## 1. Introduction

[2] Scattering of plane waves by two-dimensional (2D), perfectly conducting or penetrable objects has been treated by many authors, both for TE and TM polarization and by various numerical techniques [*Tsang*, 2001]. The scatterers are either placed in homogeneous space or can be embedded in an unbounded planar stratified medium. Surface integral equation techniques are ideally suited for homogeneous scatterers, either single scatterers or periodic arrangements of scatterers. Such integral equation techniques are based on the knowledge of appropriate Green's functions. In the past, *Rogier et al.* [1996], *Rogier and De Zutter* [2003], *Olyslager et al.* [1993] and *Rogier et al.* [2000] have contributed to this field of research, both for scattering problems as well as for guided wave problems. Typically, the equivalence principle is used to introduce unknown equivalent electric and magnetic surface currents on the boundary or boundaries of the scatterer or scatterers. These equivalent currents are then found by solving integral equations that enforce the continuity of the total tangential electric and magnetic field at the scatterer's surface [*Poggio and Miller*, 1973; *Charles et al.*, 1991; *Donepudi et al.*, 2003; *Ylä-Oijala and Taskinen*, 2005].

[3] In this paper it will be shown that it suffices to introduce an equivalent electric surface current **J**_{s}(**r**, *ω*) on the scatterer's surface, provided a suitable surface admittance operator is introduced relating this current at each point **r** of the scatterer's surface to the tangential electric fields **E**_{tan}(**r**′, *ω*) at every other point on the surface. This surface admittance operator allows to replace the medium of the scatterer by the medium of the surrounding background medium the scatterer is embedded in. The remaining field problem can then be solved by solely considering the interactions between the equivalent electric surface currents and the incident field in the sole presence of the background medium the scatterers were originally embedded in. It is shown that the admittance operator yields a highly accurate description of the behavior of the scatterer for a wide range of electromagnetic material properties. This is in particular the case when the scatterer becomes highly conductive and for small skin depths. In this paper, we will restrict ourselves to the time-harmonic TM case.

[4] The proposed surface admittance approach was originally developed to study the skin effect for high clock rate signals propagating on RF boards, packages and chips [*De Zutter and Knockaert*, 2005]. More in particular, in work by *De Zutter and Knockaert* [2005], attention was focused on the determination of the frequency-dependent resistance and inductance matrices per unit of length of a set of parallel signal lines. From a purely theoretical point of view, the technique presented by *De Zutter and Knockaert* [2005] can be expected to be applicable to scattering problems as well. However, in work by *De Zutter and Knockaert* [2005] the application of the theory is restricted to the low-frequency case (i.e., the dimensions of the considered multiconductor lines remain small with respect to the free space wavelength) and to a homogeneous background medium (homogeneous free space with or without a PEC ground plane). The main purpose of this contribution is to demonstrate the validity of the surface admittance approach for 2D TM-scattering problems. It is demonstrated that the surface admittance operator can be easily incorporated into an integral equation method, including the case of a layered background medium and/or for a periodic arrangement of the scatterers.

[5] In section 2, the theoretical background of the surface admittance operator is briefly recapitulated and a general expression in terms of the Dirichlet eigenfunctions of the scatterer's cross section is given. For the rectangular cross section, application of the method of moments (MoM) leads to a discretized form of the surface admittance operator: the surface admittance matrix. In section 3, this surface admittance matrix for the rectangle is briefly presented, but the reader is again referred to *De Zutter and Knockaert* [2005] for more details.

[6] From many possible examples and in order to keep this paper concise, we have selected three numerical examples which are presented in section 5. First, the simple scattering configuration of a coated rectangular cylinder in free space is considered and the obtained results are compared with data published by *Jin and Liepa* [1988]. In the two other examples the background medium consists of a dielectric slab in free space and the scatterers form a periodic grid. First (lossy) dielectric scatterers with rectangular cross section are considered. Next, the rectangles are replaced by nonlossy dielectric crosses. The incident wave is a plane TM wave and both reflection and transmission coefficients are calculated, as well as the total dissipated power for the case of lossy scatterers. Numerical results are compared with those obtained with a previously published technique based on a boundary integral equation approach using equivalent electric and magnetic currents [*Poggio and Miller*, 1973; *Rogier et al.*, 2000]. Before presenting the examples, section 4 starts with a brief explanation on the solution of the overall scattering problem when using a surface admittance operator. Finally, section 6 gives a brief conclusion and some prospects for further research.