PHYS 415: OPTICS Polarization (from Trebino

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PHYS 415 OPTICSPolarization (from Trebinos
lectures)
F. ÖMER ILDAY Department of Physics, Bilkent
University, Ankara, Turkey
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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 and elliptical
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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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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Polarizers take advantage of birefringence,
Brewster's angle, and total internal reflection.
Heres one approach 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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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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Dielectric polarizers
A multi-layer coating (which uses interference
well get to this later) can also act as a
polarizer.
Melles-Griot catalog
Glass
Ealing Optics catalog
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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 perpen- dicular
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 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 Cost
Calcite 106 1000 - 2000
Dielectric 103 100 - 200
Polaroid sheet 103 1 - 2
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Wave plates

• When a beam propagates through a birefringent
  medium, one polarization sees more phase delay
  than the other.
• This changes the relative phase of the x and y
  fields, and hence changing the polarization.

Polarization state

Input

Output
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Wave plates (continued)

• Wave plate output polarization state

(45-degree input polarization)
Quarter-wave plate
Half-wave plate
A quarter-wave plate creates circular
polarization, and a half-wave plate rotates
linear polarization by 90?. We can add an
additional 2mp without changing the polarization,
so the polarization cycles through this evolution
as d increases further.
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Half-wave plate
When a beam propagates through a half-wave plate,
one polarizationexperiences half of a wavelength
more phase delay than the other.

• If the incident polarization is 45 to the
  principal axes, then the output polarization is
  rotated by 90 to -45.

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Wave plates and input polarization
Remember that our wave plate analysis assumes 45
input polarization relative to its principal
axes. This means that either the input
polarization is oriented at 45, or the wave
plate is.
0 or 90 Polarizer
45 Polarizer
Wave plate w/ axes at 0 or 90
Wave plate w/ axes at 45
If a HWP, this yields 90 or 0 polarization. If
a QWP, this yields circular polarization.
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How not to use a wave plate
If the input polarization is parallel to the wave
plate principal axes, no polarization rotation
occurs!
0 or 90 Polarizer
45 Polarizer
Wave plate w/ axes at 0 or 90
Wave plate w/ axes at 45
This arrangement can, however, be useful. In
high-power lasers, we desire to keep the laser
from lasing and then abruptly allow it to do so.
In this case, we switch between this and the
previous case.
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Thickness of wave plates
When a wave plate has less than 2p relative phase
delay, we say its a zero-order wave plate.
Unfortunately, they tend to be very thin. Solve
for d to find the thickness of a zero-order
quarter-wave plate
Using green light at 500 nm and quartz, whose
refractive indices are ne no 1.5534 1.5443
0.0091, we find
d 13.7 mm
This is so thin that it is very fragile and very
difficult to manufacture.
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Multi-order wave plates
A multi-order wave plate has more than 2p
relative phase delay. We can design a
twentieth-order quarter-wave plate with 20¼ waves
of relative phase delay, instead of just ¼
d 561 mm
This is thicker, but its now 41 times more
wavelength dependent! Its also temperature
dependent due to ns dependence on temperature.
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A thick zero-order wave plate
The first plate is cut with fast and slow axes
opposite to those of the second one. The Jones
vector becomes

Now, as long as d1 d2 is equal to the thin
zero-order wave plate, this optic behaves like
the prohibitively thin one! This is ideal.
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Polarization Mode Dispersion plagues broadband
optical-fiber communications.
Imagine just a tiny bit of birefringence, Dn, but
over a distance of 1000 km
If l 1.5 mm, then Dn 10-12 can rotate the
polarization by 90º!
Newer fiber-optic systems detect only one
polarization and so dont see light whose
polarization has been rotated to the other.
Worse, as the temperature changes, the
birefringence changes, too.
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Circular polarizers

• A circular polarizer makes circularly polarized
  light by first linearly polarizing it and then
  rotating it to circular. This involves a linear
  polarizer followed by a quarter wave plate

Unpolarized input light
45 Polarizer
QWP
Circularly polarized light
45 polarized light
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Polarization Spectroscopy
The 45-polarized Pump pulse re-orients
molecules, which induces some birefringence into
the medium, which then acts like a wave plate for
the Probe pulse until the molecules re-orient
back to their initial random distribution.