## 2 Equivalent Current Reconstruction and FT Approach  
The vital piece of hardware in this work that enables phase reconstruction in the near-field domain is the EUmmWVx probe. This probe enables \(E\)-field measurements very close to the source with minimal probe-related distortions of the field at frequencies up to 110 GHz. The probe is based on the pseudo-vector probe design [28] which measures not only the magnitude of the field but also derives its polarization ellipse from different probe rotation angles.  
Based on these measurements on two planes, a PTP algorithm has been developed that permits the reconstruction of the PD on measurement planes as close as 2 mm distant from the source [21]. This algorithm also yields reconstructed phases of the \(E\) and \(H\)-fields on the measurement plane, such that equivalent current distributions can be computed using [PERSON]'s equivalence theorem [19]  
in the estimated phase of the fields. In our studies, we observed that adding a third measurement plane considerably improves the accuracy of the phase reconstruction by providing more data on the radiated fields. Therefore, a third measurement plane is added at a distance of one wavelength (\(\lambda\)) from the second measurement plane. These measurements are then used to formulate the inverse source problem as follows, with the notable addition of two lines that represent the measured amplitudes at the plane of the equivalent currents, \(|\mathbf{E}_{x}^{\prime}|\) and \(|\mathbf{E}_{y}^{\prime}|\) (also referred to as the closest measurement plane or \(z_{1}\) in the following):  
\[\left|\left.\begin{array}{c}\mathbf{E}_{x}^{\text{obs}}\\ \mathbf{E}_{y}^{\text{obs}}\\ \mathbf{E}_{z}^{\prime}\\ \mathbf{E}_{y}^{\prime}\end{array}\right|\right|=\left|\left[\begin{array}[] {cc}0&-G_{z}\\ G_{2}&0\\ -G_{y}&G_{x}\\ 0&-\frac{1}{2}I\\ \frac{1}{2}I&0\end{array}\right]\cdot\left[\begin{array}{c}\mathbf{M}_{x}\\ \mathbf{M}_{y}\end{array}\right]\right|, \tag{4}\]  
where \(\mathbf{E}_{c}^{\text{obs}}\) represent the vectors of length \(n_{\text{obs}}\) for the three \(E\)-field components \(c\in\{x,y,z\}\), with observation points on the third and second measurement planes. The matrices \(G_{x}\), \(G_{y}\), \(G_{z}\) are \(n_{\text{obs}}\times n_{\text{src}}\) matrices with entries corresponding to each pair of observation and source points. The operator \(|\cdot|\) corresponds to the element-wise magnitude. The matrix \(I\) represents the identity matrix. This inverse source problem is then solved for the equivalent currents \(\mathbf{M}_{x}\) and \(\mathbf{M}_{y}\) using the alternating gradient descent method presented by [PERSON] [31], using an initial estimate for \(\mathbf{M}_{x,0}\) and \(\mathbf{M}_{y,0}\) obtained with the original PTP algorithm [21]. Trials with different weightings of the entries in (4) did not yield noticeable improvements. The reason for this is the relatively small plane separation that was used for the second and third planes. An upper bound of the measured magnitude plus 5% of the peak value was forced on the magnitude of the currents \(\mathbf{M}_{x}\) and \(\mathbf{M}_{y}\) to stay close to the measurements while allowing some compensation of measurement noise. To reduce the number of unknowns, equivalent sources with a magnitude smaller than \(-25\,\)dB of the maximum current magnitude (as measured on the first plane) were omitted in both the inverse source problem (4) and the FT step (2). Once \(\mathbf{M}_{x}\), \(\mathbf{M}_{y}\) are determined, (2) is used to compute the field at any point of interest \(\mathbf{r}\).