Page 361 - Electromagnetics

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4.26 Consider a z-directed electric line source I(ω) located on the y-axis at y = h.
The region y < 0 contains a perfect electric conductor. Write the fields in the regions
0 < y < h and y > h in terms of the Fourier transform solution to the homogeneous
Helmholtz equation. Note that in the region 0 < y < h terms representing waves traveling
in both the ±y-directions are needed, while in the region y > h only terms traveling in
the y-direction are needed. Apply the boundary conditions at y = 0, h to determine the
spectral amplitudes. Show that the total field may be decomposed into an impressed
term identical to (4.410) and a scattered term identical to (4.413).
˜
4.27 Consider a z-directed magnetic line source I m (ω) located on the y-axis at y = h.
c
The region y > 0 contains a material with parameters ˜ (ω) and ˜µ 1 (ω), while the region
1
c
y < 0 contains a material with parameters ˜ (ω) and ˜µ 2 (ω). Using the Fourier transform
2
solution to the Helmholtz equation, write the total field for y > 0 as the sum of an
impressed field of the magnetic line source and a scattered field, and write the field for
y < 0 as a scattered field. Apply the boundary conditions at y = 0 to determine the
spectral amplitudes. Can you interpret the scattered fields in terms of images of the line
source?
4.28 Consider a TE-polarized plane wave incident on a PEC half-plane located at
y = 0, x > 0. If the incident magnetic field is given by

˜ i
˜
H (r,ω) = ˆ zH 0 (ω)e jk(x cos φ 0 +y sin φ 0 ) ,
determine the appropriate boundary conditions on the fields at y = 0. Solve for the
scattered magnetic field using the Fourier transform approach.

4.29 Consider the layered medium of Figure 4.34 with alternating layers of free space
and perfect dielectric. The dielectric layer has permittivity 4 0 and thickness while
the free space layer has thickness 2 . Assuming a normally-incident plane wave, solve
for k 0 in terms of κ , and plot k 0 versus κ, identifying the stop and pass bands. This
type of ω–β plot for a periodic medium is named a Brillouin diagram, after L. Brillouin
who investigated energy bands in periodic crystal lattices [23].
4.30 Consider a periodic layered medium as in Figure 4.34, but with each cell con-
sisting of three different layers. Derive an eigenvalue equation similar to (4.427) for the
propagation constant.

© 2001 by CRC Press LLC