d remembering that r >>r for r in the far zone, we can use the leading terms of a binomial expansion of the square root to get (6.19) Thus the Green's function may be approximated as Gr;o)≈re Here we have kept the approximation(6.19) intact in the phase of G but have used 1/RN 1/r in the amplitude of G. We must keep a more accurate approximation for the phase since k(f. r)may be an appreciable fraction of a radian. We thus have the far-zone approximation for the vector potential A-(r,o)≈p(a) Let us summarize the expressions for computing the far-zone field E(r, o) (6.21) f×E(r,o) (6.22) Ae(r, o) arri(o)ae(0, a), (6.23) Here ae is called the directional weighting function. This function is independent of r and describes the angular variation, or pattern, of the fields In the far zone e, H, f are mutually orthogonal. Because of this, and because the fields vary as e-jkr /r, the electromagnetic field in the far zone takes the form of a spherical TEM wave which is consistent with the Sommerfeld radiation condition Power radiated by time-harmonic sources in unbounded space. In$5. 2.1 we defined the power radiated by a time-harmonic source in unbounded space as the total time-average power passing through a sphere of very large radius. We found that for a Hertzian dipole the radiated power could be computed from the far-zone fields through P where E×H} is the time-average Poynting vector. By superposition this holds for any localized source. Assuming a lossless medium and using phasor notation to describe the time-harmonic ②2001 by CRC Press LLCand remembering that r  r for r in the far zone, we can use the leading terms of a binomial expansion of the square root to get R = r  1 − 2(rˆ · r ) r + r r 2 ≈ r  1 − 2(rˆ · r ) r ≈ r  1 − rˆ · r r  ≈ r − rˆ · r . (6.19) Thus the Green’s function may be approximated as G(r|r ; ω) ≈ e− jkr 4πr e jkrˆ·r . (6.20) Here we have kept the approximation (6.19) intact in the phase of G but have used 1/R ≈ 1/r in the amplitude of G. We must keep a more accurate approximation for the phase since k(rˆ · r ) may be an appreciable fraction of a radian. We thus have the far-zone approximation for the vector potential A˜ e(r,ω) ≈ µ(ω) ˜ e− jkr 4πr  V J˜i (r ,ω)e jkrˆ·r dV , which we may use in computing (6.18). Let us summarize the expressions for computing the far-zone fields: E˜(r,ω) = − jω  θˆ A˜ eθ (r,ω) + φˆ A˜ eφ(r,ω) , (6.21) H˜ (r,ω) = rˆ × E˜(r,ω) η , (6.22) A˜ e(r,ω) = e− jkr 4πr µ(ω) ˜ a˜e(θ, φ, ω), (6.23) a˜e(θ, φ, ω) =  V J˜i (r ,ω)e jkrˆ·r dV . (6.24) Here a˜e is called the directional weighting function. This function is independent of r and describes the angular variation, or pattern, of the fields. In the far zone E˜ , H˜ , rˆ are mutually orthogonal. Because of this, and because the fields vary as e− jkr /r, the electromagnetic field in the far zone takes the form of a spherical TEM wave, which is consistent with the Sommerfeld radiation condition. Power radiated by time-harmonic sources in unbounded space. In § 5.2.1 we defined the power radiated by a time-harmonic source in unbounded space as the total time-average power passing through a sphere of very large radius. We found that for a Hertzian dipole the radiated power could be computed from the far-zone fields through Pav = lim r→∞  2π 0  π 0 Sav · rˆr 2 sin θ dθ dφ where Sav = 1 2 Re  Eˇ × Hˇ ∗ is the time-average Poynting vector. By superposition this holds for any localized source. Assuming a lossless medium and using phasor notation to describe the time-harmonic