## 3 Time Domain Problem Formulation  
The EM scattering problem under consideration is formulated using the [PERSON] reciprocity theorem ([24], Section 28.2). To that end, the reciprocity theorem is applied to the scattered EM wave fields and the (causal) testing EM wave fields (denoted by superscript \(\mathsf{T}\)), which are generated by the testing magnetic-current surface density, \(\partial\mathbf{K}^{\text{T}}(\mathbf{r},t)\), whose spatial support is the surface of the aperture, \(\text{supp}(\partial\mathbf{K}^{\text{T}})=\mathcal{S}\). Then, upon applying the TD reciprocity theorem to the upper half-space, \(z>0\), while enforcing the surface boundary condition applying to (the tangential component of) the scattered field, \(\partial\mathbf{K}^{\text{s}}=\mathbf{E}^{\text{s}}\times\dot{\mathbf{i}}_{z}\), on the PEC plane, we arrive at  
\[ \begin{split}\int_{\mathbf{r}\in\mathcal{S}^{+}}\big{[}\mathbf{H}^{ \text{T}}(x,y,0^{+},t)&\overset{*}{\mathbf{\cdot}}\partial\mathbf{K}^{ \text{s}}(x,y,t)\\ &-\mathbf{H}^{\text{s}}(x,y,0^{+},t)&\overset{*}{\mathbf{ \cdot}}\partial\mathbf{K}^{\text{T}}(x,y,t)\big{]}\text{d}A=0,\end{split}  \tag{2}\]  
where \(\mathbf{\cdot}\) represents the standard inner product of two vectorial quantities ([26], Equation (1.2)), \(*\) denotes the continuous time-convolution operator ([26], Equation (1.11)), and the superscript \({}^{+}\) indicates that we approach the surface of the aperture, \(\mathcal{S}\), from above as \(z\downarrow 0\). Since both states are causal and each other's adjoint, the contribution from the bounding sphere \"at infinity\" vanishes ([26], Section 1.4.3). In the second step, the TD reciprocity theorem is applied to the lower half-space, \(z<0\), which leads to a similar relation:  
\[ \begin{split}\int_{\mathbf{r}\in\mathcal{S}^{-}}\big{[}\mathbf{H}^{ \text{T}}(x,y,0^{-},t)&\overset{*}{\mathbf{\cdot}}\partial\mathbf{K}^{ \text{s}}(x,y,t)\\ &-\mathbf{H}^{\text{s}}(x,y,0^{-},t)&\overset{*}{\mathbf{ \cdot}}\partial\mathbf{K}^{\text{T}}(x,y,t)\big{]}\text{d}A=0,\end{split}  \tag{3}\]  
where \(\overset{-}{}\) indicates that we approach the surface of the aperture from below as \(z\uparrow 0\). Recalling the definition of the excitation field, we may use Equation (1) with \(\dot{\mathbf{i}}_{z}\times\mathbf{H}^{\text{e}}(x,y,0^{+},t)=2\,\dot{\mathbf{i}}_{z}\times \mathbf{H}^{\text{i}}(x,y,0,t)\) and \(\dot{\mathbf{i}}_{z}\times\mathbf{H}^{\text{e}}(x,y,0^{-},t)=\mathbf{0}\) and with the continuity-type condition \(\dot{\mathbf{i}}_{z}\times\mathbf{H}(x,y,0^{+},t)=\dot{\mathbf{i}}_{z}\times\mathbf{H}(x,y,0^ {-},t)\) applying to all \(\mathbf{r}\in\mathcal{S}\) and \(t>0\), to combine relations (2) and (3). This yields the desired TD reciprocity relation:  
\[ \begin{split}\int_{\mathbf{r}\in\mathcal{S}}\mathbf{H}^{\text{T}}(x,y,0^ {+},t)&\overset{*}{\mathbf{\cdot}}\partial\mathbf{K}^{\text{s}}(x,y,t) \text{d}A\\ &=-\int_{\mathbf{r}\in\mathcal{S}}\mathbf{H}^{\text{i}}(x,y,0,t)& \overset{*}{\mathbf{\cdot}}\partial\mathbf{K}^{\text{T}}(x,y,t)\text{d}A.\end{split}  \tag{4}\]  
The final TD relation (4) will next be solved for the equivalent magnetic-current space-time distribution, \(\partial\mathbf{K}^{\text{s}}(x,y,t)\), as induced in the aperture. Owing to the relatively small dimension of the aperture, we may assume that the magnetic-current spatial distribution has the following form:  
\[\partial K^{\text{s}}_{x}(x,y,t) =\frac{k_{x}(t)}{2\Delta_{y}}\Lambda(x)\Pi(y), \tag{5}\] \[\partial K^{\text{s}}_{y}(x,y,t) =\frac{k_{y}(t)}{2\Delta_{x}}\Lambda(y)\Pi(x), \tag{6}\]  
where \(k_{x}(t)\) and \(k_{y}(t)\) are the magnetic-current pulse shapes (to be computed),  
\[\Lambda(x)= \begin{cases}1+x/\Delta_{x},&\text{for }\{-\Delta_{x}\leq x\leq 0 \}\\ 1-x/\Delta_{x},&\text{for }\{0\leq x\leq\Delta_{x}\}\\ 0,&\text{elsewhere}\end{cases}  \tag{7}\]is the triangular function, and  
\[\Pi(x)= \begin{cases}1,&\text{for}\ \{-\Delta_{x}\leq x\leq\Delta_{x}\}\\ 0,&\text{elsewhere}\end{cases}  \tag{8}\]  
represents the rectangular function.