40
2 Electrodynamics at Interface
Fig. 2.9 Light waves
associated to the incident
light k I in the three layer
model
j
i
'
k I
k R
k T’
k R’
k T
z
x
Each electromagnetic field is presented as
E X (r, ω, t) = E X exp(ik X · r − iωt) + c.c.
(2.62)
H X (r, ω, t) =
1
K
(k X × E X ) exp(ik X · r − iωt) + c.c
= H X exp(ik X · r − iωt) + c.c.
(2.63)
where X = I , R, T , R , or T . K = ω/c is the angular wavenumber of the light
in vacuo. The wavevector k X and the coefficients E X , H X for each wave X are
summarized as follows.
k X
E X
H X
I :
k I =
⎛
⎝
k x
0
−q i
⎞
⎠ ,
E I =
⎛
⎜
⎜
⎜
⎝
q i
n i K
I P
I S
k x
n i K
I P
⎞
⎟
⎟
⎟
⎠
,
H I =
1
K
⎛
⎝
q i I S
−n i KI P
k x I S
⎞
⎠ ,
R :
k R =
⎛
⎝
k x
0
q i
⎞
⎠ ,
E R =
⎛
⎜
⎜
⎜
⎝
q i
n i K
R P
R S
−
k x
n i K
R P
⎞
⎟
⎟
⎟
⎠
, H R =
1
K
⎛
⎝
−q i R S
n i KR P
k x R S
⎞
⎠ ,
T
: k T =
⎛
⎝
k x
0
−q
⎞
⎠ , E T =
⎛
⎜
⎜
⎜
⎝
q
n K
T P
T S
k x
n K
T P
⎞
⎟
⎟
⎟
⎠
,
H T =
1
K
⎛
⎝
q T S
−n KT P
k x T S
⎞
⎠ ,
R
: k R =
⎛
⎝
k x
0
q
⎞
⎠ ,
E R =
⎛
⎜
⎜
⎜
⎝
q
n K
R P
R S
−
k x
n K
R P
⎞
⎟
⎟
⎟
⎠
, H R =
1
K
⎛
⎝
−q R S
n KR P
k x R S
⎞
⎠ ,