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:

${\begin{aligned}R=X+iY\\L=X-iY\end{aligned}}$

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

${\begin{aligned}RR^{*}=(X+iY)(X+iY)^{*}&=XX^{*}-iXY^{*}+iYX^{*}+YY^{*}\\LL^{*}=(X-iY)(X-iY)^{*}&=XX^{*}+iXY^{*}-iYX^{*}+YY^{*}\\\end{aligned}}$

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,

${\begin{aligned}Stokes\,I&={\frac {RR^{*}+LL^{*}}{2}}=XX^{*}+YY^{*}\\Stokes\,V&={\frac {RR^{*}-LL^{*}}{2}}=i(XX^{*}-YY^{*})\end{aligned}}$

For completeness:

${\begin{aligned}Stokes\,Q&=XX^{*}-YY^{*}\\Stokes\,U&=XY^{*}-YX^{*}\\P_{linear}&={\sqrt {U^{2}+Q^{2}}}\\\Theta &={\frac {1}{2}}\tan ^{-1}{\frac {U}{Q}}\end{aligned}}$

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:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Prime causes double exponent: use braces to clarify"): {\displaystyle {\begin{aligned}XY^{*}'&=XY^{*}e^{-i(\phi (v)+{\frac {\pi }{2}})}YX^{*}'&=YX^{*}e^{i(\phi (v)+{\frac {\pi }{2}})}\end{aligned}}}

where φ(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

${\begin{aligned}RR^{*}&=XX^{*}-iXY^{(}*)'+iYX^{(}*)'+YY^{*}LL^{*}&=XX^{*}+iXY^{(}*)'-iYX^{(}*)'+YY^{*}\end{aligned}}$

Polarization Mixing Correction

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