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.

S meas = (X+ D X) × S in Limited by instrumental polarization effects and imperfect polarimeter.

polarimetric accuracy

X = æ æ æ æ æ æ I ® I I ® Q Q Q ® ® I Q U U ® ® I Q V V ® ® I Q I ® U I ® V Q Q ® ® U U V U ® ® U V V V ® ® U V æ æ æ æ æ æ Not a Mueller matrix, as it includes modulation/demodulation and calibration.

polarimetric accuracy

X = æ æ æ æ æ æ I ® I I ® Q Q Q ® ® I Q U U ® ® I Q V V ® ® I Q I ® U I ® V Q Q ® ® U U V U ® ® U V V V ® ® U V æ æ æ æ æ æ • transmission often normalized to 1.0

polarimetric accuracy

X = æ æ æ æ æ æ I ® I I ® Q Q Q ® ® I Q U U ® ® I Q V V ® ® I Q I ® U I ® V Q Q ® ® U U V U ® ® U V V V ® ® U V æ æ æ æ æ æ instrumental polarization

polarimetric accuracy

X = æ æ æ æ æ æ I ® I I ® Q Q Q ® ® I Q U U ® ® I Q V V ® ® I Q I ® U I ® V Q Q ® ® U U V U ® ® U V V V ® ® U V æ æ æ æ æ æ polarization cross-talk

polarimetric accuracy

X = æ æ æ æ æ æ I ® I I ® Q Q Q ® ® I Q U U ® ® I Q V V ® ® I Q I ® U I ® V Q Q ® ® U U V U ® ® U V V V ® ® U V æ æ æ æ æ æ polarization rotation

polarimetric accuracy

X = æ æ æ æ æ æ I ® I I ® Q Q Q ® ® I Q U U ® ® I Q V V ® ® I Q I ® U I ® V Q Q ® ® U U V U ® ® U V V V ® ® U V æ æ æ æ æ æ related to polarimetric efficiency

polarimetric accuracy

X = æ æ æ æ æ æ I ® I I ® Q Q Q ® ® I Q U U ® ® I Q V V ® ® I Q I ® U I ® V Q Q ® ® U U V U ® ® U V V V ® ® U V æ æ æ æ æ æ impact of polarized light on photometry

polarimetric accuracy

D X ≤ × × × × × ~ 10 -3 ~ 10 -3 ~ 10 -3 zero level ~ 10 -1 ~ 10 -2 ~ 10 -2 ~ 10 -2 ~ 10 -1 ~ 10 -2 ~ 10 -2 ~ 10 -2 ~ 10 -1 ~ 10 -2 ~ 10 -2 ~ 10 -2 × × × × × scale if Q,U≈0 or V≈0: DP ≤ 0.001 + 0.01

× P

polarimetric precision

doesn’t have any significance…

modulation & demodulation

n detected intensities I detector = O × S in n x 4 mOdulation matrix S meas = D × I detector 4 x n Demodulation matrix X = D × O

polarimetric efficiency

O

= æ ç è 1 1 1 0 0 1 0 0 ö ÷ ø first row of the total Mueller matrix for every modulation state i e Q = 1

polarimetric efficiency

O

= è æ ç ç ç ç 1 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 ø ö ÷ ÷ ÷ ÷ e Q = 1 2 e U = 1 2

polarimetric efficiency

O

= 1 1 1 1 1 1 æ ç ç ç ç ç è 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 ö ÷ ÷ ÷ ÷ ÷ ø e Q = 1 3 e U = 1 3 e V = 1 3

polarimetric efficiency

O

= è æ ç ç ç ç ç ç 1 1 1 1 1 1 1 1 3 3 3 3 1 1 1 1 3 3 3 3 1 1 1 1 3 3 3 3 ø ö ÷ ÷ ÷ ÷ ÷ ÷ e Q = 1 3 e U = 1 3 e V = 1 3

optimum demodulation

X = D × O ® I 4 • O is 4 x 4: D = O -1 • O is n x 4: ( ) -1 × O T optimizes the polarimetric efficiencies pseudo-inverse (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.

e k çç è æ n å i=1 2 kl ö ÷÷ ø 1 2 s 2 (S k ) = s 2 (I detector ) e k 2 e Q 2 + e U 2 + e 2 V ≤ 1 e

I

≤ 1

Del Toro Iniesta & Collados (2000)

calibration

• • • • • X = D × O = I 4 +D X 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.