The context is a quiet revolution in radiofrequency exposure science. For frequencies above 6 gigahertz, where millimeter-wave bands live, the established metric of specific absorption rate gives way to power density measures. International guidelines from the International Commission on Non-Ionizing Radiation Protection and the IEEE International Committee on Electromagnetic Safety define two tiers of limits: basic restrictions on internal dosimetric quantities such as absorbed power density, and reference levels expressed as field quantities in free space. For frequencies above 6 GHz, the ICNIRP 2020 guidelines set the reference level as the peak spatially averaged incident power density, typically averaged over a 4-square-centimeter area. Verifying that a real antenna complies with these limits requires measuring the electromagnetic field it radiates, and doing so efficiently has become a pressing engineering challenge.

The team’s starting point was a well-known inefficiency in classical antenna metrology. Spherical near-field measurements, the standard technique for characterizing an antenna’s radiation, involve collecting field data on the surface of a sphere enclosing the antenna and then transforming those data into far-field parameters through spherical wave expansion. Conventional practice samples at equal angular intervals in both the polar angle theta and the azimuthal angle phi. That scheme simplifies the mathematics, but it concentrates a dense thicket of measurement points near the two poles of the sphere, where the rings of constant theta shrink to almost nothing. The result is substantial redundancy: for the geometry studied here, equal-angle sampling demanded 11,556 points, roughly twice the number actually required by the transformation algorithm.

The researchers’ solution was to abandon the equal-angle grid altogether and instead distribute a minimal set of sampling points uniformly across the sphere’s surface. They implemented two schemes for generating these positions. The first is a spiral-based method, in which points are placed at equal intervals along a continuous spiral winding from the north pole to the south pole, maintaining nearly constant spacing between neighbors. A common realization is the Fibonacci or golden spiral, a pattern inspired by the phyllotaxis arrangements seen in sunflower seeds. The second is a particle-based method that treats each sample point as a charged particle confined to the sphere’s surface; the particles repel one another according to Coulomb’s law, and the system relaxes to a minimum-energy configuration that approximates equal-area coverage. This approach is closely related to the classical Thomson problem of finding the lowest-energy arrangement of identical charges on a sphere.

Once the sampling locations were fixed, the team faced the task of making non-grid data usable with the standard near-field-to-far-field transformation machinery, which expects data on regular angular grids. The probe measures two received signals corresponding to the theta and phi components of the radiation field; the radial component decays so much faster with distance that it is conventionally neglected. Using a vector transformation matrix, the measured signals are converted into Cartesian field components at the irregular sampling points. These components are then expressed as a series of spherical harmonics, interpolated onto the conventional equal-angle grids, and fed into the classical transformation algorithm. In effect, the method harvests the efficiency of irregular uniform sampling while retaining the computational convenience of the established pipeline.

Validation used a K-band rectangular standard gain horn antenna operating at 18 gigahertz, simulated with a measurement sphere of 300-millimeter radius. The choice of frequency reflected practical considerations of equipment availability and positioning accuracy rather than any inherent limitation of the method, which the authors note is not frequency-dependent and can be extended to other communication bands. The horn’s geometry, with a width of 104.6 millimeters, a height of 79.2 millimeters, and a length of 190.0 millimeters, was fully specified, and the required number of spherical wave modes was calculated as 53, dictating the sampling density.

The comparison between sampling schemes was strikingly consistent. Reconstructed near-field amplitude and phase distributions from the non-redundant methods closely matched those from equal-angle sampling, and the resulting far-field patterns were nearly identical except in regions where the pattern level fell below minus 50 decibels. Differences in directivity among the three strategies were less than about 0.2 decibels. More importantly for safety assessment, the team evaluated the peak values of the normal and total IPD components, along with their 4-square-centimeter spatially averaged counterparts, on evaluation planes at distances from 300 to 1,000 millimeters in 100-millimeter steps. The largest discrepancy between any sampling scheme and a full-wave simulation reference was 0.17 decibels, observed for the particle-based method at the closest distance of 300 millimeters.

Those error figures sit comfortably within the tolerances that exposure compliance work can accept, and they were achieved with 6,804 sampling points instead of 11,556, a reduction of approximately 40 percent. The practical significance is greatest for single-probe measurement systems, where every additional point on the sphere translates directly into acquisition time. Cutting the sample count by two-fifths means faster antenna characterization and faster safety assessments, a meaningful advantage as the proliferation of millimeter-wave devices multiplies the testing burden on laboratories worldwide.

The team also probed the method’s robustness against imperfect measurements, an essential test for any technique destined for real measurement ranges. They injected Gaussian white noise into both the magnitude and phase of the near-field samples, sweeping the phase noise standard deviation from 0 to 10 degrees in 2-degree increments and the relative magnitude noise from 0 to 10 percent in 2-percent steps. For each of the 49 noise combinations, 50 independent trials were run, and the highest relative errors in the IPD components were recorded. As expected, errors grew with noise level, but the maximum relative error never exceeded 8.2 percent, demonstrating numerical stability under conditions considerably noisier than typical laboratory practice.

The study builds on the authors’ earlier work, which established equal-angle spherical near-field measurement as a route to IPD assessment and later extended it to absorbed power density through a combination of spherical transformation and numerical dosimetry. It also complements alternative approaches, such as planar near-field reconstruction methods that work well for high-gain antennas and antenna arrays but are less general. By eliminating the polar redundancy that plagued the spherical approach, the new method closes a practical gap. The authors indicate that future work will extend validation to additional frequencies and antenna types, but the core result already stands: a mathematically principled way to measure fewer points, recover the same radiation pattern, and certify electromagnetic safety with confidence.

Subject of Research: Non-redundant spherical near-field sampling for efficient incident power density assessment in electromagnetic exposure safety

Article Title: Non-redundant spherical near-field sampling for efficient incident power density assessment for Electromagnetic Safety

Article References: Zheng, J., Guo, Q., & Diao, Y. (2026). Non-redundant spherical near-field sampling for efficient incident power density assessment for Electromagnetic Safety. Frontiers in Antennas and Propagation, 3, Article 1738329. https://doi.org/10.3389/fanpr.2025.1738329

Image Credits: AI Generated

DOI: 10.3389/fanpr.2025.1738329

Keywords: incident power density, spherical near-field measurement, non-redundant sampling, electromagnetic safety, millimeter-wave, 5G, antenna measurement, exposure assessment, ICNIRP, dosimetry, spiral sampling, horn antenna