Since this gives the contribution to the field in V from the fields on the surface receding to infinity, we expect that this term should be zero. If the medium has loss, then the exponential term decays and drives the contribution to zero. For a lossless medium the contributions are zero if lim rE(r,o)<∞, (69) limr[nf x H(r,o)+E(r,o]=0 To accompany(6. 8)we al lim rh(r,o)<∞, (6.11) lim r[nH(r,o)-fxE(r,o)=0 We refer to (6.9) and(611)as the finiteness conditions, and to(6.10) and(6. 12)as the Sommerfeld radiation condition, for the electromagnetic field. They show that far from the sources the fields must behave as a wave tem to the r-direction We shall se 86.2 that the waves are in fact spherical TEM waves 6.1.3 Fields in the excluded region: the extinction theorem The Stratton-Chu formula provides a solution for the field within the region V, external to the excluded regions. An interesting consequence of this formula, and one that helps us identify the equivalence principle, is that it gives the null resultH=E=0when evaluated at points within the excluded regions We can show this by considering two cases. In the first case we do not exclude the particular region Vm, but do exclude the remaining regions Vn, n# Then the electric field everywhere outside the remaining excluded regions(including at points within Vm) is,by(6.7), E(r,o)= JmxVG+2V'G-joij'Gdv+ G+(·E)VG-jd h)G]ds' drHVG In the second case we apply the Stratton-Chu formula only to Vn, and exclude all other regions. We incur a sign change on the surface and line integrals compared to the first case because the normal is now directed oppositely. By(6.7) we have E(r, o)= Jm xVG+=)dV E)V'G-joa(n'xhGdS'+ ②2001 by CRC Press LLC
Since this gives the contribution to the field in V from the fields on the surface receding to infinity, we expect that this term should be zero. If the medium has loss, then the exponential term decays and drives the contribution to zero. For a lossless medium the contributions are zero if lim r→∞ rE˜(r,ω) < ∞, (6.9) lim r→∞ r ηrˆ × H˜ (r,ω) + E˜(r,ω) = 0. (6.10) To accompany (6.8) we also have lim r→∞ rH˜ (r,ω) < ∞, (6.11) lim r→∞ r ηH˜ (r,ω) − rˆ × E˜(r,ω) = 0. (6.12) We refer to (6.9) and (6.11) as the finiteness conditions, and to (6.10) and (6.12) as the Sommerfeld radiation condition, for the electromagnetic field. They show that far from the sources the fields must behave as a wave TEM to the r-direction. We shall see in § 6.2 that the waves are in fact spherical TEM waves. 6.1.3 Fields in the excluded region: the extinction theorem The Stratton–Chu formula provides a solution for the field within the region V, external to the excluded regions. An interesting consequence of this formula, and one that helps us identify the equivalence principle, is that it gives the null result H˜ = E˜ = 0 when evaluated at points within the excluded regions. We can show this by considering two cases. In the first case we do not exclude the particular region Vm, but do exclude the remaining regions Vn, n = m. Then the electric field everywhere outside the remaining excluded regions (including at points within Vm) is, by (6.7), E˜(r,ω) = V+Vm −J˜i m × ∇ G + ρ˜i ˜ c ∇ G − jωµ˜ J˜i G dV + + n =m Sn (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G d S − − n =m 1 jω˜ c na+ nb (dl · H˜ )∇ G, r ∈ V + Vm. In the second case we apply the Stratton–Chu formula only to Vm, and exclude all other regions. We incur a sign change on the surface and line integrals compared to the first case because the normal is now directed oppositely. By (6.7) we have E˜(r,ω) = Vm −J˜i m × ∇ G + ρ˜i ˜ c ∇ G − jωµ˜ J˜i G dV − − Sm (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G d S + + 1 jω˜ c na+ nb (dl · H˜ )∇ G, r ∈ Vm.
Each of the expressions for E is equally valid for points within Vm. Upon subtraction we xV'G+ opj'gdv'+ 点人 (AXE)XV'G+Bwo-0mxm(4y (d·HVG,r∈V This expression is exactly the Stratton-Chu formula(6.7)evaluated at points within the excluded region Vm. The treatment of H is analogous and is left as an exercise. Since we may repeat this for any excluded region, we find that the Stratton-Chu formula returns the null field when evaluated at points outside V. This is sometimes referred to as the vector Ewald-Oseen extinction theorem 90. We must emphasize that the fields within the excluded regions are not generally equal to zero; the Stratton-Chu formula merely returns this result when evaluated there 6.2 Fields in an unbounded medium Two special cases of the Stratton-Chu formula are important because of their applica- tion to antenna theory. The first is that of sources radiating into an unbor region The second involves a bounded region with all sources excluded. We shall consider the former here and the latter in 86.3 Assuming that there are no bounding surfaces in(6.7)and( 6.8), except for one surface that has been allowed to recede to infinity and therefore provides no surface contribution ve find that the electromagnetic fields in unbounded space are given by JixV G+P G)dv n=(①xv+vo-/)w We can view the right-hand sides as superpositions of the fields present in the cases There(1)electric sources are present exclusively, and(2)magnetic sources are present clusively. With Pm =0 and Jm=0 we find that ⅴG-jojG)d (6.13) H=/J×vGdv. 6.14) Using V'G=-VG we can write (r ②2001 by CRC Press LLC
Each of the expressions for E˜ is equally valid for points within Vm. Upon subtraction we get 0 = V −J˜i m × ∇ G + ρ˜i ˜ c ∇ G − jωµ˜ J˜i G dV + + N n=1 Sn (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G d S − − N n=1 1 jω˜ c na+ nb (dl · H˜ )∇ G, r ∈ Vm. This expression is exactly the Stratton–Chu formula (6.7) evaluated at points within the excluded region Vm. The treatment of H˜ is analogous and is left as an exercise. Since we may repeat this for any excluded region, we find that the Stratton–Chu formula returns the null field when evaluated at points outside V. This is sometimes referred to as the vector Ewald–Oseen extinction theorem [90]. We must emphasize that the fields within the excluded regions are not generally equal to zero; the Stratton–Chu formula merely returns this result when evaluated there. 6.2 Fields in an unbounded medium Two special cases of the Stratton–Chu formula are important because of their application to antenna theory. The first is that of sources radiating into an unbounded region. The second involves a bounded region with all sources excluded. We shall consider the former here and the latter in § 6.3. Assuming that there are no bounding surfaces in (6.7) and (6.8), except for one surface that has been allowed to recede to infinity and therefore provides no surface contribution, we find that the electromagnetic fields in unbounded space are given by E˜ = V −J˜i m × ∇ G + ρ˜i ˜ c ∇ G − jωµ˜ J˜i G dV , H˜ = V J˜i × ∇ G + ρ˜i m µ˜ ∇ G − jω˜ c J˜i m G dV . We can view the right-hand sides as superpositions of the fields present in the cases where (1) electric sources are present exclusively, and (2) magnetic sources are present exclusively. With ρ˜i m = 0 and J˜i m = 0 we find that E˜ = V ρ˜i ˜ c ∇ G − jωµ˜ J˜i G dV , (6.13) H˜ = V J˜i × ∇ G dV . (6.14) Using ∇ G = −∇G we can write E˜(r,ω) = −∇ V ρ˜i (r ,ω) ˜ c(ω) G(r|r ; ω) dV − jω V µ(ω) ˜ J˜i (r ,ω)G(r|r ; ω) dV = −∇φ˜ e(r,ω) − jωA˜ e(r, ω),
p(r, o) E@)G(rr; o)dv (r,a=/A(o)(r,o)G(rr;a)dV are the electric scalar and vector potential functions introduced in$5.2. UsingJ'xV'G G=V×(JjG) we have H(r, o) vxa(oJ(r, o)G(rr; o)dv V×A(r,a). (6.16) These expressions for the fields are identical to those of (5.56)and(5.57), and thus the integral formula for the electromagnetic fields produces a result identical to that obtained using potential relations. Similarly, with p=0,J=0 we have E I xV'GdV H V'G-joe G dv E(r,) V×A H(r, o)=-Vph(r, o) φh(r,o) G(rr dv 正(au) Ah (r, o)= 2(o)J (r,o)G(rr;o)dv are the magnetic scalar and vector potentials introduced in 8 5.2 6.2.1 The far-zone fields produced by sources in unbounded space Many antennas may be analyzed in terms of electric currents and charges radiating in unbounded space. Since antennas are used to transmit information over great distances the fields far from the sources are often of most interest assume that the sources are contained within a sphere of radius rs centered at the origin. We define the far zone of the sources to consist of all observation points satisfying both r >>rs(and thus r>r) and kr >1. For points in the far zone we may approximate the unit vector R directed from the sources to the observation point by the unit vector f directed from the origin to the observation point. We may also approximate dR(4R VR=R(I+jkr)e-jkR =PikE R 4丌R 6.17) ②2001 by CRC Press LLC
where φ˜ e(r,ω) = V ρ˜i (r ,ω) ˜ c(ω) G(r|r ; ω) dV , A˜ e(r,ω) = V µ(ω) ˜ J˜i (r ,ω)G(r|r ; ω) dV , (6.15) are the electric scalar and vector potential functions introduced in § 5.2. Using J˜i ×∇ G = −J˜i × ∇G =∇× (J˜i G) we have H˜ (r,ω) = 1 µ(ω) ˜ ∇ × V µ(ω) ˜ J˜i (r ,ω)G(r|r ; ω) dV = 1 µ(ω) ˜ ∇ × A˜ e(r, ω). (6.16) These expressions for the fields are identical to those of (5.56) and (5.57), and thus the integral formula for the electromagnetic fields produces a result identical to that obtained using potential relations. Similarly, with ρ˜i = 0, J˜i = 0 we have E˜ = − V J˜i m × ∇ G dV , H˜ = V ρ˜i m µ˜ ∇ G − jω˜ c J˜i m G dV , or E˜(r,ω) = − 1 ˜ c(ω) ∇ × A˜ h(r, ω), H˜ (r,ω) = −∇φ˜ h(r,ω) − jωA˜ h(r, ω), where φ˜ h(r,ω) = V ρ˜i m(r ,ω) µ(ω) ˜ G(r|r ; ω) dV , A˜ h(r,ω) = V ˜ c (ω)J˜i m(r ,ω)G(r|r ; ω) dV , are the magnetic scalar and vector potentials introduced in § 5.2. 6.2.1 The far-zone fields produced by sources in unbounded space Many antennas may be analyzed in terms of electric currents and charges radiating in unbounded space. Since antennas are used to transmit information over great distances, the fields far from the sources are often of most interest. Assume that the sources are contained within a sphere of radius rs centered at the origin. We define the far zone of the sources to consist of all observation points satisfying both r rs (and thus r r ) and kr 1. For points in the far zone we may approximate the unit vector Rˆ directed from the sources to the observation point by the unit vector rˆ directed from the origin to the observation point. We may also approximate ∇ G = d d R e− jkR 4π R ∇ R = Rˆ 1 + jkR R e− jkR 4π R ≈ rˆ jk e− jkR 4π R = rˆ jkG. (6.17)