  The observation of magnetic ordering in atomically thin CrI$_3$ and
Cr$_2$Ge$_2$Te$_6$ monolayers has aroused intense interest in condensed matter
physics and material science. Studies of van de Waals two-dimensional (2D)
magnetic materials are of both fundamental importance and application interest.
In particular, exciton-enhanced magneto-optical properties revealed in CrI$_3$
and CrBr$_3$ monolayers have expanded the understanding of exciton physics in
2D materials. Unlike CrI$_3$ and CrBr$_3$ with out-of-plane magnetization,
CrSBr has an in-plane magnetic moment, therefore, providing a good opportunity
to study the magnetic linear dichroism and high-order magneto-optical effects.
Here, based on the many-body perturbation method within density-functional
theory, we have studied quasiparticle electronic structure, exciton, and
optical properties in CrSBr monolayer. Strongly bounded exciton has been
identified with the first bright exciton located at 1.35 eV, in good agreement
with an experiment of photoluminescence (Nat. Mater. \textbf{20}, 1657 (2021)).
Strong contrast in the optical absorption is found between the electric fields
lying along the in-plane two orthogonal directions. In accordance with a
typical and realistic experimental setup, we show that the rotation angle of
linear polarized light, either reflected or transmitted, could be comparable
with those revealed in black phosphorene. Such large linear dichroism arises
mainly from anisotropic in-plane crystal structure. The magnetic contribution
from the off-diagonal component of dielectric function to the linear dichroism
in CrSBr is negligible. Our findings not only have revealed excitonic effect on
the optical and magneto-optical properties in 2D ferromagnet CrSBr, but also
have shown its potential applications in 2D optics and optoelectronics.
