18
2 Electrodynamics at Interface
A
E · dl = {E t (z = +0) − E t (z = −0)} l +
4π
ε
ˆ
t · ∇ t P
S
z
l + o(l) = 0,
and in the limit of l → 0, the matching condition for E t is derived
E t = E t (z = +0) − E t (z = −0) = −
4π
ε
ˆ
t · ∇ t P
S
z (x, y).
(2.9)
It is noted that the tangential component of the electric field, E t , is not continuous
at the boundary z = 0, unlike the conventional boundary condition of electric field,
due to the source polarization P S .
The same argument also holds for the matching condition for H . The Stokes
theorem and Eq. (2.5) lead to
(d)
A
H · dl =
S
(∇ × H ) · (ˆ z × ˆ
t)ds =
S
1
c
∂D
∂t
+ 4π
∂P (2)
∂t
· (ˆ z × ˆ
t)ds.
Noting that H z and D t are regular at the interface, the following condition is
obtained for H t :
H t =
4π
c
ˆ
t ·
∂P S (x, y)
∂t
× ˆ
z
.
(2.10)
[Problem 2.2] Derive Eq. (2.10), after the discussion associated to Eq. (2.9).
The above equations (2.7), (2.8), (2.9), and (2.10) define the boundary conditions of
electromagnetic fields at the interface, z = 0, which involves the nonlinear source
polarization.
2.1.3 SFG Signal Emitted from Interface
Let us consider the spatial geometry of lights in Fig. 2.1, where all the wave vectors
of visible (ω 1 ), infrared (ω 2 ) and sum frequency ( = ω 1 + ω 2 ) lights are on the xz
plane. k
i (ω) designates the wave vector of light at frequency ω (= , ω 1 , ω 2 ) in the
bulk region i (= α or β). Suppose that two pump lights of ω 1 and ω 2 are incident
from the bulk region i = α onto the interface. These incident electric fields at ω 1
and ω 2 in the region i = α are expressed by
E
α
I (ω 1 ) exp(ik
α
I (ω 1 ) · r − iω 1 t) + c.c.,
2 Electrodynamics at Interface
A
E · dl = {E t (z = +0) − E t (z = −0)} l +
4π
ε
ˆ
t · ∇ t P
S
z
l + o(l) = 0,
and in the limit of l → 0, the matching condition for E t is derived
E t = E t (z = +0) − E t (z = −0) = −
4π
ε
ˆ
t · ∇ t P
S
z (x, y).
(2.9)
It is noted that the tangential component of the electric field, E t , is not continuous
at the boundary z = 0, unlike the conventional boundary condition of electric field,
due to the source polarization P S .
The same argument also holds for the matching condition for H . The Stokes
theorem and Eq. (2.5) lead to
(d)
A
H · dl =
S
(∇ × H ) · (ˆ z × ˆ
t)ds =
S
1
c
∂D
∂t
+ 4π
∂P (2)
∂t
· (ˆ z × ˆ
t)ds.
Noting that H z and D t are regular at the interface, the following condition is
obtained for H t :
H t =
4π
c
ˆ
t ·
∂P S (x, y)
∂t
× ˆ
z
.
(2.10)
[Problem 2.2] Derive Eq. (2.10), after the discussion associated to Eq. (2.9).
The above equations (2.7), (2.8), (2.9), and (2.10) define the boundary conditions of
electromagnetic fields at the interface, z = 0, which involves the nonlinear source
polarization.
2.1.3 SFG Signal Emitted from Interface
Let us consider the spatial geometry of lights in Fig. 2.1, where all the wave vectors
of visible (ω 1 ), infrared (ω 2 ) and sum frequency ( = ω 1 + ω 2 ) lights are on the xz
plane. k
i (ω) designates the wave vector of light at frequency ω (= , ω 1 , ω 2 ) in the
bulk region i (= α or β). Suppose that two pump lights of ω 1 and ω 2 are incident
from the bulk region i = α onto the interface. These incident electric fields at ω 1
and ω 2 in the region i = α are expressed by
E
α
I (ω 1 ) exp(ik
α
I (ω 1 ) · r − iω 1 t) + c.c.,