= Solution
\b[Field equations.] In a homogeneous source-free nonmagnetic medium, $\mathbf D=\epsilon\varepsilon_0\mathbf E$ and $\mathbf B=\mu_0\mathbf H$. <Maxwell's equations> become
$$
\boxed{\nabla\cdot\mathbf E=0,\quad\nabla\cdot\mathbf H=0,\quad
\nabla\times\mathbf E=-\mu_0\partial_t\mathbf H,\quad
\nabla\times\mathbf H=\epsilon\varepsilon_0\partial_t\mathbf E.}
$$
The first equation follows from $\nabla\cdot\mathbf D=0$ for spatially constant nonzero $\epsilon$. When the effective relative <permittivity> is negative below, its charge response is already included in $\mathbf D$; it must not also be counted as an extra free source.
\b[Amplitude relations and bulk dispersion.] Work with complex <phasors> proportional to $e^{ikx-Kz-i\omega t}$. For $\mathbf H=(0,H_y,0)$, the <Ampere-Maxwell law> gives
$$
(KH_y,0,ikH_y)=-i\omega\epsilon\varepsilon_0(E_x,0,E_z),
$$
so the amplitudes must obey
$$
\boxed{E_x=\frac{iK}{\omega\epsilon\varepsilon_0}H_y,\qquad
E_z=-\frac{k}{\omega\epsilon\varepsilon_0}H_y.}
$$
They are not arbitrary independent constants: the dispersion relation alone would not make arbitrary amplitudes satisfy the equations. These relations give $ikE_x-KE_z=0$, so the electric divergence vanishes; the magnetic divergence vanishes because there is no $y$ dependence. The only nonzero component of <Faraday's law of induction> is
$$
-KE_x-ikE_z=i\omega\mu_0H_y.
$$
Substitution then gives
$$
\boxed{k^2-K^2=\omega^2\epsilon\varepsilon_0\mu_0
=\frac{\omega^2\epsilon}{c^2}.}
$$
This proves that all four field equations hold for the specified <transverse magnetic polarization> when both the amplitude relations and the dispersion relation are imposed.
\b[Interface matching.] Let region 1 be vacuum above the interface, with $\epsilon_1=1$, $K=\beta_1>0$; region 2 has $\epsilon_2=\epsilon$, $K=\beta_2<0$. These signs make the <evanescent> fields decay as $z\to\pm\infty$. In the absence of <free surface> current, tangential $H_y$ is continuous, giving a common interface amplitude. Tangential $E_x$ is also continuous, hence
$$
\frac{i\beta_1H_y}{\omega\varepsilon_0}
=\frac{i\beta_2H_y}{\omega\epsilon\varepsilon_0},
\qquad \boxed{\epsilon\beta_1=\beta_2.}
$$
Normal $D_z$ is continuous as well: both sides have $D_z=-kH_y/\omega$. This does not require $E_z$ itself to be continuous; the material can have polarization surface charge. The normal <magnetic field> is zero on both sides.
\b[Existence and surface dispersion.] The opposite signs of $\beta_1$ and $\beta_2$ require $\epsilon<0$. The two bulk equations are
$$
k^2-\beta_1^2=\frac{\omega^2}{c^2},\qquad
k^2-\beta_2^2=\frac{\omega^2\epsilon}{c^2}.
$$
Use $\beta_2^2=\epsilon^2\beta_1^2$ to eliminate the decay constants. For the nontrivial interface branch,
$$
\boxed{k^2=\frac{\omega^2}{c^2}\frac\epsilon{\epsilon+1},\qquad
\beta_1^2=-\frac{\omega^2/c^2}{\epsilon+1}.}
$$
A positive $\beta_1^2$ together with real propagating $k$ gives \b[$\epsilon<-1$]. A negative <permittivity> between $-1$ and zero is insufficient. This <lossless planar surface-plasmon dispersion> describes a <surface plasmon polariton>; $\epsilon=1$ would instead be identical media and supplies no decaying bound mode.
\b[Frequency range and long wavelengths.] With $\epsilon=1-\omega_p^2/\omega^2$, the existence condition becomes
$$
\boxed{0<\omega<\frac{\omega_p}{\sqrt2}.}
$$
The upper value is a limiting <frequency> approached as $k\to\infty$, not an attained maximum at finite $k$. Substitution into the dispersion relation gives
$$
k^2=\frac{\omega^2}{c^2}
\frac{\omega_p^2-\omega^2}{\omega_p^2-2\omega^2}.
$$
For $\omega\ll\omega_p$, choosing propagation in the positive $x$ direction,
$$
\boxed{k=\frac\omega c\left[1+\frac{\omega^2}{2\omega_p^2}
+O\left(\frac{\omega^4}{\omega_p^4}\right)\right]
\simeq\frac\omega c.}
$$
It approaches the vacuum light line from larger $k$. The vacuum decay constant is then $\beta_1\simeq\omega^2/(c\omega_p)$, so confinement becomes weak at low <frequency>. This calculation assumes the stated real, lossless effective <permittivity>; dissipation would require complex propagation and decay constants.
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