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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} R = X + iY \\ L = X - iY \end{align} }

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} RR^* = (X + iY)(X + iY)^* &= XX^* - iXY^* + iYX^* + YY^* \\ LL^* = (X - iY)(X - iY)^* &= XX^* + iXY^* - iYX^* + YY^* \\ \end{align} }

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,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} Stokes \, I &= \frac{RR^* + LL^*}{2} = XX^* + YY^* \\ Stokes \, V &= \frac{RR^* - LL^*}{2} = i(XX^* - YY^*) \end{align} }

For completeness:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} }

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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} XY^{*}' &= XY^*e^{-i(\phi(v) + \frac{\pi}{2})} YX^{*}' &= YX^*e^{ i(\phi(v) + \frac{\pi}{2})} \end{align} }

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} RR^* &= XX^* - iXY^(*)' + iYX^(*)' + YY^* LL^* &= XX^* + iXY^(*') - iYX^(*') + YY^* \end{align} }

Polarization Mixing Correction

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