Page 346 - Electromagnetics

Figure 4.31: Geometry of a z-directed line source above an interface between two material
regions.

c 1/2
2
2
where k y1 = k − k and k 1 = ω( ˜µ 1 ˜ ) . From (4.404) we find that
1 x 1
∞+ j
˜ i
1 ∂E ˜ I(ω) e jk y1 (y−h)
˜ i z − jk x x
H =− = e dk x , 0 ≤ y < h.
x
jω ˜µ 1 ∂y 2π 2
−∞+ j
The scattered field obeys the homogeneous Helmholtz equation for all y > 0, and thus
may be written using (4.400) as a superposition of upward-traveling waves:
∞+ j
1
˜ s − jk y1 y − jk x x
E (x, y,ω) = A 1 (k x ,ω)e e dk x ,
z1
2π
−∞+ j
∞+ j
1 k y1
˜ s − jk y1 y − jk x x
H (x, y,ω) = A 1 (k x ,ω)e e dk x .
x1
2π ω ˜µ 1
−∞+ j
Similarly, in region 2 the scattered field may be written as a superposition of downward-
traveling waves:
∞+ j
1
˜ s jk y2 y − jk x x
E (x, y,ω) = A 2 (k x ,ω)e e dk x ,
z2
2π
−∞+ j
∞+ j
1 k y2
˜ s jk y2 y − jk x x
H (x, y,ω) =− A 2 (k x ,ω)e e dk x ,
x2
2π ω ˜µ 2
−∞+ j

2
c 1/2
2
where k y2 = k − k and k 2 = ω( ˜µ 2 ˜ ) .
x
2
2
We can solve for the angular spectra A 1 and A 2 by applying the boundary conditions
at the interface between the two media. From the continuity of total tangential electric
field we find that
∞+ j
˜
1 ω ˜µ 1 I(ω) − jk y1 h − jk x x
− e + A 1 (k x ,ω) − A 2 (k x ,ω) e dk x = 0,
2π 2k y1
−∞+ j
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