Optical Activity & Jones Matrices

Ways to actively control polarization

Pockels' Effect

Kerr Effect

Photo-elasticity

Optical Activity

Faraday Effect

Jones Matrices

Unpolarized light, Stokes Parameters, & Mueller Matrices

Prof. Rick TrebinoGeorgia Tech

www.physics.gatech.edu/

frog/lectures

The Pockels' Effect

An electric field can induce birefringence.

The Pockels' effect allows control over the polarization rotation.

+V 0

Polarizer

Electro-optic medium

Transparent electrode

Transparent electrode

Analyzer

The Pockels Effect: Electro-optic constants

363

0 /2

2 on r V V

V

V/2 is called the half-wave voltage.

where is the relative phase shift, V is the applied voltage, and r63 is the electro-optic constant of the material.

Q-switching

Q is the Quality of the laser cavity. It’s inversely proportional to the Loss.

Q-switching involves:

1. Preventing the laser from lasing until the flash lamp is finished flashing, and

2. Abruptly allowing the laser to lase.

This yields a short “giant” high-power pulse.

The pulse length is limited by the round-trip time of the laser and is usually 10 - 100 ns long.

100%

0%Time

Cav

ity L

oss

Cav

ity G

ain

Output intensity

The Q-Switch

In high-power lasers, we desire to prevent the laser from lasing until we’ve finished dumping all the energy into the laser medium. Then we let it lase. A Pockels’ cell is the way we do this.

The Pockels’ cell switches (in a few nanoseconds) from a quarter-wave plate to nothing.

Before switching After switching

Pockels’ cell as wave plate w/ axes at ±45°

0° Polarizer Mirror

Pockels’ cell as an isotropic

medium

0° Polarizer Mirror

Light becomes circular on the first pass and then horizontal on the next and is then rejected by the polarizer.

Light is unaffected by the Pockels’ cell and hence is passed by the polarizer.

The Kerr effect: the polarization rotation is proportional to the Kerr constant and E2

where: n is the induced birefringence, E is the electric field strength,K is the Kerr constant of the material.

Use the Kerr effect in isotropic media, where the Pockels' effect is zero.

The AC Kerr Effect creates birefringence using intense fields of a lightwave. Usually very high irradiances from ultrashort laser pulses arerequired to create quarter-wave rotations.

20n KE

Photo-elasticity: Strain-induced birefringence

Clear plastic drawing device

(“French curve”)

between crossed

polarizers

Strain-Induced birefringence in diamond

An artificially grown diamond with nitrogen impurities between crossed polarizers

Caused by strain associated with growth boundaries

Strain-induced birefringence in thin sections of rock

More Photo-elasticity

If there's not enough strain in a medium to begin with, you can always apply stress and add more yourself!

You can use this effect to improve the performance of polarizers.

Clear plastic between crossed polarizers

Optical Activity (also called Chirality)

Unlike birefringence, optical activity rotates polarization, but maintains a linear polarization throughout. The polarization rotation angle is proportional to the distance. Optical activity was discovered in 1811 by Arago.

Some substances rotate the polarization clockwise (dextrorotatory) and some produce a counterclockwise rotation (levorotatory).

Right vs. left-handed materials

Most naturally occurring materials do not exhibit chirality. But those that do can be left- or right-handed.

These molecules have the same chemical formulas and structures, but are mirror images of each other. One form rotates the polarization clockwise and the other rotates it counterclockwise.

Left-handed vs. right-handed molecules

The key molecules of life are almost all left-handed. Sugar is one of the most chiral substances known.

Occasionally, a molecule of the wrong chirality can cause serious illness (e.g., thalidimide) while its other enantiomer is harmless.

If you’d like to look for signs of life on other planets, look for chirality.

Principal Axes for Optical Activity

As for birefringent media, the principal axes of an optically active medium are the medium's symmetry axes.

We consider the component of light along each principal axis independently in the medium and recombine them afterward.

In media with optical activity, the principal axes correspond to circular polarizations.

Complex Principal Axes

Usually, we write the E-field in terms of its x- and y-components.But we can equally well write it in terms of its right and leftcircular components.

ˆ ˆ ˆ / 2

ˆ ˆ ˆ / 2

R x iy

L x iy

ˆ ˆˆ / 2

ˆ ˆˆ / 2

x L R

y L R i

When the principal axes of a medium are circular, as they arewhen optical activity is present, this is required. We must thendecompose linear polarization into its circular components:

Math of Optical Activity–CircularPrincipal Axes

At the entrance to an optically active medium, an x-polarized beam (R + L, neglecting the √2 in all terms) will be:

0

0

0

0

( , ) Re exp:

( , ) Re exp

( , ) Re exp:

( , ) Re exp

x

y

x

y

E z t E i kz t

E z t iE i kz t

E z t E i kz t

E z t iE i kz t

R

L

Note that this mess just adds up to x-polarized light!

Math of Optical Activity–CircularPrincipal Axes (cont’d)

In optical activity, each circular polarization can be regarded as

having a different refractive index, as in birefringence.

After propagating through an optically active medium of length d,

an x-polarized beam will be:

0

0

0

0

( , ) Re exp:

( , ) Re exp

( , ) Re exp:

( , ) Re exp

x

y

x

y

E z t E i kz t

E z t iE i kz t

E z t E i kz t

E z t i

kn d

kn d

E i kz

d

n t

kn

k d

L

R

R

L

R

L

Math of Optical Activity–CircularPrincipal Axes (continued)

0 0

0 0

0

( , ) Re exp exp

( , ) Re exp exp

( , ) Re exp exp e

x

y

x

E z t E i kz t E i kz t

E z t iE i kz t iE i k

kn dkn

z t

d

E z t E i i

L

L

L

R

R R L

R

L R

Adding up the field components, we have :

where and

so :

0

0 0

xp

( , ) Re exp exp exp

11

exp( ) exp/

exp( ) exp

y

y x

i kz t

E z t iE i i i kz t

i iE E i

i i

L

L

L

R

R

R

Polarization State :

Math of Optical Activity–CircularPrincipal Axes (continued)

/2 2

11

exp( ) exp( ) exp( )1 exp( ) exp( )exp( ) exp( ) exp( ) exp( ) exp( )

1

si

ave

ave

ave

i i i i ii

i i i i i i

L LR R

R

R

L

L

Letting and , we have :

1n( )

tan( )cos( )

Remarkably, the polarization state simplifies to linear polarization

for all values of the relative phase delay!

( 1/ 2) .x n y n The polarization is when and when

L( ) / 2k n n d R

Why does optical activity occur?

Imagine a perfectly helical molecule and a circularly polarized beam incident on it with a wavelength equal to the pitch of the helix.

One circular polarization tracks the molecule perfectly. The other doesn’t.

Magnetic field

The Faraday Effect

A magnetic field can induce optical activity.

The Faraday effect allows control over the polarization rotation.

Magneto-optic medium

Polarizer Analyzer

0 +V

The Faraday effect: the polarization rotation is proportional to the Verdet constant.

= V B d

where:

is the polarization rotation angle,

B is the magnetic field strength,

d is the distance,

V is the Verdet constant of the material.

Polarization-independent Optical IsolatorWe could use a polarizer and quarter-wave plate or a Faraday rotator, but they require polarized light.

Optical fiber

Input beam

Lens

This device spatially separates the return (reflected) beam polarizations from the input beam.

45° rotation

45° rotation

Optic axis (into page)

Optic axis (45° into

page)

To model the effect of a medium on light'spolarization state, we use Jones matrices.

Since we can write a polarization state as a (Jones) vector, we use

matrices, A, to transform them from the input polarization, E0, to the

output polarization, E1.

This yields:

For example, an x-polarizer can be written:

So:

1 0E EA

1 11 0 12 0

1 21 0 22 0

x x y

y x y

E a E a E

E a E a E

1 0

0 0x

A

0 01 0

0

1 0

0 0 0x x

xy

E EE E

E

A

~

~

~

Other Jones matrices

A y-polarizer:0 0

0 1y

A

1 0

0 1HWP

AA half-wave plate: 1 0 1 1

0 1 1 1

1 0 1 1

0 1 1 1

A half-wave plate rotates 45-degree-polarization to -45-degree, and vice versa.

A quarter-wave plate: 1 0

0QWP i

A

1 0 1 1

0 1i i

A wave plate is not a wave plate if it’s oriented wrong.

Remember that a wave plate wants ±45° (or circular) polarization.

If it sees, say, x polarization, nothing happens.

1 0 1 1

0 1 0 0

So use Jones matrices until you’re really on top of this!!!

AHWP

Wave plate w/ axes at 0° or 90°

0° or 90° Polarizer

Rotated Jones matrices

Okay, so E1 = A E0. What about when the polarizer or wave plate responsible for A is rotated by some angle, ?

Rotation of a vector by an angle means multiplication by a rotation matrix: 0 0 1 1' and 'E R E E R E

1

1 1 0 0

1 1

0 0 0

'

' ' '

E R E R E R R R E

R R R E R R E E

A A

A A A

1' R R A A

cos( ) sin( )

sin( ) cos( )R

Thus:

Rotating E1 by and inserting the identity matrix R()-1 R(), we have:

where:

Rotated Jones matrix for a polarizer

Applying this result to an x-polarizer:

cos( ) sin( ) 1 0 cos( ) sin( )

sin( ) cos( ) 0 0 sin( ) cos( )xA

1' R R A A

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

sin( ) cos( ) 0 0xA

2

2

cos ( ) cos( )sin( )

cos( )sin( ) sin ( )xA

1/ 2 1/ 245

1/ 2 1/ 2xA

1

0xA

for small angles,

Jones Matrices for standard components

To model the effect of many media on light's polarization state, we use many Jones matrices.

To model the effects of more than one component on the polarization state, just multiply the input polarization Jones vector by all of the Jones matrices:

1 3 2 1 0E EA A A

Remember to use the correct order!

A single Jones matrix (the product of the individual Jones matrices) can describe the combination of several components.

Multiplying Jones Matrices

Crossed polarizers:

0 0 1 0 0 0

0 1 0 0 0 0

y xA A

x

y z

1 0y xE EA A

0E1E

x-pol

y-pol

so no light leaks through.

Uncrossed polarizers(slightly):

0 0 1 0 0

0 1 0 0

y xA A

0E1E

rotatedx-pol

y-pol

00 0

0x x

y y x

E E

E E E

y xA A So Iout ≈2 Iin,x

Recall that, when the phases of the x- and y-polarizations fluctuate, 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:

Unfortunately, this is difficult to analyze using Jones matrices.

0

0

( , ) Re exp ( )

( , ) Re exp ( )

x x x

y y y

E z t E i kz t t

E z t E i kz t t

0

0

1

exp ( ) ( )yx y

x

Ei t t

E

In practice, the amplitudes vary, too!

Stokes Parameters

To treat fully, partially, or unpolarized light, we define Stokes parameters.

Suppose we have four detectors, three with polarizers in front of them:

#0 detects total irradiance............................................I0

#1 detects horizontally polarized irradiance..........…...I1

#2 detects +45° polarized irradiance............................I2

#3 detects right circularly polarized irradiance.....…….I3

The Stokes parameters:

S0 I0 S1 2I1 – I0 S2 2I2 – I0 S3 2I3 – I0

1/22 2 21 2 3 0Degree of polarization = S + S + S / S = 1 for polarized light

= 0 for unpolarized light

Mueller Matrices multiply Stokes vectors

We can write the four Stokes parameters in vector form:

And we can define matrices that multiply them,just as Jones matrices multiply Jones vectors.

0

1

2

3

S

SS

S

S

To model the effects of more than one medium on the polarizationstate, just multiply the input polarization Stokes vector by all of the Mueller matrices:

Sout = M3 M2 M1 Sin

Stokes vectors (and Jones vectors for comparison)

Mueller Matrices (and Jones Matrices for comparison)