Wave propagation in a weak gravitational field and the validity of the thin lens approximation

Wave effects can be important for the gravitational lensing of gravitational waves. In such a case, wave optics must be used instead of geometric optics. We consider a plane wave entering a lens object and solve numerically the wave equation for three lens models: the uniform density sphere, the singular isothermal sphere, and the Hernquist model. By comparing our numerical solutions with the analytical solutions under the thin lens approximation, we evaluate the error of this approximation. The results show that the relative error of the thin lens approximation is small if the geometrical thickness of the lens is much smaller than the distance between the lens and the observer.