Probing the chirality of a single microsphere trapped by a focused vortex beam through their orbital period

TL;DR

Problem: single-particle chirality measurements are challenging due to weak light–matter interactions. Approach: extend the Mie-Debye optical-tweezers theory to chiral nanospheres by introducing a chirality parameter $\kappa$ and analyze ring-trap dynamics of focused vortex beams with Laguerre-Gaussian profiles LG$_{0\ell}$. The key result is that the orbital period $T$ depends on $\kappa$ through the azimuthal force $Q_\phi$, with $T=\frac{2\pi\rho_{eq}\gamma}{(n_{w}/c)P\,Q_\phi(\rho_{eq},z_{eq})}$. The chiral resolution follows $\delta\kappa_\ell = |b_\ell|\delta T_\ell$, and the study identifies an optimal particle radius around $R\approx0.28\,\mu\mathrm{m}$ to maximize sensitivity, achieving precision on the order of $10^{-4}$ for $\kappa$ under realistic conditions. Significance: enables high-precision, single-particle chiroptical characterization and informs potential enantioselection of nanoscale particles.

Abstract

When microspheres are illuminated by tightly focused vortex beams, they can be trapped in a non-equilibrium steady state where they orbit around the optical axis. By using the Mie-Debye theory for optical tweezers, we demonstrate that the orbital period strongly depends on the particle's chirality index. Taking advantage of such sensitivity, we put forth a method to experimentally characterize with high precision the chiroptical response of individual optically trapped particles. The method allows for an enhanced precision at least one order of magnitude larger than that of similar existing enantioselective approaches. It is particularly suited to probe the chiroptical response of individual particles, for which light-chiral matter interactions are typically weak.

Figures (5)

• Figure 1: Left panel: Optical force field in the $\rho-z$ plane for an achiral sphere of radius $0.3\, \rm \upmu m$ in a focused vortex beam with topological charge $\ell = 4$. The purple vectors represent the radial force component $Q_\rho$ and the teal-colored vectors denote the axial force components. The axial force components are scaled by a factor of three compared to the radial force component. The solid purple- and teal-colored lines represent the vanishing of the radial and axial force components, respectively. The intersection (circle symbol) defines the cylindrical coordinates of the orbit ($\rho_\mathrm{eq}, z_\mathrm{eq}$). For comparison, the zero-force curves for a sphere with chirality index $\kappa = 0.01$ (dashed lines) and $\kappa = -0.01$ (dotted lines) are also shown. Note that the lines for zero axial force component are too close to be distinguishable, reflecting its weak dependence with $\kappa.$ The orbit coordinates corresponding to $\kappa = 0.01$ and $\kappa = -0.01$ are indicated by the square and the triangle, respectively. Right panel: Schematic representation of trapping of a sphere above the focal plane with the arrow indicating the propagation direction of the light beam.
• Figure 2: (a) Period $T$ as a function of the radius $R$ of a microsphere trapped by a vortex beam with topological charge $\ell=-4$ (orange), $\ell=4$ (red), $\ell=-5$ (cyan) and $\ell=5$ (blue). As illustrated by the inset, the rotation direction is defined by the sign of the topological charge. We calculated the periods for spheres with different chirality index $\kappa$. The solid curves correspond to the case of an achiral sphere ($\kappa =0$), while the dashed and dotted curves correspond to chirality indices of $0.01$ and $-0.01$, respectively. The shaded area, bounded by the dotted and dashed lines for each topological charge, contains the period of spheres with $\kappa$-values between the two limiting cases. In the considered interval, the period is linearly decreasing as a function of $\kappa$, as exemplified in (b) for $R=0.3\, \rm \upmu m$ with the same $\ell$-values. We performed a linear fit (black curves) with the absolute values of the slopes $b_\ell$ displayed at each curve.
• Figure 3: Minimum measurable chirality index $\delta \kappa_\ell$ scaled by the relative period uncertainty $\xi$ as a function of the topological charge for spheres with radii $0.15 \, \rm \upmu m$ (circle symbols), $0.25 \, \rm \upmu m$ (square symbols), $0.35 \, \rm \upmu m$ (triangle symbols). The connecting lines serve as visual guides.
• Figure 4: Resolution $\delta \kappa_\ell/\xi$ of the chirality index as a function of the sphere radius $R$ for the same topological charges as in Fig. \ref{['fig:T_x_r']}. Global minima are identified near $R \approx 0.28\, \rm \upmu m$.
• Figure 5: (a) Cylindrical coordinates of the stable orbit ($\rho_\mathrm{eq}$, $z_\mathrm{eq}$) and azimuthal force efficiency ($Q_\phi$) as functions of the chirality index $\kappa$. The results are shown for a sphere of radius $R=0.3\, \rm \upmu m$ and vortex beam with $\ell=4$. Each quantity is normalized by the respective value in the achiral case ($\kappa = 0$). (b) Radius of the orbit and azimuthal force efficiency for fixed chirality indices ($\kappa=\pm 0.01$) as functions of the sphere radii. As in (a), the results are shown for a vortex beam with $\ell=4$. The connection to the results for the azimuthal force in (a) is highlighted by the square ($\kappa = 0.01$) and triangle ($\kappa=-0.01$) symbol.