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!