10. Appendix

636 Appendix B
both media with the appropriate value for  (keeping in mind that the dielectric
constant for a solid is a function of the frequency of the electromagnetic
wave ˆ). As trial we assume a plane wave solution:
E(x, y, z) E0 exp[i(kxx kyy kzz ˆt)].
Since both media are isotropic we can choose, without loss of generality, the
direction of propagation of the wave to be in the yz-plane. Substituting this
plane wave solution into the wave equation we obtain the dispersion relation:
k 2 yn k2 zn (ˆ/c)2 Ân where n A ⇔ vacuum and n B ⇔ solid.
For the special case where ÂB is 0 and ÂA is 0, k 2 zB (ˆ/c)2 ÂB k 2 yB
is also
0 and kzB is purely imaginary. We will define kzB ±i· such that · is 0.
If we choose kzB i· we obtain a solution whose dependence on z varies
as exp(·z). This solution represents a surface wave since its amplitude decays
exponentially to 0 as z decreases from 0 to ∞. However, the condition that
ÂB 0 is necessary but not sufficient for a surface wave to exist.
To obtain the necessary and sufficient condition we will try to obtain the
relation between the wave vector of the wave ky and its frequency ˆ (ie the
dispersion relation). In the absence of any sources of charge and current at
the interface, Maxwell’s Equations for a nonmagnetic medium can be written
as:
div D 0; div H 0; curl E (1/c)H/t; and curl H (1/c)D/t,
where E and H are the electric and magnetic field vectors, respectively. In addition,
we have to include the constitutive equation D ÂBE for the solid.
The boundary conditions imposed on E, D and H by the Maxwell Equations
become:
zx(EAEB)0; zx(HAHB)0; z · (DADB)0 and z · (HAHB)0.
The subscripts A and B now denote the electric and magnetic fields at the
interface, lying within the vacuum and the solid, respectively. For example, EA
represents the electric field at z 0 ‰ with ‰ 0 in the limit ‰ ⇒ 0.
Again we assume plane wave solutions of the form:
E and H ∼ exp[i(kyy ˆt)] exp[ikzz]
for waves in both media but with the understanding that kzB i·, so that
these solutions represent surface waves. To determine the amplitudes of the
waves in the two media we apply the above boundary conditions. As an example,
we take the simple case of a transverse magnetic solution (a transverse
electric solution will give similar results):
Hy Hz 0 and
HxA CA exp[i(kyAy ˆt)] exp[ikzAz];
HxB CB exp[i(kyBy ˆt)] exp[ikzBz].