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# Polarization in Interferometry

Ninth Synthesis Imaging Summer School

Socorro, June 15-22, 2004

Polarization in Interferometry

Steven T. Myers (NRAO-Socorro)

Polarization in interferometry

• Physics of Polarization
• Interferometer Response to Polarization
• Polarization Calibration & Observational Strategies
• Polarization Data & Image Analysis
• Astrophysics of Polarization
• Examples
• References:
• Synth Im. II lecture 6, also parts of 1, 3, 5, 32
• ��Tools of Radio Astronomy�� Rohlfs & Wilson

Polarization in Interferometry – S. T. Myers

WARNING!

• Polarimetry is an exercise in bookkeeping!
• many places to make sign errors!
• many places with complex conjugation (or not)
• possible different conventions (e.g. signs)
• different conventions for notation!
• lots of matrix multiplications
• And be assured��
• I��ve mixed notations (by stealing slides )
• I��ve made sign errors  (I call it ��choice of convention�� )
• I��ve probably made math errors 
• I��ve probably made it too confusing by going into detail 
• But �� persevere (and read up on it later) 

DON��T PANIC !

Polarization in Interferometry – S. T. Myers

Polarization Fundamentals

Polarization in Interferometry – S. T. Myers

Physics of polarization

• Maxwell��s Equations + Wave Equation
• E•B=0  (perpendicular) ; Ez = Bz = 0 (transverse)
• Electric Vector – 2 orthogonal independent waves:
• Ex = E1 cos( k z – w t + d1 )         k = 2p / l
• Ey = E2 cos( k z – w t + d2 )         w = 2p n
• describes helical path on surface of a cylinder��
• parameters E1, E2, d  = d1 - d2 define ellipse

Polarization in Interferometry – S. T. Myers

The Polarization Ellipse

• Axes of ellipse Ea, Eb
• S0 = E12 + E22 = Ea2 + Eb2       Poynting flux
• d phase difference                t = k z – w t
• Ex = Ea cos ( t + d ) = Ex cos y + Ey sin y
• Eh = Eb sin ( t + d ) = -Ex sin y + Ey cos y

Rohlfs & Wilson

Polarization in Interferometry – S. T. Myers

The polarization ellipse continued��

• Ellipticity and Orientation
• E1 / E2 = tan a      tan 2y = - tan 2a cos d
• Ea / Eb = tan c      sin 2c = sin 2a sin d
• handedness  ( sin d > 0 or tan c > 0  right-handed)

Rohlfs & Wilson

Polarization in Interferometry – S. T. Myers

Polarization ellipse – special cases

• Linear polarization
• d  = d1 - d2 = m p     m = 0, ��1, ��2, ��
• ellipse becomes straight line
• electric vector position angle y = a
• Circular polarization
• d  = ½ ( 1 + m ) p     m = 0, 1, ��2, ��
• equation of circle Ex2 + Ey2 = E2
• orthogonal linear components:
• Ex = E cos t
• Ey = ��E cos ( t - p/2 )
• note quarter-wave delay between Ex and Ey !

Polarization in Interferometry – S. T. Myers

Orthogonal representation

• A monochromatic wave can be expressed as the superposition of two orthogonal linearly polarized waves
• A arbitrary elliptically polarizated wave can also equally well be described as the superposition of two orthogonal circularly polarized waves!
• We are free to choose the orthogonal basis for the representation of the polarization
• NOTE: Monochromatic waves MUST be (fully) polarized – IT��S THE LAW!

Polarization in Interferometry – S. T. Myers

Linear and Circular representations

• Orthogonal Linear representation:
• Ex = Ea cos ( t + d ) = Ex cos y + Ey sin y
• Eh = Eb sin ( t + d ) = -Ex sin y + Ey cos y
• Orthogonal Circular representation:
• Ex = Ea cos ( t + d ) = ( Er + El ) cos ( t + d )
• Eh = Eb sin ( t + d ) = ( Er - El ) cos ( t + d – p/2 )
• Er = ½ ( Ea + Eb )
• El = ½ ( Ea – Eb )

Polarization in Interferometry – S. T. Myers

The Poincare Sphere

• Treat 2y and 2c as longitude and latitude on sphere of radius S0

Rohlfs & Wilson

Polarization in Interferometry – S. T. Myers

Stokes parameters

• Spherical coordinates: radius I, axes Q, U, V
• S0 = I = Ea2 + Eb2
• S1 = Q = S0 cos 2c cos 2y
• S2 = U = S0 cos 2c sin 2y
• S3 = V = S0 sin 2c
• Only 3 independent parameters:
• S02 = S12 + S22 + S32
• I2 = Q2 + U2 + V2
• Stokes parameters I,Q,U,V
• form complete description of wave polarization
• NOTE: above true for monochromatic wave!

Polarization in Interferometry – S. T. Myers

Stokes parameters and polarization ellipse

• Spherical coordinates: radius I, axes Q, U, V
• S0 = I = Ea2 + Eb2
• S1 = Q = S0 cos 2c cos 2y
• S2 = U = S0 cos 2c sin 2y
• S3 = V = S0 sin 2c
• In terms of the polarization ellipse:
• S0 = I = E12 + E22
• S1 = Q = E12 - E22
• S2 = U = 2 E1 E2 cos d
• S3 = V = 2 E1 E2 sin d

Polarization in Interferometry – S. T. Myers

Stokes parameters special cases

• Linear Polarization
• S0 = I = E2 = S
• S1 = Q = I cos 2y
• S2 = U = I sin 2y
• S3 = V = 0
• Circular Polarization
• S0 = I = S
• S1 = Q = 0
• S2 = U = 0
• S3 = V = S (RCP) or –S (LCP)

Note: cycle in 180��

Polarization in Interferometry – S. T. Myers

Quasi-monochromatic waves

• Monochromatic waves are fully polarized
• Observable waves (averaged over Dn/n << 1)
• Analytic signals for x and y components:
• Ex(t) = a1(t) e i(f1(t) – 2pnt)
• Ey(t) = a2(t) e i(f2(t) – 2pnt)
• actual components are the real parts Re Ex(t), Re Ey(t)
• Stokes parameters
• S0 = I = <a12> + <a22>
• S1 = Q = <a12> – <a22>
• S2 = U = 2 < a1 a2 cos d >
• S3 = V = 2 < a1 a2 sin d >

Polarization in Interferometry – S. T. Myers

Stokes parameters and intensity measurements

• If phase of Ey is retarded by e relative to Ex , the electric vector in the orientation q is:
• E(t; q, e) = Ex cos q + Ey eie sin q
• Intensity measured for angle q:
• I(q, e) = < E(t; q, e) E*(t; q, e) >
• Can calculate Stokes parameters from 6 intensities:
• S0 = I = I(0��,0) + I(90��,0)
• S1 = Q = I(0��,0) + I(90��,0)
• S2 = U = I(45��,0) – I(135��,0)
• S3 = V = I(45��,p/2) – I(135��,p/2)
• this can be done for single-dish (intensity) polarimetry!

Polarization in Interferometry – S. T. Myers

Partial polarization

• The observable electric field need not be fully polarized as it is the superposition of monochromatic waves
• On the Poincare sphere:
• S02 �� S12 + S22 + S32
• I2 �� Q2 + U2 + V2
• Degree of polarization p :
• p2 S02 = S12 + S22 + S32
• p2 I2 = Q2 + U2 + V2

Polarization in Interferometry – S. T. Myers

Summary – Fundamentals

• Monochromatic waves are polarized
• Expressible as 2 orthogonal independent transverse waves
• elliptical cross-section  polarization ellipse
• 3 independent parameters
• choice of basis, e.g. linear or circular
• Poincare sphere convenient representation
• Stokes parameters I, Q, U, V
• I intensity; Q,U linear polarization, V circular polarization
• Quasi-monochromatic ��waves�� in reality
• can be partially polarized
• still represented by Stokes parameters

Polarization in Interferometry – S. T. Myers

Antenna & Interferometer Polarization

Polarization in Interferometry – S. T. Myers

Interferometer response to polarization

• Stokes parameter recap:
• intensity I
• fractional polarization       (p I)2 = Q2 + U2 + V2
• linear polarization Q,U     (m I)2 = Q2 + U2
• circular polarization V       (v I)2 = V2
• Coordinate system dependence:
• I independent
• V depends on choice of ��handedness��
• V > 0 for RCP
• Q,U depend on choice of ��North�� (plus handedness)
• Q ��points�� North, U 45 toward East
• EVPA   F = ½ tan-1 (U/Q)    (North through East)

Polarization in Interferometry – S. T. Myers

Reflector antenna systems

• Reflections
• turn RCP  LCP
• E-field allowed only in plane of surface
• Curvature of surfaces
• introduce cross-polarization
• effect increases with curvature (as f/D decreases)
• Symmetry
• on-axis systems see linear cross-polarization
• off-axis feeds introduce asymmetries & R/L squint
• Feedhorn & Polarizers
• introduce further effects (e.g. ��leakage��)

Polarization in Interferometry – S. T. Myers

• Paraboloid illuminated by feedhorn:

Polarization in Interferometry – S. T. Myers

Optics – telescope response

• Reflections
• turn RCP  LCP
• E-field (currents) allowed only in plane of surface
• ��Field distribution�� on aperture for E and B planes:

Cross-polarization

at 45��

No cross-polarization

on axes

Polarization in Interferometry – S. T. Myers

Polarization field pattern

• Cross-polarization
• 4-lobed pattern
• Off-axis feed system
• perpendicular elliptical linear pol. beams
• R and L beams diverge (beam squint)
• ��Antennas�� lecture by P. Napier

Polarization in Interferometry – S. T. Myers

Feeds – Linear or Circular?

• The VLA uses a circular feedhorn design
• plus (quarter-wave) polarizer to convert circular polarization from feed into linear polarization in rectangular waveguide
• correlations will be between R and L from each antenna
• RR RL LR RL form complete set of correlations
• Linear feeds are also used
• e.g. ATCA, ALMA (and possibly EVLA at 1.4 GHz)
• no need for (lossy) polarizer!
• correlations will be between X and Y from each antenna
• XX XY YX YY form complete set of correlations
• Optical aberrations are the same in these two cases
• but different response to electronic (e.g. gain) effects

Polarization in Interferometry – S. T. Myers

Example – simulated VLA patterns

• EVLA Memo 58 ��Using Grasp8 to Study the VLA Beam�� W. Brisken

Polarization in Interferometry – S. T. Myers

Example – simulated VLA patterns

• EVLA Memo 58 ��Using Grasp8 to Study the VLA Beam�� W. Brisken

Linear Polarization

Circular Polarization cuts in R & L

Polarization in Interferometry – S. T. Myers

Example – measured VLA patterns

• AIPS Memo 86 ��Widefield Polarization Correction of VLA Snapshot Images at 1.4 GHz�� W. Cotton (1994)

Circular Polarization

Linear Polarization

Polarization in Interferometry – S. T. Myers

Example – measured VLA patterns

• frequency dependence of polarization beam :

Polarization in Interferometry – S. T. Myers

Beyond optics – waveguides & receivers

• Response of polarizers
• convert R & L to X & Y in waveguide
• purity and orthogonality errors
• Other elements in signal path:
• Sub-reflector & Feedhorn
• symmetry & orientation
• Ortho-mode transducers (OMT)
• split orthogonal modes into waveguide
• Polarizers
• retard one mode by quarter-wave to convert LP  CP
• frequency dependent!
• Amplifiers
• separate chains for R and L signals

Polarization in Interferometry – S. T. Myers

Back to the Measurement Equation

• Polarization effects in the signal chain appear as error terms in the Measurement Equation
• e.g.  ��Calibration�� lecture, G. Moellenbrock:

• F = ionospheric Faraday rotation
• T = tropospheric effects
• P = parallactic angle
• E = antenna voltage pattern
• D = polarization leakage
• G = electronic gain
• B = bandpass response

Antenna i

Baseline ij (outer product)

Polarization in Interferometry – S. T. Myers

• Birefringency due to magnetic field in ionospheric plasma
• also present in radio sources!

Polarization in Interferometry – S. T. Myers

• The ionosphere is birefringent; one hand of circular polarization is delayed w.r.t. the other, introducing a phase shift:
• Rotates the linear polarization position angle

• More important at longer wavelengths:
• More important at solar maximum and at sunrise/sunset, when ionosphere is most active

• Watch for direction dependence (in-beam)
• See ��Low Frequency Interferometry�� (C. Brogan)

Polarization in Interferometry – S. T. Myers

Parallactic Angle, P

• Orientation of sky in telescope��s field of view
• Constant for equatorial telescopes
• Varies for alt-az-mounted telescopes:
• Rotates the position angle of linearly polarized radiation (c.f. F

• defined per antenna (often same over array)

• P modulation can be used to aid in calibration

Polarization in Interferometry – S. T. Myers

Parallactic Angle, P

• Parallactic angle versus hour angle at VLA :
• fastest swing for source passing through zenith

Polarization in Interferometry – S. T. Myers

Polarization Leakage, D

• Polarizer is not ideal, so orthogonal polarizations not perfectly isolated
• Well-designed systems have d < 1-5%
• A geometric property of the antenna, feed & polarizer design
• frequency dependent (e.g. quarter-wave at center n)
• direction dependent (in beam) due to antenna
• For R,L systems
• parallel hands affected as d•Q + d•U , so only important at high dynamic range (because Q,U~d, typically)
• cross-hands affected as d•I so almost always important

Leakage of q into p

(e.g. L into R)

Polarization in Interferometry – S. T. Myers

Coherency vector and correlations

• Coherency vector:
• e.g. for circularly polarized feeds:

Polarization in Interferometry – S. T. Myers

Coherency vector and Stokes vector

• Example: circular polarization (e.g. VLA)
• Example: linear polarization (e.g. ATCA)

Polarization in Interferometry – S. T. Myers

Visibilities and Stokes parameters

• Convolution of sky with measurement effects:
• e.g. with (polarized) beam E :

• imaging involves inverse transforming these

Instrumental

effects, including

��beam�� E(l,m)

coordinate transformation

to Stokes parameters

(I, Q, U, V)

Polarization in Interferometry – S. T. Myers

Example: RL basis

• Combining E, etc. (no D), expanding P,S:

2c for co-located

array

0 for co-located

array

Polarization in Interferometry – S. T. Myers

Example: RL basis imaging

• Parenthetical Note:
• can make a pseudo-I image by gridding RR+LL on the Fourier half-plane and inverting to a real image
• can make a pseudo-V image by gridding RR-LL on the Fourier half-plane and inverting to real image
• can make a pseudo-(Q+iU) image by gridding RL to the full Fourier plane (with LR as the conjugate) and inverting to a complex image
• does not require having full polarization RR,RL,LR,LL for every visibility
• More on imaging ( & deconvolution ) tomorrow!

Polarization in Interferometry – S. T. Myers

Leakage revisited��

• Primary on-axis effect is ��leakage�� of one polarization into the measurement of the other (e.g. R  L)
• but, direction dependence due to polarization beam!
• Customary to factor out on-axis leakage into D and put direction dependence in ��beam��
• example: expand RL basis with on-axis leakage
• similarly for XY basis

Polarization in Interferometry – S. T. Myers

Example: RL basis leakage

• In full detail:

��true�� signal

1st order:

D•I into P

2nd order:

D•P into I

2nd order:

D2•I into I

3rd order:

D2•P* into P

Polarization in Interferometry – S. T. Myers

Example: Linearized response

• Dropping terms in d2, dQ, dU, dV (and expanding G)
• warning: using linear order can limit dynamic range!

Polarization in Interferometry – S. T. Myers

Summary – polarization interferometry

• Choice of basis: CP or LP feeds
• ionospheric Faraday rotation F at low frequency
• parallactic angle P for coordinate transformation to Stokes
• ��leakage�� D varies with n and over beam (mix with E)
• Leakage
• use full (all orders) D solver when possible
• linear approximation OK for low dynamic range

Polarization in Interferometry – S. T. Myers

Polarization Calibration

& Observation

Polarization in Interferometry – S. T. Myers

So you want to make a polarization map��

Polarization in Interferometry – S. T. Myers

Strategies for polarization observations

• Follow general calibration procedure (last lecture)
• will need to determine leakage D (if not known)
• often will determine G and D together (iteratively)
• procedure depends on basis and available calibrators
• Observations of polarized sources
• follow usual rules for sensitivity, uv coverage, etc.
• remember polarization fraction is usually low! (few %)
• if goal is to map E-vectors, remember to calculate noise in F= ½  tan-1 U/Q
• watch for gain errors in V (for CP) or Q,U (for LP)
• for wide-field high-dynamic range observations, will need to correct for polarized primary beam (during imaging)

Polarization in Interferometry – S. T. Myers

Strategies for leakage calibration

• Need a bright calibrator!  Effects are low level��
• determine gains G ( mostly from parallel hands)
• use cross-hands (mostly) to determine leakage
• general ME D solver (e.g. aips++) uses all info
• Calibrator is unpolarized
• leakage directly determined (ratio to I model), but only to an overall constant
• need way to fix phase p-q (ie. R-L phase difference), e.g. using another calibrator with known EVPA
• Calibrator of known polarization
• leakage can be directly determined (for I,Q,U,V model)
• unknown p-q phase can be determined (from U/Q etc.)

Polarization in Interferometry – S. T. Myers

Other strategies

• Calibrator of unknown polarization
• solve for model IQUV and D simultaneously or iteratively
• need good parallactic angle coverage to modulate sky and instrumental signals
• in instrument basis, sky signal modulated by ei2c
• With a very bright strongly polarized calibrator
• can solve for leakages and polarization per baseline
• can solve for leakages using parallel hands!
• With no calibrator
• hope it averages down over parallactic angle
• transfer D from a similar observation
• usually possible for several days, better than nothing!
• need observations at same frequency

Polarization in Interferometry – S. T. Myers

Finding polarization calibrators

• Standard sources
• planets (unpolarized if unresolved)
• 3C286, 3C48, 3C147 (known IQU, stable)
• sources monitored (e.g. by VLA)
• other bright sources (bootstrap)

http://www.vla.nrao.edu/astro/calib/polar/

Polarization in Interferometry – S. T. Myers

Example: D-term calibration

• D-term calibration effect on RL visibilities :

Polarization in Interferometry – S. T. Myers

Example: D-term calibration

• D-term calibration effect in image plane :

Good D-term solution

Polarization in Interferometry – S. T. Myers

Example: ��standard�� procedure for CP feeds

Polarization in Interferometry – S. T. Myers

Example: ��standard�� procedure for LP feeds

Polarization in Interferometry – S. T. Myers

Special Issues

• Low frequency – ionospheric Faraday rotation
• important for 2 GHz and below (sometimes higher too)
• l2 dependence (separate out using multi-frequency obs.)
• depends on time of day and solar activity (& observatory location)
• external calibration using zenith TEC (plus gradient?)
• self-calibration possible (e.g. with snapshots)

Polarization in Interferometry – S. T. Myers

Special issues – continued��

• VLBI polarimetry
• follows same principles
• will have different parallactic angle at each station!
• can have heterogeneous feed geometry (e.g. CP & LP)
• harder to find sources with known polarization
• calibrators resolved!
• transfer EVPA from monitoring program

2200+420

Polarization in Interferometry – S. T. Myers

Subtleties ��

• Antenna-based D solutions
• closure quantities  undetermined parameters
• different for parallel and cross-hands
• e.g. can add d to R and d* to L
• need for reference antenna to align and transfer D solutions
• Parallel hands
• are D solutions from cross-hands appropriate here?
• what happens in full D solution (weighting issues?)

Polarization in Interferometry – S. T. Myers

Special Issues – observing circular polarization

• Observing circular polarization V is straightforward with LP feeds (from Re and Im of cross-hands)
• With CP feeds:
• gain variations can masquerade as (time-variable) V signal
• helps to switch signal paths through back-end electronics
• R vs. L beam squint introduces spurious V signal
• limited by pointing accuracy
• requires careful calibration
• relative R and L gains critical
• average over calibrators (be careful of intrinsic V)
• VLBI somewhat easier
• different systematics at stations help to average out

Polarization in Interferometry – S. T. Myers

Special Issues – wide field polarimetry

• Actually an imaging & deconvolution issue
• assume polarized beam D��•E is known
• see EVLA Memo 62 ��Full Primary Beam Stokes IQUV Imaging�� T. Cornwell (2003)
• Deal with direction-dependent effects
• beam squint (R,L) or beam ellipticity (X,Y)
• primary beam
• Iterative scheme (e.g. CLEAN)
• implemented in aips++
• see lectures by Bhatnagar & Cornwell

Polarization in Interferometry – S. T. Myers

Example: wide field polarimetry

• Simulated array of point sources

No beam correction

1D beam + squint

Full 2D beam

Polarization in Interferometry – S. T. Myers

Example: wide field polarimetry continued��

• Simulated Hydra A image

Model

Errors 1D sym.beam

Errors full beam

Panels:    I     Q

U    V

Polarization in Interferometry – S. T. Myers

Summary – Observing & Calibration

• Follow normal calibration procedure (previous lecture)
• Need bright calibrator for leakage D calibration
• best calibrator has strong known polarization
• unpolarized sources also useful
• Parallactic angle coverage useful
• necessary for unknown calibrator polarization
• Need to determine unknown p-q phase
• CP feeds need EVPA calibrator for R-L phase
• if system stable, can transfer from other observations
• Special Issues
• observing CP difficult with CP feeds
• wide-field polarization imaging (needed for EVLA & ALMA)

Polarization in Interferometry – S. T. Myers

Polarization data analysis

• Making polarization images
• follow general rules for imaging & deconvolution
• image & deconvolve in I, Q, U, V (e.g. CLEAN, MEM)
• note: Q, U, V will be positive and negative
• in absence of CP, V image can be used as check
• joint deconvolution (e.g. aips++, wide-field)
• Polarization vector plots
• use ��electric vector position angle�� (EVPA) calibrator to set angle (e.g. R-L phase difference)
• F = ½ tan-1 U/Q for E vectors ( B vectors �� E )
• plot E vectors with length given by p
• Faraday rotation: determine DF vs. l2

Polarization in Interferometry – S. T. Myers

Polarization Astrophysics

Polarization in Interferometry – S. T. Myers

Astrophysical mechanisms for polarization

• Magnetic fields
• synchrotron radiation  LP (small amount of CP)
• Zeeman effect  CP
• Faraday rotation (of background polarization)
• dust grains in magnetic field
• maser emission
• Electron scattering
• e.g. Cosmic Microwave Background
• and more��

Polarization in Interferometry – S. T. Myers

Astrophysical sources with polarization

• Magnetic objects
• active galactic nuclei (AGN) (accretion disks, MHD jets, lobes)
• protostars (disks, jets, masers)
• clusters of galaxies IGM
• galaxy ISM
• compact objects (pulsars, magnetars)
• planetary magnetospheres
• the Sun and other (active) stars
• the early Universe (primordial magnetic fields???)
• Other objects
• Cosmic Microwave Background (thermal)
• Polarization levels
• usually low (<1% to 5-10% typically)

Polarization in Interferometry – S. T. Myers

Example: 3C31

• VLA @ 8.4 GHz
• E-vectors
• Laing (1996)

3 kpc

Polarization in Interferometry – S. T. Myers

Example: Cygnus A

• VLA @ 8.5 GHz   B-vectors    Perley & Carilli (1996)

10 kpc

Polarization in Interferometry – S. T. Myers

Example: Blazar Jets

• VLBA @ 5 GHz             Attridge et al. (1999)

1055+018

Polarization in Interferometry – S. T. Myers

Example: the ISM of M51

Neininger (1992)

Polarization in Interferometry – S. T. Myers

Example: Zeeman effect

Polarization in Interferometry – S. T. Myers

Example: Zeeman in M17

Zeeman Blos: colors (Brogan & Troland 2001)  Polarization Bperp: lines (Dotson 1996

Color: optical from the Digitized Sky Survey                  Thick contours: radio continuum from Brogan & Troland (2001)                                       Thin contours: 13CO from Wilson et al. (1999)

Polarization in Interferometry – S. T. Myers

• VLBA
• Taylor et al. 1998
• intrinsic vs. galactic

15   12       8 GHz

Polarization in Interferometry – S. T. Myers

• See review of ��Cluster Magnetic Fields�� by Carilli & Taylor 2002 (ARAA)

Polarization in Interferometry – S. T. Myers

• Mapping galactic magnetic fields with FR

Han, Manchester, & Qiao (1999)

Han et al. (2002)

Filled: positive RM   Open: negative RM

Polarization in Interferometry – S. T. Myers

Example: Stellar SiO Masers

• R Aqr
• VLBA @ 43 GHz
• Boboltz et al. 1998

Polarization in Interferometry – S. T. Myers

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