Revealing the structure of hierarchical nondepolarizing systems through Fourier series emergence of angle-resolved Mueller matrices

Accurate identification of hierarchical structures in complex polarization systems is essential for material characterization, from engineered materials with multilayered architectures to complex biological specimens. Mainstream decomposition methods, such as Lu-chipman decomposition, factorize a Mueller matrix into an equivalent model based on the product of three basic elements, therefore suffer from limited accuracy on resolving physical parameters when analyzing arbitrary non-depolarizing multi-layer systems. To address these challenges, we propose a novel framework leveraging the Fourier series from angle-resolved Mueller matrices. Capturing distinctive frequency features, representing the angle dependency of Mueller matrix elements, enables accurate identification of multilayer system compositions. Notably, the integration of Vision Transformer model not only quantify classification boundaries but also substantiates the high discriminative value of the extracted Fourier features. Comprehensive validation is performed on both synthetic datasets, and experimental data collected via Mueller matrix microscope. Our approach attains 95% classification accuracy on the three-layer validation dataset and 86% on experimentally acquired data. Taking advantage of the predicted hierarchical structure, we adopt the optimization by matching the Fourier-series of related Mueller matrices. The physical parameters of each layer can be accurately resolved when samples are polarimetrically distinguishable. By integrating deep learning with physically grounded modeling, the approach ensures both high predictive accuracy and interpretability. It offers a unified framework that not only advances the theoretical understanding of polarization encoding in multilayer non-depolarizing systems but also highlights its potential for applications in fields such as biomedical diagnostics, thin-film metrology, layered nanomaterials, and optical coatings, where extracting polarimetric features from multilayered structures is critical.