Question

Dispersion Relation (Surface Plasma Wave)

3. (a) Show that the dispersion relation for surface plasma waves 2 26 eff 2 eff N

Answer 1

dielectric The fields of a surface plasma wave at metal-dielectric interface are governed by Maxwell's equation in each medium and associated boundary conditions. Boundary conditions, involve the continuity of the tangential component of the electric field and magnetic field across the interface and vanishing of these field away from the interface. consider a dielectric with relative permittivity E, for x 7o and a metal SPW with effective relative permittivity E (W) in the space aco. metal (E) when a surface plasma wave (spW) propagates at the interface with (1,7) variation as "(WE - K₂2) then its behavious is governed by Maxwell's equations. Txes - 0 at Taking worl of equation and using equation - we get - DE- 2 20 - where t = E(W) = E - up for a Lo and E = t for a 70. where & is the lattice permittivity, ap at the plasma Frequency. The continuity conditions on electromagnetic fields at the interface demand that the t and a components or variation of fields be the same in both media. Thus replacing at and in equation - we get by ike and - in respectively, we get

- («?-o?&") ES0 - 0 The well-behaved solutions are, Ē = (2xã ik ) A e dix - 1 (wt - KzZ) for xro, and Ě = (2- â ikz) A e dyte (wt - K22) for x 20. a ca where x, = K₂ - 6,W²; d = kr ² - w²t (w). The real part of a, and d must be positive for an electromagnetic wave localized to the dielectric-metal interface at x o. The continuity of Ex and € Ex yeilds (at the interface) AaB 02 Ei substituting all the values and rearranging, then we Here, - Ex and E(W) = Ceff. Therefore, we then have the required solution as, This is the required one.