Rigorous Asymptotics For The Elecric Field In TM Mode At Mid-Frequency In A Bidimensional Medium With Thin Layer

Abstract

Consider an ambient medium and a heterogeneous entity composed of a bidimensional material surrounded by a thin membrane. The electromagnetic constants of these materials are different. By analogy with biological cells, we call this entity a cell. We study the asymptotic behavior of the electric field in the transverse magnetic (TM) mode, when the thickness of the membrane tends to zero. The membrane is of thickness of hf(s), with s a curvilinear coordinate. We provide a rigorous derivation of the first two terms of the asymptotic expansion for h tending to zero. In the membrane, these terms are given explicitly in local coordinates in terms of the boundary data and of the function f, while outside the membrane they are the solutions of a scalar Helmholtz equation with appropriate boundary and transmissions conditions given explicitly in terms of the boundary data and of the above function f. We prove that the remainder terms are of order O(h^3/2). In addition, if the complex dielectric permittivity in the membrane, denoted by z_m, tends to zero faster than h, we give the difference between the exact solution and the above asymptotic with z_m = 0; it is of order O(h^{3/2}+|z_m|).