Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave

Abstract

The cross sections associated with absorption, scattering, extinction, and radiation pressure for homogenous isotropic spheres illuminated by plane waves are well known. We derive a new fundamental cross section, namely, the one which gives the time-averaged torque caused by circularly-polarized illumination. Consider a $z$-directed wave with pure circular polarization corresponding to a positive value for the $z$ projection of the photon spin. Formulation of the Maxwell stress dyad of the total (incident + scattered) field gives the following torque relative to the sphere's center, ${\ensuremath{\Gamma}}_{z}=\frac{{I}_{L}\ensuremath{\pi}{\ensuremath{\alpha}}^{2}{Q}_{\mathrm{abs}}}{\ensuremath{\omega}}$. Here ${I}_{L}$ and $\ensuremath{\omega}$ are the incident wave's irradiance and angular frequency and $\ensuremath{\alpha}$ and ${Q}_{\mathrm{abs}}$ are the sphere's radius and Mie-theoretic absorption efficiency. Consequently the effective cross section for torque is the same as that for energy absorption $\ensuremath{\pi}{\ensuremath{\alpha}}^{2}{Q}_{\mathrm{abs}}$ as might be expected since the scattered radiation is shown to have the same ratio of $z$ component of angular momentum to energy as the incident wave. This result is rigorous for stationary isotropic spheres in vacuo. It may be used to estimate the steady-state angular velocity ${\ensuremath{\omega}}_{\mathrm{sz}}$ of a sphere in a gas which is achieved when ${\ensuremath{\Gamma}}_{z}$ is balanced by the viscous-drag torque. A Rayleigh-scattering approximation for ${Q}_{\mathrm{abs}}$, which should be useful for small spheres, gives ${\ensuremath{\omega}}_{\mathrm{sz}}\ensuremath{\simeq}\frac{{I}_{L}{M}_{g}{M}^{\ensuremath{'}}{M}^{\ensuremath{'}\ensuremath{'}}}{\ensuremath{\eta}c({M}^{\ensuremath{'}2}+2)}$ where the sphere's refractive index is ${M}^{\ensuremath{'}}+{\mathrm{iM}}^{\ensuremath{'}\ensuremath{'}}$ relative to that of the gas ${M}_{g}$, $\ensuremath{\eta}$ is the viscosity of the gas, and $c$ is the speed of light. The radiation torque caused by elliptically-polarized illumination and the torque on stratified spheres are also discussed.