Quasi-geometrical Optics Approximation in Gravitational Lensing

The gravitational lensing of gravitational waves should be treated in the wave optics instead of the geometrical optics when the wave length $\lambda$ of the gravitational waves is larger than the Schwarzschild radius of the lens mass $M$. The wave optics is based on the diffraction integral which represents the amplification of the wave amplitude by lensing. We study the asymptotic expansion of the diffraction integral in the powers of the wave length $\lambda$. The first term, arising from the short wavelength limit $\lambda \to 0$, corresponds to the geometrical optics limit. The second term, being of the order of $\lambda/M$, is the leading correction term arising from the diffraction effect. By analyzing this correction term, we find that (1) the lensing magnification $\mu$ is modified to $\mu ~(1+\delta)$, where $\delta$ is of the order of $(\lambda/M)^2$, and (2) if the lens has cuspy (or singular) density profile at the center $\rho(r) \propto r^{-\alpha}$ ($0 < \alpha \leq 2$), the diffracted image is formed at the lens center with the magnification $\mu \sim (\lambda/M)^{3-\alpha}$.