polarization with the very large array

Atacama Large Millimeter/submillimeter Array

Karl G. Jansky Very Large Array

Robert C. Byrd Green Bank Telescope

Very Long Baseline Array

Polarization with the Very Large Array

Rick Perley

NRAO -- Socorro

Jansky VLA

Goals of This Presentation

• Polarimetry (especially interferometric polarimetry) is a

daunting topic.

• In this short presentation, I will introduce the subject, and

explain the ‘why, how, when, and where’ of polarimetry.

• Subjects:

– Why do polarimetry?

– What is polarimetry?

– Definitions : polarization, Stokes parameters, Stokes visibilities

– Short, intuitive explanation of how interferometers do polarimetry.

– Summary of calibration of polarimetry data.

– A nice example of imaging polarimetry (the planet Mars).

ASIAA Workshop on the Very Large Array 2

Jansky VLA Why Measure Polarization? • In short – to access extra physics not available in total intensity alone.

– Processes which generate polarized radiation:

• Synchrotron emission: Up to ~80% linearly polarized, with

no circular polarization. Measurement provides information

on strength and orientation of magnetic fields, level of

turbulence.

• Zeeman line splitting: Presence of B-field splits RCP and

LCP components of spectral lines by amount proportional to

the field strength.

– Processes which modify polarization state:

• Faraday rotation: Magneto-ionic region rotates plane of

linear polarization. Measurement of rotation, combined with

density estimate, gives B-field estimate.

• Free electron scattering: Induces a linear polarization

which can indicate the origin of the scattered radiation.

ASIAA Workshop on the Very Large Array

Jansky VLA

Example: Linear Polarization of Cygnus A • VLA @ 8.5 GHz Synchrotron emission B-vectors Perley & Carilli (1996)

• Field strengths in lobes of tens of mG, fields lie along brightness gradients.

10 kpc


Jansky VLA

M51 – magnetic fields in spiral arms


• Radio data from VLA + Effelsberg:

4.86 GHz, 15” resolution (700 pc)

• Contours show radio brightness,

dashed lines show magnetic field

orientation.

• Overlaid on optical image from

Hubble

• Clearly shows the magnetic fields

following the spiral pattern.

• From Fletcher et al. (2011)

Jansky VLA Example: Faraday Rotation for Cygnus A

– Color-coded map of the Rotation Measure – the slope of Faraday Rotation vs. l2.

– RM is proportional to the density-weighted longitudinal B-field, imbedded in the

cluster gas surrounding the radio source.

– For simple propagation:

• Dreher, Carilli

and Perley

(1987))

• RMs are very

high – source is

imbedded in a

dense, ionized,

magnetized

medium.


lB dnRM

there

here

e 8.0

Jansky VLA

Electron Scattering in S106


• A spectacular example of polarized

emission due to electron scattering.

• This is a 2.2m image of the bipolar

HII nebula S106

• Center peak is IR source IRS4.

• Contours show emission intensity

• Lines show the apparent E-field.

• Azimuthal orientation shows the

emission is scattered light from the

central source.

Aspin et al. (1990)

Jansky VLA


Example: Zeeman Splitting

Jansky VLA

What Is Polarization?

• EM Radiation is a transverse phenomenon – the E and B fields lie in a plane

perpendicular to the direction of propagation.

• The instantaneous E vector is comprised of two components which are

*completely independent*.

• The two components can be described in terms of orthogonal linear (Ex

and Ey), or opposite circular (Er and El).

• The polarization is defined by the locus of the E-vector as it passes

through a fixed plane, perpendicular to the propagation direction.

• The locus of the resultant E-vector describe an ellipse in this plane.

• For monochromatic radiation, three quantities are sufficient to fully

describe this ellipse:

– The amplitudes of the two components

– The phase difference between them.


Jansky VLA The Polarization Ellipse – monochromatic radiation

• We consider the time behavior of the E-field in a fixed perpendicular plane, with the z-axis directed along the direction of propagation towards the observer.

• For a monochromatic wave of frequency n, we can write

• These two equations describe an ellipse in the (X-Y) plane:

• The ellipse is described by three parameters:

– The amplitudes Ax, Ay, and

– The phase difference, d = dy – dx


10

Jansky VLA

Elliptically Polarized Monochromatic Wave

According to Maxwell’s

equations, an EM wave is

elliptically polarized.

In general, three parameters are

needed to describe the

ellipse.

• Ax – X-axis amplitude max

• Ay – Y-axis amplitude max

• a = atan(Ay/Ax) – an angle

describing the orientation

If the E vector is rotating:

• Clockwise, the wave is Left

Elliptically Polarized:

• Anti-clockwise, the wave is

Right Elliptically Polarized.


Jansky VLA

Example: Decomposition into Linear components


Jansky VLA

13 ASIAA Workshop on the Very Large Array

Form of Ellipse depends on two amplitudes, and

the phase difference:

Ex = Ey, d = 90 Ex = Ey, d = 0 in general:

Circular Linear Elliptical

Jansky VLA

And, in Motion:

• Showing the component (blue, red) and net (black) fields as the wave

passes a fixed (x,y) plane.

• Here, the amplitudes are equal, and the phase difference is 90 degrees.

• As seen from the right, the E-vector is rotating clockwise:

• The wave is 100% LCP.

14 ASIAA Workshop on the Very Large Array

Jansky VLA Alternate Description – The Circular Basis

• We can decompose the E-field into a circular basis, rather than a

cartesian one:

– where AR and AL are the amplitudes of two counter-rotating unit

vectors: eR (rotating counter-clockwise), and eL (rotating clockwise)

• The polarization ellipse is again described by three parameters:


15

Jansky VLA

Circular Basis Example • The polarization ellipse

(black) can be decomposed

into an X-component of

amplitude 2, and a Y-

component of amplitude 1

which lags by ¼ turn.

• It can alternatively be

decomposed into a

counterclockwise (RCP)

rotating vector of length 1.5

(red), and a clockwise

rotating (LCP) vector of

length 0.5 (blue).


Jansky VLA Decomposition into circular components

• The same ellipse, but now decomposed into oppositely-rotating circular

components.

17


Jansky VLA In a more Natural Reference Frame

• A more natural description is in a frame (x,h), rotated so the x-axis lies along the major axis of the ellipse.

• The three parameters of the ellipse are then:

– Ax : the major axis length

– Y: the position angle of this major axis, and

– tan c Ah/Ax : the axial ratio

• It can be shown that:

• The ellipticity c is signed:

c > 0 => LEP (clockwise)

c < 0 => REP (anti-clockwise)


18

Jansky VLA Stokes Parameters

• Three parameters are sufficient to describe the monochromatic EM polarization.

• It is most convenient to have the three parameters share the same units, and have easily grasped physical meanings.

• It is standard in radio astronomy to utilize the parameters defined by George Stokes (1852), and introduced to astronomy by Chandrasekhar (1946):

• Note that

• Thus – a monochromatic wave is 100% polarized.

• Note also that the Stokes parameters have units of brightness – very convenient for imaging instruments!


19

Jansky VLA

Interpreting the Meaning of I, V, Q, and U

• I is the total power.

• V is the difference between the

power in RCP and LCP components.

• Q is the difference between vertical

and horizontal power.

• U is the difference between

orthogonal components in a frame

rotated by 45 degrees.

• If V = 0, the wave is 100% Linearly Polarized

• If Q=U=0, the wave is 100% Circularly Polarized

• In general, the wave is Elliptically Polarized


20

Jansky VLA

Pure Linear Polarization: V = 0 • Presume a linearly polarized wave, of unit power.

• Then, if:

– The wave is vertically polarized:

• Q = 1 U = 0

– The wave is horizontally polarized:

• Q = -1 U = 0

– The wave is polarized at pa = 45 deg:

• Q = 0 U = 1

– The wave is polarized at pa = -45.

• Q = 0 U = -1


Jansky VLA Linear Polarization (Q, U)

• In general, we define:

– The Linearly Polarized Intensity:

– The orientation of the plane of polarization:

– Signs of Q and U tell us the orientation of the plane of polarization:

Q > 0

Q < 0 Q < 0

Q > 0

U > 0

U > 0

U < 0

U < 0


22

For this

case:

Q>0

U<0

Jansky VLA

In General: Some Illustrative Examples

Pol Ellipse I V Q U

1 1 0 0

1 -1 0 0

1 0 1 0

1 0 -1 0

1 0 0 -1

1 0 0 1

1 0

1 0


23

Jansky VLA Non-Monochromatic Radiation, and Partial Polarization

• Monochromatic radiation is nonphysical.

• No such condition can exist (although it can be closely

approximated).

• In real life, radiation has a finite bandwidth Dn – the signals are

quasi-sinusoidal, specified by an amplitude and phase for a

limited time t~1/ Dn.

• Because the amplitudes and phases of the received signals are

now variable, there are no unique, stationary values of

amplitude and phase.

• We must then use suitably defined averages of these

quantities to define the polarization characteristics.


Jansky VLA Stokes Parameters for wideband radiation

• In the quasi-monochromatic approximation, the incoming EM wave can be described, for a period Dt~1/Dn, by two amplitudes, Ap and Aq, and a phase difference, dpq.

• For the orthogonal linear, and opposite circular bases, we have:

22

22

2222

sin2

sin2cos2

cos2

LRXYYX

RLLRXYYX

RLLRYX

LRYX

AAAAV

AAAAU

AAAAQ

AAAAI

d

dd

d

• The angle brackets <> denote an average over a time much longer

than the coherence time 1/Dn.

• These four real numbers are a complete description of the polarization

state of the incoming radiation.

• They are a function of frequency, position, and time.


25

Jansky VLA

Imaging in All Four Stokes

• The four quantities defined on the last slide provide a

complete description of the polarization state of the

incoming radiation.

• To do astronomy, we want to get images of the sky – one

image for each Stokes quantity.

• How do we get these from a telescope? An

interferometer?

26


Jansky VLA

Antennas are Polarized! • Polarimetry is possible because antennas are polarized – their output

is not a function of I alone.

• It is highly desirable (but not required) that the two outputs be sensitive to two orthogonal modes (i.e. linear, or circular).

• In interferometry, we have two antennas, each with two differently polarized outputs.

• We can then form four complex correlations.

• What is the relation between these four correlations and the four Stokes’ parameters?

Polarizer RCP LCP

A generic antenna


Jansky VLA Four Complex Correlations

per Pair of Antennas

• Two antennas, each

with two polarized

outputs, produce four

complex correlations.

• What is the relation

between these four

correlations, and the

four Stokes

parameters?

L1 R1

X X X X

L2 R2

Antenna 1 Antenna 2

RR1R2 RR1L2 RL1R2 RL1L2

(feeds)

(polarizer)

(signal

transmission)

(complex

correlators)


Four complex cross-products

Jansky VLA

Stokes Visibilities

• But an interferometer doesn’t measure these values directly.

Recall that an interferometer measures the Visibility, which is

related to the image intensity by a Fourier Transform:

V (u,v) I (l,m) (a Fourier Transform Pair)

• Define the Stokes Visibilities I, Q, U, and V, to be the

Fourier Transforms of the Stokes’ brightesses: I, Q, U, and V:

• Then, the relations between these are:

• I I, Q Q, U U, V V

• Stokes Visibilities are complex functions of (u,v), while the

Stokes Images are real functions of (l,m).

• Our task is now to obtain these Stokes visibilities from the

cross-power measurements provided by an interferometer.


Jansky VLA Relating the Products to Stokes’ Visibilities

• Let ER1, EL1, ER2 and EL2 be the complex representation (phasors) of

the RCP and LCP components of the EM wave which arrives at the

two antennas.

• We can then utilize the definitions earlier given to show that the four

complex correlations between these fields are related to the desired

visibilities by (ignoring gain factors):

• So, if each antenna has two outputs whose voltages are faithful replicas

of the EM wave’s RCP and LCP components, then the simple

equations shown are sufficient.

2/)(

2/)(

2/)(

2/)(

*

2121

*

2121

*

2121

*

2121

iUQ

iUQ

VI

VI

RLRL

LRLR

LLLL

RRRR

EER

EER

EER

EER


Jansky VLA Solving for Stokes Visibilities

• The solutions are straighforward:

)(2121

2121

2121

2121

RLLR

RLLR

LLRR

LLRR

RRi

RR

RR

RR

U

Q

V

I

• For an unresolved, or partially resolved source, Q, U, and V are

much smaller than I (low polarization).

• Thus, the amplitudes of the cross-hand correlations are often much

less than the parallel hand correlations.

• V is formed from the difference of two large quantities, while Q and

U are formed from the sum and difference of small quantities.

• If calibration errors dominate (and they often do), the circular basis

favors measurements of linear polarization.


Jansky VLA For Linearly Polarized Antennas …

• We can go through the same exercise with perfectly linearly

polarized feeds and obtain (presuming they are oriented with the

vertical feed along a line of constant HA, and again ignoring

issues of gain):

• For each example, we have four measured quantities and four

unknowns.

• The solution for the Stokes visibilities is easy.

2/)(

2/)(

2/)(

2/)(

*

2121

*

2121

*

2121

*

2121

iVU

iVU

QI

QI

VHVH

HVHV

HHHH

VVVV

EER

EER

EER

EER


Jansky VLA Stokes’ Visibilities for Pure Linear Antennas

• Again, the solution in straightforward:

)(2121

2121

2121

2121

VHHV

VHHV

HHVV

HHVV

RRi

RR

RR

RR

V

U

Q

I

• Now, the Q parameter is the difference of large quantities –

trouble!

• But, V is now the difference of small quantities – that’s good.

• Bottom line here: No perfect system!


Jansky VLA

Summary: Polarimetry in Astronomy

• So, (in principle) the process is easy:

1. Collect all four cross-correlations from your interferometer.

2. Calibrate!

3. Generate the four Stokes visibilities from your four correlations.

4. Fourier transform all four.

5. Deconvolve all four

6. Combine to form polarization and position angle images

7. Think hard.

8. Publish that paper.


Jansky VLA

But The Real World is harder …

• The preceding analysis presumes:

1. Antennas fixed to the sky frame (no parallactic angle

rotation)

2. Perfectly polarized antennas

3. Perfectly calibrated data.

• Sadly, none of these things happens in the real world (although

the 1st can be arranged to happen by suitable design).

• Real antennas are imperfectly polarized – some RCP power

emerges from the LCP port, and vice versa.

• In this short introduction, I can give no detail.

• What follows is a summary.


Jansky VLA

Imperfectly Polarized Antennas

• Typically, for our wideband systems, there is about a 5%

‘leakage’ of opposite polarization into each output.

• The ‘leakage’ is characterized by a quantity labelled ‘D’, a

complex quantity which describes the amplitude and phase of

the leakage.

• We write: Vr,obs = Vr + DrVl, and a similar expression for Vl,obs

• Then, using the magic of Jones matrix algebra, and the

definitions give before, analysis shows that:


Jansky VLA Slightly Imperfect Circularly Polarized Antennas

Y

Y

2/)(

2/)(

2/)(

2/)(

1

1

1

1

2

2

*

21

*

21

*

2

*

211

1

*

21

*

2

*

211

*

2

21

21

21

21

VI

UQ

UQ

VI

ie

ie

DDDD

DDDD

DDDD

DDDD

R

R

R

R

P

P

i

i

RLRLLRRL

LRLRRLRL

LRRLLRRL

LRLRLRLR

LL

RL

LR

RR

L

R

i

LL

i

RR

e

e

D

D

2

2

tan

tan

Where:

• If |D|<<1, we can then ignore D*D products. (Typically, |D| ~ .05)

• Furthermore, as |Q| and |U| << |I|, we can ignore products between

them and the Ds. (This is dangerous for a well resolved objects …)

• And V can be safely assumed to be zero.

• These (generally reasonable) approximations then give us:


Jansky VLA ‘Nearly’ Circular Feeds (small D approximation)

• We get:

• Our problem is now clear. The desired cross-hand responses are contaminated by a term proportional to ‘I’.

• Stokes ‘I’ is typically 20 to 100 times the magnitude of ‘Q’ or ‘U’.

• If the ‘D’ terms are of order a few percent (and they are!), we must make allowance for the extra terms.

• To do accurate polarimetry, we must determine these D-terms, and remove their contribution.

• Knowing the D-terms, one can easily modify the Rs to their correct values.


38

Jansky VLA Measuring Cross-Polarization • Correction of the X-hand response for the ‘leakage’ is important, since the

leakage amplitude is comparable to the fractional polarization.

• There are two ways to proceed:

1. Observe a calibrator source of known polarization (preferably zero!)

2. Observe a calibrator of unknown polarization for a ‘long time’.

• First case (with polarization = 0).

2/

2/

2/

2/

*

2121

*

2121

21

21

VHVH

HVHV

HH

VV

DDR

DDR

R

R

I

I

I

I

• Then a single observation should suffice to measure the leakage terms.

• This is not actually correct – because the cross-hand visibility is always

the sum of two terms, the ‘D’ values must be referenced to an assumed

value (DV1 = 0, for example).


Jansky VLA Determining Source and Antenna Polarizations

• You can determine both the (relative) D terms and the calibrator

polarizations for an alt-az antenna by observing over a wide range of

parallactic angle, YP. (R. Conway and P. Kronberg invented this)

• As time passes, YP changes in a known way.

• The source polarization term then rotates w.r.t. the antenna

leakage term, allowing a separation.


Jansky VLA

Fortunately, It’s all ‘under the hood’.

• At this point, you could be forgiven if you have decided to take a vow to

never do interferometric polarimetry.

• Fortunately, the procedures to determine these antenna polarization

impurities (the ‘D’ terms) are quite straightforward.

• For alt-az systems like the VLA, you need only observe one (or more)

calibrators at least a half dozen times, spanning a parallactic angle change of

at least ~50 degrees.

• Normally, your regular phase calibrator will suffice.

• In addition, a single observation of an object of know position angle is

required to rotate the phase frame.

• Both AIPS and CASA have the necessary software built in.

• Once editing and parallel-hand calibration is done, the polarimetric

calibration is normally ‘a piece of cake’.


Jansky VLA

Illustrative Example – Emission from Mars.

• The planet Mars radiates as a blackbody, with a brightness temperature

near 200 K.

• Due to the change in refractive index between the Martian surface and the

atmosphere, the (unpolarized) emerging radiation becomes partially

polarized upon passing through the interface – increasingly so as the angle

increases.

• The observable effect is that emission away from the center of the planet

becomes increasingly linearly polarized, with the direction directed radially

away from the center.

• I show next some I, Q, U data of Mars, taken at 23 GHz.


Jansky VLA I,Q,U,V Visibilities

• It’s useful to look at the visibilities which made these images.

I Q

Amplitude

Phase


Jansky VLA Illustrative Example – Thermal Emission from Mars

• Mars emits in the radio as a black body.

• Shown are false-color coded I,Q,U,P images from Jan 2006 data at 23.4 GHz.

• V is not shown – all noise – no circular polarization.

• Resolution is 3.5”, Mars’ diameter is ~6”.

• From the Q and U images alone, we can deduce the polarization is radial, around

the limb.

• Position Angle image not usefully viewed in color.

I Q U U P


Jansky VLA Mars – A Traditional Representation

• Here, I, Q, and U are

combined to make a more

realizable map of the total

and linearly polarized

emission from Mars.

• The dashes show the

direction of the E-field.

• The dash length is

proportional to the

polarized intensity.

• One could add the V

components, to show little

ellipses to represent the

polarization at every point.


Jansky VLA

A Summary

• Polarimetry is a little complicated.

• But, the polarized state of the radiation gives valuable information into the

physics of the emission.

• Well designed systems are stable, and have low cross-polarization, making

correction relatively straightforward.

• Such systems easily allow estimation of polarization to an accuracy of 1

part in 10000.

• Beam-induced polarization can be corrected in software – development is

under way.


The National Radio Astronomy Observatory is a facility of the National Science Foundation

operated under cooperative agreement by Associated Universities, Inc.

www.nrao.edu • science.nrao.edu

polarization with the very large array

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