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

 

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 Fundamentals

 

Physics of polarization 

• Maxwell��s Equations + Wave Equation
• E•B=0  (perpendicular) ; E_z = B_z = 0 (transverse)
• Electric Vector – 2 orthogonal independent waves:
• E_x = E₁ cos( k z – w t + d₁ )         k = 2p / l
• E_y = 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 
   
   
   
   
  

 

The Polarization Ellipse 

• Axes of ellipse E_a, E_b
• S₀ = E_1² + E_2² = E_a² + E_b²       Poynting flux
• d phase difference                t = k z – w t
• E_x = E_a cos ( t + d ) = E_x cos ^y + E_y sin ^y
• E_h = E_b sin ( t + d ) = -E_x sin ^y + E_y cos ^y

 
 

Rohlfs & Wilson

 

The polarization ellipse continued�� 

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

 

Rohlfs & Wilson

 

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 E_x² + E_y² = E²
• orthogonal linear components:
• E_x = E cos t
• E_y = ��E cos ( t - p/2 )
• note quarter-wave delay between E_x and E_y !

 

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!

 

Linear and Circular representations 

• Orthogonal Linear representation:
• E_x = E_a cos ( t + d ) = E_x cos ^y + E_y sin ^y
• E_h = E_b sin ( t + d ) = -E_x sin ^y + E_y cos ^y

• Orthogonal Circular representation: 
  
• E_x = E_a cos ( t + d ) = ( E_r + E_l ) cos ( t + d )
• E_h = E_b sin ( t + d ) = ( E_r - E_l ) cos ( t + d – p/2 )
• E_r = ½ ( E_a + E_b )
• E_l = ½ ( E_a – E_b )

 

 

The Poincare Sphere 

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

 

Rohlfs & Wilson

 

Stokes parameters 

• Spherical coordinates: radius I, axes Q, U, V
• S₀ = I = E_a² + E_b²
• S₁ = Q = S₀ cos 2^c cos 2^y
• S₂ = U = S₀ cos 2^c sin 2^y
• S₃ = V = S₀ sin 2^c
• Only 3 independent parameters:
• S_0² = S_1² + S_2² + S_3²
• I² = Q² + U² + V²
• Stokes parameters I,Q,U,V
• form complete description of wave polarization
• NOTE: above true for monochromatic wave!

 

Stokes parameters and polarization ellipse 

• Spherical coordinates: radius I, axes Q, U, V
• S₀ = I = E_a² + E_b²
• S₁ = Q = S₀ cos 2^c cos 2^y
• S₂ = U = S₀ cos 2^c sin 2^y
• S₃ = V = S₀ sin 2^c
• In terms of the polarization ellipse:
• S₀ = I = E_1² + E_2²
• S₁ = Q = E_1² - E_2²
• S₂ = U = 2 E₁ E₂ cos d
• S₃ = V = 2 E₁ E₂ sin d

 

Stokes parameters special cases 

• Linear Polarization
• S₀ = I = E² = S
• S₁ = Q = I cos 2^y
• S₂ = U = I sin 2^y
• 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��

 

Quasi-monochromatic waves 

• Monochromatic waves are fully polarized
• Observable waves (averaged over Dn/n << 1)
• Analytic signals for x and y components:
• E_x(t) = a₁(t) e ^{i(f1(t) – 2pnt)}
• E_y(t) = a₂(t) e ^{i(f2(t) – 2pnt)}
• actual components are the real parts Re E_x(t), Re E_y(t)
• Stokes parameters
• S₀ = I = <a_1²> + <a_2^2>
• S₁ = Q = <a_1²> – <a_2^2>
• S₂ = U = 2 < a₁ a₂ cos d >
• S₃ = V = 2 < a₁ a₂ sin d >

 

 

Stokes parameters and intensity measurements 

• If phase of E_y is retarded by e relative to E_x , the electric vector in the orientation q is:
• E(t; q, e) = E_x cos q + E_y e^ie 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!

 

Partial polarization 

• The observable electric field need not be fully polarized as it is the superposition of monochromatic waves
• On the Poincare sphere:
• S_0² �� S_1² + S_2² + S_3²
• I² �� Q² + U² + V²
• Degree of polarization p :
• p² S_0² = S_1² + S_2² + S_3²
• p² I² = Q² + U² + V²

 

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

 

Antenna & Interferometer Polarization

 

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)

 

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��)

 

 

Optics – Cassegrain radio telescope  

• Paraboloid illuminated by feedhorn:

 

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 field pattern 

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

 

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

 

Example – simulated VLA patterns 

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

 

Example – simulated VLA patterns 

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

 

Linear Polarization 

Circular Polarization cuts in R & L

 

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

 

Example – measured VLA patterns 

• frequency dependence of polarization beam :

 

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

 

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)

 

Ionospheric Faraday Rotation, F 

• Birefringency due to magnetic field in ionospheric plasma

• also present in radio sources! 
   
   
   
   
   
   
   
  

 

Ionospheric Faraday Rotation, F 

• 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)

 

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

 

Parallactic Angle, P 

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

 

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)

 

Coherency vector and correlations 

• Coherency vector:

• e.g. for circularly polarized feeds: 
   
   
   
  

 

Coherency vector and Stokes vector 

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

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

 

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)

 

Example: RL basis 

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

 

2c for co-located

array 

0 for co-located

array

 

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!

 

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 
   
   
   
   
  

 

Example: RL basis leakage 

• In full detail:

 

��true�� signal 

1^st order:

D•I into P 

2^nd order:

D•P into I 

2^nd order:

D²•I into I 

3^rd order:

D²•P* into P

 

Example: Linearized response 

• Dropping terms in d², dQ, dU, dV (and expanding G)

• warning: using linear order can limit dynamic range! 
   
   
   
   
   
   
   
   
   
   
  

 

Summary – polarization interferometry 

• Choice of basis: CP or LP feeds
• Follow the Measurement Equation
• 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 Calibration

& Observation

 

So you want to make a polarization map��

 

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)

 

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.)

 

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 e^i2c
• 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

 

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/

 

Example: D-term calibration 

• D-term calibration effect on RL visibilities :

 

Example: D-term calibration 

• D-term calibration effect in image plane :

 

Bad D-term solution 

Good D-term solution

 

Example: ��standard�� procedure for CP feeds

 

Example: ��standard�� procedure for LP feeds

 

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)

 

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

 

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?)

 

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

 

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

 

Example: wide field polarimetry 

• Simulated array of point sources

 

No beam correction 

1D beam + squint 

Full 2D beam

 

Example: wide field polarimetry continued�� 

• Simulated Hydra A image

 

Model 

Errors 1D sym.beam 

Errors full beam 

Panels:    I     Q

                U    V

 

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 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 Astrophysics

 

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
• incident radiation with quadrupole
• e.g. Cosmic Microwave Background
• and more��

 

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)

 

Example: 3C31 

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

 

3 kpc

 

Example: Cygnus A 

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

 

10 kpc

 

Example: Blazar Jets 

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

 

1055+018

 

Example: the ISM of M51 

Neininger (1992)

 

Example: Zeeman effect

 

Example: Zeeman in M17 

m 

Zeeman B_los: colors (Brogan & Troland 2001)  Polarization B_perp: 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) 

 

Example: Faraday Rotation 

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

 

  15   12       8 GHz

 

Example: more Faraday rotation 

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

 

Example: Galactic Faraday Rotation 

• Mapping galactic magnetic fields with FR

 

Han, Manchester, & Qiao (1999)

Han et al. (2002) 

Filled: positive RM   Open: negative RM

 

Example: Stellar SiO Masers 

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