Theorem_circular_polarization_coeff_pseudoscalar_report.tex

%
% Copyright © 2018 Peeter Joot.  All Rights Reserved.
% Licenced as described in the file LICENSE under the root directory of this GIT repository.
%
\maketheorem{Circular polarization coefficients.}{thm:polarizationRewrite:940}{
The time domain representation of the field in \cref{eqn:polarization_circular:160} can be stated in terms of the total phase as
\begin{equation*}
F = \lr{ 1 + \Be_3 } \Be_1 \lr{ \alpha_\txtR e^{-I\phi} + \alpha_\txtL e^{I\phi} },
\end{equation*}
where the \textit{circular polarization coefficients} are
\begin{equation*}
\begin{aligned}
\alpha_\txtR &= \inv{2}\lr{ c_1 + I c_2 }^\dagger \\
\alpha_\txtL &= \inv{2}\lr{ c_1 - I c_2 },
\end{aligned}
\end{equation*}
where \( c_1, c_2 \) are the 0,2 grade multivector representation of the Jones vector coordinates
\begin{equation*}
\begin{aligned}
c_1 &= \alpha_1 + I \beta_1 \\
c_2 &= \alpha_2 + I \beta_2,
\end{aligned}
\end{equation*}
defined here as 0,3 complex numbers, using \( I \) as the imaginary.
} % theorem