atnf synthesis imaging school narrabri, nsw sept. 29 – oct 3, 2008 polarization in interferometry...

39
ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Upload: sabrina-matthews

Post on 17-Dec-2015

215 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

ATNF Synthesis Imaging School

Narrabri, NSW

Sept. 29 – Oct 3, 2008

Polarization in Interferometry --

II

Rick Perley(NRAO-Socorro)

Page 2: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Introduction

• In the last lecture, Dave has:– Shown the motivation for doing polarimetry– Defined the fundamental definitions that are used in

polarimetry. – Shown some examples to help illustrate these concepts.

• In this lecture, I will show we actually determine the Stokes parameters with an interferometric array.

• To start, we must first define the ‘Stokes Visibilities’, and then review antenna polarization properties.

Page 3: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Stokes Visibilities

• Recall the first lecture, where we defined the Visibility, V(u,v), and showed its relation to the sky emission:

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

• In analogy with this result, let us define the Stokes Visibilities I, Q, U, and V, to be the Fourier Transforms of the true I, Q, U, and V.

• Then, the relations between these are:

• I I, Q Q, U U, V V

• Stokes Visibilities are functions of (u,v), while the Stokes Images are functions of (l,m).

Page 4: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Antenna Polarization

• Fundamentally, antennas are polarized. – Their response is sensitive to the polarization of the EM wave

• To do polarimetry, the antenna must have two outputs which respond differently to the incoming elliptically polarized wave.

• It would be most convenient if these two outputs are proportional to either:– The two linear orthogonal Cartesian components, (EX, EY) or

– The two circular orthogonal components, (ER, EL).

• For simplicity (for now), let us assume that our antenna’s outputs are indeed perfectly linear, or perfect circular.

Page 5: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Two Orthogonal Outputs per Antenna

• To do polarimetry, we must have two outputs from each antenna which respond differently to the polarization state of the incoming EM wave:

• 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 or XLPLCP or YLP

Our Generic Sensor

Page 6: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Four Complex Correlations per Pair of Antennas

• Two antennas, each with two differently polarized outputs, produce four complex correlations.

• From these four outputs, we want to generate four Stokes Visibilities.

• How?

L1R1

X X X X

L2R2

Antenna 1 Antenna 2

RR1R2 RR1L2 RL1R2 RL1L2

(feeds)

(polarizer)

(signaltransmission)

(correlators)

Page 7: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Orthogonal, Perfectly Linear

• Assuming the antenna orientation is fixed to the sky frame, and ignoring calibration:

• From these, we can trivially invert and recover the desired Stokes Visibilities.

• Note that RXX and RYY have large amplitudes, while RXY and RYX will be small.

2

2

2

2

VU

VU

QI

QI

iR

iR

R

R

YX

XY

YY

XX

Page 8: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Orthogonal, Perfectly Linear

• These are:

• Note that:– Stokes ‘Q’ – normally a small number, is the difference of two

large values, RXX and RYY.

– Obtaining an accurate value for Q requires very good gain stability and calibration of the parallel-feed correlations.

)(XYYX

YXXY

YYXX

YYXX

RRi

RR

RR

RR

V

U

Q

I

Page 9: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Perfectly Circular

• So let us continue with our idealizations, and ask what the response is for perfectly circular feeds.

• Again assuming the antenna orientation is fixed to the sky,

• And again a trivial inversion provides our desired quantities. • Note again that the parallel hand correlations will normally be

much larger than the opposite-hand correlations.

2

2

2

2

UQ

UQ

VI

VI

iR

iR

R

R

LR

RL

LL

RR

Page 10: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Perfectly Circular Feeds

• Giving,

• Note that now, – It is Stokes ‘V’ which is the difference between two large numbers.– Accurate values for V require high stability and calibration. – The linear polarization visibilities, Q and U, are now the differences

between small values – which allows relaxation of the calibration accuracy requirements.

)(RLLR

LRRL

LLRR

LLRR

RRi

RR

RR

RR

U

Q

V

I

Page 11: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Which System is Best?

• The VLA (and EVLA) use circular polarization. • Most new arrays do not (e.g. ALMA, ATCA). • Why?

– Antenna feeds are natively linearly polarized. To convert to circular, a quadrature hybrid is needed. This adds cost, complexity, and degrades performance.

– For some high frequency systems, or for very wide bandwidth systems, wide-band quarter-wave phase shifters may not be available, or their performance may be too poor.

– Linear feeds can give perfectly good linear polarization performance, provided the amplifier/signal path gains are carefully monitored.

– Because the parallel-hand correlations with circular feeds do not respond to linear polarization, calibration using linearly-polarized sources is much simplified.

Page 12: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Antenna-Sky Rotation, and the Parallactic Angle

• The prior expressions presumed that the antenna feeds are fixed in orientation on the sky.

• This is the situation with equatorially mounted antennas. • For antennas with alt-az mounts, the feeds rotate on the sky as

they track a source. • The angle between a line of constant azimuth, and one of

constant right ascension is called the Parallactic Angle,

where A is the antenna azimuth, is the antenna latitude, and is the source declination.

• As the antenna tracks the sky, the azimuth changes, hence so does the parallactic angle.

cos/cossinsin A

Page 13: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Including Parallactic Angle (Linear)

• Presuming all antennas view the source with the same parallactic angle (not true for VLBI!), the responses from pure polarized antennas are, for linear feeds:

2

sin2cos2

sin2cos2

2sin2cos2

2sin2cos

VQU

VQU

UQI

UQI

iR

iR

R

R

PP

YX

PP

XY

PP

YY

PP

XX

• It is easy to solve these for I, Q, U, and V

• All four correlations are needed to determine Q and U, but only two are needed for I and V.

Page 14: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Including Parallactic Angle (Circular)

• For pure circularly polarized antennas, the expressions are simpler:

2

2

2

2

2

2

UQ

UQ

VI

VI

ieR

ieR

R

R

P

P

i

LR

i

RL

LL

RR

• Again, solution for the Stokes Visibilities is straightforward.• Only two of the correlations are actually needed for each of the Stokes visibilities.

Page 15: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Real Antennas

• The analysis so far has assumed that the antennas are perfectly polarized – their two outputs provide a perfect decomposition of the incoming wave into its orthogonal linear, or orthogonal circular, components.

• Sadly, this situation cannot be realized in practice, as antennas are in general cross-polarized.

• To incorporate these effects, define the antenna’s polarization parameters: and , where:

is the polarization position angle, and

is the ellipticity

for the particular polarized output of the antenna.

Page 16: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

In a more Natural Reference Frame

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

• The three parameters of the ellipse are then:

– A : the major axis length– : the position angle of this

major axis, and– tan : the axial ratio

• It can be shown that:

• The ellipticity is signed: > 0 => LEP (clockwise) < 0 => REP (anti-clockwise)

sin2sin2sin

cos2tan2tan

Page 17: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Antenna Polarization Ellipse

• We can thus describe the characteristics of the polarized outputs of an antenna in terms of its antenna polarization ellipses:R and R, for the RCP output

L and L, for the LCP output

if the antenna is equipped with nominally circularly polarized feeds,

• Or, x and x, for the ‘X’ output,

Y and Y, for the ‘Y’ output

if the antenna is equipped with nominally linearly polarized feeds.

Page 18: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

General Antenna Polarization

• The antenna’s polarization is specified by the far-field polarization ellipse resulting from a signal input into each feed.

• Note that this will in general be a function of direction!!!

Polarizerq p Reciprocity: An antenna transmits the same polarization that it receives.

pp

q

q

Page 19: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Parabolic Antenna Beam Polarization

• The beam polarization is due to the antenna/feed geometry.

• Grasp8 calculation by Walter Brisken. (EVLA Memo # 58, 2003).

• Contour intervals: – V/I = 4%,

– Q/I, U/I = 0.2%.

• Can be removed – at considerable cost – in imaging.

I V/I

Q/I U/I

Page 20: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Beam Polarization Profiles

Observation of 3C287 offset to half power, at 1485MHz

As 3C287 rotates about the beam:

• I (= R+L) is stable to ~1%

• V (= R-L) sinusoidally varies with ~8% amplitude

• EVLA feed at Az = 0• VLA feed at Az = 45

• Q, U vary with ~1.5% amplitude.

Page 21: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Generalized Interferometer Response

• We are now in a position to show the most general expression for the output of a complex correlator for an interferometer comprising arbitrarily polarized antennas to partially-polarized astronomical signals.

• This is a complex expression (in all senses of that adjective), and I will make no attempt to derive, or even justify it.

• The expression is completely general, valid for a linear system.

Page 22: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Here it is!

}2/)]cos()sin()sin()[cos(

2/)]cos()sin()sin()[cos(

2/)]sin()sin()cos()[cos(

2/)]sin()sin()cos()[cos({

qpqpqpqp

qpqpqpqp

qpqpqpqp

qpqpqpqppqpq

i

ii

i

iGR

V

U

Q

I

Rpq is the complex output from the interferometer, for polarizations

p and q from antennas 1 and 2, respectively. and are the antenna polarization major axis and ellipticity for

states p and q. I,Q, U, and V are the Stokes Visibilities

Gpq is a complex gain, including the effects of transmission and electronics

The terms shown are 4 of the 16 elements of the matrix (often erroneously termed the Mueller Matrix) relating the four complex correlations to the four Stokes Visibilities. This relation was derived by Morris, Radhakrishnan and Seielstad (1964).

Page 23: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Reduction for Simple Cases

• In general each of the four complex correlations responds to all four Stokes visibilities.

• For perfectly polarized antennas, these expressions quickly reduce to the examples already shown.

• For example, for perfectly linear feeds,

= 0, V = 0, H = /2.

• While for perfectly circular feeds,

R = -/4, L = /4

• I leave to you students the effort of actually showing this, and in the extension for including the parallactic angle.

Page 24: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

An Alternate Representation

• A cleaner and more intuitive representation of the general response is found if we adopt a circular basis, and make the substitutions:

• The terms represent the deviation of the antenna polarization from perfect circularity.

• the represent the orientation of the antenna polarization ellipse in the antenna’s frame.

• The term P is the parallactic angle.

• The C and S terms form the elements of the Jones Matrix.

LPL

RPR

LL

RR

4/

4/

Li

LL

Li

RR

Li

LL

Ri

RR

e

e

e

e

S

S

C

C

sin

sin

cos

cos

Then define:

Page 25: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

The Matrix Formulation

• After some (considerable) labor, we find:

• Where:

SPGR

21

21

21

21

RL

LR

LL

RR

R

R

R

R

R

U)/2(Q

U)/2(Q

V)/2(I

V)/2(I

i

iS

21

21

21

21

000

000

000

000

RL

LR

LL

RR

G

G

G

G

G

*

21

*

21

*

21

*

21

*

21

*

21

*

21

*

21

*

21

*

21

*

21

*

21

*

21

*

21

*

21

*

21

RLRLRLRL

LRLRLRLR

LLLLLLLL

RRRRRRRR

CCSSSCCS

CSCCCSSC

SCCSCCSS

CSSCSSCC

P

P

P

P

P

i

i

i

i

e

e

e

e

000

000

000

000

Page 26: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

What Good is all This???

• It allows easy derivation for any general case.

• Simplification One: Assume all parallactic angles equal.

• Simplification Two: Assume perfect circular systems:

• Simplification Three: Assume perfect linear systems:

P

P

i

i

e

e2

2

000

000

0010

0001

1000

0100

0010

0001

P

2/2/2/2/

2/2/2/2/

2/12/12/12/1

2/12/12/12/1

iiii

iiiiP

= 0

= /4 R = 0 L = /2

Page 27: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Case 3: Nearly circular system

• If our engineers have made nearly perfect circular polarizers, then:– All ‘C’ terms are ~ 1– All ‘S’ terms are very small: |S| << 1

• If we then ignore all 2nd order products in S (S1S2* = 0), and ignore all products between S and Q, S and U, and S and V: then:

10

01

0010

0001

*

21

1

*

2

RL

RL

SS

SSP

This leads to a well-known (and often used) approximation:

Page 28: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

‘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’.

• We must either make the S terms much smaller than this, or be able to correct for them to an accuracy better than this.

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

2/

2/

2/

2/

2*

2121

2*

2121

21

21

UQI

UQI

VI

VI

ieSSR

ieSSR

R

R

P

P

i

RLRL

i

LRLR

LL

RR

Page 29: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Some Comments

• Determination of the S (leakage, or cross-polarization) terms is normally done either by:– Observing a source of known (I,Q,U) intensity, or– Multiple observations of a source of unknown (I,Q,U),

and allowing the rotation of parallactic angle to separate the two terms.

• The latter method works well, as:– In the frame of the antenna, the cross-polarization term

is constant over time, while– The source linear polarization term has a phase which

rotates at twice the parallactic angle.

• Note that for each, the absolute value of S cannot be determined – they must be referenced to an arbitrary value.

Page 30: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Nearly Perfectly Linear Feeds

• We can follow the same exercise for nearly perfect linears, but the physical interpretation is not the same.

• In this case, assume that the ellipticity is very small ( << 1), and that the two feeds (‘dipoles’) are nearly perfectly orthogonal.

• We then define a *different* set of S-terms:

• The angles Y andXare the angular offsets from the exact

horizontal and vertical orientations, w.r.t. the antenna.

YYY

XXX

iS

iS

Page 31: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

This gives us …

• I’ll spare you the full equation set, and show only the results after the same approximations used for the circular case are employed.

2

sin2cos2

sin2cos2

2sin2cos2

2sin2cos

*

21

*

21

VQUI

VQUI

UQI

UQI

iSSR

iSSR

R

R

PPXY

YX

PPYX

XY

PP

YY

PP

XX

Page 32: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Some Comments on Linears

• The problem is the same as for the circular case: – The parallel-hand correlations are not affected by the cross-

polarization (to first order), while– the derivation of the Q, U, and V Stokes’ visibilities is

contaminated by a leakage of the much larger I visibility into the cross-hand response.

• Calibration is similar to the circular case:– If Q, U, and V are known, then the equations can be solved

directly for the Ds. – If the polarization is unknown, then the antenna rotation can

again be used (over time) to separate the polarized response from the leakage response.

Page 33: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Comments on Cross-Polarization

• Determination and removal of these ‘leakage’ terms is a well established and successful procedure.

• The origins of the cross-polarization lie in the electronics (primarily in the cryogenic components), and in the antenna structures themselves.

• These components are very stable in temperature and over time, so we expect the ‘S’ terms to be stable – as they nearly always are.

• If the first-order approximations are used to solve for the S terms, then they must be referenced to one antenna. Only the (complex) differences can be observed – which is all that are needed.

• Determining *absolute* S terms is possible – easily if the parallactic angles are very different (VLBI) – with more difficulty if they are similar.

Page 34: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

How Well Does This Work? 3C147, a strong unpolarized source …

Peak = 21241 mJy, = 0.21 mJy

Max background object = 24 mJy

Peak = 4 mJy, = 0.8 mJy

Peak at 0.02% level – but not noise limited!

I Q

Page 35: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

A Typical Baseline X-Pol Response

Before Polarization Calibration After Polarization Calibration

Am

plit

ude

Pha

se

2.1%andstable

125 degrees and stable

0.3%stable residual!

250 degree rotation!

Page 36: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

3C287 at 1465 MHZ I and V with the VLA

Peak = 6982 mJy, = 0.21 mJyMax Bckg. Obj. = 87 mJy

Peak = 6 mJy, = 0.16 mJyBackground sources falsely polarized.

False V!

5%

9%

I V

Page 37: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Removal of Polarization Beams

• Sanjay Bhatnagar has implemented an algorithm to permit removal of pointing variations and beam polarization

• In Stokes I, primary problem is variation in pointing. • Simulated results:

Before After

Page 38: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

Stokes V correction

• In Stokes ‘V’, primary problem is the beam squint. • Bhatnagar algorithm effectively removes the false polarization.• The same software will correct Q, and U images.

After CorrectionBefore Correction

Page 39: ATNF Synthesis Imaging School Narrabri, NSW Sept. 29 – Oct 3, 2008 Polarization in Interferometry -- II Rick Perley (NRAO-Socorro)

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.• 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. • We can surely do better with a little more effort…