  Control and detection of antiferromagnetic topological materials are
challenging since the total magnetization vanishes. Here we investigate the
magneto-optical Kerr and Faraday effects in bilayer antiferromagnetic insulator
MnBi$_2$Te$_4$. We find that by breaking the combined mirror symmetries with
either perpendicular electric field or external magnetic moment, Kerr and
Faraday effects occur. Under perpendicular electric field, antiferromagnetic
topological insulators (AFMTI) show sharp peaks at the interband transition
threshold, whereas trivial insulators show small adjacent positive and negative
peaks. Gate voltage and Fermi energy can be tuned to reveal the differences
between AFMTI and trivial insulators. We find that AFMTI with large
antiferromagnetic order can be proposed as a pure magneto-optical rotator due
to sizable Kerr (Faraday) angles and vanishing ellipticity. Under external
magnetic moment, AFMTI and trivial insulators are significantly different in
the magnitude of Kerr and Faraday angles and ellipticity. For the qualitative
behaviors, AFMTI shows distinct features of Kerr and Faraday angles when the
spin configurations of the system change. These phenomena provide new
possibilities to optically detect and manipulate the layered topological
antiferromagnets.
