18.4 Analysis of Polarization Images

The images in the three filters need to be compared, in order to produce the Stokes' parameters. To do so, the three images need to have the same dimensions and to be registered (have the source centered at exactly the same location in each image). In principle they should already be registered, provided the images in each filter were taken in the same visit without any changes of guide stars. If there is any displacement of the source between images, then each image can be registered using the IRAF task imlintran. When imlintran is used, the origin for the three input images should be set to the location of the source in each, so that in the output images the origin and the parameters nlines and ncolumns are identical.

To generate Stokes' parameters, the relative differences in flux between images in the different polarizing filters are used. Where the signal level is very faint, and the signal-to-noise ratio is very low, the differences will be very large but dominated by noise. If you attempt to calculate the Stokes' parameters using such data, you will obtain large and entirely spurious polarizations. To avoid this problem, it is advisable to estimate the noise in an area of the image free of sources, and then set a threshold at a value of order five to ten times this noise level. Using the IRAF task imreplace, all pixels with signals below this threshold should be set to some arbitrary value, probably close to the measured noise level. This action will cause all areas of the image where the signal level is very faint to show zero polarization.

18.4.2 Theory

If we define the intensity and statistical uncertainties (including read-noise) obtained in the three polarizers to be I₀, I₁₂₀ and I₂₄₀ and ₀, ₁₂₀, ₂₄₀ respectively, then we may obtain the total intensity I from:

and the Stokes parameters Q and U:

The statistical uncertainties are obtained by straightforward propagation of errors:

The Stokes parameters can then be combined to yield the polarized intensity, I_p:

and the degree, P, and position angle of polarization, q, using:

18.4.3 A Useful Script for Polarization Analysis

An interactive procedure to derive relevant parameters from NICMOS polarization images has been developed by Hines et al. (1997).¹ In addition to taking into account instrumental polarization, this routine corrects for flatfield uncertainties and for small shifts between the images.

However, first approximation results can be obtained with a straightforward IRAF script. The approach we follow here is the simplest possible path to determining the polarization properties from the data. It does not take into account the instrumental polarization and does not allow for systematic errors in the data. The script will yield the correct morphology but not the exact intensity of the polarization. There are more sophisticated tools available in the community, including the one referenced above, which have been developed specifically to analyze polarization images, and which will yield better results than this very simple approach. The IRAF script is included below, and is commented.

The script doesn't take into account that in NIC1 the POL120 filter has only 48% transmission, while the POL0 filter has 98% transmission. For sake of simplicity, we assume here that the polarization image at 120 degrees has been obtained using the POL0 filter with a spacecraft roll, rather than using the POL120 filter.

Figure 18.4: Polarization Script

The script assumes we started with three images, named pol0s.fits, pol120s.fits and pol240s.fits, taken in the short wavelength polarization filters. The output from the script is a set of files named pols_u.fits and pols_q.fits (the Stokes' U and Q parameters respectively), pols_i.fits (the total intensity), pols_ip.fits (the polarized intensity), pols_p.fits (the degree of polarization), and pols_theta.fits (the polarization angle). The error propagation should, in general, be more or less correct for statistical errors (all systematic errors are ignored), although for the polarization angle the error is a gross approximation. When a new version of the IRAF imfunction task is written to include error propagation, the rather messy parts of the above script which generate error arrays will no longer be needed.