# Assessing the behavior of modern solar magnetographs and spectropolarimeters

J.C. del Toro Iniesta  Instituto de Astrofísica de Andalucía (CSIC), Apdo. de
Correos 3004, E-18080 Granada, Spain<EMAIL_ADDRESS>V. Martínez Pillet
Instituto de Astrofísica de Canarias, Vía Láctea, s/n, E-28200 La Laguna,
Spain<EMAIL_ADDRESS>
###### Abstract

The design and later use of modern spectropolarimeters and magnetographs
require a number of tolerance specifications that allow the developers to
build the instrument and then the scientists to interpret the data accuracy.
Such specifications depend both on device-specific features and on the
physical assumptions underlying the particular measurement technique. Here we
discuss general properties of every magnetograph, as the detectability
thresholds for the vector magnetic field and the line-of-sight velocity, as
well as specific properties of a given type of instrument, namely that based
on a pair of nematic liquid crystal variable retarders and a Fabry-Pérot
etalon (or several) for carrying out the light polarization modulation and
spectral analysis, respectively. We derive formulae that give the detection
thresholds in terms of the signal-to-noise ratio of the observations and the
polarimetric efficiencies of the instrument. Relationships are also
established between inaccuracies in the solar physical quantities and
instabilities in the instrument parameters. Such relationships allow, for
example, to translate scientific requirements for the velocity or the magnetic
field into requirements for temperature or voltage stability. We also
demonstrate that this type of magnetograph can theoretically reach the optimum
polarimetric efficiencies of an ideal polarimeter, regardless of the optics in
between the modulator and the analyzer. Such optics induces changes in the
instrument parameters that are calculated too.

###### Subject headings:

Sun: magnetic fields – Sun: photosphere – Sun: polarimetry

††slugcomment: Submitted to ApJ

##  1\. Introduction

Spectropolarimetry and magnetography have become two of the most useful tools
in solar physics because they provide the deepest analysis one can make of
light. Solar information is encoded in the spectrum of the Stokes parameters.
We measure this spectrum and infer solar quantities from it. Recently, less
and less conceptual differences exist between spectropolarimeters and
magnetographs except for the specific devices: the formers usually include
those instruments using a scanning spectrograph; the latter usually employ a
bidimensional filtergraph like a Fabry-Pérot etalon. Some decades ago
magnetographs were only able to sample one or two wavelengths across a
spectral line; nowadays, new technologies provide a better wavelength
sampling, thus enabling the scientists to interpret the data in terms of
sophisticated inversion techniques of the radiative transfer equation, a
procedure similar to the one regularly used with spectropolarimeters. Some of
the instruments mentioned below enter this category. Modern solar
spectropolarimeters and magnetographs are often vectorial because all four
Stokes parameters of the light spectrum are measured. Longitudinal
magnetography (i.e., Stokes $I\pm V$) can be interesting for some specific
applications, but the partial analysis is usually included (if possible) as a
particular case of the more general, full-Stokes polarimetry. Some of these
modern instruments have been recently built or are currently in operation
(e.g., the Tenerife Infrared Polarimeter, TIP, Martínez Pillet et al. 1999,
Collados et al. 2007; the Diffraction-Limited Spectro-Polarimeter, DLSP,
Sankarasubramanian et al. 2003; the air-spaced Fabry-Perot based CRISP
instrument, Scharmer et al. 2008; the spectropolarimeter, SP, Lites et al.
2001, for the Hinode mission, Kosugi et al. 2007; CRISP, Narayan et al. 2008;
the Visible Imaging Polarimeter, VIP, Beck et al. 2010; the Imaging
Magnetograph eXperiment, IMaX, Martínez Pillet et al. 2011, for the Sunrise
mission, Barthol et al. 2011; and the Helioseismic and Magnetic Imager, HMI,
Graham et al. 2003, for the Solar Dynamics Observatory mission, Title 2000).
Some other are being designed and built for near future operation and missions
(e.g., the Polarimetric and Helioseismic Imager, SO/PHI, [formerly called VIM,
Martínez Pillet 2007] for the Solar Orbiter mission, Marsch et al. 2005).

The common interest of users of these instruments is centered in vector
magnetic fields (of components $B$, $\gamma$, and $\phi$) and line-of-sight
(LOS) velocities ($v_{\rm LOS}$). Some spectropolarimeters can provide
information on temperatures as well (and eventually on another thermodynamical
quantity) but that feature is not common to all of them. Therefore, assessing
the magnetograph capabilities in terms of their accuracy for retrieving
magnetographic and tachographic quantities is in order since such an analysis
can diagnose how far reaching is our current knowledge of the solar dynamics
and magnetism. The diagnostics is relevant both for the design of new
instruments in order to maximize their performances and for the analysis of
uncertainties in data coming from currently operating devices. General
considerations can obviously not be made but a few. Specifically, we here
study in Sect. 2 the detection thresholds induced by random noise on the
inferred longitudinal and transverse components of the magnetic field; in the
particular case of photon-induced noise we also find uncertainty formulas.
Both thresholds and relative uncertainties are obtained in terms of the
signal-to-noise ratio of the observations and of the polarimetric efficiencies
of the instrument. Since such efficiencies vary from instrument to instrument,
at that point, the analysis concentrates in a particular type of magnetograph,
namely that consisting of two nematic liquid crystal variable retarders
(LCVRs) as the polarization modulator and a Fabry-Pérot etalon. In Sect. 3, we
demonstrate that these polarimeters can reach the theoretically optimum
efficiencies no matter the optics behind the modulator, including the etalon.
The way for calculating the required retardances for the two LCVRs are
explained along with a number of rules and periodicities in the solutions.
Section 4 analyzes these instruments in terms of the influence of temperature
and voltage instabilities, as well as of thickness inhomogeneities
(roughness), of both the LCVRs and the etalon(s), on the final magnetographic
and tachographic measurements. Finally, Sect. 5 summarizes the results.

##  2\. The thresholding action of random noise

Most astrophysical measurements are nothing but photon counting. Their
accuracy, therefore, depends on photometric accuracy, that is, on a battle
between our ability to detect changes in the solar (stellar) physical
quantities and the noise that hide such changes. The key concept is changes:
we need to discern if a given quantity like the magnetic field strength, $B$,
or the line-of-sight (LOS) velocity, $v_{\rm LOS}$, varies among pixels:
whether or not it is greater or smaller than in the neighbor zones. The only
tool we have to gauge these changes is the observable Stokes parameter changes
that are linked to them through the response functions. Discussing response
functions is out of the scope of this paper as they have been extensively
analyzed elsewhere (e.g., Del Toro Iniesta et al., 2010; Orozco Suárez & Del
Toro Iniesta, 2007; Del Toro Iniesta, 2003; Del Toro Iniesta & Ruiz Cobo,
1996; Ruiz Cobo & Del Toro Iniesta, 1994; Landi Degl’Innocenti & Landi
Degl’Innocenti, 1977; Mein, 1971). As explained by Del Toro Iniesta & Martínez
Pillet (2010), purely phenomenological approaches (e.g., Cabrera Solana et
al., 2005) are also valid to establish a relationship between changes in the
physical quantities and the Stokes parameters. Here we shall concentrate on
noise; on how it establishes the minimum threshold below which no signal
changes can be detected. In spectropolarimetry, the customary estimate for
noise (and thus for the detection threshold) is the standard deviation of the
continuum signal because polarization is assumed to be constant at continuum
wavelengths. Therefore, the noise is calculated either over a continuum window
in a given spatial pixel or over all the spatial pixels of a map in a given
continuum wavelength sample. Both estimates should agree as it is the case in
most observations. If we call $\mathbf{S}\equiv(S_{1},S_{2},S_{3},S_{4})$ the
(pseudo-)vector of Stokes parameters and denote by $\sigma_{i}$, with
$i=1,2,3,4$, the standard deviation of each Stokes parameter, then the signal-
to-noise ratio in each parameter, $(S/N)_{i}$, is defined as the inverse of
this deviation in units of the continuum intensity:

$(S/N)_{i}=\left(\frac{S_{1}}{\sigma_{i}}\right)_{\\!\\!\\!\rm c},$ (1)

where index c refers to continuum. Thus, when we say that our observations
have, for example, a $S/N=1000$, we mean that when signals in a given (non
specified) Stokes parameter are greater than $10^{-3}S_{1,{\rm c}}$ can be
detected and this is certainly valid for that parameter but not necessarily
for the others. As a matter of fact, if noise is random (or uncorrelated with
the signal) and can be represented by a Gaussian distribution (Keller & Snik,
2009), according to Del Toro Iniesta & Collados (2000),

$\sigma_{i}=\frac{\varepsilon_{1}}{\varepsilon_{i}}\sigma_{1},\,\,\,i=1,2,3,4,$
(2)

where $\varepsilon_{i}$ stands for each one of the so-called polarimetric
efficiencies of the instrument. The polarimetric efficiencies depend in a non-
linear way on the modulation matrix elements (cf. Eq. 37) that on their turn
come from the first row of the polarimeter Mueller matrix (Del Toro Iniesta,
2003). Since all the efficiencies are necessarily less than the first (that of
the intensity), Eq. (2) means that the noise is always larger in the
polarization parameters than in the intensity. Then one can easily see that
(see Martínez Pillet et al., 1999)

$(S/N)_{i}=\frac{\varepsilon_{i}}{\varepsilon_{1}}(S/N)_{1},\,\,\,i=2,3,4,$
(3)

that is, that the signal-to-noise ratio for Stokes $S_{2}$, $S_{3}$, and
$S_{4}$ is always less than that for Stokes $S_{1}$. Let us point out here,
however, that other systematic (or instrumental) errors like those introduced
by flat fielding of images may invalidate the above equation. We are
explicitly discarding these other sources of noise from our analysis.

The Stokes parameters cannot be measured with single exposures. Instead, for
vector polarimetry, a number $N_{p}\geq 4$ of single detector shots are
recorded each providing a linear combination of all the four Stokes
parameters. The set of $N_{p}$ individual measurements constitutes a
modulation cycle that is characteristic of an instrument mode of operation.
After demodulation, that is, after solving the set of $N_{p}$ linear equations
made up of the individual exposures, the Stokes vector is measured. To
increase the signal-to-noise ratio of the measurement, many instruments use
$N_{a}$ accumulations, that is, repeat the modulation cycle $N_{a}$ times and
the corresponding polarization images are added together. Since the degree of
polarization of the incoming typical solar beam is fairly small, each single
shot usually has the same light levels and, hence, the same (photon-noise-
dominated) signal-to-noise ratio $s/n$. Thus, the signal-to-noise ratio of
$S_{1}$ is related to the single-shot $s/n$ (see Martínez Pillet, 2007, for an
illustrative description) through

$(S/N)_{1}=(s/n)\,{\varepsilon_{1}}\sqrt{N_{p}N_{a}},$ (4)

because $S_{1}$ is often retrieved from the sum of all the accumulated images
for all the polarization exposures of the given modulation scheme. An
advisable practice for characterizing the signal-to-noise ratio of an
instrument is to always refer to that of intensity and equate $S/N=(S/N)_{1}$.
This is convenient because there is only one intensity while the other
polarization Stokes parameters are three and one would need to specify which
one is meant each time. Let us remark, however, that this convention is not
universal and some authors, always interested in the polarization features,
think of and speak about some of the other three Stokes parameter signal-to-
noise ratios. Such an alternative convention makes sense if one takes into
account the differential character of polarization measurements: demodulation
implies that Stokes $S_{2,3,4}$ are essentially retrieved from image
differences; hence, any systematic error like that produced by flat fielding
is naturally mitigated (or eventually cancels out). On the contrary, the
additive character of Stokes $S_{1}$ implies that intensity noise can be
higher than simple photon noise. We shall hereafter follow the $S/N=(S/N)_{1}$
convention in the paper for the sake of simplicity in the description and in
the equations (thus, explicitly neglecting systematic errors). When people is
more interested in the other three Stokes parameter signal-to-noise ratios,
Eq. (3) provides the obvious help.

As demonstrated by Del Toro Iniesta & Collados (2000), the maximum
efficiencies that an ideal system can have are
${\mathbf{\varepsilon}}=(1,1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})$, if all the
three last ones are equal. Therefore, the relationship $(S/N)_{i}\leq
1/\sqrt{3}\,(S/N)$, $i=2,3,4$, holds for any random-noise-dominated
polarimetric system.111There may be polarimeters that are designed to measure
not all four Stokes parameters or that aim at better accuracies for given
Stokes parameters. In such cases some (never all) efficiencies can be greater
than $1/\sqrt{3}$. In what follows, however, we assume that our interest is
the same for $S_{2}$, $S_{3}$, and $S_{4}$. Or, in terms of Eq. (2),
necessarily,

$\sigma_{i}\geq\sqrt{3}\,\sigma_{1}.$ (5)

Equation (5) means that the detection threshold is bigger for the polarization
parameters than for the intensity. Detectability is smaller in polarimetry
than in pure photometry.

An important parameter describing the state of any beam of light is its degree
of polarization

$p^{2}\equiv\frac{1}{S_{1}^{2}}\sum_{i=2}^{4}S_{i}^{2}.$ (6)

Now the question naturally arises as to what is the minimum detectable degree
of polarization by a given polarimetric system. If the uncertainties in the
Stokes parameters are uncorrelated (and this should be especially true when
considering a statistics on all the pixels of an image), error propagation in
Eq. (6) gives

$\sigma_{p^{2}}^{2}=\sum_{i=1}^{4}\left(\frac{\partial p^{2}}{\partial
s_{i}}\right)^{2}\sigma_{s_{i}}^{2},$ (7)

where, for convenience we have made $s_{i}\equiv S_{i}^{2}$, $i=1,2,3,4$. Now,
since $\sigma_{s_{i}}=2S_{i}\sigma_{i}$, it is easy to see that

$\sigma_{p^{2}}^{2}=\frac{4}{S_{1}^{4}}\left(p^{4}S_{1}^{2}\sigma_{1}^{2}+\sum_{i=2}^{4}S_{i}^{2}\sigma_{i}^{2}\right)$
(8)

and, according to Eq. (2) and since $\sigma_{p^{2}}=2p\,\sigma_{p}$, one can
write

$\frac{\sigma_{p}^{2}}{p^{2}}=\left(1+\frac{1}{p^{4}}\sum_{i=2}^{4}\frac{S_{i}^{2}}{S_{1}^{2}}\frac{\varepsilon_{1}^{2}}{\varepsilon_{i}^{2}}\right)\frac{\sigma_{1}^{2}}{S_{1}^{2}}.$
(9)

Now, if all the three last efficiencies are the same and certainly less than
their maxima, we finally obtain

$\frac{\sigma_{p}}{p}\geq\frac{\sqrt{1+\frac{3}{p^{2}}}}{S/N},$ (10)

an inequality already published by Del Toro Iniesta & Orozco Suárez (2010),
Martínez Pillet et al. (2011), and Del Toro Iniesta & Martínez Pillet (2010)
without demonstration.

Our instruments are aimed at measuring magnetic fields and velocities.
Therefore, any reasonable design should include lower limits for these
quantities within the overall error budget. Detectability thresholds for the
Stokes parameters imply thresholds for the magnetograph and tachograph signals
as well. The rest of this section is devoted to estimate them. Of course, any
estimation that one can make depends not only on the instrument but also on
the inference technique. Most modern magnetographs use inversion of the
radiative transfer equation to infer values for both the magnetic field vector
and the plasma velocity. These inferences involve all four Stokes parameters
and, hence, should be more accurate than those using just one or two of them.
However, for the sake of clarity in the analytical derivation, we shall
consider errors induced in the magnetographic and tachographic formulas (11),
(12), and (17).

Using classical magnetographic formulas, the longitudinal and transverse
components of the magnetic field are given by

$B_{\rm lon}=k_{\rm lon}\frac{V_{s}}{S_{1,\rm c}}$ (11)

and

$B_{\rm tran}=k_{\rm tran}\sqrt{\frac{L_{s}}{S_{1,\rm c}}},$ (12)

where $k_{\rm lon}$ and $k_{\rm tran}$ are (model-dependent) calibration
coefficients and $V_{s}$ and $L_{s}$ are the circular and linear polarization
signals calculated as

$V_{s}\equiv\frac{1}{n_{\lambda}}\sum_{i=1}^{n_{\lambda}}a_{i}\,S_{4,i},$ (13)

where $a_{i}=1$ or $-1$ depending on whether the sample is to the blue
(including the zero shift) or the red side of the central wavelength of the
line, and

$L_{s}\equiv\frac{1}{n_{\lambda}}\sum_{i=1}^{n_{\lambda}}\sqrt{S_{2,i}^{2}+S_{3,i}^{2}}.$
(14)

In the above equations, $n_{\lambda}$ stands for the number of wavelength
samples. If we now assume that the minimum polarization signals are
$V_{s}=\sigma_{4}$ and $L_{s}=\sigma_{2}=\sigma_{3}$, the minimum detectable
thresholds are

$\delta(B_{\rm lon})=\frac{k_{\rm
lon}}{S/N}\,\frac{\varepsilon_{1}}{\varepsilon_{4}}$ (15)

and

$\delta(B_{\rm tran})=k_{\rm
tran}\sqrt{\frac{\varepsilon_{1}/\varepsilon_{2}}{S/N}}=k_{\rm
tran}\sqrt{\frac{\varepsilon_{1}/\varepsilon_{3}}{S/N}}.$ (16)

Expressions (15) and (16) give the explicit dependence of magnetic
detectability thresholds in terms of the instrument efficiencies and are very
useful in practice. For example, if we use the calibration constants for IMaX
quoted in Martínez Pillet et al. (2011), assuming that the maximum
polarimetric efficiencies have been reached, and typical signal-to-noise
ratios of 1700 ($\sim 1000$ for $S_{2},S_{3}$, and $S_{4}$), the minimum
longitudinal and transverse components of the magnetic field detectable with
that magnetograph are 5 and 80 G, respectively.

As far as the velocity is concerned, we shall assume that the Fourier
tachometer technique (Beckers & Brown, 1978; Brown, 1981; Fernandes, 1992) is
used:

$v_{\rm LOS}=\frac{2{\rm
c}\,\delta\lambda}{\pi\lambda_{0}}\,\arctan\frac{S_{1,-9}+S_{1,-3}-S_{1,+3}-S_{1,+9}}{S_{1,-9}-S_{1,-3}-S_{1,+3}+S_{1,+9}},$
(17)

where c is the speed of light, $\delta\lambda$ is the spectral resolution of
the instrument, and $\lambda_{0}$ is the central wavelength of the line;
$-9,-3,+3,+9$ stand for the sample wavelengths of the intensity, measured in
picometers with respect to $\lambda_{0}$. Let us assume that the minimum
detectable difference between symmetric wavelength samples (such as
$S_{1,-3}-S_{1,+3}$) due to LOS velocity shifts is $\sigma_{1}$. Then, if the
difference between the samples at the same flank of the line is approximated
by $~{}1/2\,\,S_{1,\rm c}$, the minimum detectable LOS velocity can be
approximated by

$\delta(v_{\rm LOS})\simeq\frac{2{\rm
c}\,\delta\lambda}{\pi\lambda_{0}}\,\arctan\frac{2}{S/N}.$ (18)

Likewise Eqs. (10), (15) and (16), this new expression (18) relates the
velocity threshold with the signal-to-noise ratio of the instrument. If we use
again IMaX values ($\delta\lambda=8.5$ pm; $\lambda_{0}=525.02$ nm) and assume
$S/N=1700$, the minimum detectable LOS velocity change is roughly 4 m s-1.

###  2.1. Uncertainties induced by photon noise

Fluctuations in the light levels due to photon statistics necessarily imply
variances in the Stokes parameters that in the end induce uncertainties in the
measured physical quantities, $B_{\rm lon}$, $B_{\rm tran}$, and $v_{\rm
LOS}$. In this section, we are going to establish a relationship between those
variances and uncertainties. Note that we discard for the moment any random
fluctuation in the instrument that will be dealt with in Sect. 4.

Error propagation in Eq. (11) easily yields

$\frac{\sigma^{2}_{B_{\rm lon}}}{B_{\rm
lon}^{2}}=\frac{\sigma_{4}^{2}}{n_{\lambda}V_{s}^{2}}+\left(\frac{1}{S/N}\right)^{2},$
(19)

because $\sigma^{2}_{V_{s}}=\sigma^{2}_{4}/n_{\lambda}$. Using now Eqs. (2),
(11), and (1), one obtains

$\frac{\sigma^{2}_{B_{\rm lon}}}{B_{\rm lon}^{2}}=\left(\frac{k_{\rm
lon}^{2}}{n_{\lambda}B_{\rm
lon}^{2}}\frac{\varepsilon_{1}^{2}}{\varepsilon_{4}^{2}}+1\right)\frac{1}{(S/N)^{2}},$
(20)

that relates the $B_{\rm lon}$ relative error with itself, the signal-to-noise
ratio of the observations, and the polarimetric efficiencies. On its turn,
error propagation in Eq. (12) gives

$\frac{\sigma^{2}_{B_{\rm tran}}}{B_{\rm
tran}^{2}}=\frac{1}{4}\left[\frac{\sigma_{L_{s}}^{2}}{L_{s}^{2}}+\left(\frac{1}{S/N}\right)^{2}\right].$
(21)

Now, since the variances for Stokes $S_{2}$ and Stokes $S_{3}$ should be
approximately the same,
$\sigma_{L_{s}}^{2}\simeq(1/2n_{\lambda})(\sigma_{2}^{2}+\sigma_{3}^{2})$ and,
using Eqs. (2), (12), and (1), Eq. (21) turns out to be

$\frac{\sigma^{2}_{B_{\rm tran}}}{B_{\rm tran}^{2}}=\left[\frac{k_{\rm
tran}^{4}}{8n_{\lambda}B_{\rm
tran}^{4}}\left(\frac{\varepsilon_{1}^{2}}{\varepsilon_{2}^{2}}+\frac{\varepsilon_{1}^{2}}{\varepsilon_{3}^{2}}\right)+\frac{1}{4}\right]\frac{1}{(S/N)^{2}},$
(22)

that again relates the relevant magnetographic quantity relative error with
itself, the photon-induced signal-to-noise ratio of the observations, and the
polarimetric efficiencies of the instrument.

If we use the values for the calibration constants quoted in Martínez Pillet
et al. (2011) for IMaX, $n_{\lambda}=5$ for this instrument, and assume that
the maximum polarimetric efficiencies are reached, then the estimated relative
errors for $B_{\rm lon}$ and $B_{\rm tran}$ induced by a photon noise of
$S/N=1700$ are of 2 and 15%, respectively, for magnitudes in either quantities
of 100 G; for magnitudes of 1000 G, the relative errors drop to 0.2 and 0.1%,
respectively.

After a similar calculation for photon-noise-induced uncertainties in the
tachographic formula (17), one gets

$\sigma_{v_{\rm
LOS}}^{2}=\frac{4c^{2}(\delta\lambda)^{2}}{\pi^{2}\lambda_{0}^{2}}\frac{2\sigma_{1}^{2}}{\Delta},$
(23)

where $\Delta=(S_{1,-9}-S_{1,+3})^{2}+(S_{1,+9}-S_{1,-3})^{2}$, and the
variances of the Stokes $S_{1}$ samples are all assumed to be $\sigma_{1}$.
Note that the slight asymmetry between Eq. (23) and Eqs. (20) and (22) is not
such as the ratio $\sigma_{1}^{2}/\Delta$ is a kind of inverse, square signal-
to-noise ratio. A numerical estimate for the IMaX instrument, and using the
FTS spectrum by Brault & Neckel (1987) to evaluate $\Delta$ for its Fe i line
at 525.02 nm, we conclude that the photon-noise-induced uncertainty is 4 m
s-1.

##  3\. An optimum vector plus longitudinal polarimeter

As explained by Martínez Pillet et al. (2004), a versatile polarimeter is
obtained through the combination of two nematic liquid crystal variable
retarders (LCVRs) with their optical axes properly oriented at 0∘and 45∘with
the Stokes $S_{2}$ positive ($X$) direction. This is so because it can provide
optimum modulation schemes for both the vectorial and the longitudinal
$(S_{1}\pm S_{4})$ polarization analyses by simply tuning the voltages that
change their retardances. The theoretical maximum efficiencies mentioned above
can in principle be reached by such an ideal polarimeter. We have assumed
these maximum efficiencies for our instruments so far. However, instrumental
effects may corrupt the measurement so that the final efficiencies are lower.
Let us see in this section what happens if some typical optical elements are
included between the modulator and the analyzer in the analysis.

The corrupting effect of the optical elements of an instrument in the final
polarization analysis is called instrumental polarization. It is well known
that those optical components acting on light after the polarization
modulation do not produce any instrumental polarization. However, nobody has
yet demonstrated whether optimum polarimetric efficiencies can still be
reached no matter the optics in between the modulator and the analyzer. In
this section we are going to show that this is the case with these two-LCVR-
based polarimeters because retardances can be fine tuned by simply changing
the acting voltages. This property certainly makes this type of polarimeters
very versatile and optimum for solar investigations. To understand the result,
let us start by demonstrating that, indeed, a polarimeter made up of two
nematic LCVRs oriented as above plus a linear analyzer can reach the optimum
polarimetric efficiencies.

According to Del Toro Iniesta (2003), the modulation matrix of any
polarimetric system consists of rows that equal the first row of the system
Mueller matrix for each of the measurements. If ${\bf R}(\theta,\delta)$
stands for the Mueller matrix of a general retarder whose fast axis is at an
angle $\theta$ with the $X$ axis and whose retardance is $\delta$, our LCVR
Mueller matrices can be described by ${\bf M}_{1}={\bf R}(0,\rho_{i})$ and
${\bf M}_{2}={\bf R}(\pi/4,\tau_{i})$, where $i=1,2,3,4$ is an index for each
of the four measurements. Hence, in our case, where the analyzer (of Mueller
matrix ${\bf M}_{4}$) is a linear polarizer at 0∘,222Dual-beam polarimeters
use a polarizing beam splitter as a double analyzer. Hence, another analyzer
at 90∘is indeed present simultaneously although the double calculation is not
necessary. such a modulation matrix, disregarding a 1/2 gain factor, is given
by (see Martínez Pillet et al., 2004)

${\bf
O}=\left(\begin{array}[]{llll}1&\cos\tau_{1}&\sin\rho_{1}\sin\tau_{1}&-\cos\rho_{1}\sin\tau_{1}\\\
1&\cos\tau_{2}&\sin\rho_{2}\sin\tau_{2}&-\cos\rho_{2}\sin\tau_{2}\\\
1&\cos\tau_{3}&\sin\rho_{3}\sin\tau_{3}&-\cos\rho_{3}\sin\tau_{3}\\\
1&\cos\tau_{4}&\sin\rho_{4}\sin\tau_{4}&-\cos\rho_{4}\sin\tau_{4}\end{array}\right).$
(24)

As explained in Del Toro Iniesta & Collados (2000), if all the three last
column elements of ${\bf O}$ have a magnitude of $1/\sqrt{3}$ (with their
signs properly altered), then the modulation is optimum and the maximum
efficiencies are reached. $|\cos\tau|=1/\sqrt{3}$ has the four independent
solutions
$\tau=54\hbox{$.\\!\\!^{\circ}$}736,125\hbox{$.\\!\\!^{\circ}$}264,234\hbox{$.\\!\\!^{\circ}$}736,$
and $305\hbox{$.\\!\\!^{\circ}$}264$. With them,
$|\sin\rho\,\sin\tau|=1/\sqrt{3}$ is equivalent to $|\sin\rho|=\sqrt{2}/2$
that has four independent solutions as well:
$\rho=45^{\circ}\/,135^{\circ}\/,225^{\circ}\/$, and $315^{\circ}\/$. The
verification of the above two equations ensures the automatic verification of
that for the third column and, therefore, we have found that several
combination of matrix elements exist that qualify $\bf O$ as the modulation
matrix of an optimum polarimetric scheme, as we aimed at demonstrating.

Real polarimeters, however, have some optics in between the modulator and the
analyzer. Very importantly, modern magnetographs like CRISP, VIP, IMaX, or
SO/PHI have one or several Fabry-Pérot etalons. Such etalons can modify the
Mueller matrix that leads to a modulation matrix like that in Eq. (24) and,
hence, we must check whether or not the resulting modulation matrix, ${\bf
O}^{\prime}$, remains optimum. To do that, let us model the most general
behavior of an etalon as a retarder ${\bf M}_{3}={\bf R}(\theta_{\rm
etalon},\delta_{\rm etalon})$. Then, the final Mueller matrix of the system is
now ${\bf F}={\bf M}_{4}{\bf M}_{3}{\bf M}_{2}{\bf M}_{1}$ and its first row
(again disregarding the gain factor) is given by $F_{11}=1$,

$\begin{array}[]{lll}F_{12}&=&M_{3,22}\cos\tau_{i}+M_{3,24}\sin\tau_{i},\\\
F_{13}&=&M_{3,22}\sin\rho_{i}\,\sin\tau_{i}+M_{3,23}\cos\rho_{i}-M_{3,24}\sin\rho_{i}\,\cos\tau_{i},\\\
F_{14}&=&-M_{3,22}\cos\rho_{i}\,\sin\tau_{i}+M_{3,23}\sin\rho_{i}+M_{3,24}\cos\rho_{i}\,\cos\tau_{i}.\end{array}$
(25)

We do not need any more matrix elements of ${\bf F}$ because the rows of the
new modulation matrix are $O^{\prime}_{ij}=F_{1j}(\tau_{i},\rho_{i})$. Now, we
only need to find out four different combinations of the first and second
retardances that are solutions for Eqs. (25) with $|F_{1k}|=1/\sqrt{3}$, where
$k=2,3,4$. Equations (25) are transcendental and, thus, have to be solved
numerically. However, before proceeding with the numerical exercise we can
realize several features in the solutions. First, the trivial cases, where
$\delta_{\rm etalon}=0$ (that is, no etalon exists or it is not birefringent)
or $\theta_{\rm etalon}=0,\pi/2$, the orthogonal directions of the analyzer
axis, are indeed trivial because the effect of ${\bf M}_{3}$ disappears and
$O^{\prime}_{ij}=O_{ij}$. Second, a number of periodicities can be deduced
from the equations structure:

  * •

If $\tau_{0}$ is a solution for the first of Eqs. (25) with
$F_{12}=1/\sqrt{3}$, then $\tau_{0}+(2k+1)\pi$, with $k$ integer, are
solutions for that equation when $F_{12}=-1/\sqrt{3}$ and vice versa.

  * •

If $\rho_{0}$ is a solution for the second or the third of Eqs. (25) with
$F_{12}=1/\sqrt{3}$, then $\rho_{0}+(2k+1)\pi$, with $k$ integer, are
solutions for that equation when $F_{12}=-1/\sqrt{3}$ and vice versa.

  * •

If $\rho_{0}$ is a solution for the second of Eqs. (25) with
$F_{12}=1/\sqrt{3}$, then $\rho_{0}+(2k+1)\pi/2$, with $k$ even integer, are
solutions for the third equation when $F_{12}=-1/\sqrt{3}$. When $k$ is odd,
then the solution for the third equation is when $F_{12}=1/\sqrt{3}$ as well.

To solve the first of Eqs. (25) let us consider the function

$f(\tau)=F_{12}-M_{3,22}\cos\tau-M_{3,24}\sin\tau,$ (26)

that has extrema where its derivative becomes zero. This occurs at $\tau_{0}$
where either $M_{3,24}=\sin\tau_{0}$ and $M_{3,22}=\cos\tau_{0}$ or
$M_{3,24}=-\sin\tau_{0}$ and $M_{3,22}=-\cos\tau_{0}$. These values imply that
$f(\tau_{\rm max})=F_{12}+1$ and $f(\tau_{\rm min})=F_{12}-1$. That is, the
maximum of the function is positive and the minimum is negative when
$|F_{12}|=1/\sqrt{3}$ (which is required for reaching optimum
efficiencies).333Note that we have selected one out of the infinite solutions
for the derivative of $f$ to be zero, but this is coherent with our neglecting
multiplicative, gain factors in the definition of Mueller matrices. Therefore,
since $f$ is continuous, Bolzano’s theorem ensures that a solution exists in
$(\tau_{\rm min},\tau_{\rm max})$ and this enables us to find that solution,
for instance, through the bisector method. This has to be done just once per
value of $F_{12}$; the other value derives from the first of the above
specified properties.

As a summary, the first of Eqs. (25) has four solutions in $[0,2\pi]$, each
two belonging to one of the signs of $F_{12}$. For each of these four
retardances, $\tau_{i}$, four possibilities are open according to the values
of $F_{13}$ and $F_{14}$. These four solutions for $\rho_{i}$ in the second
and third of Eqs. (25) can be shown to be enclosed in the following single
expression,

$\cos\rho=\pm\frac{(n\mp m)}{\sqrt{3}\,(m^{2}+n^{2})},$ (27)

where $n=M_{3,23}$ and $m=M_{3,22}\sin\tau-M_{3,24}\cos\tau$. A further
property of the solutions thus derives from Eq. (27): if $\rho_{0}$ is a
solution for the second and the third of Eqs. (25), then $2\pi-\rho_{0}$,
$\pi-\rho_{0}$, and $\pi+\rho_{0}$ are solutions as well.

Therefore, the presence of an etalon modeled as a retarder is not a problem
for the two-LCVR-based polarimeter to be optimum. No matter the possible
retardance or orientation the etalon may have, we are always able to find out
more than four combinations of $\rho$ and $\tau$ that ensure theoretical
polarimetric efficiencies for all three Stokes parameters all equal to
$1/\sqrt{3}$. In practice, these new solutions can be achieved by simply
tuning the acting voltages of the two LCVRs.

Figure 1.— Retardances of the first (bottom) and second (top) LCVRs as
functions of the angle $\theta$ of the etalon orientation with respect to the
$S_{2}$ positive direction. Only values between $0$ and $2\pi$ are displayed.
Different line types refer to different values of the etalon retardance (see
text).

Figure 1 displays the LCVR retardances in $[0,2\pi]$ that guarantee optimum
performance as functions of the orientation angle of the etalon. Values for
the second LCVR are in the top panel and those for the first one are in the
bottom panel.444Note that retardances larger than $2\pi$ may be needed for
design convenience. Their values are easily deducible from the above-mentioned
properties. Different colors correspond to different solutions; different line
types correspond to different values of the assumed etalon retardance:
0∘(dotted), 30∘(solid), 45∘(dashed), and 60∘(dashed-dotted). As commented on
before, when $\theta_{\rm etalon}=90^{\circ}\/$, both $\rho$ and $\tau$
recover the same value as if the etalon were absent. Moreover, both
retardances are periodic with $\theta$, with periods $\pi/4$ ($\rho$) and
$\pi/2$ ($\tau$). It is also interesting to note that four out of the eight
solutions for $\rho$ are equal to the other four but phase shifted by $\pi/4$.

Now, it is a little tedious but easy to demonstrate that, regardless of how
many, mirrors can be introduced in the optical path between the modulator and
the analyzer (as in real instruments) without affecting the maximum
polarimetric efficiencies, provided they all are perpendicular to the optical
axis plane. From the Mueller matrix of a single mirror, one can realize
(Collett, 1992) that the matrix of such a mirror train, no matter the angles
between them, keeps always the shape

${\bf E}=\left(\begin{array}[]{cccc}a&b&0&0\\\ b&a&0&0\\\ 0&0&c&d\\\
0&0&-d&e\end{array}\right).$ (28)

The elements of the first row in the new final Mueller matrix of the system
become $F^{\prime}_{1j}=(a+b)F_{1j}$, $j=1,2,3,4$. That is, the final
modulation matrix remains the same as before introducing the mirrors but
scaled by a gain factor that can be disregarded as we have been doing for all
the treatment. Therefore, we can conclude that optimum efficiencies can still
be achieved with as many mirrors as needed. Since a mirror is indeed a
combination of a retarder and a partial polarizer (the $d$ element is zero for
the latter; see e.g. Stenflo, 1994), the same conclusion can be reached for
whatever differential absorption effects for the orthogonal polarization
states that may be located between the modulator and the analyzer. Therefore,
if, for example, the etalon or the diffraction grating of the instrument
display different transmittances for orthogonally polarized beams the
polarimetric efficiencies can still attain maximum values.

##  4\. Instrument-induced inaccuracies

Now that we know that our magnetograph can reach optimum polarimetric
performance, let us study the behavior of this particular instrument against
instabilities in its main optical elements. Photon noise is not the only harm
for magnetographic or tachographic measurements. Instabilities of different
types like those in the temperature or in the tuning voltage of both the LCVRs
and the etalon, or roughness in their final thicknesses, can induce
inaccuracies. For single measurements, the inaccuracies can imply errors in
magnetic field or absolute wavelength calibration. For time series like those
needed in helioseismological studies, such inaccuracies may avoid detection of
some particular oscillatory modes. An assessment on such inaccuracies is
therefore in order for clearly defining design tolerances of these
instrumental quantities. Such tolerances should ensure the fulfillment of the
scientific requirements of the instrument.

###  4.1. Polarimetric inaccuracies

An important example of a scientific requirement is the polarimetric accuracy
of the system. By such we understand any of the inverse signal-to-noise ratios
for $S_{2}$, $S_{3}$, or $S_{4}$ as defined in Eq. (1). Following our general
assumption that these $(S/N)_{i}$ are intended to be the same by design,
aiming at a $S/N$ of 1700 is equivalent to require the system to have a
polarimetric accuracy of $10^{-3}$. Temperature, voltage, and other
instabilities and defects of the LCVRs lead to changes in the retardances
that, on their turn, induce modulation and demodulation changes. Such changes
can be seen as cross-talk between the Stokes parameters that drive to
covariances in the magnetographic measurements (Asensio Ramos & Collados 2008;
see also an interesting discussion on seeing-induced cross-talk in Casini et
al. 2011). Since our modulation matrix (24) is analytical, we attempt an
analytical approach to the study of the effect of these retardance changes
onto the polarimetric accuracy of the system.

Let ${\bf D}$ be the demodulation matrix of the instrument that always exists.
Thus, ${\bf OD}={\bf DO}={1\mskip-7.0mu1}$. The measured Stokes vector is then
given by ${\bf S}={\bf D}{\bf I}_{\rm meas}$, where ${\bf I}_{\rm meas}$
stands for the four intensity measurement vector of each modulation cycle. If
we linearly perturb matrix ${\bf D}$ as a consequence of a small perturbation
in the LCVR retardances ($\sigma_{\rho_{k}},\sigma_{\tau_{k}},\,k=1,2,3,4$),
then the Stokes vector becomes ${\bf S}+{\bf S}^{\prime}=({\bf D}+{\bf
D}^{\prime)}{\bf I}_{\rm meas}$, where the perturbed demodulation matrix ${\bf
D}^{\prime}$ is given by

${\bf D}^{\prime}=\sum_{k=1}^{4}\left(\frac{\partial{\bf
D}}{\partial\rho_{k}}\sigma_{\rho_{k}}+\frac{\partial{\bf
D}}{\partial\tau_{k}}\sigma_{\tau_{k}}\right).$ (29)

It is obvious that the polarimetric accuracy requirement directly implies that
none of the four elements of ${\bf S}^{\prime}={\bf D}^{\prime}{\bf I}_{\rm
meas}$ can be greater than $10^{-3}$. To fulfill that requirement, let us then
study how the perturbation in the retardances can be produced and which are
the tolerances for the instrument quantities whose fluctuations produce them.

If we call $\delta_{L}$ anyone of the retardances, we have by definition that

$\delta_{L}=\frac{\beta t}{\lambda_{0}},$ (30)

where $\beta=n_{\rm e}-n_{\rm o}$ is the birefringence, i.e., the difference
between the extraordinary and the ordinary refractive indices of the liquid
crystal, and $t$ stands for its geometrical thickness. Therefore, it is
evident that

$\frac{\sigma^{2}_{\delta_{L}}}{\delta^{2}_{L}}=\frac{\sigma^{2}_{\beta}}{\beta^{2}}+\frac{\sigma^{2}_{t}}{t^{2}},$
(31)

that is, the relative inaccuracy in the LCVR retardance is the square root of
the sum of the square relative inaccuracies in the birefringence and in the
geometrical thickness at the operating wavelength. Let us ascribe for
convenience any possible local fabrication defect in the LC like an air bubble
to the thickness inaccuracy, so that birefringence can be considered spatially
constant for the whole device.

Figure 2.— Retardance (in degree) of a specific nematic LCVR as a function of
the acting voltage (in volts; solid line) and its derivative with inverted
sign (dashed line).

Variations in the birefringence can be produced by either variations in the
LCVR temperature, the acting voltage, or both. Hence, one can write

$\sigma^{2}_{\beta}=q_{T}^{2}\sigma_{T}^{2}+q_{V}^{2}\sigma_{V}^{2},$ (32)

where $q_{T}$ and $q_{V}$ are values of the (partial) derivatives of $\beta$
with respect to $T$ and $V$ at the given values of voltage and temperature,
respectively. Since we indeed have calibrations of the $\delta_{L}$
dependences rather than those of $\beta$, we better rewrite Eq. (31) as

$\sigma^{2}_{\delta_{L}}=m_{T}^{2}\,\sigma^{2}_{T}+m_{V}^{2}\,\sigma^{2}_{V}+\frac{\delta_{L}^{2}}{t^{2}}\,\sigma^{2}_{t},$
(33)

where $m_{T}^{2}$ and $m_{V}^{2}$ have a clear meaning and can be deduced from
calibrations. According to Martínez Pillet et al. (2011), based on data by
Heredero et al. (2007),

$m_{T}=-1.16+0.305V-0.02V^{2},$ (34)

for $V<8$ volt and $0$ otherwise, and $m_{V}$ can be obtained from data like
those displayed in Fig. 2 where $\delta_{L}$ (solid line) and its derivative
(dashed line; inverted sign) are plotted as functions of the acting voltage
for a particular LCVR.

To get a numerical estimation of the value of real tolerances, that is, of the
maximum $\sigma_{T}$, $\sigma_{V}$, and $\sigma_{t}$ affordable in real
instruments in order not to have the ${\bf S}^{\prime}$-elements greater than
$10^{-3}$, we shall use IMaX parameters555The combination for IMaX retardances
was $\rho=[315,315,225,225]$ and $\tau=[305.264,54.736,125.264,234.736]$, in
degrees. Their corresponding voltages were $V(\rho)=[2.535,2.535,3.112,3.112]$
and $V(\tau)=[2.4,9.0,4.3,2.9]$, in volts. plus the analytic expressions for
${\bf D}$ and its partial derivatives.666An IDL program that evaluates such
analytic expressions is available upon request. Direct (by hand) evaluation is
so tedious that only with the help of software applications like Mathematica
such expressions can be obtained. The last term in Eq. (33) may vary spatially
and is, thus, responsible for the pixel-to-pixel variations of the retardance
but can easily be calibrated if needed. Indeed, roughness in the device
thickness is a fabrication specification and can be checked upon delivery from
the manufacturer. Thickness inaccuracies may produce locally significant
effects (e.g., Alvarez-Herrero et al., 2010). Our own estimation, using Eqs.
(29, 33) indicate that relative errors in the thickness larger than 6 % induce
perturbations of the Stokes vector that are larger than the polarimetric
accuracy. Therefore, should this be the only instrumental instability,
specific pixel-to-pixel calibration of ${\bf D}$ would be needed if the
relative roughness is larger than 6 %. Note that, since the typical
thicknesses of LCVRs are of the order of micrometers, a roughness less than 6%
may mean a stringent requirement for the manufacturer of the order of tens of
nanometers.

Using again Eqs. (29, 33), we find out that instabilities larger than
$\sigma_{T}=600$ mK or $\sigma_{V}=1.5$ mV deteriorate the polarimetric
accuracy below the required $10^{-3}$. These two tolerances and that for the
roughness have been calculated after assuming that each instability is acting
individually. According to Eq. (33), the final uncertainty in the retardance
stems from the three sources simultaneously. Hence, a safety reduction factor
of $\sqrt{3}$ (assuming all the three contribute the same) is advisable.
Therefore, in the end, the final tolerance specification for our instrument to
reach a polarimetric accuracy of $10^{-3}$ is 300 mK for temperature, 1 mV for
voltage, and a 4 % for LCVR roughness.

###  4.2. Magnetographic inaccuracies

The retardance perturbations of Eq. (33) induce changes in the maximum
polarimetric efficiencies of the instrument. Such changes necessarily imply
modifications in the magnetographic measurements. The modifications might
jeopardize the quality of the results. Imagine, for instance, that a
requirement on the repeatability of $B_{\rm lon}$ and $B_{\rm tran}$ applies
because we are interested on a measurement time series: a calculation of
tolerances in the instrument parameters ($T$, $V$, roughness, etc.) that
ensure the fulfillment of the magnetographic repeatability is in order.

Since no explicit dependence of $B_{\rm lon}$ and $B_{\rm tran}$ on
$\varepsilon_{i}$ exists, we cannot analytically gauge the induced
inaccuracies in any magnetographic measurement. Nevertheless, we can obtain a
hint on the global by studying the specific variations of $\delta(B_{\rm
lon})$ and $\delta(B_{\rm tran})$, the minimum detectable values of such
magnetograph quantities.

Error propagation in Eqs. (15) and (16) readily gives

$\sigma^{2}_{{\delta(B_{\rm lon})}}=\frac{{\delta^{2}(B_{\rm
lon})}}{4\varepsilon_{4}^{2}}\,\sigma^{2}_{\varepsilon^{2}_{4}}$ (35)

and

$\sigma^{2}_{{\delta(B_{\rm tran})}}=\frac{{\delta^{2}(B_{\rm
tran})}}{16\varepsilon_{2}^{2}}\,\sigma^{2}_{\varepsilon^{2}_{2}}=\frac{{\delta^{2}(B_{\rm
tran})}}{16\varepsilon_{3}^{2}}\,\sigma^{2}_{\varepsilon^{2}_{3}}.$ (36)

According to Del Toro Iniesta & Collados (2000), the maximum polarimetric
efficiencies that can be reached by any system are

$\varepsilon^{2}_{{\rm max},i}=\frac{\sum_{j=1}^{4}O^{2}_{ji}}{N_{p}},$ (37)

where $O_{ji}$ are the matrix elements of ${\bf O}^{\rm T}$, the transpose of
$\bf O$. For a system with a modulation matrix like that in Eq. (24), it is
easy to see that the efficiency inaccuracies ensuing the thermal instabilities
are

$\sigma^{2}_{\varepsilon^{2}_{{\rm max},1}}=0,$ (38)

as a consequence of using normalized Mueller matrices, and

$\sigma^{2}_{\varepsilon^{2}_{{\rm
max},2}}=\frac{1}{16}\sum_{j=1}^{4}\sin^{2}2\tau_{j}\,\,\sigma^{2}_{\tau_{j}},$
(39) $\displaystyle\sigma^{2}_{\varepsilon^{2}_{{\rm max},3}}$
$\displaystyle=$
$\displaystyle\frac{1}{16}\left\\{\sum_{j=1}^{4}\sin^{2}2\rho_{j}\,\,\sin^{4}\tau_{j}\,\,\sigma^{2}_{\rho_{j}}\right.+$
(40)
$\displaystyle\left.\sum_{j=1}^{4}\sin^{4}\rho_{j}\,\,\sin^{2}2\tau_{j}\,\,\sigma^{2}_{\tau_{j}}\right\\},$
$\displaystyle\sigma^{2}_{\varepsilon^{2}_{{\rm max},4}}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\left\\{\sum_{j=1}^{4}\sin^{2}2\rho_{j}\,\,\sin^{4}\tau_{j}\,\,\sigma^{2}_{\rho_{j}}\right.+$
(41)
$\displaystyle\left.\sum_{j=1}^{4}\cos^{4}\rho_{j}\,\,\sin^{2}2\tau_{j}\,\,\sigma^{2}_{\tau_{j}}\right\\},$

where $\sigma^{2}_{\tau_{j}}$ and $\sigma^{2}_{\rho_{j}}$ are the variances of
the LCVR retardances and where we have neglected the influence of possible
time-exposure differences or instabilities like those caused by a rolling-
shutter detector or a non-ideal repeatability of a mechanical shutter. Such
influences can be estimated separately for the specific instrument and
directly scale the effective exposure time (either per pixel or per frame).

Using the values for IMaX, and assuming maximum efficiencies, instabilities of
0.3 K or of 1.1 mV produce a 5 % repeatability error in the threshold for
$B_{\rm lon}$. A comparison of Eqs. (35) and (36) readily tells that the
effect is 2 times smaller on the relative repeatability error in the threshold
for $B_{\rm tran}$ but since the threshold itself is 16 times larger, the
absolute error is in the end 8 times larger as well.

###  4.3. Velocity inaccuracies

Error propagation in Eq. (17) yields

$\sigma_{v_{\rm LOS}}^{2}=\left(\frac{\partial v_{\rm
LOS}}{\partial\delta\lambda}\right)^{2}\sigma_{\delta\lambda}^{2}+\left(\frac{\partial
v_{\rm
LOS}}{\partial\lambda_{0}}\right)^{2}\sigma_{\lambda_{0}}^{2}+\sum_{k}\left(\frac{\partial
v_{\rm LOS}}{\partial S_{1,k}}\right)^{2}\sigma_{{1,k}}^{2}$ (42)

with $k=-9,-3,+3,+9$. Uncertainties in the spectral resolution come from
uncertainties in the etalon spacing and related fabrication details that
produce etalon roughness. Errors in the central wavelength come from the
etalon tuning that mostly depends on the ambient temperature, $T$, and on the
tuning voltage, $V$:

$\sigma_{\lambda_{0}}^{2}=k_{T}^{2}\sigma_{T}^{2}+k_{V}^{2}\sigma_{V}^{2},$
(43)

where $k_{T}$ and $k_{V}$ are constants that give the (linear) dependence of
$\lambda_{0}$ on $T$ and $V$. Finally, uncertainties in the Stokes $S_{1}$
samples come both from pure photon noise, as in any photometric measurement,
and from etalon tuning uncertainties. Since the (inexplicit) dependence of
$S_{1,k}$ on $\lambda_{0}$ is non linear, let us linearize it (that is,
introduce a small perturbation and take the first approximation) and write

$\sigma_{1,k}^{2}=\sigma_{1}^{2}+s_{1,k}^{2}\sigma_{\lambda_{0}}^{2},$ (44)

where we have assumed that the photometric contribution is equal for all the
samples and indeed equal to the photon noise as calculated in the continuum;
$s_{1,k}$ stand for the derivatives of the Stokes $S_{1}$ profile at the
corresponding wavelengths.

After a tedious but straightforward algebra, Eq. (42) can be recast as

$\sigma_{v_{\rm LOS}}^{2}=\frac{v_{\rm
LOS}^{2}}{(\delta\lambda)^{2}}\sigma_{\delta\lambda}^{2}+\frac{4c^{2}(\delta\lambda)^{2}}{\pi^{2}\lambda_{0}^{2}}\frac{2\sigma_{1}^{2}}{\Delta}+$
$\left[\frac{v_{\rm
LOS}^{2}}{\lambda_{0}^{2}}+\frac{4c^{2}(\delta\lambda)^{2}}{\pi^{2}\lambda_{0}^{2}}\frac{d_{1}+d_{2}}{\Delta^{2}}\right](k_{T}^{2}\sigma_{T}^{2}+k_{V}^{2}\sigma_{V}^{2}),$
(45)

where $\Delta$ is defined in Sect. 2.1,
$d_{1}=(S_{1,+9}-S_{1,-3})^{2}(s_{1,-9}^{2}+s_{1,+3}^{2})$ and
$d_{2}=(S_{1,-9}-S_{1,+3})^{2}(s_{1,+9}^{2}+s_{1,-3}^{2})$. Hence, clear
contributions to the final LOS velocity uncertainty can be discerned from the
etalon roughness, the photon noise, the etalon temperature instability, and
the etalon voltage instability.

Quantitative estimates of the various terms in Eq. (45) can be made by using
the FTS spectrum by Brault & Neckel (1987) to evaluate $\Delta$, $d_{1}$, and
$d_{2}$ for a given spectral line and a given instrument. Let assume the HMI
and SO/PHI Fe i line at $\lambda_{0}=617.3$ nm and a spectral resolution of
the etalon of $\delta\lambda=10$ pm. The first term has a clear impact on the
tachographic results: the etalon relative roughness is directly translated
into the same $v_{\rm LOS}$ relative uncertainty. In other words, we cannot
expect better accuracy in the line-of-sight velocity (when measured with the
Fourier tachometer formula) than that limited by the etalon relative
roughness; this means that a mere 0.1 pm, rms resolution uncertainty induces
10 m s-1 errors for speeds of 1 km s-1. The second term coincides with the
right-hand side of Eq. (23) and has been discussed already in Sect. 2.1. Since
the ratio between the second and the first terms within brackets is of the
order of 4$\cdot$109 for velocities of up to 5 km s-1, it is the second one
what really matters in the estimation; this means that the dependence of the
profile shape on the central wavelength of the line is really important. If we
use the same IMaX values of $k_{T}=2.52$ pm/K and $k_{V}=3.35\cdot$10-2 pm/V
for the SO/PHI etalon, Eq. (45) gives instabilities of the order of 3.3 mK or
0.25 V induce the same LOS velocity uncertainty of 4 km s-1 quoted above for
pure photon noise. Another way of seeing the same effect can be explained by
saying that a 100 m s-1 uncertainty is produced by either a 45 mK or a 3.4 V
instabilities. These uncertainties are really important when stability during
given periods of time of the instrument is required as for helioseismic
measurements. For single shots, uncertainties in temperature or voltage imply
thresholds for accurate absolute wavelength (velocity) calibration. The
importance of having included the measurement technique in this error budget
analysis is clear: should one have simply used the $k_{V}$ and $k_{T}$
calibration constants above and the Doppler formula an uncertainty of just 55
m s-1 would have been obtained. Hence, the uncertainty would have been
underestimated by a factor almost 2.

##  5\. Summary and conclusions

An assessment study on the salient features and properties of solar
magnetographs has been presented. An error budget procedure has been followed.
Special care has been devoted in including photon-induced and instrument-
induced noise as well as specific measurement technique contributions to the
final variances. We have first discussed the effect of random noise in the
measurements and deduced useful formulae –general for every device– that
provide some minimum detectable parameters like the degree of polarization of
light, the longitudinal and transverse components of the magnetic field, and
the line-of-sight velocity. The detection thresholds are given as functions of
the polarimetric efficiencies of the instrument and of the signal-to-noise
ratio of the observations. (As a proposal, we have suggested as well to use
the $S/N$ for the Stokes intensity as the signal-to-noise ratio for the
instrument.) When the random noise is photon-induced, we have calculated as
well the relative uncertainty in the magnetographic and tachographic
quantities. Secondly, an analysis is presented for those instruments based on
two nematic liquid crystal variable retarders as a polarization modulator and
a Fabry-Pérot etalon as the spectrum analyzer. Although specific for these
magnetographs, the methodology can easily be followed by others in order to
characterize their capabilities and accuracies. We have demonstrated that this
type of instrument can indeed reach theoretical maximum polarimetric
efficiencies because solutions always exist for the retardances of the two
LCVRs that ensure such efficiencies, hence optimizing the detection thresholds
and the relative uncertainties. Very remarkably, the existence of such
solutions is independent of the optics that is in between the polarization
modulator and the analyzer. Neither retarders nor partial polarizers or
mirrors (the most commonly used devices) alter that property. The LCVR optimum
retardances do depend in such pass-through optics but can be fine-tuned
according to the polarizing properties of the optics. A number of rules and
periodicity properties of the required retardances have also been deduced.
These polarimeters have modulation and demodulation matrices that are
explicitly calculated through an IDL procedure that is available upon request.
Thirdly, error propagation has yielded equations relating the variances of the
measured Stokes vector and the solar physical quantities and instrument
parameters, hence providing a bridge between scientific requirements and
instrument design specifications. The analytic character of this particular
type of instrument has also allowed the quantitative estimation of the
mentioned uncertainties. Hopefully, the discussion presented in this paper
excites (and helps) further diagnostics of other instruments.

This work has been partially funded by the Spanish Ministerio de Ciencia e
Innovación, through Projects No. AYA2009-14105-C06 and AYA2011-29833-C06, and
Junta de Andalucía, through Project P07-TEP-2687, including a percentage from
European FEDER funds.

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