Page 303 - Electromagnetics

Figure 4.20: Interaction of a uniform plane wave with a multi-layered material.

region, except region N, contains an incident-type wave of the form
˜ i
˜ i − jk ·r
E (r,ω) = E e i
0
and a reflected-type wave of the form
˜ r − jk ·r
˜ r
E (r,ω) = E e r .
0
In region n we maywrite the wave vectors describing these waves as
r
i
k = ˆ xk x,n + ˆ zk z,n , k = ˆ xk x,n − ˆ zk z,n ,
n
n
where
2
2
2
2
c
k 2 + k 2 = k , k = ω ˜µ n ˜ = (β n − jα n ) .
x,n z,n n n n
We note at the outset that, as with the single interface case, the boundaryconditions
are onlysatisfied when Snell’s law of reflection holds, and thus
(4.304)
k x,n = k x,0 = k 0 sin θ i
c 1/2
where k 0 = ω( ˜µ 0 ˜ ) is the wavenumber of the 0th region (not necessarilyfree space).
0
From this condition we have

2
k z,n = k − k 2 = τ n e − jγ n
n x,0
where
1 −1 B n
2
2 1/4
τ n = (A + B ) , γ n = tan ,
n n
2 A n
and
2
2
2
2
2
2
A n = β − α − (β − α ) sin θ i , B n = 2(β n α n − β 0 α 0 sin θ i ).
n n 0 0
Provided that the incident wave is uniform, we can decompose the fields in everyregion
into cases of perpendicular and parallel polarization. This is true even when the waves
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