1. More Polarization Polarization rotators Compensators
Prof. Rick Trebino, Georgia Tech
Polarization rotators
Wave plates
Rhombs
Compensators
Circular polarizers
Reflection & polarization
Scattering & polarization
Polarization Spectroscopy

2. Wave plates x
Wave plate z
Optic axis y
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 changes the polarization.
+45° Polarization
-45° Polarization
Polarization state: }
Input: }
Output:

3. Wave plates (continued)
Wave plate output polarization state:
(45-degree input polarization)
Quarter-wave plate
Half-wave plate
A quarter-wave plate creates circular polarization, and a half-wave plate rotates linear polarization by 90.
We can add an additional 2mp without changing the polarization, so the polarization cycles through this evolution as d increases further.

4. Half-wave plate Ex(z) nx > ny Ey(z)
When a beam propagates through a half-wave plate, one polarization experiences half of a wavelength more phase delay than the other.
Ex(z)
Ey(z)
nx > ny
In phase
180° out of phase
Half-wave plate
If the incident polarization is +45° to the principal axes, then the output polarization is rotated by 90° to -45°.

5. 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.
±45° Polarizer
0° or 90° Polarizer
Wave plate w/ axes at 0° or 90°
Wave plate w/ axes at ±45°
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.

6. How NOT to use a wave plate
If the input polarization is parallel to the wave plate principal axes, no polarization rotation occurs!
0° or 90° Polarizer
±45° Polarizer
Wave plate w/ axes at 0° or 90°
Wave plate w/ axes at ±45°
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.

7. Thickness of wave plates
When a wave plate has less than 2p 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: d
Using green light at 500 nm and quartz, whose refractive indices are ne – no = – = , we find:
d = 13.7 mm
This is so thin that it is very fragile and very difficult to manufacture.

8. Multi-order wave plates
A multi-order wave plate has more than 2p relative phase delay.
We can design a twentieth-order quarter-wave plate with 20¼ waves of relative phase delay, instead of just ¼: d
d = 1.1 mm
This is thicker, but it’s now 81 times more wavelength dependent!
It’s also temperature dependent due to n’s dependence on temperature.

9. 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:
Optic axes
Input
beam
Output
beam d1 d2
First plate
Second plate
Now, as long as d1 – d2 is equal to the thickness of the thin zero-order wave plate, this optic behaves like the really thin one! This is ideal.

10. The Babinet Compensator rotates polarization by an arbitrary amount.
Top wedge is cut with fast and slow axes opposite to those of bottom wedge.
Optic axes d1
Input
beam
Output
beam
Slide one wedge with
respect to the other,
changing d1 and/or d2. d1 d2
In practice, the two wedges are contacted.
Top wedge
Bottom wedge

11. Polarization Mode Dispersion plagues broadband optical-fiber communications.
Imagine just a tiny bit of birefringence, Dn, but over a distance of 1000 km…
Distance
Polarization state at receiver =
A “Hinge” Model for the Temporal Dynamics of Polarization Mode Dispersion
Image with permission from Misha Brodsky, Misha Boroditsky, Peter Magill, Nicholas Frigo, Moshe Tur*, AT&T Labs –Research
* Tel Aviv University
If l = 1.5 mm, then Dn ~ can rotate the polarization by 90º!
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.

12. A Lyot Filter is a wave plate between polarizers.
# Lyot filters 1 2 3 4 5
Wavelength
Transmission
Because the wave-plate polarization rotation varies with wavelength, placing one between polarizers yields wavelength-dependent transmission.
Polarizer
Polarizer
Wave plate
Plots from
Placing several Lyot filters (with factor-of-2 different-thickness wave-plates) in a row yields a narrowband filter, which transmits only a narrow range of wavelengths.

13. Circular polarizers
Quarter wave plate (QWP)
±45° Polarizer
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
Additional QWP and linear polarizer comprise a circular "analyzer."
45° Polarizer
QWP
-45° polarized light
45° Polarizer
QWP
45° polarized light
Circularly polarized light

14. The Fresnel rhomb uses total-internal-reflection phase shifts to rotate polarization
After two internal reflections, input 45° polarization rotates to
circular polarization.
54.6°
45° polarization
Circular polarization
Elliptical polarization
Unlike polarization rotation in a wave plate, this polarization rotation
is nearly wavelength independent.

15. Polarization Spectroscopy
0 polarizer
90 polarizer
Yellow filter (rejects red)
Typical molecule
Birefringence!
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.

16. Depolarization by reflection or transmission
Suppose that 45° polarization is incident on an interface, which has different parallel (x) and perpendicular (y) reflection coefficients.
Incident light fields: y
Incident polarization
Reflected polarization
(if rx >ry) x
Reflected light fields:
Unless light is purely parallel or perpendicularly polarized (or incident
at 0°), polarization rotation will occur in reflection (or transmission).

17. Fresnel Reflection and Depolarization
Reflectances
Incidence angle, qi
1.0 .5
0° ° ° °
R||
Interface: ni = 1 nt = 1.5
Fresnel reflections are a common cause of polarization rotation.
The ratio of polarization-component strengths will change on transmission through or reflection off an interface.
This effect is particularly strong near Brewster's angle.

18. Cruddy stuff depolarizes
Cruddy stuff is very non-uniform: a series of interfaces at random angles.
Crossed polarizers with a piece
of wax paper in between.
Human tissue completely depolarizes light in the visible, IR, and UV.

19. The atmosphere depolarizes light slightly.
Odd-angle interfaces between regions of warm and cool air cause air to slightly depolarize light.
Star
Cooler regions of air (with higher refractive index)
Droplets of water (i.e., clouds) completely depolarize light.

20. Glare is horizontally polarized
Puddle reflection viewed
through polarizer that
transmits only horizontally
polarized light
Puddle reflection viewed
through polarizer that
transmits only vertically
polarized light
Light reflected
into our eyes
from the puddle
reflects at about
Brewster's Angle.
So parallel
(i.e., vertical)
polarization sees
zero reflection.
Polarizer sunglasses transmit only vertically polarized light.

21. Polarizers are very useful in photography.
Without a polarizer
With a polarizer
The effect of a polarizer is probably the one “filter” effect that you can’t reproduce later using Photoshop!

22. Scattering by molecules is not spherically symmetric
Scattering by molecules is not spherically symmetric. It has a dipole pattern.
The field emitted by an oscillating dipole excited by a vertically
polarized light wave:
Direction of light excitation
E-field and electron oscillation
Emitted intensity pattern
Image from Sergei Popov
Directions of scat-
tered light E-field
No light is emitted along
direction of oscillation!
Directions of scat-
tered light E-field

23. Dipole Emission Pattern from an Antenna
Analogous to a molecule emitting light, an antenna emits a dipole pattern at much lower frequency and longer wavelength:
The pattern is somewhat distorted by the earth and nearby objects.

24. Scattering of polarized light
No light is scattered along the input field direction, i.e. with k parallel to E.
Vertically polarized
input light
Horizontally polarized
input light

25. Scattering of unpolarized light
Again, no light is scattered along the input field direction,
i.e. with k parallel to Einput.

26. Scattering in the Earth's atmosphere leads to interesting polarization properties of skylight.
Sun's rays

27. Skylight is polarized if the sun is to your side.
Right-angle scattering
is polarized
This polarizer transmits
horizontal polarization
(of which there is very little).
Polarizer transmitting vertical polarization
Multiple scattering yields some light of the other polarization.
In clouds, much multiple scattering occurs, and light there is
unpolarized.

28. Brewster's Angle Revisited
A complex trigonometric calcu- lation reveals that the reflection
coefficient for parallel-polarized
light goes to zero for Brewster's
angle incidence, tan(qi) = nt / ni
When the reflected beam makes a
right angle with the transmitted beam,
and the polarization is parallel, then
no scattering can occur, due to the
scattered dipole emission pattern.
But our right-angle assumption
implies that qi + qt = 90°. So:
Thus, ni nt qi qt
qi +qt = 90°