## 3 Numerical Results
Figure 3: Components of the effective electric current densities on a two stacked thin resistive disks resonator for different values of the radius of the disks (\(a=\lambda/2,\lambda,2\lambda\)). \(d/a=1\), \(R_{1}=100\)\(\Omega_{\text{,}}\)\(R_{2}=200\)\(\Omega_{\text{,}}\)\(\left|\text{E}_{0}\right|=1\) V/m, \(\phi_{0}=0^{\circ}\), and TE incidence: (**a**) \(a=\lambda/2\), \(\left|I_{i\phi}\right|\); (**b**) \(a=\lambda/2\), \(\left|I_{i\phi}\right|\); (**c**) \(a=\lambda\), \(\left|I_{i\phi}\right|\); (**d**) \(a=\lambda\), \(\left|I_{i\phi}\right|\); (**e**) \(a=2\lambda\), \(\left|I_{i\phi}\right|\); (**f**) \(a=2\lambda\), \(\left|I_{i\phi}\right|\). It is interesting to observe that the far scattered electric field can be expressed in closed form as a function of the unknowns in the spectral domain by means of the stationary phase method [9]:  
\[E_{s}^{\infty}(r,\theta,\phi)\overset{r\rightarrow+\infty}{\sim}\frac{e^{-\,j\,k _{0}r}}{r}F_{s}(\theta,\phi), \tag{15}\]  
with \(s=\theta,\phi\), where  
\[F_{\theta}(\theta,\phi)= -\frac{\omega\mu_{0}}{2}\cos\theta\sum_{n=-\infty}^{+\infty}e^{ \mathrm{i}\,n(\phi+\frac{\pi}{2})}\sum_{i=1}^{2}\,\tilde{J}_{i,\mathrm{D}}^{(n )}\,\left(k_{0}\sin\theta\right)e^{\mathrm{i}z_{i}k_{0}\cos\theta}, \tag{16}\] \[F_{\phi}(\theta,\phi)= -\frac{\omega\mu_{0}}{2}\sum_{n=-\infty}^{+\infty}e^{\mathrm{i}\, n(\phi+\frac{\pi}{2})}\,\sum_{i=1}^{2}\,\tilde{J}_{i,\mathrm{D}}^{(n)}\,\left(k_{0} \sin\theta\right)e^{\mathrm{i}z_{i}k_{0}\cos\theta}. \tag{17}\]  
As a result, the bistatic radar cross section (BRCS),  
\[\sigma_{\mathrm{BRCS}}(\theta,\phi)=\lim_{r\rightarrow+\infty}\frac{4\pi r^{ 2}|\underline{E}^{\infty}(r,\theta,\phi)|^{2}}{\left|\underline{E}^{\mathrm{ inc}}(r,\theta,\phi)\right|^{2}}=\frac{4\pi|\underline{E}(\theta,\phi)|^{2}}{ \left|\underline{E}_{0}\right|^{2}}, \tag{18}\]  
can be immediately evaluated once the solution of the problem is known. In Figure 5, BRCS is plotted for the cases considered in Figure 3, i.e., for \(d/a=1\) and for \(a=\lambda/2,\lambda,2\lambda\). Comparisons with CST-MWS are provided for all the examined cases. As can be clearly seen, the agreement is really very good. However, the proposed method drastically outperforms the Integral Equation solver of CST-MWS, which requires tens of thousands of cells and a few minutes to find a reasonable approximation of the solution. A resonant behavior can be observed due to the multiple reflection between the disks, i.e., the system of two stacked resistive disks works like a lossy Fabry-Perot resonator. One way to individuate the resonance frequencies on the natural modes consists in searching for the peaks of TSCS and ACS, which are defined as in the following:  
\[\sigma_{\text{TSCS}}=\frac{1}{\left|\underline{E}_{0}\right|^{2}}\int\limits_{ 0}^{\pi}\int\limits_{0}^{2\pi}\left|\underline{E}(\theta,\phi)\right|^{2}\sin \theta d\phi d\theta, \tag{19}\]  
\[\sigma_{\text{ACS}}=\frac{\text{pabs}}{\left|\underline{E}_{0}\right|^{2}/ \left(2Z_{0}\right)}=\ \frac{2\pi Z_{0}\Re\left\{R\right\}}{\left|\underline{E}_{0}\right|^{2}}\sum \limits_{n=-\infty}^{+\infty}\int\limits_{0}^{a}\left|\textbf{I}^{(n)}(\rho) \right|^{2}\rho d\ \rho, \tag{20}\]  
where \(P^{\text{abs}}\) is the power absorbed by the disks and \(\Re\left\{\cdot\right\}\) denotes the real part of a complex number. As clearly shown in [43], by means of [PERSON]'s formula [52], Weber-Schafheitlin discontinuous integral [49] and forward scattering theorem [53], even TSCS and ACS can be expressed in closed form and then quickly evaluated.