Identifying multiple images of gravitational-wave sources lensed by elliptical lensing potentials

Real astrophysical lenses typically lack axisymmetry, necessitating the study of gravitational-wave (GW) lensing by elliptical mass distributions to accurately assess detectability and waveform interpretation. We investigate strong lensing using the singular isothermal ellipsoid (SIE) model, which produces two or four images depending on the source's position relative to lens caustics. Employing a quasi-geometrical optics framework, we determine that the geometrical-optics approximation holds reliably for lens masses above approximately $10^5 \, M_\odot$ at GW frequencies relevant for ground-based detectors $(\sim 10^2 \,\text{Hz})$, though wave-optics effects become significant for lower masses or sources near caustics. Our waveform mismatch analysis demonstrates that the use of three-image templates significantly improves our ability to distinguish source signals, reducing mismatches from $O(10^{-1})$ to $O(10^{-2})$, typically by factors between 1.5 and 5 compared to the standard two-image template model. At lens masses above $10^7 \, M_\odot$, diffraction effects become negligible for ground-based detectors, resulting in an additional mismatch reduction by a factor of approximately three. These findings highlight the critical need for multi-image templates in GW searches to enhance detection efficiency and accuracy.