## 5 Numerical Results and Discussion  
This section focuses on illustrative numerical examples of RCS, which we use to study in detail the far-field backscattering properties of waveguides. We define the RCS per unit length as follows  
\[\sigma=\lim_{\rho\rightarrow\infty}\left(2\pi\rho\frac{\left|\phi^{d}\right| ^{2}}{\left|\phi^{d}\right|^{2}}\right), \tag{89}\]  
where \(\phi^{d}\) is the diffracted field defined by (88). For real \(k\), (89) is simplified by using (2), (79), and (88) as  
\[\sigma=\frac{\lambda}{2}\left|U_{(+)}(-k\cos\theta)\pm V_{(+)}(-k\cos\theta) \right|^{2} \tag{90}\]  
for \(\theta\gtrgtr 0\) with \(\lambda\) being the free-space wavelength. By using the approximate expressions in (62) and (63), we have calculated \(U_{(+)}(-k\cos\theta)\) and \(V_{(+)}(-k\cos\theta)\) involved in (90). We need to numerically invert the two sets of \(N\times N\) matrix equations to obtain physical quantities. We have verified by careful numerical experiments that the choice \(N\geq 2 kb/\pi\) in (62) and (63) generates sufficiently accurate results. Figures 2-5 show the normalized monostatic RCS \(\sigma/\lambda\) versus incidence angle \(\theta_{0}\), where the value of \(\sigma/\lambda\) is plotted in decibels [dB] by calculating \(10\log_{10}(\sigma/\lambda)\). Figure 6 shows the monostatic RCS [dB] versus normalized frequency \(kb\). Figures 7 and 8 show  
comparisons of the angular and frequency characteristics of the monostatic RCS, respectively, between E-polarization [46] and H-polarization (this paper). In order to investigate the scattering mechanism over a wide frequency range, six typical values of the normalized waveguide aperture width have been chosen as \(kb=1.57\), 3.14, 15.7, 31.4, 47.1, and 62.8. In addition, the ratio \(L_{1}/2b\) has been chosen as 0.5 (Figure 2), 1.0 (Figure 3), 3.0 (Figure 4), and 5.0 (Figure 5). For numerical computations, we have selected five different materials from the RAM study by [PERSON] et al. [6]. It is important to note here that although these material properties are fictitious, they represent a wide range of available radar absorbing materials. The material constants are \(\varepsilon_{1}\)= 8+\(i\)10, \(\mu_{1}=1+i\)0 for region I, \(\varepsilon_{2}\)= 10+\(i\)6, \(\mu_{2}=1+i\)0 for region II, \(\varepsilon_{3}\)= 15+\(i\)0, \(\mu_{3}=3+i\)15 for region III, \(\varepsilon_{4}\)= 15+\(i\)0, \(\mu_{4}=7+i\)12 for region IV, and \(\varepsilon_{5}\)= 15+\(i\)0, \(\mu_{5}=25.8+i\)10.3 for region V. The thickness of the material in region I extends from \(-L_{5}\) to \(-\infty\). The thickness of regions II, III, IV, and V is such that \(L_{1}-L_{2}=L_{2}-L_{3}=L_{3}-L_{4}=L_{4}-L_{5}=t/4\), and the total thickness of the four-layer material (regions II-V) is taken as \(kt=1.57\). Figures 2-6 include the results from one- to four-layer material loading for comparison. From Figures 2-5 for angular dependences, it is observed that the RCS exhibits large values over a certain range of the incidence angle due to the internal irradiation and reduces with an increase of the number of material layers. From these characteristics, we see that multi-layer loading gives rise to better RCS reduction over a wide range of frequencies. Another common feature in all numerical examples is that the RCS oscillates rapidly for larger \(kb\) and \(L_{1}/2b\). Comparing the characteristics for one- to five-layer cases in all numerical examples, better RCS reduction is observed in the five-layer case for all \(kb\). Next, we will examine the frequency dependences of the RCS to gain a more comprehensive understanding of the backscattering characteristics. Figure 6 presents the normalized monostatic RCS as a function of the normalized frequency \(kb\), for angles of incidence \(\theta_{0}=0^{\circ}\),30\({}^{\circ}\), and 60\({}^{\circ}\) with \(L_{1}/2b=1.0\). We have again considered five different geometries: waveguides with one-, two-, three-, four-, and five-layer material loading.