## 1 Introduction  
Surfaces with two-dimensional structures have attracted a great deal of research interests. This is due to many desirable features such as their low profile, light weight and ease of fabrication, as well as their diversified capabilities of manipulating electromagnetic (EM) waves. During the past decades, different kinds of two-dimensional EM surfaces such as frequency selective surfaces (FSS) [1; 2; 3], electromagnetic band gap (EBG) structures [4; 5; 6] and metasurfaces (MTS) [7; 8; 9] have been studied in depth, leading to diversified applications which realize various kinds of functions. To comprehensively understand EM surfaces, it is necessary to locate surface problems among overall EM studies. In respect to spatial dimensions, EM problems can be categorized according to the electrical size of structures interacting with EM fields. As shown in Figure 1, where dimensions along different directions in Cartesian coordinates are designated as \(L_{x}\), \(L_{y}\) and \(L_{z}\) respectively, all EM problems are classified into four types:  
* 3D EM problems. For structures with \(L_{x}\), \(L_{y}\) and \(L_{z}\) all comparable to wavelength \(\lambda\), they are usually discussed with three-dimensional electromagnetics, which is analyzed by general EM theory. For 3D problems, permittivity \(\varepsilon\) and permeability \(\mu\) are utilized to characterize properties of homogeneous mediums while their effective counterparts are defined for inhomogeneous ones. The corresponding mathematical tools are derived from [PERSON]'s equations. * 0D EM problems. In contrast, if all dimensions are much smaller than \(\lambda\), problems can be addressed with the zero dimension approach through description by lumped parameters. As a result, these problems can be analyzed with circuit theory. For 0D problems, structures can be represented by lumped circuit parameters, such as resistance \(R\), inductance \(L\) and capacitance \(C\). Field relations are consistent with [PERSON]'s laws. * 1D EM problems. If transverse dimensions \(L_{x}\) and \(L_{y}\) are much smaller than \(\lambda\), these one-dimensional problems can be solved with transmission line theory. For 1D problems, the characteristic impedance \(Z_{0}\) and propagation constant \(\beta\) are the principal parameters which comply with transmission line equations [10]. * 2D EM problems. When only the longitudinal dimension \(L_{z}\) becomes much smaller than \(\lambda\), characterization of these two-dimensional surfaces can be denominated as theory of surface electromagnetics (SEM). For 2D problems, the most appropriate characterization parameters are effective surface susceptibilities \(\overline{\overline{x}}_{ee}\) and \(\overline{\overline{x}}_{mm}\)[11], while related mathematical models are named as generalized boundary conditions (GBCs). This begs the question of how GBCs differ from conventional boundary conditions. In surface electromagnetics, field discontinuities across engineered surfaces are deliberately designed in order to manipulate or transform EM waves [12; 13; 14], and boundary conditions are used to describe these field discontinuities. In conventional boundary conditions, field discontinuities are attributed to current sources distributed along surfaces, as shown in Figure 2. Meanwhile, the current sources are dependent not only on the surface characteristics but also on the applied fields. Thus, the EM fields and the current sources are coupled together. As a result, it is not straightforward and sometimes even complicated to solve field discontinuities with conventional boundary conditions. To overcome this difficulty, an alternative approach to determine the field discontinuities directly from the surface characteristics exists: generalized boundary conditions (GBCs). It is worthwhile to point out that the surface characteristics here are eigen-parameters of surfaces dependent on surface geometry and material, but independent of applied fields.