ion gives the correspondence between the time-dependence of the electromagnetic wave (&omega), and the spatial variation (k);
the wavelength of the wave is given by &lambda=2&pi/k. Figure A-1 shows a plot of the bulk plasmon dispersion relation (solid line), along with the free space dispersion relation (&omega = ck). For frequencies above the plasma frequency &omegap, the metal supports propagating modes. For frequencies below &omegap, k becomes imaginary and the fields exponentially decay inside the metal. For typical metals, h&omegap/2&pi &asymp 10eV (in the ultra-violet); for semiconductors, h&omegap/2&pi < 0.5eV (in the terahertz regime). Although bulk plasmons, which can exist even deep inside a metal, do not exist for &omega < &omegap, there is an additional solution to Maxwell's equations for these frequencies: surface plasmons.
Surface plasmons are propagating waves bound to plasmon oscillations of electrons at a metal-dielectric interface. The field intensity of the wave is at a maximum at the interface and exponentially decays away from the interface. The mathematical form of surface plasmons can be obtained by solving Maxwell's equations at a metal-dielectric interface:
Assuming the metal-dielectric interface lies in the xy plane at z = 0, it can be shown that x-propagating surface waves of the following form are solutions to Maxwell's equations:
with Ex,m = Ex,d (E|| continuous), &epsilonmEz,m = &epsilondEz,d (D&perp continuous), and Hy,m = Hy,d (B|| continuous); all other field components are zero. The subscript j is used to denote fields in the metal or dielectric. The dispersion relation for surface plasmons can be obtained by inserting the equations for E and H into Maxwell's equations and enforcing the boundary conditions:
The dispersion relation, assuming &epsilonm = 1 - (&omegap/&omega)2, is shown in Fig. 1, along with the free-space and bulk plasmon dispersion relations. The surface plasmon dispersion relation always lies to the right of the free-space dispersion relations; as a result, free-