= Solution
A real transmitted normal wavenumber exists only if
$$
\sin^2\theta_I\leq\frac{\varepsilon_+}{\varepsilon_-},
$$
so the <critical angle for total internal reflection> is
$$
\boxed{\theta_I^{\rm cr}=\arcsin\sqrt{\frac{\varepsilon_+}{\varepsilon_-}}}.
$$
For $\theta_I>\theta_I^{\rm cr}$, put
$$
k_x=k_I\cos\theta_I,
\qquad k_z=k_I\sin\theta_I,
\qquad
\kappa=k_I\sqrt{\sin^2\theta_I-\frac{\varepsilon_+}{\varepsilon_-}}.
$$
For incident amplitude $E_0\widehat{\mathbf y}$, the <transverse-electric Fresnel reflection coefficient> and transmission coefficient are
$$
r=\frac{k_x-i\kappa}{k_x+i\kappa},
\qquad
t=1+r=\frac{2k_x}{k_x+i\kappa}.
$$
A boundary-matched solution is
$$
\mathbf E=\begin{cases}
\operatorname{Im}\!\left\{E_0\widehat{\mathbf y}\left[e^{i(k_xx+k_zz-\omega t)}+r e^{i(-k_xx+k_zz-\omega t)}\right]\right\},&x<0,\\
\operatorname{Im}\!\left\{E_0t\widehat{\mathbf y}\,e^{-\kappa x}e^{i(k_zz-\omega t)}\right\},&x>0,
\end{cases}
$$
with each magnetic component obtained from $\mathbf B_X=\mathbf k_X\times\mathbf E_X/\omega$, now using the complex transmitted wavevector $(i\kappa,0,k_z)$. Since $|r|=1$, the wave undergoes <total internal reflection>; the transmitted field is an <evanescent wave> that propagates parallel to the interface and decays exponentially into $x>0$.
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