= Circular-eigenmode Jones propagator
{title2=$\mathcal J(z)=e^{-\gamma^+z}P_++e^{-\gamma^-z}P_-$}
For fixed circular <eigenvectors> $\mathbf u_\pm=(1,\mp i)^{\mathsf T}/\sqrt2$, let $P_\pm=\mathbf u_\pm\mathbf u_\pm^\dagger$. A uniform optical medium with amplitude propagation coefficients $\gamma^\pm$ has the displayed <Jones matrix>. Put $\gamma=(\gamma^++\gamma^-)/2$ and $\delta=(\gamma^+-\gamma^-)/2$. Then
$$
\mathcal J(z)=e^{-\gamma z}
\begin{pmatrix}\cosh(\delta z)&-i\sinh(\delta z)\\i\sinh(\delta z)&\cosh(\delta z)\end{pmatrix}
=e^{-Kz},\qquad K=\begin{pmatrix}\gamma&i\delta\\-i\delta&\gamma\end{pmatrix}.
$$
The sum of the two modal exponentials supplies a hyperbolic cosine, not its reciprocal. The <determinant> is $e^{-2\gamma z}$, and $\mathcal J(z_1+z_2)=\mathcal J(z_1)\mathcal J(z_2)$. Real modal coefficients give <circular dichroism>; purely imaginary coefficients give lossless <circular birefringence>. Complex modal labels do not have an intrinsic numerical ordering.
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