## 5 Illustrative Examples
To quantify the deviation between the signals, we evaluated the normalized root-mean-squared error according to \(\text{ERR}=\sqrt{\sum_{k=1}^{M}\left(\tilde{f}_{k}-f_{k}\right)^{2}/M}/\Delta f\), where \(\tilde{f}_{k}\) and \(f_{k}\) represent the time samples of the FIT and CdH-MoM signals, respectively, and \(\Delta f=f_{\rm max}-f_{\rm min}\). The error corresponding to the \(E_{x}\)-field component (see Figure 3a) is about \(\text{ERR}\simeq 1.5\%\), while \(\text{ERR}\simeq 3.2\%\) for the \(E_{y}\)-field component (see Figure 3b). The higher error in the \(E_{y}\)-field component can be attributed to the postulated magnetic-current space-time distribution (see Equations (5) and (6)), which is less accurate along the relatively longer side of the aperture. Typical computational times to obtain the TD responses shown in Figure 3 are about 20 s using a non-optimized MATLAB(r) CdH-MoM implementation and about 30 min using CST Microwave Studio(r). An even more striking difference is in the number of required unknowns and accompanying memory requirements. While our dedicated CdH-MoM computational model solves the system for two unknown quantities only (see Equations (5) and (6)), the FIT model consists of about 600 thousand discretization elements. The calculations were conducted on a standard laptop with Intel(R) Core(TM) i7-10510U CPU @ 1.80 GHz. To indicate the range of applicability of the presented computational model, we increased the aperture's length along the \(x\)-direction such that \(2\Delta_{x}\) is now a fifth of the excitation pulse's spatial support \(c_{0}t_{\rm w}\), i.e., \(2\Delta_{x}=0.20\,\rm m=c_{0}t_{\rm w}/5\). In addition, we took \(\Delta_{y}=\Delta_{x}/10\), so that the model represents a relatively narrow slit in the PEC screen. The remaining parameters were kept the same. Figure 4 shows the corresponding pulse shapes of the \(E_{y}\)-field at the center of the aperture. As the length of the slit is no longer sufficiently small with respect to \(c_{0}t_{\rm w}\), discrepancies between the signals, quantified by \(\text{ERR}\simeq 5.8\%\) are apparent. While this result can still be useful for initial estimates, improving the accuracy through the incorporation of the piecewise-linear spatial expansion of the axial current density (e.g., ([28], Section 14.3)) is advisable in this case. Figure 2: Incident plane-wave pulse shape. Once the space-time distribution of the equivalent magnetic-current surface density in the aperture is known, it can be used to evaluate the scattered and, hence, total EM-fields via Equation (1). To that end, one may employ the pertaining EM wave field representations (e.g., ([24], Section 28.12)). For example, the TD scattered far-field amplitudes, \(\{E^{\text{s};\infty},\mathbf{H}^{\text{s};\infty}\}\), defined via ([24], Section 26.12)  
Figure 4: Pulse shape of \(E_{y}\)-field as induced at the center of the narrow aperture with relative dimensions \(2\Delta_{x}/c_{0}t_{\text{w}}=1/5\) and \(2\Delta_{y}/c_{0}t_{\text{w}}=1/50\). Figure 3: Electric-field pulse shapes as induced at the center of the aperture with relative dimensions \(2\Delta_{x}/c_{0}t_{\text{w}}=1/10\) and \(2\Delta_{y}/c_{0}t_{\text{w}}=1/20\). (**a**) \(E_{x}\)-field component; (**b**) \(E_{y}\)-field component.