EDP Sciences
Free access
Volume 423, Number 3, September I 2004
Page(s) 787 - 792
Section Astrophysical processes
DOI http://dx.doi.org/10.1051/0004-6361:20040212

A&A 423, 787-792 (2004)
DOI: 10.1051/0004-6361:20040212

Quasi-geometrical optics approximation in gravitational lensing

R. Takahashi

Division of Theoretical Astrophysics, National Astronomical Observatory, Mitaka, Tokyo 181-8588, Japan
    e-mail: takahasi@th.nao.ac.jp

( Received 6 February 2004 / Accepted 28 April 2004 )

The gravitational lensing of gravitational waves should be treated in wave optics instead of geometrical optics when the wave length  $\lambda$ of the gravitational waves is larger than the Schwarzschild radius of the lens mass  M. 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. Analysing 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 a cuspy (or singular) density profile at center $\rho(r) \propto
r^{-\alpha}$ ( $0 < \alpha \leq 2$), the diffracted image is formed at the lens center with magnification $\mu \sim
(\lambda/M)^{3-\alpha}$ .

Key words: gravitational lensing -- gravitational waves

© ESO 2004