Page 203 - Electromagnetics

hence
∂
˜
ψ(r, t) ↔ jωψ(r,ω). (4.4)
∂t
By virtue of (4.2), any electromagnetic field component can be decomposed into a contin-
uous, weighted superposition of elemental temporal terms e jωt . Note that the weighting
˜
factor ψ(r,ω), often called the frequency spectrum of ψ(r, t), is not arbitrary because
ψ(r, t) must obey a scalar wave equation such as (2.327). For a source-free region of
space we have
2 ∞
∂ ∂ 1
2 jωt
˜
∇ − µσ − µ ψ(r,ω) e dω = 0.
∂t ∂t 2 2π −∞
Differentiating under the integral sign we have
1 ∞ 2 2 jωt
˜
∇ − jωµσ + ω µ ψ(r,ω) e dω = 0,
2π −∞
hence by the Fourier integral theorem
˜
2 2
∇ + k ψ(r,ω) = 0 (4.5)
where

√ σ
k = ω µ 1 − j
ω
is the wavenumber. Equation (4.5) is called the scalar Helmholtz equation, and represents
the wave equation in the temporal frequency domain.

4.2 The frequency-domain Maxwell equations

If the region of interest contains sources, we can return to Maxwell’s equations and
represent all quantities using the temporal inverse Fourier transform. We have, for ex-
ample,
1 ∞ jωt
˜
E(r, t) = E(r,ω) e dω
2π
−∞
where
3 3
E
˜
ˆ ˜
ˆ ˜ jξ (r,ω)
E(r,ω) = i i E i (r,ω) = i i |E i (r,ω)|e i . (4.6)
i=1 i=1
All other field quantities will be written similarly with an appropriate superscript on the
phase. Substitution into Ampere’s law gives
1 ∞ jωt ∂ 1 ∞ jωt 1 ∞ jωt
˜
˜
˜
∇× H(r,ω) e dω = D(r,ω) e dω + J(r,ω) e dω,
2π ∂t 2π 2π
−∞ −∞ −∞
hence
1 ∞ jωt
˜
˜
˜
[∇× H(r,ω) − jωD(r,ω) − J(r,ω)]e dω = 0
2π
−∞
© 2001 by CRC Press LLC