# Construction of transparent conditions for electromagnetic waveguides

Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, Sonia Fliss, Aurélien Parigaux

► To cite this version:

Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, Sonia Fliss, Aurélien Parigaux. Construction of transparent conditions for electromagnetic waveguides. WAVES 2024 - 16th International Conference on Mathematical and Numerical Aspects of Wave Propagation, Jun 2024, Berlin, Germany. hal-05032178

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Submitted on 13 Apr 2025

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Distributed under a Creative Commons Attribution 4.0 International License
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# Construction of transparent conditions for electromagnetic waveguides

A.-S. Bonnet-Ben Dhia¹

L. Chesnel²

S. Fliss¹

A. Parigaux¹

¹POEMS (CNRS - Inria - ENSTA Paris)

²IDEFIX (Inria - EDF - ENSTA Paris)

July 1, 2024.

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# Introduction

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* We consider the Maxwell's equations in time harmonic regime in a waveguide $\Omega \subset \mathbb{R}^3$ which is unbounded in one direction:

$\mathrm{Curl} \mathbf{E} - i\omega\mu\mathbf{H} = 0$, $\mathrm{Curl} \mathbf{H} + i\omega\varepsilon\mathbf{E} = \mathbf{J}$, $\mathrm{div}(\varepsilon\mathbf{E}) = 0$, $\mathrm{div}(\mu\mathbf{H}) = 0$, in $\Omega$,

$\mathbf{v} \times \mathbf{E} = 0$, $\mathbf{v} \cdot \mu\mathbf{H} = 0$, on $\partial\Omega$.

## Hypotheses:

* $\varepsilon, \mu \ge C > 0$, $\mathrm{supp} \mathbf{J} \subset \Omega_0$, $\mathbf{J} \in L^2(\Omega_0)$ and $\mathrm{div} \mathbf{J} = 0$.

* In $\Omega_{\mathrm{ext}}$, $\varepsilon$ and $\mu$ are independent of the axial coordinate $z$.

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# Introduction

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## Objectives:

* Computation of the **outgoing solution**.

## Difficulties:

* Unbounded domain and non-decreasing solution at infinity.

* Maxwell's equations in $\mathbb{R}^3$.

## Method:

* Reduction of the problem to a **bounded** domain.

* Construction of **transparent conditions** to avoid spurious reflections.
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# Introduction

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To simplify the presentation, we work in a semi-infinite waveguide, and we denote
$\Theta$ the section of $\Omega_{ext}$.

We are interested in the second order problem for the electric field.

## Mode of the guide:

* Particular solution in $\Omega_{ext}$ of the form $E(x, y, z) = \mathcal{E}(x, y)e^{i\beta z}$.

## Two cases:

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**Homogeneous:** $\varepsilon, \mu$ constant in $\Theta$.

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**Heterogeneous:** $\varepsilon(x, y), \mu(x, y)$ vary in the section. ⚙

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# Introduction

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Two main methods available for the
**homogeneous case:**

## Perfectly Matched Layers (PMLs)

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Easy to implement.

## Modal decomposition

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More efficient near the cutoff.

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And for the **heterogeneous case**?

## Perfectly Matched Layers (PMLs)

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Work only in absence of inverse
modes.

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Inverse modes can appear for
high contrast of $\varepsilon$ and/or $\mu$.

## Modal decomposition

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Work even in presence of inverse
modes.

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Non self-adjoint transverse operator.

S. Kim. “Analysis of the non-reflecting boundary condition for the time-harmonic electromagnetic wave propagation in waveguides”. In: *J. Math. Anal. App.* vol. 453 (2017), pp. 82–103

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Table of Contents

1. Modes of the guide

* Homogeneous case

* Heterogeneous case

2. An **Electric-to-Magnetic** operator with overlap for the **homogeneous** case

* Definition of the operator

* Truncation and error estimate

* Numerical results

3. A **Poynting-to-Magnetic** operator for the **heterogeneous** case

* Definition of the operator

* Numerical results

4. Ongoing work

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# Table of Contents

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## 1 Modes of the guide

● Homogeneous case

● Heterogeneous case

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## 2 An Electric-to-Magnetic operator with overlap for the homogeneous case

● Definition of the operator

● Truncation and error estimate

● Numerical results

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## 3 A Poynting-to-Magnetic operator for the heterogeneous case

● Definition of the operator

● Numerical results

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## 4 Ongoing work

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Homogeneous case

Reminder: $\mathbf{E}(x, y, z) = \mathcal{E}(x, y)e^{i\beta z}$.

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$$ \Delta\varphi_n + (\omega^2 - \beta^2)\varphi_n = 0 $$

$$ \partial_v \varphi_n^{\text{TE}} = 0 / \varphi_n^{\text{TM}} = 0 $$

The modes form a countable family $(\beta_n, \mathcal{E}_n, \mathcal{H}_n)$, composed of TE, TM (and possibly TEM) modes.

The TE and TM modes can be expressed with the eigenelements of the
Neumann/Dirichlet scalar Laplacian.

Completeness and orthogonality result

The family of transverse fields $(\mathbf{v} \times \mathcal{E}_n)_n$ form a hilbertian basis of $(L^2(\Theta))^2 \times \{0\}$.

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Homogeneous case

Reminder: $\mathbf{E}(x, y, z) = \mathcal{E}(x, y)e^{i\beta z}$.

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Heterogeneous case

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$$
\mathbf{E} = \mathcal{E}(x,y) e^{i\beta z} = \begin{pmatrix} \mathbf{e} \\ e_z \end{pmatrix} e^{i\beta z}, \quad \mathbf{H} = \mathcal{H}(x,y) e^{i\beta z} = \begin{pmatrix} \mathbf{h} \\ h_z \end{pmatrix} e^{i\beta z}
$$

● A first formulation

$$
\forall \mathbf{e}'_l, \int_{\Theta} \mu^{-1} \operatorname{curl}(\mathbf{e}) \operatorname{curl}(\mu \mathbf{e}'_l) + \int_{\Theta} \varepsilon^{-1} \operatorname{div}(\varepsilon \mathbf{e}) \operatorname{div}(\mathbf{e}') - \omega^2 \int_{\Theta} \varepsilon \mu \mathbf{e} \cdot \mathbf{e}' = -\beta^2 \int_{\Theta} \mathbf{e} \cdot \mathbf{e}'.
$$

● A second formulation

$$
\forall \mathbf{e}'_l, \mathbf{h}'_l, \int_{\Theta} \varepsilon^{-1} \mathrm{curl} \mathbf{h} \mathrm{curl} \mathbf{h}'_l + \int_{\Theta} \mu^{-1} \mathrm{curl} \mathbf{e} \mathrm{curl} \mathbf{e}'_l - \omega^2 \int_{\Theta} (\mu \mathbf{h} \cdot \mathbf{h}'_l + \varepsilon \mathbf{e} \cdot \mathbf{e}'_l) = \beta \omega \int_{\Theta} \mathbf{e}^{\perp} \cdot \mathbf{h}'_l + \mathbf{h} \cdot \mathbf{e}'_l^{\perp}.
$$

Biorthogonality relations

$$
\text{If } \beta \neq \bar{\beta}', \quad (\mathbf{v} \times \mathcal{E}, \mathcal{H}')_{\Theta} + (\mathcal{H}, \mathbf{v} \times \mathcal{E}')_{\Theta} = 0.
$$

● M. Halla and P. Monk used a third formulation to prove discreteness.

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Heterogeneous case

Reminder: $E(x, y, z) = \mathcal{E}(x, y)e^{i\beta z}$.

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Finite element discretization of the second formulation.

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Numerical modes

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Definition of the operator

$\mathbb{H}(\text{Curl}, \Omega_L) := \{ \mathbf{\Phi} \in \mathbb{L}^2(\Omega_L) : \text{Curl} \mathbf{\Phi} \in \mathbb{L}^2(\Omega_L), \mathbf{v} \times \mathbf{\Phi} = \mathbf{0} \text{ on } \Gamma_L \}$. We suppose that $\bar{\omega}$ is such that $\beta_n \neq 0$.

Let $\mathbf{g} = (\mathbf{v} \times \mathbf{\Phi})|_{\Sigma_0}$ with $\mathbf{\Phi} \in \mathbb{H}(\text{Curl}, \Omega_L)$. We consider $\mathbf{E}$ the outgoing solution of:

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➤ From this, we can define the **Electric-to-Magnetic (EtM) operator** with an overlap $L \ge 0$:

$$ \Lambda_{\text{EtM}}^{L}(\mathbf{g}) := \mu^{-1}(\mathbf{v} \times \text{Curl} \mathbf{E})|_{\Sigma_L} \\ (= i\bar{\omega}\mathbf{v} \times \mathbf{H}|_{\Sigma_L}) $$

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A modal expression of the EtM operator

Let $g = \sum_{n=1}^{+\infty} a_n (\mathbf{v} \times \mathcal{C}_n)$ on $\Sigma_0$. Due to orthogonality relations:

$$a_n = (g, \mathbf{v} \times \mathcal{C}_n)_{\Sigma_0}.$$

We deduce the modal decomposition of the fields **E** and **H** solution of the Maxwell Problem in $\Theta \times (0; +\infty)$ with $\mathbf{v} \times \mathbf{E} = g$ on $\Sigma_0$ as follows:

$$\mathbf{E} = \sum_{n=1}^{+\infty} a_n e^{i\beta_n z} \mathcal{C}_n$$

$$\mathbf{H} = \sum_{n=1}^{+\infty} a_n e^{i\beta_n z} \mathcal{H}_n$$

So we have:

$$\Lambda_{\text{EtM}}^{L}(g) = i\omega(\mathbf{v} \times \mathbf{H})|_{\Sigma_L} = \sum_{n=1}^{+\infty} a_n i\omega e^{i\beta_n L}(\mathbf{v} \times \mathcal{H}_n).$$

* $L = 0$: Kim's operator.

* $L > 0$: Note that $e^{i\beta_n L} \underset{n \to \infty}{\longrightarrow} 0$ exponentially.

⇒ $\Lambda_{\text{EtM}}^L$ is a compact operator in appropriate space.

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Bounded domain problem

► The Maxwell's problem defined in Ω is equivalent to:

$$
\left\{
\begin{array}{l}
\text{Find } \mathbf{E} \in \mathbf{H}(\text{Curl}, \Omega_L), \text{ such as:} \\
\forall \mathbf{\Phi} \in \mathbf{H}(\text{Curl}, \Omega_L), \quad a(\mathbf{E}, \mathbf{\Phi}) = i\omega \int_{\Omega_L} \mathbf{J} \cdot \overline{\mathbf{\Phi}},
\end{array}
\right.
\qquad (\mathbf{P})
$$

where $a(\mathbf{E}, \mathbf{\Phi}) = \int_{\Omega_L} \mu^{-1} \mathbf{Curl} \mathbf{E} \cdot \mathbf{Curl} \overline{\mathbf{\Phi}} - \omega^2 \int_{\Omega_L} \varepsilon \mathbf{E} \cdot \overline{\mathbf{\Phi}} + \langle \Lambda_{\text{EtM}}^L (\mathbf{v} \times \mathbf{E})|_{\Sigma_0} \rangle |\overline{\mathbf{\Phi}}|_{\Sigma_L}$, and
$\mathbf{H}(\mathbf{Curl}, \Omega_L) = \{\mathbf{\Phi} \in L^2(\Omega_L) : \mathbf{Curl} \mathbf{\Phi} \in L^2(\Omega_L), \mathbf{v} \times \mathbf{\Phi} = 0 \text{ on } \Gamma_L\}$.

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The space $\mathbf{H}(\text{Curl}, \Omega_L)$ is not compactly embedded in $L^2(\Omega_L)$.

Theorem

Except for a countable set of real frequencies, there *exists a unique* solution to (P),
depending continuously on the source term **J**.

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Key idea of the proof: work in an *adapted divergence free subspace*.

$$
X := \left\{ \mathbf{\Psi} \in \mathbf{H}(\mathbf{Curl}, \Omega_L) \mid \forall \varphi \in H_{0,\Gamma_L}^1(\Omega_L), \omega^2 \int_{\Omega_L} \varepsilon \mathbf{\Psi} \cdot \nabla \varphi - i\omega \langle \Lambda_{\text{EtM}}^L (\mathbf{v} \times \mathbf{\Psi})|_{\Sigma_0} \rangle |\nabla_T \varphi|_{\Sigma_L} = 0 \right\}.
$$

We can show that the injection $X \subset L^2(\Omega_L)$ is compact.

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Truncation and error estimate

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For the numerics, it is necessary to truncate the EtM.

Let $N \ge 1$. In the problem, replace $\Lambda_{\text{EtM}}^{L}$ by $\Lambda_{\text{EtM}}^{L,N}$ with:

$$ \Lambda_{\text{EtM}}^{L,N} : \mathbf{v} \times \mathbf{E}|_{\Sigma_0} = \sum_{n=1}^{+\infty} a_n (\mathbf{v} \times \mathcal{E}_n) \rightarrow \sum_{n=1}^{N} a_n i \bar{\omega} e^{i \beta_n L} (\mathbf{v} \times \mathcal{H}_n). $$

Find $\mathbf{E}_N \in \mathbb{H}(\text{Curl}, \Omega_L)$, such as for all $\mathbf{\Phi} \in \mathbb{H}(\text{Curl}, \Omega_L)$:

$$ \int_{\Omega_L} \mu^{-1} \mathrm{Curl} \mathbf{E}_N \cdot \mathrm{Curl} \bar{\mathbf{\Phi}} - \bar{\omega}^2 \int_{\Omega_L} \varepsilon \mathbf{E}_N \cdot \bar{\mathbf{\Phi}} + i \bar{\omega} \langle \Lambda_{\text{EtM}}^{L,N} (\mathbf{v} \times \mathbf{E}_N|_{\Sigma_0}) | \bar{\mathbf{\Phi}} \rangle_{\Sigma_L} = \int_{\Omega_L} \mathbf{J} \cdot \bar{\mathbf{\Phi}}. $$

Theorem

If (P) is well-posed, then for $N$ large enough there exists a unique solution to the previous truncated problem. Moreover, we have the estimate:

$$ \exists C, \alpha_N > 0, \quad \|\mathbf{E} - \mathbf{E}_N\|_{\mathbb{H}(\text{Curl}, \Omega_L)} \le Ce^{-\alpha_N L} \|\mathbf{J}\|_{\mathbb{L}^2(\Omega_0)}, \\ \alpha_N \text{ is the decay rate of the } (N+1)^{\text{th}} \text{ mode (evanescent).} $$
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# Numerical results

## Finite elements: first family of Nédélec elements of order 2.

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XLIFE++

## Real part of $E_y$ for a propagating TE mode:

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Without obstacle.

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With obstacle.

If $\omega$ is such that **only one mode is propagating**, then we observe that the energy conservation relation $|R|^2 + |T|^2 = 1$ is well approximated numerically.

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Convergence with the length of the overlap

Use of PMLs to define a reference solution.

<figure><img src="image_126.png" /><figcaption>Figure: $\log |\mathbf{E} - \mathbf{E}_h^{L,N}|$ with respect to $L$.</figcaption></figure>

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Other geometries

<figure><img src="image_133.png" /><figcaption>Figure: Real part of the component *E*<sub>y</sub> of the total electric field.</figcaption></figure>

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# Extension to the heterogeneous case?

## Homogeneous case

* The modes form a Hilbertian basis

$$ \mathbf{E} = \sum_{n=1}^{+\infty} a_n e^{i\beta_n z} \mathcal{E}_n . $$

* Orthogonality relations

$$ a_n = (\mathbf{v} \times \mathbf{E} | \mathbf{v} \times \mathcal{E}_n)_{\Sigma_0} . $$

## Heterogeneous case

* Loss of the completeness of the modes.

* Loss of the orthogonality.

* But *biorthogonality* can be exploited.

We want to adapt the method developed in the elasticity context:

A. Tonnoir. "Conditions transparentes pour la diffraction d'ondes en milieu élastique anisotrope". PhD thesis. ENSTA ParisTech, 2015

V. Baronian, A.-S. Bonnet-Ben Dhia, S. Fliss, and A. Tonnoir. "Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides". In: *Wave Motion* 64 (2016), pp. 13–33
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# Table of Contents

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## 1 Modes of the guide

* Homogeneous case

* Heterogeneous case

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## 2 An Electric-to-Magnetic operator with overlap for the homogeneous case

* Definition of the operator

* Truncation and error estimate

* Numerical results

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## 3 A Poynting-to-Magnetic operator for the heterogeneous case

* Definition of the operator

* Numerical results

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## 4 Ongoing work

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Definition of the operator

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According to Kondratiev's theory, the above problem is well posed and we have:

$$
\mathbf{E} = \sum_{n=1}^{N_{\text{prop}}} a_n e^{i\beta_n z} \mathcal{E}_n + \mathbf{E}_{\text{ev}} \text{ in } \Omega_{\text{ext}}, \quad \exists \alpha_{\text{ev}} > 0, e^{\alpha_{\text{ev}} z} \mathbf{E}_{\text{ev}} \in \mathbb{L}^2(\Omega)
$$

$$
a_n = (\mathbf{v} \times \mathbf{E}, \mathcal{H}_n)_{\Sigma_0} + \frac{1}{i\omega} (\mu^{-1} \mathbf{Curl} \mathbf{E}, \mathbf{v} \times \mathcal{E}_n)_{\Sigma_0}. \quad \text{(Biorthogonality)}
$$

B. Plamenevskii and A. Poretskii. "The Maxwell system in waveguides with several
cylindrical outlets to infinity and nonhomogeneous anisotropic filling". In: St.
Petersburg Mathematical Journal 29 (2018), pp. 289–314

$(\mu^{-1} \mathbf{Curl} \mathbf{E}, \mathbf{v} \times \mathcal{E}_n)_{\Sigma_0}$ has no meaning if we only have $\mathbf{E} \in \mathbb{H}(\mathbf{Curl}, \Omega_L)$.

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Definition of the operator

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Introduce $\bar{E}_n \in H(\text{Curl}, \Omega_r)$ a lifting such that $\bar{E}_n = \begin{cases} \mathcal{C}_n & \text{on } \Sigma_0 \\ 0 & \text{on } \Sigma_r \end{cases}$.
$$ (\text{Curl}(\mu^{-1} \text{Curl E}), \bar{E}_n)_{\Omega_r} = (\mu^{-1} \text{Curl E}, \text{Curl} \bar{E}_n)_{\Omega_r} - (\mu^{-1} \text{Curl E}, \mathbf{v} \times \bar{E}_n)_{\partial \Omega_r} $$
$$ (\mu^{-1} \text{Curl E}, \mathbf{v} \times \mathcal{C}_n)_{\Sigma_r} = (\mu^{-1} \text{Curl E}, \text{Curl} \bar{E}_n)_{\Omega_r} - \bar{\omega}^2 (\varepsilon \mathbf{E}, \bar{E}_n)_{\Omega_r} $$

We obtain the approximated PtM operator for $\mathbf{E} \in H(\text{Curl}, \Omega_r)$:

$$ \begin{aligned} \Lambda_{\text{PtM}}^{L,N}(\mathbf{E}) &= \sum_{n=1}^{N_{\text{prop}}} a_n i \bar{\omega}(\mathbf{v} \times \mathcal{H}_n)|_{\Sigma_L} \\ &= \sum_{n=1}^{N_{\text{prop}}} \left( (\mathbf{v} \times \mathbf{E}, \mathcal{H}_n)_{\Sigma_0} + \frac{1}{i\bar{\omega}} (\mu^{-1} \text{Curl E}, \text{Curl} \bar{E}_n)_{\Omega_r} + i\bar{\omega}(\varepsilon \mathbf{E}, \bar{E}_n)_{\Omega_r} \right) i\bar{\omega}(\mathbf{v} \times \mathcal{H}_n)|_{\Sigma_L}. \end{aligned} $$

We neglected $\mathbf{E}_{ev} \Rightarrow$ approximation to $\mathcal{O}(e^{-\alpha_{ev}L})$.

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Numerical results

<figure><img src="image_164.png" /><figcaption>Figure: Real part of $E_y$ for the diffraction by an obstacle.</figcaption></figure>

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Numerical results

<figure><img src="image_171.png" /><figcaption>Figure: $E_x$ and $E_y$ of a TEM mode in a waveguide with a holed section.</figcaption></figure>

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Numerical results

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$\mu = 1$

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<figure><img src="image_182.png" /><figcaption>Figure: $E_x$, $E_y$ and $E_z$ of a **scattering field** in a **heterogeneous case**.</figcaption></figure>

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Table of Contents

1. Modes of the guide
   * Homogeneous case
   * Heterogeneous case

2. An Electric-to-Magnetic operator with overlap for the homogeneous case
   * Definition of the operator
   * Truncation and error estimate
   * Numerical results

3. A Poynting-to-Magnetic operator for the heterogeneous case
   * Definition of the operator
   * Numerical results

4. Ongoing work

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Ongoing work

* Find parameters such that there are inverse modes and test the PtM in cases where PMLs should not work.

* Prove the convergence of the method in the heterogeneous case.

* Test the benefits of the overlap in an iterative solver approach.

* Prove the convergence of the finite element discretization of our methods.

Thank you for your attention !

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References I

[1] V. Baronian, A.-S. Bonnet-Ben Dhia, S. Fliss, and A. Tonnnoir. “Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides”. In: *Wave Motion* 64 (2016), pp. 13–33.

[2] R. Dautray and J.-L. Lions. “Analyse mathématique et calcul numérique pour les sciences et les techniques”. In: *Collection du Commissariat a l'Energie Atomique. Serie Scientifique* tome 2 (1985), pp. 237–317.

[3] A.-S. Bonnet-Ben Dhia, L. Chesnel, and V. Pagneux. “Trapped modes and reflectionless modes as eigenfunctions of the same spectral problem”. In: *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences* 474 (2018), p. 20180050.

[4] S. Fliss. “Wave propagation in periodic media: mathematical analysis and numerical simulation”. PhD thesis. Université Paris Sud (Paris 11), 2019.

[5] M. Halla and P. Monk. *On the analysis of waveguide modes in an electromagnetic transmission line*. 2023. arXiv: 2302.11994 [math.AP].

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[6] S. Kim. "Analysis of the non-reflecting boundary condition for the time-harmonic electromagnetic wave propagation in waveguides".
In: *J. Math. Anal. App.* vol. 453 (2017), pp. 82–103.

[7] D. Pauly and N. Skrepek.
"A compactness result for the div-curl system with inhomogeneous mixed boundary conditions for bounded Lipschitz domains and some applications".
In: *ANNALI DELL'UNIVERSITA'DI FERRARA* (2022), pp. 1–15.

[8] B. Plamenevskii and A. Poretskii.
"The Maxwell system in waveguides with several cylindrical outlets to infinity and nonhomogeneous anisotropic filling".
In: *St. Petersburg Mathematical Journal* 29 (2018), pp. 289–314.

[9] B. Plamenevskii, A. Poretskii, and O. Sarafanov. "On a Method of Approximate Computing of Scattering Matrices for Electromagnetic Waveguides".
In: *Doklady Physics*. Vol. 63. 2018, pp. 414–417.

[10] A. Tonnoir. "Conditions transparentes pour la diffraction d'ondes en milieu élastique anisotrope". PhD thesis. ENSTA ParisTech, 2015.
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# Modes in the homogeneous case

Reminder: $\mathbf{E}(x, y, z) = \bar{\mathbf{E}}(x, y)e^{i\beta z}$.

Set $\bar{\mathbf{E}} = (\mathbf{e}, e_z)^t$, $\nabla = (\partial x, \partial y)^t$ and $\nabla^\perp = (\partial y, -\partial x)^t$.

## Two families of modes:

### Transverse Electric (TE)

$$ 
\begin{cases}
\mathbf{e} = \nabla^{\perp} \varphi_n^{\text{TE}}, \\
e_z = 0,
\end{cases}
$$

where

$$ 
\begin{cases}
\Delta\varphi_n^{\text{TE}} + (\omega^2 - \beta_n^{\text{TE}2}) \varphi_n^{\text{TE}} = 0 & \text{in } \Theta \\
\frac{\partial \varphi_n^{\text{TE}}}{\partial \mathbf{v}} = 0 & \text{on } \partial\Theta.
\end{cases}
$$

### Transverse Magnetic (TM)

$$ 
\begin{cases}
\mathbf{e} = \nabla \varphi_n^{\text{TM}}, \\
e_z = \frac{\omega^2 - \beta_n^{\text{TM}2}}{i\beta_n^{\text{TM}}} \varphi_n^{\text{TM}},
\end{cases}
$$

where

$$ 
\begin{cases}
\Delta\varphi_n^{\text{TM}} + (\omega^2 - \beta_n^{\text{TM}2}) \varphi_n^{\text{TM}} = 0 & \text{in } \Theta \\
\varphi_n^{\text{TM}} = 0 & \text{on } \partial\Theta.
\end{cases}
$$

The family $(\nabla\varphi_n^{\text{TE}}, \nabla^\perp\varphi_n^{\text{TM}})_n$ forms a hilbertian basis of $(L^2(\Theta))^2$.

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# A modal expression of the EtM operator

Let $g = \sum_{n=1}^{+\infty} a_n \nabla \varphi_n^{\text{TE}} + b_n \nabla^{\perp} \varphi_n^{\text{TM}}$ on $\Sigma_0$.

Due to orthogonality relations:

$$a_n = (g, \nabla \varphi_n^{\text{TE}})_{\Sigma_0}, \quad b_n = (g, \nabla^{\perp} \varphi_n^{\text{TM}})_{\Sigma_0}.$$

We suppose that $\omega$ is such that $\beta_n^{\text{TE}}, \beta_n^{\text{TM}} \neq 0$.

We deduce the modal decomposition of the fields $\mathbf{E} = (\mathbf{e}, e_z)^t$ and $\mathbf{H} = (\mathbf{h}, h_z)^t$ solution of the Maxwell Problem in $\Theta \times (0; +\infty)$ with $\mathbf{v} \times \mathbf{E} = g$ on $\Sigma_0$ as follows:

$$\mathbf{e} = \sum_{n=1}^{+\infty} a_n e^{i\beta_n^{\text{TE}}z} \nabla^{\perp} \varphi_n^{\text{TE}} - b_n e^{i\beta_n^{\text{TM}}z} \nabla^{\perp} \varphi_n^{\text{TM}},$$

$$\mathbf{e}_z = \sum_{n=1}^{+\infty} -b_n e^{i\beta_n^{\text{TM}}z} \frac{\omega^2 - \beta_n^{\text{TM}2}}{i\beta_n^{\text{TM}}} \varphi_n^{\text{TM}},$$

$$\mathbf{h} = \sum_{n=1}^{+\infty} a_n \frac{\beta_n^{\text{TE}}}{\omega} e^{i\beta_n^{\text{TE}}z} \nabla \varphi_n^{\text{TE}} + b_n \frac{\omega}{\beta_n^{\text{TM}}} e^{i\beta_n^{\text{TM}}z} \nabla^{\perp} \varphi_n^{\text{TM}},$$

$$\mathbf{h}_z = \sum_{n=1}^{+\infty} a_n e^{i\beta_n^{\text{TM}}z} \frac{\omega^2 - \beta_n^{\text{TE}2}}{i\omega} \varphi_n^{\text{TE}}.$$

So we have:

$$\Lambda_{\text{EtM}}^{L}(\mathbf{e}) = i\omega(\mathbf{v} \times \mathbf{H})|_{\Sigma_L} = \sum_{n=1}^{+\infty} -a_n i\beta_n^{\text{TE}} e^{i\beta_n^{\text{TE}}L} \nabla^{\perp} \varphi_n^{\text{TE}} + b_n \frac{i\omega^2}{\beta_n^{\text{TM}}} e^{i\beta_n^{\text{TM}}L} \nabla^{\perp} \varphi_n^{\text{TM}}.$$

Note that $e^{i\beta_n^{\text{TE},LM}}$ — 0 exponentially.

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