Optical Magnus effect on gravitational lensing

Abstract

The optical Magnus effect refers to transverse shift of a trajectory of light caused by its polarization and appears as a correction to geometrical optics at the linear order in wavelength. Here, we start from Maxwell's equations in a curved spacetime to derive the equation of motion for a wave packet of circularly polarized light, which confirms the known result involving the helicity-dependent anomalous velocity with some generalization and clarification. We then study possible consequences of the optical Magnus effect on gravitational lensing in the Schwarzschild spacetime as well as under a weak gravitational potential in an expanding spacetime. Among others, by formulating the lens equation modified to incorporate the optical Magnus effect, the Einstein ring is found impossible to emerge from a point source for any axially symmetric thin lens. Analytic solutions to the modified lens equation are also obtained for simple lens models, illuminating how image formation is affected by the optical Magnus effect.

Figures (2)

• Figure 1: Trajectories of circularly polarized light with $\lambda=+1$ incident from ${\bm{x}}\to(2.6,0,-\infty)$ in units of $r_s=1$, represented by $x$ (upper panel) and $y$ (lower panel) coordinates as functions of $z$. The solid (blue), dashed (green), and dotted (red) curves correspond to initial wavevectors of ${\bm{k}}=(0,0,1)$, $(0,0,2)$, and $(0,0,5)$, respectively. The photon sphere (orange) is also presented in the upper panel. The corresponding trajectories for $\lambda=-1$ are obtained by the replacement of $y\to-y$.
• Figure 2: Angular coordinates of image by circularly polarized light with $\lambda=+1$ for the point mass lens (upper panel) and for the singular isothermal sphere (lower panel) in units of $\theta_c=1$. The solid (blue), dashed (green), and dotted (red) curves correspond to $|\Lambda|=0.4$, 0.2, and 0.1, respectively, for a source located at $\bm\beta=(\beta,0)$ with $\beta$ varied over $[\beta_c,\infty)$. A pair of images outside and inside the critical curve (orange) appears for $\beta>\beta_c$ and their locations for $\beta=2\beta_c$ are presented by dots. They move closer to (away from) the critical curve as $\beta$ decreases (increases) and merge on the critical curve for $\beta=\beta_c$, whereas no images are formed for $\beta<\beta_c$. The corresponding images for $\lambda=-1$ are obtained by the replacement of $\theta_2\to-\theta_2$.