Inverse Optical Torques on Dielectric Nanoparticles in Elliptically Polarized Light Waves

Elliptically polarized light waves carry the spin angular momentum (SAM), so they can exert optical torques on nanoparticles. Usually, the rotation follows the same direction as the SAM due to momentum conservation. It is counterintuitive to observe the reversal of optical torque acting on an ordinary dielectric nanoparticle illuminated by an elliptically or circularly polarized light wave. Here, we demonstrate that negative optical torques, which are opposite to the direction of SAM, can ubiquitously emerge when elliptically polarized light waves are impinged on dielectric nanoparticles obliquely. Intriguingly, the rotation can be switched between clockwise and counterclockwise directions by controlling the incident angle of light. Our study suggests a new playground to harness polarization-dependent optical force and torque for advancing optical manipulations.

Figure 1

Positive and negative optical torques on dielectric triangular prisms. (a) Schematics of the triangular prism (refractive index $n+ik$) placed at the interface of air ($n=1$) and water ($n=1.337$) and under the illumination of obliquely incident light beam (wavelength 532 nm, incident angle $\theta$) with the left-handed circular polarization (LCP). The prism (side $a$, thickness $t$) rotates counterclockwise [top view, defined as the POT ($M_z>0$)] when the incident angle $\theta =0$. The NOT can emerge when the triangular prism is placed at the interface for the LCP, or immersed inside water for the elliptical polarization. (b) Schematics of the contributions from the geometric anisotropy and the SAM. ${\mathbf{S}}^{(i)}$ represents the incident Poynting vector. (c) Optical torque versus the orientation angle of the quarter-wave plate $\gamma$ when the rotation angle of the triangular prism $\phi =30°$. Red and blue curves denote the ellipticity angles $\chi =1/2\pi$ (phase delay in the $y$ direction) and $\chi =-1/2\pi$, respectively. In this case with the geometry symmetry, ${\mathbf{M}}_{\text{aniso}}=0$, and $\mathbf{M}={\mathbf{M}}_{\mathrm{SAM}}$. (d) Optical torque versus the orientation angle of the quarter-wave plate $\gamma$ when $\phi =0$. When the incident angle $\theta =40°$, ${\mathbf{M}}_{\text{aniso}}\ne 0$. The $s$ and $p$ polarizations correspond to the maximum POT and NOT, respectively. ${\mathbf{M}}_{\text{aniso}}=0$, and $\mathbf{M}={\mathbf{M}}_{\mathrm{SAM}}$ when $\theta =0$. In (c) and (d), $a=150\mathrm{nm}$, $t=150\mathrm{nm}$, and ${ϵ}_p=16$.

Figure 2

Size-, permittivity-, and absorption-dependent NOTs. (a) Optical torque versus the orientation angle of the quarter-wave plate $\gamma$ and the permittivity of particle ${ϵ}_p$. The small $\gamma$ (approaching the $p$ polarization) leads to the NOT, while the large $\gamma$ (approaching the $s$ polarization) leads to the POT. ${ϵ}_p>15$ is required to obtain the NOT for the LCP ($\gamma =45°$). $a=150\mathrm{nm}$, $t=150\mathrm{nm}$, and $\theta =40°$. (b) Optical torque versus the side $a$ and the permittivity ${ϵ}_p$ of the particle. Most of $M_z$ remain negative, while the POTs occur in the fringe with the high permittivity. $\gamma =15°$, $t=150\mathrm{nm}$, and $\theta =40°$. (c) Influence of $\mathrm{Im}({ϵ}_p)$ to the NOT. The presence of absorption significantly diminishes the NOT by introducing a POT from the absorption. $t=150\mathrm{nm}$, $\mathrm{Re}({ϵ}_p)=16$, and $\theta =40°$. In (a)–(c), the rotation angle of the triangular prism $\phi$ is 0. The contours in (a) and (c) denote $M_z=0$.

Figure 3

Size-dependent optical torques on ${\mathrm{TiO}}_2$ triangular prisms and optical field properties. (a) Optical torque $M_z$ on the ${\mathrm{TiO}}_2$ (${ϵ}_p=2{.67}^2$) triangular prism versus side $a$ and thickness $t$ in a beam with $\gamma =35°$ and $\theta =40°$. $M_z$ on ${\mathrm{TiO}}_2$ triangular prisms can remain NOTs with different sizes when $\gamma$ is small (e.g., $\gamma =15°$, see Supplemental Material [38]). The inset presents the incident Poynting vector in the top view. (b) Stream lines of the scattering Poynting vector ${\mathbf{S}}^{(s)}$ for the LCP (red lines) and RCP (blue lines) when $\theta =0$. The clockwise rotation of ${\mathbf{S}}^{(s)}$ for the LCP and counterclockwise rotation of ${\mathbf{S}}^{(s)}$ for the RCP rotate the triangular prism counterclockwise and clockwise, respectively, following the direction of the electric SAM ${\mathbf{s}}_e$. $a=200\mathrm{nm}$ and $t=150\mathrm{nm}$. (c) Incident SAM for the RCP light with $\theta =40°$. The SAM is in the same direction as the wave vector. (d) Most of [$\mathbf{S}-{\mathbf{S}}^{(i)}$] for the design labeled as the star ($a=200\mathrm{nm}$, $t=150\mathrm{nm}$) rotates counterclockwise, meaning that the prism rotates clockwise (NOT). (e) Most of [$\mathbf{S}-{\mathbf{S}}^{(i)}$] for the design labeled as the triangle ($a=164\mathrm{nm}$, $t=150\mathrm{nm}$) rotate clockwise, meaning that the prism rotates counterclockwise (POT). The rotations of Poynting vectors in (b), (d), and (e) should be seen from the far field (away from the triangular prism).

Figure 4

Control of positive and negative optical torques on triangular prisms made of ordinary materials. (a) Optical torque $M_z$ on the silicon (Si, refractive index $n+ik$, with $n=4.152$, $k=0.0518$ at the incident wavelength of 532 nm) and titanium dioxide (${\mathrm{TiO}}_2$, $n=2.67$, $k=0$) triangular prisms versus the incident angle. “Water” means that the triangular prism ($a=150\mathrm{nm}$, $t=150\mathrm{nm}$, rotation angle $\phi =0$) is fully immersed in water. The optical torque turns from positive to negative with the increment of the incident angle $\theta$ when $\gamma =15°$. When $\gamma =45°$ (LCP), the increment of the incident angle also generates NOTs on triangular prisms placed at the interface. Whereas the optical torque remains positive when the triangular prisms are fully immersed in water. (b) The optical torque on the triangular prism ($a=150\mathrm{nm}$, $t=150\mathrm{nm}$) does not reverse sign when the refractive index (RI) of the medium increases. (c) Rotation and freeze of the triangular prism versus the rotation angle of triangular prism $\phi$. The optical torques on the Si and ${\mathrm{TiO}}_2$ triangular prisms keep positive when $\gamma =45°$. The triangular prism can freeze under the balance of the NOT from ${\mathbf{M}}_{\text{aniso}}$ and the POT from ${\mathbf{M}}_{\mathrm{SAM}}$. The incident angles in (b) and (c) are 40°. In (a)–(c), 15° and 45° mean $\gamma =15°$ and $\gamma =45°$, respectively.