Synthesis Polarimetry Fundamentals for ALMA. George Moellenbrock (NRAO) 2013 June 25 ALMA Italian ARC Node

Transcription

1 Synthesis Polarimetry Fundamentals for ALMA George Moellenbrock (NRAO) 2013 June 25 ALMA Italian ARC Node

2 References Synthesis Imaging in Radio Astronomy II (1998 NRAO Summer School Book) Thompson, Moran, & Swenson Sault, Killeen, & Kesteven 1991, AT PolarizaUon CalibraUon (memo) Hamaker, Bregman, & Sault 1995

3 Synthesis Fundamentals Van- CiZert- Zernike Theorem: Ft {I } = V true = <s i s j *> s i is the direcuon- dependent EM field disturbance generated by an astronomical source as ideally sampled at antenna i It is actually a vector quanuty, s i And it must be calibrated: V obs = <x i x j *> x i = J i s i

4 PolarizaUon Bases To sample the incident vector EM field with opumum accuracy and precision, we arrange to receive dual (nominally) orthogonal polarizauons at each antenna Any orthogonal basis will suffice; we typically choose one of two standard bases: Linear: X, Y (e.g., ALMA) Circular: R, L (e.g., EVLA > 1GHz) Basis choice influences observauon pazerns and calibrauon heurisucs, in pracuce

5 CorrelaUons and Stokes Parameters - I Measured correla8ons or visibili8es are 4- vectors: V XX = I + Q V XY = U + iv V YX = U iv V YY = I Q V RR = I + V V RL = Q + iu V LR = Q iu V LL = I V Total Intensity (Stokes I) always in parallel- hands Linear PolarizaUon (Stokes Q,U): linear basis: parallel- and cross- hands circular basis: cross- hands Circular PolarizaUon (Stokes V): linear basis: cross- hands circular basis: parallel hands

6 CorrelaUons and Stokes Parameters - II Calibrated correlauon visibiliues are combined to form the Stokes visibiliues: I = (V XX + V YY )/2 = (V RR + V LL )/2 Q = (V XX V YY )/2 = (V RL + V LR )/2 U = (V XY + V YX )/2 = (V RL V LR )/2i V = (V XY V YX )/2i = (V RR V LL )/2 All four correlauons must be consistently calibrated, so the combinauons can be formed correctly Stokes visibili8es may then be inverted to form Stokes images

7 Generic Matrix CalibraUon Scalar: x = J s Vector: PolarizaUon may mix! x p = J p p J p q s p x p J q p J q q s p J pq, J qp generally small, by design, but not strictly zero

8 Factored CalibraUon J i s i = B i G i D i P i T i s i B i : bandpass G i : (electronic) gain D i : instrumental polarizauon T i : troposphere P i : parallacuc angle Approximately physical OK to imagine each antenna s received signal, s i, corrupted by terms in order from right to lel, though actual effects are not strictly quite so discrete Really a convenient factorizauon of effects that organizes the matrix algebra Consistent with hardware design (e.g., polarizer in front of amplifiers, etc.) Order is important! NotaUon in this talk: One subscript: antenna- based Two subscripts: baseline- based Zero subscripts: mnemonic or operator notauon Lower case: vector or matrix element

9 The Measurement EquaUon V obs =B G D P T V true V corr = T - 1 P - 1 D - 1 G - 1 B - 1 V obs Where: V obs = observed visibility 4- vector (correlauons) V true = true visibility 4- vector; esumated by V mod when solving V corr = corrected visibility 4- vector B, G, D, P, T = baseline- based operators each represenung pair- wise combinauons of antenna- based terms We will solve for relevant calibrauon terms with an appropriate heurisuc Each solve has the form: (V obs )= J (V mod ) () indicates parual correcuon/corrupuon by already- known terms

10 Gain- like terms: T, G, B Troposphere: T = Scalar Commutes with everything; olen absorbed in G WVR phase is a variety of T Electronic gain: G = Diagonal The ubiquitous term: typically describes many instrumental effects Bandpass: B = Diagonal: Frequency- dependent version of G ALMA Tsys is a variety of B t 0 0 t b X (ν) 0 g X 0 0 b Y (ν) 0 g Y

11 CalibraUon is a Bootstrapping Process Assume unity for all non- relevant terms Adopt priors, if available, e.g., B tsys, T wvr Solve for the dominant term, e.g., G: (B tsys - 1 V obs ) = G (T wvr Vmod) Then, using G, solve for next- most- important term, B: (B tsys - 1 V obs )= B (G T wvr V mod ) B will be a funcuon of how good net prior calibrauon is Iterate and extend, as necessary (generalized selfcal ) E.g., use B to obtain G for other sources, Umescales Revise source models, as needed (tradiuonal selfcal ) Correct the observed data (~sufficient for total intensity): V corr = (T wvr - 1 G - 1 B - 1 B tsys - 1 V obs )

12 Scalar Treatment and Poln Basis Linear basis: If calibrator has non- zero (and unknown) linear polarizauon, polarizauon- dependent gain- like solves will absorb it, e.g.: g X = g X (1 + Q/I) 0.5 g Y = g Y (1 - Q/I) 0.5 Parallel hands will be corrected for calibrator polarizauon! Cross- hand correcuon error is 2 nd order in Q : (1 Q 2 /I 2 ) 0.5 DistorUon is small (we will rely on this later) Formally, desirable to measure G on an unpolarized source, depend on g X /g Y stability, and use (unpol) T(t) Q is systemaucally Ume- dependent (due to parallacuc angle), so the apparent gain rauo provides a means of esumaung source polarizauon if sufficient sampling available, and true gain rauo is stable: g X /g Y = (g X /g Y ) (1 + Q/I) 0.5 /(1 - Q/I) 0.5 (g X /g Y ) (1 2Q/I) 0.5 Circular basis: Q V ; Usually, V = 0, so no problem

13 ParallacUc Angle, P Sky orientauon rotates in the field of view of an alt- az telescope: ψ(t) = cos b sin H(t) sin b cos δ - cos b sin δ cos H(t) b = la8tude; H(t) = Hour Angle; δ = declina8on cosψ sinψ P lin = P circ = - sinψ cosψ e - iψ 0 0 e iψ At ψ = 0 (meridian: H = 0), mechanical feed posiuon angle may be offset (linear basis)

14 P in the Linear Basis If ψ i = ψ for all antennas (small array: ~uniform b and H), the trigonometry of P greatly simplified Uniform rotauon of linear polarizauon Cross- hand rotauon (U ψ ) in quadrature w/ parallel- hand rotauon (Q ψ ): V XX = I + (Q cos2ψ + U sin2ψ) = I + Q ψ V XY = (- Q sin2ψ + U cos2ψ) + iv = U ψ + iv V YX = (- Q sin2ψ + U cos2ψ) iv = U ψ iv V YY = I (Q cos2ψ + U sin2ψ) = I Q ψ

15 P in the Circular Basis If ψ i = ψ for all antennas (small array), uniform rotauon of linear polarizauon occurs only in the cross- hands: V RR = I + V V RL = (Q + iu ) e- i2ψ V LR = (Q iu ) e +i2ψ V LL = I V

16 Instrumental PolarizaUon, D Each polarized receptor sees some of the other polarizauon: D = 1 d X (ν) d Y (ν) 1 General NotaUon: d X is the fracuon of Y polarizauon sensed by X N.B.: On- diagonal effects factored into G, B Origins: Finite impuriues in polarizers ReflecUons that return in opposite polarizauon: standing waves Asymmetry in opucs induce spurious polarizauon (e.g., squint)

17 D: ProperUes Orthogonality condiuon: (d Xi +d Yi *) = 0 Easier to achieve than purity Orthogonality esumator: (d Xi +d Yi *)/2 Purity esumator: (d Xi - d Yi *)/2 In the linear basis (d << 1.0): Real(d) = linear polarizauon orientauon error Imag(d) = ellipucity error

18 V = D P V true : D in the Linear Basis - I V XX = (I+Q ψ ) + (U ψ +iv)d Xj * + d Xi (U ψ -iv ) + d Xi (I-Q ψ )d Xj * V XY = (I+Q ψ )d Yj * + (U ψ +iv) + d Xi (U ψ -iv )d Yj * + d Xi (I-Q ψ ) V YX = d Yi (I+Q ψ ) + d Yi (U ψ +iv)d Xj * + (U ψ -iv ) + (I-Q ψ )d Xj * V YY = d Yi (I+Q ψ )d Yj * + d Yi (U ψ +iv) + (U ψ -iv )d Yj * + (I-Q ψ ) Nice symmetries: d muluplies pure cross- /parallel- hands in the parallel- /cross- hands d 2 muluplies other pure parallel- /cross- hand (c.f. antenna- based descripuon)

19 D in the Linear Basis - II Linearized, sorted: V XX = (I+Q ψ ) + (U ψ +iv)d Xj * + d Xi (U ψ -iv ) V XY = (U ψ +iv) + (I+Q ψ )d Yj * + d Xi (I-Q ψ ) V YX = (U ψ -iv ) + d Yi (I+Q ψ ) + (I-Q ψ )d Xj * V YY = (I-Q ψ ) + d Yi (U ψ +iv) + (U ψ -iv )d Yj *

20 D in the Linear Basis - III Linearized, sorted, dv ~ 0, regrouped Stokes V XX = (I+Q ψ ) + U ψ (d Xj *+d Xi ) V XY = (U ψ +iv) + I (d Yj *+d Xi ) + Q ψ (d Yj *- d Xi ) V YX = (U ψ -iv) + I (d Yi +d Xj *) + Q ψ (d Yi - d Xj *) V YY = (I-Q ψ ) + U ψ (d Yi +d Yj *) ProperUes: Constant (per baseline) complex offset proporuonal to I in cross- hands d- scaled Ume- dependent source linear polarizauon in all correlauons

21 D in the Circular Basis - I V = D P V true : V RR = (I+V) + (Q+iU)e - i2ψ d Rj * + d Ri (Q-iU )e +i2ψ + d Ri (I-V)d Rj * V RL = (I+V)d Lj * + (Q+iU)e - i2ψ + d Ri (Q-iU)e +i2ψ d Lj * + d Ri (I-V) V LR = d Li (I+V) + d Li (Q+iU)e - i2ψ d Rj * + (Q-iU )e +i2ψ + (I-V)d Rj * V LL = d Li (I+V)d Lj * + d Li (Q+iU)e - i2ψ + (Q-iU)e +i2ψ d Lj * + (I-V) Symmetries (same as linear basis): d muluplies pure cross- /parallel- hands in parallel- /cross- hands d 2 muluplies other pure parallel- /cross- hand

22 D in the Circular Basis - II Linearized, sorted: V RR = (I+V) + (Q+iU)e - i2ψ d Rj * + d Ri (Q-iU )e +i2ψ V RL = (Q+iU)e - i2ψ + (I+V)d Lj * + d Ri (I-V) V LR = (Q-iU )e +i2ψ + d Li (I+V) + (I-V)d Rj * V LL = (I-V) + d Li (Q+iU)e - i2ψ + (Q-iU)e +i2ψ d Lj *

23 D in the Circular Basis - III Linearized, sorted, dv ~ 0: V RR = (I+V) + (Q+iU)e - i2ψ d Rj * + d Ri (Q-iU )e +i2ψ V RL = (Q+iU)e - i2ψ + I (d Lj *+d Ri ) V LR = (Q-iU )e +i2ψ + I (d Li +d Rj *) V LL = (I-V) + d Li (Q+iU)e - i2ψ + (Q-iU)e +i2ψ d Lj * TradiUonally, VLA also has ignored Q,U terms in V RR, V LL Degenerate D soluuon: (d Lj - a)*+(d Ri +a*) = (d Lj *+d Ri ), so one d R arbitrarily set to zero (refant) Time- dependent closure errors in V RR, V LL for polarized sources: limits I dynamic range to K

24 Visualizing PolarizaUon Effects in the Linear Feed Basis Illustrates corrupuon by P, D, G in a single channel A 4.5h observauon of 3C286 An idealized point source model (10% linear polarizauon): I = 1.0 Q = 0.06 U = 0.08 V = 0.0 No noise Baselines to a single antenna

25 Ideal VisibiliUes: V true V XX = I + Q V XY = U + iv V YX = U iv V YY = I Q

26 V true : Real vs Ume V XX V YY V XY V YX

27 V true : Imag vs. Real V XY V YX V XX V YY

28 ParallacUc Angle: P V true V XX = I + (Q cos2ψ + U sin2ψ) = I + Q ψ V XY = (- Q sin2ψ + U cos2ψ) + iv = U ψ + iv V YX = (- Q sin2ψ + U cos2ψ) iv = U ψ iv V YY = I (Q cos2ψ + U sin2ψ) = I Q ψ

29 P: parang vs. Ume

30 P V true : Real vs. Ume V XX V YY V XY V YX

31 P V true : Real vs. parang V XX V YY V XY V YX

32 P V true : Imag vs. Real V XY V YX V XX V YY

33 Instrumental Polarization: D P V true V XX = (I+Q ψ ) + U ψ (d Xj *+d Xi ) V XY = U ψ + I (d Yj *+d Xi ) + Q ψ (d Yj *- d Xi ) V YX = U ψ + I (d Yi +d Xj *) + Q ψ (d Yi - d Xj *) V YY = (I-Q ψ ) + U ψ (d Yi +d Yj *) (linearized)

34 D P V true : Real vs. parang V XX V YY V XY V YX

35 D P V true : Real vs. parang V XY V YX only

36 D P V true : Imag vs. Real V XY V YX V XX V YY

37 D P V true : Imag vs. Real V XY V YX only

38 Gain: G D P V true V XX = g Xi g Xj * {(I+Q ψ ) + U ψ (d Xj *+d Xi )} V XY = g Xi g Yj * {U ψ + I (d Yj *+d Xi ) + Q ψ (d Yj *- d Xi )} V YX = g Yi g Xj * {U ψ + I (d Yi +d Xj *) + Q ψ (d Yi - d Xj *)} V YY = g Yi g Yj * {(I-Q ψ ) + U ψ (d Yi +d Yj *)} For this illustrauon, G contains only random, antenna- based phases that are constant in Ume (i.e., all amplitudes are 1.0)

39 G D P V true : Real vs. parang

40 G D P V true : Real vs. parang V XY V YX only

41 G D P V true : Imag vs. Real V XX V YY V XY V YX

42 G D P V true : Imag vs. Real V XY V YX only

43 P, D, G CorrupUon Sequence Real vs. Imag V true P V true D P V true G D P V true

44 P, D, G CorrupUon Sequence Real vs. Imag V true P V true V XY V YX only D P V true G D P V true

45 Solving for D (linear basis) Unpolarized source, single scan? Simple, but d s are degenerate, to first order: V XY = I (d Yj *+d Xi ) = I [(d Yj - a)*+(d Xi +a*)] for any complex number a Therefore, one d remains effecuvely unconstrained Standard convenuon is to apply a reference antenna that effecuvely enforces a = - d Xref *: d xi (d Xi +a*) d Yi (d Yi - a) (for all i) These referenced d s correct the data to some orthogonal (to first order) basis that is defined by the refant s true d X - - but not a pure one

46 Solving for D (linear basis) Polarized source, single scan (ψ = const)? SUll degenerate in linearized cross- hands: V XY = (I + Q ψ )d Yj *+(I Q ψ )d Xi = (I + Q ψ )(d Yj - b)*+(i Q ψ )(d Xi +c*) for any complex numbers b, c sausfying b = c(i + Q ψ )/(I Q ψ ) Incorrectly calibrates data with different Q ψ (other Umes, sources)

47 Solving for D (linear basis) Polarized source w/ parallacuc angle coverage MulUple, non- zero Q ψ breaks degeneracies Depends on Ume stability of D (also required for accurate transfer to other sources) ResulUng soluuon accuracy limited by: accuracy of calibrator model s Q,U systemauc bias (if any) in assumed feed posiuon angle sezng (if Q,U derived from data) systemauc biases (if any) originaung in 2 nd - order terms

48 Cross- hand phase spectrum An arufact of gain calibrauon reference antenna (refant) We do not measure absolute G and B Instead, we measure G r and B r, wherein a reference antenna s phase is fixed to zero in both polarizauons, yielding relauve phases for all other antennas Differences among antennas in each polarizauon (separately) are preserved: no effect on parallel- hand calibrauon The refant s cross- hand bandpass phase remains undetected: B G = B r G r X r And uncorrected: G r - 1 r B r - 1 B G = X r e X r = iρ X r is as interesung as any bandpass phase spectrum in the system

49 Revised factorizauon We therefore rewrite the calibrauon operator equauon: V obs = B G D P Vmod = B r G r X r D P V mod It is convenient to move X r upstream of D: = B r G r D r X r P V mod (D r = X r D X r- 1 ) D r is the instrumental polarizauon measured in the cross- hand phase frame of the gain & bandpass calibrauon reference antenna In the linear basis, X r must be determined so cross- and parallel- hands can be combined to extract correct Stokes parameters In the circular basis, X r is just a polarizauon posiuon angle offset, which can be deferred for later external calibrauon

50 Solving for X r (linear basis) (G r - 1 B r - 1 V obs ) = X r D P V mod Consider just the cross- hands: V XY = e iρ {U ψ + I (d Yj *+d Xi ) + Q ψ (d Yj *- d Xi )} V YX = e - iρ {U ψ + I (d Yi +d Xj *) + Q ψ (d Yi - d Xj *)} Average over baselines and correlauons: (<V XY > + <V YX *>)/2 = U ψ e iρ + <I(d Yj *+d Xi ) + Q ψ (d Yj *- d Xi )>e iρ = (- Q sin2ψ + U cos2ψ)e iρ + ε ε is small if d s are small and ~random Requires non- zero Q, U (c.f. desirability for D soluuon) Measurements at (at least) 3 disunct ψ sufficient to determine ρ, Q, U, ε Degeneracy: (ρ, Q, U) (ρ+π, -Q, -U) Resolvable using Q, U esumate from gain rauo Requires X r stability: a good refant for gain and bandpass (c.f. g X /g Y stability expectauons) ATCA (linears) use a calibrauon signal to monitor X r refant operauon in G r and B r solves will then merely enforce the truth) (Circular basis: polarizauon posiuon angle offset)

51 G D P V true Imag vs. Real V XY V YX only

52 (G r - 1 G) D P V true = X r D P V true Imag vs. Real V XY = e iρ {U ψ + I (d Yj *+d Xi ) + Q ψ (d Yj *- d Xi )} V YX = e - iρ {U ψ + I (d Yi +d Xj *) + Q ψ (d Yi - d Xj *)} V XY V YX only

53 <X r D P V QU > = X r <D> P V QU Imag vs. Real <V XY > = U ψ e iρ + <I (d Yj *+d Xi ) + Q ψ (d Yj *- d Xi )>e iρ = (- Q sin2ψ + U cos2ψ)e iρ + ε <V YX > = U ψ e - iρ + <I (d Yi +d Xj *) + Q ψ (d Yi - d Xj *)>e - iρ = (- Q sin2ψ + U cos2ψ)e - iρ + ε V XY V YX only

54 PolarizaUon CalibraUon Bootstrapping CalibraUon Model: Basic Solve sequence: Normal B r and G r (parallel- hands): X r, Q and U (cross- hands): V obs = B r G r D r X r P Vmod V obs = B r V I (B r - 1 V obs ) = G r V I (+esumate QU) (G r - 1 B r - 1 V obs ) = X r P V QU (+resolve amb) Revise G r, using IQU (parallel- hands): (B r - 1 V obs ) = G r (P V IQU ) D r, using IQU (cross- hands): (X r - 1 G r - 1 B r - 1 V obs ) = D r (X r P V IQU ) (IteraUon?) CorrecUon: V corr = (P - 1 X r - 1 D r - 1 G r - 1 B r - 1 V obs )

55