Surface Plasmon - Dispersion Relation

Dispersion Relation

The electric field of a propagating electromagnetic wave can be expressed

where k is the wave number and ω is the frequency of the wave. By solving Maxwell's equations for the electromagnetic wave at an interface between two materials with relative dielectric functions ε₁ and ε₂ (see figure 3) with the appropriate continuity relation the boundary conditions are

and

where c is the speed of light in a vacuum, and k_x is same for both media at the interface for a surface wave. Solving these two equations, the dispersion relation for a wave propagating on the surface is

In the free electron model of an electron gas, which neglects attenuation, the metallic dielectric function is

where the bulk plasma frequency in SI units is

where n is the electron density, e is the charge of the electron, m* is the effective mass of the electron and is the permittivity of free-space. The dispersion relation is plotted in Figure 4. At low k, the SP behaves like a photon, but as k increases, the dispersion relation bends over and reaches an asymptotic limit corresponding to the surface plasma frequency. Since the dispersion curve lies to the right of the light line, ω = k·c, the SP has a shorter wavelength than free-space radiation such that the out-of-plane component of the SP is purely imaginary and exhibits evanescent decay. The surface plasma frequency is given by

In the case of air, this result simplifies to

If we assume that ε₂ is real and ε₂ > 0, then it must be true that ε₁ < 0, a condition which is satisfied in metals. Electromagnetic waves passing through a metal experience damping due to Ohmic losses and electron-core interactions. These effects show up in as an imaginary component of the dielectric function. The dielectric function of a metal is expressed ε₁ = ε₁' + i·ε₁" where ε₁' and ε₁" are the real and imaginary parts of the dielectric function, respectively. Generally |ε₁'| >> ε₁" so the wavenumber can be expressed in terms of its real and imaginary components as

The wave vector gives us insight into physically meaningful properties of the electromagnetic wave such as its spatial extent and coupling requirements for wave vector matching.