Curvilinear framework for vector light fields

Vector beams are often regarded as non-separable superpositions of spatial and polarization degrees of freedom that satisfy the wave equation. This interpretation ties their polarization structure to their spatial shape. Here, we introduce a generalized method to construct vector beams whose structure is entirely encoded in the polarization degree of freedom. Using conformal maps, we construct orthonormal polarization bases from the geometry of the coordinates and encode them experimentally via phase-only spatial light modulators. We apply the method to four systems, elliptical, parabolic, bipolar, and dipole, which represent algebraic and transcendental families of conformal maps. Stokes polarimetry measurements confirm agreement with theoretical predictions. Our framework enables encoding and control of polarization in arbitrary conformal mapping geometries, regardless of the spatial beam shape.