4 Data reduction

The data reduction applied to AO polarimetry data consists of the removal of the detector signature and sky subtraction, which is common to IR imaging in general, followed by registration and derivation of the polarization parameters.

4.1 Removal of the detector signature

 The basic data reduction steps were performed with the "eclipse'' package (Devillard 1997). Flat-fields were acquired on the twilight sky at the beginning of each night of observation, in an exactly similar way as for the targets, at nine angles of the polarizer. The integration times were 7, 10 and 20 s for the J, H and K bands respectively; no flat field was taken with the $K_{\rm c}$ filter. The flat field images must first be processed to flag bad pixels, caused by either permanently dead pixels or ones whose sensitivity undergoes large fluctuation during the exposure. Two methods have been employed depending on the number of frames available in a cube: sky variation or median threshold.

The "sky variation'' method works on a data cube, with preferably many planes ($^\gt _\sim$20) in order to obtain reliable statistics on the variations. The standard deviation ($\sigma$) with frame number is computed for each pixel in the frame. A histogram plot of the standard deviations has a Gaussian shape representing the response to the, assumed constant, sky signal. All pixels whose response is too low (dead) or too high (noisy), compared to a central $\pm \sigma/2$ interval, are rejected. The "median threshold'' method can be applied to a small number of input frames (such as flat field data) and detects the presence of spikes above or below the local mean in each individual image independently. If the signal is assumed to be smooth enough, bad pixels are found by computing the difference between the image and its median filtered version, and thresholding it. This latter method is not as stringent as using the temporal variation, but is the only possibility when there are an insufficient number of images to calculate reliable statistics. Some bad pixels may however remain in the images after applying the bad pixel correction by either method; however the number is small and they can be manually added to the bad pixel map. Slightly different bad pixel maps were found for the different positions of the polarizer; which could be explained by a polarization sensitivity of the pixels ($\sim$1%), since the NICMOS detector sensitivity is slightly polarization dependent, or simply by the random variation of hot pixels.

Once corrected for the bad pixels, the twilight flats were normalized, then multiple exposures were averaged for the same position of the polarizer to derive the flat field maps. The target data cubes were corrected with the bad pixel map derived using the "sky variation'' method from the background sky frames and divided by the flat-field to give flat-fielded, cleaned images, where the sky contribution is still to be subtracted. All these operations were performed independently for the nine positions of the polarizer.

4.2 Sky subtraction

 The sky background can be bright in the IR and may also be polarized so it is criticical in the case of polarimetry to ensure that the uncertainties introduced by sky subtraction are minimized. Several tests were performed to determine the impact of the method of sky subtraction, in conjunction with the bad pixel correction, on the data. The first method considers one sky and a bad pixel map for each position of the polarizer; the second method a single averaged sky (all polarizer positions confounded) but individual bad pixel maps for each position; whilst the third method uses the same averaged sky and bad pixel map for all polarizer angles.

All three methods were tested (Ageorges 1999) and the results demonstrated that the largest modification of pixel values, and therefore photometry, comes from the bad pixel map used. The third method produced the largest discrepancies from the expected $\cos(2 \theta)$ curve, where $\theta$ is the polarizer position angle. The first method is clearly to be preferred since the effect of any polarization of the sky signal on the target data is correctly removed and any short term variation in sky background is subtracted.

It was found, from sky background level in the polarization calibrator data, that the sky subtraction has been successful to better than 1% (rms noise of 3.5 ADUs). For the 0 and 180$^\circ$ data, a further test of the quality of the sky subtraction was performed: the skies have been exchanged, i.e. "sky 0'' has been used for the data taken at PA 180$^\circ$ and conversely. This resulted in "photometric'' variations less than 0.05%, thus giving us further confidence in our sky subtraction method.

  

Figure 2: The photometric variation of the basic data is illustrated by the time sequence of measured counts in a region of the Homunculus taken in J band. Every 200 frames, the polarizer has been rotated by 22.5$^\circ$ and the vertical dashed lines indicate the change of polarizer position angle. The width of this curve is characteristic of the photometric variations. The discrepant point at 157.5$^\circ$ is attributable to an instrumental problem (see text). Left: global intensity variation over the full data frame but excluding a 30 $\times$ 30 pixel box centered on $\eta$ Carinae; right: variation over a 50 $\times$ 50 pixel area of the nebula, far from the saturated center of the image

  

Figure 3: Photometric variation between data taken at PA 0 and 180$^\circ$ is shown for the full frame (excluding the central source, i.e. $\eta$ Car itself) at left, and for a small area centered on the Homunculus (right)

4.3 Photometric quality

 The photometric quality of the data can be checked in two different ways: either by comparing the photometry of an object when acquired at 0$^\circ$ and at 180$^\circ$ or by plotting the measured signal against the polarizer angle where a $\cos(2 \theta)$ form should be obtained for polarized data. The latter is illustrated in Fig. 2, for J band data of the NE lobe of the Homunculus nebula around $\eta$ Carinae. The signal is plotted with time as the polarizer was rotated from 0 to 180$^\circ$;every ensemble of 200 points (within the dashed vertical lines) corresponds to frames acquired at the same position of the polarizer. The spread of points at a given polarizer angle gives a measure of the photometric variation.

The images, used to create this plot, have been overexposed on purpose in order to get as much signal as possible on the faint nebula. The central region of the images has thus been obtained outside the linear regime of the CCD. The intensity variation over this image has thus been recalculated avoiding a 30 $\times$ 30 pixels area centered on $\eta$ Car. This is represented Fig. 2 together with a plot of the intensity variation over a 50 $\times$ 50 pixels area centered on a lobe of the nebula, away from $\eta$ Car and thus obtained in the linear regime of the CCD. Figure 3, representing the photometric variation of frames acquired at 0 and 180$^\circ$, clearly illustrates the fact that the night of these observations was not photometric: there is a 0.3 mag extinction of the data acquired at 0$^\circ$ compared to that at 180$^\circ$.

In Fig. 2 it is clear that there is a discrepant point, at 157.5$^\circ$, since this does not fit into the smooth $\cos(2 \theta)$ progression of the curve. This problem, found for every source observed, was attributed to a technical problem of unknown origin; it appears from the figure that the polarizer may actually have been at an angle of 45$^\circ$. All maps taken at this polarizer angle were ignored in the subsequent derivation of polarization parameters, thus reducing the number of independent polarizer angles to 7 (0 and 180$^\circ$ being equivalent).

4.4 Derivation of polarization maps

 The polarization degree for each pixel, binned pixel area or within an aperture was determined by fitting a $\cos(2 \theta)$ curve to the variation of signal with polarizer rotation angle $\theta$ for the eight signal values (excluding the value at 157.5$^\circ$). A least-squares procedure was used with linearization of the fitting function and weighting by the inverse square of the errors (Bevington 1969). The error on the polarization was determined from the inverted curvature matrix and the error on the position angle by the classical expression (Serkowski 1962):   $\sigma_{\theta}({\rm deg.}) = 28.648 ( \sigma_{\rm p}/p )$ when $\sigma_{\rm p}/p$ was $\geq$8 or from the error distribution of $\sigma_{\theta}/\theta$ given by Naghizadeh-Khouei & Clarke (1993) when $\sigma_{\rm p}/p \leq 8$.The errors on the individual points in the images at each polarizer rotation angle take into account the number of images averaged, the read-out noise and the sky background contribution. Since the detector offset is not fixed per image it was necessary to bootstrap for the value of the sky level. A series of polarization maps were made with increasing sky contribution at a fixed polarization error per pixel. The sky signal was adopted when it produced polarization vectors which began to deviate from the expected centrosymmetric pattern (e.g. to the NE of R Mon - see Fig. 6) in the regions of lowest signal. Thus the polarization errors are not absolute errors. Applying a polarization error cut-off to the maps produces maps consistent with the expected structure (which can also be partially checked by binning the data). Figure 4 shows a typical fit to the $\cos(2 \theta)$ curve for a 8$\times$8 pixels binned region of the R Monocerotis H band image (see Table 1 and Fig. 6). The error bars on the individual points arise from the photon statistics on the object and sky frames, with read-out noise considered.

  

Figure 4: A typical fit of the observed signal as a function of polarizer rotator angle by $p \cos(2 \theta)$ for the summed counts in an aperture over the H band image of R Monocerotis (see Table 1 and Fig. 6). The derived value of linear polarization and position angle is shown by the bold line. The point at 157.5$^\circ$ was not considered in the fit

  

Figure 5: Illustration of the variation of the position of the centroid of an image on the detector while rotating the polarizer from 0 to 180$^\circ$ in steps of 22.5$^\circ$. The target was $\eta$ Car observed in $K_{\rm c}$ band and the shifts are clockwise with increasing rotator angle

It was noted in Sect. 2.3 that the rotation of the polarizer induces an image shift on the detector. Figure 5 is an illustration of the displacement observed, for images of $\eta$ Carinae in $K_{\rm c}$, while rotating the polarizer from 0$^\circ$ to 180$^\circ$ in steps of 22.5$^\circ$ (see Sect. 3 for details on the observation procedure). Since the PSF is variable in time, reproducibility is not guaranteed. However the displacements were found to agree with those in Fig. 5 for different targets (mostly unpolarized standard stars - see Table 1), and in different filters, to better than 0.5 pixel and so were adopted to register the images at different polarizer angles.

For a point source, where only the integrated polarization is of interest, the exact position of the source is not relevant provided all the signal is included in the summing aperture. However for extended sources, such as for $\eta$ Carinae and the Homunculus nebula, a polarization map which exploits the available spatial resolution is desired. It is therefore extremely important to ensure that the data are centered on the same position for all position angles observed, to avoid some smearing of the information. For unsaturated stellar images, the centroid of the point source can be used as a fiducial to shift the images to a common centre. In the case of saturated images it proved possible to obtain reliable centering by using a very large aperture for the centroid; this is then weighted by the outer (unsaturated) regions of the PSF. However if the source is polarized, and in particular if there is polarization structure across the point source then centroids at particular angles will be dependent on the source polarization. It was found that if the images were shifted to match the centroids at the 8 polarizer angles for the R Mon data, then a map with uniform, almost zero, polarization was derived, in contradiction to the known (aperture) polarimetry of this source (e.g. Minchin et al. 1991). In such a case the set of image shifts, derived from unpolarized point sources (Fig. 5), were applied to the data and the polarization maps were determined. Figure 6 shows the resulting J, H and K polarization vector maps superposed on logarithmic intensity plots; the raw data has been binned 4 $\times$ 4 pixels, i.e. 0.2''. Those shifts applied are closer to reality than those determined by the centroid of R Mon, but good to within $\pm$0.5 pixel. This might explain the difference in structure between our H band map and that of Close et al. (1997). Considerable structure across the central (almost point) source is evident. The cut-off of the maps is determined by the value of the 1$\sigma$ polarization error (4, 4 and 6% respectively for J, H and K). The structures seen in the J, H & K band maps (Fig. 6) change with wavelength, which might be an optical depth effect of the inclined disk. The striking difference between the maps in Fig. 6 and the one reproduced in Ageorges & Walsh (1997) comes from the calibration of the data. Indeed the latter were preliminary results and the first polarization maps derived with ADONIS.