Forces and torques in an optical vortex

Figure 3: (a) Computed intensity distribution in the focal plane, $z=0$ of an optical vortex of radius $R_\ell$ . (b) In-plane force distribution, $\vec {F}(\vec {r})$ , including six representative trajectories whose starting points are represented by dots. Hue indicates direction in the $(x,y)$ plane according to the inset color wheel. Color saturation indicates relative magnitude. (c) In plane torque distribution. (d) Intensity distribution in the $(x,z)$ plane along the optical axis, $y = 0$ . (e) Force distribution, with unbounded trajectories showing absence of stable axial trapping. (f) Axial torque distribution. Scale bars indicates 1  $\unit{\mu m}$ .

An optical tweezer may be transformed into a torque-exerting optical vortex (34,35,36) by imposing the helical wavefront phase profile

$$\varphi_j = \ell \theta_j \bmod 2 \pi$$ (28)

with the DOE. The winding number $\ell$ controls the beam's helicity and is referred to as the topological charge. The helical topology gives rise to destructive interference along the beam's axis. Light therefore is redistributed to a ring of radius $R_\ell$ that is proportional to $\ell$ in typical holographic implementations (37,38,39). The image in Fig. 3(a) shows the computed intensity in the focal plane, $z=0$ , for half of such a ring. In this case, the topological charge $\ell = 60$ corresponds to the radius $R_\ell = 5~\ensuremath{\unit{\mu m}}\xspace$ .

Figure 3(b) shows the in-plane component of the total optical force $\vec {F}(\vec {r})$ for a sphere with $ka = 5.9$ , $n_p = 1.46 + 10^{-5}i$ , and $n_m = 1.33$ . These values are appropriate for a 1  $\unit{\mu m}$ diameter silica sphere dispersed in water and trapped at $\lambda = 532~\unit{nm}$ . Hues indicate the direction of the force in the $(x,y)$ plane according to the inset color wheel. The saturation of the color corresponds to the magnitude of the force, with unsaturated white regions corresponding to weak forces, and brightly saturated regions corresponding to $Q_{\text{max}} = 0.007$ .

The six black dots in Fig. 3(b) show the starting points for the computed trajectories that are superimposed on the force field. These two-dimensional trajectories are calculated for particles constrained to move in the $(x,y)$ plane, and do not account for the axial component of the force. They correspond to the motion typically described in experimental studies of colloidal particles in high-index optical vortexes (40,41,37), where particles are pressed against a glass surface to prevented them from escaping along the axial direction. These representative trajectories show that particles are drawn by intensity-gradient forces to the bright ring and then are driven around the ring by phase-gradient forces (42,7) This circulatory motion is a consequence of the helical beam's orbital angular momentum, which amounts to $\ell \hbar$ per photon (43,44). The orbital angular momentum flux in a helical mode is independent of the photons' spin, and thus is independent of the light's polarization (43,45).

Although the incident laser beam is assumed to be linearly polarized, the strongly focused light field has a far more complicated spatially varying polarization. Gradients in the intensity, phase and polarization of the light can exert torques as well as forces on illuminated objects, as the in-plane torque distribution in Fig. 3(c) demonstrates. The hue in Fig. 4(c) indicates direction of the torque in the $(x,y)$ plane, and the saturation indicates the magnitude of the torque efficiency. A homogeneous isotropic sphere only experiences a torque if it absorbs light (24). The scale of the torque efficiency in Fig. 3(c) therefore is proportional to the imaginary part of $n_p$ . For the micrometer-diameter silica sphere in this calculation, $\tau_{\text{max}} = 2 \times 10^{-6}$ . The maximum rotation frequency of 0.1 Hz/W would be challenging to observe experimentally, particularly on a background of vigorous brownian motion. Nevertheless, this demonstrates that optically-induced rotation can arise even in linearly polarized optical traps, and may become an important factor for materials such as polystyrene that absorb light more strongly than silica.

The axial structure of an optical vortex, plotted in Figs. 3(d), (e) and (f), reveals its limitations as an optical trap. The axial intensity profile in Fig. 3(d) corresponds to a region around the principal focal ring, the dashed line indicating the position of the focal plane, $z=0$ . The associated force distribution in Fig. 3(e) shows that particles are driven downstream along the optical axis, and so are not axially trapped. Hues correspond to directions in the $(x,z)$ plane corresponding to the color wheel, and maximum saturation corresponds to the force scale $Q_{\text{z}} = 0.006$ .