## 1. Introduction

[2] The full wave analysis of electromagnetic radiation and interference phenomena in planar antennas, circuits and interconnections in multilayered media is most efficiently done by an integral equation formulation combined with a Spectral Domain Approach (SDA) [*Itoh*, 1980]. The incorporation of the complicated multilayered environment in an analytical manner provides more efficient and stable solutions as compared to local differential equation based methods like Finite Difference Time Domain (FDTD) [*Taflove*, 1997] or Finite EleMents (FEM) [*Volakis et al.*, 1997] that rely more on purely computational power to resolve the relevant physical phenomena. As the geometry of the structure becomes more complicated, and deviates more and more from the purely planar case, these advantages fade away, and local methods may become competitive. This occurred first with the need to model probe feeds [*Zheng and Michalski*, 1990] and short circuits in patch antennas and via's and air bridges in integrated circuits [*Tsai et al.*, 1996]. More serious problems threaten the integral equation approach when arbitrary 3-D surfaces have be analyzed. For such a case, currents are present in a continuous manner at all positions and can have arbitrary orientation. The theoretical formulation of a Mixed Potential Integral Equation (MPIE) becomes cumbersome since one is confronted with multiple scalar potentials [*Erteza and Park*, 1969] or a dyadic vector potential kernel [*Michalski and Zheng*, 1990]. These problems translate into severe numerical problems such as the need to evaluate higher order Sommerfeld integrals and an angular azimuth dependence for the components of the vector potential. Furthermore, a large number of Green's functions need to be computed and stored such that reaction integrals can be evaluated by numerical interpolation [*Gay-Balmaz and Mosig*, 1996]. The aforementioned problems are avoided when we focus on structures as the one depicted in Figure 1. This is a 3-D structure subject to the limitation that the conducting surfaces are either horizontal, parallel to the layers, or vertical, normal to the stratification of the medium. This is the only geometrical limitation, since the vertical current sheet can have arbitrary dimension, can cut through dielectric interfaces, and can connect or intersect the horizontal conductors. This kind of geometry allows to refurbish the integral equation approach by exploiting the analytical possibilities of the SDA to the utmost. The formulation of the electric field is adapted to the geometry in a hybrid dyadic-mixed potential form which avoids the problems of a full 3-D formulation. This leads to a blending of the previously separately used Electric Field Integral Equation (EFIE) [*Faché et al.*, 1992] and the Mixed Potential Integral Equation (MPIE) [*Mosig*, 1988]. The EFIE relates fields directly to currents via a dyadic Green's function, while in an MPIE charges are introduced, such that scalar and vector potential Green's functions are distinguished. We will use the EFIE for vertical field and current components and the MPIE for the horizontal or transverse components (see section 2). The discretisation of the integral equation into a set of linear equations is done with a method of moments [*Ney*, 1985], which requires evaluation of reaction integrals [*Rumsey*, 1954] which can be performed in the space domain [*Sercu et al.*, 1993] or the spectral domain [*Horng et al.*, 1992]. The MPIE formulation in the space domain is most popular since it avoids the 1/*R*^{3} spatial singularity of the dyadic formulation by a rather physical method: derivatives of the Green's functions are transfered to the expansion and test functions such that charges are obtained. The same problem is overcome in the EFIE spectral domain approach in a more numerical manner, by evaluating the reaction integrals in the spectral domain. The extra integrations improve the slow spectral decay of the Green's functions. Again, our solution of the integral equation combines both approaches with a mixed spectral-spatial domain approach (see section 3) where the evaluation of the reaction integrals is spread out over both domains. The (*z*, *z*′) dependent part of the reaction integrals can be done analytically (see section 3.4) in the spectral domain, prior to the numerical inverse Fourier transform. The remaining integrals are done in the space domain, where the MPIE retains its advantages. The proposed methods were implemented and applied to a number of problems shown in section 4 which show that most 3-D problems in multilayered media can actually be handled.