Identifying type II strongly lensed gravitational-wave images in third-generation gravitational-wave detectors

Strong gravitational lensing is a wave (GW) propagation effect that influences the inferred GW source parameters and cosmological environment. Identifying strongly-lensed images challenging as waveform amplitude magnification degenerate with shift in intrinsic mass redshift. However, even geometric-optics limit, type II cannot be fully matched by I (or unlensed) templates, especially large binary ratios orbital inclination angles. We propose to use this mismatch distinguish individual images. Using planned noise spectra of Cosmic Explorer, Einstein Telescope LIGO Voyager, we show significant fraction can distinguished from unlensed sources, given sufficient SNR ($\sim 30$). Incorporating models on population lens population, predict yearly detection rate lensed sources detectable 172.2, 118.2 27.4 for CE, ET respectively. Among these events, 33.1%, 7.3% 0.22% will distinguishable via their log Bayes factor larger than 10. conclude such events are likely appear third-generation detector catalog; our strategy significantly supplement existing strong search strategies.