Asymptotic Schwarzschild solutions in $f(R)$ gravity and their observable effects on the photon sphere of black holes

Abstract: We investigate asymptotic Schwarzschild exterior solutions in the context of modified gravity theories, specifically within the framework of $f(R)$ gravity, where the asymptotic behavior recovers the standard Schwarzschild solution of General Relativity. Unlike previous studies that rely mainly on analytical approximations, our approach combines asymptotic analysis with numerical integration of the underlying differential equations. Using these solutions, we analyze strong lensing effects to obtain the photon sphere radius and the corresponding capture parameter. Considering rings produced by total reflection, we define the photon sphere width as the difference between the first total reflection and the capture parameter; and study how it is modified in the $f(R)$ scenario. Our results show that the photon sphere width increases in the presence of $f(R)$-type modifications, indicating deviations from GR that could be observable in the strong-field regime.