Spin optics for gravitational waves lensed by a rotating object

We study gravitational lensing of gravitational waves taking into account the spin of gravitational waves coupled with a dragged spacetime made by a rotating object. We decompose the phase of gravitational waves into helicity-dependent and helicity-independent components with spin optics, analyzing waves whose wavelengths are shorter than the curvature radius of a lens object. We analytically confirm that the trajectory of gravitational waves splits depending on the helicity, generating additional time delay and elliptical polarization onto the helicity-independent part. We exemplify monochromatic gravitational waves lensed by a Kerr black hole and derive the analytical expressions of corrections in phase and magnification. The corrections are enhanced for longer wavelengths, potentially providing a novel probe of rotational properties of lens objects in low-frequency gravitational-wave observations in the future.