## 2 Spectral Functional Equations in an Angular Region Filled by Arbitrary Linear Media  
Spectral functional equations in angular regions filled by arbitrary linear media are obtained by exploiting the combination and extension of the following powerful mathematical tools developed in different contexts: first, the Bresler-Marcuvitz (BM) transverse equation theory for layered media [33; 35], and second, the characteristic Green function procedure [36; 37]. In this section, following [34], we first briefly revisit the BM theory for layered planar arbitrary linear media as a fundamental step in analyzing layered angular regions. We then apply the characteristic Green function procedure to derive solutions of the obtained system of differential equations. Finally, we provide the spectral functional equations by evaluating the solution at the faces of the angular region. In particular, the functional equations relate continuous (tangential) spectral field components defined at the two faces of the angular region. We start from the application of BM theory to [PERSON]'s equations in layered arbitrary nondispersive homogeneous linear media with tensorial constitutive relations (i.e., bianisotropic media [25; 26; 27]):  
\[ \begin{split}&\mathbf{D}=\underline{\underline{\varepsilon}}\cdot\mathbf{E}+ \underline{\underline{\zeta}}\cdot\mathbf{H}\\ &\mathbf{B}=\underline{\underline{\zeta}}\cdot\mathbf{E}+\underline{ \underline{\mu}}\cdot\mathbf{H}\end{split}  \tag{1}\]  
where the electric and magnetic fields \((\mathbf{E},\mathbf{H})\) are related to the electric and magnetic fluxes \((\mathbf{D},\mathbf{B})\) and the tensors \((\underline{\underline{\varepsilon}},\underline{\underline{\mu}},\underline{ \underline{\zeta}},\underline{\underline{\zeta}})\) are, respectively, the electric permittivity, the magnetic permeability, and the two magneto-electric coupling parameters. By assuming  
1. [label=()]
2. Cartesian coordinates \((z,x,y)\),
3. \(e^{+j\omega t}\) time-harmonic field dependence,
4. Invariant geometry along \(z\) and stratification along \(y\), and
5. Sources constituted of plane waves with \(z\)-dependence \(e^{-j\omega z}\), where \(\alpha_{o}\) depends on the skewness angle with respect to \(z\) (with \(\alpha_{o}=0\) at normal incidence on \(z\)),  
We obtain the transverse differential equations in matrix form for layered planar media:  
\[-\frac{\partial}{\partial y}\mathbf{\psi}_{y}(x,y)=\mathbf{M}_{y}(-j\alpha_{o},\frac{ \partial}{\partial x})\cdot\mathbf{\psi}_{y}(x,y) \tag{2}\]  
where \(\mathbf{\psi}_{y}\) is the four-dimensional column vector (Throughout the article, we assume the notation \(|\ |\) for vectors and not for the modulus of a vector). \[\mathbf{\psi}_{y}=|\mathbf{E}_{t}^{t},\mathbf{H}_{t}^{t}|^{t},\ \ \text{with}\ \mathbf{E}_{t}=|\mathbf{E}_{z},\mathbf{E}_{x}|^{t},\ \mathbf{H}_{t}=|\mathbf{H}_{z},\mathbf{H}_{x}|^{t}. \tag{3}\]  
Based on the nature of [PERSON]'s equations, \(\mathbf{M}_{y}(-j\alpha_{o},\frac{\partial}{\partial x})\) is a second-order four-dimensional matrix differential operator of the form  
\[\mathbf{M}_{y}(-j\alpha_{o},\frac{\partial}{\partial x})=\mathbf{M}_{yo}+\mathbf{M}_{y1} \frac{\partial}{\partial x}+\mathbf{M}_{y2}\frac{\partial^{2}}{\partial x^{2}}, \tag{4}\]  
where the explicit forms of the matrices \(\mathbf{M}_{yo},\mathbf{M}_{y1},\mathbf{M}_{y2}\) for an arbitrarily linear medium (1) are reported in Appendix A with (A2)-(A9).