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.