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A transverse electric polarization with constant permeability and conserved transverse wavenumber $k_0\sin {\theta }_0$ satisfies $e^{\prime \prime }+[{\beta }^2-({\nu }^2-1/4)/(z-z_0{)}^2]e=0$, with ${\beta }^2=k_0^2(a^2-{\sin }^2{\theta }_0)$ and ${\nu }^2=1/4+k_0^2{b}^2$. Substitution $e=\sqrt{|z-z_0|}{Z}_{\nu }(\beta |z-z_0|)$ reduces it to the Bessel equation. The profile is singular at $z_0$, so each side requires its own boundary conditions.

Use time dependence $e^{i\omega t}$ and consider fields independent of $y$, with $\mathbf{E}=\hat{\mathbf{y}}{E}_y(x,z)$. The source-free Maxwell's equations are $\nabla \times \mathbf{E}=-i\omega \mu \mathbf{H}$ and $\nabla \times \mathbf{H}=i\omega \epsilon \mathbf{E}$. The first givesSubstitute these into the $y$ component of the second equation. The TE wave equation in a stratified magnetodielectric isBecause its coefficients depend only on $z$, ${\partial }_x{E}_y$ obeys the same equation. Since it is proportional to $\mu H_z$, substitution and division by $\mu$ giveFor 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 isThe $H_z$ 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.

With permittivity and permeability depending only on $z$ and time dependence $e^{i\omega t}$, a transverse electric polarization $\mathbf{E}=\hat{\mathbf{y}}{E}_y(x,z)$ obeys the displayed equation. Maxwell's equations give $H_x=E_{y,z}/(i\omega \mu )$ and $H_z=-E_{y,x}/(i\omega \mu )$. The latter satisfies $H_{z,xx}+H_{z,zz}+({\mu }^{\prime }/\mu )H_{z,z}+[{\omega }^2\mu \epsilon +(\log \mu {)}^{\prime \prime }]H_z=0$. At normal incidence $H_z=0$; the nonzero magnetic component is $H_x$.