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| − | == Linear to Circular Conversion ==
| + | [[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:
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| − | :<math>
| + | [[Polarization Mixing Due to Feed Rotation]] ([[Polarization Mixing Correction (Old)]]) |
| − | \begin{align}
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| − | R = X + iY \\
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| − | L = X - iY
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| − | \end{align}
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| − | </math>
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| − | 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
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| − | :<math>
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| − | \begin{align}
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| − | RR^* = (X + iY)(X + iY)^* &= XX^* - iXY^* + iYX^* + YY^* \\
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| − | LL^* = (X - iY)(X - iY)^* &= XX^* + iXY^* - iYX^* + YY^* \\
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| − | \end{align}
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| − | </math>
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| − | 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,
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| − | :<math>
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| − | \begin{align}
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| − | Stokes \, I &= \frac{RR^* + LL^*}{2} = XX^* + YY^* \\
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| − | Stokes \, V &= \frac{RR^* - LL^*}{2} = i(XX^* - YY^*)
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| − | \end{align}
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| − | </math>
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| − | For completeness:
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| − | :<math>\begin{align}
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| − | Stokes \, Q &= XX^* - YY^* \\
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| − | Stokes \, U &= XY^* - YX^* \\
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| − | P_{linear} &= \sqrt{U^2 + Q^2} \\
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| − | \Theta &= \frac{1}{2}\tan^{-1}{\frac{U}{Q}}
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| − | \end{align}
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| − | </math>
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| − | 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.
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| − | 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:
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| − | :<math>\begin{align}
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| − | XY^{*}' &= XY^*e^{-i(\phi(v) + \frac{\pi}{2})}
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| − | YX^{*}' &= YX^*e^{ i(\phi(v) + \frac{\pi}{2})}
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| − | \end{align}
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| − | </math>
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| − | where φ(v) is the phase fit by the linear function. The corrected phases are shown in the lower panel of Figure 2.
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| − | When the corrected (primed) quantities are used in
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| − | :<math>\begin{align}
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| − | RR^* &= XX^* - iXY^(*)' + iYX^(*)' + YY^*
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| − | LL^* &= XX^* + iXY^(*') - iYX^(*') + YY^*
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| − | \end{align}
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| − | </math>
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| − | == Polarization Mixing Correction ==
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| − | Due to relative feed rotation between az-al mounted antennas and equatorial mounted antennas
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