Stokes Parameters - Fraction of Linear/Circular Polarization

[Silversonic]

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.

 

[mark726]

Is it because the real part doesn't affect the polarization state?

Your calculations for the fractions of linear and circular polarization are correct. The reason why you are getting different percentages is because you are using the wrong formula for calculating the fraction of linear polarization.

The correct formula is:

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

This formula takes into account the magnitude of the x and y components of the electric field, while the formula you were using (\frac{Q^{2} + U^{2}}{I^{2}}) only takes into account their squares.

To answer your question about taking the real part of the formula, it is not necessary to do so in this case because the real part will not affect the polarization state. In general, when dealing with complex numbers, we only need to take the real part if we are interested in the actual physical quantity (in this case, the electric field), rather than just the polarization state.