JSTOR

1. INTRODUCTION

We first describe the High‐Resolution Visible Spectrograph (HiVIS; § 2) and the design of the spectropolarimeter (§ 3). We then show some examples of new data reduction packages we have developed (§§ 4 and 5) and discuss the observations of unpolarized and polarized sources. We use these measurements to construct a polarization calibration model for the instrument and telescope (§ 6) and discuss the polarization calibration methods and ways in which they can be applied to future observations with the HiVIS spectropolarimeter.

2. THE AEOS SPECTROGRAPH

The Advanced Electro‐Optical System (AEOS) telescope is a 3.67 m altitude‐azimuth telescope located on Mt. Haleakala, Maui, Hawaii. The spectrograph uses an f/200 coudé optical path with seven reflections (five at ∼45° incidence) before entering the optics room. This is illustrated in the telescope optical ray trace in Figure 1, made using Zemax (industry‐standard optical design software). The optics room, shown in Figure 2, contains a visible (0.5–1.0 μm) and IR (1.0–2.5 μm) cross‐dispersed echelle spectrograph. The visible spectrograph has two gratings that cover 500–770 and 640–1000 nm at resolutions of 12,800 to 48,900. The common fore optics includes three fold mirrors, followed by a collimator, K cell (image rotator), another fold mirror, tip‐tilt correction mirror, and a reimaging spherical mirror. A movable pick‐off mirror allows switching between the visible and IR channels (Thornton 2002; Thornton et al. 2003).

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Fig. 1.— Zemax schematic of the coudé telescope shown pointing at the zenith, shortened for illustration purposes. These are the seven coudé reflections (m1 to m7) before the optics room.

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Fig. 2.— View of the AEOS optics room. The common fore optics are in the center of the figure, with the visible (bottom) and IR (top) arms of the spectrograph on either side. The light enters the room from center right after bouncing off the last coudé mirror (m7; not shown).

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The spectropolarimeter is designed for use with the visible arm of the AEOS spectrograph, which is shown in Figure 3. From the pick‐off mirror (bottom left), the beam passes through the slit in the slit mirror, reflects off fold mirror 1, the collimator, echelle, collimator, long mirror, collimator, fold mirror 2, and finally the cross‐disperser before being imaged on the CCD by a five‐lens camera. A slit‐viewer camera is focused on the slit mirror, which allows real‐time monitoring of the object. The spectropolarimeter was mounted just downstream of the slit, near the image plane. The plate scale is 0 16 pixel⁻¹. The cross‐disperser’s “red” setting was used for all observations (nominally 6375–9680 Å). The spectrograph uses two 2K × 4K Lincoln Laboratory high‐resistivity CCDs mounted side by side with a 15 pixel gap to form a 4K × 4K array. The CCD was set to rebin 2 × 2 so that the two output images are each 1024 × 2048 pixels.

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Fig. 3.— View of the AEOS optical bench. The spectropolarimetry module is mounted just downstream of the slit, near the bottom left corner of the figure.

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3. THE SPECTROPOLARIMETER

The spectropolarimetry module for the AEOS spectrograph consists of a rotating achromatic half‐wave plate and a calcite Savart plate. The Savart plate is two crossed and bonded calcite crystals that separate the incoming uncollimated beam into two parallel, orthogonally polarized beams displaced laterally by 4.5 mm. The half‐wave plate allows the rotation of the linear polarization of the incoming light to provide complete measurement of the linear polarization of the incident light. The beams are displaced in the spatial direction (along the long axis of the slit).

In a Savart plate, the two beams (extraordinary and ordinary rays) are swapped between the two crossed crystals, so both displaced rays have the same optical path length through the calcite. This design contrasts with other spectropolarimeters, which use Fresnel rhomb retarders and a Wollaston prism to separate the polarization states (Petit & Donati 2003; Goodrich & Cohen 2003; Donati 2001, 2004; Donati et al. 1999; Kawabata et al. 1999; Manset & Donati 2003; Goodrich et al. 1995).

The small deviation in the image plane splits each spectral order detected on the CCD into two orthogonally polarized orders separated by a few pixels, from which orthogonally polarized spectra are extracted separately. For example, the red cross‐disperser setting originally shows 19 orders across the CCD, but after inserting the Savart plate, it shows 38 (two polarizations for 19 orders), as illustrated in Figure 4.

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Fig. 4.— Before‐and‐after images from insertion of the spectropolarimetry module. Left: Cropped raw image of a stellar spectrum without the polarimeter (19 orders on the chip). Insertion of the spectropolarimetry module splits each order into two orthogonally polarized orders. Right: Same region of the image, with the same star and with the polarimeter in place, doubling each order (2 × 19 orders).

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These components were added to the Zemax, Inc., optical ray‐trace model of the spectrograph to assess the aberrations and determine the specifications of the optics in order to produce the proper polarized order separation on the CCD (∼25 pixels). The aberrations induced by the wave plate and Savart plate were predicted to be smaller than half a pixel, leading to insignificant image degradation. A deviation of 4.5 mm at the slit was found to separate the polarized orders by ∼24.5 pixels on the CCD. A 60 mm Savart plate produces this separation (0.075 × 60 mm).

We then purchased and tested a Savart plate (Halbo Optics, Inc.) with a 15 mm aperture, an achromatic half‐wave plate (Bolder Vision Optik, Inc.), and a motorized closed‐loop rotation stage. The half‐wave plate retardance was measured with a spectrograph and two crossed polarizers. The retardance varied roughly quadratically with wavelength at the 5% level, from 500 to 800 nm (the useful range of our polarizers), and was 0.49 waves retardance at Hα. The transmission of the Savart plate was above 95% for these wavelengths.

After calibration and testing of the half‐wave plate, the Savart plate, and the rotation stages, the instrument was assembled on the spectrograph on a movable plate, which allows the spectropolarimeter to be inserted into or removed from the beam in under a minute. We mounted a decker upstream of the slit to reduce the slit length from 15 to ∼7 to minimize stray light introduced from the displacement. This also ensures polarized order separation on the CCD. Flat fields taken after installation showed that the polarized orders were each ∼30 pixels across (∼4 ), with a separation of 7 pixels between each polarized beam.

To test the equipment on the telescope, we mounted a linearly polarizing filter just upstream of the spectropolarimeter, in order to feed 100% linearly polarized light to the spectropolarimeter, and detected 95%–98% polarization over the useful range of the polarizer (500–800 nm, orders 0–12). This shows that we can efficiently detect linearly polarized light with the spectropolarimeter.

4. OBSERVING WITH AEOS: MEASURING LINEAR POLARIZATION

The Stokes parameters Q and U fully specify the linear polarization state of light (Collet 1992). The parameter Q is the difference between horizontal and vertical polarizations, and U is the difference between +45° and −45° polarizations. Typically, the fractional Stokes parameters and , where I is the total intensity, are presented to quantify the fraction of light that is polarized. These are measured by taking the fractional difference between the orthogonally polarized spectra.

The observing sequence for the AEOS spectropolarimeter is to take spectra at 0°, 22 5, 45°, and 67 5 rotations of the half‐wave plate with respect to the axis of the Savart plate, hereafter called orientations 1, 2, 3, and 4. Incoming light that is linearly polarized at some angle to the fast axis of the half‐wave plate will have its linear polarization states rotated by twice that angle. For example, incoming light polarized at 22 5 with respect to the wave plate's axis will exit polarized at 45° (Stokes ). Since the axis of the Savart plate remains fixed, the rotation of the wave plate by 45° will move linearly polarized light from one polarized beam to the other, swapping their location on the CCD and reversing their sign in the calculation of fractional polarization ( ). This allows for the cancellation of systematic errors (derivative, misalignment, CCD response, etc.) in the Stokes Q and U calculations. The wave plate angle can be controlled to 50 .

The observing sequence is illustrated in Figure 5. A linear polarizer was mounted upstream of the spectropolarimeter and aligned with the wave plate’s axis, and a normal data set (four wave plate orientations) was obtained. Note how the bright order is swapped (top to bottom) with a 45° rotation of the wave plate, and how the 22° and 67° orientations show uniform intensity for both spectra. The rotation of the wave plate will allow the linearly polarized light to be sampled by systematically moving it from one polarized order to the other.

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Fig. 5.— Images of a 100% polarized source at four wave plate rotations. A polarizer was mounted next to the slit to show how rotation of the wave plate swaps the position of polarized light on the CCD. Successive 22 5 rotations take the light from the top order to the bottom order and back again. Since a 45° rotation moved the light from the top to the bottom, instrumental systematic errors can be removed. Imperfect alignment with the Savart plate's axis produces the slight asymmetry in the right‐hand panels.

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The linear polarization measurements in this paper are defined as follows: Let a and b be the orthogonally polarized spectra (bottom and top, respectively) with the wave plate at 0°, and let c and d be the orthogonally polarized spectra with the wave plate at 45°. A measurement of a fractional Stokes parameter q is defined as

Since a 45° rotation of the wave plate will swap the position of the polarized light incident along the original wave plate axis on the CCD (bottom to top, or vice versa) but not the unpolarized light or linearly polarized light incident at 22 5 or 67 5, this measurement is only sensitive to Stokes q. The subtraction of the two terms allows us to cancel systematic errors resulting from any misalignment of the polarized orders, since each term is an independent measurement of Stokes q. Measurement of Stokes U uses the same formula, but with the wave plate at 22 5 and 67 5. Since this is a fractional quantity, any correction applied equally to each order and chip, such as spectrophotometric calibrations, vignetting corrections, or sky‐line subtractions, will not change the polarimetry in any significant way. That is to say, no standard calibrations affect the spectropolarimetry.

A frame from the slit‐viewer camera is shown in Figure 6, giving the general geometry of the slit, decker, and room as seen by the camera. The slit is parallel to the floor, with +U parallel to the slit and +Q rotated 45° counterclockwise, pointing to the upper right. Since the image rotator was not used, the projection of the slit on the sky is a function of pointing and is not constant.

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Fig. 6.— Orientation of Q and U and the projection onto the sky as seen by the slit‐viewer camera at altitude 45°, azimuth 225°. The Stokes parameters are fixed to the instrument, whereas the projection on the sky changes with time when the image rotator is not used.

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The simultaneous imaging of orthogonal polarization states also allows for a more efficient observing sequence and a reduction of atmospheric and systematic effects. Since both polarized spectra are imaged simultaneously with identical seeing, a single polarization measurement (Q or U) can be done with a single image, and consistency between images is easy to quantify. Using the fractional polarizations measured as the difference between the orthogonally polarized spectra divided by the sum (q and u), we can calculate the degree of polarization (P) and the position angle of polarization (θ) as

The polarization angle is measured in the telescope frame, and a projection of the slit’s position angle onto the sky is done with the K cell. The AEOS image rotator was not used, so the absolute angle of the slit on the sky was not fixed. However, the orientation of the K cell mirrors in the optics room was fixed, making the polarization calibration much easier.

We have written reduction scripts in IDL to reduce the data and compute the spectropolarimetry as described above. Each spectral order is located by using a second‐order polynomial fit to the intensity maximum of a calibration‐star image. This template for the order positions is then aligned with the data. A median order profile is then obtained for each order. A least‐squares fit to this profile at each point along the order is used to calculate the intensity. Standard flat‐fielding, bias‐subtraction, cosmic‐ray removal, and wavelength‐calibration routines were also written. We found that a 180 s exposure of an star gave spectra with a signal‐to‐noise ratio S/N ∼ 50–200. The S/N changes across each order, being highest where the disperser’s efficiency is highest (e.g., the middle left of each panel in Fig. 7). For example, the bottom order in Figure 4 goes from a S/N of 180 at 649 nm in the bottom left to a S/N of 110 at 657 nm in the bottom right, and the top order goes from a S/N of 110 at 956 nm in the top left to a S/N of 60 at 968 nm in the top right.

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Fig. 7.— Scattered‐sunlight data set at azimuth 180°, altitude 65°, taken at sunset. Wave plate orientations are the same as in Fig. 5, starting with 0° in the upper left.

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We are neglecting circular polarization in this paper because the effects are typically much smaller than linear polarization, as discussed below, and we have no sensitivity to circularly polarized light.

It should be noted that many spectropolarimeters show a polarimetric “ripple” effect caused by interference between multiple reflections in the wave plate (e.g., Aitken & Hough 2001). We do not see such a ripple in our observations. We have observed to 0.3% polarimetric accuracy in a single frame at full resolution, and we are currently investigating higher S/N observations to verify the absence of this ripple.

5. SPECTROPOLARIMETRY OF SCATTERED SUNLIGHT

Scattered sunlight is highly polarized at a scattering angle of 90°. We observed scattered twilight at many different pointings from 2005 June 26 to July 7; a sample data set is shown in Figure 7. Since the twilight filled the slit, the figure shows the width of each polarized order on the CCD (∼27 pixels). Wavelength increases to the right for each order and toward the top with increasing order number. The solar Hα line is obvious in the bottom right corner of each panel. The atmospheric A and B bands appear on the bottom left side in the fourth and eighth orders.

As an illustration of the new data reduction software routines, a spectrum from one polarized order in a single twilight exposure is shown in Figure 8. The AEOS spectra are a good match to the spectra in a standard solar spectral atlas (Wallace et al. 1998).

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Fig. 8.— Solar spectrum from the bottom polarization state for the first order (650–657 nm), as reduced with the new reduction software.

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The polarization reduction package was applied to a twilight data set, and the results are shown in Figure 9. The polarization spectra have been averaged to a 200 times lower spectral resolution, for ease of plotting. The top curve is the degree of polarization, which starts at around 25% at 650 nm, rises to 70% by 700 nm, and stays flat out to 950 nm. Another interesting feature is the rotation of the plane of polarization across the CCD. Note how Stokes parameters q and u track each other. At 650 nm, the magnitude of both q and u is 10%. At 800 nm, % and . Since the angle of polarization is , this angle has rotated by 90°. The scattered‐sunlight observations are discussed in more detail below.

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Fig. 9.— Scattered‐sunlight polarization at azimuth of 180° and elevation 65°. Polarization of scattered sunlight is maximum at a 90° scattering angle and at sunset; this is the arc from north to south through the zenith. The polarization spectra have been averaged 200:1 for ease of plotting. The graph plots q (diamonds), u (triangles), and P [(Q² + U²)^1/2] (plus signs). The top curve, P, begins at 25% at 650 nm, rises to 70% by 750 nm, and stays flat to 950 nm. Stokes q and u are more sinusoidal, with a strong rotation of the plane of polarization [  tan⁻¹ (q/u)].

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6. POLARIZATION CALIBRATION: THE AEOS MODEL

The next step in instrument development was the calibration of the instrumental polarization response. Absolute polarization at all wavelengths requires very careful calibration of the telescope. Internal optics can induce or reduce polarization in the beam and also rotate the plane of polarization ( ). All of these effects are functions of wavelength, altitude, and azimuth. Since AEOS is an alt‐az telescope, the mirror positions change as the telescope tracks, inducing polarization effects with pointing. For AEOS, there are five ∼45° reflections before the coudé window, two of which change their relative orientation (see Fig. 1). The first rotation is between the tertiary mirror, m3, which is tied to the altitude axis of the telescope, while the next mirror, m4, is fixed to the mount. The second relative rotation occurs because the coudé mirror, m7, is fixed to the building, while m6, just below the primary, is tied to the mount, which rotates with azimuth.

Absolute spectropolarimetric calibration is difficult, due to a lack of spectropolarimetrc standards. Usually, one observes polarimetric standards that have a precise value averaged over some bandpass (Breger & Hsu 1982; Hsu & Breger 1982; Bastien 1988; Clarke & Naghizadeh‐Khouei 1994; Gil‐Hutton & Benavidez 2003). The calibration and creation of a telescope model is done by measuring unpolarized standard stars and polarized sources at many pointings, to calibrate the effects of the moving mirrors.

If we ignore circular polarization, a simple and self‐consistent way to describe the telescope's response is with a three‐variable system that includes (1) the difference in absorption between two orthogonal directions (mirror reflection axes), (2) the angle of those directions with respect to some fixed axis, and (3) the wavelength dependence of those two parameters, all of which are functions of altitude and azimuth. This is analogous to describing the telescope's response as an elliptical polarization at each pointing and wavelength (Collet 1992). In general, the telescope will have different responses to linearly polarized light, since the absorption coefficients of the optical components vary with wavelength and the differential absorption is a function of the angle of the incident polarization with respect to the mirror/component axes. While our models can calculate the full Mueller matrix as functions of altitude, azimuth, and wavelength, we cannot measure all of these terms. However, we can measure certain parts of the Mueller matrix, given proper sources (polarized standards, unpolarized standards, and scattered sunlight).

Unpolarized standard stars were the first source we used to calibrate the telescope. Observations of these standards at as many pointings as possible allows one to construct a model of the telescope's response to unpolarized light and to quantify the telescope‐induced polarization (but not the depolarization or plane rotation, which requires a linearly polarized standard).

Polarized sources were the last calibration source we used, and the most difficult. Using a polarized source allows one to measure the depolarization and rotation of the plane of polarization, all of which are, in principle, orientation dependent. Polarized standard stars are polarized in their continuum light, but the constant value quoted in standard references is an average polarization in a relatively line‐free, wide‐bandpass image. The degree of polarization is usually not more than a few percent, making instrument calibration difficult with these sources. A calibration method that we explore below uses twilight (scattered sunlight) as a linearly polarized source with a roughly known degree of polarization.

6.1. Flat‐Field Polarization: Fixed‐Optics Polarization

All of the moving mirrors in the AEOS telescope are located upstream of the polarizing optics. As an illustration of the polarization induced by the reflections between the halogen flat‐field lamp ( ) and the spectropolarimeter, Figure 10 shows the polarization reduction applied to a set of flat‐field frames with the four wave plate orientations. The flat‐field lamp is located just after the coudé window, on the far right side of Figure 2.

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Fig. 10.— Flat‐field polarization analysis, averaged 200:1 for clarity. The polarization (plus signs) begins at 650 nm, being dominated by u (triangles), but begins to rise as −q (diamonds) gets larger.

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The flat fields show a significant polarization that changes with wavelength. At 650 nm, the induced polarization is dominated by the +U term, but the U and −Q terms become equal at 900 nm. The position angle varies almost linearly across all wavelengths, while the degree of polarization rises from 2% at 650 nm to 6% at 950 nm. This illustrates the polarization induced by the 11 fixed reflections in the optics room.

6.2. Unpolarized Standard Stars: Telescope Mirror–induced Polarization

Many unpolarized standards were observed in 2004 November and 2005 July, totaling 19 data sets (152 spectra). A plot of q and u for all the unpolarized standard stars, where each 1000 pixel order has been rebinned to 1 data point, is shown in Figures 11 and 12. They show the induced polarization as a function of pointing and wavelength. There is a fairly consistent trend between all stars, where q goes from 0% to 3% at 650 nm, falling to between −2% and +2% at 950 nm, and u starts between 0% and −6% at 650 nm and then rises to between −2% to +3%.

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Fig. 11.— Stokes q, rebinned 1000:1, for the unpolarized standard star observations in Table 1. Most observations show similar trends with wavelength (q falling).

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Fig. 12.— Stokes u, rebinned 1000:1, for the unpolarized standard star observations in Table 1. All observations show similar trends with wavelength (u rising).

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Calibration of the induced polarization to any future observations is done by observing the unpolarized standard stars at different pointings and creating a map of the telescope's response by projecting these measurements in alt‐az space. This map can then be interpolated to any pointing one wishes to calibrate, the ultimate aim being to have a high‐resolution map of the telescope’s polarization response.

6.3. Examples of Unpolarized Standard Star Spectropolarimetry: Changes with Pointing

The telescope‐induced polarizations for the first three of the standards in Table 1 are given in Figures 13–15. Each star shows different polarization and illustrates how the induced polarization varies with wavelength and pointing, since each star is unpolarized. These stars cover declinations of −7°, +27°, and +42°, allowing us to measure the induced polarization at many different azimuths and altitudes, from far north to far south.

TABLE 1 Unpolarized Standard Star Observations

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Fig. 13.— Percentage degree of polarization for the unpolarized standard star HD 125184. The progression in time at 650 nm from the bottom curve to the top curve is 7:20, 8:00, 5:55, and 5:45 UT.

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The spectropolarimetry for HD 125184 is shown in Figure 13. It shows moderate change with wavelength (∼1.5% increase in the blue and a decrease in the red). The change in altitude is about 15° (63°–49°), and the azimuth changes by 55° (175°–230°).

HD 114710 spectropolarimetry for many different pointings is shown in Figure 14. The spectropolarimetry does not show much change with pointing, but it does show a strong change with wavelength at all pointings, varying from 1% to 6% in a roughly quadratic way. The change in azimuth was 20° (76°–44°), and the altitude changed by 30° (309°–290°).

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Fig. 14.— Same as Fig. 13, but for HD 114710. The progression in time is not very significant. The curve that appears above the rest from 800 to 950 nm is at 7:00 UT.

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Spectropolarimetry of HD 142373 (Fig. 15) shows one of the strongest changes of polarization with pointing. During the observations, the star’s elevation does not change much (∼8°, 57°–64°), but the star's azimuth changes by about 65° (40°–335°), and the star transits in the middle of the data set, between 7:20 and 7:50 UT. The blue polarization (650 nm) drops by 3%, while the 850 nm polarization goes from near zero to 3%, and the 950 nm polarization rises by only 1%.

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Fig. 15.— Same as Fig. 13, but for HD 142373. The progression in time at 650 nm from the bottom curve to the top curve is 5:45, 7:20, 8:30, and 7:50 UT.

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Using these types of measurements for unpolarized standard stars at many different pointings, we can construct a response map for the telescope.

6.4. Telescope‐induced Polarization: An All‐Sky Map

Observations of the degree of polarization for all the unpolarized standards observed, averaged to a single measurement per order, are plotted on the alt‐az plane to illustrate the induced polarization for a single order (wavelength) as a function of pointing. Figure 16 shows the measurements for a single order (640 nm). There are 19 of these sky‐map surfaces (19 orders in alt‐az) that constitute the telescope response used for calibrating the instrument. Note how the induced polarization reaches a maximum of ∼6% at high altitude in the north and has a double‐valley structure falling to around 2% in the south (180°) at lower elevations (30°). This order has the highest induced polarization, meaning that the induced polarization is strongest at shorter wavelengths for this instrument. The longest‐wavelength order is shown in Figure 17. The peak still occurs at high elevation in the north but has a much lower magnitude of 3.5% and falls to 1% at low elevations in the south. These maps illustrate the wavelength dependence of the polarization as a function of position on the sky. As seen in Figures 13–15, the errors are 0.1%–1.0%, with a dependence on wavelength and pointing, making the calibration uncertain at this level. To calibrate future spectropolarimetry, we use an interpolation of all 19 surfaces (wavelengths) to the pointing of the telescope in the middle of each data set as the induced polarization calibration applied to the measurements. The error in this calculation is then calculated as the combined error in the nearest observations used in the interpolation. We will perform more observations in the future to improve the calibration map. It should be noted that for high‐resolution line polarimetry, where one is only interested in the polarization changes across a spectral line (e.g., Hα), this calibration serves as the baseline. Since the mirror‐induced polarization effects do not change significantly over a single spectral line, the calibration process is simplified by removing the wavelength dependence of all the corrections.

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Fig. 16.— Measured polarization of unpolarized standard stars for order 0 (640 nm) in the alt‐az plane. The peak is at 6%. This is a triangulation of 19 independent observations. The wavelength dependence for a few of the points in this plot are shown in Figs. 13–15.

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Fig. 17.— Same as Fig. 16, but for the measured polarization of unpolarized standard stars for order 18 (960 nm) in the alt‐az plane, with a peak near 4%.

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6.5. Zemax Model of Telescope Polarization: Mueller Matrices

We constructed a model of the telescope’s polarization response, using Zemax software to compare the predicted effects with our observations. Zemax allows the user to propagate arbitrarily polarized light through any optical design. We wrote programs to compute the Mueller matrix of the telescope for any altitude or azimuth by tracking polarized light through our telescope design. Using the optical constants for aluminum from the Handbook of Optical Constants of Solids II (Palik 1991), we traced the polarization of the telescope to the first coudé focus, after the second fold mirror in the optics room. This allows us to get a qualitative idea of the polarization effects induced by the relative rotation of the altitude and azimuth mirror pairs (m3‐m4 and m6‐m7 in Fig. 1). Since the effect of aluminum oxide on the mirror surfaces is very significant and is variable in time, and since the model does not include many of the common fore optics, this model is only an illustrative guide to what polarization effects will be induced by the relative rotation of the mirrors (Collet 1992).

To compute the Mueller matrix, we input pure polarization states ±Q, ±U, and ±V at 400 pupil points, propagate the light through the telescope, and average the resulting polarized light at the image plane. This is computed in 5° steps in altitude and azimuth, giving 1296 independent measurements to project onto a sky map. The telescope‐induced polarization in unpolarized incident light is then just the IQ and IU Mueller matrix terms added in quadrature. The result for 600 nm light is shown in Figure 18. The induced polarization is expected to be minimal when the mirrors are crossed, and maximal when they are aligned. Since the azimuth axis has two angles where the mirrors are aligned (parallel and antiparallel), we expect a double‐ridge structure in azimuth. The altitude‐axis mirrors are crossed at 0° so that the induced polarization rises with increasing altitude.

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Fig. 18.— Zemax model of telescope‐induced polarization using bare aluminum surfaces, with the predicted induced polarization projected on the sky at 600 nm.

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This model compares favorably with the measurements in Figure 16. Since we have only obtained 19 independent measurements of unpolarized standard stars, the sparsely sampled map in Figure 16 is not expected to match this much higher resolution model. The measured polarization shows a roughly double‐ridge structure in azimuth that rises with altitude, as expected. The measured polarization is roughly half the model’s prediction, but this is entirely plausible, given that the effect of the oxide layer on the mirrors can be drastic. We have computed this model for many wavelengths, and all show this qualitative shape.

The model also gives justification for neglecting circular polarization. The circular polarization cross‐talk terms, Mueller matrix elements IV and VI, are less than 5%. The VQ and VU terms, which describe the degree to which incident circular polarization becomes measured linear polarization, are larger, typically 0%–70%, as are the QV and UV terms. Since most astronomical objects have circular polarization that is an order of magnitude smaller than linear polarization, we do not anticipate significant calibration problems with this amount of circular to linear cross talk.

The last of the model predictions is that the QI and UI terms are small (0%–20%), leading to low depolarization, and the rotation of the polarization plane does not vary much with wavelength (QU and UQ terms change by a few percent).

Since the actual polarization properties of the mirrors are very sensitive to the optical constants of the aluminum, and since the aluminum oxide coating on all mirrors is not included in our models, these predictions are not expected to be quantitatively precise. However, the predictions do give a rough guide to the altitude, azimuth, and wavelength dependence. We have yet to monitor the polarization of the telescope as a function of time, in order to test the stability of these calibrations, although we hope to do this in the coming months.

6.6. Scattered Sunlight: A Linearly Polarized Source for Calibration

At twilight, scattered sunlight is a linearly polarized source with a roughly known position angle and degree of polarization at all pointings. This assumes single scattering of incident solar radiation. Singly scattered light is polarized orthogonal to the scattering plane, with the degree of polarization proportional to (Coulson 1988). Since the scattering geometry changes with pointing, so does the degree of polarization of the scattered sunlight (which reaches a maximum 90° from the Sun). Observing scattered sunlight at many different altitudes and azimuths allows us to check the instrument’s response to linearly polarized light. This scattered light is not a perfect source, however. The degree of polarization is also a strong function of the scattering properties of the air. In particular, aerosol content (salt, dust, molecules, water, etc.), optical depth (altitude and horizon distance), and reflections from the Earth’s surface (land and ocean) can significantly change the observed linear polarization (Lee 1998; Liu & Voss 1997; Suhai & Horváth 2004; Coulson 1988; Cronin et al. 2005). Our observations were done at high altitude, surrounded by a reflective ocean surface. Since the Sun was up during some of our observations, the polarized light reflecting off the ocean surface complicated our measurements. The broad trends in the degree of polarization we measured are in agreement with the singly scattered sky polarization model, but the linear polarization observed by many researchers has been shown to be a function of time, atmospheric opacity, aerosol size distributions, and composition, all of which vary significantly from day to day (Coulson 1988; Liu & Voss 1997; Suhai & Horváth 2004). Since we have no accurate, quantitative model of the atmospheric polarization to compare with our observations, we use them only as an illustration of the types of effects one can expect from a highly polarized source.

We obtained 52 polarization measurements (416 spectra) of scattered sunlight at altitudes and azimuths of 30°, 50°, 65°, 75°, and 90° and 0°, 90°, 180°, and 270°, respectively, to create a map of the degree of polarization and position angle for the sky for comparison with other observations (Horváth et al. 1998, 2002; Coulson 1988). These spectra show varied wavelength dependencies. A plot of the degree of polarization for some of the 52 sky polarization measurements is shown in Figure 19, and the position angle is shown in Figure 20. The rotation of the plane of polarization was a very strong function of wavelength and pointing, with typically 90° of rotation from orders 0 through 18.

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Fig. 19.— Degree of polarization of scattered sunlight at various combinations of altitudes (30°, 50°, 65°, 75°, and 90°) and azimuths (0°, 90°, 180°, and 270°). The degree of polarization of the scattered sunlight is intrinsically a function of wavelength and pointing, since the scattering geometry changes, making quantitative interpretation of the HiVIS degree of polarization difficult; however, there is good correlation with theoretical skylight polarization models.

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Fig. 20.— Calculated angle of polarization [  tan⁻¹ (Q/U)] for scattered sunlight at various combinations of altitudes (30°, 50°, 65°, 75°, and 90°) and azimuths (0°, 90°, 180°, and 270°).

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As with the unpolarized standards, the scattered‐sunlight polarization measurements can be plotted on the altitude‐azimuth plane. Figure 21 shows the measured degree of polarization projected on the sky for order 14 (860–870 nm). Since scattered sunlight reaches a maximum polarization 90° away from the Sun, and since these measurements were done at sunset, the degree of polarization is maximum along azimuths of 0° and 180°, and minimum at 90° and 270°. The figure shows a very pronounced structure that is consistent with a simple single‐scattering (Rayleigh) atmosphere (Coulson 1988). The polarization is highest at high altitudes and in the north and south. However, the measured polarization seen in Figure 19 does show strong wavelength dependence, and so these surfaces change significantly with wavelength. This is entirely plausible, since the scattering properties of the atmosphere are also strong functions of wavelength.

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Fig. 21.— Measured polarization of scattered sunlight at sunset for order 14 (860–870 nm) in the alt‐az plane. At sunset, polarization is at maximum 90° off the Sun (north, south, zenith) and at minimum at the antisolar point (low altitude in the east and west).

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The strong rotation of the plane of polarization in Figure 20 is also very interesting. Since the reflections off the ocean surface are very wavelength dependent, we cannot say how much of this rotation is instrumental and how much is ocean reflection. This rotation is significantly different from that observed in other locations, but our environment is also very different (Horváth et al. 1998, 2002; Coulson 1988). The rotation is strongest when the degree of polarization is lowest, suggesting a mixture of the singly scattered light with a smaller amount reflected light (compare Figs. 19 and 20).

7. CONCLUSIONS

This was the first test of the HiVIS spectropolarimeter over broad wavelengths. We have illustrated a method for calibrating an alt‐az coudé telescope with many moving mirrors for induced polarization, efficiency of polarization detection, and rotation of the plane of polarization. Measurements of unpolarized sources and linearly polarized sources (∼1000 spectra) with known position angles at many pointings were necessary to obtain good altitude‐azimuth coverage. The AEOS telescope shows a maximum of 6% induced polarization at 640 nm and at high altitude, with values falling with wavelength and altitude to near 1%. We constructed an all‐sky map of the telescope‐induced polarization and can use it to calibrate future observations. We hope to conduct more observations of unpolarized and polarized sources in order to improve the resolution of the calibration sky maps.