Version of May 11, 2011Preprint typeset using LATEX style emulateapj v. 11/10/09

DERIVING TELESCOPE MUELLER MATRICES USING DAYTIME SKY POLARIZATION OBSERVATIONS

David M. Harrington1, J. R. Kuhn2, and Shannon Hall3

1Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI, 968222Institute for Astronomy Maui, University of Hawaii, 34 Ohia Ku St., Pukalani, HI, 96768 and

3Department of Astronomy, Whitman College, 345 Boyer Ave., Walla Walla, WA, 99362Version of May 11, 2011

ABSTRACT

Telescopes often modify the input polarization of a source so that the measured circular or linearoutput state of the optical signal can be signficantly different from the input. This mixing, or polar-ization “cross-talk”, is defined by the optical system Mueller matrix. We describe here an efficientmethod for recovering the input polarization state of the light and the full 4 × 4 Mueller matrix ofthe telescope with an accuracy of a few percent without external masks or telescope hardware mod-ification. Observations of the bright, highly polarized daytime sky using the Haleakala 3.7m AEOStelescope and a coudé spectropolarimeter demonstrate the technique.Subject headings: Astronomical Techniques, Astronomical Instrumentation, Data Analysis and Tech-

niques

1. INTRODUCTION

Spectropolarimetry is a powerful tool which may beenhanced by efficient techniques for telescope-detectorpolarization calibration. Unfortunately, it is common formodern altitude-azimuth telescopes with multiple mirrorreflections to scramble the input polarization of the lightbefore it reaches the analyzer of a coudé, Nasmyth, orGregorian polarimeter. Solar astronomers in particularhave developed a repretoire of techniques for calibratingtelescopes (Elmore et al. 1992; Kuhn et al. 1993; Giroet al. 2003; Socas-Navarro 2005a,b; Socas-Navarro et al.2005; Selbing 2005; Socas-Navarro et al. 2006; Ichimotoet al. 2008; Keller & Snik 2009; Elmore et al. 2010.Our experience with the Advanced Electro-Optical Sys-tem (AEOS) 3.67m telescope is that linearly polarizedlight can even, in some circumstances, be transformed tonearly 100% circularly polarized light. There are 7 highlyoblique reflections before polarization is analyzed at ourHigh Resolution Visible and Infra-red Spectropolarime-ter (HiVIS) (cf. Hodapp et al. 2000, Thornton et al.2003, Harrington et al. 2006, 2009, 2010, Harrington &Kuhn 2007, 2008; Harringtion & Kuhn 2009a,b).

Effective calibration of the telescope and instrumentminimizes these systematic errors and enhances the po-larimetric precision of an observation. Moving opticalelements, detector effects such as non-uniform sensitiv-ity or nonlinearity, atmospheric seeing, and transparencyvariations can all induce polarization errors that are of-ten larger than the photon noise. Strategies to minimize,stabilize and calibrate the instrument and telescope of-ten involve use of non-moving optics and emphasize rapidchopping and common-path differential techniques. Cali-bration techniques include observing unpolarized and po-larized standard stars, and use of polarized optical cal-ibration units that are designed to mimic the nominalillumination of the telescope optics during target obser-vations.

Electronic address: dmh@ifa.hawaii.eduElectronic address: kuhn@ifa.hawaii.eduElectronic address: hallsw@whitman.edu

The new technique presented here is designed to yieldthe full Mueller matrix calibration of many telescope-polarimeter systems with an accuracy of a few percent.It requires only two or more observations of the linearlypolarized sky due to solar Rayleigh scattering. The day-time sky is normally easily observable, highly polarized,and is a relatively well characterized calibration source.It is observable without using precious dark observingtime and requires no hardware calibration modificationsof the telescope. As long as multiple linear polarizationinput states are detected at each telescope pointing, thefull telescope Mueller matrix can be recovered with an ac-curacy of a few percent. This is realized by observing thesolar illuminated sky with the Sun at different zenith andazimuth positions. This paper illustrates the techniqueusing the Haleakala Air Force AEOS 3.7m telescope onMaui.

1.1. Polarization

The following discussion of polarization formalismclosely follows Collet 1992 and Clarke 2009. In theStokes formalism, the polarization state of light is de-noted as a 4-vector:

Si = [I, Q, U, V ]T (1)

In this formalism, I represents the total intensity, Qand U the linearly polarized intensity along polarizationposition angles 0◦ and 45◦ in the plane perpendicular tothe light beam, and V is the right-handed circularly po-larized intensity. Note that according to this definition,linear polarization along angles 90◦ and 135◦ will be de-noted as −Q and −U , respectively. The typical conven-tion for astronomical polarimetry is for the +Q electricfield vibration direction to be aligned North-South while+U has the electric field vibration direction aligned toNorth-East and South-West.

The normalized Stokes parameters are denoted withlower case and are defined as:

[1, q, u, v]T = [I, Q, U, V ]T /I (2)

2 Harrington et al.

The degree of polarization can be defined as a ratio ofpolarized light to the total intensity of the beam:

P =

√

Q2 + U2 + V 2

I=√

q2 + u2 + v2 (3)

For details on polarization of light and stellar spec-tropolarimetry, see Collet 1992 and Clarke 2009.

To describe how polarized light propagates throughany optical system, the Mueller matrix is constructedwhich specifies how the incident polarization state istransferred to the output polarization state. The Muellermatrix is a 4 × 4 set of transfer coefficients which whenmultiplied by the input Stokes vector (Siinput) gives theoutput Stokes vector (Sioutput):

Sioutput = MijSiinput (4)

If the Mueller matrix for a system is known, then oneinverts the matrix and deprojects a set of measurementsto recover the inputs. One can represent the individ-ual Mueller matrix terms as describing how one incidentstate transfers to another. In this work we will use thenotation:

Mij =

II QI UI V IIQ QQ UQ V QIU QU UU V UIV QV UV V V

(5)

1.2. Deriving Telescope Mueller Matrix Elements

Typical calibration schemes on smaller telescopes oftenuse fixed polarizing filters placed over the telescope aper-ture to provide known input states that are detected andyield terms of the Mueller matrix. This approach hasbeen used for solar telescopes (Socas-Navarro 2005a,b,Socas-Navarro et al. 2005, 2006). Another approach,also used in the solar observations has been to imageknown sources and use spectropolarimetric data and Zee-man effect symmetry properties to isolate and determineterms in the Mueller matrix (Kuhn et al. 1993, Elmoreet al. 2010). Night-time observations can use unpolar-ized and polarized standard stars to measure polarizationproperties of telescopes (cf. Hsu & Breger 1982, Schmidtet al. 1992, Gil-Hutton & Benavidez 2003, Fossati et al.2007). Many studies have either measured and calibratedtelescopes, measured mirror properties or attempted todesign instruments with minimal polarimetric defects (cf.Sánchez Almeida et al. 1991, Giro et al. 2003, Patat& Romaniello 2006,Tinbergen 2007, Joos et al. 2008,van Harten et al. 2009, Roelfsema et al. 2010).

1.3. The Polarized Sky as a Calibration

The observed polarization of the daytime sky is auseful calibration source. Scattered sunlight is bright,highly (linearly) polarized and typically illuminatesthe telescope optics more realistically than calibrationscreens. A single-scattering atomic Rayleigh calcula-tion (cf. Coulson 1980, 1998) is often adequate to de-scribe the skylight polarization. More realistic model-ing, involving multiple scattering and aerosol scattering,is also available using industry standard atmospheric ra-diative transfer software packages such as MODTRAN(cf. Fetrow et al. 2002, Berk et al. 2006).

Fig. 1.— The celestial triangle representing the geometry forthe sky polarization at any telescope pointing. γ is the angu-lar distance between the telescope pointing and the sun. θsis the solar angular distance from the zenith. θ is the angulardistance between the telescope pointing and the zenith. φ isthe angle between the zenith direction and the solar directionat the telescope pointing. ψ is the angle between the tele-scope pointing and the solar direction at the zenith. The lawof Cosines is used to solve for any angles needed to computedegree of polarization and position angle of polarization.

There are many atmospheric and geometric considera-tions that determine the skylight polarization at a partic-ular observatory site. The linear polarization amplitudeangle can depend on the solar elevation, atmosphericaerosol content, aerosol vertical distribution, aerosolscattering phase function, wavelength of the observationand secondary sources of illumination (cf. Horváth etal. 2002a,b, Lee 1998, Liu & Voss 1997, Suhai &Horváth 2004, Gál et al. 2001, Vermeulen et al. 2000,Pomozi et al. 2001, Cronin et al. 2005, 2006, Hegedüset al. 2007). Anisotropic scattered sunlight from re-flections off land or water can be highly polarized andtemporally variable (Litvinov et al. 2010, Peltoniemi etal. 2009, He et al. 2010, Salinas & Liew 2007, Ota etal. 2010, Kisselev & Bulgarelli 2004). Aerosol parti-cle optical properties and vertical distributions also vary(cf. Wu & Jin 1997, Shukurov & Shukurov 2006, Ver-meulen et al. 2000, Ougolnikov & Maslov 2002, 2005a,b,2009a,b, Ugolnikov et al. 2004). The polarization canchange across atmospheric absorption bands or can beinfluenced by other scattering mechanisms (cf. Boescheet al. 2006, Zeng et al. 2008 Aben et al. 1999, 2001).

Deviations from a single scattering Rayleigh modelgrow as the aerosol, cloud, ground or sea-surface scat-tering sources affect the telescope line-of-sight. Clear,cloudless, low-aerosol conditions should yield high linearpolarization amplitudes and small deviations in the po-larization direction from a Rayleigh model. Observationsgenerally support this conclusion (Pust & Shaw 2005,2006a,b, 2007, 2008, 2009, Shaw et al. 2010).

The geometry of our Rayleigh sky model is seen in Fig-ure 1. The geometrical inputs are the observers location

Mueller Matrices Using Sky Polarization 3

(latitude, longitude, elevation) and local time. The solarlocation and relevant angles from the telescope point-ing are computed from the spherical geometry in Fig-ure 1. The maximum degree of polarization (δmax) inthis model occurs at a scattering angle (γ) of 90◦. TheRayleigh sky model predicts the degree of polarization(δ) at any telescope pointing as:

δ =δmaxsin

2γ

1 + cos2γ(6)

Since the angle of polarization in the Rayleigh skymodel is always perpendicular to the scattering plane,one can derive the angle of polarization with respect tothe altitude and azimuth axes of the telescope. The lawof Cosines for the scattering plane breaks down at theZenith where θ is 0◦ and one must simply calculate thedifference in azimuth between the telescope pointing andthe solar azimuth to find the orientation of the polariza-tion. An example of the model we compared with ourobservations is shown in Figure 2 for Haleakala on Jan-uary 27th 2010.

One set of all-sky polarization measurements obtainedat the Mauna Loa Observatory, a nearby site and at verysimilar altitude to Haleakala (Dahlberg et al. 2009) areparticularly relevant. The maximum degree of polariza-tion (δmax) changed throughout the day in their observa-tions from a value of ∼60% with the sun at an elevationof 30◦, to 85% with the sun setting. Although this is asignificant change in the maximum degree of polarization(δmax) presumably due to multiple scattering, the polar-ization direction at all telescope pointings is expected tobe close to the Rayleigh model (cf. Dahlberg et al. 2009,Suhai & Horváth 2004, Pust & Shaw 2009, Pomozi etal. 2001). In our analysis below we will use the Rayleighmodel to define the input polarization angle. We believeour assumed input angle is accurate over most of the skyto

4 Harrington et al.

Fig. 3.— This Figure shows the average sky polarization withthe telescope pointed at the Zenith measured by LoVIS on January27th 2010 UT. The statistical error in the measurement is muchsmaller than the width of the line. The solid line is the averagepolarization measured for all wavelengths. The dashed line is aRayleigh Sky model prediction for the polarization at the Zenithfollowing measurements from Mauna Loa in Dahlberg et al. 2009Figure 7. The maximum degree of sky polarization in the 90◦

scattering plane (δmax) is assumed to rise linearly from 65% to85% as the sun sets from elevation 30◦ to 0◦. The scattering angle(γ) between the sun and the Zenith telescope pointing is shown inthe right hand y-axis.

in Figure 4. The average polarization shown in Figure 3was removed from the spectra leaving only residual chro-matic changes. There is a mild trend with wavelengthfor the measured polarization to decrease by about 10%from 5500Å to 7500Å. The atmospheric oxygen absorp-tion band shows a mild increase in polarization. There isalso a substantial change seen the chromatic trend seenin the final observation taken just after sunset as shownby the dashed line of Figure 4 consistent with studiesof the occultation and changing illumination (cf. Ougol-nikov & Maslov 2002, 2005a,b, 2009a,b, Ugolnikov et al.2004).

Given that AEOS telescope and LoVIS spectropo-larimeter induces polarization of less than 3% and thedepolarization is of the same order, we can use a 3 × 3Mueller matrix approximation. In this 3×3 QUV space,the degree of polarization is preserved for any input stateand the 3x3 Mueller matrix would be approximated as arotation matrix inside the Poincaré sphere.

3. TELESCOPE POLARIZATION AS ROTATION

To model the the 3 × 3 Mueller matrix as a rotationmatrix, we will use Euler angles. We scaled all our mea-sured Stokes vectors to unit length in order to removethe residual effects from changes in the sky degree ofpolarization, telescope induced polarization and depo-larization; and to put our measurements on the Poincarespheré. We denote the 3 Euler angles as (α, β, γ) and usea short-hand notation for sines and cosines where cos(γ)is shortened to cγ . We specify the rotation matrix (Rij)using the ZXZ convention as:

Rij =

(

cγ sγ 0−sγ cγ 0

0 0 1

)(

1 0 00 cβ sβ0 −sβ cβ

)(

cα sα 0−sα cα 0

0 0 1

)

(7)

Fig. 4.— This Figure shows the continuum polarization variationfrom average as a function of wavelength for all twenty LoVISZenith measurements on Jan. 27th 2010. The average value ofthe polarization, shown in Figure 3 has been removed from allobservations to highlight the change with wavelength. The solidlines show observations taken 2-3 hours before sunset. The dot-dash lines show observations taken 0.2-1 hour before sunset. Thedashed line shows a single observation taken just after sunset.

With this definition for the rotation matrix, we solvefor the Euler angles assuming a fully linearly polarizeddaytime sky as calibration input. If we denote the mea-sured Stokes parameters, Si, as (qm, um, vm) and the in-put sky Stokes parameters, Rj , as (qr , ur, 0) then the3 × 3 QUV Mueller matrix elements at each wavelengthare:

Si =

(

qmumvm

)

= MijRj =

(

QQ UQ V QQU UU V UQV UV V V

)(

qrur0

)

(8)This set of equations has 6 variables and only 5 known

quantities. We have no V input to constrain the V Q, V Uand V V terms. Nevertheless, two input and output vec-tors on the Poincare sphere are sufficient to fully specifythe full 3×3 rotation matrix. Thus, we use the fact thatthe sky polarization changes orientation with time andtake measurements at identical telescope pointings sep-arated by enough time for the solar sky illumination tochange. This yields an over-constrained solvable prob-lem for all six linear polarization terms in the Muellermatrix.

When using this rotation matrix approximation for thetelescope Mueller matrix, the Rayleigh Sky input Stokesparameters multiply each term of the rotation matrix togive a system of equations for the three Euler anlges (α,β, γ). This system of equations can be solved using anormal non-linear least-squares minimization by search-ing the (α, β, γ) space for minima in squared error. Thisdirect solution of this set of equations using standardminimization routines is subject to several ambiguitiesthat affect convergence using standard minimization rou-tines. For example, Euler angles are symmetric underthe exchange of (α, β, γ) with ( -180◦+α, -β, -180◦+γ).Solutions are also identical with multiples of 2π. As aconvenient approach to this least-squares problem, weused a different set of equations in a two-step solution.We first perform simple least squares solution directly

Mueller Matrices Using Sky Polarization 5

Fig. 5.— This Figure shows the six Mueller matrix elementsversus wavelength at a telescope pointing of 90◦ altitude, 225◦ az-imuth. These were derived from three sky measurements on Jan-uary 27th 2010 using the two-step solution. The UV term growswith increasing wavelength while the QU term decreases with wave-length. The other Mueller matrix elements remain near 0 or 1 andshow smaller chromatic variation.

for the six linear-polarization Mueller matrix elements.In the second step we fit a rotation matrix to the sixMueller matrix elements at each wavelength. As an ex-ample, Figure 5 shows the Mueller matrix elements de-rived with the Jan. 27th 2010 observations. The detailsof our methods for deriving Euler angles and an exampleof how one could plan sky calibration observations areoutlined in the Appendix.

Fig. 6.— This Figure shows the variation in Euler angles withwavelength at a telescope pointing of 90◦ altitude, 225◦ azimuth.The value for each Euler angle at 6560Å have been removed for clar-ity [α=176.5◦, β=81.6◦, γ=98.8◦]. The change with wavelength isdominated by β which corresponds to the changes in the QU andUV Mueller matrix elements from Figure 5. The statistical noiseis smaller than the width of the line.

3.1. Euler Angle Solution Properties

The Euler angles are well behaved smooth functions ofboth wavelength and telescope pointing. For instance,the Euler angle variations with wavelength calculatedfrom January 27th 2010 observations at a telescope point-ing of at 90◦ altitude and 225◦ azimuth are shown in

Figure 6. The change with wavelength is dominated byβ while smaller variation is seen in α and γ. For clarityin the Figure, the Euler angle values at 6560Å have beenremoved so all curves overlap. Similar curves are seen forother telescope pointings.

The Euler angles are also smooth functions of telescopepointing. We have observations on multiple days takenon a grid of 8 azimuths and 3 different altitudes. Figure 7shows the derived Euler angles at 6560Å as a functions ofazimuth. There are three different curves correspondingto the three different telescope altitudes. We find that γeffectively absorbs the changing azimuth while α stronglyvaries with altitude. Since there is substantial telescopeinduced rotation, these curves show significant deviationsfrom a purely geometrical rotation.

The derived Euler angles in Figure 6 are quite repeat-able when derived from calibration observations taken ondifferent days. For instance, we have two sets of sky ob-servations taken December 10th and 11th 2009 each con-sisting of thee observation sets spaced roughly one hourapart. The Euler angles derived from on different daysagree to within 0.5◦ giving an estimate of the system-atic error limits. The Mueller matrix elements derivedwith observations from different days agree to better than0.01. The changes in β with telescope azimuth shown inFigure 7 are well above the systematic noise of roughly0.5◦ and are repeatable even when using many differentcombinations of calibration data on both days. Similartrends for Euler angle variation with telescope pointingare seen at other wavelengths.

Fig. 7.— This Figure shows the variation in derived Euler anglesat 6560Å as a function of telescope pointing. The top panel shows αfor all telescope azimuths at elevations of 90◦ (solid), 75◦ (dashed)and 55◦ (dot-dashed). The middle panel shows β and the bottompanel shows γ for all azimuths at the same elevations as α usingthe same line scheme. This Figure clearly shows that α changesthe strongest with elevation while γ shows changes with azimuth.There is not much change seen in β.

3.2. Calibrating Spectropolarimetric Observations

Once telescope calibrations have been derived, observa-tions subsequently taken with the instrument at the sametelescope pointing can be de-rotated. The calibrated po-larization measurements are then oriented in a referenceframe on the sky corrected for all geometric, telescope

6 Harrington et al.

and instrument induced rotation. We have done manyexperiments where sky observations on some days areused to calibrate sky observations taken on other daysto examine the stability and repeatability of the calibra-tions.

As an example, Figure 8 shows six individual sky spec-tropolarimetric measurements taken on December 10th

and 11th 2009 at different times of the afternoon. Allobservations were taken at a telescope pointing of az-imuth 021◦ and altitude 70◦. The input linear polariza-tion orientation changes with time and there is significantchromatism and cross-talk observed in all Stokes param-eters. The three observations from each day are taken atsimilar times and the observed polarization spectra aresimilar.

Fig. 8.— This Figure shows the measured spectropolarimetrywith LoVIS while the telescope is fixed at altitude 70◦ and azimuth021◦ scaled to 100% polarization. Three observations were takenDecember 10th 2009 (solid lines) and another three observationswere taken at similar times on December 11th 2009 (dashed lines).The statistical noise is smaller than the width of the line.

In order to illustrate a general method of telescopecalibration, imagine a scenario where we derive a set ofcalibrations using observations from one day and thenuse this telescope calibration to de-rotate observationstaken on another day at an identical telescope pointing.We will treat the December 10th 2009 observations as acalibration set and derive Euler anlges for all telescopepointings and wavelengths. With these Euler angles inhand, we will then treat observations taken on December11th 2009 as our science targets. The de-rotated scienceobservations will be compared against the Rayleigh skyprediction. This scenario is exactly what a typical ob-server would perform during routine operations. Thedifference between the de-rotated December 11th obser-vations and the expected sky input Stokes parametersare shown in Figure 9. The residual error is typicallyless than 0.01 for any Stokes parameter showing that thede-rotated observations match the expected sky modelwith very high accuracy.

We get consistent results for Euler angles using manydifferent combinations of observations taken between Au-gust 2009 and March 2010. Over this time period therewere changes to the optical configuration such as re-mounting of the retarders, switching gratings and chang-ing LoVIS / HiVIS modes as would be expected during

Fig. 9.— This Figure shows the difference between the de-rotatedmeasured Stokes parameters and the predicted sky input Stokesparameters at altitude 70◦ and azimuth 021◦. The residual errorsare less than 0.01 showing very low error between measured andexpected Stokes parameters.

normal observing operations. If one derives a residual ro-tation angle between the de-rotated measurements andthe predicted sky, the residual rotation angles are al-most always less than 3◦ when using any calibration setagainst any other observation set. If one uses calibrationdata where there are no optical changes between calibra-tion and observation, this resulting angular error is muchsmaller, typically less than 0.5◦ rotation. This allows usto estimate systematic error from optical configurationchanges as a few degrees residual rotation. When observ-ing as calibrated, the errors are under 1◦ QUV rotation.

4. DISCUSSION

We have demonstrated a simple method for derivingtelescope polarization properties and Mueller matrix el-ements. By using observations of the bright, highly po-larized daytime sky and taking observations at multipletimes using identical optical configurations one can ex-tract the polarization response of the telescope.

This method can be easily implemented via leastsquares methods. A direct least-squares solution forEuler angles and a two-step solution for the linear-polarization sensitive Mueller matrix elements and a ro-tation matrix fit to these elements have been imple-mented. Examples of how to optimally prepare calibra-tion observations and solve for the telescope responseproperties are detailed in the Appendix.

The AEOS telescope and LoVIS spectropolarimeterhave over 20 reflections, many at high incidence anglesbefore the polarization analyzer. However, the major po-larization cross-talk effect is a simple rotations in QUVspace that is readily derived to an accuracy of a few per-cent from sky observations. Observations taken on mul-tiple days show we can de-rotate LoVIS sky observationsto the predicted Rayleigh sky with at least 0.5◦ accuracy.

The polarized sky is a calibration source that is bright,easily observable, highly polarized, simple to model, wellcharacterized and easy to verify with additional instru-mentation. Unpolarized and polarized standard starshave several disadvantages as calibration sources. Starsare not available at all telescope pointings and require us-ing night-time observing time for calibration. Telescopes

Mueller Matrices Using Sky Polarization 7

with complex optical trains such as AEOS require cali-bration at many pointings making the unavailability ofstars at all pointings a serious problem. Solar instru-ments are typically unable to observe stars at all butcan easily utilize this sky-based technique. Polarizedstandard stars normally have low degrees of polarizationwhich adds time or noise constraints to such calibrationefforts. This technique uses sky-light to illuminate thetelescope and instrument optics like the target observa-tions. Night-time observations benefit from the fact thatday-time observations are used to calibrate, rather thanusing precious night telescope time for calibration.

8 Harrington et al.

5. APPENDIX: SOLUTIONS FOR EULER ANGLES

There are a number of methods available to solve forEuler Angles and Mueller matrix elements given a setof sky polarization observations. Suppose we have mea-surements at different times but at identical telescopepointings. Multiplying out terms in equation 7 for theZXZ-convention we get:

Rij =

(

cαcγ − sαcβsγ sαcγ + cαcβsγ sβsγ−cαsγ − sαcβcγ −sαsγ + cαcβcγ sβcγ

sαsβ −cαsβ cβ

)

(9)

Then equating Mueller matrix elements to rotationmatrix elements could write the system of equations forthe three Euler angles as:

qm1um1vm1qm2um2vm2qm3um3vm3

=

qr1(+cαcγ − cβsαsγ) + ur1(+cγsα + cαcβsγ )qr1(−cβcγsα − cαsγ) + ur1(+cαcβcγ − sαsγ)qr1(+sαsβ) + ur1(−cαsβ)qr2(+cαcγ − cβsαsγ) + ur2(+cγsα + cαcβsγ )qr2(−cβcγsα − cαsγ) + ur2(+cαcβcγ − sαsγ)qr2(+sαsβ) + ur2(−cαsβ)qr3(+cαcγ − cβsαsγ) + ur3(+cγsα + cαcβsγ )qr3(−cβcγsα − cαsγ) + ur3(+cαcβcγ − sαsγ)qr3(+sαsβ) + ur3(−cαsβ)

(10)This system of equations can be solved using a nor-

mal non-linear least-squares minimization by searchingthe (α, β, γ) space for minima in squared error. Withthe measured Stokes vector (Si), the Rayleigh sky inputvector (Ri) and a rotation matrix (Rij) we define theerror (ǫ) as:

ǫ2(α, β, γ) =2∑

i=1

3∑

j=1

[Si − RiRij(α, β, γ)]2 (11)

For n measurements, this gives us 3 × n terms in thisleast squares problem. The Euler angles give identicalrotation matrix elements under the exchange of (α, β, γ)with ( -180◦+α, -β, -180◦+γ) as well as with additionalmultiples of 2π. This direct solution is easily solvablein principle but the ambiguities make implementing thissolution with existing software languages more cumber-some than necessary.

As an alternative method to the direct least-squares so-lution for Euler angles, we can do a two-step process thatgives the same result but is much easier to implement.First we solve a system of equations for the Mueller ma-trix elements directly that are not subject to rotationalambiguity. With the Mueller matrix elements in hand,we can then perform a simple fit of a rotation matrixelements to the derived Mueller matrix terms. This two-step process allows us to have accurate starting guessesto speed up the minimization process, resolve Euler angleambiguities and to estimate error propagation.

When deriving the Mueller matrix elements of the tele-scope, one must take care that the actual derived matri-ces are physical. For instance, there are various matrixproperties and quantities one can derive to test the physi-cality of the matirx (Givens & Kostinski 1993, Kostinskiet al. 1993, Takakura & Stoll 2009). Noise and system-atic uncertainty might give over-polarizing or unphysicalMueller matrices such as the element above that is >1.By simply using Mueller matrix elements to fit a rotation

matrix, we avoid over-polarizing.Provided the arrays are indexed properly, the normal

solution for Mueller matrix elements can be computedvia the normal least-squares method. We re-arrange thetime-varying Rayleigh sky inputs to (Rij) for i indepen-dent observations and j input Stokes parameters. Themeasured Stokes parameters (Si) become individual col-umn vectors. The unknown Mueller matrix elements arealso arranged as a column vector by output Stokes pa-rameter (Mj). If we write measured Stokes parametersas (qmi , umi, vmi) and the Rayleigh input Stokes parame-ters as (qri , uri), we can explicitly write a set of equationsfor just two Mueller matrix elements:

Si =

(

qm1qm2qm3

)

= RijMj =

(

qr1 ur1qr2 ur2qr3 ur3

)

(

QQUQ

)

(12)

We have three such equations for each set of Muellermatrix elements. With this can express the residual er-ror (ǫi) for each incident Stokes parameter (Si) with animplied sum over j as:

ǫi = Si − RijMj (13)

The normal solution of an over specified system ofequations is easily derived in a least-squares sense us-ing matrix notation. The total error E as the sum of allresiduals for m independent observations we get:

E =

m∑

i=1

ǫ2i (14)

We solve the least-quares system for the unknownMueller matrix element (Mj) by minimizing the errorwith respect to each equation. The partial derivative ofEquation 13 for ǫi with respect to Mj is just the skyinput elements Rij . Taking the partial with respect toeach input Stokes parameter we get:

∂E

∂Mj= 2

X

i

ǫi∂ǫi

∂Mj= −2

X

i

Rij

Si −

X

k

RikMk

!

= 0

(15)

We have inserted a dummy sum over the index k. Mul-tiplying out the terms and rearranging gives us the nor-mal equations:

∑

i

∑

k

RijRikMk =∑

i

RijSi (16)

Which written in matrix notation is the familiar solu-tion of of a system of equations via the normal method:

M =R

TS

RTR(17)

This simple equation is very easy to implement witha few lines of code provided your observation times arechosen to give you a range of input states for a well-conditioned inversion. The noise properties and inver-sion characteristics of this equation can be calculated inadvance of observations and optimized. We can write

Mueller Matrices Using Sky Polarization 9

the matrix A with an implied sum over i observationsfor each term as:

A = RTR =

(

qriqri qriuriqriuri uriuri

)

(18)

The solution to the equations for the three sets ofMueller matrix elements can be written as:

Mj =

(

QQUQ

)

= A−1(

qriqmiuriqmi

)

(19)

Mj =

(

QUUU

)

= A−1(

qriumiuriumi

)

(20)

Mj =

(

QVUV

)

= A−1(

qrivmiurivmi

)

(21)

As an example, if we compute the inverse of A andmultiply out A−1 for the QQ term we can write:

QQ =(qriqmi)(uriuri) − (uriqmi)(qriuri)

(qriqri)(uriuri) − (qriuri)(qriuri)(22)

In this manner, we can easily implement the usual ma-trix formalism with a time-series of daytime sky obser-vations to measure six Mueller matrix elements.

With the Mueller matrix terms in hand, solving foraccurate initial Euler angle guesses is straightforward.As an example, by using QV and UV from one can solvefor α and β directly:

tan(α) = −QV

UV(23)

sin(β) =QV

sin(α)=

−UV

cos(α)(24)

If one writes cγ as x then uses coefficients a =QQ

sαsβ

and b = cαsαsβ

one can numerically solve a quadratic for

and estimate of γ:

0 = (b2 + 1)x2 − (2ab)x + (a2 − 1) (25)

The sign ambiguity in solving for γ=cos−1(x) is re-solved by comparing the sign of the computed Muellermatrix elements with those of the rotation matrix de-rived with these estimated Euler angles. The estimateshown here only utilized QQ, QV and UV but provideda guess for minimization routines typically accurate tobetter than 2◦ in our data set.

In our two-step method, we next solved for the best fitEuler angles by doing a normal non-linear least-squaresminimization by searching the (α, β, γ) space around ourinitial guesses for minima in the summed squared error:

E(α, β, γ) =2∑

i=1

3∑

j=1

[Mij − Rij(α, β, γ)]2 (26)

By using the built-in IDL routines POWELL orAMOEBA, we can construct the error function and findthe minima. Both of these minimization routines require

estimated starting points but give the same best-fit Eu-ler angles to better than one part per million. Note thatby choosing Euler angle guesses in the proper quadrants,the derived rotation matricies avoid the ambiguity of (α,β, γ) with ( -180◦+α, -β, -180◦+γ). The differences be-tween rotation matrix elements and Mueller matrix ele-ments for our observations typically much less than 0.1.The two-step solution gives us the same Euler angles asthe direct least squares solution typically better than 0.2◦

and allows for computationally inexpensive and easy im-plementation of the least-squares minimization.

Fig. 10.— This Figure shows the squared error (ǫ2) for the two-step solution derived using Equation 26. Each panel shows a slicethrough the volume (α, β, γ) at four values of α. Error increasesfrom dark to light colors. Good solutions are found at α ∼ 0 andat α = ±180.

In order to verify this method, we computed the errorfunctions E for both the direct solution and the two-stepsolution and verified the minima with a direct search ofthe entire Euler angle space. We created a grid of Eulerangles from -180◦ to 180◦ (using 2 degree increments)and computed the error E as in Equation 11. Figure 10shows the derived errors as functions of β and γ for fourdifferent values of α. Each panel is color coded to showthe square error E increasing from dark to light. The Eu-ler angles found using the two-step method are identicalto those found with the direct solve within the crude 2◦

sampling we chose for this brute-force search. The errorfunctions are always continuous with only two minima inthe search volume showing convergence is easily achievedusing any minimization scheme.

10 Harrington et al.

6. APPENDIX: PLANNING SKY OBSERVATIONS USINGTIME DEPENDENCE FOR MODULATION

In order to use this technique efficiently, one must con-sider the noise propagation when inverting a sequenceof sky observations to derive telescope Mueller matrixproperties. There is an analogy between using the time-dependent Rayleigh sky to measure polarization proper-ties of the a telescope and the retardances chosen to cre-ate an efficient modulation scheme for polarization mea-surements. The underlying mathematics is the same andone can easily derive the requirements on the observa-tions needed to derive high-accuracy telescope Muellermatrix measurements. Effectively, one is attemptingto derive properties of a matrix through what differentgroups call demodulations, inversions or deprojections.We outline here the analogy between polarimetric modu-lation and calculating noise properties with the Rayleighsky to determine a quality observing sequence for deriv-ing telescope Mueller matrices.

Polarimeters modulate the incoming polarization statevia retardance amplitude and orientation changes. Thisretardance modulation translated in to varying inten-sities using an analyzer such as a polarizer, polarizingbeam splitter or crystal blocks such as Wollaston prismsor Savart plates. These modulation schemes can varywidely. For example, Compain et al. 1999, del ToroIniesta & Collados 2000, De Martino et al. 2003, Na-garaju et al. 2007 and Tomczyk et al. 2010 overviewoptimal schemes, error propagation and outline schemesto maximize or balance polarimetric efficiency and cre-ate polychromatic systems. There have been many im-plementations of achromatic and polychromatic designsin both stellar and solar communities (cf. Gisler et al.2003, Hanaoka 2004, Xu et al. 2006). In the notationof these studies, the instrument modulates the incomingpolarization information in to a series of measured inten-sities (Ii) for i independent observations via the modu-lation matrix (Oij) for j input Stokes parameters (Sj):

Ii = OijSj (27)

This is exactly analogous to our situation where wehave changed the matrix indices to be i independentStokes parameter measurements for j different sky inputStokes parameters:

Si = RijMj (28)

In most night-time polarimeters, instruments choose amodulation matrix that separates and measures individ-ual parameters of the Stokes vector:

Oij =

1 +1 0 01 −1 0 01 0 +1 01 0 −1 01 0 0 +11 0 0 −1

(29)

Other instruments choose only four measurements: theminimum number of exposures required to measure theStokes vector. In these schemes, one can balance theefficiency of the measurement to minimize the noise oneach Stokes parameter:

O =

1 + 1√3

+ 1√3

−1√3

1 + 1√3

−1√3

+ 1√3

1 − 1√3

+ 1√3

+ 1√3

1 − 1√3

−1√3

−1√3

(30)

One recovers the input Stokes vector from a series ofintensity measurements by inverting the modulation ma-trix (O) provided it is a square and multiplying this in-verse by the measured intensities. If the matrix is notsquare, one can simply solve the over-specified system ofequations via the normal least squares formalism:

S =O

TI

OTO(31)

Even for non-square matrices, we can define a demod-ulation matrix that captures the transfer properties ofthe modulation scheme:

Dij = [OTO]−1OT (32)

In our case, the Rayleigh sky input parameters becomethe modulation matrix (Oij = Rij) and the formalismfor noise propagation developed in many studies such asdel Toro Iniesta & Collados 2000 apply.

As an example, we will use the Jan. 27th 2010 Muellermatrices derived in Figure 5. The sky polarization ro-tated by about 33◦ during this period and the three cal-culated Rayleigh sky inputs scaled to 100% polarizationare:

Rij =

(

qr1 ur1qr2 ur2qr3 ur3

)

=

(

+0.980 +0.201+0.847 +0.534+0.716 +0.698

)

(33)

If each measurement has the same noise σ and thereare n total measurements then the noise on each demod-ulated parameter (σi) becomes:

σ2i = nσ2

n∑

j=1

D2ij (34)

And the efficiency of the observation becomes:

ei =

n

n∑

j=1

D2ij

− 12

(35)

For instance, the normal modulation sequence of equa-tion 29 used by most night time spectropolarimeters givesn=6 and e2i = [1,

13 ,

13 ,

13 ]. The efficiency balanced scheme

gives the same relative efficiencies as the normal modu-lation scheme but uses only 4 exposures instead of 6. Inthe case of our Jan. 27th observations considering onlyqu terms our demodulation matrix is:

Dij = [RTR]−1RT =

(

+1.23 +0.16 −0.48−1.49 +0.43 +1.54

)

(36)

With this demodulation matrix the efficiency for uis only 60% worse than q and we have efficiencies ofei = [0.43, 0.26] computed with Equation 35. The

Mueller Matrices Using Sky Polarization 11

Mueller matrix derived from these Rayleigh sky obser-vations will have similar noise properties between Q andU terms.

One must take care with this technique to build upobservations over a wide range of solar locations so thatthe inversion is well conditioned. The path of the sunthroughout the day will create regions of little input skyStokes vector rotation causing a poorly constrained in-version with high condition number. For instance, atour location in the tropics the sun rises and sets with-out changing azimuth until it rises quite high in the sky.We are constrained to observing in early morning andlate evening with the dome walls raised since we maynot expose the telescope to the sun. This causes inputvectors at east-west pointings to be mostly q orientedwith little rotation over many hours. Observations atother times of the year or at higher solar elevations arerequired to have a well conditioned inversion. One caneasily build up the expected sky input polarizations at agiven observing site with the Rayleigh sky polarizationequations. Then it is straightforward to determine themodulation matrix and noise propagation for a plannedobserving sequence to ensure a well-measured telescopematrix with good signal-to-noise.

12 Harrington et al.

7. APPENDIX: LOVIS SPECTROPOLARIMETERCHARACTERIZATION

In order to expand the capabilities of our spectropo-larimeter, we have adapted the optics for use at low spec-tral resolution. The polarimeter unit for the spectro-graph is a Savart plate providing dual-beam analyzingmounted behind the entrance slit with either two rotat-ing achromatic retarders or two liquid-crystal variable re-tarders (LCVRs) outlined in Harrington & Kuhn 2008;Harrington et al. 2010. We have modified the HiVISechellé housing to allow a flat mirror to be mounted by-passing the echellé without disrupting the optical path.We call this new mode LoVIS.

Fig. 11.— This Figure shows example LoVIS data when observ-ing a spectrophotometric standard star on Sept. 4th 2009. Wave-length increases from right to left. The Na D lines, Hα and theatmospheric A-band can be seen by eye near spectral pixels 2300,1400 and 900 respectively.

With a flat mirror replacing the echellé, the only dis-persive element in the spectrograph is the cross-disperserworking in 1st order. Only a single order is imaged onthe detector making the illuminated region of the devicemuch smaller. Figure 11 shows a raw data frame from aspectrophotometric standard star (HR7940) taken withLoVIS in spectropolarimetric mode using the Apogee de-tector. There is a single spectral order seen with orthogo-nally polarized spectra being displaced roughly 70 pixelsspatially by the Savart plate. Wavelength increases fromright to left with the Hα line seen near spectral pixel1400 and the 6870Å atmospheric band seen near pixel900. We have two separate cross-dispersers available onthe rotation stage blazed for 6050Å and 8600Å ( Hodappet al. 2000, Thornton et al. 2003).

The new LoVIS mode gives us much higher sensitivityand allows for much faster calibrations at spectral res-olutions of 1,000 to 3,000 depending on the slit. The1.5“ slit had spectral resolutions of 850 and 990 derivedfrom Thorium-Argone (ThAr) lines at 5700Å and 7000Å.These lines were spectrally sampled with roughly 9.5pixels per Gaussian Full-Width-Half-Max at each wave-length. The IDL reduction scripts we wrote for HiVISoutlined in Harrington & Kuhn 2008 were adapted tothis new low-resolution data. Figure 12 shows the ex-tracted spectrum from the raw data of Figure 11.

Fig. 12.— This Figure shows an example extracted spectrumof the spectrophotometric standard star HR7950 taken on Sept.4th 2009. The Hα, Na D lines and the atmospheric A-band areidentified.

This change in resolution allows us to observe muchfainter targets or observe sources at a much higher ca-dence. When observing the daytime sky with AEOS andLoVIS, we can achieve a spectropolarimetric precision of0.1% with a 5 second exposure time using the 1.5“ slit.The polarimetric images sequences finish in one minuteand the bulk of the overhead is from rotating the achro-matic wave plates in the typical 6-exposure sequence.

To calibrate this new polarimetric mode and the tele-scope polarization we used LoVIS to gather a wide rangeof sky and calibration observations from August 2009to February 2010. All observations were taken with theApogee detector described in Harrington et al. 2010.We have sky observations taken solely at the Zenith aswell as observations on a grid of altitudes and azimuths.On several occasions we did observations for 3 or 4 hourscontinuously before sunset to accumulate a wide rangeof polarization inputs at multiple pointings.

In order to decouple the LoVIS and AEOS polariza-tion measurements, we did a number of tests with ourpolarization calibration unit mounted at the slit as wellas at the entrance port to our instrument at the calibra-tion lamp unit. We did tests both without and with theimage rotator inserted in the the path. This image rota-tor is the source of most of the cross-talk in the LoVISinstrument. Removing it makes the polarization prop-erties of LoVIS much more benign as shown in Figures15 and 16 of Harrington et al. 2010. There is a trans-missive window that separates the coudé rooms from thecentral optical path to the telescope. This window isdirectly after the last fold mirror of the telescope, m7,and can be set to either BK7 or infrasil. This windowcan also have a polarimetric effect on the beam and bothwindows were also tested. Effectively we find that thepolarization properties derived for HiVIS are similar toLoVIS and that we can efficiently reproduce polarizationmeasurements with both rotating achromatic wave platesand LCVR modulators.

We also measured the induced polarization of the en-tire system when illuminated by a point-source utilizingunpolarized standard stars. As an example, we show ob-servations of many unpolarized standard stars at many

Mueller Matrices Using Sky Polarization 13

Fig. 13.— This Figure shows the telescope induced polarization.We measured quv spectra for many different unpolarized standardstars at many different pointings observed with LoVIS on Sept. 5th

2009. Each Stokes parameter is shown in percent. The inducedpolarization is at the few percent level with some dependence onwavelength.

pointings from September 5th 2009 in Figure 13. Themeasured polarization is typical of results obtained onany individual night. Consistent with our previous HiVISmeasurements from Harrington & Kuhn 2008, the in-duced polarization is around a few percent, has somemild chromatic properties and does depend on telescopefocus, the seeing and stellar location on the slit.

In summary, the polarization properties of LoVIS aresimilar to HiVIS. The induced polarization is below afew percent. The polarization calibration optics givevery pure inputs that are reproduced with both achro-matic wave plate and LCVR modulators. This new low-resolution mode works efficiently and allows us to obtaincalibration observations at a much faster cadence.

14 Harrington et al.

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