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When it comes to long-wavelength gravitational waves (GWs), diffractioneffect becomes significant when these waves are lensed by celestial bodies.Typically, the traditional diffraction integral formula neglects large-anglediffraction, which is often adequate for most of cases. Nonetheless, there arespecific scenarios, such as when a GW source is lensed by a supermassive blackhole 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 accountsfor large-angle diffraction as well. This paper explores the disparitiesbetween this exact diffraction formula and the traditional, approximate oneunder various special conditions. Our findings indicate that, under specificparameters, such as a lens-source distance of $D_{\rm LS}=0.1$\,AU and a lensmass of $M_{\rm L}=4\times10^{6}M_\odot$, the amplification factor for theexact diffraction formula is notably smaller than that of the approximateformula, differing by a factor of approximately $r_F\simeq0.806$. Thisdifference is substantial enough to be detectable. Furthermore, our studyreveals that the proportionality factor $r_F$ gradually increases from 0.5 to 1as $D_{\rm LS}$ increases, and decreases as $M_{\rm L}$ increases. Significantdifferences between the exact and approximate formulas are observable when$D_{\rm LS}\lesssim0.2$ AU (assuming $M_{\rm L}=4\times10^{6}M_\odot$) or when$M_{\rm L}\gtrsim2\times10^6M_\odot$ (assuming $D_{\rm LS}=0.1$ AU). Thesefindings suggest that there is potential to validate our general diffractionformula through future GW detections.

Testing an exact diffraction formula with gravitational wave source lensed by a supermassive black hole in binary systems