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 )

** Abstract **

The gravitational lensing of gravitational waves should
be treated in wave optics instead of geometrical optics
when the wave length
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
.
The first term, arising from the short wavelength limit
, corresponds to the geometrical optics limit.
The second term, being of the order of
, is the leading
correction term arising from the diffraction effect.
Analysing this correction term, we find that (1) the lensing
magnification
is modified to
, where
is of the order of
, and (2) if the lens has a cuspy
(or singular) density profile at center
(
), the diffracted image is formed
at the lens center with magnification
.

**Key words:**gravitational lensing

**--**gravitational waves

**©**

*ESO 2004*