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Field equations. In a homogeneous source-free nonmagnetic medium, $\mathbf{D}=ϵ{\epsilon }_0\mathbf{E}$ and $\mathbf{B}={\mu }_0\mathbf{H}$. Maxwell's equations becomeThe first equation follows from $\nabla \cdot \mathbf{D}=0$ for spatially constant nonzero $ϵ$. 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.

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 givesso the amplitudes must obeyThey are not arbitrary independent constants: the dispersion relation alone would not make arbitrary amplitudes satisfy the equations. These relations give $ik{E}_x-K{E}_z=0$, 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 isSubstitution then givesThis proves that all four field equations hold for the specified transverse magnetic polarization when both the amplitude relations and the dispersion relation are imposed.

Interface matching. Let region 1 be vacuum above the interface, with ${ϵ}_1=1$, $K={\beta }_1>0$; region 2 has ${ϵ}_2=ϵ$, $K={\beta }_2<0$. 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, henceNormal $D_z$ is continuous as well: both sides have $D_z=-k{H}_y/\omega$. 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.

Existence and surface dispersion. The opposite signs of ${\beta }_1$ and ${\beta }_2$ require $ϵ<0$. The two bulk equations areUse ${\beta }_2^2={ϵ}^2{\beta }_1^2$ to eliminate the decay constants. For the nontrivial interface branch,A positive ${\beta }_1^2$ together with real propagating $k$ gives $ϵ<-1$. A negative permittivity between $-1$ and zero is insufficient. This lossless planar surface-plasmon dispersion describes a surface plasmon polariton; $ϵ=1$ would instead be identical media and supplies no decaying bound mode.

Frequency range and long wavelengths. With $ϵ=1-{\omega }_p^2/{\omega }^2$, the existence condition becomesThe upper value is a limiting frequency approached as $k\to \infty$, not an attained maximum at finite $k$. Substitution into the dispersion relation givesFor $\omega \ll {\omega }_p$, choosing propagation in the positive $x$ direction,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.

Plasmonics studies and uses the coupling of electromagnetic waves with collective electron motion, especially at material surfaces and in small metallic structures. A surface plasmon polariton propagates along an interface while its field decays away from it; a localized surface plasmon resonance confines an oscillating response near a finite structure. Such modes can concentrate fields on scales smaller than the free-space wavelength.

For an ideal flat interface between a lossless metal of relative permittivity ${\epsilon }_m$ and a dielectric of relative permittivity ${\epsilon }_d$, the bound transverse-magnetic mode hasWith ${\epsilon }_d>0$ and ${\epsilon }_m<-{\epsilon }_d$, its parallel wave number exceeds that of a propagating wave in the dielectric. This explains why direct illumination of a perfectly flat interface needs additional momentum coupling. A grating or nearby optical field can supply that coupling. Real metals have loss: stronger confinement does not automatically imply long propagation distances.

A striking experiment used periodic arrays of tiny holes in metal films. In 1998, T. W. Ebbesen and colleagues reported transmission peaks at wavelengths much larger than individual hole diameters. One silver sample had 150-nanometre holes, 900-nanometre spacing and 200-nanometre film thickness. A spectrophotometer measured the transmitted beam, and angle-dependent spectra supported coupling to surface electronic excitations. Efficiency normalized to hole area could exceed one; this compares with the small open area, not with all incident power.^[ref]