PhysRef: justify decomposition of \OPR\rho

Doc/PhysRef/Polarized.tex

@@ -659,22 +659,32 @@ function \T{Compute::magneticR} in file \SRC{Sim/Computation}{SpecularComputatio
 %===============================================================================
 
 As any other 2$\times$2 matrix,
-the polarizer density operator
+the polarization operator
 \index{Polarization!density operator}%
 can be written as
-\begin{equation}\label{DOprPi}
+\begin{equation}\label{EdecomposePi}
+  \OPR\Pi = p_0\OPR 1 + \v{p}\Pauli,
+\end{equation}
+and the polarizer density operator
+\index{Polarization!density operator}%
+as
+\begin{equation}\label{Edecompose1rho}
+  \OPR\rho = r_0\OPR 1 + \v{r}\Pauli.
+\end{equation}
+From \cref{Drhoi} or~\cref{Drhof}, we know that $\OPR\rho=\OPR\Pi\,\OPR\Pi^+$.
+Inserting \cref{EdecomposePi}, we can conclude that $\OPR\rho$ is Hermitean,
+that $r_0$ and~$\v{r}$ are real,
+and that $|\v{r}|\le|r_0|$.
+This allow up to replace \cref{Edecompose1rho} by
+\begin{equation}\label{Edecompose2rho}
   \OPR\rho = \left(\OPR 1 + \v{P}\Pauli\right) \tau.
 \end{equation}
-By construction \cref{Drhoi} or~\cref{Drhof}, it is Hermitean.
-As the Pauli matrices are also Hermitean,
-the parameters $\tau$ and~$\v{P}$ must be real.
 We identify $\v{P}$ as the \textit{polarization vector},
 \index{Polarization!vector}%
 and $\tau$ as the \textit{mean transmission} of an unpolarized beam;
 it can take values between 0 and~1/2,
 whereas the polarization strength $P\coloneqq|\v{P}|$ may take
 values between 0 and~1.
-
 For a source flux~$I_0$, the flux after a beam polarizer has the components
 \begin{equation}
   I_\pm \coloneqq \Tr (\pm \v{\hat P} \Pauli) \OPR\rho_\si \OPR\rho_0 I_0