# Transverse Electric Guided Wave Propagation in a Plane Waveguide with Kerr Nonlinearity and Perturbed Inhomogeneity in the Permittivity Function

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Statement of the Problem

**Statement**

**1.**

#### 3.2. Problem ${\mathcal{P}}_{\alpha}$

**Statement**

**2.**

**Statement**

**3.**

**Theorem**

**1.**

#### 3.3. Problem $\mathcal{P}$

**Statement**

**4.**

**Statement**

**5.**

**Statement**

**6.**

**Theorem**

**2.**

#### 3.4. Numerical Results

#### 3.5. Proofs

#### 3.5.1. Proof of Statement 4

#### 3.5.2. Proof of Statement 5

#### 3.5.3. Proof of Statement 6

#### 3.5.4. Proof of Theorem 2

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Dispersion curves of problem ${\mathcal{P}}_{\alpha}$ (blue curve) and problem ${\mathcal{P}}_{0}$ with $\beta =0$ (red curve). Solid diamonds denote the propagation constants ${\tilde{\gamma}}_{1}\approx 1.415$ (red), ${\tilde{\gamma}}_{2}\approx 0.902$ (green) of problem ${\mathcal{P}}_{0}$ with $\beta =0$ and propagation constants ${\overline{\gamma}}_{2,1}\approx 1.141$ (brown), ${\overline{\gamma}}_{2,2}\approx 3.705$ (blue) of problem ${\mathcal{P}}_{\alpha}$.

**Figure 3.**Dispersion curves of problem ${\mathcal{P}}_{\alpha}$ (blue curve) with $\epsilon =2.405$ and problem ${\mathcal{P}}_{\alpha}^{\beta}$ (green curve) with $\epsilon =2.405+0.05cos\left(x\right)$. Solid diamonds denote the propagation constants ${\overline{\gamma}}_{2,1}\approx 1.141$ (blue), ${\overline{\gamma}}_{2,2}\approx 3.705$ (orange) of problem ${\mathcal{P}}_{\alpha}$ and propagation constants ${\widehat{\gamma}}_{2,1}\approx 1.131$ (green), ${\widehat{\gamma}}_{2,2}\approx 4.197$ (purple) of problem ${\mathcal{P}}_{\alpha}^{\beta}$.

**Figure 4.**Subfigure (

**a**): the eigenfunction of problem ${\mathcal{P}}_{\alpha}^{\beta}$ (green curve) corresponding to the propagation constant denoted by the green diamond in Figure 3. Subfigure (

**b**): the eigenmodes of problems ${\mathcal{P}}_{\alpha}^{\beta}$ (purple curve) and ${\mathcal{P}}_{\alpha}$ (orange curve) corresponding to the propagation constants denoted by the purple and orange diamonds in Figure 3.

**Figure 5.**Dispersion curves of problem ${\mathcal{P}}_{\alpha}$ (blue curve) with $\epsilon =2.405$ and problem ${\mathcal{P}}_{\alpha}^{\beta}$ (green curve) with $\epsilon =2.405+0.05cos\left(10x\right)$. Solid diamonds denote propagation constants ${\overline{\gamma}}_{2,1}\approx 1.141$ (blue), ${\overline{\gamma}}_{2,2}\approx 3.705$ (orange) of problem ${\mathcal{P}}_{\alpha}$ and propagation constants ${\widehat{\gamma}}_{2,1}\approx 1.115$ (green), ${\widehat{\gamma}}_{2,2}\approx 3.656$ (purple) of problem ${\mathcal{P}}_{\alpha}^{\beta}$.

**Figure 6.**Subfigure (

**a**): the eigenmode of problem ${\mathcal{P}}_{\alpha}^{\beta}$ (green curve) corresponding to the propagation constant denoted by the green diamonds in Figure 5. Subfigure (

**b**): the eigenmode of problem ${\mathcal{P}}_{\alpha}^{\beta}$ (purple curve) corresponding to the propagation constant denoted by the purple diamonds in Figure 5.

**Figure 7.**Dispersion curves of problem ${\mathcal{P}}_{\alpha}$ (blue curve) with $\epsilon =2.405$ and problem ${\mathcal{P}}_{\alpha}^{\beta}$ (green curve) with $\epsilon =2.405+0.05cos\left(4x\right)$. Solid diamonds denote propagation constants ${\overline{\gamma}}_{2,1}\approx 1.141$ (blue), ${\overline{\gamma}}_{2,2}\approx 3.705$ (orange) of problem ${\mathcal{P}}_{\alpha}$ and propagation constants ${\widehat{\gamma}}_{2,1}\approx 1.112$ (green), ${\widehat{\gamma}}_{2,2}\approx 3.404$ (purple) of problem ${\mathcal{P}}_{\alpha}^{\beta}$.

**Figure 8.**Subfigure (

**a**): the eigenmode of problem ${\mathcal{P}}_{\alpha}^{\beta}$ (green curve) corresponding to the propagation constant denoted by the green diamond in Figure 7. Subfigure (

**b**): the eigenmode of problem ${\mathcal{P}}_{\alpha}^{\beta}$ (purple curve) and ${\mathcal{P}}_{\alpha}$ (orange curve) corresponding to the propagation constant denoted by the purple and orange diamonds in Figure 7.

**Figure 9.**Dispersion curves of problem ${\mathcal{P}}_{\alpha}$ (blue curve) with $\epsilon =2.405$ and problem ${\mathcal{P}}_{\alpha}^{\beta}$ (green curve) with $\epsilon =2.405+0.00526(-1.6{x}^{2}+8x)$. Solid diamonds denote propagation constants ${\overline{\gamma}}_{2,1}\approx 1.141$ (blue), ${\overline{\gamma}}_{2,2}\approx 3.705$ (brown), ${\overline{\gamma}}_{3,1}\approx 6.866$ (orange) of problem ${\mathcal{P}}_{\alpha}$ and propagation constants ${\widehat{\gamma}}_{2,1}\approx 1.179$ (green), ${\widehat{\gamma}}_{2,2}\approx 3.585$ (red), ${\widehat{\gamma}}_{3,1}\approx 6.493$ (purple) of problem ${\mathcal{P}}_{\alpha}^{\beta}$.

**Figure 10.**Subfigure (

**a**): the eigenfunctions of problem ${\mathcal{P}}_{\alpha}$ (brown curve) and problem ${\mathcal{P}}_{0}$ with $\beta =0$ (green curve) corresponding to the propagation constants denoted by the green and brown diamonds in Figure 2. Subfigure (

**b**): the eigenfunction corresponding to the propagation constant of problem ${\mathcal{P}}_{\alpha}$ denoted by the blue diamonds in Figure 2.

**Figure 11.**Subfigure (

**a**): the eigenmode of problem ${\mathcal{P}}_{\alpha}^{\beta}$ (red curve) corresponding to the propagation constant denoted by the red diamonds in Figure 9. Subfigure (

**b**): the eigenmodes of problems ${\mathcal{P}}_{\alpha}^{\beta}$ (purple curve) and ${\mathcal{P}}_{\alpha}$ (orange curve) corresponding to the propagation constants denoted by the purple and orange diamonds in Figure 9.

**Figure 12.**In subfigure (

**a**) $\epsilon =2.405$, in subfigure (

**b**) $\epsilon =2.405+0.05cos\left(x\right)$, in subfigure (

**c**) $\epsilon =2.405+0.05cos\left(10x\right)$, in subfigure (

**d**) $\epsilon =2.405+0.00526(-1.6{x}^{2}+8x)$.

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**MDPI and ACS Style**

Dyundyaeva, A.; Tikhov, S.; Valovik, D. Transverse Electric Guided Wave Propagation in a Plane Waveguide with Kerr Nonlinearity and Perturbed Inhomogeneity in the Permittivity Function. *Photonics* **2023**, *10*, 371.
https://doi.org/10.3390/photonics10040371

**AMA Style**

Dyundyaeva A, Tikhov S, Valovik D. Transverse Electric Guided Wave Propagation in a Plane Waveguide with Kerr Nonlinearity and Perturbed Inhomogeneity in the Permittivity Function. *Photonics*. 2023; 10(4):371.
https://doi.org/10.3390/photonics10040371

**Chicago/Turabian Style**

Dyundyaeva, Anna, Stanislav Tikhov, and Dmitry Valovik. 2023. "Transverse Electric Guided Wave Propagation in a Plane Waveguide with Kerr Nonlinearity and Perturbed Inhomogeneity in the Permittivity Function" *Photonics* 10, no. 4: 371.
https://doi.org/10.3390/photonics10040371