Geometrical optical activity induced by a continuous distribution of screw dislocations

TL;DR

This work shows that a medium with uniform torsion, modeled as a continuous distribution of screw dislocations, supports intrinsic chirality and geometric optical activity: circular birefringence between right- and left-circular polarizations leads to a polarization rotation $\Delta\theta = \Omega\rho L$ and an effective birefringence $\Delta n = \frac{2c\Omega\rho}{\omega}$, where $\Omega = b\sigma/2$. The analysis rests on Maxwell equations in a Riemann–Cartan spacetime with a spiral metric, yielding a clean dispersion split $k_z^{(-)} - k_z^{(+)} = 2\Omega\rho$ and a broadband geometric phase gate for polarization qubits, $U(\Omega,\rho,L) = \exp[-i\,\Omega\rho L\,\sigma_3]$. The authors further present a photonic design framework and provide realistic estimates for dislocated semiconductors (e.g., GaN) and metamaterial platforms, predicting millidegree rotations over millimeter-scale paths that are detectable with modern polarimetry. An electronic analogue is developed for Dirac surface states on a cylindrical topological insulator, where torsion Mixes azimuthal and longitudinal momenta via $E_m(\kappa) = \pm ( \hbar v_F /R) \sqrt{(m - \tau\kappa)^2 + \kappa^2}$ with $\tau = \Omega R^2$, revealing a unified geometric link between torsion, optical activity, and topological electronic responses. Overall, the work provides a first-principles geometric route to engineer optical activity and polarization-phase control through defect-engineered or metamaterial-inspired torsion.

Abstract

We study light propagation in a medium with uniform torsion, modeled as a continuum of screw dislocations within the geometric theory of defects. By solving Maxwell's equations in covariant form, we show that torsion induces intrinsic chirality and circular birefringence: right- and left-circular polarizations acquire different wavenumbers, leading to a purely geometric optical activity. The polarization plane of a linearly polarized beam rotates according to the simple law $Δθ= ΩρL$, linear in the dislocation density $Ω$, propagation length $L$, and transverse coordinate $ρ$. This can be recast as an effective birefringence $Δn = 2cΩρ/ω$, providing geometric design rules for torsion-induced rotatory power. Using parameters from dislocated semiconductors, we obtain millidegree rotations over millimetre-scale paths, within reach of modern polarimetric techniques and amenable to enhancement in metamaterial platforms. We also show that the same spiral geometry implements a broadband geometric phase gate for polarization qubits and has an electronic analogue on the surface of cylindrical topological insulators, where torsion shears the Dirac cone, establishing a unified geometric link between torsion, optical activity, and topological electronic responses.

Figures (6)

• Figure 1: Schematic dispersion relation $\omega/c$ versus $k_z$ in the torsioned geometry. The dashed black line corresponds to the torsionless case ($\Omega = 0$), where right- and left-circularly polarized modes are degenerate. In the presence of uniform torsion ($\Omega > 0$), the right-circularly polarized (RCP) and left-circularly polarized (LCP) modes follow the blue and orange branches, respectively, and acquire a nonzero splitting $\Delta k_z$. The grey region ($k_z < \alpha$) indicates the single-mode regime, where only the RCP solution is propagating, while the shaded orange region highlights the birefringent regime, where both circular polarizations propagate with different wavenumbers.
• Figure 2: Rotation angle $\Delta\theta$ of a linearly polarized beam as a function of the propagation distance $L$ at fixed radius $\rho$, for three values of the torsion parameter, $\Omega = \Omega_0$, $2\Omega_0$, and $3\Omega_0$. The straight lines illustrate the linear dependence $\Delta\theta \propto \Omega L$ predicted by Eq. \ref{['eq:te']}. The grey background marks the short-path regime with weak rotation, whereas the shaded orange region emphasizes the long-path regime where the rotation accumulates and becomes experimentally relevant.
• Figure 3: Torsion-induced rotation angle $\Delta\theta$ as a function of wavelength $\lambda$ for a GaN-like medium with screw-dislocation density $\sigma = 10^9~\mathrm{cm}^{-2}$ and beam radius $\rho = 20~\mu\mathrm{m}$, for different sample thicknesses $L$. The curves are obtained from Eqs. \ref{['br']} and \ref{['eq:standard_rotation']}, using material parameters extracted from Refs. Ponce1999Vurgaftman2001.
• Figure 4: Infidelity $1-F$ of the geometric phase gate $U(\Omega,\rho,L)$ [RTF - Relative thickness fluctuation] as a function of relative fluctuations in the torsion strength $\delta\Omega/\Omega$ and sample thickness $\delta L/L$, for a target $Z$ gate ($\theta = \Omega\rho L = \pi/2$). The dashed contour marks the $1-F = 10^{-3}$ line.
• Figure 5: Dimensionless dispersion relation for Dirac surface states on a cylindrical topological insulator in the presence of torsion.The longitudinal momentum and energy are rescaled as $\kappa = k_z R$ and $E / (\hbar v_F / R)$, respectively. Panel (a) shows the torsionless case ($\tau = \Omega R^2 = 0$), where the branches $E_m^{(0)}(\kappa) = \pm \sqrt{m^2 + \kappa^2}$ form a symmetric Dirac cone. Panel (b) displays the spectrum for a finite torsion parameter ($\tau = 1$), where the dispersion becomes $E_m(\kappa) = \pm \sqrt{(m - \tau\kappa)^2 + \kappa^2}$ and the cone is sheared in momentum space. Different colors correspond to azimuthal quantum numbers $m = 0, \pm 1, \pm 2$. The torsion-induced deformation of the cone is the electronic analogue of the geometric birefringence obtained for photons in the spiral geometry.