The MkIII was designed to show K-corona signal in the measured Stokes parameter q. Until now, the calibration was done by applying a scale factor to that measured q. This simple operation was inadequate for two reasons. Each of the measured i, q, and u Stokes parameters is composed of sums of the input I, Q, and U. Furthermore, the input I, Q, and U also include polarization signals from instrument and sky.
The following steps were taken to create a calibration technique for the MkIII: build a mathematical model of the telescope and calibration optics, determine unknowns in the model, and develop a program to solve for the unknowns of the model using observations.
The mathematical technique developed by Paul Seagraves to
describe polarimeters [ Seagraves and
Elmore, 1994] and experience in its application to the Advanced Stokes
Polarimeter [ Elmore, 1988] were used to create a
polarization model for the MkIII.
For the MkIII model, there are three cases to treat: the telescope
and detection system alone, those
with the addition of a calibration opal, and those with the further addition of calibration tilt
plates mounted at an angle of 15
.
The corresponding equations are given below:
where r is the measured vector in digital units,
g is the gain,
X is the polarimeter response matrix,
R
is the barrel angle,
L is the objective lens partial polarizer matrix,
S is the sky partial polarizer matrix,
s is the input stokes vector,
P is the opal partial polarizer matrix,
R
is the orientation of the calibration tilt plates,
and T is the calibration tilt plate partial polarizer matrix.
It has been assumed that only linear polarization needs to be considered.
The matrices are 3 
3 instead of the normal 4 
4
for Mueller matrices.
The X matrix is normalized to 1 so that elements can be
compared from calibration to calibration.
Fractional intensity of the opal is set to that of a perfect
opal (10.11
).
Measured transmission of the HAO standard opal 2B was found
to be 10.1
at
800nm [ Streete, 1989].
Fractional polarization of the calibration tilt plates
was computed from theory [ Born and
Wolfe, 1975].
For two plates with an index of refraction of 1.5105 (BK7 at 820nm),
each tilted at 
, the partial polarization of the plates is .01543.
The orientation of the calibration linear polarizer produces
vertically polarized light
for channel 1 at an azimuth angle of zero.
It is assumed that the calibration opal, calibration tilt plates,
telescope and sky are
simple partial linear polarizers.
The telescope and opal polarization value, as well as their
orientations, are treated as
unknowns in the model.
Instrument constants are:
Then, for a given diode, elements of the polarimeter response matrix are:
Light leaks in the opal I data must be noted and excluded, because they can make the gain too high.
The tilt exposure is used for the remainder of the X matrix. For a given diode:
In practice, the cosine of twice the barrel angle signal
in the I
data is so small
that X
and X
are also very small,
therefore they are set to zero.
Thus light leaks in the I
data are not important,
and these data aren't
considered.
(The light leaks in the I data do not show up in the Q and U data,
regardless of scan: opal, tilt or
coronal).
To solve for X
and X
, and similarly X
and X
,
the following identity is used:
Therefore by analogy,
and
Phi (
) and the
amplitude (Amp) are found with a least squares fit between the
actual data and a generic 
wave.
Once the X matrix and gain have been determined, calibrated I, Q and U can be computed from the raw coronal i, q and u data. For a given diode:
where
and
A sample of X matrix elements as a function of diode height is shown
in the upper nine graphs of
Figure 1
.
The diagonal
elements are scale factors. The X
and X
elements
show instrument polarization.
The X
and X
elements show the mixing of Q and
V signals by the polarimeter. The lower left graph of the figure shows
the scale factor applied to all matrix elements.
Figure 2 . shows raw and calibrated coronal data at a fixed height as a function of scan azimuth. Note the negative Q level in the raw data removed through calibration.
There is still residual polarization due to the objective lens
and sky polarization.
The magnitude and
orientation of sky polarization varies with sky conditions and time of day.
The K-corona (calibrated Q) is polarized tangent
to the limb of the Sun.
With this residual polarization there is a signal in the calibrated
U data (
to tangent)
which is fixed with respect to the sky and objective lens and appears
as a two-period sine wave in a
coronal scan.
The magnitude of this polarization normalized by intensity is fit to the sine
wave.
That amount of polarization is shifted
by 
and subtracted
from the K-corona for
that scan.
In detail, one
wave,
using the method of least squares for phase and amplitude,
,
wave before it is used.
This treatment of residual polarization is effective in removing the intensity-to-polarization crosstalk from the polarimeter which showed up as negative corona at the poles and excessive corona at the equator. It also removes most of the rings and reduces the spikes in the calibrated corona.
An example of exceptionally large sky polarization is shown in Figure 3 . This observation was made shortly after the eruption of Mt. Pinatubo and was useless without fixed polarization removal. Note the large periodic signal in Q removed through this technique.
Figure 1:
The upper nine graphs show a sample of X matrix elements
as a function of diode height.
The diagonal elements are scale factors.
The X
and X
elements show instrument polarization.
The X
and X
elements show the mixing
of Q and V signals by
the polarimeter. The lower left graph shows the scale
factor applied to all matrix elements.
Back to text.
Figure 2:
Figure 2: Raw and calibrated coronal data at a fixed height as a function of scan azimuth. Note the negative Q level in the raw data removed through calibration. Back to text.
Figure 3:
Figure 3: An example of exceptionally large sky polarization. This observation was made shortly after the eruption of Mt. Pinatubo and was useless without fixed polarization removal. Note the large periodic signal in Q removed through this technique. Back to text.
