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Hubble Space Telescope
WFPC2 Polarization Calibration Theory

Version 22 January 1997.

The discussion below outlines our current understanding of the photometric calibration. This discussion is somewhat preliminanry, and comments are welcomed.

The use of Stokes vectors and Mueller matrices appears to provide an attractive framework for calibration, since problems like the pick-off mirror diattenuation are easily handled. References appear near the end of this discussion.

The Stokes Vector

The Stokes vector (I Q U V) describes the full polarization information of the incident radiation. If Ex and Ey are the components of the electric field in the X and Y directions, the Stokes vector components may be described as:

I = < Ex^2 > + < Ey^2 >

Q = < Ex^2 > - < Ey^2 >

U = < 2 Ex Ey cos(Ix - Iy) >

V = < 2 Ex Ey sin(Ix - Iy) >

Where Ix and Iy are the phases of the corresponding components of the electric field. The fractional linear polarization of a signal is:

p = (Q^2 + U^2)^(1/2) / I  

And the position angle of the electric vector of the incident wave is given by:

chi = Arctan( U / Q ) / 2

Mueller Matrices

The detected counts for any polarizer observations can be described as the product of the incident Stokes vector and a Mueller matrix M representing the total action of all optical components. In general M will be a function of the HST orientation (i.e. PA_V3), the polarizer quad used, and any filter wheel rotation (i.e. theta).


C = K (1 0 0 0) * M(PA_V3, quad, rot) * (I)
                                        (Q)
                                        (U)
                                        (V)

Observing in three different combinations of HST orientation, polarizer quad, and filter rotation will give three equations for the three unknowns I, Q, and U. (Since WFPC2 does not have any sensitivity to circular polarization, V cannot be determined.) If more than three combinations are available, the problem becomes over-constrained, and one can fit for I, Q, and U, and an estimate of the statistical errors.

The instrumental Mueller matrix can be expanded into separate matrices for various optical elements (and rotations between elements):

C = K (1 0 0 0) * M(pol) * M(polrot) * M(POM) * M(PA_V3) * (I)
                                                           (Q)
                                                           (U)
                                                           (V)

Here K represents the usual photometric calibration relating incident flux to detected counts, M(pol) is a Mueller matrix for a generic quad of the polarizer quad filter, M(polrot) is a matrix representing rotation of the polarizer quad filter and rotations of individual quads in the filter, M(POM) represents the diattenuation of the pick-off mirror, and M(PA_V3) represents rotation of the HST spacecraft.

The constant K is just the usual photometric calibration which is equal to:

K = (exp. time) * (detected count rate) / (incident flux) / (POM correction)

And can be readily computed with SYNPHOT for any given spectral filter. Such calculation should include all the usual WFPC2 components, and exclude the polarizer filters. The term (POM correction) is described below.

M(pol)

M(pol) describes the action of the polarizer filter. The general matrix for a non-ideal linear polarizer is given by Morgan, Chipman, and Torr (1990 in "Polarization Considerations for Optical Systems II", SPIE conference 1166, ed. Chipman, p. 401). Elements can be estimated from figure 3.7 in the WFPC2 Instrument Handbook (V. 4):

         ( [T(par)+T(perp)]/2   [T(par)-T(perp)]/2            0                       0            )
M(pol) = ( [T(par)-T(perp)]/2   [T(par)+T(perp)]/2            0                       0            )
         (         0                    0            sqrt[T(par)T(perp)/2]            0            )
         (         0                    0                     0             sqrt[T(par)T(perp)/2]  )

Where T(par) and T(perp) are the transmissions in the parallel and perpendicular directions, respectively, from the plot. For example, for F555W, T(par)=0.67, T(perp)=0.02, and this is:

         ( 0.345  0.325    0      0   )
M(pol) = ( 0.325  0.345    0      0   )
         (   0      0    0.082    0   )
         (   0      0      0    0.082 )

M(polrot)

M(polrot) describes the rotation between the principal axis of the WFPC2 pick-off mirror and the parallel axis of the individual quad of the polarizer filter. This includes rotation of the individual quads of the polarizer filter in the pol. quad filter, as well as the rotation of the entire filter wheel:

            ( 1          0                    0           0 )
M(polrot) = ( 0  cos(2*{theta-135})   sin(2*{theta-135})  0 )
            ( 0 -sin(2*{theta-135})   cos(2*{theta-135})  0 )
            ( 0          0                    0           1 )

Where theta is given in Table 3.10 of the WFPC2 Instrument Handbook. The angle 135 degrees is subtracted, since we need angles relative to the s-axis of the pick-off mirror. For example, for the unrotated POLQ filter on WF3 we have theta=45 degrees and the matrix is:

            ( 1   0   0   0 )
M(polrot) = ( 0  -1   0   0 )
            ( 0   0  -1   0 )
            ( 0   0   0   1 )

M(POM)

M(POM) describes the instrumental polarization introduced by the WFPC2 pick-off mirror. There are two important effects. First, the reflectance will be difference for waves with electric vectors parallel to the mirror surface (s-wave with reflectance Rs) and normal to this (p-wave with reflectance Rp). This effect is sometimes refered to as "diattenuation." Second, reflection at a metal surface will generally convert a linearly polarized wave into an elliptically polarized wave. This effect is called "linear retardance," since, in effect, the phase of the s-wave is retarded relative to the p-wave. The matrix describing these phenomena is:

         ( A   B   0   0 )
M(POM) = ( B   A   0   0 )
         ( 0   0   C  -D )  
         ( 0   0   D   C )

where:

A = [Rs+Rp]/2
B = [Rs-Rp]/2
C = sqrt[Rs*Rp/2]*cos(phi)
D = sqrt[Rs*Rp/2]*sin(phi)

[note: there is some uncertainty about the signs
on the "D" terms, but in practice it is probably
unimportant as most sources have Stokes V=0]

Values of Rs and Rp may be computed from relations given by Hass (AIP Handbook, 3rd ed., section 6g, case IV):

      a^2 + b^2 - 2a * cos(i) + cos^2(i)
Rs =  ----------------------------------
      a^2 + b^2 + 2a * cos(i) + cos^2(i)


           a^2 + b^2 - 2a * sin(i)tan(i) + sin^2(i)tan^2(i)
Rp = Rs *  ------------------------------------------------
           a^2 + b^2 + 2a * sin(i)tan(i) + sin^2(i)tan^2(i)

where:

2a^2 = [Q^2 + 4n^2k^2]^(0.5) + Q

2b^2 = [Q^2 + 4n^2k^2]^(0.5) - Q

and Q = n^2 - k^2 - sin^2(i)

Some values for n, the index of refraction, and k, the extinction coefficient of evaporated aluminum are given in the following table:

Wavelength        n           k
 1600 A         0.080       1.73
 2000           0.110       2.20
 2600           0.19        2.85
 3000           0.25        3.33
 3400           0.34        4.01
 4000           0.40        4.45
 4500           0.51        5.00
 4920           0.64        5.50
 5460           0.82        5.99
 6500           1.30        7.11
 7000           1.55        7.00
 8000           1.99        7.05
 9000           1.96        7.70

The phase retardation angle (phi) for aluminum at 45 degree incidence angle is tabluated by Beckers (1990 in "Polarization Considerations for Optical Systems II", SPIE conference 1166, ed. Chipman, p. 380):

Wavelength      (phi)
4000 A           39 deg
4500             29
5000             22
7000              7
9999              5

[Note: obviously a table with finer wavelength steps
is desirable.  We will creat one soon.  - JB]

The mirror surface consists of evaporated aluminum overcoated with 250 Angstroms of MgF2. For simplicity we will ignore the overcoating since it is extremely thin compared to the wavelength. The incidence angle for the pick-off mirror is i=47 degrees.

For example, for F555W we have i=47, n=0.82, k=5.99, Q=-35.4726, a=0.8141, b=6.0337, hence Rs=0.9425 and Rp=0.8807. Also (phi)~16 deg. Hence the matrix is:

         ( 0.9116  0.0309    0      0   )
M(POM) = ( 0.0309  0.9116    0      0   )
         (    0       0    0.619 -0.178 )
         (    0       0    0.178  0.619 )

The (POM correction) mentioned above is merely the reflectivity for unpolarized light, and is [Rs+Rp]/2 = 0.9116 for the above example.

M(PA_V3)

M(PA_V3) describes the rotation of HST, with the V3 axis used as a reference point:

           ( 1          0                   0          0 )
M(PA_V3) = ( 0  cos(2*{PA_V3+90})   sin(2*{PA_V3+90})  0 )
           ( 0 -sin(2*{PA_V3+90})   cos(2*{PA_V3+90})  0 )
           ( 0          0                   0          1 )

where PA_V3 is given in the data headers. We add 90 degrees, since we actually want the orientation relative to the s-axis of the pick-off mirror.

For example, for PA_V3=45 degrees we have:

           ( 1   0   0   0 )
M(PA_V3) = ( 0  -1   0   0 )
           ( 0   0  -1   0 )
           ( 0   0   0   1 )

The above matrices can be computed for an observed image, and then multiplied together to produce the single matrix

 M(PA_V3, quad, rot) 
which predicts the observed image as a function of I, Q, U, and V images of the target. Note that since the CCD has effectively no polarization sensitivity, we only require the top row of the final product matrix.

Given 3 or more observations of a target in different settings of (PA_V3, quad, rot), and the assumption that Stokes V=0, which is valid for most astrophysical targets, we can solve the three equations giving the observed images and obtain images of Stokes I, Q, and U, which is out final goal.

If four or more images are available, a non-linear least squares fit could be done to solve for Stokes I, Q, and U, along with some empirical estimate of the uncertainties.

References:

Beckers, J. M., 1990, in "Polarization Considerations for Optical Systems II", SPIE conference 1166, ed. Chipman, p. 380. [polarization and retardation properties of metal films]

Born, M. and Wolf, E., 1985, "Principles of Optics," (Pergammon Press), Ch, 13. [derivation of equations for properties of metal films]

Chipman, R. A. 1992, in "Polarization Analysis and Measurement" SPIE Proceedings Vol. 1746, p. 49. [short overview of Mueller matrices]

Clarke, D. 1974, in "Planets, Stars, and Nebulae Studied with Photopolarimetry," ed. T. Gehrels, (Univ. Arizona press) p. 45. [basic definitions of Stokes parameters]

Collett, E. 1993, "Polarized Light," (Marcel Dekker, Inc.: New York). [generalized Stokes vector and Mueller matrix theory, optics of metal films]

Engel, J. R., 1992, in "Polarization Analysis and Measurement" SPIE Proceedings Vol. 1746, p. 317. [general discussion of non-ideal polarimeters]

Kliger, D. S., Lewis, J. W., and Randall, C. E. "Polarized Light in Optics and Spectroscopy, " (Academic Press) 1990. (Ch. 4 and 5) [generalized Stokes vector and Mueller matrix theory, also optics of metal films]

Maymon, P. W. and Chipman, R. A., 1992, in "Polarization Analysis and Measurement" SPIE Proceedings Vol. 1746, p. 148. [general discussion of non-ideal polarimeters]

Software:

Currently there are two WWW tools available to aid in polarization calibration. The first generation tool is essentially a simulator of the WFPC2 polarization properties. The second generation tool allows conversion of observed images into maps of I, Q, U, fractional polarization, and polarization position angle, or similar quantities for a point source.

The first generation tool (simulator tool) accepts information for a model target (e.g. total intensity, fractional polarization, and polarization position angle), and circumstances of up to six observed images (PA_V3, polarizer setting, aperture used, spectral filter), and then predicts the observed counts in each image. It uses the above Mueller matrix formality. The user may optionally provide measured counts and uncertainties for each image, and the program will then estimate a chi-sqaured between the model and the observed data. In this way, the simulator tool can be used to derived the Stokes parameters for a target, though it will obviously be labor intensive to manually adjust the model and iterate.

The second generation tool (calibration tool) accepts the circumstances of three observed images (PA_V3, polarizer setting, aperture used, spectral filter), computes the appropriate Mueller matrices, solves for Stokes I, Q, and U, and then outputs simple recipes for I, Q, and U. The user may optionally input measures aperture counts for some target in each images, and then the tool will addionally return I, Q, U, fractional polarization, and polarization position angle for the target.

At this point the calibrator tool appears to work reasonably well on real data taken in the un-rotated POLQ settings. It has not been exhaustively tested, and users are encouraged to check their results against some independent calibration.

The accuracy is about 1% - 2% in fractional polarization for the cases we have tried, which exceeds the accuracy of 3% stated in the WFPC2 Instrument Handbook. There were initial indications of a problem with the rotated POLQ settings (i.e. POLQN33, POLQN18, POLQP15W), but these have now been traced to a bug in the CALWP2 (WFPC2 pipeline calibration) program. The filter, observed data, and polarization model are thought to be OK.

There is currently no "automatic" facility for estimating the uncertainties due to photon noise. It is suggested that the user try to estimate the noise in their images, add some noise to the image (or aperture counts) and the repeat the calculations to get some estimate of the effect of photon noise. In general, large coefficients (>few) in the equations for I, Q, and U will mean high sensitivity to noise, since one is then differencing several large quantities to get the polarization properties.

Examples:

We now give two examples where the polarization calibrator tool is applied to actual WFPC2 data for a stellar target:

Example of aperture photometry of star BD+64D106.

Example of aperture photometry of star G191-B2B.

There is also an example where Stokes images and a vector plot are generated for the reflection nebula surrounding R Mon:

Example showing how to generate polarization image with E-vectors for reflection nebula R Mon.

Resulting Stokes I image and E-vector field for R Mon.

On-Orbit Photometric Calibrations:

Proposal 5574 (PI Sparks) performed a minimal on-orbit calibration of the polarizers. Observations were made of unpolarized and polarized stars in several of spectral filters, with one observation per filter quad.

We have analyzed the data from proposal 5574, and find it is generally consistent with the mathematical model of the polarizers described above. Typical errors for the PC1, WF2, WF3, and WF4 apertures (unrotated POLQ filter) are 1% of I (i.e. 1% in fractional polarization) in F555W and F675W. Errors are typically 2% of I in F336W and F410M.

For reasons which are unclear, the accuracy in the rotated POLQ positions is poorer, typically 2% to 5% of I. This may only mean there is some inadequacy in the above model. We are currently working on this.

We have generated a new proposal, 6194 (PI Biretta), which expands the observations in 5477. The new proposal includes CR-SPLITs and makes observations in each quad at four different angles. This will allow us to individually characterize each quad of the pol. filter. Two of the angles were to have run prior to the Feb. 1997 service mission, but a reaction wheel failure on telescope caused parts of the observations during early Novemeber to fail. Repeats of these observations, along with the other rotations, will be rescheduled after the service mission.

However, it is fairly clear that our calibration will always rely heavily on the pre-launch lab measurements of the polarizers and other WFPC2 components. A complete on-orbit polarization calibration (crossed with all spectral filters, all POLQ rotations, all apertures, four PA_V3 settings per target) would consume some 1600 orbits, and is not likely to happen.

Future Work:

Much work remains to be done. Among important items are:

(1) Understand why results for the rotation polarizer filter (POLQN33, POLQN18, POLQP15W) are less accurate than for the unrotated settings.

(2) Futher testing of existing tools under more situations.

(3) Add a non-linear least squares solution to the tools, so that 4 or more input images can be handled.

(4) Study effects of MgF2 coatings on mirrors more throroughly.

(5) Add other WFPC2 mirrors to the optics model (<1% effect).

(6) More complete on-orbit calibration. Observations for proposal 6194 are scheduled to execute between Nov. 1996 and Jan. 1998.

(7) Software in STSDAS to automatically calibrate images based on header info.

(8) Software to remove small geometric distortions caused by the lens in the POLQ filter.