The generalized mirror image principle and its application in aperture antenna theory

Abstract

1. The generalization of the mirror image principle is described for the general case when the medium in the half-space T (x₃ > 0) is homogeneous and can contain scatterers, i.e. ideal conductors and ideal magnetics, a complement of which to the whole T is denoted as Ω⁺.

Let S be the plane x₃ = 0, and Ω^– be the image region of Ω⁺ with symmetric parameters relatively S that are not only geometrical but also physical ones, and let Ω = Ω⁺ v S v Ω^–. Then, as can be proved, there exist identities, which express the generalized mirror image principle:

*,                                (1)

where x,x₀ … is the field excited by an electrical dipole of the moment p in the region Ω. This dipole is located at the point x₀ and a prime over any vector means the mirror reflection operation with respect to S.

2. An integral equation is derived from the relation (1) and the Lorenz Lemma, being a generalization of the Feld definition [1] of the equivalent current method for the aperture antenna field calculation. Antenna radiates to a half-space filled with a non-homogeneous medium and containing scattering objects (dielectric, conducting, magnetic); in particular, it is applicable to an antenna system with a dielectric radome [2, 3].

Namely, let S₀ be an aperture in S plane, and the surface Σ (being an addition of S₀ to the whole S) has, from the side Ω⁺ of the boundary, the property of a perfect conductor (case A) or ideal magnetic (case B).

Let us denote by (E^A,H^A), (E^b,H^b) the fields, which are excited by the outside sources acting from the half-space x₃ < 0 in cases A, B, respectively, and introduce the “average field” E^c ={E^A + E^B)/2, H^c =(h^a +H^B)/2 (formally it correlates to McDonald’s model of Σ as a perfectly absorbing surface). Then as it turns out, for any point x₀ e Ω⁺ the following equation is valid

*. (2)

So

If a dielectric radome G is in homogeneous medium Ω⁺, the field (s₀,H₀) in (2) is a point-dipole field in the space containing only a closed dielectric shell G, which is symmetric with respect to S (including all geometrical and physical characteristics). In the Physical Optics approach E^A ~ E^B, H^A » H^B, that results in transforming the equality (2) into a design formula where the indices A,B,C are absent; for the complex function of the radiation pattern we obtain the following expression

*, (3)

where * is field excited under a shell G by the plane wave propagating along (—R°) and having polarization vector p.