42
2 Electrodynamics at Interface
H t = 0:
H I,x + H R,x = H T ,x + H R ,x
:
1
K
q
i (I S − R S ) =
1
K
q
(T S
− R S
)
(2.69)
H I,y + H R,y = H T ,y + H R ,y
: n
i (−I P + R P ) = n
(−T P
+ R P
)
(2.70)
B z = (μH z ) = 0: where μ = 1 is assumed in the nonmagnetic media.
H I,z + H R,z = H T ,z + H R ,z
:
k x
K
(I S + R S ) =
k x
K
(T S
+ R S
)
(2.71)
In the above six boundary conditions, we notice that Eqs. (2.67) and (2.71),
and Eqs. (2.68) and (2.70) are equivalent and thus there exist four independent
conditions. These four independent conditions are summarized as follows:
From (2.66) :
q i
n i (I P + R P ) =
q
n (T P
+ R P
)
(2.72)
From (2.67), (2.71) :
I S + R S = T S
+ R S
(2.73)
From (2.69) :
q
i (I S − R S ) = q
(T S
− R S
)
(2.74)
From (2.68), (2.70) :
n
i (−I P + R P ) = n
(−T P
+ R P
)
(2.75)
(2) Interface and Medium j
E t = 0 :
E T ,x + E R ,x = E T ,x
:
q
n K
(T P
+ R P
) =
q j
n j K
T P
(2.76)
E T ,y + E R ,y = E T ,y
:
T S
+ R S
= T S
(2.77)
D z = 0:
ε
(E T ,z + E R ,z ) = ε
j E T ,z
:
ε k x
n K
(T P
− R P
) =
ε j k x
n j K
T P
(2.78)
H t = 0:
H T ,x + H R ,x = H T ,x
:
q
K
(T S
− R S
) =
q j
K
T S
(2.79)
H T ,y + H R ,y = H T ,y
:
n
(−T P
+ R P
) = −n
j T P
(2.80)
B z = (μH z ) = 0:
H T ,z + H R ,z = H T ,z
:
k x
K
(T S
+ R S
) =
k x
K
T S
(2.81)