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Homework Help: Stokes Parameters - Fraction of Linear/Circular Polarization

  1. Dec 28, 2011 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant 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]


    3. 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.
     
  2. jcsd
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