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 BecklinNeugebauerKleinmann-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
In general, the electric vector of a polarized beam of light is described by:





E  E x i  E y j  E0 x cos(t   x ) i  E0 y cos(t   y ) j
(9)
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   E x2    E y2 
Q   E x2    E y2   IPE cos 2 cos 2  IP cos 2
U   2 E x E y cos    IPE cos 2 sin 2  IP sin 2
V   2 E x E y sin    IPE sin 2  IPV
I is the total intensity.
 characterizes the eccentricity
and V is the degree of circular
polarization.
PV  PE sin 2
(13)
The amount of linear
polarization is:
P  PE cos 2
(12)
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.
(11)
If we know the Stokes parameters we can calculate the polarization:
I
Q2  U 2
P
I
U 
2  arctan  
Q
(14)
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
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).
Uniaxial materials, at 590 nm
Material
no
beryl
1.602
Be3Al2(SiO3)6
calcite CaCO3 1.658
calomel
1.973
Hg2Cl2
ice H2O
1.309
lithium
niobate
2.272
LiNbO3
magnesium
1.380
fluoride MgF2
quartz SiO2
1.544
ruby Al2O3
1.770
rutile TiO2
2.616
peridot (Mg,
1.690
Fe)2SiO4
sapphire
1.768
Al2O3
ne
Δn
1.557
-0.045
1.486
-0.172
2.656
+0.683
1.313
+0.004
2.187
-0.085
1.385
+0.006
1.553
1.762
2.903
+0.009
-0.008
+0.287
1.654
-0.036
1.760
-0.008
1.336
-0.251
1.669
1.638
-0.031
1.960
2.015
+0.055
sodium
1.587
nitrate NaNO3
tourmaline
(complex
silicate )
zircon, high
ZrSiO4
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.
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.
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.
Interpreting the Measurements
For simplicity, assume a perfect analyzer, Tl = 0.5 and Tr = 0. Then the intensities emerging
in the principal and orthogonal planes are
1
( I  Q cos 2  U sin 2 )
2
1
 ( I  Q cos 2  U sin 2 )
2
I PP 
I OP
(16)
where  is the angle between the north celestial
pole and the principal plane. Let
I PP  I OP Q cos 2  U sin 2
R
I PP  I OP

I
(17)
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,
P  q2  u2

u
1
arctan  
2
q
(18)
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
 ( )  28.65

 ( P)
P
(19)
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.