Polarization in Interferometry
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1 Polarization in Interferometry Calibration Tutorial Ivan Martı -Vidal Onsala Space Observatory Chalmers University of Technology (Sweden) European Radio Interferometry School Dwingeloo (October 2017)
2 TUTORIAL - ALMA B5 I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
3 How we will proceed in this Tutorial There are three scripts and one measurement set provided: scriptforcalibration.py Standard calibration. scriptforpolcalibration.py Polarization calibration. ERIS2017 helper.py Some helper functions for plotting, hacking tables, etc. ERIS 2017.Polarization Tutorial.DATA.ms ALMA Band 5 (commissioning full-pol) observations of VY CMa. The script is almost ready to go. Just a few lines have to be uncommented (after discussion). The steps will be performed one by one. Results after each step will be discussed. Freedom to choose among different options (e.g., bandpass, reference antenna, etc.) and to play with (i.e., hack) the calibration tables. The script to obtain the final image of the target will be fully written by you! I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
4 ALMA Band 5 PART I: Standard Calibration Look at the schedule. Understand the observations. Putting a model to the primary flux calibrator. Check model column. Bandpass calibration Fast phase calibration (i.e., optimize coherence). Bandpass calibration. Does it vary with time? Gain calibration Fast phase & slow amplitude Absolute flux-density scaling Slow phase (for phase referencing) Check X/Y cross-phases. What happens to the ref. antenna? I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
5 ALMA Band 5 PART I: Standard Calibration Look at the schedule. Understand the observations. Putting a model to the primary flux calibrator. Check model column. Bandpass calibration Fast phase calibration (i.e., optimize coherence). Bandpass calibration. Does it vary with time? Gain calibration Fast phase & slow amplitude Absolute flux-density scaling Slow phase (for phase referencing) Check X/Y cross-phases. What happens to the ref. antenna? I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
6 ALMA Band 5: Understand I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
7 ALMA Band 5: Understand I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
8 ALMA Band 5 PART I: Standard Calibration Look at the schedule. Understand the observations. Putting a model to the primary flux calibrator. Check model column. Bandpass calibration Fast phase calibration (i.e., optimize coherence). Bandpass calibration. Does it vary with time? Gain calibration Fast phase & slow amplitude Absolute flux-density scaling Slow phase (for phase referencing) Check X/Y cross-phases. What happens to the ref. antenna? I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
9 ALMA Band 5: Flux Cal. I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
10 ALMA Band 5: Flux Cal. I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
11 ALMA Band 5 PART I: Standard Calibration Look at the schedule. Understand the observations. Putting a model to the primary flux calibrator. Check model column. Bandpass calibration Fast phase calibration (i.e., optimize coherence). Bandpass calibration. Does it vary with time? Gain calibration Fast phase & slow amplitude Absolute flux-density scaling Slow phase (for phase referencing) Check X/Y cross-phases. What happens to the ref. antenna? I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
12 ALMA Band 5: Bandpass I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
13 ALMA Band 5: Bandpass I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
14 ALMA Band 5: Bandpass I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
15 ALMA Band 5 PART I: Standard Calibration Look at the schedule. Understand the observations. Putting a model to the primary flux calibrator. Check model column. Bandpass calibration Fast phase calibration (i.e., optimize coherence). Bandpass calibration. Does it vary with time? Gain calibration Fast phase & slow amplitude Absolute flux-density scaling Slow phase (for phase referencing) Check X/Y cross-phases. What happens to the ref. antenna? I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
16 ALMA Band 5: Gain Cal. I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
17 ALMA Band 5: Gain Cal. I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
18 ALMA Band 5: Gain Cal. PART I: Standard Calibration Look at the schedule. Understand the observations. Putting a model to the primary flux calibrator. Check model column. Bandpass calibration Fast phase calibration (i.e., optimize coherence). Bandpass calibration. Does it vary with time? Gain calibration Fast phase & slow amplitude Absolute flux-density scaling Slow phase (for phase referencing) Check X/Y cross-phases. What happens to the ref. antenna? I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
19 ALMA Band 5: Gain Cal. I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
20 ALMA Band 5: Gain Cal. I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
21 ALMA Band 5 PART II: Polarization Calibration X/Y gain ratio for the Pol. calibrator (Q effect on XX/YY). Cross-delay calibration (take a look at the data) Cross-phase calibration (look at the estimated Stokes) Cross-phase calibration ambiguity Polarization leakage Amplitude cross-polarization ratios I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
22 ALMA Band 5 PART II: Polarization Calibration X/Y gain ratio for the Pol. calibrator (Q effect on XX/YY). Cross-delay calibration (take a look at the data) Cross-phase calibration (look at the estimated Stokes) Cross-phase calibration ambiguity Polarization leakage Amplitude cross-polarization ratios I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
23 ALMA Band 5: Q/U from Gain Remember the brightness matrix in linear-polarization basis: ( ) I + Qant U ant + j V S = U ant j V I Q ant where Q ant = Q sky cos 2ψ + U sky sin 2ψ I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
24 ALMA Band 5: Q/U from Gain Remember the brightness matrix in linear-polarization basis: ( ) I + Qant U ant + j V S = U ant j V I Q ant where Q ant = Q sky cos 2ψ + U sky sin 2ψ So: G x G y I + Q ant I Q ant = f (ψ). I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
25 ALMA Band 5: Q/U from Gain We estimate Q sky and U sky from G x G y vs. ψ I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
26 ALMA Band 5: Q/U from Gain We estimate Q sky and U sky from G x G y vs. ψ I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
27 ALMA Band 5 PART II: Polarization Calibration X/Y gain ratio for the Pol. calibrator (Q effect on XX/YY). Cross-delay calibration (take a look at the data) Cross-phase calibration (look at the estimated Stokes) Cross-phase calibration ambiguity Polarization leakage Amplitude cross-polarization ratios I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
28 ALMA Band 5: X-Y Delay I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
29 ALMA Band 5: X-Y Delay I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
30 ALMA Band 5 PART II: Polarization Calibration X/Y gain ratio for the Pol. calibrator (Q effect on XX/YY). Cross-delay calibration (take a look at the data) Cross-phase calibration (look at the estimated Stokes) Cross-phase calibration ambiguity Polarization leakage Amplitude cross-polarization ratios I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
31 ALMA Band 5: X-Y Phase Let s add a cross-polarization phase to the brightness matrix: S = ( I + Q ant (U ant + j V )e j ) (U ant j V )e +j I Q ant where G x = 1 and G y = e +j at all antennas. I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
32 ALMA Band 5: X-Y Phase Let s add a cross-polarization phase to the brightness matrix: S = ( I + Q ant (U ant + j V )e j ) (U ant j V )e +j I Q ant where G x = 1 and G y = e +j at all antennas. So the cross-polarization visibilities are: XY = (U sky cos 2ψ Q sky sin 2ψ + jv )e j. I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
33 ALMA Band 5: X-Y Phase Let s add a cross-polarization phase to the brightness matrix: S = ( I + Q ant (U ant + j V )e j ) (U ant j V )e +j I Q ant where G x = 1 and G y = e +j at all antennas. So the cross-polarization visibilities are: XY = (U sky cos 2ψ Q sky sin 2ψ + jv )e j. Real vs. Imag: Re(XY ) = (U sky cos 2ψ Q sky sin 2ψ) cos V sin Im(XY ) = (U sky cos 2ψ Q sky sin 2ψ) sin + V sin I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
34 ALMA Band 5: X-Y Phase Let s add a cross-polarization phase to the brightness matrix: S = ( I + Q ant (U ant + j V )e j ) (U ant j V )e +j I Q ant where G x = 1 and G y = e +j at all antennas. So the cross-polarization visibilities are: XY = (U sky cos 2ψ Q sky sin 2ψ + jv )e j. Real vs. Imag: Re(XY ) = (U sky cos 2ψ Q sky sin 2ψ) cos V sin Im(XY ) = (U sky cos 2ψ Q sky sin 2ψ) sin + V sin If we plot Re(XY ) vs. Im(XY ), we will see a straight line (as long as V is negligible) with a slope proportional to tan ( ). I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
35 ALMA Band 5: X-Y Phase I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
36 ALMA Band 5: X-Y Phase I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
37 ALMA Band 5: X-Y Phase Ambiguity We fit for Q sky, U sky, and using XY and XY as data. I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
38 ALMA Band 5: X-Y Phase Ambiguity We fit for Q sky, U sky, and using XY and XY as data. BUT notice that XY = (U sky cos 2ψ Q sky sin 2ψ + jv )e j has an ambiguity. (Q sky, U sky, ) ( Q sky, U sky, + π) I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
39 ALMA Band 5: X-Y Phase Ambiguity I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
40 ALMA Band 5 PART II: Polarization Calibration X/Y gain ratio for the Pol. calibrator (Q effect on XX/YY). Cross-delay calibration (take a look at the data) Cross-phase calibration (look at the estimated Stokes) Cross-phase calibration ambiguity Polarization leakage Amplitude cross-polarization ratios I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
41 ALMA Band 5: Leakage Effect of the Dterms on the XY cross-correlations: XY XY + D a x YY + (D b y ) XX I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
42 ALMA Band 5: Leakage Effect of the Dterms on the XY cross-correlations: XY XY + D a x YY + (D b y ) XX In terms of the brightness matrix (i.e., assuming a point source): XY U ant + jv + D a x (I Q ant ) + (D b y ) (I + Q ant ) I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
43 ALMA Band 5: Leakage Effect of the Dterms on the XY cross-correlations: XY XY + D a x YY + (D b y ) XX In terms of the brightness matrix (i.e., assuming a point source): XY U ant + jv + D a x (I Q ant ) + (D b y ) (I + Q ant ) Re-arranging terms: XY U ant + Q ant ((D b y ) D a x ) + jv + I (D a x + (D b y ) ) I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
44 ALMA Band 5: Leakage By knowing U ant and Q ant, we can solve for D x and D y for all antennas. I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
45 ALMA Band 5 PART II: Polarization Calibration X/Y gain ratio for the Pol. calibrator (Q effect on XX/YY). Cross-delay calibration (take a look at the data) Cross-phase calibration (look at the estimated Stokes) Cross-phase calibration ambiguity Polarization leakage Amplitude cross-polarization ratios I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
46 ALMA Band 5: X-Y Amp. Ratio Let s add a cross-polarization amplitude ratio to the brightness matrix: ( ) I + Qant (U ant + j V )ρ b F[V ab ] (U ant j V )ρ a (I Q ant )ρ a ρ b where G i x = 1 and G i y = ρ i. Changes in the amplitudes vs. ψ allow us to solve for ρ i. I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
47 ALMA Band 5: X-Y Amp. Ratio I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
48 FINISHED! I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
49 FINISHED! LET S PLAY NOW! I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
50 ALMA Band 5: Pol. Calib. Image I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
51 ALMA Band 5: Hacking Tables! The function finetune G allows you to change any calibration table in many different ways. Make a backup of your tables, run finetune G and repeat steps 9, 11, and 12 of the polarization calibration script. You can load the different images with the plotpolimage function to compare! Some ideas: Add an offset to the imaginary parts of the Dterms (different sign for each polarization). Same as above, but add it to the real parts. Repeat both previous items by using the same signs for both polarizations. Add an offset to the X-Y cross-polarization phase. Add a ratio to the X-Y amplitude-ratio gains. Add a cross-polarization delay. Any combination of the above. I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
52 BONUS: Leakage calibration on circular polarizers Calibrate R/L phase offset using pol. calibrator. I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
53 BONUS: Leakage calibration on circular polarizers Calibrate R/L phase offset using pol. calibrator. ( ) ( D L) V obs = D a X Vab true X H Db H a ; X = ; D a = 0 e jα Da R 1 V obs RL = ((D R a + (D L b ) ) I + P) e jα + O(D 2 ) V obs LR = ((D L a + (D R b ) ) I + P ) e jα + O(D 2 ) I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
54 BONUS: Leakage calibration on circular polarizers Calibrate R/L phase offset using pol. calibrator. ( ) ( D L) V obs = D a X Vab true X H Db H a ; X = ; D a = 0 e jα Da R 1 V obs RL = ((D R a + (D L b ) ) I + P) e jα + O(D 2 ) V obs LR = ((D L a + (D R b ) ) I + P ) e jα + O(D 2 ) Unpolarized calibrator: (D L a, D R a, e jα ) (D L a e j + jk, D R a e j + jk, e j(α ) ) Polarized calibrator: (D L a, D R a, e jα ) (D L a + jk, D R a + jk, e j(α) ) I. Martí-Vidal (Onsala Space Observatory) Interferometric Polarimetry ERIS / 31
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