Polarization Calibration

Anita RIchardsThanks to Ivan Marti-Vidal, Robert Laing, Rosita Palladino et al.

Main example: Linearly polarized continuum targetAlt-az array with circular feeds

Summary Polarization Part 1● Recap: Origins of hazards to be corrected● Polarization calibration outline

– Circular feeds, linearly polarized target● Practical approach for calibration● Polarization image products● Other cases: circular polarization or linear feeds

– Mixed linear-circular array? (like EHT) Ask Ivan!● Part 2: Laing, polarization image analysis

More information: Ivan's talk, CASA guides,

Hales 2017AJ....154...54H

Calibration Errors in Interferometric Radio Polarimetry

HazardsClose to AGN: Scintillation

Lobes Faraday rotation

Ionospheric refraction/Faraday rotationTropospheric refraction/absorption

Galactic CO etc.

Delay (timing) errors

RFI

RFI

● Above the telescope– Mostly high frequency– Mostly low frequency

Ionosphere ● Refraction by electrons● Phase delay ∝ Ne l2

– Ne atmospheric electron column density

● Electrons spiral round Earth's magnetic field– Polarization angle of

radiation Faraday rotated

● Ionospheric errors worst at n<1 GHz– Exacerbated by Solar

activity– l>20m only from

space

bannawit.pimpanuwat@postgrad.manchester.ac.uk

Ionospheric variability

Meters of delay

Signal at Onsala ~10m shorter effective path than signal entering Kutunse

Models of long-term trends used to correct delays during observations

TEC monitoring e.g. GPS can be used to correct after data recorded. Correct rapid fluctuations using astronomical standards

https://www.gpsworld.com/innovation-the-european-way/https://www.gpsworld.com/innovation-the-european-way/

Map of ionospheric delay at ~1.5 GHz calculated for sunspot number 150

Hazards

Insufficient corrections for delay tracking

Bandpass response

Atmospheric emission noiseSystem (electronic) noise (Rx etc.)

Antenna positions Pointing, FocusEfficiency (surface)

Timing and frequency information issues(station clock, local oscillator...)

● At the telescope and later

LINEAR feeds

Correlations XX, YY, XY, YX

Stokes I = (XX+YY)/2

Stokes Q = (XX – YY)/2

Stokes U = (XY – YX)/2

Stokes V = (XY – YX)/2i

Polarized intensity P = √(Q2+U2+V2)

Polarization angle c = ½ atan(U/Q) Diagrams thanks to Wikipaedia

CIRCULAR feeds

Left/Right/cross correlations LL RR LR RL

Stokes I = (RR+LL)/2

Stokes V = (RR-LL)/2

Stokes Q = (RL + LR)/2

Stokes U = (RL – LR)/2i

Polarization jargon: Rx feeds

VectorsVV isibility = f(u,v)I I mage

AAdditive baseline error

ScalarsScalars

SS (mapping I to observer polarization)

x,y image plane coordsu,v Fourier plane coordsi,j telescope pair

Libraries use Measurement Equation

Goal

Starting point

Hazards

Methods

VVijij = MMijijBBijijGGijijDDijij∫∫EEijijPPijijTTijijFFijijSSIInn (x,y) (x,y) exp[i2exp[i2pp (u (uijijx+vx+vijijy)] dxdy y)] dxdy +AAijij

Jones MatricesMMultiplicative baseline

errorBBandpass responseGGeneralised electronic

gainDDterm (pol. leakage)EE (antenna voltage

pattern)PParallactic angleTTropospheric effectsFFaraday rotation

The method behind solving the ME

Eij=ei e j†=(

Ri

Li) (R j

✳ L j✳

)=(Ri R j

✳ Ri L j✳

Li R j✳ Li L j

✳ )

● Express the correlator output as the coherency matrix of the signals from each pair of antennas ij.– Using a circular polarization basis, form outer product:

● Equivalent to ● Replace signal e from each antenna with corrupted

signal e´i = Ji ei

– Ji is a (2 x 2) Jones matrix for antenna-based terms e.g., for the complex 'gain' errors affecting amplitude and phase:

V (u , v)ij=(RR RLLR LL)

JG=(gR 00 gL

)

Example data similar to calibration T4

● 3C277.1 aka 1252+5634 and calibrators– Version with all 4 polarizations, RR, RL, LR, LL

● Calibration script run– refantmode='strict' to ensure relative R-L phase

is consistent● Apply calibration and split out one spw

– spw 1 from the full data set: small.avg.pol1.ms– RR and LL are corrected

● Polarization calibration also includes RL and LR– (Or XY and YX for linear feeds)

Target, Phase-refBandpass, Flux cal

Measurement Set

gaincalphase of calibration sources

gaincalphase & amp of cal sources

derive flux densities of cals:fluxscale

Set flux model of flux scale source: setjy

time-dependent calibration:gaincalphase of bandpass cal

frequency-dependent calibration: delay,bandpass phase & amp of bandpass cal

applycal, concatenate Measurement Setssetjy set

phase-ref flux

gaincalamp of phase-ref

per int for cal sources per scan to transfer to

target

applycal relevant tables for each field

applycal, make first images

Phase tone, Tsys, WVR, online flags...

ALREA

DY APP

LIED

Weighting to optimise imaging● Initial weights proportional to samples per integration:

– Observed using 1-s integrations– Some flagging, then average (mstransform) to 4-s– Some averaged integrations only have 1, 2 or 3 samples

● Adjust weights

Weighting● Re-weight to improve sensitivity, decrease pol. noise

– Weight by variance (statwt or applycal calwt=True)

– Additional weighting using toolkit to reduce Defford● Currently, worst polarization performance

Weighting for polarization● Re-weight to improve sensitivity, decrease pol. noise

– Weight by variance (statwt or applycal calwt=True)

– Additional weighting using toolkit to reduce Defford● Currently, worst polarization performance

Defford baselines mostly ~60 - 120 km

Polarization calibration● Appropriate for circular feeds (RR, RL, LR, LL)

● Correct cross-hand delay

– Use calibrator with bright apparent polarization● Enough signal to fit to phase slope with frequency

● Correct leakage ('D-terms')

– Need good polarization model - here, we have this.

– If not known a-priori, calculate:● At least 3 scans over 6 hr● Use parallactic angle rotation to deduce source polarization

● Origin of phase is arbitrary, so rotate pol angle (L-R phase difference) with reference to known calibrator

● Apply to target along with parallactic angle correction

Cross-hand delay corrections● 0319+4030 unpolarized● But so bright, ~10%

leakage gives best S/N in these data

● After cross-hand delay corrections, phase is flat across band

Parallel-hand delay/BP cal. relative to reference antennaLeaves a cross-hand phase slope due to that antenna

Cross-hand delay corrections● 0319+4030 unpolarized● But so bright, ~10%

leakage gives best S/N in these data

● After cross-hand delay corrections, phase is flat across band

Here, noisy channel and baseline dependent residuals not correctedMain reason for image pol. noise

Parallel-hand delay/BP cal. relative to reference antennaLeaves a cross-hand phase slope due to that antenna

Polarization calibration● Receiver feeds sensitive either to (L, R) or (X, Y)

– Gains can differ so start calibrating separately● For small fields, fainter sources, can treat independently

– Combine to make total intensity images● Actually few% leaks between polarizations

– Most QSO variably, linearly polarised● Solve for:

– Source polarization,– Receiver leakage,

● Using rotation of Rx feeds on the sky as alt-az mount tracks● Need at least 3 scans over >60o rotation for solutions

● At cm wavelengths a few bright sources e.g. 3C84 (0319+4130) are unpolarized

– Can correct leakage with a single scan, model Q=U=V=0

Parallactic angle● Caibrated parallel hands of point source (phase ref) flat● Cross hands undulate as alt-az receiver feeds rotate in tracking

– Simple analytic correction– Apply after solving for any leakage calibrator polarization

Cross hand phase not yet corrected for parallactic angle rotation'True' uncorrected cross hand amps vary Leakage does not vary with angle

Polarization angle calibration

● Polarization angle unknown for circular feeds– Origin of phase is arbitrary– Use a standard source to set conventional angle

● 3C286 dominated by core, ~10% polarised– Consistent 33o position angle

● Faraday rotation by ionosphere ∝ l2 – Serious and variable at GHz-MHz ν

● Especially when Sun active– Observe standard e.g. hourly &/or

use GPS or other estimate of ionospheric electron content

● See Robert Laing's part

3C286 5 GHz polarization vectors (perpendicular to projection of magneticfield for synchrotron radiation)

L-R phase● L-R phase in visibility plane = 2x pol. P.A.

Polarized intensity~12% total intensityQ ∝ cos(2 P.A.)U ∝ sin(2 P.A.)RL = Q + UiLR = Q − Ui

LR & RL phase ±66o

3C286 = 1331+3030Polarization Angle 33o

Polarization imaging● Apply total intensity calibration and:

– Cross-hand delay corrections KCROSS (if required)– Leakage 'D' terms– Pol. angle correction

● In tclean: stokes='IQUV' (or chosen products)– Even if you expect no circular pol., V useful diagnostic

● Interactive masking: click 'all polarizations' if you expect the same total I and polarization distributions– I may be dynamic range limited, lower noise in QUV– After cleaning, can make:

● (linear) polarized intensity image P = (Q2 + U2)0.5

● polarization angle ½atan(U/Q) (use atan2 to remove ambiguity)

De-biasing and blanking● Linear polarized intensity P = (Q2 + U2)0.5

– Q and U image noises sums to zero● But P image must be entirely positive

– P will appear too high due to 'Rician bias' ● Rician bias is complicated depending on S/N

– See e.g. Wardle & Kronberg 1974; Bon Wong Sohn 2011

– Weak polarization: Pcorrected ~ (Pobs2 - σP

2)0.5 ● Rayleigh approximation at low S/N

– CASA task immath can de-bias using parameter sigma● Or viewer estimates bias to remove for suitable images

● Also blank pol. angle image input maps at ~σP

Polarization accuracy● Leakage: We assumed 0319+4130 is unpolarized.

– Apply the polarization calibration – Image 0319+4130 and make Poli P image

● At I peak position, P / I is fractional leakage ● Polarization Angle: 1331+3030 core has constant PA

– In aperture enclosing I peak, measure Pola rms● Circular polarization

– At cm wavelengths, resolution ~10 mas or bigger, QSO cores have V~zero, so apparent V tells you circular leakage

● Measure 0319+4130 V / I before self-calibration – Using a total intensity model forces RR and LL to converge,

i.e. reduces V - gives over-optimistic ratio!

Refinements

● Even if you are not interested in polarization, it should be calibrated for high dynamic range, wide-field imaging– The I, Q, U, V primary beam responses differ

● For high accuracy on long baselines, parallactic angle rotation should be corrected to align the L and R phases – Position error ~1% interferometric beam at 1.6 GHz

● e.g. 2 mas at 200 mas resolution - significant for e.g. masers– But if leakage calibrator has unknown polarization, determine

this before correcting parallactic angle● Apply corrections/set polarized model and repeat total intensity

calibration if necessary● I is affected by leakage from V for circular feeds and (worse)

from Q for linear feeds

Circular polarization calibration

● Circular feeds: same use of calibration sources– Including polarization calibration if all 4 pol products

● Target self-calibration: cannot use total intensity model as that forces RR=LL

● Make RR,LL 2-plane image– RR and LL both strong (bright source &/or small V):

● Use model for calibration, gaintype G● If only one of RR or LL has good S/N, e.g. RR:

– Copy gain table solutions

from RR to LL (in this case)● Use toolkit ms tool

Maser polarizedspectral line

Linear Feeds (XY e.g. ALMA, ATCA)Leakage D in the Linear Basis: V= DPVtrue :Visibility V; Stokes IQUV ; parallactic angle P effect ψ

Contaminating fractions Ud etc

Cross-hands complex offset proportional to I , constant in time

Leakage in all correlations, frequency-dependent

Linear Feed Data Calibration● Observe polarization calibrator 3 times over at least 3 hr

– Significant but unknown polarization● Calibrate total intensity of bandpass and phase-ref cals

– Bandpass, time-dependent phase, amplitude gaintype T● i.e. average XX, YY (no assumptions about polarization)

● Time-dependent cal of pol. cal.: gaintype G to keep X, Y separate● Polarization calibration

– Cross-hand delay– XY phase offset– Estimate Q and U from calibration gain variation with parallactic angle– Remove parallactic angle, re-calibrate XY YX using improved QU model

● Remove residual time-dependent errors– Solve for leakage

● The good news: known feed orientation properties provide good estimate of 'true' polarization angle

● See e.g. ALMA 3C286 CASA Guide, Ivan's 2017 ERIS tutorial

RadioNet has received funding from the European Union’s Horizon 2020 research and innovation programme under grant

agreement No 730562