= Solution
When $\gamma^\pm=\alpha^\pm$ are real, $\gamma$ and $\delta$ are real. The two opposite <circular polarizations> retain their <wave phases> but are attenuated at different <wave amplitude> rates. This <circular dichroism> changes the ellipticity without rotating the principal axis of the <ellipse>.
For input $E_0\widehat{\mathbf x}$, the corrected <Jones matrix> gives
$$
\mathbf E_{\rm out}=E_0e^{-\gamma z}
\begin{pmatrix}\cosh(\delta z)\\i\sinh(\delta z)\end{pmatrix}.
$$
The common factor $e^{-\gamma z}$ describes the mea <wave amplitude> attenuation. The $\cosh(\delta z)$ factor reflects the sum of the two transmitted modal <wave amplitudes>; $\tanh(\delta z)$ fixes their imbalance and hence the signed ellipticity. The real field has components proportional to $\cosh(\delta z)\cos\omega t$ and $\sinh(\delta z)\sin\omega t$, so
$$
\boxed{\frac{\text{minor semiaxis}}{\text{major semiaxis}}=|\tanh(\delta z)|,\qquad
\frac{I(z)}{I(0)}=e^{-2\gamma z}\cosh(2\delta z)
=\frac{e^{-2\alpha^+z}+e^{-2\alpha^-z}}2.}
$$
The <ellipse>'s principal axis stays along the original input direction. For large differential optical depth it approaches the less attenuated circular <eigenmode>. A passive medium has $\alpha^\pm\geq0$, or $\gamma\geq|\delta|$, so the growing <hyperbolic function> factor does not imply net optical gain. Using the printed $\operatorname{sech}$ prefactor would give a spurious extra attenuation inconsistent with the modal laws.
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