(r,t)分joy(r,o) By virtue of(4.2), any electromagnetic field component can be decomposed into a contin- uous, weighted superposition of elemental temporal terms eJor. Note that the weighting factor y(r, a), often called the frequency spectrum of y(r, t), is not arbitrary because y(r, t)must obey a scalar wave equation such as(2. 327). For a source-free region of pace we have for do =o Differentiating under the integral sign we have 2T/[(v2-joHua +a ue)v(r, a)Jelo do =0 hence by the Fourier integral theorem he 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 Maxwells equations and represent all quantities using the temporal inverse Fourier transform. We have, for ex- (r,t) E(r, o)eor do he r, o) ∑E:(r (4.6) All other field quantities will be written similarly with an appropriate superscript on the phase. Substitution into Ampere's law gives r D(r, o)ejr do+ H(r,)-joD(r,o)一 0 2001 by CRC Press LLChence ∂ ∂t ψ(r, t) ↔ jωψ( ˜ r, ω). (4.4) 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 − µσ ∂ ∂t − µ ∂2 ∂t 2  1 2π ∞ −∞ ψ( ˜ r,ω) e jωt dω = 0. Differentiating under the integral sign we have 1 2π ∞ −∞ ∇2 − jωµσ + ω2 µ ψ( ˜ r,ω) e jωt dω = 0, hence by the Fourier integral theorem  ∇2 + k2 ψ( ˜ r,ω) = 0 (4.5) where k = ω √µ 1 − j σ 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, E(r, t) = 1 2π ∞ −∞ E˜(r,ω) e jωt dω where E˜(r,ω) = 3 i=1 ˆii E˜i(r,ω) = 3 i=1 ˆii|E˜i(r,ω)|e jξ E i (r,ω). (4.6) All other field quantities will be written similarly with an appropriate superscript on the phase. Substitution into Ampere’s law gives ∇ × 1 2π ∞ −∞ H˜ (r,ω) e jωt dω = ∂ ∂t 1 2π ∞ −∞ D˜ (r,ω) e jωt dω + 1 2π ∞ −∞ J˜(r,ω) e jωt dω, hence 1 2π ∞ −∞ [∇ × H˜ (r,ω) − jωD˜ (r,ω) − J˜(r,ω)]e jωt dω = 0