# Analytically Regularized Evaluation of the Coupling of Planar Concentric Conducting Rings
## 1 Introduction  
The canonical annular ring shape has attracted much attention during recent decades due to its relative simplicity, which has also not prevented it from having numerous practical applications. Microstrip ring antennas have been proposed since the 1980s [1], due to their compactness and greater versatility with respect to patch disc antennas, and have been the subject of intensive work [2; 3; 4; 5; 6]. A particularly attractive property of this kind of geometry is the possibility to consider concentric configurations, especially suited for multi-band operation [7; 8; 9; 10; 11; 12], even in reflect array structures [13]. Another relevant configuration where the annular ring geometry is applied is represented by grating. As a matter of fact, metal grating on a dielectric surface has many important applications, such as frequency selective surfaces [14; 15; 16], leaky wave antennas [17; 18; 19], and optical devices [20; 21]. In the above mentioned contexts it is of paramount importance that the characterization of the coupling between annular rings be accurate, since they can be very close to each other and their coupling cannot be neglected or misestimated in order to correctly predict the behavior of the device at hand. The aim of this paper is to show a full wave, accurate, and effective method to analyze annular ring coupled structures based on Method of Analytical Regularization (MAR). The focus is on the method itself, so only concentric annular rings in free space will be considered: different configurations can be analyzed by means of the same functions presented here, and medium stratification can be taken into account by modifying the Green's function, without any other change in the overall method. Maxwell equations can be solved by using many different methods in the time or frequency domain; finite difference and finite elements discretizations are the most common examples. However, in such methods high accuracy and error control are difficult to achieve due to issues related to the truncation of the investigation domain and its meshing and the consequent large matrices produced. Another possibility is to resort to [PERSON]'s functionmethods [22]; in such a case the boundary value problem is formulated as IEs, leading to various advantages. Among them, the main one is that the discretization is needed only on finite domains. Nevertheless, that is not a _panacea_, since IEs that are obtained are usually of the first kind with logarithmic-type (or higher order) singular kernels. Consequently, if Method of Moments (MoM) is directly applied, then usually ill-conditioned dense matrices are obtained. In addition, it is not possible to demonstrate the convergence of MoM in every case or even the existence of an exact solution for such IEs [23]. Similar considerations hold when IEs are of the second kind, but with strongly singular kernels, and MoM is applied blindly. The aforementioned difficulties can be overcome by transforming the IEs from the first to the second kind, with smoother kernels. In such a case, the new equations of the Fredholm type can be discretized by a Galerkin-type projection onto suitable basis functions. Matrix equations are then obtained, which are much better conditioned, so that if the \"impedance-matrix\" size is progressively increased the condition number remains small. The mentioned approach is called MAR [24; 25], and it relies on the analytical inversion of the singular part of the original IE. The identification of the operator to be inverted is the first step and many options are possible. On the other hand, its analytical inversion is very difficult and is based on very specialized functional techniques such as Wiener-Hopf or similar methods. However, what is really needed is a discretized counterpart of IEs, that is, a matrix equation, to find a solution numerically. Consequently, an approach usually adopted to deal with the above mentioned cases [24; 25] is to find a set of orthogonal eigenfunctions of the most singular part of the integral operator.