## 5 Plane Size Requirements  
In theory, (2) is only valid for infinitely large planes. However, the proper truncation of the plane for sampling the fields is achieved if all the field power is captured in the measured plane, meaning that all the fields outside this truncated area are negligible. This is valid for directional antennas or planar antenna structures when the sampling plane (measurement plane) resides in the near-field region. This section analyzes the minimum plane size required for the set of common directional antennas from Section 3. A crucial factor determining the overall characterization time and computation cost for the AUT is the grid step size for field sampling. On one hand, this step size should be small in order to provide sufficient sampling points for phase reconstruction and subsequently for near-field to far-field transformation; on the other, a large step size is favored to minimize the total measurement time. In our study, the optimum grid step size was found to be \(0.25\,\lambda\), which is the recommended step size for the PTP algorithm [21]. This step size is set to \(0.15\,\lambda\) for measurements at \(10\,\mathrm{GHz}\); therefore, the plane size is directly proportional to the number of sampling points. Simulated equivalent currents \(\mathbf{M}\) were computed from simulations using (1). Phase reconstruction was omitted in this case to reduce the number of dependent variables and to avoid correcting model inaccuracies by deviating from the actual phases. FT was then applied on a large range of plane sizes. An example of different plane sizes used as planes of equivalent sources is illustrated in Figure 2. The figure also shows the PD computed using FT to a plane further away than the measurement plane. This illustration qualitatively shows how larger plane sizes result in more accurate reconstruction by FT. As a next step, we investigated the error as function of the ratio of the captured power in the finite plane and the total power in the infinite plane. To quantify the plane size, we used the ratio of power captured by the plane compared to a very large reference plane. The power was computed by the integral of \(\mathrm{Re}(\mathbf{S}_{z})\) over the plane. The reference plane was chosen as \(80\times 80\) sample points (with a grid step size of \(0.25\,\lambda\)). The plane of equivalent currents was placed at \(z_{1}=2\,\mathrm{mm}\) and two observation planes were used--one at \(z_{\mathrm{obs}}=5\,\mathrm{mm}\) and one at \(z_{\mathrm{obs}}=150\,\mathrm{mm}\)--for all frequencies to cover a range that will likely be relevant for the compliance testing of millimeter-wave consumer devices. Note that when performing the FT from the full reference plane to the observation plane, the deviation in psPD was within \(0.4\,\mathrm{dB}\) of the simulated value for the antennas operating at and above \(30\,\mathrm{GHz}\). This deviation was within \(0.6\,\mathrm{dB}\) for the antennas operating at \(10\,\mathrm{GHz}\). These small deviations are mainly attributed to the FT model, which suffers from certain inaccuracies; e.g., the numerical error arising from the discretization of the surface currents in terms  
Figure 2: Different sizes for the plane of equivalent sources (ranging from \(4\times 4\) samples, or \(1\,\lambda^{2}\), to \(80\times 80\) samples, or \(400\,\lambda^{2}\)) used for forward transformation for the \(30\,\mathrm{GHz}\) cavity-fed dipole array overlapped on the normalized electric field distribution. The plane is at a distance of \(2\,\mathrm{mm}\) from the antenna under test (AUT) plane and \(E_{0}=72.3\,\mathrm{V/m}\). of square pulse basis functions. This error due to model imperfections was not further reduced with increasing plane size. In Figure 3, two errors are reported; (i) the error with respect to the real simulated field as the total error of the presented procedure, and (ii) the error with respect to the large reference plane, representing the error caused by plane truncation.