On the structure of the polarization-orbitalization tensor.

This research extends the recently introduced orbitalization ellipse (OE) from a scalar framework to a comprehensive electromagnetic (EM) harmonic beam model for advanced wave analysis. The investigation focuses on defining the polarization-orbitalization tensor (POT) to capture the intricate dynamics of vector light. Investigators aim to demonstrate how the geometric properties of a beam emerge from its underlying polarization components. The study seeks to establish a mathematical relationship between the superposition of individual polarization states and the total beam structure. By formalizing this tensor, the work provides a tool for simultaneous analysis of both polarization and orbitalization characteristics. The project intends to clarify how singular value decomposition (SVD) can be applied to these complex optical tensors. This objective ensures that the dual nature of the electromagnetic field is captured within a single analytical step.

The investigative process centers on the rigorous formulation of the polarization-orbitalization tensor (POT) for an electromagnetic (EM) harmonic beam. Analysts derive the total orbitalization ellipse (OE) by invoking the principle of superposition across two distinct polarization components. The methodology employs singular value decomposition (SVD) as the primary analytical tool for tensor characterization. This mathematical operation decomposes the polarization-orbitalization tensor (POT) into its fundamental singular values and vectors. The researchers apply this framework to harmonic beams to ensure the model remains consistent with electromagnetic wave theory. By transforming the scalar orbitalization ellipse (OE) into a tensor, the study accounts for the vector nature of the light field. The researchers rigorously test the superposition principle to verify that the total beam ellipse accurately reflects its constituent parts.

Results indicate that the orbitalization ellipse (OE) of an electromagnetic (EM) beam represents the direct superposition of the ellipses associated with its two polarization components in the vector field. The polarization-orbitalization tensor (POT) successfully integrates the vector properties of the light field into a unified mathematical structure. Applying singular value decomposition (SVD) to the tensor provides immediate and simultaneous access to both polarization and orbitalization characteristics. The analysis confirms that the scalar-to-vector extension maintains the geometric integrity of the original orbitalization metric. Data show that the interaction between the two polarization states determines the final shape and orientation of the total beam ellipse. The mathematical structure of the POT reveals a deep symmetry between the spatial and vector degrees of freedom in harmonic beams. These findings validate the use of tensor-based decomposition for extracting complex physical parameters from electromagnetic fields.

The extension of the orbitalization ellipse (OE) to a tensor-based model provides a more complete description of electromagnetic (EM) harmonic beams for singular optics applications. These findings suggest that the polarization-orbitalization tensor (POT) will serve as a foundational tool for future studies in singular optics. The ability to extract polarization and orbitalization data simultaneously through singular value decomposition (SVD) streamlines the characterization of complex light fields. Future research may apply this tensor framework to more diverse classes of structured light and non-harmonic wave packets. The study establishes a rigorous mathematical link between the scalar properties of light and its full vector representation. This theoretical advancement enhances the precision of optical modeling in fields ranging from microscopy to free-space communication. The authors conclude that the unified tensor approach offers a significant improvement over traditional scalar-only metrics.