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NICMOS Data Handbook > Chapter 5: Data Analysis > 5.6 Analysis of Polarization Images

5.6 Analysis of Polarization Images
The filter wheels of cameras NIC1 and NIC2 each contain three polarizing filters with unique polarizing efficiencies and position angle offsets. The original design specified that the position angle of the primary axis of each polarizer (as projected onto the detector) be offset by 120 from its neighbor, and that the polarizers have identical efficiencies. While this clean concept was not strictly achieved in NICMOS, the reduction techniques described below permit accurate polarimetry using both cameras over their full fields of view.
A complete set of polarimetric observations will contain images obtained in all three polarizers of the selected wavelength range. We assume that each image has been processed through calnica and calnicb to produce a fully reduced and (if necessary) mosaiced image in each of the three filters, with the data corrected for saturation and cosmic rays and converted to flux density using the appropriate photometric calibration constants for the polarizers.
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 may be very large but dominated by noise. If you attempt to calculate the Stokes parameters using such data, you will likely obtain large and entirely spurious polarizations. Therefore, it is not advisable to use low signal-to-noise data to calculate polarization. To avoid this problem, it is suggested 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. To further increase signal-to-noise, bin the data in each of the three images before computing the Stokes parameters. Once the parameters have been derived, clipping the Q and U images at 1 and the polarization intensity image at values < 0 or >1 will also help in increasing the signal-to-noise.
“Analysis of Polarized Light with NICMOS” by Hines, Schmidt & Schneider (2000, PASP, 112, 983) is a highly valuable reference discussing NICMOS polarimetric measurements and data analysis.
In order to reduce data obtained with a set of polarizers, three quantities are needed. They are the throughput for unpolarized light, the efficiency of the element as a polarizer, and the orientation of the polarizer. These quantities can be expressed in a polarization reduction algorithm to form a solution containing the polarization characteristics of the incoming beam (i.e., the Stokes parameters I, Q, and U).
The general form of the equation for polarimetric data reduction is expressed as
where is the emerging light intensity from the kth polarizer, Ak, Bk and Ck are the transmission coefficients, and εk is the polarizing efficiency.1 This linear equation captures the observed signal from a polarized source of intensity I and linear Stokes parameters Q and U, which describe the state of polarization for the target object. The above equation reduces to a set of three equations with three unknowns. The solution results in the Stokes parameters for the incoming light.
For NICMOS the transmission coefficients are2
where φk is the position angle of the kth polarizer relative to the NICMOS entrance aperture, tk is the fraction of light transmitted for a 100% polarized input aligned with the polarizer's axis, and lk is the fraction transmitted when the incoming light is perpendicular to the axis of the polarizer.
For NICMOS, the observed signal from a polarized source of total intensity I contains an added term in the transmission coefficients, namely 0.5 (1 + lk).
Table 5.7 and Table 5.8 below present respectively the properties of the individual polarizers for Cycle 7/7N and Cycles 11 and beyond. The Cycle 11 and beyond properties have been reduced by Batchelclor et al. (2009, PASP, 121, 153). Therefore the values in Table 5.8 and the post-NCS coefficient matrices, supersede values given in earlier versions of the Data Handbook.
Table 5.7: NIC1 & NIC2 Polarizer Properties
The resulting Cycle 7/7N coefficient matrices become3
which can be used to compute the expected (I1, I2, I3) for a given set (I, Q, U). By inverting the appropriate matrix, the Stokes parameters (I, Q, U) can be computed from a set of observations (I1, I2, I3).
Table 5.8: NIC1 & NIC2 Polarizer Properties for Cycle 11 and Beyond
The resulting coefficient matrices for post-NCS data become
The errors on the Stokes parameters are determined by straightforward propagation of errors,
where Sk represents an incoming Stokes vector, and defines the set of three observed intensities. Therefore,
where aij represents the elements of the inverted transmission coefficients matrix.
The Stokes parameters can then be combined to yield the polarized intensity,
as well as the degree of polarization P and the position angle of polarization θp, where
Because the polarizers are non-ideal, I, Q, and U are correlated when calculating P and θp. Therefore, covariance must be taken into account when calculating the errors.
For the degree of polarization, the covariance is
The covariance for the position angle is
An interactive IDL program to derive relevant parameters from NICMOS polarization images has been developed. The IDL program reads three images taken with three polarizers from NIC1 or NIC2 produces five images as output. The output images are: and, two images representing the Stokes parameters,, the total intensity,, the degree of polarization, and, the polarization angle.
Polarization vectors or contour maps can be superimposed over the intensity image. The program is available from the STScI NICMOS Web site under software tools, and is described in more detail in Mazzuca & Hines, NICMOS ISR 99-004, “User’s Guide to Polarimetric Imaging Tools.” For news and updates on this tool, see the NICMOS Web page:

Polarizer efficiency is defined as where Spar and Sperp are the respective measured signals for a polarizer oriented parallel and perpendicular to the axis of a fully polarized beam.

For further detailed information on the derivation of the coefficient matrices, see Hines, D.C. “Imaging Polarimetry with NICMOS”, VLT Conference 1998.

Further detailed information on calibration methodologies and transformations between different polarizers, see Mazzuca, Sparks, Axon, NICMOS ISR 98-017, “Methodologies to Calibrating NICMOS Polarimetry Characteristics, 1998 (computed from equation 27) of Mazzuca et al 1998.

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