16. More About Polarization

Polarization control

Wave plates

Circular polarizers

Reflection & polarization

Scattering & polarization

Active polarization control

Polarization state:

11 }

When a beam propagates through a birefringent medium, one polarization experiences more phase delay than the other.

If both polarizations are present, this has the effect of changing the relative phase of the xand y fields, and hence rotating the polarization.

Birefringence for polarization control

Suppose we illuminate a slab of birefringentmaterial with a wave that has equal parts of ordinary and extraordinary polarization:

x

y Ein

Input:

0

0

( , ) Re exp ( )

( , ) Re exp ( )

x

y

E z t E j kz t

E z t E j kz t

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

Azimuthalpolarization

Radial polarization

Here are a few examples.

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

Cruddy stuff depolarizes

Cruddy stuff is very non-uniform: a series of interfaces at random angles.

Crossed polarizers with a piece of wax paper in between.

Fresnel Reflection and DepolarizationFresnel reflections are a common cause of polarization rotation.

This effect is particularly strong near Bewster'sangle.

For angles between ~35° and ~85°, R > R||

Glare is polarizedWindow 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 km…

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! (recall: in fibers, = 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 scat-tered light E-field

Directions of scat-tered light E-field

No light is emitted along direction of oscillation!

Direction of light excitationE-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

unpolarized

polarized scattered

light!

polarized scattered

light!

Again, no light is scattered along the input field direction, i.e. with koutparallel to Einput.

We should therefore expect the blue sky to be polarized in certain

directions (at right angles to 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

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( ) ti

i

nn

ni

nt

i i

t

i +t = 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 i + t = 90°. So:

direction of motion of oscillating molecules at the surface (along the direction of the E-field in the transmitted beam)

Active polarization control: Pockels Effect

Friedrich Carl Alwin Pockels(1865 - 1913)

Pockels discovered that, for certain materials, applying an electric field can cause them to become birefringent, or change the existing birefringence.

voltage on

voltage off

An example: BaTiO3 has a cubic lattice, but an applied voltage distorts the lattice into a tetragonal shape.

The Pockels Effect: Electro-optic constants30

0 / 2

2 ijn r V V

V

is the relative phase shift between the two polarization axes V is the applied voltage rij is the “electro-optic tensor” of the material. i,j = (x,y,z) are indices that depend on the crystal orientation

V/2 is referred to as the “half-wave voltage”.

potassium di-hydrogen phosphate (KDP)

quartz

barium titanateBaTiO3

lithium niobateLiNbO3

r41 = 8.6r63 = 10.6

r41 = 0.2r63 = 0.9

r33 = 23r13 = 8.0r42 = 820

r33 = 30.8r13 = 8.6r42 = 28r22 = 3.4

Non-zero elements of the electro-optic tensor, for some materials that are often used. The units are 10-12 meters/volt.

In most materials, the Pockels Effect does not exist (i.e., rij = 0)

Consider a material that possesses “inversion symmetry”, which means that reflecting the position of every atom through a givencentral point doesn’t change the crystal.

Examples: any liquid or gas, amorphous solids with random atomic positionsmany crystalline solids (e.g. silicon or diamond)

If applying an electric field causes a change in the refractive index in proportion to the field:

n Ethen applying the opposite electric field must cause the same change in the index: n E

This can only be true if = 0. Thus there is no Pockels effect in materials with inversion symmetry.

A Pockels CellIf we add polarizers, the Pockels' effect allows control over the amplitude of the wave.

a commercial Pockels cell

Applications of Pockels cells

• creating an amplitude-modulated or phase-modulated laser beam

• picking one pulse out of a train of pulses

Input pulse train

Electro-optic modulator

voltage pulse

Output single pulse

• switching energy out of a laser cavity - this is known as “Q-switching”. It is the way many high power lasers work.

Pockels effect: phase modulationSuppose we start with an ordinary input wave from a laser:(polarized along one of the principle axes of the device)

0 j tinE E e

sin0

0 1 sin

j tj tout

j t

E E e e

E e j t

and the output wave is:

Suppose the voltage applied to the electro-optic material is a sinusoidally oscillating voltage, with small amplitude and frequency .

Then the phase acquired by the light wave is: sin t

2

1 V

where

0

0

12 2

2 2

j t j t j tout

j t j tj t

E E e e e

E e e e

output light has new frequency components! This is known as ‘sideband generation’.

This is because ex ≈ 1 + x

for |x| << 1