# Stokes Parameters - Fraction of Linear/Circular Polarization

## Homework Statement

The electric field of an electromagnetic wave is given by;

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

Identify the polarization state.

## Homework Equations

$I = |E_{x}|^{2} + |E_{y}|^{2}$

$Q = |E_{x}|^{2} - |E_{y}|^{2}$

$U = |E_{a}|^{2} - |E_{b}|^{2}$

$V = |E_{l}|^{2} - |E_{r}|^{2}$

$I^{2} = Q^{2} + U^{2} + V^{2}$

Fraction of Linear Polarization = $\frac{\sqrt{Q^{2} + U^{2}}}{I}$

Fraction of Circular Polarization = $\frac{\sqrt{V}}{I}$

## 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

$E_{x} = \frac{-5}{\sqrt{13}}E_{0}e^{i(kz-wt)}$

$E_{y} = \frac{3i}{\sqrt{13}}E_{0}e^{i(kz-wt)}$

I obtain

$I = E_{0}^{2}$

$Q = \frac{-5}{13}E_{0}^{2}$

$U = 0$

$V = \frac{12}{13}E_{0}^{2}$

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 = $\frac{Q^{2} + U^{2}}{I^{2}}$

Fraction of Circular Polarization = $\frac{V^{2}}{I^{2}}$

?

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.