opticsi 09 polarization i

8/2/2019 OpticsI 09 Polarization I

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Polarization

Linear, circular, and elliptical polarization

Mathematics of polarization

Uniaxial crystals

Birefringence

Polarizers

Prof. Rick TrebinoGeorgia Tech

www.physics.gatech.

edu/frog/lectures


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45 Polarization

0

0

( , ) Re exp[ ( )]

( , ) Re exp[ ( )]

x

y

E z t E i kz t

E z t E i kz t

Here, the complexamplitude,E0, isthe same for eachcomponent.

So thecomponents arealways in phase.

~


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Arbitrary-Angle Linear Polarization

0

0

( , ) Re cos( ) exp[ ( )]

( , ) Re sin( ) exp[ ( )]

x

y

E z t E i kz t

E z t E i kz t

Here, the y-component is in phase

with thex-component,but has different magnitude.

x

y

E-fieldvariation overtime (andspace)


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Define the polarization state of a field as a 2D vector

Jones vectorcontaining the two complex amplitudes:

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

(alternatively normalize this vector to unity magnitude)

Specifically:

0 linear (x) polarization:Ey

/Ex

= 0

90 linear (y) polarization:Ey/Ex=

45 linear polarization:Ey/Ex= 1

Arbitrary linear polarization:

The Mathematics of Polarization

sin( )tan( )

cos( )

y

x

E

E

x

y

EE

E

1

y

x

x

EE

E

E

~ ~

~ ~

~ ~


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Circular (or Helical) Polarization

The resulting E-field rotatescounterclockwise around thek-vector (looking along k).

0

0

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

x

y

E z t E kz t E z t E kz t

Here, the complex amplitude ofthey-component is -i times the

complex amplitude of thex-component.

So the components are always90 out of phase.

0

0

( , ) Re exp ( )( , ) Re exp ( )

x

y

E z t E i kz t

E z t E i kz t

-i

Or, more generally,


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Right vs. Left Circular (or Helical) Polarization

The resulting E-field rotatesclockwise around thek-vector (looking along k).

0

0

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

x

y

E z t E kz t E z t E kz t

Here, the complex amplitude

of the y-component is +i times

the complex amplitude of thex-component.

So the components are always

90 out of phase, but in the

other direction.

x

yE-field variationover time (andspace)

kz-t = 0

kz-t = 90

0

0

( , ) Re exp ( )

( , ) Re exp ( )

x

y

E z t E i kz t

E z t iE i kz t

Or, more generally,


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Unequal arbitrary-relative-phasecomponents yield elliptical polarization

The resulting E-field canrotate clockwise or counter-clockwise around the k-vector (looking along k).

x

yE-field variationover time (andspace)

0 0x yE Ewhere and are arbitrary

complex amplitudes.

0

0

( , ) Re exp ( )

( , ) Re exp ( )

x x

y y

E z t E i kz t

E z t E i kz t

0

0

( , ) cos( )

( , ) cos( )

x x

y y

E z t E kz t

E z t E kz t

Or, more generally,

0 0.

x yE Ewhere


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Circular polarization has an imaginary Jones vector y-component:

Right circular polarization:

Left circular polarization:

Elliptical polarization has both real and imaginary components:

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

The mathematics of circular andelliptical polarization

1x

y

EE

E i

/y x

E E i

/y x

E E a bi

/y x E E i


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When the phases of the x- and y-polarizationsfluctuate, we say the light is unpolarized.

where x(t) and y(t) are functions that vary on a time scale slower than1/, but faster than you can measure.

The polarization state (Jones vector) will be:

As long as the time-varying relative phase, x(t) y(t), fluctuates, the light

will not remain in a single polarization state and hence is unpolarized.

( , ) Re exp ( )

( , ) Re exp ( )

x x x

y y y

E z t A i kz t t

E z t A i kz t t

1

exp ( ) ( )y

x y

x

Ai t t

A

In practice, theamplitudes vary,too!


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Light with very complex polarizationvs. position is also unpolarized.

Light that has passed through cruddy stuff is often unpolarized

for this reason. Well see how this happens later.

The polarization vs. position must be unresolvable, or else, weshould refer to this light as locally polarized.


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An intentionally spatially varyingpolarization can be interesting.

Imagine a donut-shaped beam with radial polarization.

k

Lens

Focus

Longitudinalelectric fieldat the focus!

This type of beam can also focus to a smaller spot than a beam

with a spatially uniform polarization!


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A complex polarization spatialdependence

An opticalvortex

x

y


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Birefringence

The molecular "springconstant" can be

different for differentdirections.


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Birefringence

The x- and y-

polarizationscan seedifferentrefractiveindex curves.


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Uniaxial crystals have an optic axis

Light with any other polarization must be broken down into itsordinary and extraordinary components, consideredindividually, and recombined afterward.

Uniaxial crystals have one

refractive index for light polarizedalong the optic axis (ne) andanother for light polarized ineither of the two directionsperpendicular to it (no).

Light polarized along the opticaxis is called the extraordinaryray, and light polarizedperpendicular to it is called theordinary ray. These polarization

directions are the crystalprincipal axes.


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Birefringence can separate the twopolarizations into separate beams

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

no

ne

o-ray

e-ray


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Calcite

Calcite is one of the most birefringent materials known.


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Birefringent Materials


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How to make a polarizer

Perpendicular polarization

Incidence angle, i

1.0

.5

00 30 60 90

R

T

Parallel polarization

Incidence angle, i

1.0

.5

00 30 60 90

R

T

Well use birefringence, and well take advantage of the Fresnel

equations we derived last time.

When ni > nt, theres a steep variation in reflectivity with incidenceangle. This corresponds to a steep variation with refractive index.


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Polarizers take advantage of birefringence,Brewster's angle, and total internal reflection.

Snells Law separates the beams at the entrance. The perpendicular

polarization then goes from high index (1.66) to low (1.55) andundergoes total internal reflection, while the parallel polarization istransmitted near Brewster's angle.

If the gap is air, its called the Glan-Taylor polarizer.

The Glan-Thompson polarizer uses two prisms of calcite (with paralleloptical axes), glued together with Canada balsam cement (n = 1.55).

no = 1.66

Cementn = 1.55

ne = 1.49ne = 1.49

Optic axis

Optic axis

Alternatively,

the optical axispoints out of thescreen, and thepolarizations arealso rotated.


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Combine two prisms of calcite, rotated so that the ordinary

polarization in the first prism is extraordinary in the second (andvice versa).

The perpendicular polarization goes from high index (no) to low(ne) and undergoes total internal reflection, while the parallelpolarization is transmitted near Brewster's angle.

Even better, rotate the second prism:The Nicol polarizer.

no = 1.66 no = 1.66

ne = 1.49

Optic axis

Optic axis (out of page)

BrewstersAngle

TIR


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RealPolarizers

Air-spaced polarizers


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Wollaston Polarizing Beam Splitter

The ordinary and extraordinary rays have different refractive indicesand so diverge.

The Wollaston polarizing beam splitter uses two rotated

birefringent prisms, but relies only on refraction.


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The Pile-of-Plates Polarizer

After numerous Brewster-angle transmissions,mainly the parallel polarization remains.

Unfortunately, only about 10% of the perpendicular polarizationis reflected on each surface, so you need a big pile of plates!

Unpolarizedinput light


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Dielectric polarizers

A multi-layer coating (which uses

interference; well get to this later) canalso act as a polarizer. It still uses thesame basic ideas of TIR and Brewsters

angle.

Glass


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Wire Grid Polarizer

The light can excite electrons to move along the wires, which thenemit light that cancels the input light. This cannot happenperpendicular to the wires. Such polarizers work best in the IR.

Polaroid sheet polarizers use the same idea, but with long polymers.

Input lightcontains bothpolarizations


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Wire grid polarizer in the visible

Using semiconductor fabrication techniques, a wire-grid polarizer was

recently developed for the visible.

The spacing is less than 1 micron.


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The wires need not be very long.

Extinction coefficient > 10,000 Transmission > 99%

Hoya has designed a wire-grid polarizer for telecom applicationsthat uses small elongated copper particles.


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The Measure of a Polarizer

The ideal polarizer will pass 100% of the desired polarization and

0% of the undesired polarization.It doesnt exist.

The ratio of the transmitted irradiancethrough polarizers oriented parallel

and then crossed is the Extinction ratioor Extinction coefficient. Wed like theextinction ratio to be infinity.

Type of polarizer Ext. Ratio Transmission Cost

Calcite: 106 > 95% $1000 - 2000

Dielectric: 103 > 99% $100 - 200

Polaroid sheet: 103 ~ 50% $1 - 2

0 Polarizer

0 Polarizer

0 Polarizer

90 Polarizer


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Visible wire-gridpolarizer

performance

Transmission

Extinction ratio

A polarizers performance

can vary with wavelengthand incidence angle.

The overall transmission

is also important, as isthe amount of the wrongpolarization in eachbeam.

opticsi 09 polarization i

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