View source for Polarization Calibration

== Linear to Circular Conversion ==
At EOVSA’s linear feeds, in the electric field the linear polarization, X and Y, relates to RCP and LCP (R and L) as:

:<math>
\begin{align} 
R = X + iY \\
L = X - iY 
\end{align}
</math>

In terms of autocorrelation powers, we have the 4 polarization products XX*, YY*, XY* and YX*, where the * denotes complex conjugation.  The quantities RR* and LL* are then

:<math>
\begin{align} 
RR^* = (X + iY)(X + iY)^* &= XX^* - iXY^* + iYX^* + YY^* \\
LL^* = (X - iY)(X - iY)^* &= XX^* + iXY^* - iYX^* + YY^* \\
\end{align}
</math>

One problem is that there is generally a non-zero delay in Y with respect to X.  This creates phase slopes in XY* and YX* from which we can determine the delay very accurately.  As a check,

:<math>
\begin{align} 
Stokes \, I &= \frac{RR^* + LL^*}{2} = XX^* + YY^* \\
Stokes \, V &= \frac{RR^* - LL^*}{2} = i(XX^* - YY^*) 
\end{align}
</math>

For completeness:

:<math>\begin{align} 
Stokes \, Q &= XX^* - YY^* \\
Stokes \, U &= XY^* - YX^* \\
P_{linear} &= \sqrt{U^2 + Q^2} \\
\Theta &= \frac{1}{2}\tan^{-1}{\frac{U}{Q}}
\end{align}
</math>

When I plot the quantities I, V, R and L as measured (Figure 1) for geosynchronous satellite Ciel-2, the results look reasonable, except that there are parts of the band where R and L are mis-assigned, and others where they do not separate well.

The problem is that residual phase slope of Y with respect to X, caused by a difference in delay between the two channels.  This can be seen in the upper panel of Figure 2, which shows the uncorrected phases of XY* and YX*.  To correct the phases, the RCP phase was fit by a linear least-squares routine, and then the phases were offset by &pi;/2 for both XY* and YX* according to:

:<math>\begin{align}
XY^{*}' &= XY^*e^{-i(\phi(v) + \frac{\pi}{2})} 
YX^{*}' &= YX^*e^{ i(\phi(v) + \frac{\pi}{2})}
\end{align}
</math>

where &phi;(v) is the phase fit by the linear function. The corrected phases are shown in the lower panel of Figure 2.

When the corrected (primed) quantities are used in

:<math>\begin{align}
RR^* &= XX^* - iXY^(*)' + iYX^(*)' + YY^*
LL^* &= XX^* + iXY^(*') - iYX^(*') + YY^*
\end{align}
</math>

== Polarization Mixing Correction ==
Due to relative feed rotation between az-al mounted antennas and equatorial mounted antennas