Novel Quantity for Probing Matter Perturbations Below the Fresnel Scale in Gravitational Lensing of Gravitational Waves

Gravitational lensing of gravitational waves provides a powerful probe of the mass density distribution in the universe. Wave optics effects, such as diffraction, make the lensing effect sensitive to the structure around the Fresnel scale, which depends on the gravitational wave frequency and is typically sub-Galactic for realistic observations. Contrary to this common lore, we show that wave optics can, in principle, probe matter perturbations even below the Fresnel scale. This is achieved by introducing a new quantity derived from the amplification factor, which characterizes the lensing effect, and analyzing its correlation function. Our results demonstrate that this quantity defines an effective Fresnel scale: a characteristic scale that can be arbitrarily small, even when observational frequencies are bounded. In practice, the effective Fresnel scale is constrained by the observation time $T$ and is suppressed by a factor of $1/\sqrt{fT}$ relative to the standard Fresnel scale at frequency $f$. Nevertheless, it remains significantly smaller than the conventional Fresnel scale for $fT \gg 1$; for instance, in one-year observations of mHz GWs, the effective Fresnel scale can be as small as 1 pc. This approach opens new avenues for probing the fine-scale structure of the universe and the nature of dark matter.