Structured-light propagation in a medium with uniform torsion: polarization textures, geometric birefringence, and beam-resolved optical activity

Abstract

We investigate finite-width optical-beam propagation in a medium with uniform torsion described by the geometric theory of a continuous distribution of screw dislocations. Starting from the Riemann--Cartan framework that yields torsion-induced circular birefringence for local plane waves, we construct a minimal paraxial beam model in which the same contortion-driven helicity splitting remains explicit. We show that uniform torsion breaks the degeneracy between the two circular-polarization sectors and induces a geometric rotation of the polarization that scales with both the propagation distance and the radial position in the beam. As a consequence, a finite-width beam develops spatially varying polarization textures across its transverse profile, naturally described by the Stokes parameters. We introduce beam-level observables based on the integrated Stokes vector, the transverse inhomogeneity of the polarization texture, and the number of resolved radial polarization domains, thereby connecting the torsion parameter to experimentally accessible beam diagnostics. The paper combines two complementary levels of description: an analytic short-distance regime, used to isolate the geometric mechanism, and full paraxial propagation including diffraction, used to test the robustness of the predicted textures. Within the cylindrically symmetric minimal model, the most robust structured-light signature of uniform torsion is beam-resolved polarization structuring, whereas strong orbital-angular-momentum conversion is not expected without additional azimuthal structure. We also identify the geometric ingredient required for genuine torsion-assisted spin--orbit conversion beyond the minimal radial model: an effective azimuthal geometric connection.

Figures (10)

• Figure 1: Local Stokes maps for the torsional $q$-plate extension in the short-distance, diffraction-neglected regime, for a Gaussian input beam with homogeneous linear polarization, $q=1$, $\Gamma_0=1.2$, and $\zeta=1.0$. Panel (a) shows $S_1(x/w_0,y/w_0)$, panel (b) shows $S_2(x/w_0,y/w_0)$, panel (c) shows $S_3(x/w_0,y/w_0)$, and panel (d) shows the local polarization angle $\psi=\frac{1}{2}\arg(S_1+iS_2)$, with polarization-direction arrows overlaid. Unlike the minimal radial model, in which the torsion-induced texture depends only on the radial coordinate, the present extension displays explicit azimuthal modulation generated by the locally rotating polarization frame $U_q(\phi)$. The appearance of a nontrivial $S_3$ pattern indicates that the beam acquires local ellipticity, while the angular structure of $\psi$ shows that the polarization rotation is no longer purely radial. This figure, therefore, provides the real-space signature of the azimuthal geometric connection that enables torsion-assisted spin--orbit conversion.
• Figure 2: Three-dimensional visualization of the local Stokes maps for the torsional $q$-plate extension in the short-distance, diffraction-neglected regime, for a Gaussian input beam with homogeneous linear polarization, $q=1$, $\Gamma_0=1.2$, and $\zeta=1.0$. Panel (a) shows $S_1(x/w_0,y/w_0)$, panel (b) shows $S_2(x/w_0,y/w_0)$, panel (c) shows $S_3(x/w_0,y/w_0)$, and panel (d) shows the local polarization angle $\psi=\frac{1}{2}\arg(S_1+iS_2)$. The floor projections reproduce the corresponding transverse maps, while the three-dimensional surfaces emphasize the azimuthally modulated structure induced by the locally rotating polarization frame. In particular, the quadrupolar structure of $S_3$ becomes especially transparent in this representation. The apparent discontinuities in panel (d) are due to the principal-branch representation of the angle variable $\psi$, rather than to any additional physical singularity.
• Figure 3: Orbital-angular-momentum content in the torsional $q$-plate extension for a Gaussian input beam with homogeneous linear polarization. Panel (a) shows the OAM spectrum $P_m$ for the minimal radial model ($q=0$), where the torsion-induced phase depends only on the radial coordinate and the azimuthal content remains concentrated in the initial mode. Panel (b) shows the corresponding spectrum for the torsional $q$-plate extension with $q=1$, for which symmetry-allowed sidebands $m\rightarrow m\pm q$ appear. Panel (c) displays the evolution of the dominant spectral weights $P_0$, $P_{+1}$, and $P_{-1}$ as functions of $\zeta$, showing the transfer of weight from the initial azimuthal sector into the sidebands. Panel (d) shows the spin--orbit conversion efficiency $\eta_{\rm SO}=P_{+1}+P_{-1}$, together with the short-distance perturbative guide $\eta_{\rm SO}\propto (\Gamma_0\zeta)^2$. The figure demonstrates that the azimuthal geometric connection activates a genuine spin--orbit conversion channel absent in the purely radial minimal theory.
• Figure 4: Spin--orbit conversion map for the torsional $q$-plate extension in the short-distance, diffraction-neglected regime, for $q=1$. Panel (a) shows the conversion efficiency $\eta_{\rm SO}=P_{+1}+P_{-1}$ in the $(\Gamma_0,\zeta)$ plane, with white contours marking representative efficiency levels. Panel (b) displays $\eta_{\rm SO}(\zeta)$ for selected values of the torsional strength $\Gamma_0$, showing how the conversion sets in more rapidly as $\Gamma_0$ increases. Panel (c) presents the universal collapse of the same data when plotted against the combined variable $\chi=\Gamma_0\zeta$, demonstrating that the local conversion dynamics are controlled primarily by this product. Panel (d) shows the corresponding spectral partition into the carrier and sideband sectors, $P_0$, $P_{+1}$, and $P_{-1}$, together with the short-distance perturbative guide $\eta_{\rm SO}\propto \chi^2$. The figure shows that, within the local torsional $q$-plate model, the onset and strength of spin--orbit conversion obey a simple scaling law and can be summarized by a compact operating diagram in the $(\Gamma_0,\zeta)$ plane.
• Figure 5: Transverse Stokes maps for a Gaussian input beam ($\Gamma_0=1.2$, $\zeta=1.0$): (a) intensity profile $S_0$, (b) linear-polarization Stokes parameter $S_1$, (c) linear-polarization Stokes parameter $S_2$, and (d) local polarization angle $\psi$ with polarization-direction arrows overlaid. The intensity remains close to the Gaussian envelope, whereas the polarization sector develops a torsion-induced radial texture in the diffraction-neglected short-distance regime. The angle map in panel (d) is shown on the principal branch of $\frac{1}{2}\arg(S_1+iS_2)$.