Post on 23-Mar-2020
16. More About Polarization
Polarization control
Wave plates
Circular polarizers
Waves with more complicated polarization
Reflection & polarization
Scattering & polarization
Birefringent materials have more than one refractive index
A special case of birefringence is a uniaxial crystal, where two of the three indices are the same.
A light wave with polarization along the optic axis experiences one value for n: the extraordinary index newhile orthogonal polarizations experience the other value: the ordinary index no.
Most materials are not birefringent. But sometimes birefringence can be induced.
Birefringence requires some form of anisotropy at the atomic or molecular level. This can be due to anisotropy in the electron binding, or to more macroscopic features.
Materials with random atomic or molecular structure (e.g., glasses, liquids, gases) do not exhibit birefringence.
However, external factors, such as applied mechanical stress, can lead to a net orientation, and therefore can induce birefringence in otherwise non-birefringent materials.
This can be observed when the object is placed between crossed polarizers. This is known as ‘stress birefringence’.
Input polarization state: 11 }
If both polarizations are present, this has the effect of changing the relative phase of the xand y fields, and hence altering the polarization.
Birefringence for polarization control
Input:
0
0
( , ) Re exp ( )
( , ) Re exp ( )
x
y
E z t E j kz t
E z t E j kz t
Suppose we illuminate a slab of birefringent material with a wave that has equal parts of ordinary and extraordinary polarization:
x
y Ein
optic axis
ˆ ˆinE E x E y
Wave plates
Output:
0
0
( , ) Re exp ( )
( , ) Re exp ( )
x
y
o
e
E z t E j kz t
E
kn d
kz n dt E j kz t
}
0
0
exp( )exp( )
12exp
o
e
o e
E jkn dE jkn d
j n n d
x
y Ein
thickness d
The output wave acquires a phase that is different for the two polarization components:
Here, is the wavelength in empty space.
A device that changes the polarization of a light wave in this manner is called a ‘wave plate’.
Wave plate output polarization state:
12exp
o ej n n d
A quarter-wave plate creates circular polarization from linear polarization, and a half-wave plate rotates 45° linear polarization to its orthogonal state.
(assuming 45-degree input polarization)
2 2 exp o e o en n d j n n d
output
polarization state
“Quarter-wave plate”
0 1 45° linear/2 j right circular
“Half-wave plate”
1 45° linear3/2 j left circular2 1 45° linear
We can add an additional 2m without changing the polarization, so the polarization cycles through this evolution as d increases further.
Wave plates (continued)
Half-Wave PlateWhen 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 optic axis, then the output polarization is rotated by 90°. If the incident polarization is parallel or perpendicular to the optic axis of the plate, then no polarization rotation occurs.
+45° polarization at input
-45° polarization at output
Vertical (green): 4 cycles
Horizontal (blue):3.5 cycles
Half-wave plate for arbitrary angle linear input polarization
Polarization state:
1tan }
0
0
( , ) Re cos exp ( )
( , ) Re sin exp ( )
x
y
E z t E j kz t
E z t E j kz t
x
y
input
o ek n n d For a half-wave plate,
so the output state is:
11 1
tantan tan
je
output
If the incident polarization is at an angle to the optic axis, then the output polarization remains linear, and is rotated to .
Circular polarizersUnpolarized input light
Circularly polarized light
linear polarizer
quarter-wave plate
A circular polarizer makes circularly polarized light by first linearly polarizing it and then rotating it to circular. This uses a linear polarizer followed by a quarter wave plate
Light beams can have complicated polarization dependence
An optical vortex
x
y
Azimuthal polarization
Radial polarization
Here are a few examples.
Polarization can also be different for different frequencies in a beam.
• white light is split into red, green, blue by dichroic mirrors• liquid crystal displays (LCDs) impose images on each of the three color beams• three colors recombined in a dichroic beam combiner• lens projects image onto external screen
How does the LCD projector work?
Blue light and red light are reflected.
Green light is transmitted.
This requires the green light to have a different polarization
from the red and blue components.
Dichroic beam combiner
Depolarization by reflection or transmissionSuppose that 45° polarization is incident on an interface, which has different parallel (x) and perpendicular (y) reflection coefficients.
x
y Incidentpolarization
Reflectedpolarization(if rx >ry)
Incident light fields:
0
0
( , ) Re exp ( )
( , ) Re exp ( )
x
y
E z t E j kz t
E z t E j kz t
Unless light is purely parallel or perpendicularly polarized (or incident at 0°), some polarization rotation will occur (also true for transmitted light).
Reflected light fields:
0
0
( , ) Re exp ( )
( , ) Re exp ( )y
x
y
xE z t E j kz t
E z
r
t E jr kz t
Most commonly encountered stuff depolarizes
Typical stuff, like a piece of clear plastic, is very non-uniform: a series of interfaces at random angles.
crossed polarizers
plastic baggie
Fresnel Reflection and DepolarizationFresnel reflections are a common cause of polarization rotation.
This effect is particularly strong near Bewster'sangle, but is notable over a wide range of angles.
For a wide range of angles, R is distinctly larger than R||.
Glare is polarized because of the Brewster angle effect
Window reflection viewed through polarizer that transmits
only s polarized light
Polarizing sunglasses transmit only vertically polarized light, because for objects like puddles on the ground or car windows, the glare is largely horizontally polarized (s polarized).
Window reflection viewed through polarizer that transmits
only p polarized light
Brewster's Angle RevisitedA trigonometric calculation reveals that the reflectioncoefficient for parallel-polarizedlight goes to zero for Brewster'sangle incidence, tan(i) = nt / ni
sin( ) sin( )i i t tn n
sin( ) sin(90 ) cos( )
i i t i
t i
n nn
tan( )i t in n
ni
nt
i i
t
i +t = 90°
Consider what happens when the reflected beam makes a right angle with the transmitted beam, and the polarization is parallel
No scattering can occur, due to the scattered dipole emission pattern.
But our right-angle assumption implies that i + t = 90°. So:
direction of motion of oscillating molecules at the surface (along the direction of the E-field in the transmitted beam)
Depolarization by unintended birefringence (polarization mode dispersion)
Imagine an optical fiber with just a tiny bit of birefringence, n, but over a distance of 1000 km…
Why is this bad?
Many fiber-optic systems detect only one polarization and so don’t see light whose polarization has been rotated by /2.
Worse, as the temperature changes, the birefringence changes, too.
12exp j n d
Distance
Polarization state at
receiver=
Because d is large, even n as small as 10-12 can rotate the polarization by 90º! (fiber systems typically use = 1.5 m)
Pipes containing fiber optic cables
Scattering by molecules is not sphericallysymmetric. It has a "dipole pattern."
The field emitted by an oscillating dipole excited by a verticallypolarized light wave:
Directions of scattered light E-field
No light is emitted along direction of oscillation!
Direction of light excitationE-field and electron oscillation
Emitted intensity pattern
Directions of scattered light E-field
Scattering of polarized light
No light is scattered along the input field direction, i.e. with kout parallel to Einput.
Vertically polarized input light
Horizontally polarized input light
Scattering of unpolarized lightAgain, no light is scattered along the input field direction, i.e. with kout parallel to Einput.
Vertically polarized
scattered light
Partially polarized
Unpolarized
Vertically polarized
Horizontally polarized
scattered light
Unpolarized input light
We should therefore expect the blue sky to be polarized in certain directions: in particular, at right angles to the direction of the sun.
Skylight is polarized in certain directions
This polarizer transmitshorizontal polarization(of which there is little).
In clouds, light is scattered multiple times. So the light emerging from a cloud has its polarization randomized.
Right-angle scattering is polarized
vertical polarizer horizontal polarizer
Don’t use a polarizer on a wide-angle lens.A polarizer on a wide-angle lens necessarily sees both polarized
and unpolarized regions of the sky.
It’s difficult to imagine this effect being useful!