All optical symbol classification based on diffractive optics has recently emerged as a promising technique. To enable classification of different symbols, the phase on single or multiple planes should be trained by data set with the target to maximize the field intensity within a few specific size-limited areas. While the trained optical network gives promising classification accuracy, its physical insight for the phase distribution is not clear, which makes it uncertain whether the performance can be further improved. In this work, the physical principle for optimal phase distribution on a single layer optical symbol classifier is revealed. The optical intensity within a specific area on the output plane is used as the optimization target, and the related orthonormal modes with respect to the output target can be found. The field on the tunable phase plane may be decomposed into multiple orthonormal modes and the optimal phase distribution is found by a simple iterative algorithm with the weighted summation of the orthonormal modes. The method is applicable when there are multiple input fields and multiple target areas. Numerical simulations as well as experimental validations have been conducted to verify that proposed optimal phase has very high power concentration within the target areas, which are over 25% higher than the results of the trained phase through gradient descent algorithm. The proposed optimal phase is also quite robust with respect to the misalignment. While the theory is implemented on a single plane optical symbol classifier, it can be extended to the multiplane case.

We report an ultra-compact and process-compatible vertical grating coupler on a silicon-on-insulator platform synthesized using a cascaded inverse-design framework. Our methodology employs a joint optimization strategy that combines the beam propagation impact of a hybrid L-shaped and strip grating. By integrating a boundary-scaling algorithm with explicit fabrication constraints into the gradient-based loop, we ensure strict adherence to an 80 nm or 100 nm minimum feature size while achieving structural robustness independent of initial configurations. Comparing with the metalens approach, which has its size to be physically constrained by its constitutive unit-cell dimensions, the proposed method scaled the focusing length down to 2.5 µm for a total footprint of 12.5 × 14.14 µm2. The resulting design exhibits promising performance across the C- and L-bands, yielding a peak efficiency of -3.36 dB and a 1-dB bandwidth of 59 nm with 80 nm minimum feature size. This work provides a practical and scalable pathway for high-density photonic integration.

Wave propagation inside optical waveguides is characterized by their eigen modes and eigenvalues. To obtain these important information, numerical algorithms convert the Maxwell equations into the linear equations, and hence, the waveguide mode problem becomes the matrix eigenvalue problem, whose efficiency is dependent on the matrix size. In this work, an orthonormal mode expansion based full vectorial mode solver is proposed with the matrix size significantly reduced. The full vectorial operator is decomposed into a Hermitian part and a perturbation part. The Hermitian part possesses orthonormal eigen modes, which can be used as the basis to compute the waveguide modes. The perturbation part constitutes a matrix with a greatly reduced size, which is determined by the number of orthonormal modes. As the matrix size shrinks, the computational resource and time to find the eigen modes and eigenvalues can be greatly saved by 2 orders of magnitude while maintaining a comparable accuracy.

The diffractive deep neural networks (D2NNs) have been widely adopted due to their advantages of ultra-low latency, low power consumption, and highly parallel optical computing capability. However, its hardware implementation faces several challenges, among which the alignment of multiple diffractive planes remains a major obstacle to practical deployment. To address this issue, we propose a parallel subnetwork-filtered diffractive deep neural networks (PSF-D2NNs) architecture to enhance the robustness of conventional D2NNs against unavoidable multi-plane misalignment and other system errors during the experimental implementation. The robustness of the proposed network is enhanced through two strategies. Firstly, a dataset incorporating random misalignment errors of the modulation planes is introduced. Secondly, phase filtering factors are employed to smooth the modulation phase. The proposed network demonstrates strong performance enhancement with respect to misalignment both in simulations and experiments. In simulation, PSF-D2NNs achieve a misalignment tolerance up to 26 pixels with an MSE of 5.441 × 10-5 and an SSIM of 0.7741, while the experimental tolerance exceeds 13 pixels with an MSE of 3.350 × 10-6 and an SSIM of 0.9760.

Optical waveguides serve as the fundamental basis for optical integration. Their optical characteristics arise from the mode field distributions and propagation constants, which determine the beam evolution inside the waveguides. Solving the modes of the optical waveguides is usually achieved by the transformation of the Maxwell equations into the linear equations, and hence, the computational complexity lies in the size of the matrix for the eigenvalue problem. Generally, the matrix size is determined by the grid size and therefore, it becomes quite time consuming to solve the two-dimensional full vectorial waveguide modes. In this work, a perfectly matched layer (PML) mode expansion method is proposed to find the full vectorial modes. The pseudo modes related to the PML boundary, which can serve as the basis functions to expand the waveguide modes, have explicit analytical formulations for the mode fields and propagation constants. The matrix size of the linear equation is reduced to the number of pseudo modes and henceforth, the computational cost is significantly reduced. Since the PML modes significantly reduce the reflection at the boundary, the accuracy for waveguide mode evaluation is highly guaranteed. Even for the leaky modes, the proposed full vectorial mode solver gives accurate predictions both for the mode fields and the propagation constants. The proposed method is expected to significantly reduce the computational cost for the design and analysis of the integrated photonic system while maintaining an acceptable accuracy.

Current broadband non-line-of-sight (NLOS) imaging techniques in the visible-light band face challenges in complex physical modeling and instrumentation. To address this, we propose a novel visible-light broadband NLOS imaging method based on an all-forward optical neural networks (AF-ONN). The system leverages the AF-ONN architecture to directly map the scattered light fields to hidden object images across the visible spectrum ($400-700 ~\text{nm}$). The system architecture is as follows: A collimated illumination source irradiates an out-of-view object, and its reflected light randomly scatters by the wall to form dynamic speckle patterns, which are captured by a camera. The camera is connected to an amplitude-only SLM, which outputs zero-phase images that serve as input to the AF-ONN. Crucially, both data (speckles) and error signals (the difference between the reconstructed image and the target image) are strictly forward-propagating, which does not require error backpropagation (BP) for ONN optimization. The mean squared error (MSE) between the output and target fields serves as the loss function, and standard gradient descent is used for iterative phase optimization through forward-propagating errors. Once trained, the AF-ONN enables real-time, high-fidelity reconstruction of hidden objects from input speckle images within the operational bandwidth. Totally 11 frequency points within the broadband spectrum were tested with the good performance of imaging. Compared to existing optical NLOS methods, our approach achieves, efficient full-visible-spectrum NLOS reconstruction while significantly reducing computational complexity without backpropagation. This breakthrough provides a new solution for high-resolution, real-time concealed imaging, opening new possibilities for the applications in autonomous navigation, remote sensing, and security surveillance.

Optical waveguide structure is characterized by its full vectorial modes and corresponding eigenvalues, whose calculation requires extensive computational resources. In this paper, a full vectorial mode solver based on the non-Hermitian adiabatic perturbation method accompanied by a scalar mode solver is proposed. The full vectorial mode solver gradually converts the initial modes generated by the scalar mode solver to the full vectorial modes by the non-Hermitian adiabatic perturbation theory, which incorporates the waveguide index gradient contribution and the modal non-orthogonality during the conversion. While saving about 3/4 of the computational time compared with the conventional full vectorial mode solver, the proposed method is quite accurate for the waveguides with low and high index contrast, as well as with large and small cross sections. The effective index discrepancy for the first six modes between the proposed method and the full vectorial mode solver is less than 4e-9 for a weakly guided thick waveguide and is less than 0.17 for a strongly guided thin waveguide. The newly proposed non-Hermitian perturbation method can be also potentially applied in other fields with non-Hermitian eigen mode problems.

Diffractive deep neural networks (D2NN) have been widely applied as a novel method for wavefront shaping and beam manipulation. However, achieving high-precision alignment across multiple planes remains a significant challenge. In this paper, we propose a phase-filtered diffractive deep neural network (PF-D2NN), which introduces a phase filtering operator during phase optimization for the modulation layers to enhance the robustness of the network. A back-propagation (BP) algorithm is specifically designed for the phase optimization of the PF-D2NN. Both simulations and experiments validate that the proposed PF-D2NN is quite robust with respect to the alignment error. The experiment results show that even when conventional D2NN fail to produce clear images with the misalignment over 5 pixels, the proposed network continues to deliver clear images for various targets.

Nonlinear frequency division multiplexing (NFDM) offers a possible solution to the fiber nonlinearity-induced signal distortion in the optical fiber networks. However, the application of NFDM is challenged during the information retrieval after perturbed propagation. The paper proposes a novel, to the best of our knowledge, nonlinear frequency domain neural network (NN)-based equalizer that exploits the phase relationship between modulated information on different eigenvalues. Similar parameters are utilized in a time-domain NN to address the overall complexity issue of the receiver. Such configurations enhance performance and reduce the computational complexity. In the dual-polarization NFDM (DP-NFDM) transmitting systems, the proposed NN-based equalizers successfully maintain a low bit error rate (BER) after up to 2800 km transmission, demonstrating an effective approach to address the perturbation issue in NFDM. (c) 2025 Optica Publishing Group. All rights, including for text and data mining (TDM), Artificial Intelligence (AI)training, and similar technologies, are reserved.

Optical waveguides, serving as the fundamental structure to manipulate light propagation, have eigenmodes and eigenvalues. Full-vectorial modes contain full-scale modal information, particularly for waveguides with strong index contrast. Conventional algorithms to compute full-vectorial eigenmodes and eigenvalues usually require time-consuming eigenvalue computation, which becomes computationally intensive in case the waveguide cross-section becomes large. In this work, a full-vectorial mode computation algorithm is proposed based on adiabatic perturbation theory. Starting from scalar modes computed for the same optical waveguide structure, one is able to include the impact of the index profile gradient and obtain the full-vectorial mode profiles and propagation constants by the adiabatic perturbation theory, which does not need further eigenvalue computation. The method is applied to optical waveguides with weak and strong index contrast. High accuracy is observed with ultra-low computational cost.

Eigen mode and eigen value evaluation is crucial in optical waveguide design and is quite time consuming. It is known that the perturbation method can compute the modes efficiently for an index perturbed structure based on the known eigen modes and thereby avoids the time consuming eigen value decomposition process. However, the perturbation method can only be applied to the waveguide structures with small index variations, and hence it poses a significant constraint to the applicability of the method to the optical waveguide inverse design problem, which might have large index variations. In this paper, an adiabatic perturbation theory is proposed to tackle this problem. The eigen modes and the propagation constants of the optical waveguides are evaluated gradually by adding the index variation with respect to the previous step. While keeping the index variation to a small value within a step, the optical waveguide structure achieves a significant index change in total and thereby enables the application of the perturbation theory to the optical waveguide inverse design problem.

A learning and perturbation based multi-stage pre-compensation method is proposed. The nonlinear compensation performance is enhanced by introducing amplitude and phase factors in each stage of pre-compensation, which are obtained by the gradient descent algorithm.

Polarization sensitivity has been a major issue in Brillouin scattering-based optical fiber sensing systems. Randomization of the polarization state of the pump is one of the ways to circumvent the problem. However, there could exist a residual degree of polarization (DOP) for the pump after polarization randomization, and hence, a model to characterize the polarization evolution in Brillouin scattering with a partially polarized pump is essential for the performance evaluation. In this work, a comprehensive theoretical model to characterize the beam variation with the partially polarized pump wave and Stokes wave is proposed, which is based on a set of stochastic differential equations (SDEs). The polarized part of the pump wave and the Stokes wave, as well as the total powers of the waves, are incorporated in the coupled SDE simultaneously, which enables the comprehensive simulation of the polarization evolution in the fiber. It is revealed in the study that the DOPs of the pump wave and the Stokes wave affect the gain stability and should be reduced simultaneously by polarization scrambling to ensure a fixed Brillouin gain without fluctuations.

The analytical Gaussian noise (GN) model can estimate the nonlinear noise power for the systems with multiple identical spans. In this paper, an analytical solution of the GN model for hybrid heterogeneous links is proposed.

Polarization dependent loss (PDL) has become a major source of linear impairment in the polarization division multiplexing systems. In this paper, an out-of-band noise correlation matrix detection and whitening method is proposed. The out-of-band noise is detected after the signal is filtered at the receiver side and the correlation matrix is obtained afterwards. A noise whitening matrix obtained by singular values decomposition (SVD) is multiplied on the signal with the in-band noise. With the signal transfer matrix estimated by the pilot signals, the maximum likelihood detection method is implemented afterwards. Simulations have been performed to evaluate the performance of the proposed noise whitening method. The Q-factor of the proposed whitening method and the noise whitening method without considering the received noise correlation are compared, and the results show that the proposed PDL mitigation method can be considered as a feasible solution to improve the system performance under PDL.

In this chapter, multimode fiber amplifiers, including the multimode erbium-doped fiber amplifiers and multimode fiber Raman amplifiers (MRFAs) are discussed. The basic physical models for the multimode fiber amplifiers are provided along with the optimization tools to reduce the modal gain excursion and to broaden the amplification bandwidth. A design example for the multimode MFRA is provided to illustrate the application of the model and the tools.