In the spherical near-field measurement process, EMF data are commonly collected using an equal-angle-interval sampling scheme, where the measurements are taken at regular angular intervals in both θ and ϕ directions. The equal-angle sampling method for θ and ϕ simplifies the mathematical processing in near-field to far-field transformation, facilitating the direct application of spherical coordinate system data. However, a limitation of this method arises from the excessive number of samples that are required as θ approaches 0° or 180°, as illustrated in Figure 1. This leads to a redundancy in the measurements, where numerous data points are collected in regions of the near-field that may not significantly contribute to the far-field pattern reconstruction.

The classical spherical near-field to far-field transformation (Hansen, 1988) can be utilized to convert near-field data into antenna far-field radiation parameters. The signal received by the probe can be expressed in the form of a spherical wave expansion, as shown in Equation 1: w A , χ , θ , ϕ = v ∑ μ = ± 1 ∑ s = 1 2 ∑ n = 1 N ∑ m = − n n T s m n · e i m ϕ d μ m n θ e i μ χ P s μ n k A (1) where A (in meters) represents the radius of the measurement sphere, ν (in volts) denotes the transmitted signal from the antenna under test (AUT), χ denotes the rotation angle of the probe, and e i m ϕ d μ m n θ e i μ χ is the rotation coefficient of the spherical basis function. In spherical near-field sampling, the rotation angle χ of the probe is commonly chosen to be χ = 0 ° and χ = 90 ° , corresponding to two scans of the entire measurement sphere.

For equal-angle sampling, the number of required sampling points is determined by the angular coverage of the measurement sphere. Specifically, the θ angle coverage, ranging from 0 ° to 180 ° , necessitates at least N + 1 samples. Meanwhile, the ϕ angle coverage, ranging from 0 ° to 360 ° , requires at least 2 N + 1 samples. Thus, the total number of samples is given by 2 2 N + 1 N + 1 sampling points. This is approximately twice the number of samples that are actually required for the transformation algorithm, which is estimated to be 2 N N + 2 (Hansen, 1988).

The goal of non-redundant uniform spherical sampling is to optimize the measurement process by minimizing unnecessary data points while maintaining the accuracy of the far-field reconstruction. The first step is to determine the number of sampling points. The minimum number of sampling points is calculated based on the operating frequency and the measurement sphere radius. Equation 2 determines the highest order N of the spherical wave modes, where k is the wave number, λ is the wavelength, and r t is the radius of the smallest sphere enclosing the AUT. To ensure computational accuracy, the number of sampling points should be close to or greater than N 1 as stated in Equation 3. N = k r t + 10 (2) N 1 = 2 N N + 2 (3)

In this study, we utilized both spiral-based and particle-based sampling methods to generate the locations of the non-redundant spherical sampling points, although other uniform spherical sampling methods could also be applied.

The particle-based uniform sampling on a sphere is a method for generating a set of points that are evenly distributed over the surface of a sphere. It models the sample points as charged particles confined to the sphere surface. Each particle repels the others according to Coulomb’s law, and the system evolves until it reaches a state of minimum potential energy. The resulting equilibrium yields a quasi-uniform distribution of points that approximate equal-area coverage, although an exact uniform distribution is impossible for arbitrary numbers of points. This technique is closely related to the classical Thomson problem (Thomson, 2009), which seeks the lowest-energy configuration of identical charges on a spherical surface. The spiral-based method determines the points locations following a continuous spiral trajectory from the north pole to the south pole of the sphere. Instead of relying on equal-angle sampling or energy minimization, this method constructs smooth curves that winds around the sphere while maintaining nearly constant spacing between neighboring points. Points are then placed at equal intervals along this spiral path. A common implementation is the Fibonacci spiral or golden spiral method, inspired by phyllotaxis patterns found in nature (e.g., sunflower seeds) (González, 2010). The two algorithms used for the generation of the N 1 spherical sampling points were implemented by (Semechko, 2021).

Once the sampling locations are determined, measurements are conducted at these positions using a probe. The probe measures the received signals w A , 0 ° , θ , ϕ and w A , 90 ° , θ , ϕ , which correspond to the w θ and w ϕ components of the radiation field from the AUT, respectively. Due to the significantly greater distance decay of the E r component compared to E θ and E ϕ , it is common practice to neglect the E r component. Using the vector transformation matrix T , the measurement signals corresponding to the E x , E y and E z components at the N 1 sampling points are obtained, as shown in Equation 4. w x A , θ , ϕ w y A , θ , ϕ w z A , θ , ϕ = T · w A , 0 ° , θ , ϕ w A , 90 ° , θ , ϕ w r A , θ , ϕ (4)

Based on the spherical harmonic function expansion, the field Cartesian components can be expressed as a series of spherical harmonics, as shown in Equation 5: f θ , ϕ = ∑ l = 0 ∞ ∑ m = − l l f l m Y l m θ , ϕ (5) where f l m represents the coefficients of the expansion, and Y l m θ , ϕ are the spherical harmonic functions. We next interpolate the w x A , θ , ϕ , w y A , θ , ϕ and w z A , θ , ϕ on to the corresponding equal angle interval sampling grids, denoted as w x ′ A , θ , ϕ , w y ′ A , θ , ϕ and w z ′ A , θ , ϕ , and these data are then transformed to yield w θ ′ and w ϕ ′ at equal angular intervals. Utilizing the spherical near-field to far-field transformation algorithm, the known w θ ′ and w ϕ ′ components at equal-angle grids are applied in the classical spherical near-field to far-field transformation, as used in previous works (Hansen, 1988; Diao and Hirata, 2021a).

With the w θ ′ and w ϕ ′ components at equal-angle grids, the coefficients T s m n can be determined by solving (1). If an infinitesimal electric dipole with a probe response constants, as expressed in Equations 6, 7, are used, the transverse electric field components E θ and E ϕ are related to the received signal w θ ′ and w ϕ ′ of the electric probe by Equations 8, 9: P s 1 n k A = 6 8 i − s 2 n + 1 R s n 3 k A (6) P s , − 1 , n k A = − 6 8 i s 2 n + 1 R s n 3 k A (7) E θ A , θ , ϕ = 2 k 6 π η w θ ′ A , θ , ϕ (8) E ϕ A , θ , ϕ = 2 k 6 π η w ϕ ′ A , θ , ϕ (9) where R s n 3 k A is defined by Equation 10 R s n 3 k A = h n 1 k A + i 1 k A d d d A k A h n 1 k A , (10) with h n 1 k A represents the spherical Hankel function of the first kind.

Similarly, the magnetic field components H θ and H ϕ are related to the magnetic probe received signal. Details regarding this technique can be found in (Hansen, 1988; Diao and Hirata, 2021a). Finally, the free-space normal component of the IPD, defined as Equation 11: IPD n = 1 2 Re E × H * · n (11) where n is the unit norm vector of the evaluation plane, and norm of the IPD, defined as Equation 12: IPD tot = 1 2 Re E × H * (12) can be computed on an evaluation plane at a distance d from the antenna aperture.

In this study, a K-band rectangular standard gain horn antenna, as shown in Figure 2, was simulated. The operating frequency was set to 18 GHz, and the geometric parameters of the antenna are listed in Table 1. While the method was validated using this specific antenna at 18 GHz, it is important to note that the approach is not inherently frequency-dependent and can be extended to other communication frequencies. The choice of 18 GHz in this study was motivated by practical considerations related to experimental validation, as it allows a balance between measurement equipment availability, and positioning accuracy.

The measurement surface was selected as a sphere with a radius of 300 mm, enclosing the entire antenna. The results of using equal-angle sampling and non-redundant uniform sampling were compared. For equal-angle sampling, the maximum number of modes required, calculated using (2), is 53. This leads to 54 sampling points in the θ direction, resulting in a total of 11,556 sampling points on the measurement sphere. The sampling point locations are shown in Figure 3a. As observed, near the upper and lower poles of the measurement sphere, the sampling points are dense, introducing significant redundancy. Using the spiral- and particle-based spherical sampling methods, a total of 6,804 uniformly distributed sampling point positions were obtained, as depicted in Figure 3b for spiral-based sampling and Figure 3c for particle-based sampling.

The IPD was compared at distances d ranging from 300 mm to 1,000 mm with a 100 mm interval, between the uniform and non-uniform sampling methods. The IP D n and IP D tot , as well as the peak 4 cm² spatially averaged values IP D n , 4 cm 2 and IP D tot , 4 cm 2 , as specified in the guidelines and standards, were evaluated and reported.

Figure 4a shows the amplitude and phase distributions of the E θ and E ϕ using the equal-angle sampling scheme. Figures 4b,c present the reconstructed distributions using non-redundant uniform sampling generated by the spiral-based and particle-based methods and further interpolated onto the same θ and ϕ grids as described in Section 2.2. As seen, the reconstructed distributions closely match the results obtained using equal-angle sampling.

Subsequently, the E θ and E ϕ components, treated as measurement data, are input into the near-field to far-field transformation algorithm to calculate the antenna’s far-field parameters. The difference in directivity between the three computed strategies is less than approximately 0.2 dB. Figure 5 compares the antenna patterns. Figure 5a shows the pattern obtained with equal-angle interval sampling, while Figures 5b,c show the patterns obtained using the spiral-based and particle-based uniform sampling methods, respectively. As seen, the antenna patterns highly resemble the patterns of the equal-angle sampling results, except for difference where the value is lower than −50 dB.

Figures 6a,b compare the computed peak IP D n and IP D tot values over distances ranging from 300 to 1,000 mm. As the main beam points in the +z direction, the peak values of IP D n and IP D tot are nearly identical for all sampling schemes. The largest difference in IP D n and IP D tot between the equal-angle sampling approach and full-wave simulation is 0.13 dB, observed at d = 300 mm. For the spiral-based sampling approach, the largest difference is 0.15 dB at d = 800 mm, and for the particle-based sampling approach, the largest difference is 0.17 dB, occurring at d = 300 mm.

Figures 7a,b compare the computed peak 4-cm² averaged IP D n , 4 cm 2 and IP D tot , 4 cm 2 . The largest difference between the equal-angle sampling approach and the full-wave simulation is 0.12 dB, observed at d = 300 mm. For the spiral-based sampling approach, the largest difference is 0.14 dB at d = 400 mm, and for the particle-based sampling approach, the largest difference is 0.16 dB at d = 300 mm.

These results validate the proposed approach, achieving a more than 40% reduction in the number of measurement points while maintaining accuracy in the assessment of the IPD values.

To evaluate the stability of the proposed methods, we applied the Gaussian white noises to both the magnitude and phase of the near-field samples. The standard deviation of the phase noise was changed from 0° to 10° in 2° increments, while the relative standard deviation of the magnitude noise was varied from 0% to 10% in 2% increments. For each noise combination, 50 independent trials were conducted, then the highest resultant relative errors in the IPD tot and IPD tot , 4 cm 2 were reported, as shown in Figure 8. As expected, the relative errors increased with higher noise levels in both magnitude and phase. The maximum relative error was only 8.2%, demonstrating the robustness and numerical stability of the proposed method against measurement noise.