At present, few-mode fiber (FMF) communication technology, which is popular, is based on mode division multiplexing (MDM). The technology uses the new freedom of mode as an independent channel for information transmission. It can break through the capacity limit of traditional single-mode optical fiber communication and become the key to Tbit/s or even Pbit/s optical fiber communication. FMF fusion splicing is inevitable in the MDM system based on FMF. Therefore, the accurate analysis of the misalignment tolerance of FMF fusion splicing is of great significance for evaluating the fusion-splicing quality and optimizing the matching parameters of FMF, as well as ensuring the reliable and efficient operation of long-distance and large-capacity FMF links. At present, the tolerance of the coupling loss of the fundamental mode LP₀₁ to the transverse offset, rotation angle, and fiber parameter misalignment are the main focus of a priori research. For the FMF that supports multiple spatial modes with coupling between spatial modes, the traditional single-mode fiber LP₀₁ coupling loss theoretical is no longer suitable for analyzing coupling characteristics and misalignment tolerance of FMF fusion splicing. Given the current situation, it is of significance to study the tolerance characteristics of the high-order spatial mode coupling efficiency on the fusion splicing parameters of different transverse offsets and rotation angles under different parameters of FMF.

We propose a theoretical model based on the Laguerre-Gaussian mode to analyze the misalignment tolerance of FMF fusion splicing. Laguerre-Gaussian mode is utilized to approximate the LP_mn mode field distribution of each spatial mode. The coupling efficiency model between LP_i mode (transmitting FMF) and LP_j mode (receiving FMF) at the FMF fusion splicing is calculated by using the power transmission coefficient. The mathematical model can be estimated as a function of Gaussian waist radius ω, transverse migration d, and angle misalignment θ of the transmitting and receiving FMF. Based on the coupling efficiency model, the tolerance of the spatial mode coupling efficiency at the fusion point to different transverse migrations, rotation angles, and other fusion splicing parameters is analyzed under the welding conditions of different normalized cutoff frequencies and core radius parameters of FMF.

The fusion splicing of six-mode fiber (LP₀₁, LP_11a, LP_11b, LP_21a, LP_21b, and LP₀₂ modes) is taken as an example. The numerical analysis results show that under the fusion splicing conditions of different fiber parameters V and a, the coupling efficiency between different spatial modes presents different distribution rules for the tolerance of parameters d and θ. Figures 3 and 4 show the variation curves of the self-coupling efficiency of LP₀₁ mode and the mutual coupling efficiency between LP₀₁ mode and high-order mode with V (normalized cutoff frequency) and a (the core radius of the optical fiber) of FMF parameters under different fusion splicing transverse offsets (0-2 μm), respectively. The analysis shows that the self-coupling efficiency of LP₀₁ increases and then decreases with the increase of V and a. The greater the difference of V and a between transmitting and receiving FMF, the lower the tolerance of LP₀₁ self-coupling efficiency to the transverse offset. Meanwhile, the influence of a on the coupling efficiency tolerance to the transverse offset is greater than that of fiber V. The coupling efficiency between LP₀₁ and the high-order mode also has a similar variation rule. Figures 5 and 6 present the analysis results of the tolerance of spatial mode coupling efficiency to fusion-splicing angle misalignment. Since the field distribution of fundamental mode LP₀₁ is axisymmetric, the self-coupling efficiency of LP₀₁ presents consistent distribution characteristics under different angle misalignments. Similarly, due to the axisymmetric distribution of the LP₀₂ field, the η16 is not affected by the angle misalignment. As V₂ increases, the value decreases and then increases. The coupling efficiency between LP₀₁ and degenerate modes LP_11a, LP_11b, LP_21a, and LP_21b is affected by sinθ and cosθ factors, and the coupling efficiency is symmetrically distributed relative to the welding angle misalignment. Similarly, the variation trend of coupling efficiency under the condition of a is greater than that of V. Figure 7 shows the efficiency distribution of the self-coupling of LP_21a modes and the efficiency distribution of the mutual coupling between LP_21a and LP₀₂ mode under different FMF parameters and fusion splicing parameters. Similarly, the coupling efficiency between high-order modes shows regular changes under different angle misalignments and transverse offset fusion. Therefore, it is necessary to strictly control the transverse offset and angle misalignment according to different parameters of FMF fusion splicing to achieve the ideal coupling efficiency.

In this study, a general theoretical analysis model of misalignment tolerance for FMF fusion splicing based on Laguerre-Gaussian mode is proposed given that the traditional single-mode fiber LP₀₁ coupling loss theoretical model is no longer suitable for analyzing the coupling characteristics and misalignment tolerance of FMF fusion splicing. Taking six-mode fiber fusion splicing as an example, the tolerance of the mode coupling efficiency to different transverse offsets and rotation angles of the fusion parameters is analyzed under the welding conditions of different normalized cutoff frequencies and core radius of FMF. The numerical results show that the theoretical model can be used to analyze the tolerance of the spatial mode coupling efficiency to different parameters such as transverse offsets and rotation angles. This model provides a theoretical basis for evaluating the welding quality of FMF and the alignment design and optimization of fusion splicing parameters.