Page 351 - Electromagnetics

√ √
2 2
Setting the integrand to zero and using k − k = k − k x k + k x ,wehave
x

f cos −1 k x k E 0 h(k x )
˜
√ (k x + k x0 ) = k + k x . (4.420)
2 jπ h(−k x0 )
k − k x
The left member has a branch point at k x = k while the right member has a branch point
at k x = −k. If we choose the branch cuts as in Figure 4.30 then since f is regular in
the region k xi > the left side of (4.420) is regular there. Also, since h(k x ) is regular in
the region k xi < , the right side is regular there. We assert that since the two sides are
equal, both sides must be regular in the entire complex plane. By Liouville’s theorem
[35] if a function is entire (regular in the entire plane) and bounded, then it must be
constant. So

f cos −1 k x ˜ h(k x )
k E 0
√ (k x + k x0 ) = k + k x = constant.
k − k x 2 jπ h(−k x0 )
We may evaluate the constant by inserting any value of k x . Using k x =−k x0 on the right
we find that

f cos −1 k x ˜
k E 0
√ (k x + k x0 ) = k − k x0 .
k − k x 2 jπ
Substituting k x = k cos ξ and k x0 = k cos φ 0 we have
√ √
˜
E 0 1 − cos φ 0 1 − cos ξ
f (ξ) = .
2 jπ cos ξ + cos φ 0
√
Since sin(x/2) = (1 − cos x)/2, we may also write
˜ sin φ 0 sin ξ
E 0 2 2
f (ξ) = .
jπ cos ξ + cos φ 0
Finally, substituting this into (4.415) we have the spectral representation for the field
scattered by a half-plane:
˜
E 0 (ω) sin φ 0 sin ξ
˜ s 2 2 − jkρ cos(φ±ξ)
E (ρ, φ, ω) = e dξ. (4.421)
z
jπ C cos ξ + cos φ 0
The scattered field inversion integral in (4.421) may be rewritten in such a way as to
separate geometrical optics (plane-wave) terms from diffraction terms. The diffraction
terms may be written using standard functions (modified Fresnel integrals) and for large
values of ρ appear as cylindrical waves emanating from a line source at the edge of the
half-plane. Interested readers should see James [92] for details.

4.14 Periodic fields and Floquet’s theorem
In several practical situations EM waves interact with, or are radiated by, structures
spatially periodic along one or more directions. Periodic symmetry simplifies field com-
putation, since boundary conditions need only be applied within one period, or cell,of
the structure. Examples of situations that lead to periodic fields include the guiding of
waves in slow-wave structures such as helices and meander lines, the scattering of plane
waves from gratings, and the radiation of waves by antenna arrays. In this section we
will study the representation of fields with infinite periodicity as spatial Fourier series.

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