= Solution
Use time dependence $e^{i\omega t}$ and consider fields independent of $y$, with $\mathbf E=\widehat{\mathbf y}E_y(x,z)$. The source-free <Maxwell's equations> are $\nabla\times\mathbf E=-i\omega\mu\mathbf H$ and $\nabla\times\mathbf H=i\omega\varepsilon\mathbf E$. The first gives
$$
H_x=\frac{\partial_zE_y}{i\omega\mu},
\qquad H_z=-\frac{\partial_xE_y}{i\omega\mu}.
$$
Substitute these into the $y$ component of the second equation. The <TE wave equation in a stratified magnetodielectric> is
$$
\boxed{E_{y,xx}+E_{y,zz}-\frac{\mu'}\mu E_{y,z}+\omega^2\mu\varepsilon E_y=0.}
$$
Because its coefficients depend only on $z$, $\partial_xE_y$ obeys the same equation. Since it is proportional to $\mu H_z$, substitution and division by $\mu$ give
$$
\boxed{H_{z,xx}+H_{z,zz}+\frac{\mu'}\mu H_{z,z}
+\left[\omega^2\mu\varepsilon+(\log\mu)''\right]H_z=0.}
$$
For a conserved transverse wavenumber $k_x$, replace $\partial_x^2$ by $-k_x^2$ to obtain the corresponding ordinary <Helmholtz equations>.
If “propagating along the $z$ axis” is read as strict normal incidence with no $x$ dependence, then $H_z=0$ identically. The nonzero magnetic component is $H_x$, and its equation is
$$
H_x''-\frac{\varepsilon'}{\varepsilon}H_x'+\omega^2\mu\varepsilon H_x=0.
$$
The $H_z$ equation above applies to the oblique <transverse electric polarization> used in the next part and also includes the zero normal-incidence solution. This distinguishes the named component from the usual transverse magnetic field.
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