1. Optical Activity & Jones Matrices
Prof. Rick Trebino
Georgia Tech
Ways to actively control polarization
Pockels' Effect
Kerr Effect
Photo-elasticity
Optical Activity
Faraday Effect
Jones Matrices
Unpolarized light, Stokes Parameters, & Mueller Matrices
frog/lectures

2. The Pockels' Effect +V An electric field can induce birefringence.
Electro-optic medium
Transparent electrode
Polarizer
Analyzer +V
The Pockels' effect allows control over the polarization rotation.

3. The Pockels Effect: Electro-optic constants
where Dj is the relative phase shift, V is the applied voltage, and r63 is the electro-optic constant of the material.
Vl/2 is called the half-wave voltage.

4. 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.
100% 0%
Time
Cavity Loss
Cavity Gain
Output intensity
This yields a short “giant” high-power pulse.
The pulse length is limited by the round-trip time of the laser and is usually ns long.

5. 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
0° Polarizer
Mirror
0° Polarizer
Mirror
Pockels’ cell as wave plate w/ axes at ±45°
Pockels’ cell as an isotropic medium
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.

6. The Kerr effect: the polarization rotation is proportional to the Kerr constant and E2
where:
Dn 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 light
wave. Usually very high irradiances from ultrashort laser pulses are
required to create quarter-wave rotations.

7. Photo-elasticity: Strain-induced birefringence
Clear plastic drawing device (“French curve”) between crossed polarizers
Image from

8. Strain-Induced birefringence in diamond
An artificially grown diamond with nitrogen impurities between crossed polarizers
The diamond is grown as [1,1,1]sectors(right, lower left, upper left) from a tiny seed (in the centre). At the sector interfaces there is a slight crystal-lattice misalignment, which allows for the incorporation of nitrogen (slightly absorbing, see intensity image). The misalignment together with the presence of nitrogen causes a build-up of strain. The strain from one boundary dominates half of each sector (j -image).
Images from
Caused by strain associated with growth boundaries

9. Strain-induced birefringence in thin sections of rock

10. More Photo-elasticity
If there's not enough strain in a medium to begin with, you can always apply stress and add more yourself!
Clear plastic between crossed polarizers
carbon.cudenver.edu/.../CLARIstyle_03.html
You can use this effect to improve the performance of polarizers.

11. 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.
Jpeg from
Some substances rotate the polarization clockwise (dextrorotatory) and some produce a counterclockwise rotation (levorotatory).

12. 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.
Wikipedia
Dextrorotation [1] is the property of rotating plane polarized light clockwise. Laevorotation is the opposite of dextrorotation.

13. 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.
If you’d like to look for signs of life on other planets, look for chirality.
Occasionally, a molecule of the wrong chirality can cause serious illness (e.g., thalidimide) while its other enantiomer is harmless.

14. 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.

15. 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 left
circular components.
When the principal axes of a medium are circular, as they are
when optical activity is present, this is required. We must then
decompose linear polarization into its circular components:

16. Math of Optical Activity–Circular Principal Axes
At the entrance to an optically active medium, an x-polarized beam (R + L, neglecting the √2 in all terms) will be:
Note that this mess just adds up to x-polarized light!

17. Math of Optical Activity–Circular Principal 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:

18. Math of Optical Activity–Circular Principal Axes (continued)

19. Math of Optical Activity–Circular Principal Axes (continued)

20. 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.

21. The Faraday Effect A magnetic field can induce optical activity.
Magneto-optic medium
Polarizer
Analyzer +V
Magnetic field
The Faraday effect allows control over the polarization rotation.

22. The Faraday effect: the polarization rotation is proportional to the Verdet constant.
b = V B d
where:
b is the polarization rotation angle,
B is the magnetic field strength,
d is the distance,
V is the Verdet constant of the material.

23. Polarization-independent Optical Isolator
We could use a polarizer and quarter-wave plate or a Faraday rotator, but they require polarized light.
Input beam
Optical fiber
Lens
Optic axis (45° into page)
This device spatially separates the return (reflected) beam polarizations from the input beam.
Optic axis (into page)
45° rotation
The polarization independent isolator is made of three parts, an input birefringent wedge (with its ordinary polarization direction vertical and its extra-ordinary polarization direction horizontal), a Faraday rotator, and an output birefringent wedge (with its ordinary polarization direction at 45 degrees, and its extra-ordinary polarization direction at -45 degrees).
Light travelling in the forward direction is split by the input birefringent wedge into its vertical (0 degrees) and horizontal (90 degrees) components, called the ordinary-ray (o-ray) and the extra-ordinary ray (e-ray) respectively. The Faraday rotator rotates both the o-ray and e-ray by 45 degrees. This means the o-ray is now at 45 degrees, and the e-ray is at -45 degrees. The output birefringent wedge then recombines the two components.
Light travelling in the backward direction is separated into its o-ray at 45 degrees and e-ray at -45 degrees by the birefringent wedge. The Faraday Rotator again rotates both the rays by 45 degrees. Now the o-ray is at 90 degrees, and the e-ray is at 0 degrees. Instead of being focused by the second birefringent wedge, the rays diverge.
Typically collimators are used on either side of the isolator. In the transmitted direction the beam is split and then combined and focused into the output collimator. In the isolated direction the beam is split, and then diverged, so it does not focus at the collimator.
The figure shows the propagation of light through a polarization independent isolator. The forward travelling light is shown in blue, and the backward propagating light is shown in red. The rays were traced using an ordinary refractive index of 2, and an extra-ordinary refractive index of 3. The wedge angle is 7 degrees.
45° rotation

24. To model the effect of a medium on light's polarization 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: ~ ~ ~

25. Other Jones matrices A y-polarizer: A half-wave plate:
A half-wave plate rotates 45-degree-polarization to -45-degree, and vice versa.
A quarter-wave plate:

26. A wave plate is not a wave plate if it’s oriented wrong.
Wave plate w/ axes at 0° or 90°
0° or 90° Polarizer
Remember that a wave plate wants ±45° (or circular) polarization.
If it sees, say, x polarization, nothing happens.
AHWP
So use Jones matrices until you’re really on top of this!!!

27. Rotated Jones matrices
Okay, so E1 = A E0. What about when the polarizer or wave plate responsible for A is rotated by some angle, q ?
Rotation of a vector by an angle q means multiplication by a rotation matrix:
where:
Rotating E1 by q and inserting the identity matrix R(q)-1 R(q), we have:
Thus:

28. Rotated Jones matrix for a polarizer
Applying this result to an x-polarizer:
for small angles, e

29. Jones Matrices for standard components

30. 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:
A single Jones matrix (the product of the individual Jones matrices) can describe the combination of several components.
Remember to use the correct order!

31. Multiplying Jones Matricesx y z
Multiplying Jones Matrices
x-pol
y-pol
Crossed polarizers:
so no light leaks through.
rotated
x-pol
y-pol
Uncrossed polarizers
(slightly):
So Iout ≈ e2 Iin,x

32. Recall that, when the phases of the x- and y-polarizations fluctuate, the light is "unpolarized."
where qx(t) and qy(t) are functions that vary on a time scale slower than
1/w, but faster than you can measure.
The polarization state (Jones vector) will be:
Unfortunately, this is difficult to analyze using Jones matrices.
In practice, the amplitudes vary, too!

33. Stokes Parameters S0 º I0 S1 º 2I1 – I0 S2 º 2I2 – I0 S3 º 2I3 – I0
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 º I S1 º 2I1 – I S2 º 2I2 – I0 S3 º 2I3 – I0
= 1 for polarized light
= 0 for unpolarized light

34. 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.
To model the effects of more than one medium on the polarization
state, just multiply the input polarization Stokes vector by all of the
Mueller matrices:
Sout = M3 M2 M1 Sin

35. Stokes vectors (and Jones vectors for comparison)

36. Mueller Matrices (and Jones Matrices for comparison)