A curvilinear framework for vector light fields

TL;DR

This work proposes a geometry-driven framework to construct vector beams whose structure is encoded in polarization through curvilinear coordinates defined by conformal maps. The authors derive orthonormal polarization bases aligned with four representative geometries and implement them experimentally using phase-only spatial light modulators and a quarter-wave plate, validating the predictions with Stokes polarimetry. By linking polarization distributions to the geometry of coordinate maps, the method provides a general path to generate vector fields beyond traditional spatial modal bases, with controllable polarization inhomogeneity quantified by the Vector Quality Factor. The approach is applicable to arbitrary conformal maps, offering a versatile toolkit for applications in imaging, communication, and photonic information processing that require precise, geometry-informed polarization control.

Abstract

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 our method to four systems, elliptical, parabolic, bipolar, and dipole, that represent algebraic and transcendental families of conformal maps. Stokes polarimetry measurements confirm agreement with theoretical predictions.

Figures (7)

• Figure 1: Curvilinear coordinate systems considered: (a) elliptical coordinates, showing confocal ellipses and hyperbolas; (b) parabolic coordinates, with orthogonal parabolas opening in opposite directions; (c) bipolar coordinates, with intersecting and non-intersecting circular arcs; and (d) dipole coordinates, defined by orthogonal families of non-concentric circles.
• Figure 2: Experimental setup. Laser, HeNe laser source; MO, microscope objective; PH, pinhole, L1--L3, lenses; SLM, Spatial Light Modulator; QWP1--QWP2, quarter-wave plate; LP, linear polarizer; CMOS, sensor. The dashed box outlines the Stokes polarimetry module used for polarization state reconstruction.
• Figure 3: Experimental results for the elliptical coordinate system. (a) Reconstructed polarization distribution aligned with hyperbolic coordinate curves for the semi-focal distance $a=0.5 ~ \mathrm{mm}$. (b)–(e) Corresponding spatial distributions of the Stokes parameters $S_0$, $S_1$, $S_2$, and $S_3$. (f)–(i) Reconstructed polarization distributions for $a \in \left\{ 0.0, 0.5, 0.75, 1.0 \right\} ~ \mathrm{mm}$, in that order. Insets show the corresponding theoretical predictions.
• Figure 4: Experimental results for the parabolic coordinate system. (a) Reconstructed polarization distribution aligned with coordinate curves corresponding to parabolas opening to the right. (b)–(e) Spatial distributions of the Stokes parameters $S_0$, $S_1$, $S_2$, and $S_3$. Insets show the corresponding theoretical predictions.
• Figure 5: Experimental results for the bipolar coordinate system. (a) Reconstructed polarization distribution aligned with coordinate curves corresponding to non-concentric circles for semi-focal distance $a=0.5 ~\mathrm{mm}$. (b)–(e) Corresponding spatial distributions of the Stokes parameters $S_0$, $S_1$, $S_2$, and $S_3$. (f)–(i) Reconstructed polarization distributions for $a \in \left\{0.0, 0.5, 0.75, 1.0 \right\} ~ \mathrm{mm}$, in that order. Insets show the corresponding theoretical predictions.