Green’s Functions Slash Simulation Time for Horn Antennas Near Reflectors

Antennas rarely operate in the tidy, empty space of textbook problems. Radar level gauges peer down at tank floors, motion sensors sit centimeters from walls, and next-generation wireless devices must radiate inches above metal chassis and layered circuit boards. In all of these situations, the reflecting surface sits so close to the antenna that the familiar far-field approximation collapses, and engineers are forced to grapple with the full, tangled structure of the near field. A new study published in Results in Engineering offers a mathematically rigorous and computationally lean way out of this bind, using dyadic Green’s functions to compute the radiation of pyramidal horn antennas above layered, inhomogeneous media with striking efficiency.

The research team, led by Adnan M. Taha and including Mahdi Ghafourivayghan, Konstantin Burlakov, Sergey Shabunin, and Mohammad Alibakhshikenari, set out to address a persistent bottleneck in computational electromagnetics. Commercial full-wave solvers such as ANSYS HFSS, CST, and FEKO discretize an entire three-dimensional volume into a finite-element mesh, and their cost balloons when reflected waves must be tracked near a boundary. When the antenna operates in its near or intermediate zone, the amplitude and phase of the field cannot be inferred from a simple aperture distribution; every field component must be computed numerically, which dramatically inflates processing time and memory demands. For applications like radar sensors, subsurface radiolocation, and level gauges, where the far-zone condition is simply never met, this is more than an inconvenience.

The core of the new method is the Green’s tensor function, a mathematical object that encodes how an infinitesimal electric or magnetic current source radiates through a specified environment. In unbounded homogeneous space, the Green function takes a comparatively simple form. In a layered medium, however, it must absorb the thickness, number, and electromagnetic parameters of every layer, including permittivity, permeability, and conductivity. The authors build these properties into characteristic functions g(z, z’) and f(z, z’), which solve Sturm-Liouville-type differential equations and automatically satisfy all boundary conditions at each interface. Crucially, the modal conductances are recalculated recursively through the stack using an equivalent transmission-line model, in which spectral components of the field are mapped onto voltages and currents in equivalent circuits. Once this analytical groundwork is laid, the field can be evaluated only where it is actually needed, rather than everywhere in space.

To model a real aperture antenna, the team begins with the Huygens element, a pair of orthogonally crossed electric and magnetic dipoles that together reproduce the radiation of an elementary patch of wavefront. The electric field is obtained by integrating the external electric and magnetic currents over the source region, weighted by the appropriate Green’s tensor components. Because a Huygens element radiates a curved, non-uniform wavefront rather than a plane wave, its reflection from a nearby interface behaves nothing like the textbook plane-wave diffraction problem, and the Green’s-function framework captures this distinction exactly. The formulation also yields closed-form far-zone expressions via the saddle-point method, providing a convenient analytical check: the radial field component vanishes, and the radiation maximum points toward the interface, exactly as physics demands.

The extension from the elementary Huygens source to a practical pyramidal horn is where the study claims its principal advance over the authors’ earlier work. The horn flares in both principal planes, combining the properties of E-plane and H-plane sectoral horns, and its aperture field carries a quadratic phase variation that reflects the true curvature of the wavefront. The researchers derive a rigorous aperture integration in which the path-length difference across the aperture is expanded binomially, and the resulting integrals are evaluated in terms of cosine and sine Fresnel integrals. This captures interference structure near the reflector that a simple Huygens source fundamentally cannot represent, making the tool genuinely design-ready rather than a purely theoretical exercise.

Numerical stability is handled through a documented three-region spectral truncation strategy. The integration domain in wave-number space is split into an evanescent region of exponentially decaying near-field contributions, a narrow singularity gap around the branch point where split-path integration is applied, and a propagating region of oscillatory far-field radiation. Convergence analysis shows the truncation error decays exponentially with the upper limit, achieving one percent accuracy well within the adopted defaults, while the singularity treatment remains stable for gap sizes down to ten to the minus twenty-fifth. A parametric robustness study across horn heights from half a wavelength to two wavelengths, frequencies from 0.5 to 3 gigahertz, and substrate permittivities from 1 to 10 confirms that the chosen settings sit on a stable plateau and need no geometry-specific retuning.

The validation against ANSYS HFSS is the study’s most persuasive evidence. On identical observation grids, the proposed method required 30.9 seconds of wall-clock time against 88 seconds for HFSS, a roughly threefold speed-up, and used 200 megabytes of memory against 749 megabytes, a nearly fourfold reduction. The speed advantage stems from the region-of-interest strategy: the spectral integral is evaluated only at the user-specified observation points, whereas the finite-element solver must discretize the entire computational volume. Error budgets quantify the agreement precisely. The L2-norm relative error for the electric field magnitude is 2.78 percent, with maximum deviation of 2.29 percent, while the phase component shows an L2 error of 4.50 percent, a figure the authors attribute to the notorious sensitivity of phase near field nulls.

Even more telling is how the error scales with antenna height above the interface. At a separation of half a wavelength, deep in the reactive near field, the L2 error against HFSS is about 2.6 to 2.9 percent. At three-quarters of a wavelength it drops to roughly 0.65 percent, and at one and a half wavelengths it falls below 0.01 percent as the observation domain moves into the Fresnel region. Notably, the error shows only weak dependence on frequency and substrate permittivity, with higher permittivity actually marginally reducing the error at close separations because stronger field confinement diminishes the relative weight of the evanescent spectral tail. For radar-sensor and level-gauge designers, these numbers suggest the method can be trusted across a decade of bandwidth without recalibration.

The comparison also exposed an interesting divergence between the two approaches at a perfect electric conductor boundary, where the Green’s-function method delivered cleaner wave patterns than the finite-element reference. The authors point out that HFSS results depend on mesh symmetry and on the correct selection of the bounding box around the analyzed object; an asymmetric mesh around a structure symmetric with respect to the reference plane can introduce spurious field asymmetry. The analytical method, by contrast, is electrodynamically exact and automatically enforces all boundary conditions, eliminating such numerical artifacts. In benchmark runs over a dielectric boundary, the conventional solver took roughly twice as long at a comparable spatial step.

The authors are careful to delineate the method’s limits. Its computational advantage is specific to planar, stratified reflecting surfaces; for finite-sized, curved, or laterally inhomogeneous reflectors, edge diffraction and geometric scattering fall outside the stratified-media kernel, and full-wave solvers remain the appropriate tools. Substrates containing vias, patches, or etched patterns would require augmenting the transmission-line recursion with periodic or moment-method treatments. Within its domain, however, the framework offers a unified and efficient foundation that extends naturally to printed and slot antennas, transmitarrays, and reflectarrays, whose multiple functional layers are handled natively by stratified-media Green’s functions. As near-field applications multiply, from high-resolution electromagnetic imaging and secure short-range communication to over-the-air testing of 5G and 6G arrays, tools that trade brute-force meshing for analytical rigor are likely to become indispensable to the antenna community.

Subject of Research: Electromagnetic near-field analysis of pyramidal horn antennas above layered reflecting media using dyadic Green's functions

Article Title: Analysis of pyramidal horn antennas near a reflecting surface using Green’s functions

Article References: Taha, A. M., Ghafourivayghan, M., Burlakov, K., Shabunin, S., & Alibakhshikenari, M. (2026). Analysis of pyramidal horn antennas near a reflecting surface using Green’s functions. Results in Engineering, 32, Article 113033. https://doi.org/10.1016/j.rineng.2026.113033

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113033

Keywords: pyramidal horn antenna, Green's functions, near-field, reflecting surface, stratified media, ANSYS HFSS, electromagnetic modeling, radar sensors, spectral integration, computational electromagnetics, 5G, aperture antennas