Stokes Parameters - Fraction of Linear/Circular Polarization

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Homework Statement



The electric field of an electromagnetic wave is given by;

E = [itex]Re(\frac{1}{\sqrt{13}}E_{0}(2\widehat{x}+ 3i\widehat{y})e^{i(kz-wt)})[/itex]

Identify the polarization state.

Homework Equations



[itex] I = |E_{x}|^{2} + |E_{y}|^{2} [/itex]

[itex] Q = |E_{x}|^{2} - |E_{y}|^{2} [/itex]

[itex] U = |E_{a}|^{2} - |E_{b}|^{2} [/itex]

[itex] V = |E_{l}|^{2} - |E_{r}|^{2} [/itex]


[itex] I^{2} = Q^{2} + U^{2} + V^{2} [/itex]

Fraction of Linear Polarization = [itex]\frac{\sqrt{Q^{2} + U^{2}}}{I}[/itex]

Fraction of Circular Polarization = [itex]\frac{\sqrt{V}}{I}[/itex]


The Attempt at a Solution



I won't go through the full-workings out because it'll take my days to write it, but my main concern is the formula for the fractions of linear and circular polarization.

Taking

[itex] E_{x} = \frac{-5}{\sqrt{13}}E_{0}e^{i(kz-wt)}[/itex]

[itex]E_{y} = \frac{3i}{\sqrt{13}}E_{0}e^{i(kz-wt)} [/itex]

I obtain

[itex] I = E_{0}^{2}[/itex]

[itex]Q = \frac{-5}{13}E_{0}^{2} [/itex]

[itex]U = 0 [/itex]

[itex]V = \frac{12}{13}E_{0}^{2} [/itex]

My answers tell me it is 85% circularly polarized and 15% linearly polarized.

But shoving the values for U, Q and I in the "fraction of linear polarization formula" we obtain 5/13, and similarly the "fraction of circular polarization" we obtain 12/13, which aren't the same as the percentages given in the answer. However if I square the value gotten in those formulas I get the answer given, so should I have this instead;

Fraction of Linear Polarization = [itex]\frac{Q^{2} + U^{2}}{I^{2}}[/itex]

Fraction of Circular Polarization = [itex]\frac{V^{2}}{I^{2}}[/itex]

?

I can't find formulas anywhere in my text-books or on the internet that will tell me the actual answer, so I need this forum's help.


I also don't understand why we don't take the real part of the formula first before deciding what the x and y-components of the electric field are.
 

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