Optical multiplexing is crucial for high-capacity communication, traditionally relying on orthogonal channels (e.g., space, wavelength, polarization, mode division multiplexing). However, this orthogonality constraint limits capacity. Breaking this paradigm opens the door to non-orthogonal multiplexing, offering significant potential for increasing information capacity. The inverse transmission matrix method, effective for demultiplexing orthogonal signals, fails for non-orthogonal channels sharing the same polarization, wavelength, and spatial position. Deep learning has shown promise in optics, particularly in improving orthogonal multiplexing performance through scattering media. However, its application to non-orthogonal multiplexing in multimode fibers (MMFs) remains unexplored. This research investigates the feasibility of using deep learning to achieve non-orthogonal optical multiplexing over an MMF, aiming to overcome the limitations of traditional methods and significantly enhance multiplexing capacity. The core challenge lies in finding an effective demultiplexing method for non-orthogonal channels, which lack the inherent separability of orthogonal signals. This work proposes a solution leveraging the power of deep learning to decode the complex interactions within a non-orthogonal multiplexing system.

Extensive research has explored various optical multiplexing techniques, focusing on orthogonal channels to simplify demultiplexing. Space division multiplexing (SDM) leverages the spatial modes of an optical fiber to increase capacity. Wavelength division multiplexing (WDM) uses different wavelengths for parallel transmission. Polarization division multiplexing (PDM) utilizes the two orthogonal polarization states of light. Mode division multiplexing (MDM) exploits the multiple spatial modes of a multimode fiber. However, the inherent orthogonality of these techniques ultimately restricts capacity. The transmission matrix method has been successfully used for demultiplexing orthogonal signals, even through scattering media like MMFs. However, this method fails for non-orthogonal signals, as the individual signals cannot be disentangled from a single intensity measurement. Recent advances have demonstrated the potential of deep learning to solve inverse problems in optics and improve the performance of orthogonal multiplexing. Deep neural networks have been successfully trained to reconstruct images transmitted through scattering media, but these applications primarily focus on orthogonal input channels. This research seeks to bridge this gap and extend the capabilities of deep learning to the realm of non-orthogonal optical multiplexing.

This study employs a speckle light field retrieval network (SLRnet), a deep neural network, to achieve non-orthogonal optical multiplexing over an MMF. The SLRnet learns the complex nonlinear mapping between multiple non-orthogonal input light fields, encoded with information in amplitude and phase, and the resulting single-shot speckle intensity output at the distal end of the MMF. The input-output relationship of a single channel through the MMF is described by a transmission matrix (Equation 1). For multiple non-orthogonal inputs (Equation 3), the output intensity is a complex function (Equation 4), making it impossible to retrieve individual input signals using the inverse transmission matrix method (Fig. 1a). The SLRnet solves this ill-posed problem by learning the inverse mapping (Equation 5), expressed as Equation (6), where the network minimizes the difference between its output and the actual input signals in the training set. The trained network then can retrieve information from the output speckle intensity (Equation 7), even for input signals outside the training set. The architecture of the SLRnet (Fig. 2b) is a variant of U-Net, incorporating a fully connected layer and residual blocks with skip connections to improve performance and robustness. The network is trained using a supervised learning approach with pairs of output speckles and corresponding input wavefronts. The network outputs predicted amplitude and phase information for each input channel. Information in non-orthogonal channels is orthogonally labelled during training, enabling the network to decode the signals. In the experiments, two input light channels were used, with each channel encoded with amplitude and phase information (Fig. 2a). Twelve combinations of polarization states were used to validate robustness. Pearson Correlation Coefficient (PCC) and Structural Similarity Index Measure (SSIM) were used to quantify the fidelity of the retrieved information. Experiments were performed using 1m and 50m MMFs, demonstrating successful non-orthogonal multiplexing even with the same polarization, wavelength, and input spatial region (Figs. 3-5). Furthermore, the SLRnet's performance was assessed on diverse datasets, including images from CelebA, random binary data, natural scene images from ImageNet, and historical Muybridge recordings (Fig. 6), highlighting the network's generalization capability. The experimental setup used a monochromatic laser (532nm), a spatial light modulator (SLM) for amplitude and phase modulation, a wave plate for polarization control, and a CMOS camera to capture the output speckles.

The study's key findings demonstrate the successful implementation of non-orthogonal optical multiplexing using a deep learning approach. The SLRnet achieved remarkably high fidelity in retrieving multiplexed information, reaching a Pearson correlation coefficient (PCC) of approximately 0.98 for various scenarios. This fidelity remained consistent across different combinations of polarization states, including linear, circular, and elliptical polarizations, and even when input channels shared the same polarization, wavelength, and spatial position. The network's ability to perform well under diverse input conditions highlights the robustness of the method. The results obtained with the 1m MMF were superior to those with the 50m MMF, which is likely due to increased environmental sensitivity in the longer fiber. Importantly, the SLRnet successfully demultiplexed information from a variety of input sources, such as general natural scene images, random binary data, and images not belonging to the training dataset, showcasing the network's excellent generalization capabilities. The achieved fidelity in many cases was remarkably high, often exceeding 0.90 for both SSIM and PCC. Furthermore, when information was encoded only in the phase dimension, an even higher fidelity of approximately 0.945 for PCC was observed, underscoring the potential benefits of optimized information encoding strategies. In comparison to the existing literature on orthogonal optical multiplexing, the results obtained in this study demonstrate comparable or even superior performance in terms of optical degrees of freedom, fidelity, and spatial channel numbers. These findings collectively establish the effectiveness and robustness of the proposed deep learning-based non-orthogonal optical multiplexing method.

The successful demonstration of high-fidelity non-orthogonal optical multiplexing through an MMF using the SLRnet addresses a critical limitation of traditional multiplexing techniques. By overcoming the orthogonality constraint, this approach offers the potential for significantly increasing the capacity of optical communication systems. The high fidelity of retrieved information, even in challenging scenarios with identical polarization, wavelength, and spatial position, validates the power of deep learning to disentangle complex signal interactions within the MMF. The ability of the SLRnet to generalize well to diverse datasets suggests a high degree of robustness and adaptability to real-world conditions. The findings have implications beyond optical communications. The ability to accurately retrieve non-orthogonal information could lead to innovations in various fields, such as sensing and imaging systems, where disentangling overlapping signals is crucial. The ability to achieve high-fidelity demultiplexing of uncorrelated digital data suggests a significant step towards high-capacity non-orthogonal information transmission.

This research presents a novel approach to optical multiplexing based on deep learning. The developed SLRnet successfully achieves non-orthogonal multiplexing over MMFs with high fidelity, overcoming the limitations of traditional orthogonal methods. This approach expands the possibilities of information capacity in optical systems. Future work could focus on incorporating physically-informed models into the neural network architecture, increasing training data diversity, and exploring alternative network architectures, such as transformers, to further enhance performance and reduce training data requirements. The exploration of additional optical degrees of freedom, such as wavelength and orbital angular momentum, could further increase the multiplexing capacity.

The current study is limited to the multiplexing of two channels. Scaling to a larger number of channels would require a substantially larger training dataset, posing a challenge for data acquisition. While the network demonstrated robustness to variations in polarization states, it might be less robust to other environmental factors. The performance with the 50m MMF was less optimal than with the 1m MMF, suggesting the need for further improvements to handle the increased scattering and environmental influences in longer fibers. The generalization ability of the trained model to completely unseen data remains to be thoroughly evaluated. The study focuses on a specific type of MMF and wavelength. Further investigations are needed to determine the generality of the approach to different fiber types and wavelengths.