# Treatment of Mode Coupling in Step-Index Multimode Microstructured Polymer Optical Fibers by the Langevin Equation

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## Abstract

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## 1. Introduction

_{1}can be easily reduced with larger or more densely spaced holes in the cladding. The transmission of light along the microstructured optical fibers is influenced by differential mode coupling, modal attenuation, and modal dispersion [18]. Mode coupling is the process of energy transfer between neighboring modes during their propagation along the optical fiber. Mode coupling is mostly induced by intrinsic random perturbations of the fiber, such as refractive index variations, microbends, and stresses [19,20,21].

_{0}to the fiber axis, a ring can be imaged behind the output end of a short fiber—the ring diameter is related to that initial launch angle ${\theta}_{0}$. As the fiber is “lengthened” (replaced by longer and longer fibers), the edges of this ring become blurred and the ring morphs gradually into a disk. This is due to effects of mode coupling accumulating with distance from the input end and causing the angular power distribution, initially narrowly centered around θ = θ

_{0}, to gradually widen and shift towards θ = 0°. At the coupling length L

_{c}, the distribution, even of the highest order guiding mode, has shifted its midpoint to zero degree, where the equilibrium mode distribution (EMD) is achieved. By lengthening the fiber to beyond the value known as ${z}_{s}$, the angular light distribution becomes fixed and centered (the disk is brightest in its center). This is a steady-state distribution (SSD) that is independent of the launch conditions except for the overall brightness: normalized to its peak value, the SSD is one and the same whatever the launch angle(s). By employing the power flow equation [19], these patterns have been predicted as a function of the launch conditions and fiber length in SI mPOF. In this paper, we report for the first time on the application of the Langevin equation in the treatment of mode coupling in SI mPOF. This way, by solving a stochastic differential equation (the Langevin equation), we show that one can successfully treat a mode coupling in multimode SI mPOF caused by its intrinsic random perturbations.

## 2. The Langevin Equation

_{n}depends only on the value θ

_{n−}

_{1}at the preceding position z

_{n−}

_{1}. To solve the Langevin Equation (3), the fiber length z = z

_{f}is divided into N length steps k:

_{n+}

_{1}at fiber length z

_{n+}

_{1}is determined by the following discretized Langevin equation:

_{0}, ω

_{1},…, ω

_{N−}

_{1}are independent random numbers with Gaussian distribution, zero mean <ω

_{n}> = 0, and variance <ω

_{n}ω

_{n′}> = 2δ

_{nn′}. This way, one obtains θ

_{N}= θ(z

_{f}). By calculating a large number of representations of ω

_{n}, and averaging in appropriate intervals ∆θ, one obtains <θ(z

_{f})>.

## 3. Numerical Results and Discussion

_{0}is the RI of the core, $\mathsf{\Lambda}$ is the pitch, d is the hole diameter of the cladding,${a}_{eff}=\mathsf{\Lambda}/\sqrt{3}$ [25], $\lambda $ is the operating wavelength, and fitting parameters ${A}_{i}$ ($i=1$ to 4) are given as:

^{2}/m [19]. In the calculations, we used a drift coefficient W = (0.0051 ± 0.0005) rad/m, which was determined by averaging the rate of switching from the ring to the disk output field pattern for low- and high-order modes (incidental angles) [22]:

_{r}is a drift coefficient of the rth mode. In Equation (9), drift coefficients W

_{r}(r = 1, 2) for modes with launch angles θ

_{0}= 5°and 10°were averaged. We performed a Monte Carlo sampling of $5\times {10}^{5}$ representations of the ω

_{n}in Equation (5) in intervals ∆θ = 0.2°, where k = 0.0001 m was used.

_{0}= 0°, 5°, and 10° are compared with our previously reported results obtained by solving the power flow equation [19]. There is a high degree of agreement between these results, with mean square errors below 1%. The radiation patterns in the short fiber (z = 2 m) in Figure 3b indicate that distributions of low-order modes have shifted towards θ = 0°. Higher-order mode coupling can be observed after longer fiber lengths. It is not until a fiber’s coupling length L

_{c}of 39 m that all the mode distributions have shifted their midpoints to zero degree (from the initial value of θ

_{0}at the input fiber end), producing the EMD in Figure 3c. The coupling continues beyond the L

_{c}mark until all distribution widths equalize and SSD is reached at length z

_{s}in Figure 3d: z

_{s}= 102 m.

^{®}Core™ i3 CPU 540 at 3.07 GHz computer for the Langevin equation and the power flow equation was 1.8 min and 2.7 min, respectively. The numerical solution of the power flow equation is more complex than the solution of the Langevin equation.

_{c}lies in the fact that at fiber lengths shorter than L

_{c}, the pulse spreading is linear with length, while after establishing the EMD at length L

_{c}, it has a z

^{1/2}dependence. Therefore, the shorter length L

_{c}is more desirable since it results in a slower bandwidth decrease [26].

## 4. Conclusions

_{c}the pulse spreading is linear with length, while after establishing the EMD at length L

_{c}, it has a z

^{1/2}dependence. Therefore, the shorter length L

_{c}is more desirable since it results in a slower bandwidth decrease in SI mPOFs. This is significant because mode coupling has an impact on the vast majority of fiber-based applications.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Cross-section of multimode SI mPOF, (

**b**) refractive-index profile of the referent multimode SI mPOF.

**Figure 3.**Normalized output angular power distribution calculated by solving the Langevin equation for launch angles ${\theta}_{0}$ = 0°(∎), 5°(•), and 10°(▲) and normalized output angular power distribution calculated by solving the power flow equation [19] for launch angles ${\theta}_{0}$ = 0°( ), 5°(- - -), and 10° (• • •), for fiber length (

**a**) z = 2 m; (

**b**) z = 15 m; (

**c**) $z\equiv {L}_{c}=39\text{}\mathrm{m}$; (

**d**) $z\equiv {z}_{s}$ = 102 m ($\mathsf{\Lambda}=3\text{}\mathsf{\mu}\mathrm{m}$ and d = 2 µm).

$\mathit{i}\mathbf{=}\mathbf{1}$ | $\mathit{i}\mathbf{=}\mathbf{2}$ | $\mathit{i}\mathbf{=}\mathbf{3}$ | $\mathit{i}\mathbf{=}\mathbf{4}$ | |
---|---|---|---|---|

${a}_{i0}$ | 0.54808 | 0.71041 | 0.16904 | −1.52736 |

${a}_{i1}$ | 5.00401 | 9.73491 | 1.85765 | 1.06745 |

${a}_{i2}$ | −10.43248 | 47.41496 | 18.96849 | 1.93229 |

${a}_{i3}$ | 8.22992 | −437.50962 | −42.4318 | 3.89 |

${b}_{i1}$ | 5 | 1.8 | 1.7 | −0.84 |

${b}_{i2}$ | 7 | 7.32 | 10 | 1.02 |

${b}_{i3}$ | 9 | 22.8 | 14 | 13.4 |

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**MDPI and ACS Style**

Savović, S.; Li, L.; Savović, I.; Djordjevich, A.; Min, R.
Treatment of Mode Coupling in Step-Index Multimode Microstructured Polymer Optical Fibers by the Langevin Equation. *Polymers* **2022**, *14*, 1243.
https://doi.org/10.3390/polym14061243

**AMA Style**

Savović S, Li L, Savović I, Djordjevich A, Min R.
Treatment of Mode Coupling in Step-Index Multimode Microstructured Polymer Optical Fibers by the Langevin Equation. *Polymers*. 2022; 14(6):1243.
https://doi.org/10.3390/polym14061243

**Chicago/Turabian Style**

Savović, Svetislav, Linqing Li, Isidora Savović, Alexandar Djordjevich, and Rui Min.
2022. "Treatment of Mode Coupling in Step-Index Multimode Microstructured Polymer Optical Fibers by the Langevin Equation" *Polymers* 14, no. 6: 1243.
https://doi.org/10.3390/polym14061243