26
2 Electrodynamics at Interface
The term for the surface integral on the side,
side
B · n dσ , can be neglected in the
limit of infinitesimal thickness, as B(r) is regular. Therefore,
B z = B z (z = +0) − B z (z = −0) = 0.
(2.7)
Equation (2.8) We integrate the Maxwell equation of (a) (Eq. (2.2)) in the volume V ,
V
(∇ · D)dr = −4π
V
∇ ·
P
S (x, y)δ(z)
dr
and apply the Gauss’ theorem to both sides of this equation. Thus the following
relation for the surface integrals is obtained,
σ
D · ndσ = −4π
σ
P
S (x, y)δ(z)
· n dσ.
The surface integrals on σ are explicitly represented as follows,
{D z (z = +0) − D z (z = −0)} l
2
+
side
D · n dσ
= −4π
⎡
⎣
P
S
z δ(z = +0) − P
S
z δ(z = −0)
l
2
+
side
P
S (x, y)δ(z) · n dσ
⎤
⎦ .
(2.26)
In the left-hand side of Eq. (2.26), the surface integral on the side planes is neglected
in the limit of infinitesimal thickness. This is because the tangential component of
D, D t , has no singularity in the interface (see the discussion in Appendix A.1). In the
right-hand side of Eq. (2.26), the surface integral on the top and bottom planes has no
contribution as there is no nonlinear polarization P S outside the interface, whereas
the integral on the side planes could remain finite due to the singular component of
P (2) (δ-function in Eq. (2.6)) in the interface. Consequently, Eq. (2.26) becomes
{D z (z = +0) − D z (z = −0)} l
2
= D z l
2
= −4π
side
P
S (x, y)δ(z) · n dσ
= −4π
y+l/2
y−l/2
P
S
x
x +
l
2
, y
− P
S
x
x −
l
2
, y
dy