Field Equivalence Principle

Suppose that the entire space is divided into two non-overlapping partitions by a closed surface S (Fig. 7). Let us denote the interior region by V₁, and the exterior by V₂. According to the field equivalence principle (a formal representation of the Huygens-Fresnel principle³³), if one is only interested in the electromagnetic fields in the exterior region (), the geometry and sources in the interior can be removed and replaced by a set of surface electric and magnetic currents impressed on the boundary of the two regions³⁴. More rigorously, the field equivalence principle can be expressed by an illustrative form known as the Kottler’s formulation³³:

(a) The entire space is divided into two non-overlapping partitions. (b) In order to calculate the electromagnetic fields in the exterior region, geometry and sources of the interior can be replaced by a set of surface electric and magnetic currents impressed on the boundary.

where, and represent total electric and magnetic field intensities, ω is radian frequency, ε and μ are electric permittivity and magnetic permeability of the background medium, k is the wavenumber, and is the scalar Green’s function of free-space. Also, primed and unprimed variables denote source and observation spaces [see Fig. 7]. Detailed justification of the above relations can be found in ref. ³³. It is noteworthy that, in Eqs (2) and (3), if is chosen in the interior region (V₁), the calculated fields will be identical to zero (extinction theorem)³³. In other words, the aforementioned relations are only valid for obtaining the fields outside the closed surface S.

By comparing volume and surface integrals of Eqs (2) and (3), one can define a set of equivalent surface currents [see Fig. 7(b)]

These surface current densities do not necessarily have any physical counterparts. To recap, the whole sources and geometries inside the surface can be replaced by a set of fictitious equivalent surface current densities to calculate the filed at an observation point in the exterior region.

Yet, there are some practical issues in solving real-world problems by this method. First of all, we are interested in problems involving open boundaries in lots of scenarios including Fig. 1. For instance, radiation characteristics of an aperture antenna. It is a very well-known concept in antenna engineering³³ that, as the aperture dimension of a radiator becomes larger (typically larger than a wavelength), calculating the integrals in Eqs (2) and (3) only on the finite aperture of the antenna gives a very accurate prediction of the radiated fields, particularly, the main lobe of the radiation pattern.

Secondly, the total field intensities at the boundary S are usually the unknowns we are looking for and we do not have them in the first place to calculate the equivalent currents. However, it might be possible to calculate an acceptable local approximation of total fields given the inherent characteristic of the problem. Due to the large radii of curvature of the surface, compared to the wavelength, the conformal metasurface array can be approximated by locally flat inclusions experiencing proper plane wave excitation¹⁹.