Specifying Polarisation Properties For Radio Telescopes

Transcription

1 Specifying Polarisation Properties For Radio Telescopes Bruce Veidt NRC Herzberg Astronomy and Astrophysics, Penticton, BC, Canada PAF Workshop, Sydney, November 2017

2 Outline Motivation: Polarised sky and PAFs Confusing aspects of polarisation Introduction of a refined measure of polarisation quality/purity for radio telescopes: IXR Where does the approximation come from? Comments on relative error inequalities 1

3 Motivation: Polarised Sky Courtesy T. Landecker, map made by M. Wolleben with data from Wolleben et al (north) and Testori et al (south) 2

4 Motivation: Polarisation Properties of PAFs Veidt, Hovey, Burgess, Smegal, Messing, Willis, Gray, & Dewdney, Demonstration of a Dual-Polarized Phased-Array Feed, IEEE Trans. Ant.Propagat., 2011, vol. 59, pp

5 Polarisation Can Be Confusing Sign of circular polarised component (classical physics vs. IEEE) Factor of 1/2 (Measurement Equation) or 1 (AIPS etc.) in equations for interferometer response Co/cross-polarised response (engineers) vs. Stokes parameters and leakage terms (astronomers) Mueller matrix representation vs. Jones matrix SKA Reference Design: 40 db polarisation purity 4

6 A New Figure of Merit Antenna engineers (from a heritage of communications systems with polarised waves) often quantify polarisation performance by co- and cross-polarised response However, cross-polarisation isolation or discriminations can be dependent upon coordinate system used Problem recognized for a number of years: GRASP has major-minor-xpd for circular polarisation More recently, Carozzi & Woan have proposed the Intrinsic Cross-polarisation Ratio or IXR Pontoppidan, GRASP8 Technical Description, 2002 Carozzi & Woan, A Fundamental Figure of Merit for Radio Polarimeters, IEEE Trans. Ant. Propagat., vol. 59, pp ,

7 IXR Development Carozzi & Woan show that for a non-exactly-orthogonal crossed dipole, the traditional cross-polarisation ratios are dependent upon coordinate system alignment e.g. align with one dipole vs. alignment with mid-angle Define IXR in terms of maximum and minimum gains: IXR ( gmax + g min g max g min ) 2 This will be put in terms of a Jones matrix description of receiving system 6

8 Jones Formalism Transmission matrix J at each frequency and direction angle [ v A v B ] V rx = J telescope E sky = [ J Ax J Bx J Ay J By ] [ e x e y feed port Jones plane-wave voltages matrix fields Want to solve for E sky = J 1 telescope V rx Therefore the matrix conditioning of J is critical ] 7

9 Conditioning of Jones Matrix Condition number based on Singular-Value Decomposition: κ(j) = σ max(j) σ min (J) Re-write in terms of the condition number (κ) of the Jones matrix: IXR = ( κ(j)+1 κ(j) 1) 2. Consider the two extreme cases: 1. Ideal conditioning: κ (J) = 1 and IXR. 2. Poor conditioning: κ (J) and IXR 1. This means that we want IXR to be as large as possible. See Wikipedia for an image which is helpful in understanding SVD. 8

10 Relative Error Relation Carozzi & Woan present an error relation based on matrix norms ( ): E E ( 1 + ) ( 2 J +... IXR J + V ) V This relation comes from linear algebra theory Places limits on relative error of estimated sky fields E E error in knowledge of the radiation pattern J J voltage signal V V Where does the error multiplier come from? based on relative and measurement of the 9

11 Approximate Error Multiplier Condition number in terms of IXR: κ exact (J) = IXR + 1 = IXR IXR IXR Multiply numerator by series expansion of the reciprocal of the denominator: ( ) ) κ (J) 1 + ( IXR IXR IXR IXR +... Drop all but the first two terms: κ approx (J) IXR Confirmed by T. Carozzi, , Jan

12 Compare with Exact Relation κ approx (J) IXR = IXR + 2 IXR Compare with the exact relation: κ exact (J) = IXR + 1 IXR 1 The term is κ (J) an error multiplication factor: E sky E sky κ (J) ( J J + V ) rx V rx 11

13 Compare Exact and Approximate Error Multipliers The approximate error multiplication factor does not work well for small IXRs However, do hope to work at the right of the plot 12

14 An Example: 10-m Reflector with Dipole Feed 13

15 Plot of Error Multiplier Large errors where one polarisation component 0 14

16 IXR or Condition Number Which should we be using, IXR or Condition Number? Really need to know the error multiplier Measure/simulate radiation pattern J κ (J) IXR κ (J) IMHO using condition number is more direct Some people may be more comfortable with IXR since it is like cross-pol ratios set maximum condition number and calculate IXR min 15

17 Other Issues Carozzi & Woan actually propose three variations of IXR: Jones Mueller Interferometer Mueller variant has an error relationship based on a Stokes vector representation of the radiation field s = [I, Q, U, V ] T I (and error I) dominate other terms and this may skew the error analysis 16

18 Conclusions Carozzi & Woan are to be commended for developing a better metric of polarisation performance Using error relationships, the impact of low or high IXR numbers readily understood SKA polarisation purity now defined in terms of IXR [better than 15 db raw, 40 db processed] But still not well known in mainstream antenna engineering community or by radio astronomers Should consolidate to one IXR IMHO Jones variant, because that is what we measure in the antenna range 17

19 Conclusions (II) More work needed on understanding error relationship in Mueller case Knowing the condition number (or IXR) of the Jones matrix for a telescope helps in the development of an error budget E sky E sky κ (J) ( J J + V ) rx V rx 18

20 Thank you 19

21 More Carozzi & Woan define the total error of the Stokes vector as s = I 2 + Q 2 + U 2 + V 2. The total relative error is then s / s where s = I 2 + Q 2 + U 2 + V 2 = I 1 + ( ) 2 Q + I ( U I ) 2 + ( V I ) 2. Radio astronomy signals usually weakly polarized ({Q, U, V } I) so that s = I. 20

22 More (II) Returning to s, rearrange as s = I 1 + ( ) 2 Q + I ( U I ) 2 + ( V I ) 2. Again, the I-component is dominant so that s = I and the relative error is s s = I I. Unfortunately this does not give an estimate of the relative error in the polarised components. 21