1. Polarization Linear, circular, and elliptical polarization
Mathematics of polarization
Uniaxial crystals
Birefringence
Polarizers
Prof. Rick Trebino
Georgia Tech

2. 45° Polarization
Here, the complex amplitude, E0, is the same for each component.
So the components are always in phase. ~

3. Arbitrary-Angle Linear Polarizationx y
E-field variation over time (and space) a
Here, the y-component is in phase
with the x-component,
but has different magnitude.

4. The Mathematics of Polarization
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: ~ ~ ~ ~ ~ ~

5. Circular (or Helical) Polarization
Or, more generally,
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.
The resulting E-field rotates
counterclockwise around the
k-vector (looking along k).

6. Right vs. Left Circular (or Helical) Polarizationx y
E-field variation over time (and space)
kz-wt = 0°
kz-wt = 90°
Or, more generally,
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, but in the
other direction.
The resulting E-field rotates
clockwise around the
k-vector (looking along k).

7. Unequal arbitrary-relative-phase components yield elliptical polarizationx y
E-field variation over time (and space)
where
Or, more generally,
The resulting E-field can rotate clockwise or counter-clockwise around the k-vector (looking along k).

8. The mathematics of circular and elliptical polarization
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.

9. When the phases of the x- and y-polarizations fluctuate, we say 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:
As long as the time-varying relative phase, qx(t) - qy(t), fluctuates, the light will not remain in a single polarization state and hence is unpolarized.
In practice, the amplitudes vary, too!

10. Light with very complex polarization vs. 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, we
should refer to this light as locally polarized.

11. An intentionally spatially varying polarization can be interesting.
Imagine a donut-shaped beam with radial polarization.
Focus
Longitudinal electric field at the focus!
Lens
This type of beam can also focus to a smaller spot than a beam with a spatially uniform polarization!

12. A complex polarization spatial dependence…
An optical vortex y x

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

14. Birefringence
The x- and y-polarizations can see different refractive index curves.

15. Uniaxial crystals have an optic axis
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.
Light with any other polarization must be broken down into its ordinary and extraordinary components, considered individually, and recombined afterward.

16. Birefringence can separate the two polarizations into separate beamsno ne
o-ray
e-ray
Due to Snell's Law, light of different polarizations will bend by different amounts at an interface.

17. Calcite
Calcite is one of the most birefringent materials known.

18. Birefringent Materials

19. How to make a polarizer R R T T
We’ll use birefringence, and we’ll take advantage of the Fresnel equations we derived last time.
Perpendicular polarization
Incidence angle, qi
1.0 .5
0° ° ° ° R T
Parallel polarization
Incidence angle, qi
1.0 .5
0° ° ° ° R T
When ni > nt, there’s a steep variation in reflectivity with incidence angle. This corresponds to a steep variation with refractive index.

20. Polarizers take advantage of birefringence, Brewster's angle, and total internal reflection.
The Glan-Thompson polarizer uses two prisms of calcite (with parallel optical axes), glued together with Canada balsam cement (n = 1.55).
no = 1.66
Cement n = 1.55
ne = 1.49
Optic axis
Alternatively, the optical axis points out of the screen, and the polarizations are also rotated.
Snell’s Law separates the beams at the entrance. The perpendicular polarization then goes from high index (1.66) to low (1.55) and undergoes total internal reflection, while the parallel polarization is transmitted near Brewster's angle.
If the gap is air, it’s called the Glan-Taylor polarizer.

21. Even better, rotate the second prism: The Nicol polarizer.
no = 1.66
ne = 1.49
Optic axis
Optic axis (out of page)
Brewster’s Angle
TIR
Combine two prisms of calcite, rotated so that the ordinary polarization in 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 parallel polarization is transmitted near Brewster's angle.
Image from Wikipedia

22. Real Polarizers Air-spaced polarizers
Photographs taken from a Lambda Research Optics advertisement
Air-spaced polarizers

23. Wollaston Polarizing Beam Splitter
The Wollaston polarizing beam splitter uses two rotated birefringent prisms, but relies only on refraction.
The ordinary and extraordinary rays have different refractive indices and so diverge.

24. The Pile-of-Plates Polarizer
After numerous Brewster-angle transmissions, mainly the parallel polarization remains.
Unpolarized input light
Unfortunately, only about 10% of the perpendicular polarization is reflected on each surface, so you need a big pile of plates!

25. Dielectric polarizers
A multi-layer coating (which uses interference; we’ll get to this later) can also act as a polarizer. It still uses the same basic ideas of TIR and Brewster’s angle.
Ealing Optics catalog and Melles Griot catalog
and
Glass

26. Wire Grid Polarizer Input light contains both polarizations
The light can excite electrons to move along the wires, which then emit light that cancels the input light. This cannot happen perpendicular to the wires. Such polarizers work best in the IR.
Polaroid sheet polarizers use the same idea, but with long polymers.

27. 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.

28. The wires need not be very long.
Hoya has designed a wire-grid polarizer for telecom applications that uses small elongated copper particles.
HOYA's CUPO is designed as a versatile glass polarizer for the telecommunication industry. CUPO maximizes your device performance at 0.2mm thickness and operates at 1310nm, 1480nm and 1550nm.
HOYA has developed this glass polarizer using light absorption of copper particles created by an electron plasma resonance mechanism. This polarizer contains elongated copper particles aligned along a common axis. This feature produces a high performance polarization with high isolation and low insertion loss. With thickness as low as 0.2mm, it is compact enough for the most miniaturized optical components.
HOYA's CUPO shows no performance degradation in most Telcordia environmental conditions. CUPO is particularly suited for applications in optical communication where high reliability is required.
Features
High Isolation
Low Loss
High Reliability
0.2mm Thickness
Applications
Optical Isolators
Liquid Crystal Switches
Circulators
Product Specifications
Wavelength (nm) Isolation (dB)> 40> 40> 40Insertion Loss (dB)< 0.1< 0.1< 0.1Refractive Index Reflectance % (AR coating both sides) Thermal PropertiesTransformation Temperature (Tg)477°CSag Temperature (Ts)545°CCoefficient of Thermal Expansion63.4 x 10-7/°CMechanical PropertiesYoung's Modulus62.4 GPaShear Modulus25.5 GPaPoisson's Ratio0.221Knoop Hardness433 kgf/mm2Physical PropertiesSpecific Gravity2.31
Extinction coefficient > 10, Transmission > 99%

29. 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.
0° Polarizer
0° Polarizer
90° Polarizer
Type of polarizer Ext. Ratio Transmission Cost
Calcite: > 95% $
Dielectric: > 99% $
Polaroid sheet: ~ 50% $1 - 2

30. Visible wire-grid polarizer performance
Transmission
A polarizer’s performance can vary with wavelength and incidence angle.
Extinction ratio
The overall transmission is also important, as is the amount of the wrong polarization in each beam.
Stephen Arnold
Lightwave Enterprises Inc., West Henrietta, NY, USA
Eric Gardner, Douglas Hansen, Ray Perkins
MOXTEK, Inc., Orem, Utah, USA