PHYS 415: OPTICS Polarization (from Trebino’s lectures)

www.bilkent.edu.tr/~ilday

PHYS 415: OPTICS

Polarization (from Trebino’s lectures)

F. ÖMER ILDAY

Department of Physics,Bilkent University, Ankara, Turkey


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 complex amplitude, E0 is the same for each component.

So the components are always in phase.

~


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 the x-component,

but has different magnitude. x

y

E-field variation over time (and space)


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

Jones vector—containing 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

1

x

y

yx

x

EE

E

EE

EE


Circular (or Helical) Polarization

The resulting E-field rotatescounterclockwise around the k-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 the x-component.

So the components are always 90° out of phase.

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,


Right vs. Left Circular (or Helical) Polarization

The resulting E-field rotatesclockwise around the k-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 amplitudeof the y-component is +i times the complex amplitude of the x-component.

So the components are always90° out of phase, but in the other direction.

x

yE-field variation over time (and space)

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,


Unequal arbitrary-relative-phase components yield elliptical polarization

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

x

yE-field variation over time (and space)

0 0x yE E

where 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


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 and elliptical polarization

1x

y

EE

E i

/y xE E i

/y xE E a bi

/y xE E i


When the phases of the x- and y-polarizations fluctuate, 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 ( ) ( )yx y

x

Ai t t

A

In practice, the amplitudes vary, too!


Light with very complex polarizationvs. position is also unpolarized.

Light that has passed through “cruddy stuff” is often unpolarized for this reason. We’ll see how this happens later.

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


Birefringence

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


Birefringence

The x- and y-polarizations

can see different refractive

index curves.


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 extraordinary ray, 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 bend by different amounts at an interface.

no

ne

o-ray

e-ray


Calcite

Calcite is one of the most birefringent materials known.


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

Here’s one approach:

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


Polarizers

Air-spaced polarizers


Wollaston Polarizing Beam Splitter

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

The Wollaston polarizing beam splitter uses two rotatedbirefringent prisms, but relies only on refraction.


Dielectric polarizers

A multi-layer coating (which uses interference; we’ll get to this later) can also act as a polarizer.

Ealing Optics catalog

Melles-Griot catalog

Glass


Wire Grid Polarizer

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

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

Input light contains both polarizations


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.


The Measure of a Polarizer

The ideal polarizer will pass 100% of the desired polarization and 0% of the undesired polarization.

It doesn’t exist.

The ratio of the transmitted irradiance through polarizers oriented parallel and then crossed is the Extinction ratio or Extinction coefficient. We’d like the extinction ratio to be infinity.

Type of polarizer Ext. Ratio Cost

Calcite: 106 $1000 - 2000

Dielectric: 103 $100 - 200

Polaroid sheet: 103 $1 - 2

0° Polarizer

0° Polarizer

0° Polarizer

90° Polarizer


Wave plates

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

This changes the relative phase of the x and y fields, and hence changing the polarization.

Polarization state:

0

0

( , ) Re exp

( , ) Re exp

x

y

E z t E i kz t

E z t E i kz t

1

1

exp( )

exp( )

1

2exp

o

e

o e

ikn d

ikn d

i n n d

Output:

}

}

0

0

( , ) Re exp

( , ) Re exp

o

y e

xE z t E i kz t

E

kn d

knz t E i dkz t

Wave plate

+45° Polarization

Optic axis

-45° Polarization

x

y

z

Input:


Wave plates (continued)

Wave plate output polarization state:

1

2exp o ei n n d

0 1 45 Linear/2 Left Circular

1 45 Linear3 /2 Right Circular2 1 45 Linear

Output2 2 exp

Polarization Stateo e o e

i

i

n n d i n n d

A quarter-wave plate creates circular polarization, and a half-wave plate rotates linear polarization by 90.

We can add an additional 2m without changing the polarization, so the polarization cycles through this evolution as d increases further.

Quarter-wave plate

Half-wave plate

(45-degree input polarization)


Half-wave plate

If the incident polarization is +45° to the principal axes, then the output polarization is rotated by 90° to -45°.

When a beam propagates through a half-wave plate, one polarizationexperiences half of a wavelength more phase delay than the other.


Wave plates and input polarization

Remember that our wave plate analysis assumes 45° input polarization relative to its principal axes. This means that either the input polarization is oriented at 45°, or the wave plate is.

Wave plate w/ axes at 0° or 90°

±45° Polarizer

Wave plate w/ axes at

±45°

0° or 90° Polarizer

If a HWP, this yields 45° polarization.

If a QWP, this yields circular polarization.

±

If a HWP, this yields 90° or 0° polarization.

If a QWP, this yields circular polarization.


How not to use a wave plate

If the input polarization is parallel to the wave plate principal axes, no polarization rotation occurs!

Wave plate w/ axes at 0° or 90°

Wave plate w/ axes at

±45°

0° or 90° Polarizer ±45° Polarizer

This arrangement can, however, be useful. In high-power lasers, we desire to keep the laser from lasing and then abruptly allow it to do so. In this case, we switch between this and the previous case.


Thickness of wave plates

When a wave plate has less than 2 relative phase delay, we say it’s a zero-order wave plate. Unfortunately, they tend to be very thin.

Solve for d to find the thickness of a zero-order quarter-wave plate:

2

2o en n d

4 o e

dn n

Using green light at 500 nm and quartz, whose refractive indices are ne – no = 1.5534 – 1.5443 = 0.0091, we find:

This is so thin that it is very fragile and very difficult to manufacture.

d

d = 13.7 m


Multi-order wave plates

A multi-order wave plate has more than 2 relative phase delay.

We can design a twentieth-order quarter-wave plate with 20¼ waves of relative phase delay, instead of just ¼:

240

2o en n d

4141

4 zero ordero e

d dn n

This is thicker, but it’s now 41 times more wavelength dependent!

It’s also temperature dependent due to n’s dependence on temperature.

d

d = 561 m


A thick zero-order wave plate

The first plate is cut with fast and slow axes opposite to those of the second one.

The Jones vector becomes:

1 2 1 2

1 1

2 2 2exp expo e e o o ei n n d i n n d i n n d d

First plate Second plate

Now, as long as d1 – d2 is equal to the thin zero-order wave plate, this optic behaves like the prohibitively thin one! This is ideal.

Inputbeam

Optic axesx

y

zOutputbeam

d1 d2


Polarization Mode Dispersion plagues broadband optical-fiber communications.

Imagine just a tiny bit of birefringence, n, but over a distance of 1000 km…

Newer fiber-optic systems detect only one polarization and so don’t see light whose polarization has been rotated to the other.

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

1

2exp i n d

Distance

Polarization state at receiver

=

If = 1.5 m, then n ~ 10-12 can rotate the polarization by 90º!


Circular polarizers

A circular polarizer makes circularly polarized light by first linearly polarizing it and then rotating it to circular. This involves a linear polarizer followed by a quarter wave plate

Unpolarized input light

Circularly polarized light

QWP45°

Polarizer

45° polarized light

Quarter wave plate (QWP)

±45° Polarizer

Additional QWP and linear polarizer comprise a circular "analyzer."

45° PolarizerQWP

-45° polarized light


Polarization Spectroscopy

The 45°-polarized Pump pulse re-orients molecules, which induces some birefringence into the medium, which then acts like a wave plate for the Probe pulse until the molecules re-orient back to their initial random distribution.

0 polarizer

90 polarizer

Yellow filter (rejects red)

PHYS 415: OPTICS Polarization (from Trebino’s lectures)

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