## Appendix A  
\(\mathbb{A}_{e}[fE_{i}]\) is obtained by exploiting (5), (9), and (11) as follows:  
\[ \begin{split}\mathbb{A}_{e}[fE_{i}]&=j\sigma\left( \frac{k\zeta}{4}\right)^{2}\sum\limits_{n}\\ \iint\limits_{n-thbar}\left[H_{0}^{(2)}(k|\mathbf{r}_{o}-\mathbf{ r}|)\right]^{2}d\mathbf{r}\end{split}  \tag{2}\]  
In the hypothesis that \(a<<|\mathbf{r}_{o}-\mathbf{r}|_{n}\), it can be assumed that the Hankel function is almost constant when \(\mathbf{r}\) varies within the \(n-\)th bar and coincides with its value calculated at the center of the bar, that is, \(H_{0}^{(2)}(k|\mathbf{r}_{o}-\mathbf{r}|)\approx H_{0}^{(2)}(k|\mathbf{r}_{o }-\mathbf{r}_{n}|)\), such that the double integral is obtained by multiplying this value by the area of the circle \(\pi a^{2}\). It results:  
\[ \begin{split}\mathbb{A}_{e}[fE_{i}]&=j\sigma\pi \left(\frac{ka\zeta}{4}\right)^{2}\sum\limits_{n}\\ &\left[H_{0}^{(2)}(k|\mathbf{r}_{o}-\mathbf{r}_{n}|)\right]^{2} \end{split}  \tag{3}\]  
The computation of \(\mathbb{A}_{e}[\mathbb{A}_{i}[fE_{i}]]\) is slightly more complex. \[ \begin{split}\mathbb{A}_{e}[\mathbb{A}_{i}[fE_{i}]]& =\sigma\frac{k^{4}\zeta^{2}}{64}\iint\limits_{\Omega}H_{0}^{(2)}(k| \mathbf{r}_{o}-\mathbf{r}|)\cdot\\ \sum\limits_{n}\iint\limits_{n^{th}bar}H_{0}^{(2)}\left(k|\mathbf{ r}-\mathbf{r}^{\prime}|\right)H_{0}^{(2)}\left(k|\mathbf{r}_{o}-\mathbf{r}^{ \prime}|\right)d\mathbf{r}^{\prime}d\mathbf{r},\end{split}  \tag{4}\]  
applying the same arguments as before, we approximate the Hankel function of the argument \(k|\mathbf{r}_{o}-\mathbf{r}^{\prime}|\) with its value at the \(n-\)th bar center and put it outside the integrals  
\[ \begin{split}\mathbb{A}_{e}[\mathbb{A}_{i}[fE_{i}]]& =\sigma\frac{k^{4}\zeta^{2}}{64}\sum\limits_{n}H_{0}^{(2)}(k|\mathbf{r}_{o}- \mathbf{r}_{n}|)\cdot\\ \iint\limits_{\Omega}H_{0}^{(2)}(k|\mathbf{r}_{o}-\mathbf{r}|) \iint\limits_{n^{th}bar}H_{0}^{(2)}\left(k|\mathbf{r}-\mathbf{r}^{\prime}| \right)d\mathbf{r}^{\prime}\;d\mathbf{r}.\end{split}  \tag{5}\]  
where the summation and external integral have been exchanged. We focused on the integral of the \(n^{th}\) bar. It is a function of \(\mathbf{r}\), that is, \(b_{n}(\mathbf{r})\). It is convenient to refer to local polar coordinates \((R,\theta)\) centered on the \(n-\)th bar, where \(\mathbf{R}=\mathbf{r}-\mathbf{r}_{n}\) and \(\mathbf{R}^{\prime}=\mathbf{r}^{\prime}-\mathbf{r}_{n}\) (see Figure 2). In this way:  
\[b_{n}(\mathbf{r})=\int\limits_{0}^{2\pi}\int\limits_{0}^{a}H_{0}^{(2)}\left(k| \mathbf{R}-\mathbf{R}^{\prime}|\right)R^{\prime}dR^{\prime}d\theta^{\prime}. \tag{6}\]  
Two cases must be considered:  
1. \(R>a\)It is straightforward to see that, in this case, the following results:  
\[b_{n}(\mathbf{r})=\frac{2\pi a}{k}H_{0}^{(2)}(k|\mathbf{r}-\mathbf{r}_{n}|)J_{1}( ka)\] (A6)  
where the addition theorem for the Bessel function of integer order (formula 9.1.79 in [34])  
\[H_{0}^{(2)}\big{(}k|\mathbf{R}-\mathbf{R}^{\prime}|\big{)}=\] \[\sum_{m=0}^{\infty}J_{m}(kR^{\prime})H_{m}^{(2)}(kR)\cos m(\theta- \theta^{\prime});\]  
and the formula 5.52.1 in [35])  
\[\int\limits_{0}^{a}J_{0}(kR^{\prime})R^{\prime}dR^{\prime}=\frac{a}{k}J_{1}( ka)\]  
have been exploited. 2. \(R<a\) In this case, the addition theorem provides two different expressions \[H_{0}^{(2)}\big{(}k|\mathbf{R}-\mathbf{R}^{\prime}|\big{)}=\] \[ \begin{cases}\sum_{m=0}^{\infty}J_{m}(kR^{\prime})H_{m}^{(2)}(kR) \cos m(\theta-\theta^{\prime})&R^{\prime}<R\\ \sum_{m=0}^{\infty}J_{m}(kR)H_{m}^{(2)}(kR^{\prime})\cos m(\theta-\theta^{ \prime})&R^{\prime}>R\end{cases} \] (A7) Again, substituting such expressions in (A5), the integral on \(\theta^{\prime}\) makes null the terms of the sum \(m\
eq 0\), and only the term \(m=0\) survives.