Scalar Quasi-Normal Modes in Black Hole Gravitational Lensing

We investigate the excitation of quasi-normal modes (QNMs) in gravitational lensing by a Schwarzschild black hole using a scalar field model. By employing a time-domain mode-sum method, we analyze the complex interplay between an incident burst signal and the black hole spacetime. We find that the incident waves can non-resonantly excite a substantial number of high-$l$ modes, with amplitudes for modes as high as l=20 remaining significant compared to the fundamental l=0 mode. We confirm through QNM template fitting that the late-time behaviors of these excited modes are indeed QNMs. After passing through the black hole, we find that the lensed waves form a highly directional and coherent Gaussian beam whose cross-sectional intensity profile is well-described by a Gaussian profile. Unlike spherical waves, this beam's amplitude does not decrease with distance from the black hole but remains nearly constant in the near-field region. Moreover, due to the superposition of numerous QNMs, oscillations largely cancel each other out. The lensed temporal waves do not exhibit typical oscillatory patterns.