Polarimetry

What is polarization?

Linear polarization refers to photons with their electric vectors always aligned in the same direction (below). Circular polarization is when the tip of the electric vector of a photon describes a circle as it propagates – or equivalently if the electric vector traces a helix around the direction of propagation.

Why do we care about polarization?

Processes that lead to significant polarization include: Reflection from solid surfaces, e.g., moon, terrestrial planets, asteroids Scattering of light by small dust grains, e.g., interstellar polarization Scattering by molecules, e.g., in the atmospheres of the planets Scattering by free electrons, e.g., envelopes of early-type stars Zeeman effect, e.g., in radio-frequency HI and molecular emission lines Strongly magnetized plasma, e.g., white dwarfs Synchrotron emission, e.g., supernova remnants, AGN

The Egg Nebula is a protoplanetary nebula, that is a star that has ejected its outer shells and is evolving into a planetary. The bright blue

lobes are lit up by scattered light, as can be seen from the uniform direction of the polarization vectors.

Orion Becklin-Neugebauer-Kleinmann-Low region: finding the energy source through polarimetry in a heavily obscured region.

Antonucci & Miller used spectropolarimetry of NGC 1068 to establish the unified theory of AGN: note the broad, polarized H beta line that is scattered

over the rim of the obscuring “torus”

Polarization is also characteristic of non-thermal emission, e.g. this map of the Crab Nebula at 20cm (Velusamy 1985). Note how the

pattern is very different from scattering, perhaps tracing a toroidal magnetic field.

Interstellar polarization arises through scattering by elongated and aligned grains

)9()cos()cos( 00

jtEitEjEiEE yyxxyx

)11(2sinsin2

2sin2sin2coscos2

2cos2cos2cos22

22

VEyx

Eyx

Eyx

yx

IPIPEEV

IPIPEEU

IPIPEEQ

EEI

)12(2cos EPP

)13(2sin EV PP

In general, the electric vector of a polarized beam of light is described by:

and it traces an ellipse in space as the light propagates. = x - y is the phase difference between the x and y vibrations. The ellipse is described by the Stokes parameters:

I is the total intensity. characterizes the eccentricity and V is the degree of circular polarization.

The amount of linear polarization is:

The angle of linear polarization is characterized by . It comes into Q and U multiplied by 2 because linear polarization is degenerate over 180 degrees.

)14(arctan2

22

Q

U

I

UQP

I

If we know the Stokes parameters we can calculate the polarization:

The Stokes parameters are a convenient way to describe polarization because, for incoherent light, the Stokes parameters of a combination of several beams of light are the sums of the respective Stokes parameters for each beam. A polarization analyzer is needed to make polarization measurements. It is a device that divides a beam of light in half, one half polarized in the principal plane of the analyzer and the other polarized in the orthogonal plane.

A grid of very finely spaced wires makes an analyzer because the wires absorb the electric vectors of photons where they are parallel to the wires:

How it works:

A real example: a wire grid polarization analyzer or polarizer

Here are some wire grid polarizers. Plastic polaroid film (familiar in sunglasses) works on a similar principle: start with polyvinyl alcohol plastic doped with iodine. The sheet is stretched during its manufacturing so the molecular chains are aligned, and these chains are rendered conductive by electrons freed from the iodine dopant.

A simple polarimeter would just put a few of these into a photometer filter wheel (at different angles) and measure sequentially. However, it would not be able to reach very low levels of polarization. Why not??

Here are some wire grid polarizers. Plastic polaroid film (familiar in sunglasses) works on a similar principle: start with polyvinyl alcohol plastic doped with iodine. The sheet is stretched during its manufacturing so the molecular chains are aligned, and these chains are rendered conductive by electrons freed from the iodine dopant.

A simple polarimeter would just put a few of these into a photometer filter wheel (at different angles) and measure sequentially. However, it would not be able to reach very low levels of polarization. Why not?? Because we would be trying to get our signal as the difference between two large numbers – always a bad procedure unless there is no other choice

Uniaxial materials, at 590 nm

Material no ne Δn

beryl Be3Al2(SiO3)6

1.602 1.557 -0.045

calcite CaCO3 1.658 1.486 -0.172

calomel Hg2Cl2

1.973 2.656 +0.683

ice H2O 1.309 1.313 +0.004

lithium niobate LiNbO3

2.272 2.187 -0.085

magnesium fluoride MgF2

1.380 1.385 +0.006

quartz SiO2 1.544 1.553 +0.009

ruby Al2O3 1.770 1.762 -0.008

rutile TiO2 2.616 2.903 +0.287

peridot (Mg, Fe)2SiO4

1.690 1.654 -0.036

sapphire Al2O3

1.768 1.760 -0.008

sodium nitrate NaNO3

1.587 1.336 -0.251

tourmaline (complex silicate )

1.669 1.638 -0.031

zircon, high ZrSiO4

1.960 2.015 +0.055

We can make a “better” analyzer using birefringence, as with the calcite below. It has a substantial difference in the index of refraction for two orthogonal polarizations (relative to the crystal axis).

http://en.wikipedia.org/wiki/Berylhttp://en.wikipedia.org/wiki/Calcitehttp://en.wikipedia.org/wiki/Calomelhttp://en.wikipedia.org/wiki/Icehttp://en.wikipedia.org/wiki/Lithium_niobatehttp://en.wikipedia.org/wiki/Lithium_niobatehttp://en.wikipedia.org/wiki/Magnesium_fluoridehttp://en.wikipedia.org/wiki/Magnesium_fluoridehttp://en.wikipedia.org/wiki/Quartzhttp://en.wikipedia.org/wiki/Rubyhttp://en.wikipedia.org/wiki/Rutilehttp://en.wikipedia.org/wiki/Peridothttp://en.wikipedia.org/wiki/Sapphirehttp://en.wikipedia.org/wiki/Sodium_nitratehttp://en.wikipedia.org/wiki/Sodium_nitratehttp://en.wikipedia.org/wiki/Tourmalinehttp://en.wikipedia.org/wiki/Zirconhttp://en.wikipedia.org/wiki/Birefringencehttp://en.wikipedia.org/wiki/Birefringencehttp://en.wikipedia.org/wiki/Birefringencehttp://en.wikipedia.org/wiki/Birefringence

We can combine different pieces of a birefringent crystal with their axes in different directions to make various kinds of prism that separate light into two polarizations. This one is a Glan-Thompson prism that rejects one direction by

total internal reflection.

This one is a Wollaston prism.

Here is a polarimeter based on a Wollaston prism. We can take the signal as the difference in outputs of detectors A and B. Since they

won’t be exactly the same, we need to rotate the prism, swap detectors, or……..?

Here is a polarimeter based on a Wollaston prism. We can take the signal as the difference in outputs of detectors A and B. Since they

won’t be exactly the same, we need to rotate the entire instrument on the telescope, or even better rotate the telescope!

But that sounds pretty awkward.

Manipulating polarized light: If we shift, or retard the electric vector by half the wavelength, we can rotate the plane of the polarization. If we rotate the retarder, then for a change of angle of , the plane of polarization changes by 2. Retarders can be made readily from birefringent crystals.

Here is an implementation, SPOL. The half-wave-retarder is the “rotating waveplate.” It is put directly in the beam from the telescope to avoid extra polarization that occurs in all off-axis reflections. After that, reflections do not matter. So this instrument gives us a pair of spectra and we can change the polarization for these spectra by rotating the waveplate, even reversing the roles of the two beams out of the Wollaston prism. We can calibrate by putting an analyzer into the beam ahead of the rotating waveplate and measuring the result.

All this might be clearer

from this schematic

diagram.

Note that the grating is

strongly polarizing, so this

design is critical for good

performance.

The key is having a retarder that does nothing to the beam other than retard it – no beam motion or transmission changes with rotation. A mechanical waveplate is pretty

good, but something that does not move would be better. There are certain crystals that retard depending on the applied voltage.

Birefringence can also be induced in a crystal by stressing it.

http://www.hindsinstruments.com/PEM_Components/Technology/principlesOfOperation.aspx

Photoelastic modulators vibrate the crystal at its resonant frequency (about 50kHz is typical) so large forces are not required. Two in series can be used to produce a modulation at the difference frequency, in the Hz range.

http://www.hindsinstruments.com/PEM_Components/Technology/principlesOfOperation.aspx

Circular Polarization

• Similar approaches can measure circular polarization, since a quarter-wave retarder converts it to linear – and the linear can be measured as above.

)16()2sin2cos(

2

1

)2sin2cos(2

1

UQII

UQII

OP

PP

Interpreting the Measurements For simplicity, assume a perfect analyzer, Tl = 0.5 and Tr = 0, where Tl. Is the transmittance for unpolarized light. And Tr is that is the transmittance with two analyzers crossed. Then the intensities emerging in the principal and orthogonal planes are

where is the angle between the north celestial pole and the principal plane. Let

)17(2sin2cos

I

UQ

II

IIR

OPPP

OPPP

We can determine the polarization through measurements at a number of values of φ. For φ=0, we get R0 = Q/I = q, while for φ=45o, we get R45 = U/I = u. Then,

)18(arctan2

1

22

q

u

uqP

It is convenient to use a diagram of q vs. u, with angles in 2θ, to represent polarization measurements. For example, different measurements can be combined vectorially on this diagram.

Error Analysis

Error analysis for polarimetry is generally straightforward, except when it comes to the position angle for measurements at low signal to noise. Assume that the standard deviations of q, u, and P are all about the same. Then the uncertainty in the polarization angle is

)19()(

65.28)(P

P

Thus, nominally a measurement at only one standard deviation level of significance (that is, a non-detection) achieves a polarization measurement within 28.65o. This high accuracy is non-physical – the probability distribution for θ at low signal to noise does not have the Gaussian distribution assumed in most error analyses (e.g., Wardle and Kronberg 1974). Similarly, P is always positive and hence does not have the Gaussian distribution around zero assumed in normal error analysis.