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Abstract
Few-mode fiber (FMF) supporting many modes with weak-coupling is highly desired in mode division multiplexing (MDM) systems. The multi-parameter design of FMF becomes comparably difficult, inaccurate and time-consuming when it comes for complex fiber structures and many high order modes. In this work, we demonstrate a machine learning method using neural network to inversely design the desired FMF based on multiple-ring structure. By using the minimum index difference between adjacent modes as the weak-coupling optimization aim, we realize the inverse design of 4-ring step-index FMFs for supporting 4, 6 and 10 -mode operation, and 6-ring step-index FMF for supporting 20-mode operation. This method provides high-accuracy, high-efficiency and low-complexity for fast and reusable design of optical fibers, including particularly weak-coupling FMF in this work. It can be widely extended to a lot of fibers and has great potential for instantaneous applications in the optical fiber industry.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
To overcome the capacity crunch of optical communications based on single-mode fiber (SMF), different modes in few-mode fiber (FMF) can be employed for mode division multiplexing (MDM) [1–6]. Due to the coupling between modes, Multiple-Input-Multiple-Output (MIMO) processing will be needed for strongly coupled systems, which significantly increases the cost and power consumption [7–11]. Therefore, weakly coupled system based on weakly coupled FMF would be more preferable in high speed MDM communications, due to its reduced complexity in signal processing [12–16]. Particularly, MIMO-less MDM communication can also be achieved for short reach optical interconnections [17–20].
Weakly coupled FMFs have two important parameters: one is the number of independent modes that can be accommodated, and the other is the mode effective index difference between different modes ($\Delta {n_{eff}}$). The model crosstalk of the fiber is related closely to the minimum value of $\Delta {n_{eff}}\; $[21]. To better support MIMO-less MDM, one can use degenerate modes in groups for multiplexing [22,23]. Generally, when the minimum index difference between adjacent modes $\Delta {n_{eff,min}}$ is larger than $1 \times {10^{ - 3}}$, the coupling between mode groups in FMF can be approximately neglected so as to realize weakly coupled MDM.
The design of weakly coupled FMF is to obtain large value of $\Delta {n_{eff,min}}$ with large number of modes. Conventional methods are based on parameter-sweeping forward design, which is quite limited due to the difficulty for supporting multi-parameter optimization and complex structure. Such methods are generally very time-consuming especially for complex structures which need very fine meshes for high precision. Furthermore, researchers have exploited genetic algorithm (GA) and particle swarm optimization (PSO) in designs of optical fibers [24–26]. These optimized algorithms can’t guarantee the best convergence for complex structures. At the same time, the results are basically not reusable and refreshed simulation will be needed each time for a new design, makes them time-consuming.
Presently, the demonstrated FMFs are all based on the conventional forward design methods. Simple step-index fibers are firstly used [27,28], which is difficult to enlarge refractive index of higher-order modes. High index level and graded index profile are usually utilized, which leads to highly increased fabrication difficulty and cost [21]. Multiple step-index ring structure is thus more preferable and several FMFs have been demonstrated with limited weak-coupling performance [29]. The challenge for the ring-assisted FMF is to optimize the parameters of each ring to increase the number of modes and reduce mode crosstalk simultaneously. The optimization involves many parameters, including the radius and refractive index values of each ring. The objective of this work is to overcome this challenge via machine learning aided inverse design method, which can effectively replace the complex physical design process and achieve design automation.
In this paper, we propose an inverse design method based on neural network (NN) to optimize the structural parameters of ring-assisted FMFs that can accommodate 4, 6, 10 and even 20 modes (groups), and ultimately realize weak coupling with effective index difference between adjacent modes ${\Delta }{n_{eff}}$ greater than $1 \times {10^{ - 3}}$. By training the NN, the structural parameters of the optical fiber are calculated instantaneously according to the distribution of the effective refractive index value with low complexity of computation. This method provides high-accuracy, high-efficiency and low-complexity for fast and reusable design of optical fibers, including particularly weak-coupling FMF in this work. It can be widely extended to a lot of fibers and has great potential for instantaneous applications in the optical fiber industry.
2. Inverse design process and neural network structure
2.1. FMF inverse design process
The entire design process of FMF is shown in Fig. 1, which can be divided into two steps, forward design and inverse one. Neural networks can bypass the complex relationships in physical processes by using simple linear operations and non-linear operations in combination to fit the mapping between two sets. In the forward design process, we input different fiber structure parameters into the simulation software to generate the data set. The size of the data set needs to be carefully selected. Too large data set wastes training time, and too small data set is difficult to obtain accurate fitting results. After obtaining the suitable data set, we need to normalize the parameters and input them into the NN. Afterwards, we train the NN by continuously adjusting the weight of each layer and comparing the error values. In the inverse design process, we set a set of parameters (optimization aims) that meet the target performance, input them into the trained NN, and predict the corresponding structural parameters. Finally, we put these structural parameters into simulation software to evaluate the accuracy of the NN.
2.2. Data set for training
The FMF in this work has multiple ring core structure with step index profile, as shown in Fig. 2(a). The data set for training is thus obtained based on such kind of ring-core structure, including 4-ring-core and 6-ring-core in this work, with the consideration of good performance of FMF weak coupling optimization and low cost of practical fabrication [30].
Fig. 2. The parameters of ring-assisted FMF. (a) 6-ring-core FMF structure (b) schematic diagram of four LP modes of 4-ring-core FMF; (c) schematic diagram of twenty LP modes of 6-ring-core FMF;
Figure 2(b) and (c) present the schematic index profiles of the 4-ring-core and 6-ring-core FMFs. The cladding of FMF is the background material which is silica glass with the refractive index of 1.445 at 1550 nm. The fiber diameter is set at 125 um for compatibility to standard fibers. Then, the fiber design parameters includes the ring radius and the refractive index of each ring.
Our goal is to make $\Delta {n_{eff,min}}\; $as large as possible. Therefore, we firstly need to obtain enough data set for training, which is based on finite element method simulation used commercial software (like Lumerical or COMSOL). Take the 6-ring 20-mode FMF as the example, we put different parameter group [${r_1}$, $ {r_2}$, $ {r_3}$, $ {r_4}$, $ {r_5}$, ${r_6}$, $ \Delta {n_1}$, $ \Delta {n_2}$, $ \Delta {n_3}$, $ \Delta {n_4}$, $ \Delta {n_5}$, $ \Delta {n_6}$] into the simulation software (Lumerical), where ${r_i}$ is the core radius, $\Delta {n_i}$ is the refractive index difference between the ith ring and the cladding (background silica glass), i = 1∼6. Then, we can obtain the effective refractive index values for the desired modes as the data set for training. For 20-mode FMF, it would be [${n_{eff,1}},{n_{eff,2}},\; {n_{eff,3}},{n_{eff,4}}, \ldots \ldots ,{n_{eff,20}}$], where ${n_{eff,i}}$ is the effective index of the ith mode. To generate a data set, with M=46656 32-dimensional elements, we set certain ranges of the ring radius and refractive index difference so as to support 20 modes and provide a wide range of effective index. By considering the data set more effective and the fiber structure more reasonable, the range is set as shown in Table 1. The initial range was chosen by meeting the following conditions: 1) the FMF’s structure with parameters in range should support enough number of modes that we aim to optimize. 2) the corresponding weak-coupling performance parameters should include at least one set of data that meeting the minimum index difference between adjacent modes close to 0.001. This simulation took about a week.
Table 1. The structure parameter ranges of 6-ring-core FMF
2.3. Neuro networking structure and optimization algorithm
Then we use this data set to train the NN (shown in Fig. 3). The structure of NN is very significant because it determines the accuracy of mapping. Deep neural network is a good choice to fit complex mappings between input and output. We set up the NN structure by Keras, an open source artificial NN library written in Python. For the accuracy of the NN, the selection of the scale of the NN is very important. So we chose three hidden layers where each layer has 300 neurons, as shown in Fig. 3(b). What ‘s more, the choice of activation function and loss function is also a key step in building a NN [31]. We employ two activation functions, relu and sigmoid, and the relu function is used in the first three layers. The sigmoid function is applied in the last layer because it is convenient to compress data between 0 and 1 and converge in forward propagation. The loss function is used to characterize the relationship between model output and training samples. There are three common loss functions: hinge function, mean square error (MSE), and cross entropy loss function. The MSE function is the mean square error between the model and the sample. The ideal NN model is obtained by continuously reducing the MSE value. Similarly, the cross-entropy loss function can also be used in classification tasks, which is a smoothing function. After comparing the performance of these two functions, we finally use the cross-entropy loss function as the loss function.
Fig. 3. The inverse design frame of NN. (a) The FMF modes; (b) NN structure; (c) 6-ring FMF structure.
The choice of optimizer is also an important procedure. In machine learning, there are many optimization methods that can be used to find the optimal solution of a model by minimizing the loss function. Gradient descent is a commonly used optimizer, of which stochastic gradient descent (SGD) is the most widely used. But this method is easy to fall into a local optimal solution. Therefore, we use an adaptive learning rate optimization algorithm to continuously improve the operation speed and accuracy by updating the learning rate. By comparison, the optimizer is Adam algorithm with good universal performance.
We can preliminarily determine whether the training effect of NN is perfect by checking the output relationship between the actual value and the predicted value. Then we formulated an objective set [${n_{eff,1}},{n_{eff,2}},\; {n_{eff,3}},{n_{eff,4}}, \ldots \ldots ,{n_{eff,20}}$] based on the objective of $\Delta {n_{eff,min}}$, and put the set into the trained NN. Through the prediction of the NN, we can get the core radius and refractive index difference set [${r_1}$, ${r_2}$, $ {r_3}$, $ {r_4}$, $ {r_5}$, ${r_6}$, $\Delta {n_1}$, $ \Delta {n_2}$, $ \Delta {n_3}$, $ \Delta {n_4}$, $ \Delta {n_5}$, $ \Delta {n_6}$] corresponding to the target value. In order to verify the accuracy, we input the core parameter set into Lumerical to get a prediction set. The criterion for judging the perfection of the whole design is whether the difference between objective set and prediction set is small.
3. Results and analysis
According to the above principle, we firstly optimized a 4-ring FMF with four modes for weak coupling MDM [30]. This result preliminarily proves the feasibility of the method of NN based reverse design to obtain the structure of FMF with weakly coupling. We then further investigate the optimization of FMFs with 6, 10 and even 20 modes, based on 4 and 6 -ring structures.
Taking the optimization of a 6-ring FMF with 20 modes as an example, the specific performance of this method is described in detail below. After training the NN with the data set obtained above, we draw up 100 ${\Delta }{n_{eff,min}}$ data in the range of $({8.42 \times {{10}^{ - 4}},\; 10.47 \times {{10}^{ - 4}}} )$, and respectively formulate 100 sets of $[{{n_{eff,1}},{n_{eff,2}},{n_{eff,3}}, \ldots ,{n_{eff,20}}} ]$ as the test data set to verify the accuracy of NN. They are not exactly the same as the data in the initial data set. The result will be accurate when ${n_{eff}}$ is in the range of performance parameters that can be generated by the range of structural parameters. These are actual data. Obviously, ${\Delta }{n_{eff,min}}$ is a second-order parameter because there is no one-to-one mapping relationship between ${\Delta }{n_{eff,min}}$ and FMF’s structure, while ${n_{eff,i}},\; i = 1,\; \ldots ,\; 20$, which is directly related to the NN, is a first-order parameter. For NN, the accuracy of the first parameter prediction is often greater than that of the second parameter. Although the first parameter is not the direct data to meet the weak coupling requirements, what we only need is that the second parameter can reach the target value. Predicted data can be obtained by the above analysis. In order to evaluate the prediction accuracy of multiple sets of data, we use correlation graphs to compare actual data with predicted data. In Fig. 4, we use five of the nineteen sets of data to display the results. The abscissa is the actual value and the ordinate is the predicted value. What we want is that the predicted value is infinitely close to the actual value. So the blue proportional line is our goal and dots represent our predictions. Most of the dots are on the blue line. The correlation coefficient of the ${n_{eff,i}}$ prediction is in the range of $({0.9953,\; 0.9996} )$, indicating good predictability. And the correlation of ${\Delta }{n_{eff}}$ (index difference between different modes) is above $0.99$ (shown in the Fig. 4). It can be observed that the predicted performance coincides well with the original target performance. It allows us to make a judgement that the predicted structure can reproduce ${n_{eff,i}}$ and ${\Delta }{n_{eff,min}}$ in good agreement with actual data.
To obtain better weakly-coupled characteristics, the standard is not only ${\Delta }{n_{eff,min}}$, but also the average and standard deviation of ${\Delta }{n_{eff}}$ of twenty modes. According to the above standards, we formulate a group of ${\Delta }{n_{eff}}$(shown in Table 2) as predicted value. The relative errors of the predicted value of ${n_{eff,mn}}$ are less than $0.0017\%$, compared with the actual value. At the same time, the relative errors of the predicted value of $\Delta {n_{eff}}$ are shown in the Table 3, the predicted structural parameter and its corresponding mode curve are shown in the Fig. 5 and Fig. 6. As shown in the Fig. 7, predict $\Delta {n_{eff,min}}$ is $10.495 \times {10^{ - 4}}$ . Besides, the average value is $1.4 \times {10^{ - 3}}$ and standard deviation is $4.603 \times {10^{ - 4}}$. The above results show that NN provides a new method for inverse design of FMFs with robust reliability.
Table 2. The relative errors of the predicted value of $\Delta {\boldsymbol {n}_{\boldsymbol {eff}}}$
In the actual manufacturing process, the dopant diffusion will make the multiple-ring FMF’s sharp edges round off, which means the refractive index of material near the edge of high refractive index ring will decrease, while that near the edge of low refractive index ring will increase. Considering this effect, we can adjust the refractive index of the edge of different rings. For easier operation, we flexibly convert this change into changing the radius of different rings, which means the edge is totally diffused. Besides, combined with the influence of preform doping and fiber drawing accuracy, we analyzed the tolerance of the structural parameters. We set the deviation of r and $\Delta n$ of the predicted FMF (shown in Fig. 5) from 1% to 5%, and find the effective refractive index distribution of the corresponding deviation. As shown in Fig. 8, with the increase of deviation, the $\Delta {n_{eff,min}}$ decreases a little bit. The most important criterion of weakly-coupled FMF is $\Delta {n_{eff,min}} > 1 \times {10^{ - 3}}$. We can reach this standard until the deviation exceeds 1.4%. As for a larger deviation of 5%, the value of $\Delta {n_{eff,min}}$ has a 15% of decrease to $8.74 \times {10^{ - 4}}$. In addition, we calculate the impact of the variation of a single parameter on ${\Delta }{n_{eff,min}}$, while holding other parameters fixed, as shown in Fig. 9. We found that the outer core radius ${r_6}$ has the strongest role in the ${\Delta }{n_{eff,min}}$, which cause high $\Delta {n_{eff}}$ between higher order modes. Meanwhile, ${r_5}$, $d{n_3}$ and $d{n_4}$ have the secondary dominant effects after ${r_6}$. Nevertheless, the sensitivity of $\Delta {n_{eff,min}}$ to the structural parameters of the first and second ring, was relatively small.
Fig. 9. The effective refractive index difference distribution of different deviation of single parameter.
Table 3 presents the final model characteristics of the designed 6-ring 20-mode fiber at 1550 nm wavelength. The refractive index ranges from $1.453$ to $1.481$. The dispersion ranges from $1.427$ to $34.366$ ps/nm/km. The optimization for wideband operation is not included in this work, but it can also be realized by this method given suitable date sets and optimization aims.
Table 3. The result of 20-mode FMF based on 6-ring structure
For 4-ring FMF, actually, we also carried out the inverse design for weak coupling optimization of the 6-mode and 10-mode FMFs. And these experiments are implemented to test the feasibility and reusability of this proposed method in more modes. The results are shown in Fig. 10. The predicted performance matches the target performance very well. And the correlation of $\Delta {n_{eff}}$ (index difference between different modes) of 6-mode FMF and 10-mode FMF are above $0.99$. The actual $max\Delta {n_{eff,min}}$ of 6-mode FMF is $1.23 \times {10^{ - 3}}$ and actual $max\Delta {n_{eff,min}}$ of 10-mode FMF is $1.27 \times {10^{ - 3}}$. Besides, the relative errors of the predicted value of ${n_{eff,mn}}$ are less than $0.0010\%$ and $0.0006\%$ respectively, compared with the actual value. For $max\Delta {n_{eff,min}}$, the predicted structure parameters of 6 and 10 modes are shown in Fig. 10.
Different FMFs for supporting different number of modes can be utilized for versatile applications, such as 4-mode FMF for compatible with optical interconnection multisource agreement (MSA) like 4-channel parallel QSFP, PSM4 and so on.
In addition, a larger $\Delta {n_{eff,min}}$ can be obtained by reasonably adjusting the range of structural parameters in the data set, such as introducing a low-index trench structure [21]. As shown in Fig. 11, the $max\Delta {n_{eff,min}}$ of 6-mode FMF is $3.135 \times {10^{ - 3}}$ and $max\Delta {n_{eff,min}}$ of 10-mode FMF is $1.906 \times {10^{ - 3}}$, much larger than former structure. The relative errors of predicted and actual data are less than $0.0019\%$ and $0.0018\%$ respectively. And the predicted structure parameters are shown in Fig. 12. We further applied this structure to 20-mode FMFs. By training NN and formulating proper predict data set, the $max\Delta {n_{eff,min}}$ can increase from $10.495 \times {10^{ - 4}}$ to $11.14 \times {10^{ - 4}}$. The predicted structural parameters are shown in Fig. 12. As shown plotted in Fig. 13, the blue dots are marked as our target value and blue line represents our prediction $\Delta {n_{eff}}$. The predicted $\Delta {n_{eff}}$ distribution is close to actual data and the relative errors of the predicted ${n_{eff,mn}}$ are less than $0.0025\%$, compared with the actual value. Besides, the average value is $1.8 \times {10^{ - 3}}$ and standard deviation is $5.72 \times {10^{ - 4}}$.
Fig. 11. The effective refractive index difference distribution and predicted structure parameter of (a) four and (b) ten modes.
We also set the deviation of r and $\Delta n$ of the predicted FMF (shown in Fig. 13) from 1% to 5%, and the effective refractive index distribution are shown in Fig. 14. We can maintain $\Delta {n_{eff,min}} > 1 \times {10^{ - 3}}$ until the deviation exceeding 3.2%. As for a larger deviation of 5%, the value of $\Delta {n_{eff,min}}$ is 15% decreased from $8.75 \times {10^{ - 4}}$.
Table 4 presents the characteristics of the designed 6-ring trench structure 20-mode fiber at 1550 nm wavelength. The refractive index ranges from $1.446$ to $1.481$. The dispersion ranges from $2.459$ to $41.938$ ps/nm/km. However, the ${A_{eff,min}}$ decreases from 146 $ \mu {m^2}\; $to 104 $\mu {m^2}$ with the increased $\Delta {n_{eff,min}}$. The above results show that using NN to inversely design the structure of FMF also present good properties in a specific more complex structure, and can be used to achieve better weak-coupling performance.
Table 4. The result of 20-mode FMF based on 6-ring trench structure
By using NN in inverse design, we do not need to know the specific and complex mathematical or physical relationship between the structure and optical performance of FMFs. NN can automatically establish the mapping between input and output, which greatly facilitates our design. Besides, this method has a high universal applicability, because as long as there is a certain correlation between input and output, we can get an accurate prediction value by adjusting the weight of the NN for different objectives without refreshed simulation, making the established model reusable. This method can also be extended to more applications as the dispersion and the loss by obtaining a data set and adjusting the weight of the NN. The trained NN can reasonably and quickly predict FMF’s structures satisfied the performance parameters, which are corresponding to arbitrary combination of values in the range of each ring structure parameters. Compared with the traditional method based on physical principal [32,33], the NN can get a relatively high accuracy result in a very short time. What’s more, for any given specific value of optimized aims, the optimized optical fiber structure can be inversely designed theoretically by this method.
4. Conclusion
As a conclusion, we proposed a novel machine learning method in this work to inversely design and optimize weakly coupled FMF. It has been demonstrated that neural network is an efficient tool to predict the FMF structure for the desired optimization aims. In this way, we have successfully designed 4-ring and 6-ring FMFs for supporting 4, 6, 10, and 20 -mode MDM communication with weak coupling optimization. A large $\Delta {n_{eff,min}}$ ($> {10^{ - 3}}$) for 20-mode weak coupling FMF is obtained. We believe this proposed method would be of great value in the FMF fabrication and MDM communication.
Funding
National Key Research and Development Program of China (2018YFB1801004); National Natural Science Foundation of China (61675128, 61875049, 61875124, 61935011); Science, Technology and Innovation Commission of Shenzhen Municipality (JCYJ20180507183418012, KQJSCX20180328165451777).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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