# Full-Vectorial Fiber Mode Solver Based on a Discrete Hankel Transform

## Abstract

**:**

## 1. Introduction

## 2. Coupled Set of 1D Wave Equations

## 3. The Discrete Hankel Transform

## 4. Transformation of the Differential Operators

- 1.
- 2.
- The common derivative operator $\partial /\partial \rho $, which occurs as $\partial /\partial \rho \phantom{\rule{0.166667em}{0ex}}\mathrm{ln}\left(\u03f5\right)$ in Equations (12) and (13) and additionally in Equation (12) as a term $\partial /\partial \rho \phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{A}_{\rho}\partial /\partial \rho \phantom{\rule{0.166667em}{0ex}}\mathrm{ln}\left(\u03f5\right)$.

#### 4.1. Transformation of the Bessel Operator

#### 4.2. Transformation of the Derivative Operator

## 5. Implementation and Validation of the DHT-Based Mode Solver

#### 5.1. Step-Index Fiber Design

#### 5.2. Graded-Index Fiber Design

## 6. Conclusions

## 7. Outlook

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Validation of Equation (44) regarding the 0th-order DHT of the differential operator ${\rho}^{-1}\partial /\partial \rho $. (

**a**): Sampled Gaussian and its analytical derivative as well as its derivative computed via the 0-th order DHT and the discussed transformation rule of the ${\rho}^{-1}\partial /\partial \rho $ operator. (

**b**): Mean relative deviation of the derivative computed via the 0-th order DHT with respect to its analytical counterpart. The dashed $1/{N}^{2}$ trend line is not fitted but only a guide to the eye.

**Figure 2.**Results of the DHT-based mode solver regarding the HE${}_{11}$ and TE${}_{01}$ modes of a step-index fiber design. (

**a**): Computed effective refractive indices for a varying number of spatial mesh points i.e., Fourier–Bessel coefficients. (

**b**): Deviation of the computed effective refractive indices with respect to their analytically computed counterparts. (

**c**): Typical required time to compute the HE${}_{11}$ mode and its propagation constant, either with a noisy guess for each call of the solver or a recycled and correspondingly interpolated guess from the previous call. The dashed $1/{N}^{2}$ trend line is not fitted but only a guide to the eye.

**Figure 3.**Results of the DHT-based mode solver regarding the HE${}_{11}$ mode in a graded-index fiber design, either with or without the terms that scale with $\partial /\partial \rho \phantom{\rule{0.166667em}{0ex}}\mathrm{ln}\left(\u03f5\right)$ in Equations (12) and (13). (

**a**): Absolute deviation from the analytically computed propagation constant of the linearly polarized modes. (

**b**): Absolute deviation from the propagation constants computed by a 2D FFT-based full-vectorial mode solver. (

**c**): Required computational times. As before, the dashed $1/{N}^{2}$ trend lines are not fitted but only guides to the eye.

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**MDPI and ACS Style**

Steinke, M.
Full-Vectorial Fiber Mode Solver Based on a Discrete Hankel Transform. *Photonics* **2021**, *8*, 439.
https://doi.org/10.3390/photonics8100439

**AMA Style**

Steinke M.
Full-Vectorial Fiber Mode Solver Based on a Discrete Hankel Transform. *Photonics*. 2021; 8(10):439.
https://doi.org/10.3390/photonics8100439

**Chicago/Turabian Style**

Steinke, Michael.
2021. "Full-Vectorial Fiber Mode Solver Based on a Discrete Hankel Transform" *Photonics* 8, no. 10: 439.
https://doi.org/10.3390/photonics8100439