## 1 Introduction  
Accurate prediction and reduction of the radar cross section (RCS) is essential for the analysis of electromagnetic wave scattering from objects and radar absorbing materials (RAM) [1; 2; 3; 4; 5; 6]. In recent years, there has been a lot of interest in multi-layer radar absorbers as a means of predicting and reducing the RCS of objects [7; 8]. Meanwhile, simple geometric elements like plates, shells, spheres, and edges can be used to model complex objects like aircrafts or vehicles. In the past, scattering and diffraction properties of these simple geometric elements have been analyzed to understand how to predict or reduce the RCS of complex objects. One important geometry in this context is the open-ended metallic waveguide cavity [1; 2; 3; 4; 5; 6; 9], which is commonly encountered in various engineering applications, such as microwave filters, antennas, and transmission lines. In addition, it can be used as a base model for duct structures, such as jet engine intakes, and for the analysis of surface cracks on more complex objects. A variety of high-frequency and numerical methods, as well as hybrid ray-numerical methods [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22], have been used to study the scattering from cavities of various shapes, but these methods may not provide uniformly valid solutions for any cavity size. The Wiener-Hopf technique is known to be a powerful and rigorous method for the analysis of wave scattering and diffraction problems involving canonical geometries [23; 24; 25; 26; 27]. It has been successfully applied to a wide range of problems, including those involving semi-infinite media, planar interfaces, and layered media [26; 27]. The technique is based on the use of the Wiener-Hopf equation, which arises from the requirement that the fields in two half-spaces satisfy certain boundary conditions. The solution of the Wiener-Hopf equation consists of decomposing the associated symbolic function into its positive and negative frequency components, which allows the problem to be reduced to a set of algebraic equations that can be solved to obtain the unknown field components. The advantage of the Wiener-Hopf technique is that it can handle problems involving arbitrary angles of incidence and polarization states, and it can also be used to study problems with multiple scattering or diffraction orders. In addition, the Wiener-Hopf technique has been applied to solve problems in various fields of physics and engineering, such as electromagnetism, acoustics, and seismology. It allows the determination of unknown fields and scattering coefficients by using complex variable analysis and contour integration. Although the method can be computationally intensive, it provides exact solutions for a wide range of problems and is highly valuable in the analysis of wave phenomena. Especially for the analysis of plane wave scattering by half-planes, parallel plate waveguides, and waveguide cavities, the Wiener-Hopf technique has been widely used to obtain strong solutions [28; 29; 30; 31; 32; 33; 34; 35; 36; 37]. In the previous papers, the authors have performed a rigorous RCS analysis of various two-dimensional cavities formed by finite parallel-plate waveguides [38; 39; 40; 41; 42] and semi-infinite parallel-plate waveguides [43; 44; 45] using the Wiener-Hopf technique. Our final solutions have been proven to be effective over a wide range of frequencies and can be used to validate other commonly used numerical methods and high-frequency ray techniques. In our very recent paper [46], we have considered a semi-infinite parallel-plate waveguide with five-layer material loading that can form cavities and rigorously analyze the E-polarized plane wave diffraction using the Wiener-Hopf technique. It has been verified that our final solution is uniformly valid for arbitrary waveguide dimensions and any homogeneous material layers.