Page 337 - Electromagnetics

and thus
ω 2
2 2
k = = k .
z
c 2
Substitution of k z = k into (4.390) gives the time-domain representation of the elemental
component
1 ∞ jω(t+z/c)
φ(z, t) = e dω.
2π
−∞
Finally, using the shifting theorem (A.3) along with (A.4), we have
z

φ(z, t) = δ t + , (4.391)
c
which we recognize as a uniform plane wave propagating in the −z-direction with velocity
c. There is no variation in the directions transverse to the direction of propagation and
the surface describing a constant argument of the δ-function at anytime t is a plane
perpendicular to the direction of propagation.
We can also consider the elemental spatial component in tandem with a single sinu-
soidal steady-state elemental component. The phasor representation of the elemental
spatial component is
ˇ
φ(z) = e jk z z = e jkz .
This elemental term is a time-harmonic plane wave propagating in the −z-direction.
Indeed, multiplying by e j ˇωt and taking the real part we get
φ(z, t) = cos( ˇωt + kz),
which is the sinusoidal steady-state analogue of (4.391).
Manyauthors choose to define the temporal and spatial transforms using differing
sign conventions. The temporal transform is defined as in (4.1) and (4.2), but the spatial
transform is defined through
∞

z jk z z
ψ (x, y, k z , t) = ψ(x, y, z, t)e dz, (4.392)
−∞
1 ∞ z − jk z z
ψ(x, y, z, t) = ψ (x, y, k z , t)e dk z . (4.393)
2π
−∞
This employs a wave traveling in the positive z-direction as the elemental spatial com-
ponent, which is quite useful for physical interpretation. We shall adopt this notation in
§ 4.13. The drawback is that we must alter the formulas from standard Fourier transform
tables (replacing k by −k) to reflect this difference.
In the following sections we shall show how a spatial Fourier decomposition can be
used to solve for the electromagnetic fields in a source-free region of space. Byemploying
the spatial transform we mayeliminate one or more spatial variables from Maxwell’s
equations, making the wave equation easier to solve. In the end we must perform an
inversion to return to the space domain. This maybe difficult or impossible to do
analytically, requiring a numerical Fourier inversion.

4.13 Spatial Fourier decomposition of two-dimensional fields
Consider a homogeneous, source-free region characterized by ˜ (ω), ˜µ(ω), and ˜σ(ω).
We seek z-independent solutions to the frequency-domain Maxwell’s equations, using

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