44
2 Electrodynamics at Interface
From (2.86) :
q i
n i (I P + R P ) =
q j
n j T P
(2.92)
From (2.87), (2.91) :
I S + R S = T S
(2.93)
From (2.89) :
q
i (I S − R S ) = q
j T S
(2.94)
From (2.88), (2.90) :
n
i (−I P + R P ) = −n
j T P
(2.95)
Actually Eqs. (2.92), (2.93), (2.94), (2.95) are derived from Eqs. (2.72), (2.73),
(2.74), (2.75) and (2.82), (2.83), (2.84), (2.85). There are a total of eight independent
boundary conditions in the three-layer model, which are two sets among Eqs. (2.72),
(2.73), (2.74), (2.75), (2.82), (2.83), (2.84), (2.85) and (2.92), (2.93), (2.94), (2.95).
We choose Eqs. (2.72), (2.73), (2.74), (2.75) and (2.92), (2.93), (2.94), (2.95) in the
following. These conditions are utilized to determine the relations among I P , I S ,
R P , R S , T P
, T S
, R P
, R S
, T P and T S .
A.3.3 Solution of Boundary Equations
Using Eqs. (2.92), (2.93), (2.94), (2.95), following two relations are readily derived:
R P =
ε i q j − ε j q i
ε i q j + ε j q i I P ,
(2.96)
R S =
q i − q j
q i + q j I S .
(2.97)
P component By using Eq. (2.96), Eqs. (2.72) and (2.75) are converted to
T P
+ R P
=
n q i
n i q
1 +
ε i q j − ε j q i
ε i q j + ε j q i
I P =
2n n i q i q j
q (ε i q j + ε j q i )
I P , (2.98)
T P
− R P
=
n i
n
1 −
ε i q j − ε j q i
ε i q j + ε j q i
I P =
n i
n
2ε j q i
ε i q j + ε j q i I P .
(2.99)
Equations (2.98) and (2.99) allow us to relate the P component of the electric field
in the interface layer (T P
, R P
) to the P component of the incident field (I P ).
Using these equations along with Eqs. (2.64), the following relations for the x and z
components are derived:
E T ,x + E R ,x =
q
n K
(T P
+ R P
) =
q
n K
2n n i q i q j
q (ε i q j + ε j q i )
I P
2 Electrodynamics at Interface
From (2.86) :
q i
n i (I P + R P ) =
q j
n j T P
(2.92)
From (2.87), (2.91) :
I S + R S = T S
(2.93)
From (2.89) :
q
i (I S − R S ) = q
j T S
(2.94)
From (2.88), (2.90) :
n
i (−I P + R P ) = −n
j T P
(2.95)
Actually Eqs. (2.92), (2.93), (2.94), (2.95) are derived from Eqs. (2.72), (2.73),
(2.74), (2.75) and (2.82), (2.83), (2.84), (2.85). There are a total of eight independent
boundary conditions in the three-layer model, which are two sets among Eqs. (2.72),
(2.73), (2.74), (2.75), (2.82), (2.83), (2.84), (2.85) and (2.92), (2.93), (2.94), (2.95).
We choose Eqs. (2.72), (2.73), (2.74), (2.75) and (2.92), (2.93), (2.94), (2.95) in the
following. These conditions are utilized to determine the relations among I P , I S ,
R P , R S , T P
, T S
, R P
, R S
, T P and T S .
A.3.3 Solution of Boundary Equations
Using Eqs. (2.92), (2.93), (2.94), (2.95), following two relations are readily derived:
R P =
ε i q j − ε j q i
ε i q j + ε j q i I P ,
(2.96)
R S =
q i − q j
q i + q j I S .
(2.97)
P component By using Eq. (2.96), Eqs. (2.72) and (2.75) are converted to
T P
+ R P
=
n q i
n i q
1 +
ε i q j − ε j q i
ε i q j + ε j q i
I P =
2n n i q i q j
q (ε i q j + ε j q i )
I P , (2.98)
T P
− R P
=
n i
n
1 −
ε i q j − ε j q i
ε i q j + ε j q i
I P =
n i
n
2ε j q i
ε i q j + ε j q i I P .
(2.99)
Equations (2.98) and (2.99) allow us to relate the P component of the electric field
in the interface layer (T P
, R P
) to the P component of the incident field (I P ).
Using these equations along with Eqs. (2.64), the following relations for the x and z
components are derived:
E T ,x + E R ,x =
q
n K
(T P
+ R P
) =
q
n K
2n n i q i q j
q (ε i q j + ε j q i )
I P