## 1. Introduction

[2] Time-harmonic electromagnetic fields in the presence of heterogeneous media can efficiently be analyzed using surface integral equations. Within each homogeneous subregion the field can be represented in terms of unknown electric and magnetic currents defined over the boundary of that subregion. A direct imposition of the interface conditions yields systems of coupled integral equations in the densities of these electric and magnetic currents. There are two main disadvantages of the formulations based on such equations. First, there are two unknown vector functions to be determined over all the interfaces and, second, all these unknown functions are to be determined simultaneously, which makes the size of the resultant matrix equation in the numerical computation very large. A more efficient alternative is to use surface integral equations satisfied by only one unknown function per interface.

[3] A formulation using a single unknown surface function for the scattering of a plane wave by a penetrable homogeneous long cylinder of arbitrary cross section was first presented by *Maystre and Vincent* [1972]. Later, a recursive procedure making possible the analysis of plane-wave scattering from a two-dimensional layered structure using an integral equation involving a single unknown function only over the outer surface was developed [*Maystre*, 1978] for applications to periodical structures in optical gratings. *Marx* [1982] extended the construction of a single surface integral equation to the three-dimensional electromagnetic scattering of time-harmonic and, also, general time-varying fields from homogeneous dielectric bodies. *Glisson* [1984] showed how to derive such a single integral equation for time-harmonic fields based on equivalence theorems. Recently, general recursive procedures for two-dimensional systems of heterogeneous bodies and complex nested structures, incorporating the properties of invariance to translation and rotation of the reduction operators, have been presented and numerically implemented [*Swatek and Ciric*, 1998, 2000a, 2000b]. As well, reduced surface integral equations have been derived for the analysis of two-dimensional eddy-current fields in solid conductors [*Ciric and Curiac*, 2005] and of Laplacian fields in the presence of layered dielectric structures [*Ciric*, 2006].

[4] The aim of this paper is to present the formulation of reduced vector integral equations for the three-dimensional electromagnetic wave scattering from arbitrarily shaped layered bodies. Expressions in a matrix form of the integral operators involved, adequate for computer implementation, and a detailed analysis of the high efficiency of the numerical solution of field problems using the proposed formulation will be presented in subsequent papers.