a system with optics between the modulator and the analyzer

a system with optics between the modulator and the analyzer jose carlos del toro iniesta (SPG, IAA-CSIC) (based on a paper with valentín martínez pillet, IAC, currently NSO ApJS 1,, 1)

previous considerations as time passes by, less and less conceptual differences between spectropolarimeters and magnetographs the former: scanning spectrographs the latter: (bidimensional) tunable imaging filtergraphs some decades ago, magnetographs only sampled one or two wavelengths today s technology enables better sampling and eventual use of inversion codes to interpret the data (like with those from spectropolarimeters) modern instruments are vectorial longitudinal analysis: eventual common interest: B, γ, ϕ and v LOS (an assessment is in order) others may be interested on T and, eventually other quantities

noise in action (thresholds; i) we only measure photons everything depends on photometric accuracy (systematic errors ideally absent) a battle between our ability to detect changes and the noise that hide the changes noise: limiting factor (S 1, S, S 3, S 4 ) (I, Q, U, V ) S iv,b,, i usually estimated as the standard deviation of the continuum signal because polarization is assumed constant @ continuum wavelengths calculated either over a continuum window in a given pixel or over all the spatial pixels of a map in a given continuum wavelength sample signal-to-noise ratio: S/N = 1 has no precise meaning some Stokes parameter has signals greater than 1-3 S 1,c (S/N) i = S1 i c

noise in action (thresholds; ii) if noise is uncorrelated with the signal and distributed with a Gaussian (keller & snik 9; del toro iniesta & collados ) i = 1 i 1, i =, 3, 4 since ε i ε 1, then σ i σ 1, i =, 3, 4 and it s easy to see that (martínez pillet et al. 1999) (S/N) i = i 1 (S/N) 1, i =, 3, 4 S/N is greater for S 1 but systematic errors (like flat-fielding) may invalidate it (but we are discarding systematic effects) Stokes parameters cannot be measured with single exposures (S/N) 1 = " 1 (s/n) p N p N a advice (not a universal convention): S/N = (S/N) 1 there is only one intensity; any other Stokes parameter (S/N) i can easily be obtained from the above equation (S/N) i or polarimetric accuracy have a precise meaning (are the same)

noise in action (thresholds; iii) polarimetric efficiencies are the link between instrument and signal since ε i (1, 1/ 3, 1/ 3, 1/ 3) (del toro iniesta & collados ) then (S/N) i 1/ 3 (S/N), i =, 3, 4 (for an optimum polarimeter with the same efficiencies) detectability is smaller in polarimetry than in pure photometry minimum detectable thresholds (for optimum polarimeters) should be included within a global error budget degree of polarization magnetic field components and v LOS inference techniques must be taken into account modern instruments use inversions; we shall use simpler formulae p 1 S 1 4 i= S i p p 1+ 3 p S/N

noise in action (thresholds; iv) classical magnetographic formulae where B lon = k lon V s S 1,c V s 1 n B tran = k tran minimum polarization signals are σ 4 and σ = σ3 (B lon )= k lon S/N nx a i S 4,i, i=1 L s 1 n " 1 " 4 (B tran )=k tran if S/N = (S/N) 1 = 17 (1 for S, S3, and S4) then δ(blon) = 5 G and δ(btran) = 8 G for IMaX (martínez pillet et al. 11) s nx i=1 s L s S 1,c q S,i + S 3,i " 1 /" S/N = k tran s " 1 /" 3 S/N

noise in action (thresholds; v) Fourier tachometer (beckers & brown 1978; brown 1981; fernandes 199) v LOS = c arctan S 1, 9 + S 1, 3 S 1,+3 S 1,+9 S 1, 9 S 1, 3 S 1,+3 + S 1,+9 where c is the speed of light, δ(λ) is the spectral resolution and λ is the central wavelength of the line; +9, +3, -3, -9 are the sample wavelengths in pm wrt λ if S1,+9 - S1+3 S1,-9 - S1,-3 1/ S1,c, then (v LOS ) c arctan S/N with IMaX values [δ(λ) = 8.5 pm; λ = 55. nm], and a S/N = 17, δ(vlos) = 4 ms -1

noise in action (uncertainties) error propagation in the magnetographic formulae B lon B lon = k lon n B lon 1 4 +1 1 (S/N) B tran B tran = k 4 tran 8n B 4 tran 1 + 1 3 + 1 4 1 (S/N) and in the Fourier tachometer formula where v LOS = 4c ( ) 1, =(S 1, 9 S 1,+3 ) +(S 1,+9 S 1, 3 ) assuming IMaX values, a noise of S/N = 17 induces a %, a 15 % (for 1 G), and 4 ms -1 (FTS)

an optimum polarimeter (i) let s optimize the efficiencies a modulator made up of two nematic LCVRs can be optimum (martínez pillet et al., 4) if axes are at º and 45º with S > direction ideally maximum efficiencies can be reached for both vector and longitudinal analyses instrumental polarization may corrupt efficiencies we show that ideal efficiencies can be reached by simply tuning voltages we first demonstrate that the ideal result can be reached for the single polarimeter

modulation and demodulation M1 M M4 measurements are linear combinations of the Stokes parameters B @ I 1 I I 3... I n 1 C A = B @ O 11 O 1 O 13 O 14 O 1 O O 3 O 4 O 31 O 3 O 33 O 34............ O 11 O 1 O 13 O 14 1 B C @ A we have to demodulate while optimizing S/N, that is, while optimizing the efficiencies I = OS ) S = DI S 1 S S 3 S 4 1 C A

an optimum polarimeter (ii) M1 M M4 (ε 1,ε,ε 3,ε 4 ) (1,1/ 3,1/ 3,1/ 3) (del toro iniesta & collados, ) nematic LCVRs especially good (martínez pillet et al., 1999) optimum theoretical modulation with four measurements: M 1 = R(,ρ); M = R(π/4,τ); M 4 = L(); M M 4 M M 1 O ij = (1, cos τ i, sin ρ i sin τ i, - cos ρ i sin τ i ) optimum longitudinal modulation (I ± V): M 1 = R(,); M = R(π/4,±π/); M 4 = L()

an optimum polarimeter (iii) the etalon, a retarder: M3 = R(ϑ,δ) M = M4 M3 M M1 Oij = M1j (τi,ρi) M1j (τi,ρi) = ±1/ 3, j =,3,4, are transcendental equations with solution and M11 = 1 optimum polarimetric efficiencies can be achieved trivial cases: ϑ =,π/; δ = M1 M M4

an optimum polarimeter (iii)

an optimum polarimeter (iv) M1 M M4 a train of mirrors (no matter the number and the relative angles) has a mueller matrix like (collet 199) E = all modulation matrix elements turn out to be multiplied by (a+b) no effect on the result! since mirrors are retarders plus partial polarizers, any differential absorption effect is included calibration necessary for non-ideal instruments B @ a b b a c d d f 1 C A

an optimum polarimeter (iv) M1 M M3 E M4 a train of mirrors (no matter the number and the relative angles) has a mueller matrix like (collet 199) E = all modulation matrix elements turn out to be multiplied by (a+b) no effect on the result! since mirrors are retarders plus partial polarizers, any differential absorption effect is included calibration necessary for non-ideal instruments B @ a b b a c d d f 1 C A

instrument-induced inaccuracies photon noise is not the only harm instabilities like those in T or in V and roughness in the thickness of both the LCVRs and the etalon(s) retardance changes modulation and demodulations changes inaccuracies in the vector magnetic field wavelength tuning, spectral resolution, and profile shape changes inaccuracies in the LOS velocity single modulation cycles (single measurements) absolute errors in B or v LOS time series inability to detect given oscillatory modes

magnetographic inaccuracies efficiency variances can be seen as functions of retardance variances (del toro iniesta & collados ) " max,i = retardance variances can be written as functions of birefringence, thickness, and wavelength variances L = t ) P 4 j=1 O ji N p ) " max,i L L birefringence variance is a sum of thermal and voltage variances L = m T T + m V V + L t t + = + t = f ( j, t + L j ) the last term hardly introduces any effect within the limited wavelength range of a line. the relative differences are negligible the last-but-one term may vary spatially and is responsible of pixel-to-pixel variations thickness is a fabrication specification; inaccuracies may produce significant effects (alvarez-herrero et al. 1) rel. errors larger than 6% induce poor polarimetric accuracy (6% ~ tens of nm!). calibration is needed

magnetographic inaccuracies efficiency variances can be seen as functions of retardance variances (del toro iniesta & collados ) " max,i = retardance variances can be written as functions of birefringence, thickness, and wavelength variances L = t ) P 4 j=1 O ji N p ) " max,i L L birefringence variance is a sum of thermal and voltage variances L = m T T + m V V + L t t + = + t = f ( j, t + L j ) IMaX instabilities larger than.3 K and 1 mv and a roughness of 4 % induce a polarimetric accuracy less than 1-3 and 5 % and.5 % repeatability errors in the detection thresholds for Blon and Btran, respectively

velocity inaccuracies velocities error propagation gives v LOS = c arctan S 1, 9 + S 1, 3 S 1,+3 S 1,+9 S 1, 9 S 1, 3 S 1,+3 + S 1,+9 v LOS = f (v LOS, ) + g(v LOS,,, S 1,i, s 1,i )(k T T + k V V )+h(,, S 1,i ) 1 assume λ = 6173 Å and δλ = 1 må (SO/PHI) a roughness instability inducing σ δλ = 1 må produces σ v = 1 ms -1 for speeds of 1 ms -1! (and is linear in v) imagine temperature and noise contribute equally. then, σ T /σ I = 5.7 and S/N =17 σ T = 1 mk pure photon noise of σ I = 1-3 I c induces σ v = 7 ms -1 uncertainties larger than 45 mk or 3.4 V produce σ v > 1 ms -1 (and this can be an issue for global helioseismology)

conclusions (i) an assessment study of the salient features and properties of solar magnetographs and spectropolarimeters both photon-induced and instrument-induced noise as well as the specific measurements techniques have been taken into account useful formulae have been derived that provide detection thresholds (p, B lon, B tran, v LOS ) as functions of ε i and the S/N relative uncertainties have been deduced for the magnetographic and tachographic quantities when noise is induced by photons

conclusions (ii) an analysis of instruments based on nematic LCVRs and an etalon has been presented (can easily be extended) this type of instrument can reach theoretical maximum efficiencies no matter the optics in between the modulator and the analyzer optimum detection thresholds and relative uncertainties LCVR optimum retardances depend on the pass-trough optics but can easily tuned up error propagation has yielded relationships between the variances of Stokes parameters and solar physical quantities and instrument parameters a bridge between scientific requirements and instrument specifications has been provided