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

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, ${\Gamma }_z=\frac{I_L\pi {\alpha }^2{Q}_{\mathrm{abs}}}{\omega }$. Here $I_L$ and $\omega$ are the incident wave's irradiance and angular frequency and $\alpha$ 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 $\pi {\alpha }^2{Q}_{\mathrm{abs}}$ 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 ${\omega }_{\mathrm{sz}}$ of a sphere in a gas which is achieved when ${\Gamma }_z$ is balanced by the viscous-drag torque. A Rayleigh-scattering approximation for $Q_{\mathrm{abs}}$, which should be useful for small spheres, gives ${\omega }_{\mathrm{sz}}\simeq \frac{I_L{M}_g{M}^{\prime }{M}^{\prime \prime }}{\eta c(M^{\prime 2}+2)}$ where the sphere's refractive index is $M^{\prime }+{\mathrm{iM}}^{\prime \prime }$ relative to that of the gas $M_g$, $\eta$ 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.