Matching of Asymptotic Expansions for Wave Propagation in Media with Thin Slots I: The Asymptotic Expansion

Abstract

In this series of two articles, we consider the propagation of a time harmonic wave in a medium made of the junction of a half-space (containing possibly scatterers) with a thin slot. The Neumann boundary condition is considered along the boundary on the propagation domain, which authorizes the propagation of the wave inside the slot, even if the width of the slot is very small. We perform a complete asymptotic expansion of the solution of this problem with respect to the small parameter ε/λ, the ratio between the width of the slot, and the wavelength. We use the method of matched asymptopic expansions which allows us to describe the solution in terms of asymptotic series whose terms are characterized as the solutions of (coupled) boundary value problems posed in simple geometrical domains, independent of ε/λ: the (perturbed) half-space, the half-line, a junction zone. In this first article, we derive and analyze, from the mathematical point of view, these boundary value problems. The second one will be devoted to establishing error estimates for truncated series.

On the Well-Posedness, Stability and Accuracy of an Asymptotic Model for Thin Periodic Interfaces in Electromagnetic Scattering Problems

“…[17] (see for instance also Ref. [35], [22] and [25] for general results on this topic), we can also prove an optimal estimate of E δ − (E 0 + δ E 1 ), that is valid in any (fixed) domain that does not intersect a small neighbourhood of Γ, namely in any (γ > 0 being given, possibly arbitrarily small)…”

“…[26], [23] and [29]). In section 2, after having rewritten the transmission conditions in an adequate form via the introduction of a boundary operator G ω (whose main properties are described in proposition 7), we establish the variational formulation of our transmission problem (see (21,22,23,24)) in an appropriate function framework (see (25,27)). In section 3, we construct the appropriate (Helmholtz like) decomposition of the variational space (see section 3.1 for the main statements and section 3.2 for their proof).…”

Enhancement of Electromagnetic Fields Caused by Interacting Subwavelength Cavities

“…The interaction of light and rough metallic surfaces gives rise to fascinating phenomena, such as transmission of light through subwavelength apertures [9,15,13,16], or such as Surface Enhancement Raman Scattering [5,6,7,11]. The optical excitation of resonant modes can lead to a concentration and localization of energy in volumes much smaller than λ 3 , where λ is the wavelength of the incident light.…”

Matching of asymptotic expansions for waves propagation in media with thin slots II: The error estimates

“…In [10], we derived formal asymptotic expansions of the exact solution using the technique of matched asymptotic expansions (see, for instance, [5,8,16,24]). Remark 1.1.…”

“…The outline of the paper is as follows. In Section 2.1, we first recall the results obtained in [10]. Namely, we give three different asymptotic expansions in three regions: the half-space (also called the far-field zone), the slot, and the near-field zone which is a transition zone between the slot and the half-space.…”