Rigorous Asymptotics for the Electric Field in Tm Mode at Mid-frequency in a Bidimensional Medium with a Thin Layer

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(θ), with θ 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(h3/2). In addition, if the complex dielectric permittivity in the membrane, denoted by zm, tends to zero faster than h, we give the difference between the exact solution and the above asymptotic with zm = 0; it is of order O(h3/2 + |zm|). Introduction We study in this paper the behavior of the solution of Helmholtz equation in a bidimensional medium in transverse magnetic (TM) mode (see Balanis and Constantine [3]). The medium is made out of three materials: a central region surrounded by a thin membrane of thickness hf(θ), with θ curvilinear coordinate and a third material, which is not assumed to be thin; see Fig. 1. This assemblage is submitted to a field of pulsation ω; after proper scalings, ω is included in the complex dielectric permittivity, which may be different in the three materials. By analogy with the biological cell, we call this entity a cell in an environment. In this article, we show that as the thickness of the membrane tends to zero, i.e as h tends to zero, the electric field tends to the solution of a Helmholtz equation with an appropriate transmission condition at the boundary between the cell and the ambient medium. This work is a sequel to the author’s former article in the static case [10]. Let us give now the precise notations. Let Ω be a bidimensional bounded domain composed of three subdomains: a bounded domain Oc surrounded by a thin membrane Oh with small thickness hf , and an exterior domain Oe,h: Ω = Oc ∪ Oh ∪Oe,h. Date: ... 1991 Mathematics Subject Classification. 34E05, 34E10,35J05. 1