Eigenmode reciprocity in k-space

Abstract

Until the mid 1960’s the problem of reciprocity in electromagnetics had been developing in two separate, and seemingly unrelated directions. As early as 1896 Lorentz [91] had demonstrated that if two independent current distributions, J ₁(r) and J ₂(r), generated electromagnetic fields, E ₁(r), H ₁(r) and E ₂(r), H ₂(r) respectively, in free space, then

$$ \int {{E_1}(r) \cdot {J_2}(r){d^3}r = \int {{E_2}(r) \cdot {J_1}(r){d^3}r} } $$

((2.1))

and this was recognized as an expression of the ‘interchangeability’ of transmitting and receiving antennas. This, or an equivalent formulation,

$$\nabla \cdot ({E_1} \times {H_2} - {E_2} \times {H_1}) = 0$$

((2.2))

became to be known as the Lorentz reciprocity theorem, and will be discussed in some detail in Chap. 4. The theorem was used, inter alia, to deduce the properties of transmitting antennas if their properties as receiving antennas were known. Eckersley [51], for instance, used the theorem to deduce the radiation pattern of a transmitting antenna as modified by an imperfectly conducting ground below it, by solving the simpler problem of its response as a receiving antenna.