May 01 LASERS 51

Polarized Light• Vectors

– what they are and how they apply to light waves• Linearly polarized light

– Vertical, horizontal, other directions– Components of linear polarization in other directions– Law of Malus and linear polarizers

• Circular and elliptical polarization• Unpolarized light

– Random polarization• Production of polarized light• Birefringence and crystals

– waveplates

May 01 LASERS 51

Vectors

• Many physical quantities have both a strength (magnitude) and a direction– velocity (magnitude is speed), force, electric field

• Objects with a strength and direction are called vectors– symbolized by an arrow with length of arrow indicating direction– sometimes location of base of arrow is significant (where a force is

applied) other times only the strength and direction are importantForce applied

by support

velocity Force of gravity

May 01 LASERS 51

Addition (or superposition) of vectors• More than one vector of the same type can act on one object

(e.g. several forces)– In this case, vectors must be added

• Vectors are added by placing the tail of the second vector at the tip of the first– result is the vector from the tail of the first to the tip of the second– same result is obtained if first and second vector are reversed– vectors to be added called components, sum called resultant

Resultant vectorBlue vector

added firstRed vector added first

May 01 LASERS 51

Components of a vector

• The sum of the two forces on the car gives the resultant force

• The resultant is the component of the force of gravity along the ramp– The force from the ramp cancels the force of gravity

perpendicular to the ramp

May 01 LASERS 51

Components in any given direction

• To find a vector’s component in a given direction:– draw a line in the given direction through the tail of the vector– drop a perpendicular from the tip of the vector to the line– the component in the given direction is the vector along the given

direction to the point where the perpendicular intersects– length of component always less than length of vector

vector

Component of vector in direction of line

Direction line

May 01 LASERS 51

Independent vectors• A vector has no component in a direction perpendicular to

the direction of the vector• Two vectors which are perpendicular to each other are

called independent

direction

Independent vectors

vector

Vector has zero component in this direction

May 01 LASERS 51

Linear polarization• Light is a transverse wave

– electric field and magnetic field vibrate perpendicular to the propagation direction ( fields have no component along propagation direction)

– For a horizontal propagation direction, the electric field can vibrate in the horizontal plane or the vertical direction

– can vibrate in any other transverse direction also

Electric field for vertical polarization

Electric field for horizontal polarization

May 01 LASERS 51

What does this diagram mean?Electric Field at P

P• This means that at P there is an electric field

– An electric charge placed at P would experience a force– The direction of the force is indicated by the arrow– The strength of the force is indicated by the length of

the arrow

• The diagram says nothing about electric fields at other points– There is no field indicated at the tip of the arrow or any

other place along it

May 01 LASERS 51

Properties of linearly polarized light• Polarization direction is always perpendicular to direction of

propagation– Any two perpendicular directions may be chosen as fundamental

directions (possibly horizontal and vertical for beams propagating horizontally)

• Electric field vector oscillates in plane of polarization– Sometimes we symbolize it by a single arrow or double headed

arrow or a line, but if we follow the change in vector with time it will go positive and negative

x

y

x

y

Electric field at peak for vertical polarization

Electric field at peak for horizontal polarization

May 01 LASERS 51

Superposition of polarization vectors• Since the electric field is a vector, two light

waves traveling in the same direction can be added by adding their electric field vectors– This just like what we did in interference,

except there we didn’t talk about the directionof the field vector only its size and phase

Resultant(at one time and position)beam 1

vertical

beam 2horizontal

Just as in interference, the result of adding these two vibrating fields depends on the phase of their vibrations

May 01 LASERS 51

Superposition of two in-phase, linearly polarized waves

• Two linearly polarized waves in phase add to give another linearly polarized wave with a different polarization plane– New plane is found by vector addition of two components

• A linearly polarized wave can be thought of (and is) a sum of two other linearly polarized waves– Can be resolved into any two orthogonal directions

resultant

At peak After going through zero

Negative peakAfter peak

May 01 LASERS 51

Law of Malus

• A beam polarized in the pass direction is transmitted through the linear polarizer

• A beam polarized perpendicular to the pass direction is not transmitted

• Several different types of linear polarizer will be discussed later

• The beam exiting from the linear polarizer is always polarized in the pass direction

Pass direction

Input beam

Output beam

May 01 LASERS 51

• The input polarization is resolved into two components, along and perpendicular to the pass direction

• The component along the pass direction is transmitted, the perpendicular component is not

Pass direction

Input beam

Output beam

Input polarization direction

Θ

Component along pass direction

Output polarization direction

What if the beam is not polarized either parallel to or perpendicular to the pass direction?

Law of Malus

Simple trigonometry gives, Pout=Pin*cos2(Θ)

May 01 LASERS 51

Circular polarization• There are other ways that the electric field can

vibrate in a light wave– The vibration must be transverse, i.e. perpendicular to

the propagation direction– Field direction and magnitude must repeat after a

wavelength• In circular polarization, the electric field maps out

a circular pattern– Either observe at fixed point or wave frozen in time

Propagation direction

Path of electric field vector

May 01 LASERS 51

Out-of-phase components—circular polarization

• Two components equal in amplitude, but 90° out of phase– x is at its peak when y is zero, and vice-versa– resultant traces out a circle as components oscillate– changing sign of one component gives opposite rotation

May 01 LASERS 51

Elliptical polarization• If the x and y components are not equal amplitude, the

path of the resultant is an ellipse

etc.

• If the amplitudes are equal, but the phase difference is not 90° the polarization is also elliptical

x component is at maximum, but y is not zero

May 01 LASERS 51

Change of polarization with phase between x and y components

• Many polarizations can be obtained from the same x and y components just by changing phase between them– all possible polarizations can be inscribed in a rectangle

max.x

field

max

. yfie

ld

Phase = 0° Phase = 45° Phase = 90° Phase = 180°

May 01 LASERS 51

Unpolarized (natural) light• Polarized light is predictable

– If you know light is circularly polarized you know what its electric field vector will be at any time or place

– The phase difference between the x and y components is fixed• If the phase between the two components is unpredictable,

rapidly changing in time, the light is unpolarized– Unpolarized light is a mixture of linearly polarized components in

all possible directions, as well as all possible circular and elliptical polarizations

– unpolarized light originates in natural (thermal) sources• Partially polarized light can be thought of as a mixture of

polarized and unpolarized light– no device exists that can separate the two however

May 01 LASERS 51

Coherent light is always polarized!!!• If a light wave is perfectly coherent then the x and y

components both have known and constant phases– Since we know the phase of x and y separately for all time, we

also know their difference– may be linear, circular, or elliptical, but stays constant

Whoa? What about unpolarized lasers???• Coherence is an ideal, the phase of a laser eventually (in a

coherence time) “forgets” is past– coherence time, or coherence length, varies greatly between

different laser types– during a coherence time polarization of a laser stays constant– to acknowledge this state of affairs, a laser without a definite

polarization is often called randomly polarized (confusing terminology, but its all we have for now)

May 01 LASERS 51

Production of polarized light—scattering

• Microscopically, light interacts with materials by setting their electrons in motion– the electrons then reradiate producing

absorption, reflection, scattering, and refractive index

– skylight is partially polarized

Scattered light, partially polarized

-

Single scaterer

electron

incident

scattered

scatterers

Incident, unpolarized light

Force on the electron is transverse to propagation direction, thus only one polarization emitted at right angles

May 01 LASERS 51

Production of polarized light—reflection• Reflected ray is partially polarized

in the direction out of the paper

ray

• Refracted ray is partially polarized in the plane of the paper

• Reflected ray and refracted ray are generated by microscopic radiators also

incidentglassair refracted

ray

refle

cted

ray

• When refracted and reflected ray are at 90° the reflected ray is completely polarized (Brewster’s angle)

• Refracted ray is partially polarized in the plane of the paper• Reflected ray and refracted ray are generated by

microscopic radiators also

May 01 LASERS 51

Polarization by reflection• s-polarization

perpendicular to plane of incidence– from German word

for perpendicular– sometimes called σ

• p-polarization parallel to plane of incidence– sometimes called π

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90

Angle (degrees)

Refle

ctiv

ity

s polarizationp polarization

both reflectivitieshigh at grazing incidence

both reflectivitiesequal at normalincidence

n=1.5

Brewster'sangle

Sometimes easiest to remember as “skip” and “pass”

May 01 LASERS 51

Polarizing pile of plates

Unpolarized

light

• Invented by Arago in 1812• 15% of s polarization rejected at each surface

– in principle the p-polarization is completely transmitted– in practice it is difficult to reduce the loss below a few

tenths of a percent• Still in use for some applications (e.g. CO2 laser at

10.6 µm) Vertically (p) polarized light

May 01 LASERS 51

Polarization by dichroic crystals• Some natural crystals

(e.g. tourmaline) have absorption coefficients that are much larger for one linear polarization than another– electrons are free to

vibrate only along one axis

– circular dichroic polarizers also exist

• Polaroid sheet invented by E. Land in 1932– microscopic polarizing crystals in nitrocellulose sheet– stretched so all the crystals line up on the same axis

• “Dichroic” has other unrelated meanings (its not my fault!!)

May 01 LASERS 51

Production of polarized light—law of Malus• What happens when polarized, or unpolarized light is

incident on a linear polarizer?

polarization of incident light

direction of polarizeramplitude of

transmitted light

• Light is “resolved” into components along and perpendicular to the polarizer direction

• Parallel component transmitted, pependicular component rejected– transmitted amplitude=Eincident*cosΘ– transmitted intensity=Iincident*cos2Θ– holds true even if incident polarization is elliptical or random

Θ

Eincident

May 01 LASERS 51

Production of polarized light by thin film polarizers

s-polarized component

p-polarized component

Input beam

• Glass has a multilayer dielectric coating (similar to antireflection coating)

• Angle is close to Brewster’s angle (makes coating design easier)

• All s-polarized light is reflected not just 15%

May 01 LASERS 51

Birefringence (Double refraction)

• Electrons in many crystals have different forces on them in different directions– these crystals are said to be anisotropic– an=not, iso=same, tropic=direction, thus not the

same in every direction• As a result, the index of refraction depends on

the polarization– result is refractive index depends on polarization– Speed of light depends on polarization

May 01 LASERS 51

Optic axis in anisotropic crystals• In an anisotropic crystal light going in one or two

special directions has the same index of refraction independent of its polarization– these special directions are called the optic axes– optic axis is a direction in the crystal not one particular line– crystals with two optic axes are biaxial– only uniaxial crystals will be discussed here

• Direction of optic axis is closely tied to crystal structure

May 01 LASERS 51

Propagation in a uniaxial crystal• Along optic axis, the light

propagates with a single refractive index, the ordinary index, no

• For other propagation directions, there are two indices– Resolve the light into two linearly polarized components

• one polarized perpendicular to the optic axis• one in plane of optic axis and propagation direction

– polarization perpendicular to optic axis has index no

– The other polarization propagates with a different index of refraction called the extraordinary index, ne

• Extraordinary index depends on the direction of propagation– Perpendicular to optic axis it differs the most from no

– Smoothly approaches no as direction approaches optic axis

optic

axispropagationdirection

ordinarypolarization

extraordinarypolarization

May 01 LASERS 51

Uniaxial crystals—refraction• Consider a light ray incident at normal

incidence on the surface of a uniaxial crystal– A light ray with a polarization

perpendicular to the optic axis is called an ordinary ray

– A light ray with the other linear polarization is called an extraordinary ray

ordinary

ray

extraordinary

ray

• At the surface, the ordinary ray obeys Snell’s law, it doesn’t refract because the incidence angle is zero– The extraordinary ray bends at the surface (except in the special

case that the optic axis is parallel to the surface)– At any angle of incidence the ordinary ray obeys Snell’s law– The extraordinary ray does not in general obey Snell’s law

May 01 LASERS 51

Birefringent polarizers—Nicol prism

• Invented by William Nicol in 1828– according to Jenkins & White he didn’t understand how it worked

• Start with single crystal of calcite– cut down ends 3° from natural angle (to 68°)– cut apart along diagonal– cement back together with Canada Balsam

• O-ray of calcite has lower index than e-ray, undergoes TIR at interface, e-ray is transmitted

May 01 LASERS 51

Other birefringent polarizers

• Nicol prisms are simple, but have disadvantages– relatively small acceptance angle (~28°)– input is not at normal incidence– cemented optics cannot be used with high-power lasers or

in the UV

• Several variations exist– Glan-Thompson is the most popular and overcomes all the

difficulties listed above (but not all at once!)

• Other polarizers separate the two components– Rochon prism, Wollaston prism

May 01 LASERS 51

Interference of polarized light—waveplates• Incident polarization is resolved

into o and e polarizations inside crystal– shown with optic axis in plane of

plate, but same principles if not– directions along o and e are called

fast axis and slow axis

• Inside crystal the two waves propagate independently– each has its own index and possibly

its own direction

direction ofoptic axis

t

birefringentplate

incident, vertically

polarized beam

Θ

• After emerging from crystal, recombine the two waves using the principle of superposition– keep track of phase difference in crystal– continues to propagate in the normal way after exiting crystal

λ/ray-o ofdelay phase tno=

λ/ray-e ofdelay phase tne=λ/)(difference phase tnn eo −=

Phases given in waves!

May 01 LASERS 51

Quarter wave plate• Phase difference is a quarter wave• If incident light polarized at 45° to

fast axis the o and e components are equal amplitudes– emergent light is circularly polarized

Fast axis

Slow axis

• Phase difference is a quarter wave• If incident light polarized at 45° to fast axis the o and e

components are equal amplitudes– emergent light is circularly polarized

• If incident light is circularly polarized, output is linearly polarized– right-hand circular comes out parallel to fast axis, left to slow

• Other polarizations result in elliptic output, but unpolarized light comes out unpolarized!

May 01 LASERS 51

Waveplates (cont.)• Quarter wave plate in which the phase delay is exactly 1/4

λ is called a zero-order plate– only works exactly for one wavelength (even neglecting

dispersion!!), but close to a quarter wave for other wavelengths– must be very thin, can be mounted on substrate for structural

stability– if phase delay is n waves + 1/4 λ acts exactly the same at λ, but

goes out of phase very quickly as λ changes

• Half-wave plate, 90° phase difference– for linearly polarized input at 45° to fast axis, emergent light is

linearly polarized, but rotated 90°• Babinet-Soliel compensator, arrangement of birefringent

plates that can produce a variable phase delay

May 01 LASERS 51

Analysis of unknown polarization• If a linear polarizer is rotated and the transmission goes to

zero at some angle, then input is linearly polarized, DONE.• If no there is no variation with polarizer rotation, light is

circularly polarized, unpolarized, or a mixture of these– to distinguish between these, put in λ/4 plate before polarizer– if light was circular, it will now be linear, detect with polarizer– if it was unpolarized, it will still be unpolarized, ie no variation

with polarizer– if there is now a variation with the polarizer but the minima don’t

go to zero, then the light is partially polarized – degree of polarization defined as

MINMAX

MINMAX

IIII

+−

=γ

May 01 LASERS 51

Analysis of unknown polarization (cont.)• If there is a variation with the linear polarizer (and no λ/4

plate) the light must be at least partially polarized but might also be elliptical

• Insert λ/4 plate with fast axis along direction of maximum transmission– for elliptically polarized light, the phase difference between major

and minor axes is also λ/4, but the two components are unequal amplitude, therefore, the λ/4 plate will convert this to linear polarization, but an angle to the original maximum, detect with polarizer

– If the polarizer show that the light is not linear even with the λ/4 plate inserted, the light is not completely polarized, degree ofpolarization defined as before, there are still two possibilities

• if minimum at same angle as before, mixture of linear polarization and unpolarized, if minimum at different angle then mixture of linear and elliptical

May 01 LASERS 51

Optical rotation• Some materials exhibit the phenomenon of optical activity-

– plane polarized light (at any angle) remains plane polarized, but its angle of polarization rotates as it goes through the material

– Note differences between this behavior and that of a waveplate, in optical activity: input polarization doesn’t matter, rotation increases with thickness of the material, output polarization isalways linear

• Optical activity can be induced in some materials due to a magnetic field, Faraday effect– This is the only one of the multitude of polarization effects we

have examined which is not reversible• By reversible I mean that the direction of propagation can be reversed if the

output and input polarizations are switched

– This effect is the basis of optical diodes and optical isolators