Generalized Current Green's Function Formalism for Electromagnetic Radiation by Composite Systems

Abstract

We provide an explicit geometric generalisation of the antenna current Green's function (ACGF) formalism from the perfect electric conducting (PEC) to generic interacting N-body systems composed of arbitrarily shaped coupled PEC and dielectric objects, with the main emphasis on the mathematical foundations and the rigorous construction of the Green's function using distributional limits. Starting from mainly reciprocity, surface equivalence theorems, and other typical regularity conditions, we carefully construct the current Green's function by employing a combination of methods including Riemannian geometry, distribution theory, and functional analysis. The formalism outlined here for composite domains turns out to be more complicated than the PEC-only formulation due to the former's need to explicitly account for the coupling interaction between the magnetic and electric degrees of freedom. The approach is developed for extremely general systems, and use is made of Riemannian geometry to avoid working with specific or concrete configurations, hence retaining high generality in our final conclusions. While the ACGF tensor's matrix representations depend on the coordinate system on the manifolds supporting the electromagnetic boundary conditions, we focus here on providing coordinate-independent integral expressions for the induced current. With the ACGF it is possible to theoretically treat arbitrary N-body coupled PEC-dielectric configurations as spacefrequency linear systems with an exact and rigorous response function being the current Green's function itself. While the derivation is very general, it still leaves open questions regarding whether the ACGF can be constructed for nonreciprocal systems or using volume integral equations.

Fundamental Spacetime Representations of Quantum Antenna Systems

“…The forward Green's function is the standard retarded radiation Green's function G(x − x , t − t ) [49]. Then, we have the following two fundamental relations [28,90]:…”

“…For simplicity, we assume perfect-electric conducting (PEC) antennas where the magnetic field does not contribute to the current Green's function. Another radiating Green's functions similar to G(x, x , t − t ) is needed in expressions such as (A2) in order to obtain the magnetic field B(x, t), which is always present in any radiation problem beside the electric field; see [89][90][91] for more details and applications. The relation (A1) captures the antenna's fundamental Mode A, where a source field E ex (x, t) produces a radiating current via the current Green's function F(x, x , t − t ), which in turn generates the radiated fields E(x, t) and B(x, t) throughout the region exterior to the surface S. To complete the antenna-based wireless communication system, a third Mode C, where a receiving (Rx) antenna, placed at some distance from S, interacts with the radiated field in order to produce an observable receive port current J rx (x) by means of the formula…”

“…where F rx (x, x , t − t ) is the receive current Green's function, which is generally different from the transmit (Tx) current Green's function F(x, x , t − t ). The three relations and Green's functions in (A1)-(A3) fully describe the antenna system in classical electromagnetic theory [92], with obvious generalization to magnetic field interactions included in essentially the same logic [90].…”

Fundamental Spacetime Representations of Quantum Antenna Systems

“…The forward Green's function is the standard retarded radiation Green's function G(x − x , t − t ) [47]. We then have the following two fundamental relations [27], [84]:…”

“…The relation (81) captures the antenna's fundamental Mode A, where a source field E ex (x, t) produces a radiating current via the current Green's function F(x, x , t − t ), which in turn generates the radiated fields E(x, t) and B(x, t) throughout the region exterior to the surface S. To complete the antennabased wireless communication system, a third Mode C, where a receiving (Rx) antenna, placed at some distance from S, interacts with the radiated field in order to produce an observable receive port current J rx (x) by means of the formula (83) where F rx (x, x , t − t ) is the receive current Green's function, generally different from the transmit (Tx) current Green's function F(x, x , t − t ). The three relations and Green's functions in (81), (82), and (83) fully describes the antenna system in classical electromagnetic theory [86], with obvious generalization to magnetic field interactions included in essentially the same logic [84].…”