10. Appendix

Solution to Problem 6.11 637
The continuity of the tangential component of H at z 0 implied by the
equation: zx(HA HB) 0 means that kyA kyB (which we will now denote
as k ||) and CA CB.
The corresponding solution for E can be obtained from the equation:
curl H (1/c)D/t. Substituting the solution of Ey F exp[i(k ||y
ˆt)] exp[ikzz] and Ez G exp[i(k ||y ˆt)] exp[ikzz] for both media A and
B into the boundary conditions we obtain the following relations:
kzCn (ˆÂn/c)Fn and k ||Cn (ˆÂn/c)Gn, where n A or B.
The continuity of Ey at z 0 implies that
FA FB or kzA/ÂA kzB/ÂB,
while the continuity of Ez at the interface is trivially satisfied and generates
no additional nontrivial equation.
When this result is substituted back into the two expressions: k 2 || k2 zA
(ˆ/c) 2 ÂA and k 2 || k2 zB (ˆ/c)2 ÂB to eliminate kzA and kzB one obtains an
expression containing k || only:
ˆ
2 ÂAÂB k
c
2 || (ÂA ÂB)
By taking the square root of both sides of this equation we obtain the dispersion
of the surface wave:
ˆ
ÂAÂB
k ||
c (ÂA ÂB)
Since we have assumed ÂB 0 and ÂA 0, ÂAÂB is 0. In order that k ||
be real we find that another condition, ÂA ÂB 0, has to be satisfied. If
A is vacuum, then ÂA 1 and we find that the medium B must satisfy the
condition that ÂB 1.
For metals the dielectric function corresponding to the free electrons can
be written as (see Problem 6.3):
 1 4appleNe2
mˆ2 1 ˆ2 p
ˆ2 where ˆp is the bulk plasmon frequency of the metal. In analogy to the bulk
plasmon oscillation, the frequency ˆsp at which a long wave length plasma oscillation
can exist on the surface of the metal is known as the surface plasmon
frequency. This frequency is given by:
ˆsp
4appleNe 2
2m
ˆp
√2
We can obtain the surface plasmon dispersion by replacing ÂA with 1 and
ÂB with the expression for the dielectric function of the metal in the surface
electromagnetic wave dispersion. Notice that in the special case of k || ∞,
ÂB 1 and the frequency of the surface electromagnetic wave is equal
to ˆsp. In this limit the photon (or electromagnetic wave in free space) and