A General Model for Linearly Polarized Optical Vector Beams

TL;DR

This work develops a comprehensive framework for modeling linearly polarized vector beams with spatially inhomogeneous polarization by introducing a complex scalar potential $V({\bf x},t)$ and an associated Lagrangian density. Polarization inhomogeneities are incorporated by adding a spatially varying polarization angle $\gamma$ to the phase, making dynamical phase and polarization equivalent in the energy description, with the potential arising from a field-symmetry argument via Noether's theorem. The resulting model recovers known paraxial vector-beam equations and extends to non-paraxial, time-dependent regimes, while providing a momentum-density expression that correctly includes polarization-gradient contributions through $\mathbf{P}_V=k_0^{-1}\rho(\nabla_X\phi+\nabla_X\gamma)$. This unified approach clarifies the role of polarization in beam dynamics, offers a consistent transport framework, and has implications for understanding optical momentum and beam steering in complex vector fields.

Abstract

We propose an approach for deriving a broad class of propagation models for inhomogeneously, linearly polarized ``vector'' beams. Our formulation leverages a complex scalar potential along with an appropriately constructed Lagrangian energy density. Importantly, we show that polarization inhomogeneities can be included by simple addition of a spatially dependent polarization angle to the complex potential phase. Thus, phase and polarization are seen to be equivalent from an energy perspective. As part of our development, we also show how the complex scalar potential arises naturally when considering polarization angle as a field symmetry during construction of the Lagrangian. We further show that the definition of linear momentum density in terms of the complex potential holds a distinct advantage over the conventional definition for inhomogeneously polarized beams.