definitions for polarimetry
Frans Snik
Sterrewacht Leiden
polarimetric sensitivity
The noise level in Q/I, U/I, V/I above which a
real polarization signal can be detected.
Due to “random” effects not directly expressible
as a Mueller matrix:
• fundamentally limited by photon noise
• detector noise
• seeing (for temporal modulation)
• diferential aberrations (for spatial modulation)
• etc.
polarimetric accuracy
Quantification of how measured Stokes
parameters (with sufficient S/N) relate to the
real Stokes parameters.
Smeas = (X+DX)×Sin
Limited by instrumental polarization effects and
imperfect polarimeter.
polarimetric accuracy
æ
ç
ç
X=
ç
ç
è
I®I Q ®I U®I V ®I
I® Q Q ® Q U ® Q V ® Q
I®U Q ®U U ®U V ®U
I® V Q ® V U® V V ® V
Not a Mueller matrix, as it includes
modulation/demodulation and calibration.
ö
÷
÷
÷
÷
ø
polarimetric accuracy
æ
ç
ç
X=
ç
ç
è
I®I Q ®I U®I V ®I
I® Q Q ® Q U ® Q V ® Q
I®U Q ®U U ®U V ®U
I® V Q ® V U® V V ® V
transmission
• often normalized to 1.0
ö
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÷
÷
÷
ø
polarimetric accuracy
æ
ç
ç
X=
ç
ç
è
I®I Q ®I U®I V ®I
I® Q Q ® Q U ® Q V ® Q
I®U Q ®U U ®U V ®U
I® V Q ® V U® V V ® V
instrumental polarization
ö
÷
÷
÷
÷
ø
polarimetric accuracy
æ
ç
ç
X=
ç
ç
è
I®I Q ®I U®I V ®I
I® Q Q ® Q U ® Q V ® Q
I®U Q ®U U ®U V ®U
I® V Q ® V U® V V ® V
polarization cross-talk
ö
÷
÷
÷
÷
ø
polarimetric accuracy
æ
ç
ç
X=
ç
ç
è
I®I Q ®I U®I V ®I
I® Q Q ® Q U ® Q V ® Q
I®U Q ®U U ®U V ®U
I® V Q ® V U® V V ® V
polarization rotation
ö
÷
÷
÷
÷
ø
polarimetric accuracy
æ
ç
ç
X=
ç
ç
è
I®I Q ®I U®I V ®I
I® Q Q ® Q U ® Q V ® Q
I®U Q ®U U ®U V ®U
I® V Q ® V U® V V ® V
related to polarimetric efficiency
ö
÷
÷
÷
÷
ø
polarimetric accuracy
æ
ç
ç
X=
ç
ç
è
I®I Q ®I U®I V ®I
I® Q Q ® Q U ® Q V ® Q
I®U Q ®U U ®U V ®U
I® V Q ® V U® V V ® V
ö
÷
÷
÷
÷
ø
impact of polarized light on photometry
polarimetric accuracy
æ
ç
-3
~
10
DX ≤ ç
ç ~ 10-3
ç
-3
è ~ 10
-1
~ 10
~ 10-2
-2
~ 10
-2
~ 10
zero level
if Q,U≈0 or V≈0:
DP ≤ 0.001 + 0.01×P
-1
~ 10
~ 10-2
-2
~ 10
-2
~ 10
-1
~ 10
~ 10-2
-2
~ 10
-2
~ 10
ö
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ø
scale
polarimetric precision
doesn’t have any significance…
modulation & demodulation
Idetector = O×Sin
n detected intensities
n x 4 mOdulation matrix
Smeas = D× Idetector
4 x n Demodulation matrix
X = D×O
polarimetric efficiency
æ 1 1 0 0 ö
O= ç
÷
è 1 -1 0 0 ø
first row of the total Mueller matrix for every modulation state i
eQ = 1
polarimetric efficiency
æ
ç
O= ç
ç
ç
è
eQ = 1
2
1 1 0
1 -1 0
1 0 1
1 0 -1
eU = 1
2
0
0
0
0
ö
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÷
÷
÷
ø
polarimetric efficiency
æ
ç
ç
ç
O= ç
ç
çç
è
eQ = 1
3
1 1 0 0
1 -1 0 0
1 0 1 0
1 0 -1 0
1 0 0 1
1 0 0 -1
eU = 1
3
ö
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÷
÷
÷
÷
÷÷
ø
eV = 1
3
polarimetric efficiency
æ
ç
ç
O= ç
ç
ç
ç
è
eQ = 1
1
1
3
1
1
1
3
-1
1 -1
3
1 -1
3 -1
3
eU = 1
3
1
3
3 -1
3
-1
3
3
3
3
1
1
eV = 1
3
ö
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÷
÷
÷
ø
3
optimum demodulation
X = D×O ® I4
• O is 4 x 4:
D=O
• O is n x 4:
(
-1
)
-1
D = O ×O ×O
T
T
pseudo-inverse
optimizes the polarimetric efficiencies
(for one wavelength?)
Del Toro Iniesta & Collados (2000)
polarimetric efficiency
Describes how efficiently a certain modulation
scheme measures a the Stokes parameters w.r.t.
the random noise.
æ n ö2
2
ek = ççnåDkl ÷÷
è i=1 ø
-1
s (Sk ) =
2
s (Idetector )
e +e +e ≤ 1
2
Q
2
U
2
V
eI ≤ 1
2
e
2
k
Del Toro Iniesta & Collados (2000)
calibration
X = D×O = I4 + DX
• Instrumental polarization issues make that
modulation matrix O is unknown (at some level).
• This is the matrix that needs to be calibrated.
• Calibration is applied through demodulation
matrix D.
• ΔX describes calibration accuracy.
• See Asensio Ramos & Collados (2008) for random
error propagation.