## 2 Mathematical Formulation
### Quasi-Quadratic Approximation of the Scattered Field  
To separate the contribution of the bars from that of the dielectric whose permittivity and shape are assumed to be known, the contrast is represented as follows:  
\[\chi(\mathbf{r})=\chi_{0}-jf(\mathbf{r}),\quad\text{with}\quad\mathbf{r}\in\Omega \tag{4}\]  
where \(\chi_{0}=\varepsilon_{r}-1\) is constant over \(\Omega\), and  
\[f(\mathbf{r})=\frac{\sigma\zeta}{k}\sum_{n=1}^{N}\phi(|\mathbf{r}-\mathbf{r}_ {n}|), \tag{5}\]  
where \(\sigma\) is the bar's conductivity, \(\zeta\) is the free space impedance, \(k\) is the free space wavenumber, and  
\[\phi(r)= \begin{cases}1&r\leqslant a\\ 0&r>a\end{cases}  \tag{6}\]  
Using (4), the quadratic approximation of the scattered field in (3) can be recast as:  
\[ \begin{split} E_{s}=\chi_{0}\mathbb{A}_{e}[E_{i}]+\chi_{0}^{2} \mathbb{A}_{e}[\mathbb{A}_{i}[E_{i}]]-j\mathbb{A}_{e}[fE_{i}]\\ -j2\chi_{0}\mathbb{A}_{e}[\mathbb{A}_{i}[fE_{i}]]-\mathbb{A}_{e}[f \mathbb{A}_{i}[fE_{i}]].\end{split}  \tag{7}\]  
The _quasi_ quadratic approximation proposed in this paper consists of neglecting the last term in (7). In this way, mutual scattering between the bars is neglected, retaining the interaction between the bars and the surrounding dielectric. Furthermore, the first two terms on the right-hand side of (7) represent the quadratic approximation of the scattered field when no bars are present. Such terms can be calculated, provided that the dielectric shape and permittivity are known, and subtracted from the actual (measured/simulated)  
Figure 1: Geometry of the problem and reference system. The investigation domain is the rectangle of size \(W\times P\) placed at distance \(D\) from the measurement line, whose length is \(L\). scattered field. By denoting such a difference by \(\hat{E}_{s}\), the problem can be formulated as the solution to the following equation:  
\[\hat{E}=-j\mathbb{A}_{e}[fE_{i}]-j2\chi_{0}\mathbb{A}_{e}[\mathbb{A}_{i}[fE_{i}]] \tag{8}\]  
where the relationship between the data, at the first member, and the unknown, i.e., the function \(f(\mathbf{r})\), is linear. The explicit computation of (8) is now in order. By exploiting the Green's function expression in the two-dimensional case, the following results are obtained:  
\[\mathbb{A}_{e}[u] =-j\frac{k^{2}}{4}\iint\limits_{\Omega}H_{0}^{(2)}(kR_{e})u( \mathbf{r})d\mathbf{r} \tag{9}\] \[\mathbb{A}_{i}[u] =-j\frac{k^{2}}{4}\iint\limits_{\Omega}H_{0}^{(2)}(kR_{i})u( \mathbf{r})d\mathbf{r}\] (10) \[E_{i} =-\frac{k^{2}_{5}}{4}H_{0}^{(2)}(kR_{e}) \tag{11}\]  
where \(R_{e}=|\mathbf{r}-\mathbf{r}_{o}|\), with \(\mathbf{r}_{o}\in\Sigma\), is the distance between a source/observation and a point inside the investigation domain, \(R_{i}=|\mathbf{r}-\mathbf{r}^{\prime}|\), with \(\mathbf{r}^{\prime}\in\Omega\), is the distance between two points of the investigation domain; \(u(\mathbf{r})\) stands for either \(E_{i}\) or \(fE_{i}\); \(H_{0}^{(2)}\) is the Hankel function of zero order and second kind, and the unitary current source is assumed. After some non-trivial passages, as reported in Appendix A, the following results are obtained:  
\[\hat{E}(k,\mathbf{r}_{o})=\sum\limits_{n=1}^{N}C_{n}(k)\Big{[}H_{0}^{(2)}(k| \mathbf{r}_{o}-\mathbf{r}_{n}|)\Big{]}^{2} \tag{12}\]  
Equation (12) appears relatively simple, as the Hankel functions act as a sort of pulse centered on the unknown bar positions, and the coefficients \(C_{n}\) do not depend on the observation point. Their expression is:  
\[ \begin{split} C_{n}(k)&=\pi\sigma\zeta^{2}\bigg{(} \frac{ka}{4}\bigg{)}^{2}\left\{1-j2\chi_{0}\bigg{(}\frac{ka}{2}\bigg{)}^{2} \cdot\right.\\ &\left.\left[1-\bigg{(}\frac{ka}{2}\bigg{)}^{2}-\frac{j}{\pi}(1- 2\log(ka))\right]-j\frac{\chi_{0}}{2}A_{n}\right\}\end{split}  \tag{13}\]  
where  
\[ \begin{split} A_{n}&=\int\limits_{0}^{2\pi}\frac{( kR_{n})^{2}}{2}\Big{[}J_{0}(kR_{n})H_{0}^{(2)}(kR_{n})+\\ J_{1}(kR_{n})H_{1}^{(2)}(kR_{n})\Big{]}d\theta\end{split}