## Appendix A Time Domain Admittance Array  
Since the \(x\)- and \(y\)-components of the induced magnetic-current surface density are in a small rectangular aperture mutually uncoupled, the TD admittance array has only non-zero diagonal terms, viz. \[\underline{\boldsymbol{\mathcal{Y}}}(t)=\left[ \begin{array}{cc}Y_{xx}(t)&0 \\ 0&Y_{yy}(t)\end{array} \right]. \tag{1}\]  
The elements of the admittance array can be for the piecewise-linear space-time distribution of the induced magnetic-current density (see Equations (5) and (6) with (20) and (21)) obtained analytically in the following form:  
\[ \begin{array}{ll}Y_{xx}(t)=\frac{Y_{0}}{c_{0}\Delta_{x}\Delta_{y}x\Delta_{ y}}\Big{[}&\Psi(3\Delta_{x}/2,\Delta_{y},t)-\Psi(3\Delta_{x}/2,-\Delta_{y},t)\\ &-3\,\Psi(\Delta_{x}/2,\Delta_{y},t)+3\,\Psi(\Delta_{x}/2,-\Delta_{y},t)\\ &+3\,\Psi(-\Delta_{x}/2,\Delta_{y},t)-3\,\Psi(-\Delta_{x}/2,-\Delta_{y},t)\\ &-\Psi(-3\Delta_{x}/2,\Delta_{y},t)+\Psi(-3\Delta_{x}/2,-\Delta_{y},t)\Big{]}, \end{array}  \tag{2}\]  
and \(Y_{yy}(t)\) is given by a similar TD expression that can be obtained upon replacing \(x\) with \(y\) (and vice versa) in Equation (2). The TD function, \(\Psi(x,y,t)\), follows as the inverse of the complex-slowness integral (cf. ([28], (G.3))):  
\[\Psi(x,y,s)=\frac{c_{0}^{2}}{8\pi^{2}}\int_{\kappa\in\mathbb{S}_{0}}\frac{ \exp(s\kappa x)}{s^{3}\kappa^{3}}\Omega_{0}^{2}(\kappa)\mathrm{d}\kappa\int_{ \sigma\in\mathbb{S}_{0}}\frac{\exp(s\sigma y)}{s\sigma}\frac{\mathrm{d}\sigma }{\gamma(\kappa,\sigma)} \tag{3}\]  
for \(\{s\in\mathbb{R}_{i}s>0\}\), \(\{x\in\mathbb{R}_{i}x\
eq 0\}\), \(\{y\in\mathbb{R};y\
eq 0\}\), and recall that \(\gamma(\kappa,\sigma)=(1/c_{0}^{2}-\kappa^{2}-\sigma^{2})^{1/2}\). Furthermore, \(\mathbb{K}_{0}\) and \(\mathbb{S}_{0}\) are integration paths that run along the imaginary axes in the complex-slowness \(\kappa\)- and \(\sigma\)-planes, respectively, and that are indented to the right around their origins with small semi-circular arcs of vanishingly small radii (cf. ([28], Figure 10)). The inversion of \(\hat{\Psi}(x,y,s)\) can be carried out using the CdH procedure as closely described in ([28], Appendix G). First, the integrand in the integral with respect to \(\sigma\) is continued analytically into the complex \(\sigma\)-plane, while keeping \(\mathrm{Re}(\gamma)\geq 0\). This implies the horizontal branch cuts extending along \(\{\mathrm{Im}(\sigma)=0,\Omega_{0}(\kappa)<|\mathrm{Re}(\sigma)|<\infty\}\). Consequently, using [PERSON]'s lemma and [PERSON]'s theorem [24] (p. 1054), path \(\mathbb{S}_{0}\) is replaced with the loop around the branch cut, a parametric form of which is \(\sigma(u)=-u\Omega_{0}(\kappa)\mathrm{sgn}(y)\pm\mathrm{i}0\) for all \(\{1\leq u<\infty\}\) with \(\mathrm{sgn}(y)=|y|/y\). In addition, the contribution from the simple pole at \(\sigma=0\) must be accounted for when \(y\geq 0\). In the integral around the branch cut, we introduce the new variable of integration, \(u\), while the integration around the pole is readily carried out analytically. The thus-transformed inner integral with respect to \(\sigma\) is next substituted back in Equation (3). Proceeding further in a similar manner with the integration in the complex \(\kappa\)-plane, we, after some lengthy, yet straightforward algebra, arrive at integral expressions that can beevaluated analytically. Transforming finally the result of integration to the TD via standard tables ([33], Section 29), we end up with the TD original of Equation (A3) in the following form ([28], (cf. Equations (G.24), (G.25), and (G.30)):  
\[