1. Introduction

In recent decades of radar absorbing material (RAM) research, the technology has evolved from basic absorption capabilities to advanced levels, transitioning from isotropic materials to anisotropic variants, and innovating from single-layer to multi-layer coatings [1,2]. Notably, the application of multi-layered anisotropic materials in modern stealth technologies is expanding. Consequently, a comprehensive exploration of the electromagnetic (EM) scattering characteristics of such materials has become a focal research area in electronic warfare, garnering significant attention from the scholarly community [3–19].

Teitler and Henvis initiated the analysis of reflection properties in layered anisotropic medium [3]. Morgan et al. developed a specialized numerical algorithm for scattering and transmission in layered heterogeneous anisotropic medium [4]. Chew and his team delved into the propagation mechanisms of EM waves within anisotropic medium [5–9]. Qiu introduced an innovative approach using vector wave functions (VWFs) to address the EM scattering and radiation problems of multi-layered anisotropic sphere [10]. Bao employed Effective Medium Theory (EMT) for detailed numerical simulations of layered biaxial anisotropic materials [11]. Liu and colleagues presented a numerical methodology for solving EM scattering problems in multi-layered generalized anisotropic medium [12–14]. Dreher proposed a concise analytical strategy based on the discrete modal matching method (DMMM), applicable to circular and non-circular waveguides and strip lines in anisotropic medium [15]. Balasubramanian extended the homogeneous theory to multi-layered anisotropic thin plate, enhancing its predictive capabilities for EM scattering [16,17]. Zheng analyzed the reflection and transmission properties of multilayered fully anisotropic medium (MFAM) using the transfer-matrix method (TMM) [9]. Lastly, Hong investigated the non-leaky radiation model of dipole sources within multi-layered anisotropic cylindrical medium [18].

Overall, the aforementioned research methodologies can be broadly categorized into analytical approach and numerical simulation, primarily focusing on targets with simpler structures. However, for complex radar targets such as satellites and aircraft, especially those exhibiting electrically large dimensions in the microwave frequency region, traditional techniques face limitations in handling the scale of unknowns. Particularly when transitioning from single-layer to multi-layer coatings on target surface, these methods may no longer meet the practical engineering demands under constrained computational resources. In contrast, physical optics (PO) based on the principles of high-frequency field locality offers a viable solution. By computing the induced EM current model on the coated target surface and subsequently summing the radiated fields vectorially, the total scattered field from the coated target can be derived. This approach ensures computational accuracy while significantly reducing the required computational resources and time, finding widespread application in predicting the scattering characteristics of electrically large targets [19–25]. Therefore, adopting high-frequency methodologies for EM scattering modeling of multi-layered anisotropic mediums (MLAMs)-coated radar targets, while meeting engineering precision requirements, appears to be a technically sound and feasible choice.

In the context of contemporary radar stealth technology, [20], [24], and [25] respectively examined the propagation behaviors of incident wave on metal target coated with single-layer anisotropic medium/plasma. Through the integration of PO method, the far-field scattering characteristics of such coated targets can be rapidly and accurately computed. Nevertheless, from an engineering standpoint, single-layer coated structures tend to have narrower absorption bands, posing design constraints and falling short of the lightweight and broadband specifications sought in stealth technology. In contrast, multi-layered configurations not only offer greater design flexibility but also provide robust technical foundations for the development of broadband absorption structures.

In light of these considerations, the paper focuses on developing an efficient modeling approach for the EM scattering of electrically large, complex targets coated with MLAMs. Initially, a high-precision three-dimensional geometric model of the complex target is constructed. The model is discretized using moderately sized triangular facets. While the number of facets should not be excessively large, it is crucial to ensure that the model fits accurately, allowing for the precise representation of the geometric features of each component of the target. The radar observation range is selected, and ray tracing and depth culling of the model are performed to obtain the surface regions for computation. The regions where the target's surface is coated with medium material are identified. For the ideal conductive regions without medium coating, surface reflection coefficients can be directly obtained. For regions coated with single/multiple layers of anisotropic medium materials, the propagation characteristics of incident waves within MLAMs are comprehensively analyzed. Utilizing the spectral domain method (SDM) in conjunction with the saddle point evaluation (SPE) based on asymptotic algorithm, the spatial domain solution of the EM field on the surfaces of the MLAMs, along with its associated reflection and transmission coefficients, are successfully derived. Subsequently, leveraging these solutions, the induced EM current model on the target's outermost surface is constructed. Through radiation integration technique, the far-field scattering data of the target is acquired. Validated with examples of planar structure and the Misty satellite target, the methodology demonstrates accuracy comparable to full-wave solution while offering notable computational efficiency.

Furthermore, to achieve the objective of precise control and intelligent modulation of the target's radar cross-section (RCS) by minimizing coating costs to obtain maximum RCS reduction. The paper integrates the proposed EM algorithm with scattering sources decomposition technique based on spatial ray tracing and diversity [26–28]. Using the Su-57 aircraft as a case, the primary scattering regions of the target within radar threat ranges are identified, informing the design of absorptive material coatings. Simulation outcomes indicate that targeted coatings on regions of pronounced scattering not only significantly diminish the overall EM scattering of the target but also highlight the superior absorption properties of MLAMs over their single-layer counterparts. In summary, this research provides an efficient and innovative technical solution for predicting the scattering characteristics, stealth design, and performance evaluation of radar targets.

2. EM scattering modeling of targets coated with MLAMs

2.1 Algorithm scheme

Figure 1 illustrates the solution for simulating the high-frequency EM scattering characteristics of electrically large, complex targets coated with MLAMs.

 

Fig. 1. The solution for simulating the high-frequency EM scattering characteristics of electrically large, complex targets coated with MLAMs.

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The specific steps are as follows:

Step 1: Model Preprocessing. Firstly, obtain high-precision three-dimensional geometric model of complex target using geometric modeling software. Divide the model into moderate-sized triangular elements to accurately describe the geometric structural features of various components of the target. Then, based on the set radar observation range, perform ray path tracing and model depth blanking to obtain the surface region to be computed.

Step 2: Construction of Target Surface Equivalent EM Current Model. Select the surface region of the target for medium coating, and for ideal conductive regions without medium coating, directly obtain surface element reflection coefficients and surface equivalent currents. For single/multiple-layer of anisotropic medium coating scenarios, employ spectral domain full-wave analysis method (SDFWAM) for systematic analysis and calculation, including: S1, derivation of plane-wave spectrum form of EM field in each layer of anisotropic medium; S2, introduction of spectral domain boundary conditions, establishment of spectral domain field coefficient equations, and solving the equations to obtain spectral domain solutions of the scattering field; S3, calculation of spectral domain integration of the scattering field using SPE to obtain the asymptotic solution of the scattering field in spatial domain, thereby deriving analytical expressions of reflection and transmission coefficients of the medium plate; S4, calculation of EM field on the outer surface of the coating medium (total field), thereby obtaining the equivalent EM current on the coating plane.

Step 3: High-Frequency Method for Scattering Field Calculation. Utilize the high-frequency local field principle and tangent plane approximation to transform the coating plane's equivalent EM current into a target surface element induced EM current model in the ray-based coordinate system, further utilize induced EM current radiation integration to obtain the target scattering far field, thus completing the construction of the PO calculation method for EM scattering of multi-layer anisotropic medium-coated targets.

Step 4: Intelligent Control of Radar Target RCS. Combine the proposed algorithm with ray tracing and ray clustering-based scattering source decomposition technology to obtain the distribution of strong scattering sources of high-frequency targets. This serves as a guideline for selecting target coating regions, achieving optimal RCS of targets at minimal material cost, and realizing precise control and intelligent adjustment of RCS of radar targets.

The proposed method effectively reveals the variation laws of the scattering field of multi-layer medium-coated targets with major physical parameters (including coating thickness, material EM parameters, incident angle, etc.), filling the gap in the field of EM scattering modeling of multi-layer anisotropic medium-coated targets. Compared with numerical methods, it has high computational efficiency and provides an effective solution for rapid estimation of the radar characteristics of real targets, target detection and recognition, and radar target stealth design and performance evaluation.

The physical model of the layered anisotropic medium studied in this paper is depicted in Fig. 2.

 

Fig. 2. Schematic of the layered anisotropic medium physical model. (a) Layered anisotropic medium model. (b) Incident and reflection directions

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The coordinate system $Oxyz$ has its $xoy$ plane defined as the surface of the first layer of anisotropic medium, marking the interface between air and the first anisotropic medium layer. Here, $\hat{i}$ represents the unit vector in the direction of the incident plane wave, $\hat{r}$ denotes the unit vector in the direction of specular reflection, $\theta$ signifies the angle between $\hat{i}$ and the plane's normal, which aligns with the $\hat{z}$-axis, and $\varphi$ is the angle between the projection of $\hat{i}$ onto the plane and the $\hat{x}$-axis.

Assume that between the (n + 1)-th substrate layer and the 0-th air layer, there are n layers of anisotropic medium coatings. The relative permittivity tensor and relative permeability tensor for the i-th layer of medium are designated as ${\bar{\bar{\varepsilon }}^{(i )}}$ and ${\bar{\bar{\mu }}^{(i )}}$ respectively. The thickness of the coating is denoted as ${d_i}$. The coating medium is assumed to be uniaxial electric anisotropic, with the relative permittivity tensor and permeability tensor of the i-th layer of medium set as:

It should be emphasized that the proposed high-frequency EM scattering modeling method for electrically large complex targets coated with MLAMs is fully applicable to the accurate and rapid modeling of EM scattering from magnetic materials. Due to space limitations and to avoid overly verbose mathematical formulations, this paper opts to analyze dielectric anisotropic materials, with the understanding that similar computations can be applied to magnetic materials.

Recognizing the Fourier transform relationship between spatial and spectral domain fields, for uniaxial anisotropic medium, the eigenvalues of the plane wave spectra within the medium for types I and II are designated as ${k_x}$ and ${k_y}$ respectively. For wave vectors propagating in the positive $\hat{z}$-direction, four distinct solutions, ${\pm} {k_{Iz}}$ and ${\pm} {k_{IIz}}$, exist with values ${k_{Iz}} = \sqrt {k_0^2{\varepsilon _ \bot } - k_x^2 - k_y^2}$ and ${k_{IIz}} = \sqrt {k_0^2{\varepsilon _{/{/}}} - k_x^2\frac{{{\varepsilon _{/{/}}}}}{{{\varepsilon _ \bot }}} - k_y^2}$, where ${k_0}$ represents the free-space wave number. By extracting the plane wave spectrum vector ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} _{I,II}} ={-} {k_x}\hat{x} - {k_y}\hat{y} - {k_{I,IIz}}\hat{z}$ corresponding to ${k_{Iz}}$ and ${k_{IIz}}$, the form of the plane wave spectrum for Type I and II waves propagating along the $\hat{z}$-axis can be derived as:

According to the plane wave spectrum expansion method [5], any EM field within an anisotropic medium can be expressed as a superposition of plane waves with continuous eigenvalues ${k_x}$ and ${k_y}$, given by:

In the layered model, the EM field within the i-th layer can be represented as:

The i-th anisotropic layer has two EM boundaries along the axial direction: one with the (i-1)-th medium layer and another with the (i + 1)-th medium layer. Thus, for the entire layered model, it is understood to contain “n + 1” EM boundaries.

In Eq. (10), d represents the thickness of the medium layer. For the n EM boundaries from $z = {z_1}$ to $z = {z_n}$, their boundary conditions are given by:

From the above equation, we can derive 4n equations. Adding the 2 equations determined by the (n + 1)-th boundary condition, we can form a total of 4n + 2 equations, matching the number of unknown coefficients.

Through the derivation from the above equation, we obtain:

Additionally, it should be noted that when i = 1, $\varepsilon _{ \bot ,/{/}}^{({i - 1} )}$ and $k_{Iz,IIz}^{({i - 1} )}$ correspond to the permittivity and wave vector components in air, respectively, i.e., $\varepsilon _{ \bot ,/{/}}^{(0 )} = 1$ and $k_{Iz,IIz}^{(0 )} = k_z^{(0 )} = \sqrt {k_0^2 - k_x^2 - k_y^2}$.

In summary,

Hence, the relationship for the EM field on the surface of the n-th layer can be expressed as:

At this point, the EM field relationship on the outermost surface of the medium layer can be expressed as:

Given the coefficients $\zeta _\textrm{ + }^{(0 )}$ and $\xi _\textrm{ + }^{(0 )}$, the scattering coefficients $\zeta _ - ^{(0 )}$ and $\xi _ - ^{(0 )}$ can be derived from the aforementioned equation. Thus, the scattering field in free space can be represented as:

Solving the spectral domain integral of the scattering field yields the corresponding spatial scattering field, as follows:

In this paper, SPE is employed to approximate the aforementioned complex integral. The angle between the incident wave direction and the plane normal is denoted as ${\theta _i}$, while the angle between the projection of the incident wave direction onto the plane and the local coordinate system's $\hat{x}$-axis direction is represented as ${\varphi _i}$. Similarly, the angles for the scattering wave direction and its projection on the plane with respect to the local coordinate system are denoted as ${\theta _r}$ and ${\varphi _r}$, respectively. By taking the derivative of the phase of the integrand and eliminating spurious saddle points, the saddle point for the scattering field is determined to be:

At this juncture, the eigenvalues of the plane wave spectrum corresponding to the saddle point are determined to be:

Substituting the above expressions into the spectral domain solution, the asymptotic representations of the scattered field in the layered model in the spatial domain are:

In the Cartesian coordinate system, it can be defined as:

It can be deduced that:

When introducing the saddle point, the reflectance matrix $\tilde{\bar{\bar{R}}}$ defined in the spectral domain can be reduced to its physical counterpart in the spatial domain $\bar{\bar{R}}$ [20], namely:

Thus, the reflection field on the outer surface of the layered anisotropic medium model can be represented by the reflection coefficient matrix $\bar{\bar{R}}$.

The PO method relies on three pivotal assumptions: Firstly, induced currents, excited by incident wave, exist solely within the illuminated region of the target. In this research, graphical hardware acceleration technique was employed to extract real-time occlusion results from the target. Leveraging GPU rendering capabilities and optimized depth buffer techniques, the PO-illuminated regions were rapidly discerned [29]. Secondly, the wavelength of the incident wave is significantly smaller than the geometric dimensions of the target. Thirdly, the tangent plane approximation necessitates that the curvature radius of the target is much larger than the wavelength. Based on these premises, the surface field on the illuminated surface is approximated as the field of an infinite PEC plate coated with a uniform dielectric layer, subsequently defining the target's outermost surface's equivalent EM current model.

The scattering from a thin dielectric-coated conductive target can be equivalently represented as a radiation problem from equivalent surface electric current source ${\vec{J}_S}$ and equivalent surface magnetic current source ${\vec{J}_M}$. It is worth noting that in our research, the thickness of the MLAMs relative to the electrically large target still falls within the category of thin-layer coatings, within the applicable range of the equivalent principle. Therefore, there is no need to be concerned about the cumulative effect of the medium thickness on the derivation process of the algorithms mentioned above. Figure 3 illustrates the model for an ideal conductive target coated with anisotropic medium. Here, $\hat{u}$, $\hat{v}$, and $\hat{n}$ denote the direction vectors of the dielectric's optical axes. The target is positioned within the $Oxyz$ coordinate system, and the external surface of the target of any arbitrary shape is discretized using planar triangular elements. The vector corresponding to the center of the triangular element ABC in Fig. 3 is denoted as ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c}$.

 

Fig. 3. Schematic of an ideal conductive target coated with anisotropic dielectric.

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At each coated triangular element ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _c}$, the equivalent EM current is:

In Eq. (30), s represents the illuminated portion of the scattering object, encompassing both the direct illumination from the incident wave and the additional reflections on the target surface. After obtaining the equivalent EM current, the Stratton-Chu far-field scattering integral yields the scattering field for this triangular element:

In the equation, $\hat{s}$ denotes the unit vector in the direction of the scattered wave, ${\vec{r}^{\prime}}$ represents the position vector on the scattering object's surface, and r is the radar distance. The scattering contributions from each facet are vectorially summed to obtain the overall scattering field of the electrically large, complex target coated with MLAMs.

To validate the accuracy and efficiency of the aforementioned PO method, the paper adopts the method of moments – finite element method (MoM-FEM) hybrid full-wave numerical technique from the FEKO software as a benchmark. The root mean square error (RMSE) serves as the evaluative metric to quantify the discrepancies between the proposed approach and the reference data. The RMSE is defined as follows:

In Eq. (32), θ represents the monostatic angle, while $RC{S_{proposed}}$ and $RC{S_{reference}}$ denote the calculated results from the proposed method and the full-wave numerical solution, respectively.

The computational platform is equipped with an Intel Core i7 – 12700 chipset, operating at a clock speed of 2.10 GHz, with a total memory of 64 GB.

2.2 Validation example – square plate

As a prototypical open-structure challenge, the EM scattering evaluation of anisotropic dielectric-coated planar structures offers foundational theoretical support for reducing the RCS of complex targets. The monostatic scattering characteristics of composite targets, comprised of a PEC square plate with electric dimensions of 5λ and coated with single-, double-, or triple-layered anisotropic medium, were computed. The coating parameters are delineated in Table 1. Radar scan parameters are configured as θ = 0° – 90°, with an incident angle φ = 0° and VV polarization. The results of the monostatic scattering simulations are illustrated in Fig. 4. Comparative data between the two methodologies in terms of CPU computation time and memory requirements are provided in Table 2.

 

Fig. 4. Monostatic RCS of square plate under different medium-coating conditions. (a) Single-layer, RMSE = 2.02 dB. (b) Double-layer, RMSE = 1.37 dB. (c) Triple-layer, RMSE = 2.11 dB. (d) Comprehensive Comparison.

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Table 1. Coating parameters

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Table 2. Comparison of CPU calculation time and memory requirements

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The simulation outcomes demonstrate a high level of consistency between the proposed algorithm and the full-wave solution. Particularly under wide-angle scanning conditions, the two-exhibit notable congruence, with a maximum root mean square error of 2.11 dB falling within acceptable error margins. Further computational resource comparisons underscore that the algorithm introduced in this research, while maintaining precision, significantly diminishes both computational time and memory overhead compared to conventional numerical approaches, highlighting its considerable potential for real-world engineering applications. Figure 4(d) delineates the scattering distinctions of the target across three coating scenarios. The research discerns that the absorptive medium layer induces a marked attenuation effect on the target's scattering behavior. Variations in medium parameters and the number of coating layers manifest distinct impacts on EM scattering, with the attenuation effects being particularly pronounced in the triple-layer medium. Such insights furnish profound theoretical guidance and pivotal perspectives for the stealth design and performance evaluation of radar targets.

2.3 Validation example – Misty satellite

In the realm of space defense systems, the significance of satellite stealth technology is self-evident. While it shares similarities with stealth technologies employed on ground armored vehicles, ships, and aircraft, the research and application of this technology for satellites present heightened challenges, given the unique manufacturing conditions and the space environment they operate in. The Misty satellite, publicly reported, is an example of an orbiting stealth satellite. Its structure primarily comprises two components: Component 1, termed the “stealth screen”, possesses an inverted cone shape, while Component 2, positioned external to the “stealth screen”, assumes a cylindrical shape. The satellite's stealth capability is largely attributed to its inflatable shield design [30]. When EM waves emitted by ground radars encounter the conical inflatable shield, they are either absorbed by its exterior or refracted in alternative directions, substantially reducing the return echo energy.

This section focuses on such complex targets with significant electrically large dimensions, like the Misty satellite, simulating the scattering from its MLAMs-coated surface and contrasting it with the full-wave solution. Figure 5 illustrates a geometric schematic of this model. Radar scan parameters are set as θ = 90° – 180° with an incident angle φ=0° and VV polarization. Initially, the scattering result for the target under ideal conducting condition is presented, as depicted in Fig. 6. The simulation aligns closely with the full-wave solution, manifesting a root mean square error of 1.35 dB, further affirming the reliability of the proposed methodology under ideal conductive scenario.

 

Fig. 5. Schematic of the Misty target model. (a) Target electrical dimension annotation. (b) Geometric schematic of the Misty model

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Fig. 6. Monostatic RCS of the Misty under ideal conducting condition, RMSE = 1.35 dB.

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The parameters for the MLAMs coating on the target surface are listed in Table 3. Figure 7 and Fig. 8 present the simulation results for the double-layer and triple-layer anisotropic medium-coated target surfaces, respectively, contrasted with the full-wave solutions.

 

Fig. 7. Monostatic RCS of the Misty under double-layer anisotropic medium coating, RMSE = 1.39 dB.

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Fig. 8. Monostatic RCS of the Misty under triple-layer anisotropic medium coating, RMSE = 1.97 dB.

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Table 3. Coating parameters

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The comparison between the proposed algorithm and the MoM-FEM method in terms of CPU computation time and memory requirements is detailed in Table 4.

Table 4. Comparison of CPU calculation time and memory requirements

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Comparative analysis reveals that for complex, electrically large targets such as the Misty satellite, the method proposed in this research aligns closely with the full-wave solution across a broad angular range. Within the 165° scanning span, potential errors might arise from faint scattering mechanisms like creeping waves on the target's surface. The overall root mean square error remains modest at 2 dB, falling within an acceptable range. Notably, while ensuring precision, the computational time and memory consumption of the algorithm proposed in this paper are notably superior to conventional numerical methods. As discerned from the figures, the chosen medium parameters exert a marked attenuating effect on the target's EM scattering. Moreover, with an increase in the number of coating layers, this attenuation intensifies. Particularly, under the context of a three-layer medium coating, the target's scattering diminishes to below -10 dB within the 135° to 180° range.

3. Optimal coating region selection based on scattering source decomposition technique

When radar detects a target, its EM characteristics are embedded within the radar echoes of that target [31]. In the high-frequency domain, the overall scattering of a target can be seen as the coherent sum of multiple localized scattering sources [32,27]. The complete EM scattering characteristics of radar targets are primarily represented by the superposition of strong EM scattering sources distributed on the target, with each scattering source having a precise “one-to-one” correspondence with the scattering structures on the target. Traditional measurement or simulation outcomes often only provide the overall EM scattering echoes of the target, thereby limiting the comprehensive acquisition of detailed radar target characteristic information. In practical applications such as radar stealth design, a meticulous analysis of the specific sources contributing to the dominant scattering of the target is paramount. This aids in discerning the particular components or structures within the echo that contribute to strong scattering, facilitating precise coating with specific materials. Addressing this challenge, our research integrates scattering source decomposition techniques based on ray tracing and zone identification with quantitative methodologies from high-frequency EM theory. This approach decomposes the high-frequency target's overall echo scattering into contributions from multiple scattering sources, successfully isolating and quantitatively characterizing the strong scattering sources of the target. It accurately identifies the primary scattering origins and regions, establishing a clear correspondence between scattering sources and the physical components of the target.

Beginning with the CAD geometric model of the target, this method effectively segments the EM current on the complex target surface into detailed zones, guided by the subdivision of its physical components and the numbering of surface regions. Moreover, by breaking down the overall echo scattering of the target in the high-frequency spectrum into contributions from individual scattering sources, it unveils the internal architecture of the target's dominant scattering sources. This provides a distinct identification of primary scattering origins and regions, further solidifying the correspondence with the target's physical components. In specific experiments, the scattering contributions from all surface elements of each solid component are aggregated to the scattering contribution of the local scattering source, wherein all rays incident on the component are combined to obtain the scattering contribution of this scattering source. The theoretical foundation of this approach stems from the locality principle of radar targets in the high-frequency region. This methodology not only furnishes a robust tool for delving into the influence of local target structures on overall echo characteristics and the intrinsic mechanisms of RCS properties but also offers a comprehensive data model and pivotal foundational knowledge for signal detection design, data processing, and stealth technology applications. A schematic of the scattering source decomposition process for high-frequency radar target is illustrated in Fig. 9.

 

Fig. 9. Flowchart the decomposition process of scattering sources for high-frequency radar target.

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Given incident and observation angles, spatial ray tracing is performed on the target surface to determine the paths of all rays. These rays are then categorized and grouped based on their respective paths. The propagation of these rays in space is described using transmission matrices, and the energy carried by each ray is determined using the PO algorithm. The scattering contribution of each ray subset is obtained by superposing energies from rays of the same category.

Based on high-frequency approximations, the EM scattering from electrically large target can be approximated as the sum of responses from multiple independent scattering sources.

Here, ${r_k}$ represents the k-th ray that has been traced.

Upon a meticulous ranking of the ray subsets based on their scattering contributions, we identified the dominant scattering sources at the given observation angle. This curated ensemble of independent scattering sources provides a comprehensive depiction of the target's overall scattering characteristics. Notably, a distinct correspondence is established between these selected scattering sources and the target's local geometric structures, laying a robust foundation for the subsequent quantitative analysis and prioritization of these sources.

Harnessing this ray-decomposition technique, we gain profound insights into the distribution and characteristics of strong scattering sources and their associated structures within high-frequency targets. Such a nuanced understanding offers pivotal guidance for the directional coating of target surface materials, aiming to achieve a maximized reduction in RCS in the most economical and efficient manner.

4. Example analysis

Figure 10 depicts the geometric model of the Su-57 aircraft, with 3-D dimensions of 20.76 m × 14.12 m × 3.3 m. Based on the theoretical mechanisms of high-frequency EM scattering, the decomposition results of the aircraft components are illustrated in Fig. 11.

 

Fig. 10. Schematic of aircraft target model.

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Fig. 11. Decomposed schematic of the aircraft target model.

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In the conditions of zenith angle θ = 100°, incident wave frequency f = 10 GHz, and azimuthal scan range φ = −60 to 60°, with VV polarization. Figure 12 presents a comparison between the scattering from the PEC target as computed by the proposed algorithm and that by the RL-GO algorithm in FEKO, with a root mean square error of 1.40 dB. The extracted results for dominant scattering sources are shown in Fig. 13.

 

Fig. 12. Monostatic RCS of the Su-57 under ideal conducting condition, RMSE = 1.35 dB.

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Fig. 13. Extracted dominant scattering sources from the target. (a) Results of dominant scattering source screening. (b)-(g): Comparative analysis of RCS between dominant scattering structures and the target's global RCS.

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In the extracted dominant scattering sources, component number 9 corresponds to the target's right main wing, component 10 to the left main wing, 13 to the left lateral wing, 21 to the right lateral wing, while numbers 26 and 27 represent the enclosed intakes on the left and right sides of the target, respectively. Within an azimuthal range of ±30°, which encompasses the nose cone scanning region, the primary scattering contributions from the target originate from the lower lateral intakes, exhibiting pronounced peak scattering around ±28°. As the azimuthal angle varies, the scattering contribution from the target's lateral wings peaks around ±47.5°, while the main wings provide the most significant scattering contribution within the observation range at approximately ±49.5°.

To achieve radar stealth, techniques such as shaping and radar absorbing paint (RAP) applications are commonly employed. Shaping techniques modify the aircraft's external structure to reduce its RCS within specific incidence angles, thereby diminishing its detectability in radar threat zones. Within the aforementioned observation range, the overall scattering contribution of the aircraft remains below 0 dB, underscoring its superior shaping for stealth. However, to further enhance stealth capabilities, one can employ insights from the target's scattering structures under ideal conductive conditions to apply radar-absorbent materials selectively.

Specifically, we chose to apply single and double-layer radar-absorbent coatings to surfaces represented by strong scattering structures such as components 9, 10, 13, 21, 26, and 27. Figure 14 illustrates a schematic of the localized application of radar-absorbent materials on these dominant scattering structures. Figure 15 presents the monostatic scattering results of the target under ideal conductive conditions and when coated with either single or double-layered anisotropic medium within a broad scanning angle. Details of the coating medium parameters are provided in Table 5.

 

Fig. 14. Schematic of the medium-coated surface on the strong scattering structures. The highlighted region in red indicates the area with medium coating

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Fig. 15. Monostatic RCS of the Su-57 under various coating conditions, 1-degree smoothing.

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Table 5. Parameters of anisotropic medium coatings on the Target's dominant scattering structures

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Through comparative analysis, it becomes evident that accurately determining the strong scattering structure distribution of a target within specific observation ranges using the scattering source analysis method and subsequently applying targeted material coating can significantly reduce the target's RCS across the entire observation domain. It is noteworthy that MLAMs coatings demonstrate outstanding attenuation in the target's EM scattering, surpassing the efficacy of single-layer coating. Particularly at azimuth angles of positive and negative 25°, the dual-layer dielectric coating exhibits pronounced scattering attenuation effects.

In summary, leveraging the scattering source analysis method to obtain insights into the target's strong scattering structure distribution, combined with the efficient RCS estimation algorithm proposed in this research, and further employing targeted absorptive material coatings on these strong scattering structures, not only substantially reduces the costs associated with stealth design but also markedly enhances the stealth capabilities of the target. This approach not only furnishes robust technical support for stealth design but also offers invaluable foundational knowledge for research and applications in related domains.

5. Conclusion

This paper introduces a high-frequency EM modeling approach applicable to electrically large complex targets coated with MLAMs, aiming to efficiently estimate the RCS of coated targets. Through comprehensive numerical comparisons, the paper validates the superior accuracy and performance of the proposed algorithm. Furthermore, by integrating this modeling approach with scattering source decomposition technique, the paper offers scientific guidance for the precise directional coating of medium materials on target surfaces. In-depth investigation reveals that precise coating of strong scattering structures on targets can effectively reduce the overall EM scattering effects of the targets, with multi-layer medium significantly outperforming single-layer medium in suppressing target RCS. This research provides cutting-edge technological methods and solid theoretical support for the rapid estimation of EM scattering characteristics, stealth design, and performance evaluation of radar targets.

However, avenues for further exploration remain. At high frequencies, the scattering behavior of targets is influenced by various scattering mechanisms. This paper primarily focuses on the scattering contributions of target model elements. Future research should expand to consider various complex mechanisms, such as diffraction effects on edges and creeping wave scattering on curved surfaces. Moreover, further research is urgently needed on high-frequency EM scattering modeling methods for inhomogeneous medium materials.