16. More About Polarization

Polarization control

Wave plates

Circular polarizers

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:

1

1

}

If both polarizations are present, this has the

effect of changing the relative phase of the x

and 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

ω

ω

Suppose we illuminate a slab of birefringent

material with a wave that has equal parts of

ordinary and extraordinary polarization:

x

yEin

optic axis

ˆ ˆinE E x E y= +r

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

ω

ω }

[ ]

0

0

exp( )

exp( )

1

2exp

o

e

o e

E jkn d

E jkn d

j n n d

→

− −

%

%

πλ

x

y Ein

thickness d

The output wave acquires a phase

that is different for the two polarization

components:

Here, λ 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: [ ]

1

2exp

− − o ej n n d

πλ

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

π πλ λ

output

polarization state

“Quarter-wave

plate”

0 1 45° linear

π/2 −j right circular

“Half-wave

plate”

π −1 −45° linear

3π/2 j left circular

2π 1 45° linear

We can add an additional 2mπ without changing the polarization, so the

polarization cycles through this evolution as d increases further.

Wave plates (continued)

Half-Wave Plate

When a beam propagates through a half-wave plate, one polarization

experiences 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 polarizationPolarization state:

1

tan

α}( ) [ ]{ }

( ) [ ]{ }0

0

( , ) Re cos exp ( )

( , ) Re sin exp ( )

= −

= −%

%

x

y

E z t E j kz t

E z t E j kz t

α ω

α ω

x

y

αinput

[ ]− =o ek n n d πFor a half-wave plate,

so the output state is:

( )11 1

tantan tan

= = −−

je π αα α−α

output

If the incident polarization is at an angle α to the optic axis,

then the output polarization remains linear, and is rotated to −α.

Circular polarizers

Unpolarized 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 transmission

Suppose that 45° polarization is incident on an interface, which has

different parallel (x) and perpendicular (y) reflection coefficients.

x

y Incident

polarization

Reflected

polarization

(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

ω

ω

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

= −

= −

ω

ω%

%

Most real stuff depolarizes

Real 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 Depolarization

Fresnel reflections are

a common cause of

polarization rotation.

This effect is particularly

strong near Bewster's

angle.

For a wide range of

angles, R > R||

Glare is polarized

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

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 kmF

Many fiber-optic systems detect only one polarization and so don’t see

light whose polarization has been rotated by π/2.

Worse, as the temperature changes, the birefringence changes, too.

1

2exp j n d

πλ

∆

Distance

Polarization

state at

receiver

=

Because d is large, even ∆n as small as 10-12 can rotate

the polarization by 90º! (recall: in fibers, λ = 1.5 µm)

Pipes containing

fiber optic cables

Scattering by molecules is not spherically

symmetric. It has a "dipole pattern."

The field emitted by an oscillating dipole excited by a vertically

polarized light wave:

Directions of scat-

tered light E-fieldDirections of scat-

tered light E-fieldNo light is emitted along

direction of oscillation!

Direction of light excitation

E-field and electron oscillation

Emitted intensity pattern

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 light

Again, 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 (at right angles to the sun).

Sun's

rays

Atmosphere

Skylight is polarized in certain

directions

This polarizer transmits

horizontal 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!

Brewster's Angle Revisited

A trigonometric calculation

reveals that the reflection

coefficient for parallel-polarized

light goes to zero for Brewster's

angle incidence, tan(θi) = nt / ni

sin( ) sin( )i i t tn nθ θ=

sin( ) sin(90 )

cos( )

i i t i

t i

n n

n

θ θ

θ

= −

=

o

tan( ) ti

i

n

nθ =

ni

nt

θi θi

θtθi +θt = 90°

When the reflected beam makes a

right angle with the transmitted beam,

and the polarization is parallel, then

no scattering can occur, due to the

scattered dipole emission pattern.

But our right-angle assumption

implies that θθθθi+ θθθθ

t= 90°. So:

direction of motion of

oscillating molecules at

the surface (along the

direction of the E-field in

the transmitted beam)