A MULTI-RESOLUTION HOMOGENIZATION (MRH) ANALYSIS FOR THE FIELDS NEAR THE INTERFACE BETWEEN COMPLEX LAMINATES Vitaliy Lomakin (1) , Ben Zion Steinberg (2) and Ehud Heyman (3) Faculty of Engineering, Tel-Aviv University, Tel-Aviv, 69978, Israel. E-mail: (1) vitaliy@eng.tau.ac.il, (2) steinber@eng.tau.ac.il, (3) heyman@eng.tau.ac.il ABSTRACT The multiresolution homogenization theory (MRH) for EM fields in multi-scale laminates is extended here to address lateral interfaces between complex laminates. The scattered field is found by matching the spectra (generally discrete and continuous) on both sides of the interface. Special attention is given to the boundary layer (a surface state) field near the interface, which is described by higher order evanescent modes that are generated by the micro-scale heterogeneities. Far from the interface the field is described asymptotically by effective vertical modes whose horizontal propagation is described in terms of rays that refract at the interface via an effective Snell’s law. INTRODUCTION Multi-resolution homogenization (MRH) is a systematic theory for deriving effective macro-scale formulations for source- excited electromagnetic fields in complex laminates. It smoothes out the micro-scale heterogeneities while retaining their effect on the macro scale (physical) observables [1]–[3]. The theory does not rely on local periodicities or on weak perturbation. It also can handle scale continuum and therefore is amenable for treating ultra wideband (UWB) fields [4]. The MRH has been derived originally for strictly stratified media, but it is being extended here to complex laminates configurations with lateral variations. For configurations with smooth variation, we have previously formulated a theory of “vertical effective modes and horizontal rays” [5], wherein the horizontal (lateral) propagation of the effective vertical modes is described by a two dimensional modal-ray equation that accounts for the lateral variations of the medium as expressed by the effective measures of the complex vertical stratification. In the present paper we explore the fields in the vicinity of a lateral interface separating two complex laminates as depicted in Figure 1. Using the rigorous construct of the MRH, the source-excited field on each side of the interface is described by the corresponding effective spectrum (generally consisting of both discrete and continuous spectra) and the scattered field is then determined by matching the spectra on the two sides. Special attention is given to the boundary layer field near the interface, which is described by higher order evanescent modes that are generated by the micro-scale heterogeneities and therefore generates a surface state. Far from the interface the fields is reduced asymptotically to a sum od effective vertical modes whose horizontal propagation is described in terms rays that are refracted at the interface via an effective Snell’s law. FORMULATION OF THE PROBLEM To explore the interface problem we consider the simplest configuration where the medium on one side of the interface (x< 0) is a homogenous slab while on the other side (x> 0) the medium is a complex laminate (see Fig. 1). The coordinate along the stratification axis is taken to be z, while the transversal coordinates are ρ =(x, y). Both slabs reside in 0 <z<d and are surrounded by free space at z< 0 and z>d. The relative constitutive parameters of the homogeneous slab are ε 1 ,μ 1 while in the complex laminate, they are diagonal tensors whose components ε z ,ε t ,μ z ,μ t (the subscripts z and t denote the z and transverse components, respectively) are multi-scale functions of z, comprising of both macro and micro scales. To demonstrate the theory for the 3D electromagnetic field in a simple format we consider an excitation by a vertically polarized current dipole J =ˆ zJ 0 δ(r - r 0 ) residing at r 0 =(x 0 ,y 0 ,z 0 ), x 0 < 0 in the uniform medium. The resulting transverse magnetic (TM) field is fully described by the scalar potential that satisfies h ∂ 2 ∂z 2 + ∇ 2 t + k 2 0 ε 1 μ 1 i G (1) (r, r 0 )= -ε 1 J 0 δ(r - r 0 ), x< 0 (1a) h ∂ ∂z 1 ε t (z) ∂ ∂z + 1 ε z (z) ∇ 2 t + k 2 0 μ t (z) i G (2) (r, r 0 )=0, x> 0 (1b) with the continuity condition G (1) | 0- = G (2) | 0+ and ε -1 1 ∂ x G| 0- = ε -1 z (z)∂ x G| 0+ at the interface x =0. Here ∇ 2 t = ∂ 2 x + ∂ 2 y and the superscripts (1, 2) correspond to the solution in the region x ≶ 0, respectively. Once G is found and properly parameterized, the electromagnetic field can be readily found by applying certain ∇ operations (see e.g. [6]).