The Field of a Microwave Dipole Antenna in the Vicinity of the Horizon

In this paper a method is developed for determining the electromagnetic field produced by a microwave antenna at points on the horizon, and on either side of it, where neither the ray theory nor the normal mode theory can be used conveniently. The theory is developed for a condition of standard atmospheric refraction, by use of a space in which the earth is flattened and the rays are curved. This allows us to make a simple derivation of the ray theory, valid in the optical region, and of the normal mode theory, suited for the shadow zone. For the intermediate region centered around the horizon we use the original integral for the potential to obtain expressions for the field under the restriction of maximum absorption, which for typical ground conditions applies to wave‐lengths less than about a meter. Three cases are treated in which the transmitter, or receiver, are either situated on the ground or are elevated several natural units of height. For an elevated transmitter and receiver the Hertzian potential ψ due to a point source at the origin is in the vicinity of the horizon given by $${\text{|Ψ|=[1/2(rr̄)}}^{\frac{1}{2}}]\left|{\text{(2/π}}^{\frac{1}{2}}{\mathrm{)e}}^{\text{iπ/4}}{\displaystyle {\int }_0^{\infty }}\hspace{0.17em}\exp {\text{[−i(2τ}}^{\frac{1}{2}}{\mathrm{t+t}}^2{\text{)]dt−[(2/3)}}^{\frac{1}{3}}{\mathrm{\Omega /\pi }}^{\frac{1}{2}}{\mathrm{]e}}^{\text{−iπ/12}}\mathrm{F(p)}\right|$$ , with $$\begin{array}{c}{\text{p=(3/2)}}^{\frac{2}{3}}{\mathrm{(x-x̄),\hspace{0.17em}\Omega }}^2{\mathrm{=(z}}_1^{-\frac{1}{2}}{\mathrm{+z}}_2^{-\frac{1}{2}}\mathrm{),}\\ {\mathrm{\tau =(x-x̄)}}^2{\mathrm{/\Omega }}^2{\mathrm{,\hspace{0.17em}x̄=z}}_1^{\frac{1}{2}}{\mathrm{+z}}_2^{\frac{1}{2}}\end{array}$$ , where x denotes the horizontal distance r expressed in natural units, z₁ and z₂ the heights of transmitter and receiver in natural units, r̄ the distance of receiver from transmitter when the former is on the horizon. F(p) (see Eq. (68) below) has been evaluated, and is given in Table IV, while the integral in (A) can be expressed in terms of the tabulated Fresnel integrals. In the limit of very short wave‐lengths the field on the horizon approaches the value 1/(2r̄) which would result from the diffraction of the direct ray only by a straight edge placed at the point of tangency of the horizon with the earth. A comparison of the field obtained from (A) with exact values computed by van d. Pol and Bremmer, using the ray theory and the normal mode theory, is shown in Figs. 6 and 7. When the transmitter is at zero elevation and the receiver is elevated several units of height, the potential in the vicinity of the horizon is given by $${\mathrm{|\Psi }}_v\mathrm{|=}\frac{{\text{(3/2πa}}_e{)}^{\frac{1}{3}}{\lambda }^{\frac{1}{3}}}{{\mathrm{\pi (rr̄)}}^{\frac{1}{2}}}\left|\frac{{\mathrm{G(p)\epsilon }}_1}{{\mathrm{(\epsilon }}_1{\text{−1)}}^{\frac{1}{2}}}\right|$$ , $${\mathrm{|\Psi }}_h\mathrm{|=}\frac{{\text{(3/2πa}}_e{)}^{\frac{1}{3}}{\lambda }^{\frac{1}{3}}}{{\mathrm{\pi (rr̄)}}^{\frac{1}{2}}}\left|\frac{\mathrm{G(p)}}{{\mathrm{(\epsilon }}_1{\text{−1)}}^{\frac{1}{2}}}\right|$$ for vertical polarization and horizontal polarization, respectively. Here ε₁ denotes the complex dielectric constant, a_c the effective radius of the earth, and λ the wave‐length. G(p) is given in Eq. (78) and is shown in Fig. 4. A comparison of (B) with exact values obtained by van d. Pol and Bremmer is shown in Fig. 8. When both the transmitter and receiver are at zero elevation, it is found that the potential can be expressed as the sum of the surface wave appropriate for a flat ground and an integral depending on the radius of the earth. At great distances, the two terms tend to cancel out. Under conditions of maximum absorption this leads to $${\mathrm{|\Psi }}_v\mathrm{|=}\frac{2}{r^2{k}_0}\left|\frac{{\epsilon }_1^2\mathrm{g(p\prime )}}{{\mathrm{(\epsilon }}_1\text{−1)}}\right|{\mathrm{;\hspace{0.17em}\hspace{0.17em}|\Psi }}_h\mathrm{|=}\frac{2}{r^2{k}_0}\left|\frac{\mathrm{g(p\prime )}}{{\mathrm{(\epsilon }}_1\text{−1)}}\right|$$ , $${\text{p′=(3/2)}}^{\frac{2}{3}}{\text{x,\hspace{0.17em}\hspace{0.17em}g(p′)=1−(2e}}^{\text{iπ/4}}{\text{/3π}}^{\frac{1}{2}}{\mathrm{)(p\prime )}}^{\frac{3}{2}}\mathrm{H(p\prime )}$$ .H (p′) is given in (84) and g(p′) is shown in Fig. 5, where it is compared with results obtained previously by van d. Pol and Bremmer using the normal mode theory. For points on the horizon Eq. (A) reduces to $$\text{|Ψ|=(1/2r̄)|1−(0.684−0.183i)Ω|}$$ , while in (B) and (C) we put |G(0)| = 2.13.