The imaged areas do not completely fill the detector: in the X direction the regions begin at around pixel 35; in the Y direction the SKY #2 O ray region begins at about pixel 30 and there is a vertical gap of about 27 pixels between each region. The actual projected size of the imaged regions depends of course on the camera scale employed (0.292'' per pix or 0.584'' per pix).
A single image is sufficient to determine a single Stokes parameter, either Q or U. See for example Clarke and Grainger 1971 for a full description of Stokes parameters. The normalised Stokes parameter is then given by:
( OBJECT O ray - OBJECT E ray )/( OBJECT O ray + OBJECT E ray )
The corresponding sky (or the mean of the two sky regions for the same ray) must be subtracted from the OBJECT image before determination of the Stokes parameter.
In order to measure the second Stokes parameter, and hence determine the linear polarization, a minimum of one independent image is required at a rotation angle of 45 degrees with respect to the first. Since there is no Half Wave plate to instrumentally rotate the input plane of polarization, the whole instrument has to be rotated by 45 degrees. Of course this means that the field no longer has the same orientation on the sky and polarimetry of large extended objects cannot be accomplished with a simple sequence at two position angles separated by 45 degrees.
The linear polarization, p, and position angle of the electric vector, theta, are given by
p = sqrt(Q*Q + U*U)
theta = 0.5*tan-1(U/Q).
The optics of the instrument itself and also the telescope
possess an intrinsic polarization so that when a truely zero
polarization external source is observed, some non-zero polarization
at some orientation is detected. Of course this instrumental
polarization has a fixed orientation in the telescope and the
instrument. If the instrument is rotated, then the plane of
polarization of the
instrument will also rotate on the sky. Thus the correction should be
with respect to the instrument. However since SOFI is mounted at an
altazimuth focus, then the telescope and instrument change their
angular alignment, and the telescope and instrument polarization
directions differ. For the most precise work it is necessary to
measure the instrumental polarization at the same rotator angles
as the source and as close as possible in time. However given an
accurate measurement of the instrumental polarization degree, it
should be possible to predict the applicable instrumental
polarization angle for a given observation at any rotator angle.
So far it has been assumed that the analyser is 100% efficient, i.e. there is no cross-talk between the two output beams. Any deviation from a perfect polarizer will manifest as less than 100% polarization being measured for a perfectly polarized input source. In practice the polarizing efficiency of the instrument (this will include the analyser and the instrument optics, plus even any polarization sensitivity of the detector) can be determined by observing a source of known polarization (polarized standard). Comparison of the measured value, corrected for the instrumental polarization, with the observed value of polarization, provides the polarizing efficiency (inverse of efficiency corrects the measured polarization to true value). In addition any offset between the measured value of the position angle of polarization and the standard value provides a rotation angle correction, which can be added to the observed position angle data. In practice polarized standard stars have at most 5% linear polarization. However since they are bright this is not a problem. Note however that the polarization is wavelength dependent and the value should refer to the magnitude of the filter in use. Appendix 1 provides a list of polarized standards for use in the IR. There are also some highly polarized IR reflection nebulae which are suitable as polarized standards but since the polarization is often highly structured, then care has to be taken to use exactly the same aperture definition as a basis for comparison (listed in Appendix 2).
Instrumental polarization determination
SOFI observations of the star HD 94851, which is an HST
zero polarization standard (Turnshek et al 1990) with a
V band polarization of 0.057% were obtained by J.-G. Cuby on
1997 December 20 with 0.292''/pixel scale in the J and K'
filters. Double-correlated sampling with 20 subintegrations
and a total integration time of 1.182s were made.
A series of exposures with the rotator shifted by 22.5
degrees were obtained. For the J filter a series from
POSANG -359.82 to -179.8 degrees (CROTA1 -449.82 to -269.80)
with the star at the same position on the detector; one
additional exposure was made at POSANG -179.80 with the
position of the star shifted on the detector. For the K'
filter a series from POSANG -179.79 to -359.75 degrees
(CROTA1 -269.79 to -449.76) with the star at the same
position on the detector; one additional exposure was
made at POSANG -179.79 and another at -359.76 with the
position of the star shifted on the detector. These
measurements oversample the polarization curve but are very
useful for obtaining an accurate determination of the
The data frames were bias corrected and flat fielded using exposures of the dome with the Wollaston analyser. The mean sky frame was formed from the two sky regions and subtracted from flat fielded object sub-image. The counts in the star image were summed over a circular aperture of 12 pixels radius (3.50'' since the 0.292'' pixel scale was used), subtracting the residual sky in an annulus from 5.84 to 8.76'' radius around the star image. All the data were reduced together using the iraf.stecf.impol package and routine hstpolpoints; the results are presented below. In computing formal errors, an e/ADU conversion factor of 5.9 was used and a read-out noise of 12e. Errors are 1sigma.
|HD 94851 Polarization Measurements|
||Normalised Q||Normalised U||Linear Poln. (%)||Poln. PA (deg.)|
It appears that the measured instrumental polarization of SOFI is significant and larger at J than K. The position angle, PA=0 is equivalent to POSANG of -359.8 degrees, may be different at the three-sigma level between the two wavelengths.
Origin of instrumental polarization
Since the instrument has a simple design with no fold mirrors,
then the chief contributor to the instrumental polarization
is expected to be the tertiary mirror (Nasmyth feed),
where light is reflected by 45 degrees to the Nasmyth
focus. If so then it is expected that the instrumental
polarization might be sensitive to the exact instrument
rotator angle, although the 45 degree reflection angle
is not significantly changed. A calculation of the
polarization induced by a 45 degree reflection for
aluminium coated mirror at J and K' was made using
interpolated values of the refractive index for
Al from the American Institute of Physics Handbook
(page 6-125; values 1.91 -i10.8 at J and 2.59 -i17.5
at K'). The predicted contribution to the instrumental
(linear) polarization is 0.7% at J and 0.8% at K'. The
reflection by the mirror however also produces some
retardance (conversion of linear to circular polarization);
see e.g. Born and Wolfe 1985 for details. If the mirror was dusty
then the results could be different than for pure Aluminium.
These low values of the polarization
induced by the tertiary mirror indicate that
good polarimetry (to a percent or slightly better)
can be readily performed with an alt-azimuth telescope
in the near IR at least.
Rather than all the instrumental polarization being caused by the 45degree reflection at the Nasmyth mirror, it is more probable that the Wollaston analyser does not provide 100% transmission for unpolarized radiation in the two beams. This is equivalent to saying that the polarizing efficiency is less than 100%. If for example a 100% polarized beam were incident on the Wollaston, then with a polarizing efficiency less than 100% a small fraction would be detected in the output E ray. For a detection of 1.4% polarization from an unpolarized source, the polarizing efficiency is 97.2%, if all the losses occur for the E ray. The manufacturers data for the SOFI polarizer should be consulted to check this estimate. The effect of subtracting (in quadrature) the polarization induced by the 45 degree reflection on the Nasmyth mirror has only a small effect on this estimate.
Ageorges & Walsh (1999) report SOFI K' polarization measurements of the highly polarized reflection nebula OH231.8 +4.2 around the Mira star QX Puppis. Comparison of the polarization in various apertures with previous results shows an effect in the direction of a lower than 100% polarizing efficiency of the Wollaston analyser; however the results are not conclusive to within the errors of the measurements.
Ageorges, N, Walsh, J. R., 2000, A&A, in press (astro-ph/0004103)
Born, M. Wolfe, E., 1985, Principles of Optics, New York, Pergamon Press
Clarke, D., Grainger, 1971 Polarized Light and its Measurement Oxford, Pergamon
Turnshek, D., Bohlin, R. C., Williamson, R. L., Lupie, O. L., Koornneef, J., 1990, AJ, 99, 1243
Appendix 1. IR Polarization Standards
of polarized and unpolarized standards are available at UKIRT.