15. Polarization

Linear, circular, and elliptical polarization

Mathematics of polarization

Uniaxial crystals

Birefringence

Polarizers

Polarization Notation near an interfaceParallel ("p") Perpendicular ("s")polarization polarization

These are only defined relative to an interface between two media. But even when there is no interface around, we still need to consider the polarization of light waves.

Polarization of a light waveWe describe the polarization of a light wave (without any interface nearby) according to how the E-field vector varies in a projection onto a plane perpendicular to the propagation direction.

For convenience, the propagation direction is generally assumed to be along the positive z axis.

x

y E-field variation over time (and space)

x

y

In these diagrams, the propagation direction is out of the page at you.

Here are two possibilities:

0 ˆ( , ) exp ( )

E r t E x j kz t 0 ˆ( , ) exp ( )

E r t E y j kz t

45° Polarization

Here, the complex amplitude, E0 is the same for each component.

So the components are always in phase.

0

0

( , ) Re exp ( )

( , ) Re exp ( )

x

y

E z t E j kz t

E z t E j kz t

ˆ ˆ( , ) x yE z t E x E y and the total wave is:

Arbitrary-Angle Linear Polarization

Here, the y-component is in phasewith the x-component, but has different magnitude. x

y

E-field variation over time (and space)

0

0

( , ) Re cos exp ( )

( , ) Re sin exp ( )

x

y

E z t E j kz t

E z t E j kz t

ˆ ˆ( , ) x yE z t E x E y and (as always) the total wave is:

The Mathematics of Polarization

For many purposes, we only care about the relative values:

x

y

EE

E1

yx

x

EEE E

sin tancos

y

x

EE

Specifically: 0° linear (x) polarization: Ey /Ex = 090° linear (y) polarization: Ey /Ex = 45° linear polarization: Ey /Ex = 1Arbitrary linear polarization:

Define the polarization state of a field as a 2D vector—“Jones vector” —containing the two complex amplitudes:

10 0

1

Jones vectors - a common mistake

NOTE: the Jones vector contains the complex amplitudes only. Its components do not depend on x,y,z, or t.

j kz tx

j kz ty

E eE

E e

This is wrong!

Circular (or Helical) Polarization

Here, the complex amplitude of the y-component is -j times the complex amplitude of the x-component.

0

0

( , ) cos( )( , ) sin( )

x

y

E z t E kz tE z t E kz t

or, in complex notation:

So the components are always 90° out of phase.

The resulting E-field rotates clockwise around the k-vector (looking along k). This is called a right-handed rotation.

0

0

( , ) Re exp ( )

( , ) Re exp ( )

x

y

E z t E j kz t

E z t jE j kz t

Right vs. Left Circular (or Helical) Polarization

Here, the complex amplitudeof the y-component is +j times the complex amplitude of the x-component.

x

yE-field variation over time (and space)

kz-t = 0°kz-t = 90°

0

0

( , ) cos( )( , ) sin( )

x

y

E z t E kz tE z t E kz t

or, more generally:

So the components are always 90° out of phase, but in the other direction.

0

0

( , ) Re exp ( )

( , ) Re exp ( )

x

y

E z t E j kz t

E z t jE j kz t

The resulting E-field rotates counterclockwise around the k-vector (looking along k). This is a left-handed rotation.

Note: In this drawing, the z axis is coming out of the screen at you. So you are looking in the oppositedirection from the k-vector, which is why it rotates clockwise according to the arrow - but we refer to this as a counter-clockwise rotation.

Circular Polarization - the movie

Question: is this cartoon showing right-handed or left-handed circular polarization?

Unequal Arbitrary-Relative-Phase Components yield "Elliptical Polarization"

The resulting E-field can rotate clockwise or counter-clockwise around the k-vector.

x

yE-field variation over time (and space)

0

0

0 0

( , ) Re exp ( )

( , ) Re exp ( )

where and are arbitrary

complex amplitudes.

x x

y y

x y

E z t E j kz t

E z t E j kz t

E E

0

0

0 0

( , ) cos( )( , ) cos( )

where

x x

y y

x y

E z t E kz tE z t E kz t

E E

or, in complex notation:

Circular polarization has an imaginary Jones vector y-component:

Right circular polarization:

Left circular polarization:

The Mathematics of Circular andElliptical Polarization

1

x

y

EE

E j

/ y xE E j

/ y xE E j

/ y xE E a jb

For elliptical polarization, the two components have different amplitudes, and may even be complex:

We can calculate the eccentricity and tilt of the ellipse if we feel like it.

A clockwise rotation, when looking along the propagation direction.

counterclockwise rotation.

An example 11

E

j

What is the polarization of this wave?

0 ˆ ˆ, 1 j kz tE z t E x j y e

This Jones vector is equivalent to:

41 2 jj e Using we find, at z = 0:

0

0

cos

2 cos 4

x

y

E E t

E E t

t = 0 0 0 E E

Ex Ey

t = /4 0 2 2 0E

t = /2 00 E

t = 3/4 0 02 2 2 E E

t = 0 0 E E

x

y

left-handed elliptical polarization

A polarizer is a device which filters out one polarization component

Such polarizers are commonly used for long-wave infrared radiation, because the wire spacing has to be much smaller than the wavelength (and so it is easier to manufacture if the wavelength is longer).

The light can excite electrons to move along the wires. These moving charges then emit light that cancels the input light. This cannot happen if the E-field is perpendicular to the wires, since the current can only flow along the wires.

Polymer-based polarizersA polymer is a long chain molecule. Some polymers can conduct electricity (i.e., they can respond to electric fields similar to the way a wire does).

This is how polarized sunglasses work.

The light can excite electrons to move along the wires, just as in the case of the polymer chains.

Crossed polarizers block light

Blocking both x and y polarizations means that you have blocked everything.

Inserting a third polarizer between the two crossed ones can allow some light to leak through.

no sunglasses

sunglasses

Why sunglasses are polarized

Brewster’s angle revisited

Birefringence

The x- and y-polarizations can see different refractive index curves.

Hence, the refractive index of a material can depend on the orientation of the material relative to the polarization axis!

The molecular "spring constant" can be different for different directions.

Uniaxial crystals have an optic axis

Light with any other polarization must be broken down into its ordinary and extraordinary components, considered individually, and recombined afterward.

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 extraordinaryray, and light polarized perpendicular to it is called the ordinary ray. These polarization directions are the crystal “principal axes.”

Birefringence can separate the twopolarizations into separate beams

Due to Snell's Law, light of different polarizations will refract by different amounts at an interface.

no

ne

o-ray

e-ray

Calcite is one of the most birefringent materials known.

Birefringent Materials

Calcite, CaCO3

Some polarizers use birefringence.

For example, a Wollaston prism:

Combine two calcite prisms, rotated so that the ordinary polarization in the first prism is extraordinary in the second (and vice versa).

The ordinary ray in the first prism becomes the extraordinary ray in the second one. Since ne < no, the E-ray is refracted away from the normal to the interface. The opposite happens for the O-ray.

opticaxis

opticaxis

E-ray

O-ray