Gradient-index, or GRIN, lenses have long fascinated engineers because they bend and focus waves not through curved surfaces, as conventional optics does, but through spatially varying dielectric properties distributed throughout the volume of the material. Instead of a shaped piece of glass, a GRIN lens is essentially a structured medium whose effective refractive index changes gradually from point to point. When a wave propagates through such a medium, its wavefront is progressively reshaped, allowing the lens to collimate, focus, or steer beams with remarkable freedom. For compact beam-steering systems, particularly those needed in satellite communications, radar, and next-generation wireless networks, this kind of wavefront control in a flat or modestly sized package is extremely attractive.

The difficulty has always been the inverse problem. Forward design is straightforward: given a specified permittivity distribution, electromagnetic solvers can predict exactly how waves will behave. Inverse design asks the opposite question. Given a desired field transformation, what volumetric permittivity distribution produces it? For most practical devices, this question has been answered through voxel-by-voxel topology optimization, in which the design volume is divided into thousands or millions of tiny cells, each of which is iteratively adjusted. Every iteration demands a full-wave electromagnetic simulation, and a converged design can require thousands of such simulations. On ordinary computing hardware, this process can stretch from days into weeks, and the resulting designs are often highly resonant, meaning they perform well only in a narrow frequency band and are exquisitely sensitive to fabrication tolerances.

The Siena and Inkbit team took a fundamentally different route. Rather than optimizing an arbitrary voxel map, they restricted the search space to smooth GRIN profiles appropriate for propagation-based devices. This constraint is not a limitation but a strategic advantage. Smooth, slowly varying index profiles are exactly the regime in which geometrical optics provides an accurate and computationally cheap description of wave behavior. By optimizing the geometrical optics propagation equations directly, the researchers sidestepped the need for repeated full-wave simulations entirely. The optimization problem collapses from a monstrous high-dimensional search into a tractable one that runs in minutes on a standard computer. The physics of propagation, rather than raw computational power, does the heavy lifting.

The mathematical logic behind the approach is elegant. Geometrical optics treats waves as rays that follow paths determined by the local refractive index, and the accumulated phase along each ray dictates how the wavefront evolves. By parameterizing the permittivity distribution with a modest number of smooth basis functions, the designers created a differentiable mapping from the lens parameters to the resulting field transformation. Gradient-based optimization then adjusts those parameters until the predicted output field matches the target, whether that target is a focused spot, a collimated beam, or, in the demonstrated case, a set of independently steered beams corresponding to different feed positions. Because the optimization operates on smooth profiles, the resulting designs are inherently broadband: a gradual index variation does not suddenly stop working when the frequency shifts, unlike resonant structures tuned to a single operating point.

To prove that the method delivers real hardware rather than pretty simulations, the team fabricated a multi-feed GRIN lens using additive manufacturing. The internal architecture is based on a three-dimensional gyroid, one of the most celebrated geometries in modern materials science. A gyroid is a triply periodic minimal surface, a structure that divides space into two intertwined, self-connected labyrinths with no straight edges or flat faces. By controlling the local density and dimensions of the gyroid lattice, the effective dielectric constant of the material can be tuned continuously across the lens volume, translating the smooth permittivity profile computed by the optimizer into a physical object. Additive manufacturing is the enabling technology here, since no conventional machining process could produce such an intricate, spatially graded internal scaffold.

The experimental results are striking. The fabricated lens supports multiple feed ports, each of which launches a beam in a different direction, effectively turning a single passive structure into a multi-beam former. As the feed is switched, the beam scans continuously across a ±35-degree range, and the measured performance holds across a 33 percent fractional bandwidth, a figure that comfortably encompasses many practical communication and sensing bands. The researchers report low scan loss, meaning the beam does not weaken appreciably as it moves away from boresight, and excellent beam fidelity, meaning the radiated pattern retains its shape and sidelobe behavior across the scanning range. These are precisely the attributes that matter in real phased-array and multibeam antenna systems, where scan loss and pattern distortion have historically limited performance.

The significance of the demonstration extends beyond the specific device. Multi-beam formers are workhorses of satellite ground stations, 5G and future 6G base stations, and high-throughput communication links, where a single aperture must serve many users or satellites simultaneously. Conventional solutions rely on phased arrays with hundreds of individually controlled elements, each requiring amplifiers, phase shifters, and calibration, or on quasi-optical systems such as Rotman lenses and reflectarrays that involve their own design compromises. A passively operated GRIN lens that steers beams simply by selecting among fixed feeds offers a dramatically simpler architecture with no active electronics in the beam-forming path, reducing cost, power consumption, and failure modes.

Equally important is what the method implies for the design workflow of metamaterial devices in general. The gap between an electromagnetic specification and a printable file has been the central obstacle to scalable wavefront engineering. Topology optimization, for all its power, produces designs whose complexity is entangled with the simulation cost of producing them, and whose narrowband resonant character often clashes with real-world bandwidth requirements. By anchoring the inverse problem in geometrical optics and smooth profiles, the new framework establishes what the authors describe as a direct pathway from electromagnetic functionality to manufacturable metamaterial devices. The optimization is fast enough to be iterative in a design office, and the output is, by construction, compatible with additive manufacturing processes that can realize graded effective media.

The work was carried out within the framework of the Huawei–University of Siena Joint Laboratory, a collaboration that reflects the growing industrial appetite for beam-steering hardware as wireless systems push toward higher frequencies and denser networks. The article, authored by Ilir Gashi, Scott Twiddy, Zachary Nelson, Stefano Maci, and Matteo Albani, with Twiddy and Nelson contributing equally, was published open access on 26 September 2026. As additive manufacturing continues to mature, allowing ever finer control over graded internal architectures, the combination of physics-informed inverse design and freeform fabrication points toward a future in which antennas and lenses are no longer designed component by component but computed as a whole, with the desired electromagnetic behavior specified first and the material emerging as the answer. For a field that has spent decades choosing between the speed of approximate models and the accuracy of exhaustive simulation, a method that achieves wideband, experimentally verified performance in minutes of computation marks a genuine turning point.

Subject of Research: Inverse design of gradient-index metamaterials for wideband multi-beam forming

Article Title: Inverse design of metamaterials for wideband multi-beam former

Article References: Gashi, I., Twiddy, S., Nelson, Z., Maci, S., & Albani, M. (2026). Inverse design of metamaterials for wideband multi-beam former. Communications Engineering. https://doi.org/10.1038/s44172-026-00787-1

Image Credits: AI Generated

DOI: 10.1038/s44172-026-00787-1

Keywords: metamaterials, inverse design, gradient-index lens, beam steering, additive manufacturing, gyroid, multi-beam former, electromagnetics, geometrical optics, antenna engineering, topology optimization, wavefront engineering

The research, led by Fatemeh Mehdipour of the Karlsruhe Institute of Technology together with colleagues including Thomas Koch and Roland Dittmeyer, set out to answer a deceptively simple question: can the design freedom of metal 3D printing produce a catalyst support that converts more pollution without choking the engine? Conventional ceramic monoliths are thermally robust but brittle, and the extrusion process that makes them restricts their interiors to simple square or hexagonal channels. That geometry produces largely laminar flow, modest heat and mass transfer, and susceptibility to maldistribution, all of which limit how efficiently harmful molecules reach the catalytic surface. Cold-start conditions, before the catalyst reaches its light-off temperature, compound the problem.

The team selected three TPMS architectures as the building blocks of their experimental carriers: the Schwarz-primitive, the gyroid, and the Schwarz-diamond surfaces. These are mathematical minimal surfaces that repeat periodically in three dimensions, producing smooth, continuous, fully interconnected channel networks with no sharp junctions. Such geometries can pack a large surface area into a small volume while promoting uniform flow distribution, offering a way to soften the classic trade-off between catalytic surface area and pressure drop. Each monolith was designed as a cylinder 50 millimetres tall and 23.88 millimetres across, sized to slot into a sample holder on an engine test bench, with unit cell size, orientation, and even axial size gradients varied systematically to isolate each design parameter’s influence.

Twenty-seven monoliths were fabricated from gas-atomised 316L stainless steel powder using laser powder bed fusion, with 0.2 millimetre walls guided by prior work showing that gas-tight thin walls in the 200 to 300 micrometre range are achievable with the process. Post-processing was deliberately gentle: ultrasonic cleaning flushed loose powder from the internal channels, and wire electrical discharge machining, rather than force-based cutting, separated the parts from the build plate to avoid distorting the thin-walled circular cross-sections. The finished samples measured within a few hundredths of a millimetre of their nominal dimensions, confirming that the intricate internal lattices could be manufactured reproducibly.

Coating the printed structures was arguably the riskiest step. Washcoating, the deposition of a catalytic ceramic layer onto a substrate, has historically been tricky on additively manufactured surfaces because of roughness, wettability, and adhesion concerns. The researchers prepared a stable suspension of platinum supported on gamma-alumina, with a median particle size near 200 nanometres, and dipped each monolith twelve times, spin-drying between immersions before calcining at 500 degrees Celsius. Drop tests and ultrasonic exposure showed relative washcoat losses below 2.5 percent for nearly all geometries, evidence that the rough printed surface actually helps by mechanically interlocking the ceramic layer. Scanning electron microscopy revealed porous, interconnected washcoat morphologies that should aid gas diffusion, along with an axial thickness profile that was thickest near the monolith ends, likely a consequence of slurry depletion and redistribution during drying.

Pressure-drop screening on longer, uncoated 150 millimetre samples showed that specific surface area alone does not dictate flow resistance. The Schwarz-diamond lattice with 5 millimetre cells, which packs the highest surface area of the set, also produced the highest pressure drop, while the open Schwarz-primitive structure with 10 millimetre cells flowed more freely than even the straight-channel hexagonal reference. In between, the gyroid geometries occupied a promising middle ground, their smooth continuous passages balancing gas-surface contact against hydrodynamic resistance.

The decisive test came on a three-cylinder 2.19-litre compressed natural gas engine operating at 1500 revolutions per minute and 150 newton-metres of load, with the air-fuel equivalence ratio swept from 0.9 to 1.4. Exhaust gas was routed through a bypass containing the washcoated monoliths, and inlet and outlet concentrations of methane, carbon monoxide, and nitrogen oxides were measured by Fourier-transform infrared spectroscopy and an exhaust gas analyser. Under stoichiometric conditions, the two best TPMS configurations exceeded 90 percent CO conversion, with the compact Schwarz-diamond SD(5,s) sample reaching approximately 99 percent. Normalising conversion by washcoat loading revealed that catalyst utilisation, not merely the amount of platinum deposited, differed substantially between topologies and was strongly influenced by unit cell orientation.

Equally striking was evidence of three-way catalytic behaviour. Methane and nitrogen oxide conversions peaked near stoichiometric operation, and under fuel-rich conditions the researchers observed ammonia forming inside the monolith, a signature of nitrogen oxides being reduced over platinum with CO and hydrogen acting as reductants. Because no external ammonia was introduced, its appearance confirmed that the printed carriers supported not just oxidation but reductive chemistry, extending their functional repertoire beyond simple CO cleanup. The team is careful to note that these results rest on one engine-tested sample per geometry, so they represent comparative trends within a proof of concept rather than statistically validated rankings.

When conversion and pressure drop were mapped together, the gyroid structure with 10 millimetre cells emerged as the standout, delivering higher washcoat-normalised CO conversion than the straight-channel reference while imposing lower pressure drop per unit length. The compact 5 millimetre designs bought their exceptional conversion with a clear backpressure penalty, whereas the most open Schwarz-primitive structures flowed beautifully but underused their catalyst. The authors conclude that performance is governed by the interplay of specific surface area, topology, effective tortuosity, flow distribution, and catalyst accessibility, and they propose TPMS architectures as a tunable design platform for application-specific catalytic supports. Before such printed converters reach production vehicles, they caution, long-term durability, thermal ageing, vibration resistance, benchmarking against commercial substrates, and techno-economic assessment will all need to be demonstrated. But the core message stands: a catalyst carrier generated from pure mathematics, printed layer by layer in steel, survived an engine’s exhaust and cleaned it remarkably well.

Subject of Research: Additively manufactured washcoated TPMS structures as catalytic converter carriers tested under real engine exhaust conditions

Article Title: Washcoated additively manufactured TPMS structures as catalytic converters for engines: A proof of concept

Article References: Mehdipour, F., Heinrich, J. E., Yu, Z., Kutscherauer, M., Tucker, M. R., Rubin, M., Wagner, U., Koch, T., & Dittmeyer, R. (2026). Washcoated additively manufactured TPMS structures as catalytic converters for engines: A proof of concept. Results in Engineering, 32, Article 113039. https://doi.org/10.1016/j.rineng.2026.113039

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113039

Keywords: additive manufacturing, TPMS, catalytic converter, gyroid, laser powder bed fusion, washcoating, CNG engine, exhaust aftertreatment, pressure drop, platinum catalyst, three-way catalysis, 316L stainless steel