= Solution
Write $\gamma^\pm=-i\beta^\pm$, $\overline\beta=(\beta^++\beta^-)/2$ and $\chi=(\beta^+-\beta^-)z/2$. Then $\gamma z=-i\overline\beta z$ and $\delta z=-i\chi$. With $\cosh(-i\chi)=\cos\chi$ and $\sinh(-i\chi)=-i\sin\chi$, the corrected matrix becomes
$$
\boxed{\mathcal J(z)=e^{i\overline\beta z}
\begin{pmatrix}\cos\chi&-\sin\chi\\\sin\chi&\cos\chi\end{pmatrix}.}
$$
The scalar is a common propagation <wave phase> and the real <rotation matrix> rotates the input <linear polarization> through $\chi$. There is no attenuation and no generated ellipticity: the field map is a <unitary matrix>. The two circular <eigenmodes> accumulate different <wave phases>, giving <circular birefringence> and <optical activity>.
The correct unfactored matrix remains finite where $\cos\chi=0$. In contrast, the printed $\operatorname{sech}$ form would give $e^{i\overline\beta z}\sec^2\chi$ times the rotation matrix, including unphysical divergent gain in this stipulated lossless case. This provides a direct physical check on the correction in part (d).
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