Polarization in Interferometry
A basic introduction
Ivan Mart´ı-Vidal
Nordic ALMA Regional Center Onsala Space Observatory Chalmers University of Technology (Sweden)
European Radio Interferometry School (2015) ???
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 1 / 26 ???
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 1 / 26 • Get familiar with some basics of polarimetry.
I The different states of polarization. I The Stokes parameters.
• Understand radioastronomical polarizers.
I Linear dipoles and quarter waveplates. I Polarization and interferometry: the Measurement Equation.
• Learn the basic calibration procedures.
I Calibration with the Measurement Equation. I The effects of cross delay (phase) and leakage. • Calibrate and process real observations (MERLIN C band).
Goals of this lecture
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 2 / 26 • Understand radioastronomical polarizers.
I Linear dipoles and quarter waveplates. I Polarization and interferometry: the Measurement Equation.
• Learn the basic calibration procedures.
I Calibration with the Measurement Equation. I The effects of cross delay (phase) and leakage. • Calibrate and process real observations (MERLIN C band).
Goals of this lecture
• Get familiar with some basics of polarimetry.
I The different states of polarization. I The Stokes parameters.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 2 / 26 • Learn the basic calibration procedures.
I Calibration with the Measurement Equation. I The effects of cross delay (phase) and leakage. • Calibrate and process real observations (MERLIN C band).
Goals of this lecture
• Get familiar with some basics of polarimetry.
I The different states of polarization. I The Stokes parameters.
• Understand radioastronomical polarizers.
I Linear dipoles and quarter waveplates. I Polarization and interferometry: the Measurement Equation.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 2 / 26 • Calibrate and process real observations (MERLIN C band).
Goals of this lecture
• Get familiar with some basics of polarimetry.
I The different states of polarization. I The Stokes parameters.
• Understand radioastronomical polarizers.
I Linear dipoles and quarter waveplates. I Polarization and interferometry: the Measurement Equation.
• Learn the basic calibration procedures.
I Calibration with the Measurement Equation. I The effects of cross delay (phase) and leakage.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 2 / 26 Goals of this lecture
• Get familiar with some basics of polarimetry.
I The different states of polarization. I The Stokes parameters.
• Understand radioastronomical polarizers.
I Linear dipoles and quarter waveplates. I Polarization and interferometry: the Measurement Equation.
• Learn the basic calibration procedures.
I Calibration with the Measurement Equation. I The effects of cross delay (phase) and leakage. • Calibrate and process real observations (MERLIN C band).
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 2 / 26 Polarization of light
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 2 / 26 Electromagnetic waves
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 3 / 26 Electromagnetic waves
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 4 / 26 A random orientation of E~
E~ as seen on the wave-front plane
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 4 / 26 A random orientation of E~
E~ as seen on the wave-front plane
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 4 / 26 A (less) random orientation of E~
E~ as seen on the wave-front plane
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 5 / 26 A (less) random orientation of E~
E~ as seen on the wave-front plane
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 5 / 26 A (less) random orientation of E~
E~ as seen on the wave-front plane
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 6 / 26 A (less) random orientation of E~
E~ as seen on the wave-front plane
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 6 / 26 A (less) random orientation of E~
E~ as seen on the wave-front plane
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 7 / 26 A (less) random orientation of E~
E~ as seen on the wave-front plane
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 7 / 26 Polarization modes • LINEAR
• CIRCULAR
• ELLIPTIC (i.e., LINEAR + CIRCULAR)
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 7 / 26 I How much polarized vs. unpolarized light do we have? I What is the strength and direction of the linear polarization? I How much circular polarization do we have?
• We need four quantities to fully describe the polarization state:
• The Stokes parameters: I , Q, U, and V
√ Q2 + U2 1 U • Linear polarization: I = , θ = arctan p I 2 Q p 2 2 2 2 • Unpolarized intensity: Iu = I − Q − U − V
The Stokes parameters
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 8 / 26 I How much polarized vs. unpolarized light do we have? I What is the strength and direction of the linear polarization? I How much circular polarization do we have? • The Stokes parameters: I , Q, U, and V
√ Q2 + U2 1 U • Linear polarization: I = , θ = arctan p I 2 Q p 2 2 2 2 • Unpolarized intensity: Iu = I − Q − U − V
The Stokes parameters
• We need four quantities to fully describe the polarization state:
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 8 / 26 • The Stokes parameters: I , Q, U, and V
√ Q2 + U2 1 U • Linear polarization: I = , θ = arctan p I 2 Q p 2 2 2 2 • Unpolarized intensity: Iu = I − Q − U − V
The Stokes parameters
• We need four quantities to fully describe the polarization state:
I How much polarized vs. unpolarized light do we have? I What is the strength and direction of the linear polarization? I How much circular polarization do we have?
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 8 / 26 √ Q2 + U2 1 U • Linear polarization: I = , θ = arctan p I 2 Q p 2 2 2 2 • Unpolarized intensity: Iu = I − Q − U − V
The Stokes parameters
• We need four quantities to fully describe the polarization state:
I How much polarized vs. unpolarized light do we have? I What is the strength and direction of the linear polarization? I How much circular polarization do we have? • The Stokes parameters: I , Q, U, and V
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 8 / 26 p 2 2 2 2 • Unpolarized intensity: Iu = I − Q − U − V
The Stokes parameters
• We need four quantities to fully describe the polarization state:
I How much polarized vs. unpolarized light do we have? I What is the strength and direction of the linear polarization? I How much circular polarization do we have? • The Stokes parameters: I , Q, U, and V
√ Q2 + U2 1 U • Linear polarization: I = , θ = arctan p I 2 Q
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 8 / 26 The Stokes parameters
• We need four quantities to fully describe the polarization state:
I How much polarized vs. unpolarized light do we have? I What is the strength and direction of the linear polarization? I How much circular polarization do we have? • The Stokes parameters: I , Q, U, and V
√ Q2 + U2 1 U • Linear polarization: I = , θ = arctan p I 2 Q p 2 2 2 2 • Unpolarized intensity: Iu = I − Q − U − V
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 8 / 26 Polarizers in Radio Astronomy
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 8 / 26 I Linear polarizers (horizontal / vertical linear polarization). I Circular polarizers (left / right circular polarization).
• Polarizing receivers (polarizers). The signal is split coherently into two orthogonal polarization states.
Detecting source polarization
• The Stokes parameters describe the polarization state of light. But how do we measure them?
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 9 / 26 I Linear polarizers (horizontal / vertical linear polarization). I Circular polarizers (left / right circular polarization).
Detecting source polarization
• The Stokes parameters describe the polarization state of light. But how do we measure them?
• Polarizing receivers (polarizers). The signal is split coherently into two orthogonal polarization states.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 9 / 26 Detecting source polarization
• The Stokes parameters describe the polarization state of light. But how do we measure them?
• Polarizing receivers (polarizers). The signal is split coherently into two orthogonal polarization states.
I Linear polarizers (horizontal / vertical linear polarization). I Circular polarizers (left / right circular polarization).
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 9 / 26 Linear polarizers
Decomposing linear pol. with linear polarizers (no phase offset)
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 10 / 26 Linear polarizers
Decomposing circular pol. (left) with linear polarizers (90◦ offset)
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 10 / 26 Linear polarizers
Decomposing circular pol. (right) with linear polarizers (270◦ offset)
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 10 / 26 Linear polarizers
Decomposing elliptical pol. (right) with linear polarizers (generic phase offset)
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 10 / 26 Linear polarizers
2 2 • I = |Ex | + |Ey |
2 2 • Q = |Ex | − |Ey |
∗ • U = 2 Re(Ex Ey )
∗ • V = 2 Im(Ex Ey )
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 10 / 26 Circular polarizers
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 11 / 26 Circular polarizers
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 11 / 26 Circular polarizers
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 11 / 26 Circular polarizers
Decomposing linear pol. with circular polarizers (phase offset gives inclination)
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 11 / 26 Circular polarizers
Decomposing elliptical pol. with circular polarizers (R/L amplitude difference)
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 11 / 26 Circular polarizers
2 2 • I = |El | + |Er |
2 2 • V = |El | − |Er |
∗ • Q = 2 Re(El Er )
∗ • U = −2 Im(El Er )
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 11 / 26 Advanced formulation: The Measurement Equation
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 11 / 26 • For baseline AB, the coherency matrix is E AB = E~A(E~B )H • In the x-y polarization basis, the coherency matrix for baseline AB is:
D A B ∗ED A B ∗E Ex (Ex ) Ex (Ey ) E AB = D A B ∗ED A B ∗E Ey (Ex ) Ey (Ey )
• We also define the brightness matrix, B. For x-y polarizers, it is
I + QU + j V ! B = U − j V I − Q
• The coherency matrix is related to the Fourier transform of the brightness matrix!
AB E = F [B]|(u,v)
The Measurement Equation. Coherency matrix
• Electric field seen by antenna A: E~A.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 12 / 26 • In the x-y polarization basis, the coherency matrix for baseline AB is:
D A B ∗ED A B ∗E Ex (Ex ) Ex (Ey ) E AB = D A B ∗ED A B ∗E Ey (Ex ) Ey (Ey )
• We also define the brightness matrix, B. For x-y polarizers, it is
I + QU + j V ! B = U − j V I − Q
• The coherency matrix is related to the Fourier transform of the brightness matrix!
AB E = F [B]|(u,v)
The Measurement Equation. Coherency matrix
• Electric field seen by antenna A: E~A. • For baseline AB, the coherency matrix is E AB = E~A(E~B )H
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 12 / 26 • We also define the brightness matrix, B. For x-y polarizers, it is
I + QU + j V ! B = U − j V I − Q
• The coherency matrix is related to the Fourier transform of the brightness matrix!
AB E = F [B]|(u,v)
The Measurement Equation. Coherency matrix
• Electric field seen by antenna A: E~A. • For baseline AB, the coherency matrix is E AB = E~A(E~B )H • In the x-y polarization basis, the coherency matrix for baseline AB is:
D A B ∗ED A B ∗E Ex (Ex ) Ex (Ey ) E AB = D A B ∗ED A B ∗E Ey (Ex ) Ey (Ey )
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 12 / 26 • The coherency matrix is related to the Fourier transform of the brightness matrix!
AB E = F [B]|(u,v)
The Measurement Equation. Coherency matrix
• Electric field seen by antenna A: E~A. • For baseline AB, the coherency matrix is E AB = E~A(E~B )H • In the x-y polarization basis, the coherency matrix for baseline AB is:
D A B ∗ED A B ∗E Ex (Ex ) Ex (Ey ) E AB = D A B ∗ED A B ∗E Ey (Ex ) Ey (Ey )
• We also define the brightness matrix, B. For x-y polarizers, it is
I + QU + j V ! B = U − j V I − Q
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 12 / 26 The Measurement Equation. Coherency matrix
• Electric field seen by antenna A: E~A. • For baseline AB, the coherency matrix is E AB = E~A(E~B )H • In the x-y polarization basis, the coherency matrix for baseline AB is:
D A B ∗ED A B ∗E Ex (Ex ) Ex (Ey ) E AB = D A B ∗ED A B ∗E Ey (Ex ) Ey (Ey )
• We also define the brightness matrix, B. For x-y polarizers, it is
I + QU + j V ! B = U − j V I − Q
• The coherency matrix is related to the Fourier transform of the brightness matrix!
AB E = F [B]|(u,v)
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 12 / 26 • The visibility matrix (i.e., voltage cross-correlations) is:
D A B ∗ED A B ∗E vx (vx ) vx (vy ) AB H V = v~A (v~B ) = D A B ∗ED A B ∗E vy (vx ) vy (vy )
• Since v~i = Ji E~i ,
D A B ∗ED A B ∗E H Ex (Ex ) Ex (Ey ) AB ~ ~ H H V = JAEA EB JB = JA JB D A B ∗ED A B ∗E Ey (Ex ) Ey (Ey )
Coherency matrix and Visibility matrix.
• Voltage for antenna A with an x-y polarizer is: v~A = JAE~A, where E~A is the electric field in the x-y base and JA is the Jones matrix that calibrates antenna A.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 13 / 26 • Since v~i = Ji E~i ,
D A B ∗ED A B ∗E H Ex (Ex ) Ex (Ey ) AB ~ ~ H H V = JAEA EB JB = JA JB D A B ∗ED A B ∗E Ey (Ex ) Ey (Ey )
Coherency matrix and Visibility matrix.
• Voltage for antenna A with an x-y polarizer is: v~A = JAE~A, where E~A is the electric field in the x-y base and JA is the Jones matrix that calibrates antenna A. • The visibility matrix (i.e., voltage cross-correlations) is:
D A B ∗ED A B ∗E vx (vx ) vx (vy ) AB H V = v~A (v~B ) = D A B ∗ED A B ∗E vy (vx ) vy (vy )
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 13 / 26 Coherency matrix and Visibility matrix.
• Voltage for antenna A with an x-y polarizer is: v~A = JAE~A, where E~A is the electric field in the x-y base and JA is the Jones matrix that calibrates antenna A. • The visibility matrix (i.e., voltage cross-correlations) is:
D A B ∗ED A B ∗E vx (vx ) vx (vy ) AB H V = v~A (v~B ) = D A B ∗ED A B ∗E vy (vx ) vy (vy )
• Since v~i = Ji E~i ,
D A B ∗ED A B ∗E H Ex (Ex ) Ex (Ey ) AB ~ ~ H H V = JAEA EB JB = JA JB D A B ∗ED A B ∗E Ey (Ex ) Ey (Ey )
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 13 / 26 Let us remember the classical interferometer equation: Z AB ∗ − 2π j (u α+v δ) dα dδ Vobs = GA GB I (α, δ) e λ α,δ z
The Measurement Equation. A full Stokes formalism
For a source with a generic structure, the visibility matrix for antennas A and B (with no direction-dependent calibration) will be
Z 2π j dα dδ H AB − λ (u α+v δ) Vobs = JA B e (JB ) , α,δ z
p where( α, δ) are the (normalized) sky coordinates in the source plane, and z = 1 − α2 − δ2.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 14 / 26 The Measurement Equation. A full Stokes formalism
For a source with a generic structure, the visibility matrix for antennas A and B (with no direction-dependent calibration) will be
Z 2π j dα dδ H AB − λ (u α+v δ) Vobs = JA B e (JB ) , α,δ z
p where( α, δ) are the (normalized) sky coordinates in the source plane, and z = 1 − α2 − δ2.
Let us remember the classical interferometer equation: Z AB ∗ − 2π j (u α+v δ) dα dδ Vobs = GA GB I (α, δ) e λ α,δ z
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 14 / 26 Jones calibration matrices. Examples
jφx (t) ! Ax (t) e 0 • Gain, G = jφy (t) 0 Ay (t) e
! ejτx (ν−ν0) 0 • Delay, K = 0 ejτy (ν−ν0)
jφx (ν) ! Ax (ν) e 0 • Bandpass, B = jφy (ν) 0 Ay (ν) e
The Jones matrices are multiplicative, e.g.: J = G × B × K, but care must be taken, since matrices generally do not commute.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 15 / 26 Polarization calibration
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 15 / 26 Polarization calibration
• Parallactic angle.
• Polarization leakage.
• Cross-Delay/phase.
• Amplitude offset.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 16 / 26 Polarization calibration I. Parallactic angle cos φ − sin φ ejφ 0 P = P = xy sin φ cos φ rl 0 e−jφ • Is the rotation of the antenna mount axis w.r.t. the sky. • Is deterministic. It’s good to apply it before the phase (and delay/rate) calibration.
• It does not commute with the gains for linear polarizers.
• In VLBI, it also mixes Vxx and Vyy with Vxy and Vyx .
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 17 / 26 Polarization calibration II. Leakage
1 Dx (ν) Dxy = Dy (ν) 1 • Is caused by cross-talking between the polarizer channels • Each leaked signal is modified by an amplitude and a phase.
• Introduces spurious ellipticity and linear polarization. LIN. + LEAK CIRC. + LEAK
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 18 / 26 Polarization calibration III. Cross-hand delay/phase
1 0 ! Kc = 0 ej(τc (ν−ν0)+φc )
• Is caused by a delay between the polarizer channels at the reference antenna.
• In linear polarizers, introduces ellipticity.
• In circular polarizers, just rotates the PA of the linear polarization. OFFSET: 0◦ OFFSET: 45◦
LINEAR:
CIRCULAR:
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 19 / 26 Polarization calibration IV. Amplitude bias ! 1 0 Ga = 0 Ac
• Is caused by different Tsys , gain and/or bandpass between polarizer channels. • In linear polarizers, introduces spurious linear polarization. • In circular polarizers, introduces ellipticity.
• Not quite used per se, but implicit in the gain calibration.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 20 / 26 • STEP 2: Calibrate the leakage using an unpolarized source.
I If all calibrators are polarized, solve for leakage and source polarization simultaneously. I Need good parallactic-angle coverage. • STEP 3 (optional): Refine the calibration (calibration “cross-talk”) • STEP 4: Image each Stokes parameter separately. Combine images: (Q, U) → (Ip, θ).
Calibration strategy
The right order for matrix product is: J = (Ga Kc ) × D × P cal −1 −1 −1 obs i.e.: V = P × D × (Ga Kc ) × V
• STEP 1 (optional): Calibrate the cross-delay using a strong polarized source.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 21 / 26 • STEP 3 (optional): Refine the calibration (calibration “cross-talk”) • STEP 4: Image each Stokes parameter separately. Combine images: (Q, U) → (Ip, θ).
Calibration strategy
The right order for matrix product is: J = (Ga Kc ) × D × P cal −1 −1 −1 obs i.e.: V = P × D × (Ga Kc ) × V
• STEP 1 (optional): Calibrate the cross-delay using a strong polarized source. • STEP 2: Calibrate the leakage using an unpolarized source.
I If all calibrators are polarized, solve for leakage and source polarization simultaneously. I Need good parallactic-angle coverage.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 21 / 26 • STEP 3 (optional): Refine the calibration (calibration “cross-talk”) • STEP 4: Image each Stokes parameter separately. Combine images: (Q, U) → (Ip, θ).
Calibration strategy
The right order for matrix product is: J = (Ga Kc ) × D × P cal −1 −1 −1 obs i.e.: V = P × D × (Ga Kc ) × V
• STEP 1 (optional): Calibrate the cross-delay using a strong polarized source. • STEP 2: Calibrate the leakage using an unpolarized source.
I If all calibrators are polarized, solve for leakage and source polarization simultaneously. I Need good parallactic-angle coverage.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 21 / 26 • STEP 3 (optional): Refine the calibration (calibration “cross-talk”) • STEP 4: Image each Stokes parameter separately. Combine images: (Q, U) → (Ip, θ).
Calibration strategy
The right order for matrix product is: J = (Ga Kc ) × D × P cal −1 −1 −1 obs i.e.: V = P × D × (Ga Kc ) × V
• STEP 1 (optional): Calibrate the cross-delay using a strong polarized source. • STEP 2: Calibrate the leakage using an unpolarized source.
I If all calibrators are polarized, solve for leakage and source polarization simultaneously. I Need good parallactic-angle coverage.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 21 / 26 • STEP 4: Image each Stokes parameter separately. Combine images: (Q, U) → (Ip, θ).
Calibration strategy
The right order for matrix product is: J = (Ga Kc ) × D × P cal −1 −1 −1 obs i.e.: V = P × D × (Ga Kc ) × V
• STEP 1 (optional): Calibrate the cross-delay using a strong polarized source. • STEP 2: Calibrate the leakage using an unpolarized source.
I If all calibrators are polarized, solve for leakage and source polarization simultaneously. I Need good parallactic-angle coverage. • STEP 3 (optional): Refine the calibration (calibration “cross-talk”)
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 21 / 26 Calibration strategy
The right order for matrix product is: J = (Ga Kc ) × D × P cal −1 −1 −1 obs i.e.: V = P × D × (Ga Kc ) × V
• STEP 1 (optional): Calibrate the cross-delay using a strong polarized source. • STEP 2: Calibrate the leakage using an unpolarized source.
I If all calibrators are polarized, solve for leakage and source polarization simultaneously. I Need good parallactic-angle coverage. • STEP 3 (optional): Refine the calibration (calibration “cross-talk”) • STEP 4: Image each Stokes parameter separately. Combine images: (Q, U) → (Ip, θ).
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 21 / 26 Calibration strategy
The right order for matrix product is: J = (Ga Kc ) × D × P cal −1 −1 −1 obs i.e.: V = P × D × (Ga Kc ) × V
• STEP 1 (optional): Calibrate the cross-delay using a strong polarized source. • STEP 2: Calibrate the leakage using an unpolarized source.
I If all calibrators are polarized, solve for leakage and source polarization simultaneously. I Need good parallactic-angle coverage. • STEP 3 (optional): Refine the calibration (calibration “cross-talk”) • STEP 4: Image each Stokes parameter separately. Combine images: (Q, U) → (Ip, θ).
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 21 / 26 THANKS
SUMMARY
• We have reviewed basic concepts of polarization.
I Modes of polarization. I Stokes parameters. • We have discussed about the different kinds of polarizers in radioastronomical receivers.
I Linear polarizers (X-Y). I Circular polarizers (R-L). • We have studied how to deal with polarization in interferometric observations.
I The Measurement Equation. I The matrices for polarization calibration. I Calibration effects on X-Y vs. R-L polarizers. I Overview of calibration procedure.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 22 / 26 SUMMARY
• We have reviewed basic concepts of polarization.
I Modes of polarization. I Stokes parameters. • We have discussed about the different kinds of polarizers in radioastronomical receivers.
I Linear polarizers (X-Y). I Circular polarizers (R-L). • We have studied how to deal with polarization in interferometric observations.
I The Measurement Equation. I The matrices for polarization calibration. I Calibration effects on X-Y vs. R-L polarizers. I Overview of calibration procedure. THANKS
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 22 / 26 TUTORIAL
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 22 / 26 Full-pol calibration − MERLIN C-band − 3C277.1
• Load the data into CASA. Inspect visibilities. • Flag bad data. • Calibrator phases and bandpass. Inspect solutions. • Set primary flux calibrator. Inspect model. • Calibrator amplitudes. Bootstrap flux-density calibration.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 23 / 26 1 0 ! 1 DL! obs true H H a I V = Da XV ab X Db ; X = ; Da = jα R 0 e Da 1 obs R L ∗ −jα 2 VRL = ((Da +( Db ) ) I + P) e + O(D ) obs L R ∗ ∗ jα 2 VLR = ((Da +( Db ) ) I + P ) e + O(D )
I Unpolarized calibrator: L R jα L j∆ R −j∆ j(α−∆) (Da , Da , e ) → (Da e + jK, Da e + jK, e )
I Polarized calibrator: L R jα L R j(α) (Da , Da , e ) → (Da + jK, Da + jK, e )
Full-pol calibration − MERLIN C-band − 3C277.1
• Calibrate phases (long time average for phase referencing). • Calibrate leakage from phase calibrator. Understand solutions. • Calibrate R/L phase offset using pol. calibrator.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 24 / 26 I Unpolarized calibrator: L R jα L j∆ R −j∆ j(α−∆) (Da , Da , e ) → (Da e + jK, Da e + jK, e )
I Polarized calibrator: L R jα L R j(α) (Da , Da , e ) → (Da + jK, Da + jK, e )
Full-pol calibration − MERLIN C-band − 3C277.1
• Calibrate phases (long time average for phase referencing). • Calibrate leakage from phase calibrator. Understand solutions. • Calibrate R/L phase offset using pol. calibrator. 1 0 ! 1 DL! obs true H H a I V = Da XV ab X Db ; X = ; Da = jα R 0 e Da 1 obs R L ∗ −jα 2 VRL = ((Da +( Db ) ) I + P) e + O(D ) obs L R ∗ ∗ jα 2 VLR = ((Da +( Db ) ) I + P ) e + O(D )
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 24 / 26 Full-pol calibration − MERLIN C-band − 3C277.1
• Calibrate phases (long time average for phase referencing). • Calibrate leakage from phase calibrator. Understand solutions. • Calibrate R/L phase offset using pol. calibrator. 1 0 ! 1 DL! obs true H H a I V = Da XV ab X Db ; X = ; Da = jα R 0 e Da 1 obs R L ∗ −jα 2 VRL = ((Da +( Db ) ) I + P) e + O(D ) obs L R ∗ ∗ jα 2 VLR = ((Da +( Db ) ) I + P ) e + O(D )
I Unpolarized calibrator: L R jα L j∆ R −j∆ j(α−∆) (Da , Da , e ) → (Da e + jK, Da e + jK, e )
I Polarized calibrator: L R jα L R j(α) (Da , Da , e ) → (Da + jK, Da + jK, e )
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 24 / 26 Full-pol calibration − MERLIN C-band − 3C277.1
• Split the target data. • CLEAN stokes I of target. Self-calibrate phase. • CLEAN stokes I of target. Self-calibrate amplitude. • CLEAN all stokes parameters of target. • Construct the polarization image. • Sanity check: CLEAN the polarization calibrator.
I. Mart´ı-Vidal (Onsala Rymdobservatorium) Polarization in Interferometry ERIS (2015) 25 / 26