Question

Surface Plasmons

An electromagnetic wave that can propagate along the surface of a metal complicates the observation of ordinary (bulk) plasmons. Let the metal be contained in the half space z > 0, z < 0 being vacuum. Assume that the electric charge density rho appearing in Maxwell’s equations vanishes both inside and outside the metal. (This does not preclude a surface charge density concentrated in the plane z = 0.) The surface plasmon is a solution to Maxwell’s equations of the form:
Ex = Aeᶦᵠˣe⁻ᴷᶻ, Ex = 0, E, Beᶦᵠˣe⁻ᴷᶻ, z> 0;
Ex = Ceᶦᵠˣe⁻ᴷᶻ, Ex = 0, E, Deᶦᵠˣe⁻ᴷᶻ z < 0; =
q, K, K' real, K, K' positive.
Assuming the usual boundary conditions (E║ continuous, (∈E)┴ continuous) and using the Drude results (1.35) and (1.29) find three equations relating q, K, and K′ as functions of to. ω

Answer 1

The question involves analyzing surface plasmons at a metal-vacuum boundary by applying Maxwell's equations and using boundary conditions to relate the wavevector components to frequency.

Step-by-step explanation:

The question pertains to the field of electrodynamics, specifically focusing on the propagation of surface plasmons at the interface between a metal and a vacuum. Surface plasmons are electromagnetic waves that can travel along the surface of a material, and their behavior is described by Maxwell's equations.

When considering a metal-vacuum interface, with the metal occupying the half space z > 0, Maxwell's equations can be used to determine the behavior of electromagnetic fields at the boundary. The conditions of the problem outline the form of the electric (E) and magnetic (B) fields, with the recognition that the usual boundary conditions for Maxwell's equations apply, necessitating the continuity of the parallel component of the electric field (E║) and normal component of the electric displacement field ((∈E)┴).

By integrating these conditions along with the permittivity results defined by the Drude model, one can derive expressions that relate the wavevector components q, K, and K' as functions of the angular frequency ω. This allows for the characterization of surface plasmons, which are important in various applications such as sensors and imaging techniques.