Abstract
Keywords
1. Introduction
Multimode optical fibers (MMFs) have been actively studied over the last decade due to their high potential for overcoming the capacity crunch of optical communications with the technology referred to as mode division multiplexing (MDM)^{[1]}. Currently, the main limiting factor of the modern systems, relying on singlemode fibers (SMFs), is the intensitydependent Kerr nonlinearity. Coarsely, methods for mitigation of nonlinear effects in SMFs can be divided into two categories: optical^{[2–4]} and digital^{[5–12]}. The former implies utilization of optical devices and relevant physical effects for lowering the impact of nonlinearities, whereas the latter relies on digital signal processing (DSP) and implementation of numerical algorithms, compensating for the corresponding impairments.
Amongst DSP techniques, digital backpropagation^{[9–12]} (DBP) stands out, as it is also capable of treating all other deterministic effects. The main idea behind DBP is solving the nonlinear Schrödinger equation^{[13]} in the direction opposite to that of the propagating light, which turns out to be possible due to the prior knowledge on the transmitting system, i.e., fiber specifications. Typically, DBP is accomplished with the help of the splitstep Fourier method^{[13]}, which has to be tuned so that it can operate within reasonable time boundaries without sacrificing performance. This tradeoff between computational effort and accuracy has been recently canceled with the help of deepneural networks (DNNs)^{[14,15]}.
In addition to the challenges posed by SMFs, MMFs propose their own, mostly originating from the nature of multimode interactions that arise in media, such as intermodal crossphase modulation and fourwave mixing, Kerr and Raman nonlinearities^{[16–22]}. For instance, interchannel cross talk represents a serious limitation for implementation of MMFs in telecommunication^{[23–26]} that is known to be successfully mitigated by multimode DBP (MMDBP)^{[25,26]}. SSF is naturally extendable to the multimode case, but its computational complexity scales as $O({N}^{4})$ for $N$ modes^{[27]}.
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It is noteworthy that no standards for MDM receiver operation principles have been established yet, and not every spatial demultiplexing scenario would be appropriate for conventional MMDBP schemes. In particular, demultiplexing may refer to utilization of bulky optics^{[28]}, physical mode separation^{[29]}, or implementation of numerical algorithms^{[30–33]}. The latter is mostly given by the technique referred to as intensityonly mode decomposition (MD), which implies separation of modes, based on the nearfield intensity distributions of multimode signals captured by a digital camera. This approach is deemed promising due to the simplicity of experimental realization of the receiver part and its reconfigurability. However, existing MMDBP methods operate in terms of independent mode signals, i.e., assuming that spatial demultiplexing is performed prior to DBP, whereas modern cameras are not able to provide temporal exposure of multimode signals, and, as a consequence, known MMDBP methods cannot be applied directly to the systems equipped with such a receiver.
To the best of our knowledge, DNNs have not been utilized as a tool for MMDBP, despite the fact that it is known that they can capture nonlinear interactions in MMFs^{[34,35]}, and, analogous to the case of SMFs, can provide fast and accurate calculation. In this Letter, we turn the tables and apply MD algorithms after DNNbased MMDBP, thus considering the DBP problem for entangled multimode signals defined by the nearfield intensity speckles. With the help of numerical simulations, we provide a proof of concept and demonstrate that such a recovery of initial modal content can take place in the case of severe nonlinearities. Additionally, we show that typical noise levels of modern cameras do not have remarkable influence on the performance of the proposed MMDBP. We consider the case of six linearly polarized (LP) modes, and following the naming convention, we call our fiber a fewmode fiber (FMF).
2. Methods
2.1. The main idea
The flow chart illustrating our approach is displayed in Fig. 1(a). Given the output nearfield intensity pattern, we want to recover the modal content of the input multimode signal. We propose utilizing two neural networks: one for MMDBP and another one for MD. Mode configurations at the fiber output are arranged as multimode beam speckles that are fed to MMDBP DNN, which results in those at the fiber input. The final stage is represented by MDDNN, which provides modal content of the initial multimode signal. In order to train the MMDBP network, we need pairs of input and output beam speckles that require numerical simulation of multimode signal propagation. Details on the MDDNN architecture lie beyond the scope of the current work, as this part is purely technical and can be given by any other method^{[30,31]}. In what follows, under MDDNN, we understand an appropriately trained vision transformer^{[33]}.
Figure 1.(a) Flow chart of the proposed method. A nearfield beam speckle, captured at the fiber output, is fed to MMDBP DNN, resulting in the recovered input pattern. MDDNN takes this input pattern and yields real and imaginary parts of complex mode coefficients. (b) Schematic of the proposed MMDBP DNN. We use a residual neural network (ResNet) based autoencoder that compresses information acquired from the speckles and maps it onto a vector from the latent feature space. The decoder maps it back to the speckle space. Three middle blue blocks denote fully connected (FC) layers with 512, 1024, and 512 neurons, respectively. After each FC layer, we also place a dropout layer. BatchNorm stands for batch normalization. We use the rectified linear unit (ReLu) as the activation function. (c), (d) Detailed structure of the encoder and the decoder
2.2. Nonlinear light propagation in FMFs
Given the complex electrical field modal envelope ${A}_{p}(z,t)$ of the mode $p$, pulse propagation in FMFs fulfills the generalized nonlinear multimode Schrödinger equation (GMMNLSE)^{[27,36]},
For obtaining the training data, we generate random amplitudes for Gaussian envelopes ${A}_{p}(t)$ with $p=1,2,\dots ,6$ and simulate multimode light propagation by solving Eq. (1) with the help of the massively parallel algorithm (MPA)^{[27]} on graphics processing unit. Then the output intensity distribution at the receiver is given by
It is worth noting that even MPA^{[27]} is computationally expensive and timeconsuming, while the main goal of this work is to prove consistency of the proposed MMDBP approach. Therefore, without loss of generality, we set simulation specifications so that the nonlinear interaction becomes colossal. Consideration of such scenarios, which are probably unsuitable for communication, allows us to observe severe interchannel cross talk on shorter distances. Detailed simulation specifications can be found in Table 1.

Table 1. Parameters Used in Simulations of Nonlinear Light Propagation in FMFs
2.3. The network
From the versatile pool of suitable DNN architectures, we opt for the model given by a residual neural network (ResNet) based autoencoder [see Figs. 1(b)–1(d)]. Input $64\times 64$ images are mapped by the encoder onto vectors from the latent feature space, which are processed by three dense layers and transformed to the output $64\times 64$ pictures by the decoder.
We quantify the network’s reconstruction quality by two values. The first one is the correlation,
Final estimation of the network’s performance is given by the mean squared error (MSE) between the genuine $\{\mathrm{Re}\text{\hspace{0.17em}}{A}_{i},\mathrm{Im}\text{\hspace{0.17em}}{A}_{i}\}$ and those obtained by MDDNN,
We train the network in two stages. At the first one, we use 150,000 image pairs and train our DNN for 100 epochs with the AdamW optimizer^{[38]}. The learning rate is set to $5\times {10}^{4}$, which is halved every 20 epochs. The loss function is chosen as MSE between two images. It turns out that after this stage, the network’s output carries visually unnoticeable reconstruction artifacts. In other words, despite high values of the correlation between genuine input speckles and those resulting from MMDBP, there is high discrepancy between the corresponding outputs of MDDNN. These artifacts can be considered and treated as noise. Therefore, the second stage is aimed at rectifying this problem by transfer learning of the MMDBP network for an extra 50 epochs with another 40,000data set by applying the MDDNN to ground truth and output speckles and calculating the loss function,
3. Results and Discussion
We validate our MMDBP model with an unfamiliar 1000data set of images. For each sample, we calculate the correlation and MSE. It is noteworthy that calculation of these metrics implicitly benchmarks our networks against MPA by formulation, since this algorithm has been utilized for collecting the training samples. The resulting averaged metrics achieve decent values: $\overline{C}=0.992$ and $\mathrm{MSE}=2.3\times {10}^{4}$. Figure 2(a) demonstrates an MMDBP example. The left panel shows the fiber output beam speckle, whereas the middle and the right panels display the true and the recovered input speckles, respectively. The resulting correlation is $C=0.995$. The impact of nonlinear interactions is reflected in Fig. 2(c), where we demonstrate energy redistribution among the modes in the case of the sample propagation shown in Fig. 2(a).
Figure 2.(a) Example of an output, initial, and recovered speckle, respectively, in the absence of noise [see Eq. (
However, random receiver noise often plays a crucial role in MD. There is always a tradeoff between noise levels and the number of decomposable modes^{[30,31]}. Therefore, investigation of the noise influence on MMDBP performance is of paramount importance. According to the most common way, such noise can be modeled as additive white Gaussian noise^{[30]},
Varying $\sigma $ yields different noise levels that we quantify by the signaltonoise ratio (SNR). We put our MMDBP network to the test for SNR values lying between 10 and 90 dB. We follow the same pipeline using the same validation 1000data set. The only difference is that inputs of the network are subjected to the noise procedure according to Eq. (6). We analogously calculate the quality metrics for each value of SNR. Figure 3 illustrates the behavior of the averaged MSE between the decomposed and original complex coefficients, as well as the average value of the correlation coefficient. Slightly above $\mathrm{SNR}=20\text{\hspace{0.17em}}\mathrm{dB}$, one can observe a threshold that defines the applicability of the current network architecture trained on the noisefree data. However, in some cases, the network performs well even at 10 dB [see Fig. 2(b)]. Taking into account that modern cameras provide SNR levels of 40–60 dB, our simulations reveal high potential for a realistic implementation of the proposed approach.
Figure 3.(a) MSE and (b) C versus SNR. Each point represents averaging over 1000 samples from the validation data set.
Herein, we proposed an alternative approach to the tedious task of compensation of intermodal interactions in FMFs relying only on the nearfield intensity distribution obtained at the fiber output. As a proof of concept, we provided a model trained for DBP of six consequent LP modes propagating through a stepindex fiber in the strongly nonlinear regime. This number of modes represents a typical value for the physically separable channels used for MMDBP. We have numerically shown that this technique might be a decent alternative to the existing pipeline of nonlinearity compensation methods in the case of FMFs. We hope that this principle of MMDBP might be useful for intensityonly receiverbased FMF communication systems.
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