34
2 Electrodynamics at Interface
The above equations (2.47), (2.48), (2.50), and (2.51) prove the relation of
Eq. (2.19) for both P and S polarizations in either phase of α or β.
Appendix
A.1 Singularity of Source Polarization
The boundary conditions of electromagnetic fields in Sect. 2.1 are affected by the
singularity of the nonlinear source polarization at the interface. The singularity is a
consequence of taking the limit of L → 0 for the interface thickness L. Here we
present some supplementary argument on the singularity issue.
Figure 2.4 show a picture of the three-layer model with a finite thickness L. The
thickness should be large enough to define the electric field and polarization in the
layer, while it is small compared to the scale of the light wavelengths. The nonlinear
source polarization P (2) inside the interface layer is given by
P
(2) (x, y, z) =
⎧
⎨
⎩
P S (x, y)
L
−
L
2
< z <
L
2
0
(otherwise)
(2.52)
Note that the limit of L → 0 recovers Eq. (2.6). In deriving Eq. (5.18) in Sect. 2.1
and Problem 2.1, we used the assumption that the tangential component of the
electric displacement, D t , has no singularity at the limit, though the nonlinear source
polarization, P
(2)
t , may have a singular component of δ(z). In the following we
explain these subtle assumptions. 2
Let us clarify the assumption by deriving the boundary condition of Eq. (2.8).
The Gauss divergence theorem is applied to Eq. (2.2) in the green rectangular box
in Fig. 2.4, and
Fig. 2.4 Geometry of the
three layers with a finite
thickness L of the interface.
The cross section of the
rectangular box is illustrated
in green
z
x (=t)
L
l/2
-l/2
l
-L/2
L/2
2 This problem is motivated by Ref. [2]. The author is grateful to Profs. Tony Heinz and Ron Shen
for clarifying this issue.