Gravitational lensing of gravitational waves: Rotation of polarization plane

Similar to the light, gravitational waves traveling in multiple paths may arrive at the same location if there is a gravitational lens on their way. Apart from the magnification of the amplitudes and the time delay between the gravitational wave rays, gravitational lensing also rotates their polarization planes and causes the gravitational wave Faraday rotation. The effect of the Faraday rotation is weak and can be ignored. The rotation of the polarization plane results in the changes in the antenna pattern function, which describes the response of the detector to its relative orientation to the gravitational wave. These effects are all reflected in the strain, the signal registered by the interferometers. The gravitational wave rays in various directions stimulate different strains, mainly due to different magnification factors, the phases and the rotation of the polarization plane. The phase difference mainly comes from the time delay. Moreover, the rotation of the polarization plane seemingly introduces the apparent vector polarizations, when these strains are compared with each other. Because of the smallness of the deflection angles, the effect of the rotation is also negligible.

Figure 1

Geometry of a Schwarzschild lens. S is the position of the binary star system, and the interferometer is at O. L represents the gravitational lens, and the thick dashed line is the optical axis. The vertical squares represent the observer, lens and source planes, from the left to the right. $\beta$ is the misalignment angle between the optical axis and the line connecting O to S. Two GW rays 1 and 2 are emitted from S, initially in the directions of ${\overrightarrow{\mathit{l}}}_1$ and ${\overrightarrow{\mathit{l}}}_2$, respectively. After passing the lens plane, their directions change, given by ${\overrightarrow{\mathit{l}}}_1^{\prime }$ and ${\overrightarrow{\mathit{l}}}_2^{\prime }$, forming angles ${\theta }_{\pm }$ with the optical axis. The deflection angles are ${\alpha }_1$ and ${\alpha }_2$, respectively.