# Mathematical and numerical analysis of the modes of a heterogeneous electromagnetic waveguide.

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. Mathematical and numerical analysis of the modes of a heterogeneous electromagnetic waveguide.. WAVES 2025 - Conference on Mathematics of Wave Phenomena, Feb 2025, Karlsruhe, Germany. hal-05032193

HAL Id: hal-05032193

https://hal.science/hal-05032193v1

Submitted on 13 Apr 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

# Mathematical and numerical analysis of the modes of a heterogeneous electromagnetic waveguide

A.-S. Bonnet-Ben Dhia¹

L. Chesnel²

S. Fliss¹

A. Parigaux¹

¹POEMS (CNRS - Inria - ENSTA Paris)

²IDEFIX (Inria - EDF - ENSTA Paris)

Conference on Mathematics of Wave Phenomena
February 24, 2025.
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

# Introduction

<figure><img src="image_9.png" /></figure>

* We consider Maxwell's equations in time harmonic regime in infinite waveguide.

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

## Subject of my PhD Thesis:

* Construction of transparent conditions.

## Two main methods

PML (L) Nicolet et al. 2008) and EtM (L) Kim 2017).

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

* For analysis and/or implementation, modes play a key role.

<figure><img src="image_12.png" /></figure>
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

# Introduction

## Modes of the guide:

* Particular solutions in $\Omega_{\infty} = S \times \mathbb{R}$ of the form $E(x, y, z) = \mathcal{E}(x, y)e^{i\beta z}$.

<figure><img src="image_16.png" /></figure>

$$ \text{CurlE} - i\partial\mu\mathbf{H} = 0, \quad \text{CurlH} + i\partial\varepsilon\mathbf{E} = 0, \quad \text{div}(\varepsilon\mathbf{E}) = 0, \quad \text{div}(\mu\mathbf{H}) = 0, \quad \text{in } \Omega_{\infty} \\ \nu \times \mathbf{E} = 0, \quad \nu \cdot \mu\mathbf{H} = 0, \quad \text{on } \partial\Omega_{\infty} $$

**Hyphthesis:** $\varepsilon, \mu \in L^{\infty}(S)$ such that $\varepsilon, \mu \ge C > 0$.

## Two situations:

<figure><img src="image_17.png" /></figure>

**Homogeneous case:** $\varepsilon, \mu$ constant in $S$.

<figure><img src="image_18.png" /></figure>

**Heterogeneous case:** $\varepsilon(x, y), \mu(x, y)$ vary in $S$.

<figure><img src="image_19.png" /></figure>
<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

## Recap in the homogeneous case

$\varepsilon = \mu = 1$ and $\omega > 0$ given. Reminder: $\mathbf{E} = e^{i\beta z}\mathcal{E}(x, y)$.

<figure><img src="image_24.png" /></figure>

$$ \Delta\varphi_n + \lambda_n \varphi_n = 0 \quad (S) \\ \partial_v \varphi_n^{\text{TE}} = \lambda_n^{\text{TM}} \varphi_n = 0 \quad (\partial S) $$

where $\lambda_n := \omega^2 - \beta_n^2$

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:

$$ \mathcal{E}_n^{\text{TE}} = \begin{pmatrix} \nabla^{\perp} \varphi_n^{\text{TE}} \\ 0 \end{pmatrix}, \quad \mathcal{E}_n^{\text{TM}} = \begin{pmatrix} \nabla \varphi_n^{\text{TM}} \\ \frac{\lambda_n^{\text{TM}}}{i\beta_n^{\text{TM}}} \varphi_n^{\text{TM}} \end{pmatrix}, \quad \text{where } \nabla^{\perp} = \begin{pmatrix} \partial_y \\ -\partial_x \end{pmatrix}. $$

## Completeness and orthogonality result

The family of transverse fields $(\hat{\mathbf{z}} \times \mathcal{E}_n)_n$ forms a hilbertian basis of $(L^2(S))^2 \times \{0\}$.

<figure><img src="image_25.png" /></figure>
<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>

## Recap in the homogeneous case

$\varepsilon = \mu = 1$. Reminder: $\mathbf{E} = e^{i\beta z} \mathcal{E}(x, y)$.

<figure><img src="image_30.png" /></figure>

$$ \varepsilon = \mu = 1 $$

Distribution of $\beta$ in the complex plane

<figure><img src="image_31.png" /></figure>
<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /></figure>

## Recap in the homogeneous case

$\varepsilon = \mu = 1$. Reminder: $\mathbf{E} = e^{i\beta z} \mathcal{E}(x, y)$.

$$ \omega_n^2 = \lambda_n + \beta^2. $$

<figure><img src="image_36.png" /><figcaption>Dispersion curves.</figcaption></figure>
<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /></figure>

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>

## Recap in the homogeneous case

<figure><img src="image_41.png" /></figure>

Vector representation of the transverse components of the modes $\mathcal{E}_n$.

<figure><img src="image_42.png" /></figure>
<figure><img src="image_43.png" /></figure>

<figure><img src="image_44.png" /></figure>

<figure><img src="image_45.png" /></figure>

<figure><img src="image_46.png" /></figure>

## Recap in the homogeneous case

<figure><img src="image_47.png" /><figcaption>Vector representation of the transverse components of the modes $\mathcal{E}_n$.</figcaption></figure>

<figure><img src="image_48.png" /></figure>
<figure><img src="image_49.png" /></figure>

<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /></figure>

# Introduction

We now focus on the **heterogeneous case**, i.e. $\varepsilon$ and/or $\mu$ vary in the cross-section.

## Two possible points of view:

* $\beta \in \mathbb{R} \to (\hat{\omega}_n^2(\beta)) \in (\mathbb{R}_+)^N$: a self-adjoint problem.

<figure><img src="image_53.png" /></figure>

Bermúdez and Pedreira 1992

* $\hat{\omega} \in \mathbb{R}_+ \to (\beta_n(\hat{\omega})) \in \mathbb{C}^N$: more challenging.

## Many questions arise:

* How to prove that the modes form a **discrete set**?

* Does orthogonality or biorthogonality between modes still hold?

* Do **inverse modes** (with opposite signs for group and phase velocities) exist?

* What about the completeness of the tangential component of the modes?

<figure><img src="image_54.png" /></figure>
<figure><img src="image_55.png" /></figure>

<figure><img src="image_56.png" /></figure>

<figure><img src="image_57.png" /></figure>

<figure><img src="image_58.png" /></figure>

# Table of Contents

1 Mathematical analysis of the modes in a heterogeneous guide

• Formulation in $\mathcal{C}$

• Formulation in $e$

• Formulation in $(e, h)$

2 Inverse Modes

• Numerical illustration of an inverse mode

• Weak perturbation of the homogeneous case

• Mechanisms of inverse mode emergence: a 2D toy model

3 Conclusion

<figure><img src="image_59.png" /></figure>
<figure><img src="image_60.png" /></figure>

<figure><img src="image_61.png" /></figure>

<figure><img src="image_62.png" /></figure>

<figure><img src="image_63.png" /></figure>

# Table of Contents

## 1 Mathematical analysis of the modes in a heterogeneous guide

* Formulation in $\mathcal{C}$

* Formulation in $e$

* Formulation in $(e, h)$

## 2 Inverse Modes
* Numerical illustration of an inverse mode
* Weak perturbation of the homogeneous case
* Mechanisms of inverse mode emergence: a 2D toy model

## 3 Conclusion

<figure><img src="image_64.png" /></figure>
<figure><img src="image_65.png" /></figure>

<figure><img src="image_66.png" /></figure>

<figure><img src="image_67.png" /></figure>

<figure><img src="image_68.png" /></figure>

# Maxwell equations for the modes

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

For a mode, we have:

$\mathbf{Curl} \mathbf{E} = e^{i\beta z} \mathbf{Curl}_{\beta} \mathcal{C}$ and $\mathrm{div}(\varepsilon \mathbf{E}) = e^{i\beta z} \mathrm{div}_{\beta}(\varepsilon \mathcal{C})$,

where: $\mathbf{Curl}_{\beta} \mathcal{C} = \begin{pmatrix} \partial_y e_z - i\beta e_y \\ i\beta e_x - \partial_x e_z \\ \partial_x e_y - \partial_y e_x \end{pmatrix}$ and $\mathrm{div}_{\beta} \varepsilon \mathcal{C} = \partial_x \varepsilon e_x + \partial_y \varepsilon e_y + i\beta \varepsilon e_z = 0$.

$$ \mathbf{curl} e_z - i\beta \mathbf{e}^{\perp} = i\omega \mu \mathbf{h}, $$

$$ \mathbf{curl} \mathbf{e} = i\omega \mu h_z, $$

$$ \mathbf{curl} h_z - i\beta \mathbf{h}^{\perp} = -i\omega \varepsilon \mathbf{e}, $$

$$ \mathbf{curl} \mathbf{h} = -i\omega \varepsilon e_z, $$

$$ \mathrm{div}(\varepsilon \mathbf{e}) = -i\beta \varepsilon e_z, \quad \mathrm{div}(\mu \mathbf{h}) = -\beta \mu h_z. $$

From these equations, different formulations can be derived.

<figure><img src="image_69.png" /></figure>
<figure><img src="image_70.png" /></figure>

<figure><img src="image_71.png" /></figure>

<figure><img src="image_72.png" /></figure>

<figure><img src="image_73.png" /></figure>

## Formulation in $\mathcal{C} = (e, e_z)$

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

Second-order Maxwell equation:

$$ \begin{gathered} \mathrm{Curl}_{\beta} (\mu^{-1} \mathrm{Curl}_{\beta} \mathcal{C}) - \varepsilon \omega^2 \mathcal{C} = 0 \quad \text{in } S, \\ \hat{\mathbf{z}} \times \mathcal{C} = 0 \quad \text{on } \partial S, \end{gathered} $$

$$ V_N(\beta) := \{ (\mathbf{e}, e_z) \mid \mathbf{e} \in \mathbf{H}(\mathrm{curl}, S); e_z \in H_0^1(S); \mathrm{div}_{\beta}(\varepsilon \mathcal{C}) = 0; \boldsymbol{\tau} \cdot \mathbf{e} = 0 \text{ on } \partial S \}. $$

Variational formulation in $\mathcal{C}$

$$ \left\| \begin{array}{l} \text{Find } \beta \in \mathbb{C} \text{ and } \mathcal{C} \in V_N(\beta) \text{ such that for all } \mathcal{C}' \in V_N(\beta), \\ \displaystyle \int_S \mu^{-1} \mathrm{Curl}_{\beta} \mathcal{C} \cdot \mathrm{Curl}_{-\beta} \overline{\mathcal{C}'} - \omega^2 \varepsilon \mathcal{C} \cdot \overline{\mathcal{C}'} = 0. \end{array} \right. $$

## Two methods to avoid the dependence of the space in beta:

* Change of variable. Halla and Monk 2024

<figure><img src="image_74.png" /></figure>

* Exact penalization of the divergence-free condition. Bamberger and Bonnet 1990

<figure><img src="image_75.png" /></figure>

<figure><img src="image_76.png" /></figure>
<figure><img src="image_77.png" /></figure>

<figure><img src="image_78.png" /></figure>

<figure><img src="image_79.png" /></figure>

<figure><img src="image_80.png" /></figure>

## Formulation in $\mathcal{C} = (e, e_z)$

$$V_N(\beta) := \{ (\mathbf{e}, e_z) \mid \mathbf{e} \in \mathbf{H}(\mathrm{curl}, S); e_z \in H_0^1(S); \mathrm{div}_\beta(\varepsilon \mathcal{C}) = 0; \boldsymbol{\tau} \cdot \mathbf{e} = 0 \text{ on } \partial S \}.$$

### Variational formulation in $\mathcal{C}$

$$ \left| \int_S \mu^{-1} \mathrm{Curl}_\beta \mathcal{C} \cdot \mathrm{Curl}_{-\beta} \overline{\mathcal{C}'} - \omega^2 \varepsilon \mathcal{C} \cdot \overline{\mathcal{C}'} = 0. \right. \quad (1) $$

$$V_N := \{ (\mathbf{e}, e_z) \mid \mathbf{e} \in \mathbf{H}(\mathrm{curl}, S); e_z \in H_0^1(S); \mathrm{div}(\varepsilon \mathbf{e}) \in L^2(S); \boldsymbol{\tau} \cdot \mathbf{e} = 0 \text{ on } \partial S \}.$$

### Penalized variational formulation in $\mathcal{C}$

$$ \left| \int_S \mu^{-1} \mathrm{Curl}_\beta \mathcal{C} \cdot \mathrm{Curl}_{-\beta} \overline{\mathcal{C}'} - \omega^2 \varepsilon \mathcal{C} \cdot \overline{\mathcal{C}'} + \sigma \mathrm{div}_\beta(\varepsilon \mathcal{C}) \mathrm{div}_{-\beta}(\varepsilon \overline{\mathcal{C}'}) = 0, \right. \quad (2) $$

where $\sigma$ is any positive bounded function.

### Equivalence?

$$ \bullet (1) \Rightarrow (2) \quad \check{\} $$

$$ \bullet (2) \Rightarrow (1) \quad ? $$

$$ \left| \begin{array}{l} \text{Find } \phi \in H_0^1(S) \text{ such that} \\ \sigma \mathrm{div}(\varepsilon \nabla \phi) + (\omega^2 - \beta^2 \sigma \varepsilon) \phi = 0. \end{array} \right. $$

<figure><img src="image_81.png" /></figure>
<figure><img src="image_82.png" /></figure>

<figure><img src="image_83.png" /></figure>

<figure><img src="image_84.png" /></figure>

<figure><img src="image_85.png" /></figure>

Formulation in $\mathcal{C} = (\mathbf{e}, e_z)$

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

$$V_N := \{ (\mathbf{e}, e_z) \mid \mathbf{e} \in \mathbf{H}(\mathrm{curl}, S); e_z \in H_0^1(S); \mathrm{div}(\varepsilon \mathbf{e}) \in L^2(S); \mathbf{r} \cdot \mathbf{e} = 0 \text{ on } \partial S \}.$$

$$ \int_S \mu^{-1} (\mathrm{curl} \mathbf{e} \, \mathrm{curl} \overline{\mathbf{e}'} + \nabla e_z \cdot \overline{\nabla e_z'}) + \sigma \mathrm{div}(\varepsilon \mathbf{e}) \mathrm{div}(\varepsilon \overline{\mathbf{e}'}) \\ + i\beta \int_S \mu^{-1} (-\mathbf{e} \cdot \overline{\nabla e_z'} + \overline{\mathbf{e}'} \cdot \overline{\nabla e_z}) + \sigma \varepsilon (e_z \mathrm{div}(\varepsilon \overline{\mathbf{e}'}) - \mathrm{div}(\varepsilon \mathbf{e}) \overline{e_z'}) \\ + \beta^2 \int_S \mu^{-1} \mathbf{e} \cdot \overline{\mathbf{e}'} + \sigma \varepsilon^2 e_z \overline{e_z'} - \omega^2 \int_S \varepsilon \mathcal{C} \cdot \overline{\mathcal{C}'} = 0. $$

Coercive + Compact.

discreteness of $\beta'$s

For a given $\omega$, the analytic Fredholm theorem implies the **discreteness of $\beta'$s** if **injectivity** holds for one $\beta \in \mathbb{C}$.

<figure><img src="image_86.png" /></figure>

Ok if the dispersion curves are not flat.

<figure><img src="image_87.png" /></figure>
<figure><img src="image_88.png" /></figure>

<figure><img src="image_89.png" /></figure>

<figure><img src="image_90.png" /></figure>

<figure><img src="image_91.png" /></figure>

Dispersion curves are not flat !

Change of perspective: fix $\beta \in \mathbb{R}$ — self-adjoint problem in $\omega$. Set $\mu_+ := \sup_S \mu$ and $\varepsilon_+ := \sup_S \varepsilon$. Using min-max formula in suitable spaces $V_N(\varepsilon)$ and $V_T(\mu)$, successively one proves:

$$ \omega_n^2(\beta, \varepsilon, \mu) = \sup_{\mathcal{H}^1, \dots, \mathcal{H}^{n-1} \in \mathbb{L}^2(S)} \inf_{\mathcal{H} \in [\mathcal{H}^1, \dots, \mathcal{H}^{n-1}]_{\tilde{V}_N(\varepsilon)}^{\perp}} \frac{\int_S \mu^{-1} |\text{Curl}_\beta \mathcal{H}|^2}{\int_S \varepsilon |\mathcal{H}|^2} \ge \omega_n^2(\beta, \varepsilon, \mu_+), $$

$$ \omega_n^2(\beta, \varepsilon, \mu_+) = \sup_{\mathcal{H}^1, \dots, \mathcal{H}^{n-1} \in \mathbb{L}^2(S)} \inf_{\mathcal{H} \in [\mathcal{H}^1, \dots, \mathcal{H}^{n-1}]_{\tilde{V}_T(\mu_+)}^{\perp}} \frac{\int_S \varepsilon^{-1} |\text{Curl}_\beta \mathcal{H}|^2}{\int_S \mu_+ |\mathcal{H}|^2} \ge \omega_n^2(\beta, \varepsilon, \mu_+). $$

Growth of eigenvalues compared to the homogeneous case

$$ \omega_n(\beta, \varepsilon, \mu) \ge \omega_n(\beta, \varepsilon_+, \mu_+) \xrightarrow{\beta \to +\infty} +\infty. $$

<figure><img src="image_92.png" /></figure>
<figure><img src="image_93.png" /></figure>

<figure><img src="image_94.png" /></figure>

<figure><img src="image_95.png" /></figure>

<figure><img src="image_96.png" /></figure>

## Formulation in $\mathcal{G} = (\mathbf{e}, e_z)$

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

$$ V_N := \{(\mathbf{e}, e_z) \mid \mathbf{e} \in \mathbf{H}(\mathrm{curl}, S); e_z \in \mathbf{H}_0^1(S); \mathrm{div}(\varepsilon \mathbf{e}) \in \mathbf{L}^2(S); \mathbf{r} \cdot \mathbf{e} = 0 \text{ on } \partial S\}. \\ \int_S \mu^{-1} (\mathrm{curl} \mathbf{e} \ \mathrm{curl} \overline{\mathbf{e}'} + \nabla e_z \cdot \nabla e_z') + \sigma \mathrm{div}(\varepsilon \mathbf{e}) \mathrm{div}(\varepsilon \overline{\mathbf{e}'}) \\ + i\beta \int_S \mu^{-1} (-\mathbf{e} \cdot \nabla \overline{e_z} + \overline{\mathbf{e}'} \cdot \nabla e_z) + \sigma \varepsilon (e_z \mathrm{div}(\varepsilon \overline{\mathbf{e}'}) - \mathrm{div}(\varepsilon \mathbf{e}) \overline{e_z'}) \\ + \beta^2 \int_S \mu^{-1} \mathbf{e} \cdot \overline{\mathbf{e}'} + \sigma \varepsilon^2 e_z \overline{e_z'} - \overline{\omega^2} \int_S \varepsilon \mathcal{G} \cdot \mathcal{G}' = 0. $$

## Discreteness

For $\beta$ large enough, we have **injectivity**. So the modes form a **discrete set**.

## Drawbacks of this formulation:

* 3 scalar unknowns.

* Quadratic eigenvalue problem in $\beta$.

<figure><img src="image_97.png" /></figure>
<figure><img src="image_98.png" /></figure>

<figure><img src="image_99.png" /></figure>

<figure><img src="image_100.png" /></figure>

<figure><img src="image_101.png" /></figure>

# Formulation in e

Second-order Maxwell equation +

$$i\beta e_z = -\varepsilon^{-1} \operatorname{div}(\varepsilon \mathbf{e})$$

$$X_N(\varepsilon) := \{\mathbf{e} | \operatorname{curl}(\mathbf{e}) \in L^2(S), \operatorname{div}(\varepsilon \mathbf{e}) \in L^2(S) \text{ and } \mathbf{v}^{\perp} \cdot \mathbf{e} = 0 \text{ on } \partial S\}$$

$$\tilde{X}_N(\mu) := \{\mathbf{e} | \operatorname{curl}(\mu\mathbf{e}) \in L^2(S), \operatorname{div}(\mathbf{e}) \in L^2(S) \text{ and } \mathbf{v}^{\perp} \cdot \mu\mathbf{e} = 0 \text{ on } \partial S\}.$$

## Variational formulation in e

$$\begin{aligned} & \text{Find } \mathbf{e} \in X_N(\varepsilon) \text{ and } \beta \in \mathbb{C} \text{ such that for all } \mathbf{e}' \in \tilde{X}_N(\mu), \\ & \int_S \mu^{-1} \operatorname{curl}(\mathbf{e}) \operatorname{curl}(\mu\mathbf{e}') + \int_S \varepsilon^{-1} \operatorname{div}(\varepsilon\mathbf{e}) \operatorname{div}(\mathbf{e}') - \bar{\omega}^2 \int_S \varepsilon \mu \mathbf{e} \cdot \bar{\mathbf{e}}' = -\beta^2 \int_S \mathbf{e} \cdot \mathbf{e}'. \end{aligned}$$

Note the unusual space for test functions.

* **Homogeneous case:** we recover the previous results.

* **Heterogeneous case:** coercive + compact or T-coercive + compact.

**Drawback:** not appropriate for numerics.

<figure><img src="image_102.png" /></figure>
<figure><img src="image_103.png" /></figure>

<figure><img src="image_104.png" /></figure>

<figure><img src="image_105.png" /></figure>

<figure><img src="image_106.png" /></figure>

## Formulation in (e, h)

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

$$ \frac{i}{\omega} \mathrm{curl}(\varepsilon^{-1} \mathrm{curl} \mathbf{h}) - i\omega\mu\mathbf{h} = i\beta\mathbf{e}^{\perp} \\ -\frac{i}{\omega} \mathrm{curl}(\mu^{-1} \mathrm{curl} \mathbf{e}) + i\omega\varepsilon\mathbf{e} = i\beta\mathbf{h}^{\perp}. $$

$$ \mathbb{H}_N := \{ \mathbf{e} \mid \mathrm{curl} \, \mathbf{e} \in L^2(S) \text{ and } \boldsymbol{\tau} \cdot \mathbf{e} = 0 \text{ on } \partial S \}. $$

## Variational formulation in (e, h)

Find $(\mathbf{e}, \mathbf{h}) \in \mathbb{H}_N \times \mathbb{H}(\mathrm{curl}, S)$ and $\beta \in \mathbb{C}$ such that for all $(\mathbf{e}', \mathbf{h}') \in \mathbb{H}_N \times \mathbb{H}(\mathrm{curl}, S)$,

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

**Open question:** how to prove discreteness with this formulation?

* We use this formulation for numerical computations.

<figure><img src="image_107.png" /></figure>
<figure><img src="image_108.png" /></figure>

<figure><img src="image_109.png" /></figure>

<figure><img src="image_110.png" /></figure>

<figure><img src="image_111.png" /></figure>

# Distribution of the $\beta$'s in the complex plane

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

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

Numerical results:

<figure><img src="image_112.png" /></figure>

XLIFE++

<figure><img src="image_113.png" /></figure>

Finite element discretization of the third formulation.

<figure><img src="image_114.png" /></figure>
<figure><img src="image_115.png" /></figure>

<figure><img src="image_116.png" /></figure>

<figure><img src="image_117.png" /></figure>

<figure><img src="image_118.png" /></figure>

Numerical modes

<figure><img src="image_119.png" /></figure>

Vector representation of the transverse components of the modes $\mathcal{E}_n$.

<figure><img src="image_120.png" /></figure>
<figure><img src="image_121.png" /></figure>

<figure><img src="image_122.png" /></figure>

<figure><img src="image_123.png" /></figure>

<figure><img src="image_124.png" /></figure>

# Table of Contents

1 Mathematical analysis of the modes in a heterogeneous guide

* Formulation in $\mathcal{C}$

* Formulation in $e$

* Formulation in $(e, h)$

2 Inverse Modes
* Numerical illustration of an inverse mode
* Weak perturbation of the homogeneous case
* Mechanisms of inverse mode emergence: a 2D toy model

3 Conclusion

<figure><img src="image_125.png" /></figure>
<figure><img src="image_126.png" /></figure>

<figure><img src="image_127.png" /></figure>

<figure><img src="image_128.png" /></figure>

<figure><img src="image_129.png" /></figure>

# Definition of inverse modes

Recall the definition of phase and group velocities of a propagating mode:

$$v_{\phi} = \frac{\omega}{\beta}, \quad v_{G} = \frac{\partial \omega}{\partial \beta}.$$

Inverse mode: modes such that $v_{\phi}v_{G} < 0$.

## Existence of inverse mode:

<figure><img src="image_130.png" /></figure>

Homogeneous case: none.

<figure><img src="image_131.png" /></figure>

Heterogeneous case: Omar and Schunemann 1987

<figure><img src="image_132.png" /></figure>

* Physically relevant.

* Prevents the use of PMLs.

<figure><img src="image_133.png" /></figure>
<figure><img src="image_134.png" /></figure>

<figure><img src="image_135.png" /></figure>

<figure><img src="image_136.png" /></figure>

<figure><img src="image_137.png" /></figure>

Animation of $\beta$'s in the complex plane

$v_{\phi} = \frac{\omega}{\beta}$, $v_{G} = \frac{\partial\omega}{\partial\beta}$

<figure><img src="image_138.png" /></figure>

$\epsilon = 10$

$\epsilon = 0.5$

<figure><img src="image_139.png" /></figure>

Animation of $\beta$ in the complex plane.

<figure><img src="image_140.png" /></figure>
<figure><img src="image_141.png" /></figure>

<figure><img src="image_142.png" /></figure>

<figure><img src="image_143.png" /></figure>

<figure><img src="image_144.png" /></figure>

Dispersion curves

<figure><img src="image_145.png" /></figure>

$$ \varepsilon = 10 $$

$$ \varepsilon = 0.5 $$

<figure><img src="image_146.png" /></figure>

Illustration of an inverse mode among the dispersion curves

<figure><img src="image_147.png" /></figure>
<figure><img src="image_148.png" /></figure>

<figure><img src="image_149.png" /></figure>

<figure><img src="image_150.png" /></figure>

<figure><img src="image_151.png" /></figure>

# Weak perturbation of the homogeneous case

We note:

$$ \delta(\mathbf{x}) := \frac{\sup_S \mu}{\mu(\mathbf{x})} - \frac{\varepsilon(\mathbf{x})}{\inf_S \varepsilon}. $$

Using min-max formula:

## Theorem: region of absence of inverse modes

Suppose $\|\delta\|_{\infty} < 1$. There exist $C(\varepsilon, \mu) > 0$ such that if

$$ \beta > C(\varepsilon, \mu)\omega_n(\beta) $$

then

$$ v_{\phi} v_G = \frac{\omega_n(\beta)}{\beta} \frac{\partial \omega_n(\beta)}{\partial \beta} > 0. $$

<figure><img src="image_152.png" /></figure>
<figure><img src="image_153.png" /></figure>

<figure><img src="image_154.png" /></figure>

<figure><img src="image_155.png" /></figure>

<figure><img src="image_156.png" /></figure>

Weak perturbation of the homogeneous case

$$C(\epsilon, \mu) \sim_{\|\delta\|_{\infty} \to 0} \tilde{c} \|\delta\|_{\infty}.$$

<figure><img src="image_157.png" /></figure>

<figure><img src="image_158.png" /></figure>

Two regions of absence of inverse modes, with $C(\epsilon_1, \mu_1) < C(\epsilon_2, \mu_2)$.

Absence of inverse mode in the homogeneous case

If $\epsilon$ and $\mu$ are constant, then $C(\epsilon, \mu) = 0$, so there is **no inverse mode**.

<figure><img src="image_159.png" /></figure>
<figure><img src="image_160.png" /></figure>

<figure><img src="image_161.png" /></figure>

<figure><img src="image_162.png" /></figure>

<figure><img src="image_163.png" /></figure>

# Mechanisms of inverse mode emergence: a 2D toy model

A simplification of the previous problem:

$$A(\beta)u = \lambda(\beta)u,$$

where $\lambda(\beta) = \omega^2(\beta)$.

We define:

$$A(\beta) = A_0 + \beta A_1 + \beta^2 A_2,$$

with:

$$A_0 = \begin{pmatrix} \lambda_0^+ & 0 \\ 0 & \lambda_0^- \end{pmatrix}, \quad A_1 = \begin{pmatrix} 0 & a \\ a & 0 \end{pmatrix} \quad \text{and} \quad A_2 = \begin{pmatrix} b & 0 \\ 0 & c \end{pmatrix}$$

and

$$0 < \lambda_0^- \le \lambda_0^+, \quad a \in \mathbb{R} \quad \text{and} \quad b, c > 0.$$

We can compute **analytically** the two eigenvalues $\lambda^{\pm}(\beta)$ of $A(\beta)$. **Inverse modes** exist in two situations:

* if $A_0$ has a **double eigenvalue** ($\lambda_0^- = \lambda_0^+$) and $a \neq 0$.

* if $A_0$ has **two distinct eigenvalues** and $a^2 > (\lambda_0^+ - \lambda_0^-)c$.

<figure><img src="image_164.png" /></figure>
<figure><img src="image_165.png" /></figure>

<figure><img src="image_166.png" /></figure>

<figure><img src="image_167.png" /></figure>

<figure><img src="image_168.png" /></figure>

# Dispersion curves of the toy problem

## Two situations:

Double eigenvalue of $A_0$

<figure><img src="image_169.png" /></figure>

Parameters:
$\lambda_0^+ = \lambda_0^- = a = c = 1$ and $b = 2$.

Simple eigenvalues of $A_0$

<figure><img src="image_170.png" /></figure>

Parameters:
$\lambda_0^+ = 3$, $\lambda_0^- = 2$, $a = 3$, $b = 2$ and $c = 1$.
<figure><img src="image_171.png" /></figure>

<figure><img src="image_172.png" /></figure>

<figure><img src="image_173.png" /></figure>

<figure><img src="image_174.png" /></figure>

Comparison of toy model and numerical dispersion

<figure><img src="image_175.png" /></figure>

Parameters: $\lambda_0^+ = 9.12$, $\lambda_0^- = 8.52$, $a = 1$, $b = 0.89$ and $c = 0.22$
<figure><img src="image_176.png" /></figure>

<figure><img src="image_177.png" /></figure>

<figure><img src="image_178.png" /></figure>

<figure><img src="image_179.png" /></figure>

# Table of Contents

1 Mathematical analysis of the modes in a heterogeneous guide

● Formulation in $\mathcal{C}$

● Formulation in $e$

● Formulation in $(e, h)$

2 Inverse Modes

● Numerical illustration of an inverse mode

● Weak perturbation of the homogeneous case

● Mechanisms of inverse mode emergence: a 2D toy model

3 Conclusion

<figure><img src="image_180.png" /></figure>
<figure><img src="image_181.png" /></figure>

<figure><img src="image_182.png" /></figure>

<figure><img src="image_183.png" /></figure>

<figure><img src="image_184.png" /></figure>

# Conclusion

To go further with modes:

* Biorthogonality relations

Reduce and general biorthogonality relations

$$ \text{If } \beta^2 \neq \beta'^2, $$

$$ (\hat{z} \times \mathcal{C}, \mathcal{H}')_S = 0, $$

$$ \text{If } \beta \neq \beta', $$

$$ (\hat{z} \times \mathcal{C}, \mathcal{H}')_S + (\mathcal{H}, \hat{z} \times \mathcal{C}')_S = 0. $$

* Design of *modal transparent conditions*.

<figure><img src="image_185.png" /></figure>

Thank you for your attention !

<figure><img src="image_186.png" /></figure>
<figure><img src="image_187.png" /></figure>

References I

<figure><img src="image_188.png" /></figure>

Bamberger, A. and A.-S. Bonnet (1990). “Mathematical analysis of the guided modes of an optical fiber”. In: *SIAM Journal on Mathematical Analysis* 21.6, pp. 1487–1510.

<figure><img src="image_189.png" /></figure>

Baronian, V. et al. (2016). “Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides”. In: *Wave Motion* 64, pp. 13–33.

<figure><img src="image_190.png" /></figure>

Bermúdez, A. and D. G. Pedreira (1992). “Mathematical analysis of a finite element method without spurious solutions for computation of dielectric waveguides”. In: *Numerische Mathematik* 61.1, pp. 39–57.

<figure><img src="image_191.png" /></figure>

Bonnet-Ben Dhia, A.-S., L. Chesnel, and V. Pagneux (2018). “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, p. 20180050.

<figure><img src="image_192.png" /></figure>

Bonnet-Ben Dhia, A.-S. and G. Legendre (2011). “An alternative to Dirichlet-to-Neumann maps for waveguides”. In: *Comptes Rendus Mathematique* 349.17-18, pp. 1005–1009.

<figure><img src="image_193.png" /></figure>
<figure><img src="image_194.png" /></figure>

References II

<figure><img src="image_195.png" /></figure>

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

<figure><img src="image_196.png" /></figure>

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

<figure><img src="image_197.png" /></figure>

Halla, M. and P. Monk (2024). “On the analysis of modes in a closed electromagnetic waveguide”. In: *Comptes Rendus. Mathématique* 362.5, pp. 469-479.

<figure><img src="image_198.png" /></figure>

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

<figure><img src="image_199.png" /></figure>

Nicolet, A. et al. (2008). “Geometrical transformations and equivalent materials in computational electromagnetism”. In: *COMPEL-The international journal for computation and mathematics in electrical and electronic engineering* 27.4, pp. 806-819.

<figure><img src="image_200.png" /></figure>
<figure><img src="image_201.png" /></figure>

References III

<figure><img src="image_202.png" /></figure>

Omar, A. S. and K. Schunemann (1987). "Complex and backward-wave modes in inhomogeneously and anisotropically filled waveguides". In: *IEEE transactions on microwave theory and techniques* 35.3, pp. 268-275.

<figure><img src="image_203.png" /></figure>

Pauly, D. and N. Skrepek (2022). "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*, pp. 1-15.

<figure><img src="image_204.png" /></figure>

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

<figure><img src="image_205.png" /></figure>

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

<figure><img src="image_206.png" /></figure>

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

<figure><img src="image_207.png" /></figure>