Transfer and approximation of the impedance for time-harmonic Maxwell’s system in a planar domain with thin contrasted multi-layers

Abstract

We consider the diffraction of time-harmonic electromagnetic waves by a perfectly conducting planar obstacle, coated by thin contrasted multi-layers of dielectric materials. A partial Fourier transform is used to derive an impedance condition that links the tangential components of the electric and magnetic fields, and transfers the boundary condition from the obstacle’s face to the other face of dielectric multi-layers.

We first determine the exact formula and construct approximations of the impedance condition for one dielectric thin layer; after that, we extend the obtained result to multi-layers.

Keywords

Time-harmonic Maxwell’s system scattering thin dielectric layers impedance operator

1. Introduction

The scattering of time-harmonic electromagnetic or acoustic waves by obstacles involves always their structures, which have in most cases thin geometry at least in one space dimension; we mention for example the case of thin slots (see, e.g., [11–13]), and thin layers.

We highlight the fact that, in the numerical approximation of the solution, of the classical problem for perfect conductor coated by a thin dielectric material, numerical instabilities appear (see [1]), due to the thinness of the thin shell. An impedance boundary condition is widely used to simplify the mathematical and numerical complexities in the solution of the scattering problem.

The impedance condition was formulated for the first time by Leontovich (see [15]), it was presented initially as the boundary condition modelling the penetration of a wave in an imperfectly conducting metal barrier. However, it was verified that it also models other situations, such as a perfect conductor coated with a thin dielectric layer which is the present study interest (see, e.g., [6,8,10]), or the perfect conductor with a rough surface (see, e.g., [9,18]).

In the literature, several authors gave different approximations of the impedance operator. Engquist and Nédélec (see [7]) derived approximations of order two with respect to the thickness of coating, while Bendali and Lemrabet in [4] gave approximate boundary condition up to the third order, via an asymptotic expansion. Senior and Volakis (see, e.g., [17]) have introduced a different way to construct that artificial condition, which consists in the approximation of the reflection coefficient, that is given exactly after solving explicitly the diffraction problem. Bartoli and Bendali have presented in [2] a study based on some techniques of domain decomposition methods to set the Helmholtz problem outside the obstacle. Bendali and Lemrabet in [5] have investigated the problem with a suitable scaling and an adapted writing of the curl operator inside the thin shell. However, after facing some difficulties related to the numerical simulation, Bendali et al. have shown in [3] how to overcome them through the development of efficient Padé-like approximation impedance boundary condition and presented some numerical procedures developed to efficiently solve the related unusual boundary value problems.

The work in this paper can be seen as a contribution to the previous studies (see [3–5]). We have specifically considered a perfectly conducting metal coated with contrasted dielectric thin multi-layers. The main idea is to present a way to transfer the impedance boundary condition from a face of the thin layer to the other one, and once we have the transfer formula we can transfer it through multiple thin layers, then we set transmission problem as a scattering problem in only the exterior domain.

We consider a plane geometry for which an explicit expression of the impedance boundary condition can be achieved by using Fourier transform; after that, we give some approximations in two cases, which can serve as a guide to obtain approximate conditions in a non planar geometry, where the calculation of the exact impedance boundary condition is not always possible.

The paper is organized as follows:

In Section 2 we present the time-harmonic Maxwell system which will be expressed inside the thin layers as an abstract first order differential system.

In Section 3 we define the impedance operator of the thin layer and use it to introduce an equivalent problem set in the exterior domain only.

In Section 4, the solution of a boundary problem in a thin planar layer is obtained by Fourier transform. The explicit solution allows us to transfer an impedance condition set on a face of the layer to another impedance condition set on the other face.

In Section 5 we consider the simple case of a thin slab and apply the transfer formula to give approximate impedance boundary condition up to order three with respect to the thickness.

The last section is devoted to the case of three thin layers. The use of the transfer formula three times leads to the exact impedance condition. A tensor notation is used to give the approximate impedance conditions up to order three in compact form.

2. Maxwell system

Let Ω be the exterior unbounded domain in which the wave propagates (see Fig. 1), its boundary is a smooth bounded surface $Γ = \partial Ω$ , which represents the face of the scatterer composed by a perfect conductor coated with a thin layer of dielectric material denoted $Ω^{δ}$ , $\begin{matrix} Ω^{δ} = {x \notin Ω; dist (x, Γ) < δ}, δ > 0 . \end{matrix}$ The interface between the thin layer $Ω^{δ}$ and the perfect conductor is denoted $Γ^{δ} = {x \notin Ω; dist (x, Γ) = δ}$ , which is parallel to the surface Γ. The parameter δ describing the thickness of the shell $Ω^{δ}$ is small enough compared with the other dimensions of the scatterer, and $dist (x, Γ)$ is the distance between the point x and the surface Γ. The vector n is the unit normal outwardly directed to $Ω^{δ}$ , which is also normal to $Γ^{δ}$ at the point where it intersects this surface.

Fig. 1.

Configuration of the domain for the general case.

The electromagnetic wave propagation is governed by the following Maxwell’s system: $\begin{matrix} (1) & \{\begin{matrix} ▽ \times E + \partial_{t} H = 0, ▽ \times H - \partial_{t} E = 0, & in Ω, \\ ▽ \times E + μ \partial_{t} H = 0, ▽ \times H - ε \partial_{t} E = 0, & in Ω^{δ}, \end{matrix} \end{matrix}$ the constants ε and μ characterize the electric permittivity and the magnetic permeability respectively inside the thin shell.

The time dependence is harmonic with a frequency $ω > 0$ , so that the electromagnetic field is of the form (see [16]) $\begin{matrix} (2) & \{\begin{matrix} E (t, x) = Re (E (x) e^{- i w t}), \\ H (t, x) = Re (H (x) e^{- i w t}), \end{matrix} \end{matrix}$ which allows us to write the time-harmonic Maxwell system $\begin{matrix} (3) & \{\begin{matrix} ▽ \times E - i ω H = 0, ▽ \times H + i ω E = 0, & in Ω, \\ ▽ \times E - i ω μ H = 0, ▽ \times H + i ω ε E = 0, & in Ω^{δ} . \end{matrix} \end{matrix}$ The system of equations (3) has to be complemented by the following conditions:

Transmission conditions on the exterior boundary Γ $\begin{matrix} (4) & \{\begin{matrix} {[E \times n]}_{Γ} = 0, \\ {[n \times H]}_{Γ} = 0, \end{matrix} \end{matrix}$ where ${[\cdot]}_{Γ}$ denotes the jump across the boundary Γ.

Dirichlet boundary condition on $Γ^{δ}$ $\begin{matrix} (5) & E \times n = 0 on Γ^{δ} . \end{matrix}$

The Silver–Müller radiation condition (see [14]) $\begin{matrix} (6) & CR \infty (H - H^{inc}, E - E^{inc}) : = lim_{| x | \to \infty} | x | ((H - H^{inc}) \times \frac{x}{| x |} - (E - E^{inc})) = 0, \end{matrix}$ where $(E^{inc}, H^{inc})$ is the incident wave. For simplicity, it is assumed to be a plane wave propagating in the inverse direction of the normal vector.

Note that we can express Eq. (3) as an abstract first order differential equations (see [3]),

\begin{matrix} (7) & \partial_{z} Y (z) = M Y (z), \end{matrix}

with

\begin{matrix} Y = [\begin{matrix} E_{T} \\ n \times H \end{matrix}], M = [\begin{matrix} 0 & - i k μ A \\ \frac{1}{i k μ} B & 0 \end{matrix}], \end{matrix}

and

\begin{array}{l} A (ε, μ) = 1 + \frac{1}{k^{2} ε μ} \nabla_{Γ^{z}} {div}_{Γ^{z}}, \\ B (ε, μ) = k^{2} ε μ - \nabla_{Γ^{z}} \times {curl}_{Γ^{z}}, \end{array}

where

\nabla_{Γ^{z}}

{div}_{Γ^{z}}

and

{curl}_{Γ^{z}}

are the surfacic gradient, surfacic divergence and scaler surfacic curl, respectively. Note that ε is the relative permittivity, μ is the relative permeability and

k = ω / c

is the wave number with c the speed of the electromagnetic wave in the vacuum.

3. Problem statement

The system of equations (3) and conditions (4)–(6) describe a scattering problem of electromagnetic plane waves by a perfectly conducting obstacle coated with thin dielectric layers.

To avoid the numerical instability mentioned in [1] and [2], we are going to replace the above scattering problem by another problem which will be called impedance problem set in the exterior domain only. The impedance problem is described through a relation linking the tangential components of the electromagnetic fields.

3.1. Impedance operator

For a sufficiently smooth function φ defined on Γ, let $(E, H)$ be the solution of the following boundary value problem: $\begin{matrix} \{\begin{matrix} ▽ \times E - i ω μ H = 0 & in Ω^{δ}, \\ ▽ \times H + i ω ε E = 0 & in Ω^{δ}, \\ E \times n = 0 & on Γ^{δ}, \\ n \times H = φ & on Γ . \end{matrix} \end{matrix}$ We define the impedance operator by the mapping $\begin{matrix} S : φ ⟼ S φ = E \times n |_{Γ} . \end{matrix}$ Let us emphasize the fact that the impedance operator S depends on δ, the thickness of the thin layer.

Making use of the impedance operators and the transmission conditions (4) we get the following impedance boundary problem set in the exterior domain $\begin{matrix} \{\begin{matrix} ▽ \times E - i ω H = 0 & in Ω, \\ ▽ \times H + i ω E = 0 & in Ω, \\ E \times n = S (n \times H) & on Γ, \\ + CR \infty & (radiation condition at \infty) . \end{matrix} \end{matrix}$ We will see later that the impedance operator S can be written as $S = G^{- 1} F$ , where G and F are differential operators on the boundary Γ, the impedance condition is $G (E \times n) = F (n \times H)$ .

As we are interested in planar geometry, a Fourier transform can be used to calculate the exact impedance operator. Approximate impedance boundary conditions at different orders with respect to δ are then obtained through a power series expansion of S with respect to δ. This results will be of great help in searching impedance boundary approximations in an other geometry when the exact impedance is not reachable (see [2]).

4. Transfer of the impedance condition through a slab

Let $Γ_{- s}$ and $Γ_{- t}$ be the faces of the slab. The impedance boundary condition on $Γ_{- t}$ is written as $\begin{matrix} G_{- t} E_{T} (- t) = F_{- t} (n \times H) (- t), \end{matrix}$ where $G_{- t}$ and $F_{- t}$ are operators that can be obtained from the given boundary condition on $Γ_{- t}$ ; for instance, if $Γ_{- t} = Γ^{δ}$ , then from (5), $F_{- t} = 0$ and $G_{- t}$ is the identity operator.

We are going to get the impedance boundary conditions on $Γ_{- s}$ as $\begin{matrix} G_{- s} E_{T} (- s) = F_{- s} (n \times H) (- s), \end{matrix}$ where the operators $G_{- s}$ and $F_{- s}$ are expressed as functions of $G_{- t}$ and $F_{- t}$ .

We call this process transfer of the impedance boundary condition.

Now the transfer of the impedance boundary condition through multiple layers can be done by repeating the above process and taking into account the transmission conditions on the different interfaces.

Let $\begin{matrix} Ω (- t, - s) = {(x, z) \in R^{2} \times R; - t < z < - s}, \end{matrix}$ be the thin layer limited between the two parallel planes $z = - t$ and $z = - s$ , $s, t > 0$ .

Let $n = (0, 0, 1)$ be the normal outwardly directed to $Ω (- t, - s)$ , and let $V (x, z) = (V_{T} (x, z), V_{n} (x, z))$ be a vector field decomposed on his tangential and normal components $V_{T}$ and $V_{n}$ , respectively.

We recall that the Fourier transform for a function $f : R^{2} \to R$ is given by $\begin{matrix} \hat{f} (ξ) = \int_{R^{2}} e^{- i x . ξ} f (x) d x . \end{matrix}$

Theorem 4.1.
The exact impedance boundary condition on the surface $Γ_{- s} = R^{2} \times {- s}$ with the partial Fourier transform is given by $\begin{matrix} (8) & G_{- s} (ξ) {\hat{E}}_{T} (ξ, - s) = F_{- s} (ξ) (n \times \hat{H}) (ξ, - s), \end{matrix}$ with $\begin{array}{l} G_{- s} (ξ) & = [G_{- t} (ξ) cos (α (t - s)) + sin (α (t - s)) F_{- t} (ξ) Q (ξ)], \\ F_{- s} (ξ) & = [- G_{- t} (ξ) sin (α (t - s)) + cos (α (t - s)) F_{- t} (ξ) Q (ξ)] P (ξ), \end{array}$ where $α^{2} = k^{2} ε μ - | ξ |^{2}$ , $G_{- t}$ and $F_{- t}$ are the corresponding operators to G and F in the boundary condition ( 5 ) through Fourier transform, P and Q are operators depending on the physical characteristics ε and μ and are defined by $\begin{matrix} (9) & \{\begin{matrix} P (ξ) = P (ε, μ) (ξ) = \frac{i k μ}{α} (1 - \frac{1}{k^{2} ε μ} ξ (ξ \cdot)), \\ Q (ξ) = Q (ε, μ) (ξ) = \frac{1}{i k μ α} (k^{2} ε μ - (ξ \times n) (ξ \times n) \cdot), \end{matrix} \end{matrix}$ where $(ξ \cdot)$ and $((ξ \times n) \cdot)$ denote the scalar product with ξ and $(ξ \times n)$ .
Proof.
Applying the partial Fourier transform to Eq. (7) we get $\begin{matrix} (10) & \{\begin{matrix} D_{z} {\hat{E}}_{T} (ξ, z) = - i k μ (1 - \frac{1}{k^{2} ε μ} ξ (ξ \cdot)) (n \times \hat{H}) (ξ, z), \\ D_{z} (n \times \hat{H}) (ξ, z) = \frac{1}{i k μ} (k^{2} ε μ - (ξ \times n) (ξ \times n) \cdot) {\hat{E}}_{T} (ξ, z) . \end{matrix} \end{matrix}$ By combining these two latter equations, we see that ${\hat{E}}_{T}$ and ${\hat{H}}_{T}$ satisfy the following second order differential equation $\begin{matrix} D_{z}^{2} {\hat{U}}_{T} + α^{2} {\hat{U}}_{T} = 0, \end{matrix}$ where $α^{2} = k^{2} ε μ - | ξ |^{2}$ .

Therefore ${\hat{E}}_{T}$ and ${\hat{H}}_{T}$ are given by $\begin{matrix} (11) & \{\begin{matrix} {\hat{E}}_{T} (ξ, z) = A (ξ) cos (α z) + B (ξ) sin (α z), \\ {\hat{H}}_{T} (ξ, z) = C (ξ) cos (α z) + D (ξ) sin (α z) . \end{matrix} \end{matrix}$ Taking the derivative of (11) with respect to z, we get $\begin{matrix} (12) & \{\begin{matrix} D_{z} {\hat{E}}_{T} (ξ, z) = - α A (ξ) sin (α z) + α B (ξ) cos (α z), \\ D_{z} (n \times \hat{H}) (ξ, z) = - α (n \times C) (ξ) sin (α z) + α (n \times D) (ξ) cos (α z) . \end{matrix} \end{matrix}$ Substituting (11) into (10), we have $\begin{matrix} (13) & \{\begin{matrix} D_{z} {\hat{E}}_{T} (ξ, z) & = & - i k μ (n \times C) (ξ) cos (α z) + \frac{i}{k ε} ξ (ξ \cdot n \times C) (ξ) cos (α z) \\ - i k μ (n \times D) (ξ) cos (α z) + \frac{i}{k ε} ξ (ξ \cdot n \times D) (ξ) sin (α z), \\ D_{z} (n \times {\hat{H}}_{T}) (ξ, z) & = & \frac{k ε}{i} A (ξ) cos (α z) + \frac{k ε}{i} B (ξ) sin (α z) \\ - \frac{1}{i k μ} (ξ \times n) [(ξ \times n) \cdot A (ξ)] cos (α z) \\ - \frac{1}{i k μ} (ξ \times n) [(ξ \times n) \cdot B (ξ)] sin (α z) . \end{matrix} \end{matrix}$ Identifying the coefficients of sin and cos we find that $\begin{matrix} (14) & \{\begin{matrix} A = P (n \times D), \\ B = - P (n \times C), \end{matrix} \{\begin{matrix} (n \times C) = - Q B, \\ (n \times D) = Q A . \end{matrix} \end{matrix}$ Let’s remark that P and Q defined in (9) satisfy $\begin{matrix} (15) & P (ξ) Q (ξ) = Q (ξ) P (ξ) = I . \end{matrix}$ Using the formulas (9) we can write (11) as $\begin{matrix} (16) & \{\begin{matrix} {\hat{E}}_{T} (ξ, z) & = & A (ξ) cos (α z) + B (ξ) sin (α z), \\ (n \times {\hat{H}}_{T}) (ξ, z) & = & - Q B (ξ) cos (α z) + Q A (ξ) sin (α z), \\ P (n \times {\hat{H}}_{T}) (ξ, z) & = & - B (ξ) cos (α z) + A (ξ) sin (α z) . \end{matrix} \end{matrix}$ Therefore, by using the values of the electromagnetic field on the surface $Γ_{- s} = R^{2} \times {- s}$ , we obtain $\begin{matrix} (17) & \{\begin{matrix} A (ξ) = cos (α s) {\hat{E}}_{T} (ξ, - s) - sin (α s) P (n \times {\hat{H}}_{T} (ξ, - s)), \\ B (ξ) = - sin (α s) {\hat{E}}_{T} (ξ, - s) - cos (α s) P (n \times {\hat{H}}_{T} (ξ, - s)) . \end{matrix} \end{matrix}$ Recall that the impedance boundary condition on the surface $Γ_{- t} = R^{2} \times {- t}$ is $\begin{matrix} G_{- t} (ξ) {\hat{E}}_{T} (ξ, - t) = F_{- t} (ξ) (n \times \hat{H}) (ξ, - t) . \end{matrix}$ Replacing ${\hat{E}}_{T} (ξ, - t)$ and $(n \times \hat{H}) (ξ, - t)$ by their values from (16), we get $\begin{array}{l} G_{- t} (ξ) (A (ξ) cos (α t) - B (ξ) sin (α t)) \\ = F_{- t} Q (ξ) (- B (ξ) cos (α t) - A (ξ) sin (α t)) . \end{array}$ Plugging the values of $A (ξ)$ and $B (ξ)$ given by (17) in the above formula we get the impedance boundary condition on the surface $Γ_{- s} = R^{2} \times {- s}$ : $\begin{matrix} G_{- s} (ξ) {\hat{E}}_{T} (ξ, - s) = F_{- s} (ξ) (n \times \hat{H}) (ξ, - s), \end{matrix}$ with $\begin{array}{l} G_{- s} (ξ) & = [G_{- t} (ξ) cos (α (t - s)) + sin (α (t - s)) F_{- t} (ξ) Q (ξ)], \\ F_{- s} (ξ) & = [- G_{- t} (ξ) sin (α (t - s)) + cos (α (t - s)) F_{- t} (ξ) Q (ξ)] P (ξ), \end{array}$ where, $G_{- t}$ and $F_{- t}$ correspond to the operators given through the boundary condition on $Γ_{- t}$ . □

5. Application for a single thin layer

Theorem 5.1.
For a slab of thickness δ (see Fig. 2 ), $\begin{matrix} Ω^{δ} = {(x, z) \in R^{2} \times R; - δ < z < 0}, \end{matrix}$ the boundary condition $E \times n = 0$ on the surface $Γ^{δ} = R^{2} \times {- δ}$ leads to the following approximate impedance boundary condition of order three: $\begin{matrix} (18) & \begin{matrix} E_{T ∣ Γ} ≃ & - i k μ δ ({(1 - \frac{δ^{2}}{3} k^{2} ε μ - \frac{δ^{2}}{3} \vec{Δ_{Γ}})}^{- 1} \\ + \frac{1}{k^{2} ε μ} \nabla_{Γ} {(1 - \frac{δ^{2}}{3} k^{2} ε μ - \frac{δ^{2}}{3} Δ_{Γ})}^{- 1} {div}_{Γ}) {(n \times H)}_{∣ Γ}, \end{matrix} \end{matrix}$ where $Δ_{Γ}$ and $\vec{Δ_{Γ}}$ are respectively the surfacic scalar Laplacian and the surfacic vectorial Laplacian on Γ.

Fig. 2.
Configuration of the domain for the planar geometry with one single layer.
Proof.
The boundary condition $E \times n = 0$ on the surface $Γ^{δ} = R^{2} \times {- δ}$ gives $\begin{matrix} G_{- δ} E_{T} (- δ) = F_{- δ} (n \times H) (- δ), \end{matrix}$ with $G_{- δ} = I$ and $F_{- δ} = 0$ , then the transfer formula (8) leads to $\begin{matrix} {\hat{E}}_{T} (ξ, 0) = - tan (α δ) P (ξ) (n \times \hat{H}) (ξ, 0), \end{matrix}$ which represents the exact impedance boundary condition.

For δ small enough and ξ fixed, we have $\begin{matrix} tan (α δ) ≃ α δ + \frac{1}{3} α^{3} δ^{3}, \end{matrix}$ and therefore $\begin{matrix} {\hat{E}}_{T} (ξ, 0) ≃ - δ (1 + \frac{δ^{2}}{3} α^{2}) α P (ξ) (n \times \hat{H}) (ξ, 0), \end{matrix}$ which can be written as $\begin{matrix} (19) & \begin{matrix} {\hat{E}}_{T} (ξ, 0) ≃ & - i k μ δ ((1 + \frac{δ^{2}}{3} k^{2} ε μ - \frac{δ^{2}}{3} | ξ |^{2}) I \\ - \frac{1}{k^{2} ε μ} ξ (1 + \frac{δ^{2}}{3} k^{2} ε μ - \frac{δ^{2}}{3} | ξ |^{2}) (ξ \cdot)) (n \times \hat{H}) (ξ, 0) . \end{matrix} \end{matrix}$ By taking the inverse Fourier transform we obtain $\begin{matrix} (20) & (1 - \frac{δ^{2}}{3} (k^{2} ε μ + \vec{Δ_{Γ}})) E_{T ∣ Γ} ≃ - i k μ δ (1 + \frac{1}{k^{2} ε μ} \nabla_{Γ} ({div}_{Γ})) {(n \times H)}_{∣ Γ}, \end{matrix}$ where $\vec{Δ_{Γ}}$ is the surfacic vectorial Laplace Beltrami operator on Γ defined by $\begin{matrix} (21) & \vec{Δ_{Γ}} = \nabla_{Γ} {div}_{Γ} - \nabla_{Γ} \times {curl}_{Γ} . \end{matrix}$ As the inverse of $(1 - \frac{δ^{2}}{3} k^{2} ε μ - \frac{δ^{2}}{3} \vec{Δ_{Γ}})$ is equal to $(1 + \frac{δ^{2}}{3} (k^{2} ε μ + \vec{Δ_{Γ}}))$ at order two with respect to δ we get $\begin{array}{l} E_{T ∣ Γ} & ≃ - i k μ δ (1 + \frac{δ^{2}}{3} (k^{2} ε μ + \vec{Δ_{Γ}})) (1 + \frac{1}{k^{2} ε μ} \nabla_{Γ} ({div}_{Γ})) {(n \times H)}_{∣ Γ} \\ = - i k μ δ [(1 + \frac{δ^{2}}{3} (k^{2} ε μ + \vec{Δ_{Γ}})) + \frac{1}{k^{2} ε μ} (1 + \frac{δ^{2}}{3} (k^{2} ε μ + \vec{Δ_{Γ}})) \nabla_{Γ} ({div}_{Γ})] {(n \times H)}_{∣ Γ}, \end{array}$ but as $\begin{matrix} \vec{Δ_{Γ}} \nabla_{Γ} ({div}_{Γ}) = (\nabla_{Γ} {div}_{Γ} - \nabla_{Γ} \times {curl}_{Γ}) \nabla_{Γ} ({div}_{Γ}) = \nabla_{Γ} {div}_{Γ} \nabla_{Γ} {div}_{Γ} = \nabla_{Γ} Δ_{Γ} {div}_{Γ}, \end{matrix}$ we get $\begin{matrix} (k^{2} ε μ + \vec{Δ_{Γ}}) \nabla_{Γ} ({div}_{Γ}) = \nabla_{Γ} (k^{2} ε μ + Δ_{Γ}) {div}_{Γ}, \end{matrix}$ and then $\begin{array}{l} E_{T ∣ Γ} ≃ & - i k μ δ ({(1 - \frac{δ^{2}}{3} k^{2} ε μ - \frac{δ^{2}}{3} \vec{Δ_{Γ}})}^{- 1} \\ + \frac{1}{k^{2} ε μ} \nabla_{Γ} {(1 - \frac{δ^{2}}{3} k^{2} ε μ - \frac{δ^{2}}{3} Δ_{Γ})}^{- 1} {div}_{Γ}) {(n \times H)}_{∣ Γ} . \end{array}$ □
Remark 5.2.

If we keep only the terms of order 1 in (18) we obtain the approximate condition $\begin{matrix} (22) & E_{T ∣ Γ} ≃ - i k μ δ [1 + \frac{1}{k^{2} ε μ} \nabla_{Γ} {div}_{Γ}] {(n \times H)}_{∣ Γ}, \end{matrix}$ given by Engquist–Nédélec (see [7]).

The approximate condition (18) is a particular case of that in Bendali et al. (see [3]) which contains curvature terms.

6. Application for three contrasted thin layers

In this section, we will extend the results obtained in the case of a single thin layer to the case when the domain $Ω^{δ}$ is the superposition of three contrasted thin layers of dielectric materials $Ω_{j}^{δ}$ , $1 ⩽ j ⩽ 3$ (see Fig. 3). The thin layers are parallel to each other and characterized by relative dielectric permittivity $ε_{j}$ and relative magnetic permeability $μ_{j}$ , $1 ⩽ j ⩽ 3$ . In this case we must add the transmission conditions $[E \times n] = 0$ and $[H \times n] = 0$ on the interfaces $Γ_{j}^{δ}$ , $j = 1, 2$ .

Fig. 3.

Configuration of the domain for planar geometry with three thin layers.

6.1. Notation

For simplicity in writing some of the formulas in the case of multiple thin layers, we shall introduce the following notations.

Let A be a ring, and let $X, Y, Z \in A^{n}$ . Then, we define

The linear form $T_{1} (X)$ by $\begin{matrix} (23) & \begin{matrix} T_{1} (X) : & R^{N} \to A \\ δ \mapsto T_{1} (X) (δ) = \sum_{i = N}^{i = 1} δ_{i} X_{i} . \end{matrix} \end{matrix}$

The bilinear form $T_{2} (X, Y)$ by $\begin{matrix} (24) & \begin{matrix} T_{2} (X, Y) : & {(R^{N})}^{2} \to A \\ (δ, η) \mapsto T_{2} (X, Y) (δ, η) = \sum_{i = N}^{i = 1} δ_{i} η_{i} X_{i} Y_{i} + \sum_{1 ⩽ j < i ⩽ N} 2 δ_{i} η_{j} X_{i} Y_{j} . \end{matrix} \end{matrix}$

The trilinear form $T_{3} (X, Y, Z)$ by $\begin{matrix} (25) & \begin{matrix} T_{3} (X, Y, Z) : & {(R^{N})}^{3} \to A \\ (δ, η, ζ) \mapsto T_{3} (X, Y, Z) (δ, η, ζ) = \sum_{i, j, k} {(T_{3} (X, Y, Z))}_{i, j, k} δ_{i} η_{j} ζ_{k}, \end{matrix} \end{matrix}$ where $\begin{matrix} {(T_{3} (X, Y, Z))}_{i, j, k} = \{\begin{matrix} X_{i} Y_{i} Z_{i}, & if i = j = k, \\ 3 X_{i} Y_{i} Z_{k} & if i = j > k, \\ 3 X_{i} Y_{j} Z_{j} & if i > j = k, \\ 6 X_{i} Y_{j} Z_{k} & if i > j > k, \\ 0 & else. \end{matrix} \end{matrix}$

6.2. Impedance boundary condition for three thin layers

Let $β_{j}$ , $j = 1, 2, 3$ be positive real numbers satisfying $β_{1} + β_{2} + β_{3} = 1$ . We denote the jth thin layer by $Ω_{j}^{δ} = Ω (- t_{j}, - t_{j - 1})$ , $t_{j} = δ \sum_{l = 1}^{l = j} β_{l}$ with the convention $t_{0} = 0$ , and $Γ_{j}^{δ}$ the interface between the jth and the $(j + 1) th$ thin layers (see Fig. 3).

Lemma 6.1.
The exact impedance boundary condition for the three thin layers is given by $\begin{matrix} (26) & G {\hat{E}}_{T} (ξ, 0) = F (n \times \hat{H}) (ξ, 0), \end{matrix}$ where $\begin{array}{l} G = & 1 - tan (α_{3} β_{3} δ) P_{3} tan (α_{2} β_{2} δ) Q_{2} - tan (α_{3} β_{3} δ) P_{3} tan (α_{1} β_{1} δ) Q_{1} \\ - tan (α_{2} β_{2} δ) P_{2} tan (α_{1} β_{1} δ) Q_{1}, \end{array}$ and $\begin{array}{l} F = & tan (α_{1} β_{1} δ) P_{1} + tan (α_{2} β_{2} δ) P_{2} + tan (α_{3} β_{3} δ) P_{3} \\ - tan (α_{3} β_{3} δ) P_{3} tan (α_{2} β_{2} δ) Q_{2} tan (α_{1} β_{1} δ) P_{1}, \end{array}$ and $\begin{matrix} P_{j} = P (ε_{j}, μ_{j}) and Q_{j} = Q (ε_{j}, μ_{j}), j = 1, 2, 3 . \end{matrix}$
Proof.
We apply recursively, the transfer formula (8). Taking into account the continuity through the interfaces between the thin layers we get the requested result. □

6.2.1. First order approximation

Theorem 6.2.
A first order approximation of the impedance boundary condition for three thin layers is given by $\begin{matrix} (27) & E_{T ∣ Γ} ≃ - i k δ (T_{1} (μ) (β) + \frac{1}{k^{2}} T_{1} (\frac{1}{ε}) (β) \nabla_{Γ} ({div}_{Γ})) {(n \times H)}_{∣ Γ} . \end{matrix}$
Proof.
An approximation of (26) at the first order with respect to δ is $\begin{matrix} {\hat{E}}_{T} (ξ, 0) ≃ - (α_{1} β_{1} δ P_{1} + α_{2} β_{2} δ P_{2} + α_{3} β_{3} δ P_{3}) (n \times \hat{H}) (ξ, 0) . \end{matrix}$ Replacing $α_{j} P_{j}$ with $i k (μ_{j} - \frac{1}{k^{2} ε_{j}} ξ (ξ \cdot))$ , we get $\begin{matrix} {\hat{E}}_{T} (ξ, 0) ≃ - δ i k (T_{1} (μ) (β) - \frac{1}{k^{2}} T_{1} (\frac{1}{ε}) (β) ξ (ξ \cdot)) (n \times \hat{H}) (ξ, 0), \end{matrix}$ with $T_{1} (μ) (β) = \sum_{l = 1}^{l = 3} μ_{l} β_{l}$ and $T_{1} (\frac{1}{ε}) (β) = \sum_{l = 1}^{l = 3} \frac{1}{ε_{l}} β_{l}$ .

Hence, by the inverse Fourier transform, we find the first order approximation of the impedance boundary condition $\begin{matrix} E_{T ∣ Γ} ≃ - i k δ (T_{1} (μ) (β) + \frac{1}{k^{2}} T_{1} (\frac{1}{ε}) (β) \nabla_{Γ} ({div}_{Γ})) {(n \times H)}_{∣ Γ} . \end{matrix}$ □
Remark 6.3.

This first order approximation is that given in [7] for multi-layers.

The result in Theorem 6.2 generalizes the one obtained in Theorem 5.1 for one single layer. If the layers are not contrasted ( $ε_{j} = ε, μ_{j} = μ$ for $j = 1, 2, 3$ ) we have the same approximation.

6.2.2. Third order approximation

Theorem 6.4.

A third order approximate impedance boundary condition for three thin layers is given by $\begin{matrix} (28) & \begin{matrix} E_{T ∣ Γ} ≃ & - i k δ (T_{1} (μ) (β) {(1 - δ^{2} r_{1} - \frac{δ^{2}}{3} r_{2} \vec{Δ_{Γ}})}^{- 1} \\ - \frac{1}{k^{2}} T_{1} (\frac{1}{ε}) (β) \nabla_{Γ} {(1 - δ^{2} s - \frac{δ^{2}}{3} r_{3} Δ_{Γ})}^{- 1} {div}_{Γ}) {(n \times H)}_{∣ Γ}, \end{matrix} \end{matrix}$ with $\begin{matrix} r_{1} = k^{2} \frac{T_{3} (μ, μ, ε) (β, β, β)}{T_{1} (μ) (β)}, r_{2} = \frac{T_{3} (μ, μ, \frac{1}{μ}) (β, β, β)}{T_{1} (μ) (β)}, r_{3} = \frac{T_{3} (\frac{1}{ε}, \frac{1}{ε}, ε) (β, β, β)}{T_{1} (\frac{1}{ε}) (β)}, \end{matrix}$ and $\begin{matrix} s = \frac{1}{T_{1} (\frac{1}{ε}) (β)} (T_{3} (\frac{1}{ε}, μ, ε) + T_{3} (μ, \frac{1}{ε}, ε) - T_{3} (μ, μ, \frac{1}{μ})) (β, β, β) . \end{matrix}$

Proof.

An approximation of (26) at order 3 with respect to δ is $\begin{matrix} (29) & G {\hat{E}}_{T} (ξ, 0) ≃ F (n \times \hat{H}) (ξ, 0), \end{matrix}$ where $\begin{array}{l} G = & 1 - δ^{2} β_{3} β_{2} (α_{3} P_{3}) (α_{2} Q_{2}) - δ^{2} β_{3} β_{1} (α_{3} P_{3}) (α_{1} Q_{1}) \\ - δ^{2} β_{2} β_{1} (α_{2} P_{2}) (α_{1} Q_{1}), \end{array}$ and $\begin{array}{l} F = & β_{1} (α_{1} P_{1}) + β_{2} (α_{2} P_{2}) + β_{3} (α_{3} P_{3}) + \frac{δ^{2}}{3} β_{1}^{3} α_{1}^{2} (α_{1} P_{1}) + \frac{δ^{2}}{3} β_{2}^{3} α_{2}^{2} (α_{2} P_{2}) \\ + \frac{δ^{2}}{3} β_{3}^{3} α_{3}^{2} (α_{3} P_{3}) - δ^{2} β_{3} β_{2} β_{1} (α_{3} P_{3}) (α_{2} Q_{2}) (α_{1} P_{1}), \end{array}$ and $\begin{matrix} P_{j} = P (ε_{j}, μ_{j}) and Q_{j} = Q (ε_{j}, μ_{j}), j = 1, 2, 3 . \end{matrix}$

By replacing the operators $P_{j}$ and $Q_{j}$ with their expressions and using the notations introduced previously, we can write the impedance condition (29) as $\begin{array}{l} {\hat{E}}_{T} (ξ, 0) ≃ & - i k δ (T_{1} (μ) (β) (1 + \frac{δ^{2}}{3} r_{1} - \frac{δ^{2}}{3} r_{2} | ξ |^{2} I) \\ - \frac{1}{k^{2}} ξ T_{1} (\frac{1}{ε}) (β) (1 + \frac{δ^{2}}{3} s - \frac{δ^{2}}{3} r_{3} | ξ |^{2}) (ξ \cdot)) (n \times \hat{H}) (ξ, 0), \end{array}$ with $\begin{matrix} r_{1} = k^{2} \frac{T_{3} (μ, μ, ε) (β, β, β)}{T_{1} (μ) (β)}, r_{2} = \frac{T_{3} (μ, μ, \frac{1}{μ}) (β, β, β)}{T_{1} (μ) (β)}, r_{3} = \frac{T_{3} (\frac{1}{ε}, \frac{1}{ε}, ε) (β, β, β)}{T_{1} (\frac{1}{ε}) (β)}, \end{matrix}$ and $\begin{matrix} s = \frac{1}{T_{1} (\frac{1}{ε}) (β)} (T_{3} (\frac{1}{ε}, μ, ε) + T_{3} (μ, \frac{1}{ε}, ε) - T_{3} (μ, μ, \frac{1}{μ})) (β, β, β) . \end{matrix}$ We can verify that $\begin{matrix} [ξ (ξ \cdot) + (ξ \times n) ((ξ \times n) \cdot)] = | ξ |^{2} I, (ξ \cdot) ξ = | ξ |^{2} . \end{matrix}$ As we have at second order with respect to δ $\begin{matrix} (1 + \frac{δ^{2}}{3} r_{1} - \frac{δ^{2}}{3} r_{2} | ξ |^{2} I) ≃ {(1 - \frac{δ^{2}}{3} r_{1} + \frac{δ^{2}}{3} r_{2} | ξ |^{2} I)}^{- 1}, \end{matrix}$ and $\begin{matrix} (1 + \frac{δ^{2}}{3} s - \frac{δ^{2}}{3} r_{3} | ξ |^{2}) ≃ {(1 - \frac{δ^{2}}{3} s + \frac{δ^{2}}{3} r_{3} | ξ |^{2})}^{- 1}, \end{matrix}$ the approximate impedance condition of order three can be written $\begin{array}{l} {\hat{E}}_{T} (ξ, 0) ≃ & - i k δ (T_{1} (μ) (β) {(1 - δ^{2} r_{1} + \frac{δ^{2}}{3} r_{2} | ξ |^{2} I)}^{- 1} \\ - \frac{1}{k^{2}} ξ T_{1} (\frac{1}{ε}) (β) {(1 - δ^{2} s + \frac{δ^{2}}{3} r_{3} | ξ |^{2})}^{- 1} (ξ \cdot)) (n \times \hat{H}) (ξ, 0) . \end{array}$ By taking the inverse Fourier transform we get the following third order approximate impedance boundary condition: $\begin{array}{l} E_{T ∣ Γ} ≃ & - i k δ (T_{1} (μ) (β) {(1 - δ^{2} r_{1} - \frac{δ^{2}}{3} r_{2} \vec{Δ_{Γ}})}^{- 1} \\ - \frac{1}{k^{2}} T_{1} (\frac{1}{ε}) (β) \nabla_{Γ} {(1 - δ^{2} s - \frac{δ^{2}}{3} r_{3} Δ_{Γ})}^{- 1} {div}_{Γ}) {(n \times H)}_{∣ Γ} . \end{array}$ □

Remark 6.5.

For N contrasted layers ( $N ⩾ 4$ ), of characteristics $ε_{j}$ , $μ_{j}$ , $j = 1, 2, \dots, N$ , the approximate impedance formula still holds.

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