High-Precision Modal Analysis of Multimode Waveguides from Amplitudes via Large-Step Nonconvex Optimization

TL;DR

This work tackles the problem of fully reconstructing modal content in multimode waveguides from amplitude-only AF and FF measurements. It introduces a deterministic phase-retrieval framework based on nonconvex optimization with a power-normalization constraint and a large-step AdaMax update, enabling recovery of both modal power and relative phase (MRPD) from a predefined modal basis. The approach accounts for twin-image ambiguity, leverages Wirtinger gradient descent in complex space, and demonstrates machine-precision modulus and phase accuracy for up to 93 modes under favorable SNR, while showing robustness to noise through increased sampling density. Compared with SPGD and a prior hybrid method, it achieves higher accuracy and lower computational cost, highlighting its potential for scalable, noninvasive modal analysis and suggesting broader applicability to inverse problems in electromagnetics.

Abstract

Optimizing multimodal waveguide performance depends on modal analysis; however, existing methods focus predominantly on modal power distribution (MPD) and, limited by experimental hardware and conditions, exhibit low accuracy, poor adaptability, and high computational cost. This work presents a novel framework for comprehensive modal analysis (recovering both power and relative phase) using aperture field (AF) and far field (FF) amplitude measurements. We formulate the modal analysis as a nonconvex optimization problem under a power-normalization constraint and, inspired by recent advances in deep learning, introduce a large-step strategy to solve it. Our method retrieves both the MPD and the modal relative-phase distribution(MRPD). The effectiveness of the proposed method is validated through visualization of the nonconvex optimization process via its loss landscape. Under noiseless conditions, analysis results of $93$ electromagnetic modes indicate that the relative amplitude accuracy $\mathrm{MRE_{Modulus}}$, and the phase accuracy $\mathrm{MAE_{Phase}}$, both reach the level of machine precision. Through noise simulations of the AF and environmental background, the operational principles of the method are demonstrated under signal-to-noise ratio (SNR) conditions ranging from $10~\mathrm{dB}$ to $60~\mathrm{dB}$. Experiments further confirm that error suppression is effectively achieved by increasing the number of sampling points, thereby maintaining high accuracy and strong robustness. Within a unified evaluation framework, the absolute amplitude error $\mathrm{MAE_{Modulus}}$, and the phase error $\mathrm{MAE_{Phase}}$, are as low as $1.633\times10^{-8}$ and $0$, respectively. The accuracy is significantly superior to existing methods, while also exhibiting higher computational efficiency.

Figures (10)

• Figure 1: Schematic diagram of phase ambiguity induced by twin-image ambiguity. The legend indicates the true complex coefficients (GT, Ground Truth) and their twin counterparts generated in the AF and the FF. The upper plot demonstrates that the modulus remains unaffected by the twin images; the lower plot, using the TE11 mode as the phase reference, reveals that the some twin phases in AF and FF are symmetric about the $\pm \pi / 2$ axis relative to the GT phase.
• Figure 2: Schematic diagram of the modal analysis method based on large-step nonconvex optimization. The workflow begins with acquiring amplitude-only measurements from the AF and FF, which serve as labels, and obtaining the field distributions for all potential modes through simulation. The unknown complex coefficients are randomly initialized and then iteratively refined in a nonconvex optimization loop. In each iteration, the current coefficients are used to calculate two loss components: a main loss ($f_{\mathrm{main}}^3$) based on the difference between the measured and simulated amplitudes, and a power normalization loss ($f_{\mathrm{norm}}^3$). An advanced optimizer with a large-step-size strategy minimizes the total loss ($f_{\mathrm{tot},3}$) until the coefficients converge to their true values. Black arrows indicate the workflow process, while red arrows represent explanatory relationships or updates between modules.
• Figure 3: The Gaussian-smoothed projection results of optimization trajectories with different step sizes and optimizers are visualized on the PCA plane, together with the corresponding loss landscape.The trajectories evolve from a shared start point (red dot) toward an endpoint (green star), with the red-to-orange color gradient marking iterative progress towards the ground truth (gold star). The white arrows indicate the direction of optimization. The underlying loss landscape is depicted by the colored isograph. The trajectories with larger step sizes exhibit a tendency to traverse across local minima.
• Figure 4: PCA of parameter evolution across $10^5$ iterations. The bar chart shows the individual explained variance for individual principal component, while the red line indicates the cumulative explained variance. The first two principal components account for 78.3% of the total variance, capturing the dominant directions of parameter evolution.
• Figure 5: Comparison of retrieved and ground truth modal complex coefficients for 93 modes. The retrieved complex coefficients (blue scatters) and their ground truth values (red scatters) are plotted on the complex plane. All coefficients are referenced to the fundamental TE11 mode, whose phase is fixed at zero (orange circle). The position of each point is determined by its normalized modulus and relative phase.