the ordinary polarization in the first prism is
extraordinary in the second (and vice versa). The
perpendicular polarization goes from high index
(no) to low (ne) and undergoes total internal
reflection, while the parallel polarization is
transmitted near Brewster's angle.
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Real Polarizers
Air-spaced polarizers
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Wollaston Polarizing Beam Splitter
The Wollaston polarizing beam splitter uses two
rotated birefringent prisms, but relies only on
refraction.
The ordinary and extraordinary rays have
different refractive indices and so diverge.
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The Pile-of-Plates Polarizer

• After numerous Brewster-angle transmissions,
  mainly the parallel polarization remains.

Unfortunately, only about 10 of the
perpendicular polarization is reflected on each
surface, so you need a big pile of plates!
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Dielectric polarizers
A multi-layer coating (which uses interference
well get to this later) can also act as a
polarizer. It still uses the same basic ideas of
TIR and Brewsters angle.
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Wire Grid Polarizer
Input light contains both polarizations
The light can excite electrons to move along the
wires, which then emit light that cancels the
input light. This cannot happen perpendicular to
the wires. Such polarizers work best in the
IR. Polaroid sheet polarizers use the same idea,
but with long polymers.
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Wire grid polarizer in the visible
Using semiconductor fabrication techniques, a
wire-grid polarizer was recently developed for
the visible.
The spacing is less than 1 micron.
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The wires need not be very long.
Hoya has designed a wire-grid polarizer for
telecom applications that uses small elongated
copper particles.
Extinction coefficient gt 10,000
Transmission gt 99
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The Measure of a Polarizer
The ideal polarizer will pass 100 of the desired
polarization and 0 of the undesired
polarization. It doesnt exist. The ratio of the
transmitted irradiance through polarizers
oriented parallel and then crossed is the
Extinction ratio or Extinction coefficient.
Wed like the extinction ratio to be infinity.
Type of polarizer Ext. Ratio Transmission
Cost Calcite 106 gt
95 1000 - 2000 Dielectric 103
gt 99 100 - 200 Polaroid sheet
103 50 1 - 2
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Visible wire-grid polarizer performance
A polarizers performance can vary with
wavelength and incidence angle.
The overall transmission is also important, as is
the amount of the wrong polarization in each beam.