etric field deco ition. Field sym source distributions through a symmetry decomposition of the sources and fields. Con sider the general impressed source distributions (J, Jm). The source set #(xy,z)=5[H(x,y,2)+f(x,y,-) J(,y,z)=5[(x,,2)+(x,y,-) J,(x, y, z) 2 [(x,y,z) Jm(x, y, z) 2 Jm.(,y,z) I(x, y, z) (x,y,z)-my(x,y,-2) learly of even symmetric type while the source set (a,y,)=1 [(x,y,x)-J(x,y,-2 1 J(x [(x,y.z)-(x,y,-2) (x,y,x)=2(x,y.3)+(,y-], Jm(,y, z)=5Jm(x, y, z)+Jmx(x, y, -2) m(,y,x)=5[m(x,y,x)+ J is of the odd symmetric type. SinceJ'=J+J and Jm= Jin +Jm, we can decompose any source into constituents having, respectively, even and odd symmetry with respect to a plane. The source with even symmetry produces an even field set, while the source with odd symmetry produces an odd field set. The total field is the sum of the fields Planar symmetry for frequency-domain fields. The symmetry conditions intro- duced above for the time-domain fields also hold for the frequency-domain fields. Because both the conductivity and permittivity must be even functions, we combine their effects and require the complex permittivity to be even. Otherwise the field symmetries and urce decompositions are identical Example of symmetry decomposition: line source between conducting planes. Consider a z-directed electric line source lo located at y =h, x=0 between conducting planes at y=td, d>h. The material between the plates has permeability i(o) and complex permittivity E(o). We decompose the source into one of even symmetric type ith line sources lo/2 located at y= th, and one of odd symmetric type with a line ②2001Symmetric field decomposition. Field symmetries may be applied to arbitrary source distributions through a symmetry decomposition of the sources and fields. Con￾sider the general impressed source distributions (Ji , Ji m). The source set J ie x (x, y,z) = 1 2 J i x (x, y,z) + J i x (x, y, −z)  , J ie y (x, y,z) = 1 2 J i y (x, y,z) + J i y (x, y, −z)  , J ie z (x, y,z) = 1 2 J i z (x, y,z) − J i z (x, y, −z)  , J ie mx (x, y,z) = 1 2 J i mx (x, y,z) − J i mx (x, y, −z)  , J ie my (x, y,z) = 1 2 J i my (x, y,z) − J i my (x, y, −z)  , J ie mz(x, y,z) = 1 2 J i mz(x, y,z) + J i mz(x, y, −z)  , is clearly of even symmetric type while the source set J io x (x, y,z) = 1 2 J i x (x, y,z) − J i x (x, y, −z)  , J io y (x, y,z) = 1 2 J i y (x, y,z) − J i y (x, y, −z)  , J io z (x, y,z) = 1 2 J i z (x, y,z) + J i z (x, y, −z)  , J io mx (x, y,z) = 1 2 J i mx (x, y,z) + J i mx (x, y, −z)  , J io my (x, y,z) = 1 2 J i my (x, y,z) + J i my (x, y, −z)  , J io mz(x, y,z) = 1 2 J i mz(x, y,z) − J i mz(x, y, −z)  , is of the odd symmetric type. Since Ji = Jie + Jio and Ji m = Jie m + Jio m , we can decompose any source into constituents having, respectively, even and odd symmetry with respect to a plane. The source with even symmetry produces an even field set, while the source with odd symmetry produces an odd field set. The total field is the sum of the fields from each field set. Planar symmetry for frequency-domain fields. The symmetry conditions intro￾duced above for the time-domain fields also hold for the frequency-domain fields. Because both the conductivity and permittivity must be even functions, we combine their effects and require the complex permittivity to be even. Otherwise the field symmetries and source decompositions are identical. Example of symmetry decomposition: line source between conducting planes. Consider a z-directed electric line source ˜I0 located at y = h, x = 0 between conducting planes at y = ±d, d > h. The material between the plates has permeability µ(ω) ˜ and complex permittivity ˜ c(ω). We decompose the source into one of even symmetric type with line sources ˜I0/2 located at y = ±h, and one of odd symmetric type with a line