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:

$R=X+iY$

$L=X-iY$

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

$RR^{*}=(X+iY)(X+iY)^{*}=XX^{*}-iXY^{*}+iYX^{*}+YY^{*}$

$LL^{*}=(X-iY)(X-iY)^{*}=XX^{*}+iXY^{*}-iYX^{*}+YY^{*}$

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,

$Stokes\,I={\frac {RR^{*}+LL^{*}}{2}}=XX^{*}+YY^{*}$

$Stokes\,V={\frac {RR^{*}-LL^{*}}{2}}=i(XX^{*}-YY^{*})$

For completeness:

${\begin{aligned}Stokes\,Q=XX^{*}-YY^{*}\\Stokes\,U=XY^{*}-YX^{*}\end{aligned}}$

$P_{linear}={\sqrt {U^{2}+Q^{2}}}$

$\Theta ={\frac {1}{2}}\tan ^{-1}{\frac {U}{Q}}$

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 π/2 for both XY* and YX* according to:

Polarization Mixing Correction

Due to relative feed rotation between az-al mounted antennas and equatorial mounted antennas