# Asymptotic Modeling of Optical Fibres: Annular Capillaries and Microstructured Optical Fibres

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Three-Dimensional Model

**∇**in Cartesian coordinates reads

#### 2.2. Non-Dimensionalization

#### 2.3. Final Asymptotic Equations

^{th}order the following mass and momentum equations

^{th}order read

^{th}order continuity equation, i.e., Equation (10a).

#### 2.3.1. Leading-Order Model for the Transverse Flow

#### 2.3.2. The Case of a Circular Tube

#### 2.3.3. The Cross-Plane Problem and the Complex Variable Formulation

#### 2.3.4. The Generalised Elliptical Pore Model (GEPM)

#### 2.3.5. Fibre Temperature Profile and Glass Viscosity

## 3. Results

#### 3.1. Solution Methodology

#### 3.2. Annular Capillaries: Slow Drawing Ratios (SDRs)

#### 3.3. Annular Capillaries: High Drawing Ratios (HDRs)

#### 3.4. Holey Fibres (HFs)

^{®}, an FEM-based commercial software. Ansys PolyFlow

^{®}is a general-purpose computational fluid dynamic (CFD) software suitable for simulating extrusion, thermoforming, glass moulding, fibre drawing, polymer, and glass forming processes. Furthermore, it possesses an extensive library of visco-elastic models. This FEM software features robust iterative solvers based on incomplete lower–upper (ILU) matrix factorization schemes. In addition, it is equipped with robust re-meshing techniques that are needed to relocate internal nodes in the case of displacement of boundary nodes. Those features are very useful in simulating the fibre drawing process where large deformations of the boundaries occur [53]. The initial hole sizes and configuration can be found in Figure 1 of the manuscript of Luzi et al. [47]. The drawing conditions are: ${T}_{peak}=$ 1890 (${}^{\xb0}$C), DR 20-187, and ${p}_{H}$= 0 (mbar). In Figure 9a, we overlap the experimental results, the previous FEM simulations, and the outcome of the present computations. Figure 9a exhibits an excellent agreement between the current computations and the experimental results. The shape, size, and positions of the holes at the end of the drawing are well predicted, and the maximum discrepancy between simulations and experiments is only 0.17%. Afterward, we consider a thirty-six-hole cross-section with internal pressurization, whose geometrical details of the shape are given in the manuscript of Frosz et al. [52]. ${T}_{peak}=$ 1973 (${}^{\xb0}$C), ${p}_{H}$ = 173 (mbar), and DR 10-42, as indicated in Table 1. Figure 9b displays an excellent agreement between the final fibre cross-section obtained experimentally and the one computed numerically. In particular, the hole sizes and locations in the fibre cross-section are correctly predicted by the numerical simulations.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MOFs | Microstructured optical fibres |

SCFs | Suspended-core fibres |

HCFs | Hollow-core fibres |

TIR | Total internal reflection |

FEM | Finite element method |

GEPM | Generalized elliptical pore model |

DR | Drawing ratio |

SDRs | Slow drawing ratios |

HDRs | High drawing ratios |

HFs | Holey fibres |

SEM | Scanning electron microscope |

CFD | Computational fluid dynamics |

ILU | Incomplete lower–upper |

PBG | Photonic band gap |

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**Figure 2.**Fibre temperature distribution according to Equation (32) with ${T}_{M}=$ 2050 (${}^{\xb0}$C).

**Figure 3.**(

**a**) Evolution of the axial component of the velocity ${\overline{u}}_{0}$ in the drawing direction $\overline{x}$. (

**b**) Dimensionless square root of the cross-sectional area $\overline{\chi}$ against $\overline{x}$. (

**c**) Evolution of the inner and outer surfaces during the drawing process. DR 36-1 and ${T}_{peak}=$ 2050 (${}^{\xb0}$C).

**Figure 4.**Final external diameter of the capillary. Comparison between experiments and numerical simulations for (

**a**) DR 36-1, (

**b**) DR 54-15, and (

**c**) DR 72-2.

**Figure 5.**Final air-filling fraction of the capillary. Comparison between experiments and numerical simulations for (

**a**) DR 36-1, (

**b**) DR 54-15, and (

**c**) DR 72-2.

**Figure 6.**(

**a**) Final external diameter and (

**b**) final air-filling fraction of the capillary. Comparison between experiments and numerical simulations.

**Figure 7.**Final external diameter of the capillary without inner pressurization for two different drawing ratios: (

**a**) DR 1-102 and (

**b**) DR 3-306. (

**c**) Final external diameter of the capillary with inner pressurization for the drawing ratio DR 2-204.

**Figure 8.**Final air-filling fraction of the capillary without inner pressurization for two different drawing ratios: (

**a**) DR 1-102 and (

**b**) DR 3-306. (

**c**) Final air-filling fraction of the capillary with inner pressurization for the drawing ratio DR 2-204.

**Figure 9.**(

**a**) Overlap of the results obtained with the present model, experimental SEM image, and FEM simulations by Luzi et al. [27]. (

**b**) Overlap of the results of the present model (red circles) and the experimental SEM image of a thirty−six−hole final cross-section by Frosz et al. [52]. The length of the scale bar is 10 ($\mathsf{\mu}$m).

Parameter | Symbol | Value | Units |
---|---|---|---|

Hot zone length ${}^{a,c}$ | L | 0.12 | m |

Density ${}^{b}$ | $\rho $ | 2200 | kg m${}^{-3}$ |

Surface tension ${}^{c}$ | $\gamma $ | 0.25 | N m${}^{-1}$ |

Initial external radius ${}^{b,c}$ | ${h}_{20}$ | 1 × 10${}^{-2}$ | m |

Initial internal radius ${}^{b,c}$ | ${h}_{10}$ | 3.65 × 10${}^{-3}$ | m |

Drawing ratio | DR 36-1 | ||

Feed speed ${}^{b,c}$ | ${W}_{0}$ | 6 × 10${}^{-5}$ | m s${}^{-1}$ |

Draw speed ${}^{b,c}$ | ${W}_{1}$ | 1.67 × 10${}^{-2}$ | m s${}^{-1}$ |

Drawing ratio | DR 54-15 | ||

Feed speed ${}^{b,c}$ | ${W}_{0}$ | 9 × 10${}^{-5}$ | m s${}^{-1}$ |

Draw speed ${}^{b,c}$ | ${W}_{1}$ | 2.5 × 10${}^{-2}$ | m s${}^{-1}$ |

Drawing ratio | DR 72-2 | ||

Feed speed ${}^{b,c}$ | ${W}_{0}$ | 1.2 × 10${}^{-4}$ | m s${}^{-1}$ |

Draw speed ${}^{b,c}$ | ${W}_{1}$ | 3.33 × 10${}^{-2}$ | m s${}^{-1}$ |

Drawing ratio | DR 1-102 | ||

Feed speed ${}^{b,c}$ | ${W}_{0}$ | 1.67 × 10${}^{-5}$ | m s${}^{-1}$ |

Draw speed ${}^{b,c}$ | ${W}_{1}$ | 1.7 × 10${}^{-1}$ | m s${}^{-1}$ |

Drawing ratio | DR 2-204 | ||

Feed speed ${}^{b}$ | ${W}_{0}$ | 3.33 × 10${}^{-5}$ | m s${}^{-1}$ |

Draw speed ${}^{b}$ | ${W}_{1}$ | 3.4 × 10${}^{-1}$ | m s${}^{-1}$ |

Drawing ratio | DR 3-306 | ||

Feed speed ${}^{b}$ | ${W}_{0}$ | 5.00 × 10${}^{-5}$ | m s${}^{-1}$ |

Draw speed ${}^{b}$ | ${W}_{1}$ | 5.1 × 10${}^{-1}$ | m s${}^{-1}$ |

Drawing ratio | DR 20-187 | ||

Feed speed ${}^{d}$ | ${W}_{0}$ | 3.33 × 10${}^{-4}$ | m s${}^{-1}$ |

Draw speed ${}^{d}$ | ${W}_{1}$ | 3.12 × 10${}^{-1}$ | m s${}^{-1}$ |

Drawing ratio | DR 10-42 | ||

Feed speed ${}^{e}$ | ${W}_{0}$ | 1.67 × 10${}^{-4}$ | m s${}^{-1}$ |

Draw speed ${}^{e}$ | ${W}_{1}$ | 7.0 × 10${}^{-1}$ | m s${}^{-1}$ |

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**MDPI and ACS Style**

Luzi, G.; Klapper, V.; Delgado, A.
Asymptotic Modeling of Optical Fibres: Annular Capillaries and Microstructured Optical Fibres. *Fibers* **2023**, *11*, 104.
https://doi.org/10.3390/fib11120104

**AMA Style**

Luzi G, Klapper V, Delgado A.
Asymptotic Modeling of Optical Fibres: Annular Capillaries and Microstructured Optical Fibres. *Fibers*. 2023; 11(12):104.
https://doi.org/10.3390/fib11120104

**Chicago/Turabian Style**

Luzi, Giovanni, Vinzenz Klapper, and Antonio Delgado.
2023. "Asymptotic Modeling of Optical Fibres: Annular Capillaries and Microstructured Optical Fibres" *Fibers* 11, no. 12: 104.
https://doi.org/10.3390/fib11120104