Gravitational wave echoes as probes of the maximum mass of strange stars in quadratic curvature-matter coupled gravity

Gravitational wave astronomy provides a powerful avenue for probing exotic compact stars. Recent analyses of GW170817 have reported possible post-merger gravitational wave echoes with a significance of $4.2\sigma$ and a dominant frequency near $72,\text{Hz}$. Such echoes may arise from ultracompact remnants possessing photon spheres that partially trap gravitational perturbations. In general relativity, photon-sphere formation requires the stellar compactness to lie between one-third and four-ninths, a condition difficult to achieve with realistic equations of state. Here, we investigate this possibility in quadratic curvature gravity with non-minimal matter coupling, considering strange stars described by the MIT bag model equation of state. By solving the modified Tolman-Oppenheimer-Volkoff equations, we obtain mass-radius relations and identify configurations capable of supporting photon spheres and gravitational wave echoes. The modified Buchdahl limit permits more compact stellar solutions, while photon-sphere constraints restrict the viable parameter space. We find that increasing the bag constant reduces the maximum mass and echo time, shifting the echo frequency toward the kHz regime. The echo constraints impose more stringent maximum mass-radius limits than hydrostatic equilibrium, suggesting revised maximum mass bounds for strange stars. We further compute the critical impact parameter and eikonal quasi-normal modes, demonstrating how modified gravity parameters systematically alter the photon-sphere radius and associated observational signatures. These results establish post-merger strange stars as potential GW echo sources and highlight echoes, optical shadows, and QNMs as complementary probes of modified gravity for next-generation detectors such as the Einstein Telescope and Cosmic Explorer.