Fast Simulation of Electromagnetic Scattering for Radar-Absorbing Material-Coated 3D Electrically Large Targets

Abstract

In this paper, a modified Shooting and Bouncing Ray (SBR) method based on high-order impedance boundary conditions (HOIBCs) is proposed to analyze the electromagnetic (EM) scattering from electrically large three-dimensional (3D) conducting targets coated with radar-absorbing material (RAM). In addition, the edge diffraction field of coated targets is included in the calculation to improve the accuracy of the calculation. Firstly, the SBR method based on the bidirectional tracing technique is presented. It is concluded that the calculation of the scattered field of the coated targets requires the determination of the reflection coefficients on the coated surface. The reflection coefficients of the coated targets are then derived using HOIBC theory. Finally, the equivalent edge current (EEC) of the impedance wedge is derived by integrating the UTD solutions for the impedance wedge diffraction with the impedance boundary conditions. The simulation results show that the proposed method improves computational efficiency compared to MLFMA while maintaining accuracy. Furthermore, the RCS characteristics of targets coated with different RAMs, different coating thicknesses and with different angles of incidence were compared, as well as the RCS results of coated targets with those of conventional perfect electrical conductor (PEC) targets.

1. Introduction

In recent years, with the rapid development of materials science and stealth technology, radar-absorbing materials (RAMs) [1,2,3,4] have been widely used in both military and civilian applications to effectively reduce the radar cross-section (RCS) of conductive targets. Therefore, previous studies in electromagnetic (EM) scattering calculations have been limited to investigating the scattering characteristics of PEC targets [5,6,7,8,9,10], which is no longer sufficient. The efficient calculation of scattering from RAM-coated conductor surfaces has become a major focus of current research.

The study of scattering problems with RAM-coated targets, which can be regarded as the conducting targets coated with RAM, typically employs two main approaches: analytical methods [11,12] and integral equation methods (IEMs) [13,14,15]. Analytical methods are based on the wave equation for EM fields and involve the derivation of rigorous series solutions using the boundary conditions of the target. Monzon [11] proposed an analytical method to solve the EM scattering of 3D rotationally coated targets. Geng et al. [12] applied analytical methods to analyze the EM scattering of a dielectric-coated impedance sphere. However, due to the difficulty of deriving analytical expressions for complex RAM-coated targets, this method is limited to simple shapes such as cylinders and spheres for modeling EM scattering. In contrast, IEMs can be applied to arbitrarily shaped targets. For example, the method of moments (MoM) is used to analyze the EM scattering from gaps on a coated flat plate [13]. Wang et al. [14] implemented the MoM to calculate the EM scattering from arbitrarily shaped 3D coated targets. In addition, based on the MoM, the EM scattering of multilayer coated targets can be calculated, which enables accurate modeling of arbitrarily shaped multilayer 3D targets [15]. By using the MoM, these methods transform the problem into matrix equations, allowing for accurate numerical solutions. However, IEMs require consideration of the electric and magnetic field distributions within the coated dielectric layer, making the computational process highly complex and difficult to implement programmatically.

To address the above issues, the Leontovich impedance boundary condition (LIBC) method [16,17,18] has been applied. This method models the RAM as a very thin dielectric coating on a metal surface, and establishes the relationship between the tangential electric and magnetic field vectors through the equivalent surface impedance of the coated target, thus allowing the calculation of the external scattered field without considering the internal field distribution of the object. As a result, several researchers have proposed using the LIBC method to calculate the EM scattering of 3D coated targets, which greatly simplifies the calculation process and improves the computational efficiency. Pelosi et al. [19] introduced a physical optics (PO) method based on the LIBC to calculate the electromagnetic scattering of coated flat plate structures. In [20], a hybrid MoM/FEM approach using the LIBC algorithm was applied to calculate the EM scattering characteristics of conductor targets coated with dielectric materials. Yu [21] proposed a hybrid MoM-PO algorithm that incorporated the LIBC method to calculate the EM scattering of anisotropic and isotropic coated 3D targets.

However, the LIBC method is limited by the dielectric parameters of the coated target and the curvature of the target surface:

• $\left|N\right|\gg 1$, $N=\sqrt{{\epsilon }_r{\mu }_r}$ is the refractive index of the medium.
• $\beta d\ll 1$ or $\alpha d\ll 1$, where $d$ represents the thickness of the dielectric coating, $\alpha$ is the attenuation constant, and β is the phase constant.
• $\lambda <a$, where $\lambda$ represents the wavelength of the incident wave and $a$ denotes the minimum radius of curvature of the surface.
• $\mathrm{Im}(N)k_{0}a\ge 2.3$, where $k_0$ represents the propagation constant in free space.

If the coating medium does not meet the required conditions, the accuracy of the LIBC method cannot be guaranteed. To overcome this problem, the high-order impedance boundary condition (HOIBC) method [22,23,24] has been introduced, which incorporates higher-order derivative terms. The HOIBC method allows a more accurate representation of the relationship between the electric and magnetic fields on the target surface, thereby improving the accuracy of the calculations. Stupfel [25,26] applied the HOIBC method to solve time-harmonic EM scattering problems on curved, infinitesimally thin frequency-selective surfaces (FSSs) over a wide range of incident angles. Galdi et al. [27] used HOIBC to analyze the high-frequency asymptotic two-dimensional (2D) scattering of truncated aperture-generated EM fields on planar conducting surfaces coated with multiple layers of homogeneous bi-anisotropic media. In addition, an integral equation method based on HOIBC has been proposed to solve scattering problems for objects coated with low-refractive-index materials [28]. Zhou et al. [29] developed a hybrid method combining an efficient iterative method of moments–physical optics (EI-MoM-PO) method with equivalent dipole moments using HOIBC to study the EM scattering of objects with isotropic or anisotropic surface coatings.

Despite these advances, these methods face challenges such as low computational efficiency and high memory requirements when applied to electrically large targets. While the PO algorithm offers improved computational efficiency, it accounts only for single reflections of EM waves, limiting its accuracy for complex targets. Consequently, these methods are unsuitable for calculating the EM scattering of electrically large targets. Moreover, complex structured targets often feature multiple edge structures, and edge diffraction effects caused by these structures can significantly influence scattering results. Therefore, it is crucial to include edge diffraction effects when analyzing such targets. To address this issue, Syed [30] proposed the physical diffraction theory (PTD) for isotropic impedance structures, but it used the inconsistent diffraction coefficients from geometric diffraction theory and lacked an effective treatment for the shadow boundary transition. Li [31] presented a simplified form of the consistent geometric diffraction theory (UTD) for oblique incidence impedance wedges based on numerical matching methods, enabling the fast solution of impedance wedge diffraction. However, they did not derive the diffraction coefficients for arbitrary oblique incidence angles and wedge angles.

In this paper, the RCS characteristics of RAM-coated electrically large targets are investigated using the HOIBC-based SBR-EEC method. First, a combination of HOIBC with the SBR method is proposed. By introducing the HOIBC method, we overcome the limitations of LIBC methods and achieve a more accurate description of the impact of the dielectric properties of the coating medium on the target scattering calculation. In addition, compared to the MoM algorithm in the aforementioned studies, the proposed method enables more efficient calculation of RCS for electrically large targets. Furthermore, in contrast to the PO algorithm in those studies, the proposed method is more accurate when calculating scattering from complex targets, as the SBR algorithm considers multiple reflections of EM waves. Then, by deriving the incremental length diffraction coefficients (ILDCs) for the impedance wedge edges and the equivalent edge currents (EECs) from the UTD solution for the impedance edges, we introduce the edge wave field to correct the physical optical field in the SBR algorithm. This modification addresses the limitation of the SBR algorithm, which cannot calculate the diffraction fields at the edge of the wedge. Compared to the PTD method for impedance structures, the proposed method provides higher calculation accuracy. Finally, the RCS characteristics of targets coated with different RAMs, different coating thicknesses and with different angles of incidence were compared, as well as the RCS results of coated targets with those of conventional perfect electrical conductor (PEC) targets.

The remainder of this paper is organized as follows: in Section 2, the reflection coefficients on the coated surface are derived based on the HOIBC, and the fast solution for the EM scattering model of electrically large RAM-coated targets is presented. In Section 3, the accuracy and efficiency of the proposed method are evaluated by comparison with the results obtained using MLFMA, and the RCS of the coated targets are discussed. The conclusion is presented in Section 4.

2. Efficient and Accurate Method for EM Scattering of the Coated 3D Target

2.1. Fast Solution for EM Scattering of Coated 3D Electrically Large Targets Based on SBR

Currently, the main methods for calculating electromagnetic scattering from targets are high-frequency asymptotic methods and low-frequency numerical methods. Low-frequency numerical methods provide relatively accurate results by rigorously solving Maxwell’s equations, but their computational efficiency and memory requirements have long been bottlenecks. In contrast, high-frequency approximation algorithms have certain advantages in terms of computational efficiency and low memory requirements, making them more suitable for rapid estimation of EM scattering from electrically large targets. Among these methods, the SBR algorithm combines the strengths of GO and PO. It uses GO to handle shadowing and multiple scattering of the target while applying PO integration to compute the far-field scattering.

As shown in Figure 1a, the incident ray initially illuminates the surface $T_0$; after the first-order reflection it reaches $T_1$, and after the second-order reflection it illuminates $T_2$. The total scattered field is the superposition of the scattered fields from all illuminated surfaces. Compared to the traditional PO method, the SBR algorithm is better suited for complex and electrically large targets. In the SBR method, higher-order rays typically illuminate multiple triangular facets. To enhance the efficiency of ray tracing, this paper employs a bidirectional ray tracing technique to determine whether a facet is illuminated, as shown in Figure 1b. The detailed procedure can be found in reference [32].

To calculate the PO integral, the Modified Equivalent Current Approximation (MECA) method was proposed in [33]. When applying the MECA method, it is essential to consider the Fresnel reflection coefficients at the air–medium interface. By calculating the equivalent electromagnetic currents at the medium surface, the scattered field can be obtained by integration. First, the incident field is decomposed into two components: horizontal polarization waves (TE) and vertical polarization (TM). As shown in Figure 2, in the case of a TE-polarized wave incident obliquely on the medium surface, the local coordinate system $\left[{\widehat{k}}_i,{\widehat{e}}_{TM},{\widehat{e}}_{TE}\right]$ of the incident wave is constructed using the incident wave direction together with the unit vectors of the TM and TE polarization directions. Similarly, the local coordinate system of the reflected wave is defined as $\left[{\widehat{s}}_{TM},{\widehat{k}}_s,{\widehat{s}}_{TE}\right]$. Therefore, the incident fields ${\overrightarrow{E}}^{inc}$ and reflected fields ${\overrightarrow{E}}^s$ can be expressed as follows:

$$\begin{array}{l}{\overrightarrow{E}}^{inc}={\overrightarrow{E}}_{TE}^{inc}+{\overrightarrow{E}}_{TM}^{inc}=E_{TE}^{inc}{\widehat{e}}_{TE}+E_{TM}^{inc}{\widehat{e}}_{TM}\\ {\overrightarrow{E}}_{TE}^s=E_{TE}^{inc}{R}_{TE}{\widehat{e}}_{TE}, {\overrightarrow{H}}_{TE}^s={\displaystyle \frac{1}{\eta }}{E}_{TE}^{inc}{R}_{TE}\left({\widehat{k}}_s\times {\widehat{e}}_{TE}\right)\\ {\overrightarrow{E}}_{TM}^s=E_{TM}^{inc}{R}_{TM}{\widehat{s}}_{TM}=E_{TM}^{inc}{R}_{TM}\left({\widehat{k}}_s\times {\widehat{e}}_{TE}\right), {\overrightarrow{H}}_{TM}^s=-{\displaystyle \frac{1}{\eta }}{E}_{TM}^{inc}{R}_{TM}{\widehat{e}}_{TM}\end{array}$$

(1)

The total electromagnetic field at the boundary is the sum of the incident field and the reflected field, which can be expressed as follows:

$$\begin{array}{l}{\overrightarrow{E}}^{tol }={\overrightarrow{E}}^{inc}+R_{TE}{E}_{TE}^{inc}{\widehat{e}}_{TE}+R_{TM}{E}_{TM}^{inc}\left({\widehat{k}}_s\times {\widehat{e}}_{TE}\right)\\ {\overrightarrow{H}}^{tol}={\displaystyle \frac{1}{\eta }}{\widehat{k}}_i\times {\overrightarrow{E}}^{inc}+{\displaystyle \frac{1}{\eta }}\left[R_{TE}{E}_{TE}^{inc}\left({\widehat{k}}_s\times {\widehat{e}}_{TE}\right)-R_{TM}{E}_{TM}^{inc}{\widehat{e}}_{TE}\right]\end{array}$$

(2)

where R_TE and R_TM represent the reflection coefficients for the TE wave and TM wave, respectively. Substituting (2) into the boundary conditions yields the equivalent electromagnetic currents on the target surface:

$$\begin{array}{l}{\overrightarrow{J}}^{MECA}=\widehat{n}\times {{\overrightarrow{H}}^{\mathrm{tot} }|}_s={{\displaystyle \frac{1}{\eta }}\left\{E_{TE}^{inc}\cos {\theta }_{inc}\left(1-R_{TE}\right){\widehat{e}}_{TE}+E_{TM}^{inc}\left(1-R_{TM}\right)\left(\widehat{n}\times {\widehat{e}}_{TE}\right)\right\}|}_s\\ {\overrightarrow{M}}^{MECA}=-\widehat{n}\times {{\overrightarrow{E}}^{\mathrm{tot} }|}_s={\left\{E_{TE}^{inc}\left(1+R_{TE}\right)\left({\widehat{e}}_{TE}\times \widehat{n}\right)+E_{TM}^{inc}\cos {\theta }_{inc}\left(1+R_{TM}\right){\widehat{e}}_{TE}\right\}|}_s\end{array}$$

(3)

Therefore, the scattered EM field can be expressed as follows:

$$\begin{array}{l}{\overrightarrow{E}}_i^s={\displaystyle \frac{\mathrm{j}}{2\lambda }}{\displaystyle \frac{\exp (-\mathrm{j}kR)}{R}}\left[{\overrightarrow{E}}_i^a-\eta {\overrightarrow{H}}_i^a\times \widehat{s}\right]\\ {\overrightarrow{H}}_i^s=-{\displaystyle \frac{\mathrm{j}}{2\lambda }}{\displaystyle \frac{\exp (-\mathrm{j}kR)}{R}}\left[{\overrightarrow{H}}_i^a-{\displaystyle \frac{1}{\eta }}\widehat{s}\times {\overrightarrow{E}}_i^a\right]\end{array}$$

(4)

The specific expressions for ${\overrightarrow{E}}_i^a$ and ${\overrightarrow{H}}_i^a$ are as follows:

$$\begin{array}{l}{\overrightarrow{E}}_i^a=\left(\widehat{s}\times {\overrightarrow{M}}_i^{MECA}\right){\displaystyle {\int }_{s_i}\exp }\left[\mathrm{j}k(\widehat{s}-\widehat{i})\cdot {\overrightarrow{r}}_i^{\prime }\right]d{s}_i\\ {\overrightarrow{H}}_i^a=\left(\widehat{s}\times {\overrightarrow{J}}_i^{MECA}\right){\displaystyle {\int }_{s_i}\exp \left[\mathrm{j}k(\widehat{s}-\widehat{i})\cdot {\overrightarrow{r}}_i^{\prime }\right]}d{s}_i\end{array}$$

(5)

The specific expression for the total scattered field of the target is as follows:

$${\overrightarrow{E}}^{total }={\displaystyle \sum _{1,2,3,\dots }{\displaystyle \sum _i^n{\overrightarrow{E}}_{i,po}}}$$

(6)

Here, the first summation symbol represents the superposition of the scattered fields from all ray orders, and the second summation symbol represents the superposition of the scattered fields from the illuminated triangular facets in the illuminated regions during each ray tracing order. n denotes the number of illuminated triangular facets in the illuminated regions during the ray tracing process.

From (4)–(6), it can be concluded that the calculation of the EM scattering characteristics of a coated 3D target requires knowledge of the reflection coefficients R_TE and R_TM on the coated surface. In the following sections, we will present the method for calculating these reflection coefficients based on higher-order impedance boundary conditions.

2.2. High-Order Impedance Boundary Conditions for Coated 3D Targets

The LIBC gives the relationship between the tangential electric field and magnetic field vectors through the equivalent surface impedance of the dielectric coated target, and the scattered field outside the target can be calculated without examining the field distribution inside the target, which facilitates the use of the surface electromagnetic field integral formula to efficiently solve the electromagnetic scattering of dielectric thin film-coated targets. However, its accuracy is low and it can only be used to analyze media coatings with large refractive indices. Therefore, in order to improve the applicability of the LIBC method, the HOIBC is proposed, which introduces higher-order derivatives of the EM field components into the LIBC model.

Consider an infinitely large metallic conductor plate model coated with a dielectric layer of thickness $d$. The permittivity and permeability parameters of the dielectric layer are $\left({\epsilon }_1,{\mu }_1\right)$. A plane incident wave with an incident frequency $f$ and an incident angle $\theta$ illuminates this structure, as shown in Figure 3.

The electric field $E_x\left(z\right)$ at any point within the dielectric layer can be expressed using the Taylor series expansion of the electric field $E_x\left(z_0\right)$ at the dielectric surface $z_0$ as follows:

$$E_x(z)=\sum _{n=0}^{\infty }\frac{{\left(z-z_0\right)}^{2n}}{\left(2n\right)!}{\partial }_z^{2n}{E}_x\left(z_0\right)+\sum _{n=0}^{\infty }\frac{{\left(z-z_0\right)}^{2n+1}}{\left(2n+1\right)!}{\partial }_z^{2n+1}{E}_x\left(z_0\right)$$

(7)

The component $E_x(z)$ of the electric field intensity $\overrightarrow{E}$ at any point within the dielectric satisfies the Helmholtz equation. Therefore, ${\partial }_z^2{E}_x\left(z_0\right)=-\left(k_1^2+{\nabla }_t^2\right)E_x\left(z\right)$, and ${\partial }_z^{2n}{E}_x\left(z_0\right)={\left(-1\right)}^n{\left(k_1^2+{\nabla }_t^2\right)}^n{E}_x\left(z\right)$. Hence, substituting them into the right-hand side of (7) we can obtain that

$$\sum _{n=0}^{\infty }\frac{{\left(z-z_0\right)}^{2n}}{\left(2n\right)!}{\partial }_z^{2n}{E}_x\left(z_0\right)=\cos \left[k_1\left(z-z_0\right)\sqrt{1+t_1}\right]E_x\left(z_0\right)$$

(8)

where $k_1=\omega \sqrt{{\epsilon }_1{\mu }_1}$ and $t_1={\nabla }_t^2/k_1^2=\left({\partial }_x^2+{\partial }_y^2\right)/k_1^2$. For the second term on the right-hand side of (7), we also have ${\partial }_z^{2n}{E}_x\left(z_0\right)={\left(-1\right)}^n{\left(k_1^2+{\nabla }_t^2\right)}^n{\partial }_z{E}_x\left(z\right)$. According to Maxwell’s equations, we can obtain that ${\partial }_z{E}_x\left(x\right)=-j({\partial }_{yz}^2{H}_z\left(z\right)-{\partial }_z^2{H}_y\left(z\right))/\omega {\epsilon }_1$. Similarly, the component $H_y\left(z\right)$ of the magnetic field intensity $\overrightarrow{H}$ satisfies the Helmholtz equation, and with $\nabla \cdot \overrightarrow{H}=0$, we obtain ${\partial }_z{E}_x\left(z\right)=-j\left(k_1^2{H}_y\left(z\right)-{\partial }_{xy}^2{H}_x\left(z\right)+{\partial }_x^2{H}_y(z_0)\right)/\omega {\epsilon }_1$. Further, we obtain that

$$\begin{array}{l}{\displaystyle \sum _{n=0}^{\infty }\frac{{\left(z-z_l\right)}^{2n+1}}{\left(2n+1\right)!}{\partial }_z^{2n+1}{E}_x\left(z_0\right)}\\ ={\displaystyle \frac{j}{\omega {\epsilon }_1{\mu }_1}}{\displaystyle \frac{\sin \left[k_1\left(z_0-z\right)\sqrt{1+t_1}\right]}{\sqrt{1+t_1}}}\cdot \left[k_1^2{H}_y\left(z_0\right)-{\partial }_{xy}^2{H}_x\left(z_0\right)+{\partial }_x^2{H}_y\left(z_0\right)\right]\end{array}$$

(9)

Finally, we obtain

$$\begin{array}{ll}\widehat{n}\times \overrightarrow{E}\left(z\right)& =\cos \left[k_1\left[z_0-z\right]\sqrt{1+t_1}\right]\cdot \\ & \left[\widehat{n}\times \overrightarrow{E}\left(z_0\right)+{\displaystyle \frac{j{\eta }_1}{k_1^2}}{\displaystyle \frac{\tan \left[k_1\left(z_0-z\right)\sqrt{1+t_1}\right]}{\sqrt{1+t_1}}}\cdot \left(k_1-L_R\right){\overrightarrow{H}}_{\tan }\right]\end{array}$$

(10)

$$\begin{array}{ll}{\overrightarrow{H}}_{\tan }(z)& =\cos \left[k_1\left(z_0-z\right)\sqrt{1+t_1}\right][{\overrightarrow{H}}_{\tan }\left(z_0\right)\\ & +{\displaystyle \frac{j}{{\eta }_1{k}_1^2}}{\displaystyle \frac{\tan \left[k_1\left(z_0-z\right)\sqrt{1+t_1}\right]}{\sqrt{1+t_1}}}\cdot \left(k_1^2+L_D\right)\left[\widehat{n}\times \overrightarrow{E}\left(z_0\right)\right]\end{array}$$

(11)

where $k_1=\omega \sqrt{{\epsilon }_1{\mu }_1}$, ${\eta }_1=\sqrt{{\mu }_1/{\epsilon }_1}$, and the subscript tan denotes the tangential component. The operators $t_1$, $L_R$ and $L_D$ are defined as follows:

$$t_1={\nabla }_{tg}^2/k_1^2=\left({\partial }_x^2+{\partial }_y^2\right)/k_1^2$$

(12)

$$L_R\left(\overrightarrow{V}\right)=\nabla \times \left[\widehat{n}\left[\widehat{n}\cdot \left(\nabla \times \overrightarrow{V}\right)\right]\right], L_D\left(\overrightarrow{V}\right)={\nabla }_t{\nabla }_t\cdot \overrightarrow{V}$$

(13)

Here, $\overrightarrow{V}$ represents any arbitrary vector in the xy-plane. When the position z is at the surface of the conductor, i.e., $z_0-z=d$, (10) and (11) can be written as follows:

$$\widehat{n}\times \overrightarrow{E}\left(z\right)=\cos \left(k_{1}d\sqrt{1+t_1}\right)\left[\widehat{n}\times \overrightarrow{E}\left(z_0\right)+T_R{\overrightarrow{H}}_{\tan }\left(z_0\right)\right]$$

(14)

$${\overrightarrow{H}}_{\tan }\left(z\right)=\cos \left(k_{1}d\sqrt{1+t_1}\right)\left[T_D\left[\widehat{n}\times \overrightarrow{E}\left(z_0\right)\right]+{\overrightarrow{H}}_{\tan }\left(z_0\right)\right]$$

(15)

$$T_R={\displaystyle \frac{j{\eta }_1}{k_1^2}}P\left(k_1^2-L_R\right), T_D={\displaystyle \frac{j}{{\eta }_1{k}_1^2}}P\left(k_1^2+L_D\right)$$

(16)

with

$$P={\displaystyle \frac{\tan \left(k_{1}d\sqrt{1+t_1}\right)}{\sqrt{1+t_1}}}$$

(17)

From the boundary conditions at the surface of the conductor, we know that the term $\widehat{n}\times \overrightarrow{E}\left(z\right)=0$ on the left-hand side of (14); thus, we have

$$\widehat{n}\times \overrightarrow{E}\left(z_0\right)=-{\displaystyle \frac{j{\eta }_1}{k_1^2}}P\left(k_1^2-L_R\right){\overrightarrow{H}}_{\tan }\left(z_0\right)$$

(18)

Equation (18) is the HOIBC of the coated 3D target, and it shows the relationship between the electric and magnetic fields on the outer surface of the coated target. To solve for the reflection coefficients of the coated target, it is necessary to approximate the pseudo-differential operator $P$ in the equation.

2.3. Reflection Coefficient of the Coated 3D Targets Based on HOIBC

P can be approximated in terms of rational fractions:

$$P={\displaystyle \frac{a_0+a_1{t}_1}{1+b_1{t}_1}}$$

(19)

The Taylor series expressions on the left and right sides of (19) with respect to $t_1$ take equal values at the zero point and their first-order derivatives, leading to the Taylor approximation:

$$\begin{array}{l}{a}_0=\tan \left(k_{1}d\right)\\ a_1=\frac{k_{1}d-\tan \left(k_{1}d\right)+k_{1}d{\tan }^2\left(k_{1}d\right)}{2}\\ b_1=0\end{array}$$

(20)

Substituting $t_1=\left(L_D+L_R\right)/k_1^2$ into P, (18) can be simplified to

$$\hat{n}\times \overrightarrow{E}\left(z_0\right)=-j{\eta }_1[a_0+a_1\frac{L_D}{k_1^2}-(a_0+a_1\left)\frac{L_R}{k_1^2}\right]{\overrightarrow{H}}_{\tan }\left(z_0\right)$$

(21)

According to (32), the vertical polarization reflection coefficient $R_{TM}$ and the horizontal polarization reflection coefficient $R_{TE}$ can be expressed as follows [24]:

$$\begin{array}{l}{R}_{TM}\left(\theta \right)={\displaystyle \frac{-\delta \left(\theta \right)\cos \theta +i{\sigma }_{TM}\left(\theta \right)}{\delta \left(\theta \right)\cos \theta +i{\sigma }_{TM}\left(\theta \right)}}\\ R_{TE}\left(\theta \right)={\displaystyle \frac{-\delta \left(\theta \right)\cos \theta +i{\sigma }_{TE}\left(\theta \right)}{\delta \left(\theta \right)\cos \theta +i{\sigma }_{TE}\left(\theta \right)}}\end{array}$$

(22)

with

$$\begin{array}{l}\delta \left(\theta \right)=1-b_1{\displaystyle \frac{{\sin }^2\theta }{N_1^2}}\\ {\sigma }_{TM}\left(\theta \right)={\displaystyle \frac{{\eta }_1}{{\eta }_0}}\left[a_0-\left(a_0+a_1\right){\displaystyle \frac{{\sin }^2\theta }{N_1^2}}\right]\\ {\sigma }_{TM}\left(\theta \right)={\displaystyle \frac{{\eta }_1}{{\eta }_0}}\left[a_0-a_1{\displaystyle \frac{{\sin }^2\theta }{N_1^2}}\right]\end{array}$$

(23)

2.4. Equivalent Edge Current of the Impedance Wedge

The perturbation method is usually used to calculate the diffraction coefficients of the impedance wedge. However, in the process of solving the spectral function coefficients, this method approximately satisfies the boundary conditions, which leads to the problem of poor accuracy of the calculation results in the case of large-angle deviation from vertical incidence or grazing incidence. On the other hand, it is necessary to perform the integration operation, which reduces the calculation speed and is not conducive to solving the scattered field of the electrically large coated target. Therefore, this paper introduces a method based on numerical matching technology to quickly calculate the diffraction coefficients of the impedance wedge structure, thereby obtaining its equivalent edge current, which is used to solve the scattering problem of electrically large coated targets.

A local coordinate system is established on the upper surface of the impedance wedge, as shown in Figure 4, where ${\overline{\eta }}^{\pm }$ represents the normalized surface impedance of the two wedge faces.

The expression for the incident electromagnetic field is as follows:

$$\begin{array}{l}{\overrightarrow{E}}^i\left({\overrightarrow{r}}^{\prime }\right)={\widehat{e}}^i\cdot e^{-\mathrm{j}{k}_0\widehat{i}\cdot {\overrightarrow{r}}^{\prime }}=\left(e_x\widehat{x}+e_y\widehat{y}+e_z\widehat{z}\right)\cdot {\mathrm{e}}^{-\mathrm{j}{k}_0\widehat{i}\cdot {\overrightarrow{r}}^{\prime }}\\ Z_0{\overrightarrow{H}}^i\left({\overrightarrow{r}}^{\prime }\right)={\widehat{h}}^i\cdot e^{-\mathrm{j}{k}_0\widehat{i}\cdot {\overrightarrow{r}}^{\prime }}=\left(h_x\widehat{x}+h_y\widehat{y}+h_z\widehat{z}\right)\cdot {\mathrm{e}}^{-\mathrm{j}{k}_0\widehat{i}\cdot {\overrightarrow{r}}^{\prime }}\end{array}$$

(24)

In the ray-based coordinate system, the components of the incident electric field can be expressed as follows:

$$\left[\begin{array}{l}{E}_{{\beta }^i}^i\left({\overrightarrow{r}}^{\prime }\right)\\ E_{{\varphi }^i}^i\left({\overrightarrow{r}}^{\prime }\right)\end{array}\right]=\left[\begin{array}{l}\widehat{t}\cdot {\widehat{e}}^i\\ \left(\widehat{i}\times \widehat{t}\right)\cdot {\widehat{e}}^i\end{array}\right]\cdot {\displaystyle \frac{e^{-\mathrm{j}{k}_0\widehat{i}\cdot {\overrightarrow{r}}^{\prime }}}{\sin {\beta }_0}}$$

(25)

where $\widehat{t}$ is the unit vector along the wedge edge direction and ${\beta }_0$ is the angle between the incident wave vector and the wedge edge.

The numerical matching technique is derived directly from the coupled integral equations and improves the enforcement of boundary conditions by implementing numerical matching on the impedance wedge surface. The spectral function of an impedance wedge with an arbitrary wedge angle under oblique incidence of a plane wave can be expanded as follows:

$$S_{e,h}\left(\alpha \right)={\sigma }_{{\varphi }_0}\left(\alpha \right){\mathsf{\Lambda }}_{e,h}\left(\alpha \right){\mathsf{\Psi }}_{e,h}\left(\alpha \right)$$

(26)

where ${\sigma }_{{\varphi }_0}\left(\alpha \right)$ is the meromorphic function and the Maliuzhinets function ${\mathsf{\Psi }}_{e,h}\left(\alpha \right)$ is expressed as follows:

$$\begin{array}{r}{\mathsf{\Psi }}_{e, h}(\alpha )={\mathsf{\Psi }}_{\mathsf{\Phi }}\left(\alpha +{\displaystyle \frac{n\pi }{2}}+{\displaystyle \frac{\pi }{2}}-{\theta }_{e, h}^{+}\right){\mathsf{\Psi }}_{\mathsf{\Phi }}\left(\alpha -{\displaystyle \frac{n\pi }{2}}-{\displaystyle \frac{\pi }{2}}+{\theta }_{e, h}^{-}\right)\cdot \\ {\mathsf{\Psi }}_{\mathsf{\Phi }}\left(\alpha +{\displaystyle \frac{n\pi }{2}}-{\displaystyle \frac{\pi }{2}}+{\theta }_{e, h}^{+}\right){\mathsf{\Psi }}_{\mathsf{\Phi }}\left(\alpha -{\displaystyle \frac{n\pi }{2}}+{\displaystyle \frac{\pi }{2}}-{\theta }_{e, h}^{-}\right)\end{array}$$

(27)

In this expression, ${\theta }_{e, h}^{\pm }$ represents the Brewster angles corresponding to the electric and magnetic polarizations for the two wedge faces, satisfying $\sin {\theta }_e^{\pm }=\sin {\beta }_0/{\overline{\eta }}^{\pm }$ and $\sin {\theta }_h^{\pm }={\overline{\eta }}^{\pm }\sin {\beta }_0$. The expression of ${\mathsf{\Psi }}_{\mathsf{\Phi }}\left(\alpha \right)$ is

$${\mathsf{\Psi }}_{\mathsf{\Phi }}(\alpha )=\exp \left[{\displaystyle \frac{\mathrm{j}}{4n\pi }}{\displaystyle {\int }_0^{\alpha }{\displaystyle {\int }_{-\mathrm{j}\infty }^{\mathrm{j}\infty }\tan }}\left({\displaystyle \frac{v}{2n}}\right){\displaystyle \frac{dudv}{\cos (v-u)}}\right]$$

(28)

${\mathsf{\Lambda }}_{e,h}\left(\alpha \right)$ is the auxiliary function and its expression is

$$\left[\begin{array}{c}{\mathsf{\Lambda }}_e(\alpha )\\ {\mathsf{\Lambda }}_h(\alpha )\end{array}\right]=\left[\begin{array}{cc}{B}_1^e(\alpha )& B_2^e(\alpha )\\ B_2^h(\alpha )& B_1^h(\alpha )\end{array}\right]\left[\begin{array}{c}{E}_z^i\\ \eta H_z^i\end{array}\right]$$

(29)

with

$$\begin{array}{l}{B}_1^{e,h}(\alpha )={\displaystyle \frac{c_0^{e,h}+c_1^{e,h}\tan (\alpha /8)\cos \beta +c_2^{e,h}{\tan }^2(\alpha /8){\cos }^2\beta }{1+d_1^{e,h}\tan (\alpha /8)\cos \beta +d_2^{e,h}{\tan }^2(\alpha /8){\cos }^2\beta }}\\ B_2^{e,h}(\alpha )={\displaystyle \frac{c_1^{e,h}\tan (\alpha /8)\cos \beta +c_2^{e,h}{\tan }^2(\alpha /8){\cos }^2\beta }{1+d_1^{e,h}\tan (\alpha /8)\cos \beta }}\end{array}$$

(30)

The expressions for each coefficient are as follows:

$$\begin{gathered} c_0^{e,h}={\displaystyle \frac{1}{{\mathsf{\Psi }}_{e,h}\left(n\pi /2-{\varphi }_i\right)}} \\ \begin{array}{l}{c}_1^e=0.261+\mathrm{j}0.231,c_2^e=0.182-\mathrm{j}0.354\\ d_1^e=0.873-\mathrm{j}0.841,d_2^e=0.205-\mathrm{j}0.008\\ c_1^h=0.322-\mathrm{j}0.0611,c_2^h=-0.453-\mathrm{j}0.087\\ d_1^h=0.243+\mathrm{j}0.002,d_2^h=0.031+\mathrm{j}0.002\end{array} \end{gathered}$$

After obtaining the spectral function, it is substituted into the Sommerfeld integral. By applying the Pauli–Clemmow steepest descent method, a uniform asymptotic UTD solution for the diffraction field can be obtained:

$$\begin{array}{ll}\left[\begin{array}{c}{E}_z^d\\ Z_0{H}_z^d\end{array}\right]& =-{\displaystyle \frac{v_1{\mathrm{e}}^{-\mathrm{j}{k}_0\rho \sin {\beta }_p}{\mathrm{e}}^{-\mathrm{j}{k}_{a}z\cos {\beta }_h}}{2n\sqrt{\rho \sin {\beta }_0}}}\cdot \\ & \left[\begin{array}{cc}{\mathsf{\Psi }}_{\mathrm{e}}\left({\alpha }_1\right){\mathsf{\Lambda }}_{\mathrm{e}}\left({\alpha }_1\right)& {\mathsf{\Psi }}_{\mathrm{e}}\left({\alpha }_2\right){\mathsf{\Lambda }}_{\mathrm{e}}\left({\alpha }_2\right)\\ {\mathsf{\Psi }}_{\mathrm{h}}\left({\alpha }_1\right){\mathsf{\Lambda }}_{\mathrm{h}}\left({\alpha }_1\right)& {\mathsf{\Psi }}_{\mathrm{h}}\left({\alpha }_2\right){\mathsf{\Lambda }}_{\mathrm{h}}\left({\alpha }_2\right)\end{array}\right]\left[\begin{array}{c}{P}_1\\ P_2\end{array}\right]\end{array}$$

(31)

where ${\alpha }_1=-\pi +{\displaystyle \frac{n\pi }{2}}-\varphi$, ${\alpha }_2=\pi +{\displaystyle \frac{n\pi }{2}}-\varphi$, $v_1=e^{-j\pi /4}/\sqrt{2\pi k_0}$, and $\rho =s\sin {\beta }_0$. Based on the incident angle and the corresponding illumination conditions of the wedge faces, $P_1$ and $P_2$ have three possible values.

When the incident angle satisfies $0<{\varphi }_0\le n\pi -\pi$, we have

$$\begin{array}{l}{P}_1=\cot {\displaystyle \frac{\pi +\varphi -{\varphi }_0}{2n}}-\cot {\displaystyle \frac{\pi +\varphi +{\varphi }_0}{2n}},\\ P_2=\cot {\displaystyle \frac{\pi -\varphi +{\varphi }_0}{2n}}F\left\{k_0\rho \sin {\beta }_0\left[1+\cos \left(\varphi -{\varphi }_0\right)\right]\right\}-\cot {\displaystyle \frac{\pi -\varphi -{\varphi }_0}{2n}}F\left\{k_0\rho \sin {\beta }_0\left[1+\cos \left(\varphi +{\varphi }_0\right)\right]\right\}\end{array}$$

When the incident angle satisfies $n\pi -\pi <{\varphi }_0\le \pi$, we have

$$\begin{array}{l}{P}_1=\cot {\displaystyle \frac{\pi +\varphi -{\varphi }_0}{2n}}-\cot {\displaystyle \frac{\pi +\varphi +{\varphi }_0}{2n}}F\left\{k_0\rho \sin {\beta }_0\left[1+\cos \left(\varphi +{\varphi }_0-2n\pi \right)\right]\right\}\\ P_2=\cot {\displaystyle \frac{\pi -\varphi +{\varphi }_0}{2n}}-\cot {\displaystyle \frac{\pi -\varphi -{\varphi }_0}{2n}} F\left\{k_0\rho \sin {\beta }_0\left[1+\cos \left(\varphi +{\varphi }_0\right)\right]\right\}\end{array}$$

When the incident angle satisfies $\pi <{\varphi }_0\le n\pi$, we have

$$\begin{array}{l}{P}_1=\cot {\displaystyle \frac{\pi +\varphi -{\varphi }_0}{2n}}F\left\{k_0\rho \sin {\beta }_0\left[1+\cos \left(\varphi -{\varphi }_0\right)\right]\right\}-\cot {\displaystyle \frac{\pi +\varphi +{\varphi }_0}{2n}}F\left\{k_0\rho \sin {\beta }_0\left[1+\cos \left(\varphi +{\varphi }_0-2n\pi \right)\right]\right\}\\ P_2=\cot {\displaystyle \frac{\pi -\varphi +{\varphi }_0}{2n}}-\cot {\displaystyle \frac{\pi -\varphi -{\varphi }_0}{2n}}\end{array}$$

The transition function is given by $F\left(x\right)=2\mathrm{j}\sqrt{x}{e}^{\mathrm{j}x}{\displaystyle {\int }_{\sqrt{x}}^{\infty }{e}^{-\mathrm{j}{t}^2}}dt$.

The diffraction field can be represented in the following standard form:

$$\left[\begin{array}{c}{E}_{\beta }^d\\ E_{\varphi }^d\end{array}\right]=-A_1(s)\left[\begin{array}{cc}{D}_{\beta \beta }& D_{\beta {\varphi }^i}\\ D_{\varphi {\beta }^i}& D_{\varphi {\varphi }^i}\end{array}\right]\left[\begin{array}{c}{E}_{\beta }^i\left(Q_E\right)\\ E_{{\varphi }^i}^i\left(Q_E\right)\end{array}\right]$$

(32)

Here, $A\left(s\right)=e^{-\mathrm{j}{k}_{0}s}/\sqrt{s}$, $E_{{\beta }^i}^i\left(Q_E\right)={\displaystyle \frac{e_z}{\sin {\beta }_0}}$, $E_{{\varphi }^i}^{\mathrm{i}}\left(Q_E\right)={\displaystyle \frac{-h_z}{\sin {\beta }_0}}$, and the diffraction coefficient $\overline{\overline{D}}$ matrix for the impedance wedge is as follows:

$$\begin{array}{l}{D}_{\beta \beta }={\displaystyle \frac{v_1}{2n}}\left[{\psi }_e\left({\alpha }_1\right)B_1^e\left({\alpha }_1\right)P_1+{\psi }_e\left({\alpha }_2\right)B_1^e\left({\alpha }_2\right)P_2\right]\\ D_{\beta {\varphi }^i}={\displaystyle \frac{v_1}{2n}}\left[{\psi }_e\left({\alpha }_1\right)B_2^e\left({\alpha }_1\right)P_1+{\psi }_e\left({\alpha }_2\right)B_2^e\left({\alpha }_2\right)P_2\right]\\ D_{\varphi {\beta }^i}={\displaystyle \frac{v_1}{2n}}\left[{\psi }_h\left({\alpha }_1\right)B_2^e\left({\alpha }_1\right)P_1+{\psi }_h\left({\alpha }_2\right)B_1^e\left({\alpha }_2\right)P_2\right]\\ D_{\varphi {\varphi }^i}={\displaystyle \frac{v_1}{2n}}\left[{\psi }_h\left({\alpha }_1\right)B_1^e\left({\alpha }_1\right)P_1+{\psi }_h\left({\alpha }_2\right)B_1^e\left({\alpha }_2\right)P_2\right]\end{array}$$

(33)

According to the incremental length diffraction coefficient (ILDC) of the edge wave, the contribution of the PO diffraction coefficient needs to be subtracted. The PO diffraction coefficient matrix ${\overline{\overline{D}}}^{PO+}$ for the upper wedge face can be expressed as follows:

$$\begin{array}{l}{D}_{\beta {\beta }^i}^{PO+}=p[-\sin \beta \sin {\varphi }_i+{\eta }_{+}^2\sin \varphi \sin \beta -{\eta }_{+}\left({\sin }^2\beta -\sin \varphi \sin {\varphi }_i-\cos {\varphi }_i\cos \varphi {\cos }^2\beta \right)]\\ D_{\beta {\varphi }^i}^{PO+}=p[-{\eta }_{+}\cos \beta \left(\sin \varphi \cos {\varphi }_i-\cos \varphi \sin {\varphi }_i\right)+\cos \beta \sin \beta \left(\cos {\varphi }_i+\cos \varphi \right)]\\ D_{\varphi {\beta }^i}^{PO+}=p[-{\eta }_{+}^2\sin \beta \cos \beta \left(\cos {\varphi }_i+\cos \varphi \right)+{\eta }_{+}\left(\sin \varphi \cos \beta \cos {\varphi }_i-\cos \varphi \cos \beta \sin {\varphi }_i\right)]\\ D_{\varphi {\varphi }^i}^{PO+}=p[\sin \beta \sin {\varphi }_i-{\eta }_{+}^2\sin \varphi \sin \beta -{\eta }_{+}\left({\sin }^2\beta -\sin \varphi \sin {\varphi }_i-\cos {\varphi }_i\cos \varphi {\cos }^2\beta \right)]\end{array}$$

(34)

where $v_2={\left(1+{\eta }_{+}\sin \beta \sin \varphi \right)}^{-1}{\left({\eta }_{+}+\sin \beta \sin \varphi \right)}^{-1}$ and $p=v_1{v}_2\sin {\varphi }_i/\left(\cos \varphi +\cos {\varphi }_i\right)$. The PO diffraction coefficient matrix ${\overline{\overline{D}}}^{PO-}$ for the lower wedge face can be obtained from the diffraction coefficient of the upper wedge face by applying the transformation ${\eta }_{+}\to {\eta }_{-},{\varphi }_i\to n\pi -{\varphi }_i,\varphi \to n\pi -\varphi ,\beta =\pi -\beta$.

The total PO diffraction coefficient of the edge is the sum of the PO diffraction coefficients of the two wedge faces, i.e., ${\overline{\overline{D}}}^{PO}={\delta }^{+}{\overline{\overline{D}}}^{PO+}+{\delta }^{-}{\overline{\overline{D}}}^{PO-}$, where δ^± is the illumination factor, as shown in Figure 5.

The ILDCs are the UTD diffraction coefficients with the contribution of the PO diffraction coefficients subtracted. Therefore, they can be expressed as ${\overline{\overline{D}}}^f=\overline{\overline{D}}-{\overline{\overline{D}}}^{PO}$. Thus, the expression for the equivalent edge electromagnetic currents can be derived as follows:

$$\begin{array}{l}{I}_e^i\left(z^{\prime }\right)=-{\displaystyle \frac{2}{Z_0}}\sqrt{{\displaystyle \frac{2\pi }{k_0}}}{\mathrm{e}}^{-\mathrm{j}{\displaystyle \frac{\pi }{4}}}\left(D_{\beta {\beta }^{\prime }}^f{E}_{{\beta }^{\prime }}^i\left(z^{\prime }\right)+D_{\beta {\varphi }^i}^f{E}_{{\varphi }^{\prime }}^i\left(z^{\prime }\right)\right)\\ I_m^f\left(z^{\prime }\right)=2\sqrt{{\displaystyle \frac{2\pi }{k_0}}}{\mathrm{e}}^{-\mathrm{j}{\displaystyle \frac{\pi }{4}}}\left(D_{\varphi {\beta }^{\prime }}^f{E}_{{\beta }^{\prime }}^i\left(z^{\prime }\right)+D_{\varphi {\varphi }^{\prime }}^i{E}_{{\varphi }^{\prime }}^i\left(z^{\prime }\right)\right)\end{array}$$

(35)

Using the electromagnetic current radiation integral formula, the radiation field of the equivalent edge electromagnetic currents can be obtained:

$${\overrightarrow{E}}_s^f=\frac{\mathrm{j}{k}_0\exp (-\mathrm{j}{k}_{0}R)}{4\pi R}{ʃ}_l[\eta \hat{s}\times (\hat{s}\times \hat{t}\left)I_e^f\right(z^{\prime })+(\hat{s}\times \hat{t}\left)I_m^f\right(z^{\prime }\left)\right]\exp (j{k}_0\hat{s}·{\overrightarrow{r}}^{\prime })d{z}^{\prime }$$

(36)

$$\begin{gathered} \mathrm{The} \mathrm{total} \mathrm{scattered} \mathrm{field} \mathrm{of} \mathrm{the} \mathrm{target} \mathrm{is} \mathrm{the} \mathrm{superposition} \mathrm{of} \mathrm{the} \mathrm{PO} \mathrm{field} \mathrm{and} \\ \mathrm{the} \mathrm{wedge} \mathrm{diffraction} \mathrm{field}, \mathrm{i}.\mathrm{e}.,{\overrightarrow{E}}_s={\overrightarrow{E}}_s^{PO}+{\overrightarrow{E}}_s^f \end{gathered}$$

(37)

For better understanding, Figure 6 represents the flowchart for the method proposed in this paper.

3. Results

In this section, first, our method is used to simulate the EM scattering of the RAM-coated target, and the results are compared with those obtained using the MLFMA algorithm. In addition, the RCS characteristics of the RAM-coated target with different dielectric parameters, coating thicknesses, and EM wave incidence angles are discussed.

3.1. Validation of the Effectiveness of the SBR-EEC Method

Figure 7 shows the monostatic RCS simulation of a dihedral reflector using the SBR-EEC algorithm proposed in this paper. The dihedral reflector has an edge length of 0.3 m, and the number of subdivided facets is 22,484. The parameters of the dielectric coating are ${\epsilon }_r=4-1.5\mathrm{j}$ and ${\mu }_r=2-\mathrm{j}$, and the thickness is $d=0.04\lambda$. The incident wave frequency is set to 10 GHz, with VV and HH polarization modes. The simulation results are compared with those obtained using the MLFMA algorithm, the PO method, and the traditional SBR method without considering edge diffraction, and the root mean square error (RMSE) of these methods compared to MLFMA is shown in

Table 1. RMSE method compared to MLFMA in Figure 7.

	PO	SBR	Our Method
VV polarization	23.38	1.99	1.08
HH polarization	23.56	1.73	1.037

. As shown in Figure 7, since the PO algorithm only considers the single reflection of the incident wave, it results in larger computational errors compared to the MLFMA algorithm. For the traditional SBR algorithm, although multiple reflections of the incident wave are considered, leading to higher accuracy than the PO algorithm, it neglects the effect of edge diffraction fields during the calculation of dihedral angles. As a result, its computational results exhibit significant errors near 0° and 90° when compared to the MLFMA method. Compared to the PO and traditional SBR algorithms, the method proposed in this paper not only fully considers multiple reflections of incident waves but also introduces the edge diffraction fields using the EEC method to correct the scattering fields obtained by the SBR algorithm, thereby improving the accuracy of the computational results.

To investigate the impact of edge structures on the accuracy of EM scattering field calculations, we analyzed the monostatic scattering characteristics of a cube using MLFMA, the HOIBC-based SBR algorithm, which does not consider edge diffraction, and the algorithm proposed in this paper. The cube has an edge length of 0.3 m, and the number of subdivided facets is 67,452. The parameters of the dielectric coating are ${\epsilon }_r=4-1.5\mathrm{j}$ and ${\mu }_r=2-\mathrm{j}$, and the thickness is $d=0.04\lambda$. The simulation uses an incident wave frequency of 10 GHz, with an incident elevation angle ranging from 0° to 90° and an azimuth angle of 0°. The polarization modes include VV and HH polarization. The RMSE of these methods compared to MLFMA is shown in Table 2. As shown in Table 2, since the HOIBC-SBR method does not consider the edge diffraction of the cube, it results in larger errors compared to the MLFMA method. The algorithm proposed in this paper improves the traditional HOIBC-SBR by incorporating the edge diffraction field using the EEC method, thereby correcting the scattering field calculations and making the results more similar to those of MLFMA. Therefore, the effect of edge diffraction is significant on the accuracy of electromagnetic scattering fields, and it is necessary to consider the effects of edge diffraction fields when analyzing the electromagnetic scattering characteristics of targets.

Table 2. RMSE method compared to MLFMA in Figure 8.

	SBR	Our Method
VV polarization	4.14	0.91
HH polarization	4.37	2.78

Table 2. RMSE method compared to MLFMA in Figure 8.

	SBR	Our Method
VV polarization	4.14	0.91
HH polarization	4.37	2.78

Figure 8. Monostatic RCS of the cube. (a) HH polarization. (b) VV polarization.

Figure 8. Monostatic RCS of the cube. (a) HH polarization. (b) VV polarization.

3.2. Influence of RAMs with Different Dielectric Parameters on Target RCS

To further the effectiveness of the proposed algorithm, a composite model consisting of several cubes was created. The maximum dimensions of the model are $0.3 \mathrm{m} \times 0.5 \mathrm{m} \times 0.4 \mathrm{m}$ and the surface of the model is divided into 7058 facets. The model was coated with two different RAMs, RMA1 and RMA2. For RMA1, the relative permittivity is ${\epsilon }_r=6-3\mathrm{j}$, the relative permeability is ${\mu }_r=1$, and the thickness is $d=0.24\lambda$. For RMA2, the relative permittivity is ${\epsilon }_r=1-1\mathrm{j}$, the relative permeability is ${\mu }_r=1$, and the thickness is $d=0.24\lambda$. The incident wave frequency is set to f = 6 GHz, with HH polarization and VV polarization. The model was analyzed using three different methods: the MLFMA algorithm, the SBR algorithm based on LIBC, and our proposed SBR-EEC algorithm based on HOIBC. Figure 9 shows the backscattering RCS results of the model, where Figure 9a,c show the backscattering RCS for RMA1, and Figure 9b,d show the backscattering RCS for RMA2. The RMSE of these methods compared to MLFMA is shown in

Table 3. RMSE method compared to MLFMA in Figure 9.

	RAM1		RAM2
	SBR + LIBC	Our Method	SBR + LIBC	Our Method
HH polarization	4.51	4.99	11.48	4.85
VV polarization	4.25	3.59	5.44	3.49

.

As shown in Figure 9a,c, when the dielectric parameters are relatively large, that is, the material exhibits a high refractive index, both the LIBC and HOIBC algorithms maintain high accuracy, in line with theoretical predictions. In contrast, as shown in Figure 9b,d, when the dielectric parameters are relatively small, corresponding to a lower refractive index, the LIBC algorithm exhibits larger errors while the HOIBC algorithm retains high accuracy. This shows that the influence of the dielectric parameters on the accuracy of HOIBC is less significant compared to LIBC, underscoring the broader applicability of the HOIBC algorithm across varying dielectric conditions.

The proposed method is used to evaluate the monostatic RCS of the aircraft model depicted in Figure 10, which has a length of 15 m and a width of 9 m, with a total of 20,282 triangular facets. The evaluation is performed at f = 10 GHz for two different RAMs: (a) ${\epsilon }_r=6-3\mathrm{j}$ and ${\mu }_r=1$, denoted as RAM1, with a coating thickness of $d=0.24\lambda$ and a refractive index of $\left|N_1\right|=2.59$; (b) ${\epsilon }_r=1-1\mathrm{j}$ and ${\mu }_r=1$, denoted as RAM2, with a coating thickness of $d=0.24\lambda$ and a refractive index of $\left|N_2\right|=1.19$. The incident angles of the plane EM wave are ${\theta }_i=0~{360}^{\circ }$ and ${\phi }_i=0^{\circ }$ on the xoz-plane. The monostatic RCS for VV and HH polarization is shown in Figure 11a and Figure 11b, respectively.

Analyzing the monostatic RCS results of the aircraft in the xoz-plane shown in Figure 11, both RAMs effectively reduce the RCS for horizontally and vertically polarized incident electric fields, and the RCS of the aircraft is highest when the EM wave is incident from the direction of the underside or the top of the aircraft. Compared to the PEC case, the application of these two RAMs significantly reduces the RCS in the nose and tail region, while the RCS in other angular regions is also reduced to some extent. In particular, RAM2 performs better than RAM1 in reducing the RCS. This suggests that materials with lower refractive indices are more effective in reducing the RCS of the target than those with higher refractive indices for the same thickness and permeability.

For cases when the incident angles of the plane electromagnetic wave are ${\theta }_i={90}^{\circ }$ and ${\phi }_i=0^{\circ }\sim {360}^{\circ }$ on the xoy-plane, the monostatic RCS for VV and HH polarization is shown in Figure 12a and Figure 12b, respectively. Analyzing the monostatic RCS results of the aircraft in Figure 12, the polarization mode of the incident electric field does not influence the RCS reduction performance of the RAMs. Compared to the PEC case, the aircraft coated with RAMs significantly reduces the RCS in the nose and tail regions, while the reductions in other angular regions are relatively moderate. Notably, under the same conditions, RAM2 with lower refractive indices achieve better RCS reduction for the aircraft.

3.3. Influence of RAMs with Different Thicknesses on Target RCS

In the theory of impedance boundary conditions, the equivalent relative surface impedance indicates that the accuracy of the algorithm is influenced not only by the dielectric parameters but also by the thickness of the coated medium. To further validate the effectiveness of the proposed algorithm, a composite model consisting of several cubes created in Section 3.2 is used to investigate the effect of different thicknesses of RAM on its RCS. RAMs characterized by dielectric parameters ${\epsilon }_r=1-2\mathrm{j}$ and ${\mu }_r=1$, with thicknesses $d=0.015\lambda$ and $d=0.075\lambda$, are applied to the model. The incident wave frequency is set to f = 6 GHz, with VV polarization and HH polarization. The model is analyzed using three different methods: the MLFMA algorithm, the SBR algorithm based on LIBC, and our proposed SBR-EEC algorithm based on HOIBC. Figure 13 shows the backscattering RCS results of the model, where Figure 13a,c show the backscattering RCS for $d=0.015\lambda$, and Figure 13b,d show the backscattering RCS for $d=0.075\lambda$. The RMSE of these methods compared to MLFMA is shown in

Table 4. RMSE method compared to MLFMA in Figure 13.

	RAM1		RAM2
	SBR + LIBC	Our Method	SBR + LIBC	Our Method
HH polarization	4.97	4.98	7.68	4.67
VV polarization	4.84	4.73	7.148	4.88

.

The results in Figure 13 and Table 4 show that for the same dielectric parameters, the accuracy of LIBC decreases as the coating thickness increases. When $d=0.015\lambda$, the accuracy of LIBC is comparable to that of HOIBC. This indicates that LIBC is suitable for situations involving thinner coatings, which is consistent with theoretical predictions. In contrast, HOIBC maintains high accuracy regardless of coating thickness, demonstrating that it is not only more accurate but also applicable to a wider range of conditions than LIBC.

In Figure 14, the aircraft illuminated in Figure 10 is used to investigate the effect of different RAM thicknesses on its RCS. RAMs characterized by dielectric parameters ${\epsilon }_r=1-2\mathrm{j}$ and ${\mu }_r=1$, with thicknesses $d=0.015\lambda$, $d=0.1\lambda$, and $d=0.24\lambda$, are applied to the model. The incident angles of the plane EM wave are ${\theta }_i=0~{360}^{\circ }$ and ${\phi }_i=0^{\circ }$ on the xoz-plane, and the frequency of the incident wave is f = 10 GHz. The monostatic RCS for VV and HH polarization is shown in Figure 14a and Figure 14b, respectively. It is observed that for the same dielectric parameters, the RCS reduction performance of RAMs improves as the coating thickness increases. Significant RCS reductions are observed in the nose and tail regions, with moderate reductions in other regions. Furthermore, this trend is observed for both horizontally and vertically polarized incident electric fields.

For cases when the incident angles of the plane electromagnetic wave are ${\theta }_i={90}^{\circ }$ and ${\phi }_i=0^{\circ }\sim {360}^{\circ }$ on the xoy-plane, the monostatic RCS for VV and HH polarization is shown in Figure 15a and Figure 15b, respectively. Analyzing the monostatic RCS results of the aircraft in Figure 15, it is evident that the polarization mode of the incident electric field does not affect the RCS reduction performance of the RAMs. Similarly, for the same dielectric parameters, the RCS reduction performance of the RAMs improves as the coating thickness increases. Significant RCS reductions are observed in the nose and tail regions, with moderate reductions in other regions.

3.4. Influence of Different Incidence Angles on the Coated Target’s RCS

In the example above, the effect of radar-absorbing materials on the monostatic EM scattering of the target was investigated. However, when studying EM scattering characteristics, it is equally important to investigate the bistatic RCS characteristics of the target. Therefore, in this example, we focus on analyzing the differences in bistatic EM scattering characteristics between coated and metallic targets under different incident angles.

To validate the effectiveness of the proposed algorithm, a model consisting of two balls is used to investigate the effect of the bistatic RCS. RAM characterized by dielectric parameters ${\epsilon }_r=2-3\mathrm{j}$ and ${\mu }_r=1$, with thicknesses $d=0.02\lambda$, is applied to the model. The incident wave frequency is set to f = 3 GHz, with VV polarization and HH polarization. The receiving angle is set as ${\theta }_s=0\sim {360}^{\circ }$, ${\phi }_s=0^{\circ }$, and the incident angle is ${\theta }_i=0^{\circ }$, ${\phi }_s=0^{\circ }$. The computational results, as shown in Figure 16, demonstrate that the results of the algorithm proposed in this paper agree well with those obtained using MLFMA, thus verifying the effectiveness of our algorithm in calculating the bistatic RCS for coated targets.

Figure 17, Figure 18, Figure 19 and Figure 20 compare the effect of the RAM on the bistatic RCS of the aircraft under different incident angles and polarization conditions. The dielectric parameters of the RAM are ${\epsilon }_r=1-2\mathrm{j}$ and ${\mu }_r=1$, with a coating thickness set to $d=0.24\lambda$. The frequency of the incident wave is f = 10 GHz, and the polarization modes are HH polarization and VV polarization. The receiving angle is set as ${\theta }_s=0\sim {360}^{\circ }$, ${\phi }_s=0^{\circ }$, and the incident angles are ${\theta }_i={30}^{\circ }$, ${\phi }_s=0^{\circ }$ (Figure 17), ${\theta }_i=-{30}^{\circ }$, ${\phi }_s=0^{\circ }$ (Figure 18), ${\theta }_i=-{120}^{\circ }$, ${\phi }_s=0^{\circ }$ (Figure 19), and ${\theta }_i={120}^{\circ }$, ${\phi }_s=0^{\circ }$(Figure 20), respectively.

From the bistatic RCS results of the aircraft in Figure 17 and Figure 19, it can be seen that when EM waves are incident on the nose region, the application of RAMs significantly reduces the RCS in the nose region compared to the PEC case. However, the reduction in the RCS near the tail region is relatively smaller. Similarly, the bistatic RCS results in Figure 18 and Figure 20 show that when EM waves are incident on the tail region, the radar-absorbing materials reduce the RCS near the tail to a lesser extent, while having a negligible effect on the nose region.

3.5. Influence of Different Incidence Frequencies on the Coated Target’s RCS

In practical applications, it is also necessary to pay attention to the EM scattering characteristics of targets under different frequency conditions. Therefore, in this example, we focus on analyzing the effects of different frequencies on the EM scattering characteristics of targets coated with radar absorbing materials. To verify the generalization of the proposed algorithm, the more complex target, which is shown in Figure 21, is used to evaluate the monostatic RCS at different incident wave frequencies.

The frequency of the incident wave is set to 6 GHz, 8 GHz, and 10 GHz, with polarization modes of HH polarization and VV polarization. The incident angles of the plane EM wave are ${\theta }_i=-{90}^{\circ }~{90}^{\circ }$ and ${\phi }_i=0^{\circ }$ on the xoz-plane, and the RAMs are set as RAM1 and RAM2, the same as those in Section 3.2.

Figure 22, Figure 23 and Figure 24 show the monostatic RCS of the ship model under VV polarization and HH polarization when the incident wave frequencies are 6 GHz, 8 GHz, and 10 GHz, respectively. It can be observed that as the frequency increases, both types of radar-absorbing materials effectively reduce the RCS of the target, with RAM2 demonstrating better RCS reduction performance than RAM1. Additionally, under the same frequency conditions, the RCS reduction effect of RAM on vertically polarized incident waves is superior to that on horizontally polarized incident waves.

4. Conclusions

This study proposes a high-frequency EM scattering computation method for electrically large 3D RAM-coated targets based on HOIBC. The method combines the SBR algorithm with the EEC for impedance edges. Various examples are used to analyze the effects of different dielectric parameters, coating thicknesses, and EM wave incidence angles on the RCS reduction performance of RAMs. Compared to the LIBC method, the proposed approach overcomes the limitations of LIBC in dealing with low-refractive-index materials and thick coatings, thus offering a broader applicability. In addition, our method incorporates the contributions of multiple scattering and edge diffraction, which effectively improves the accuracy of the high-frequency algorithms, making it more suitable for practical engineering applications.