Abstract
Abstract
When it comes to long-wavelength gravitational waves (GWs), diffraction effect becomes
significant when these waves are lensed by celestial bodies. Typically, the traditional
diffraction integral formula neglects large-angle diffraction, which is often adequate for most of
cases. Nonetheless, there are specific scenarios, such as when a GW source is lensed by a
supermassive black hole in a binary system, where the lens and source are in close proximity,
where large-angle diffraction can play a crucial role. In our prior research, we have introduced
an exact, general diffraction integral formula that accounts for large-angle diffraction as
well. This paper explores the disparities between this exact diffraction formula and the
traditional, approximate one under various special conditions. Our findings indicate that, under
specific parameters — such as a lens-source distance of D
LS = 0.1 AU and a lens mass of
M
L = 4 × 106
M
⊙ — the amplification factor for the exact diffraction formula is
notably smaller than that of the approximate formula, differing by a factor of approximately
rF
≃ 0.806. This difference is substantial enough to be detectable. Furthermore, our study
reveals that the proportionality factor rF
gradually increases from 0.5 to 1 as D
LS
increases, and decreases as M
L increases. Significant differences between the exact and
approximate formulas are observable when D
LS ≲ 0.2 AU (assuming M
L
= 4 × 106
M
⊙) or when M
L ≳ 2 × 106
M
⊙ (assuming D
LS = 0.1 AU). These findings suggest that there is potential to validate our general
diffraction formula through future GW detections.