  Characterization of strain in two-dimensional (2D) crystals is important for
understanding their properties and performance. Using first-principles
calculations, we study the effects of uniaxial strain on the Raman-active modes
in monolayer MoS$_2$. We show that the in-plane $E'$ mode at 384 cm$^{-1}$ and
the out-of-plane $A_1'$ mode at 403 cm$^{-1}$ can serve as fingerprints for the
uniaxial strain in this 2D material. Specifically, under a uniaxial strain, the
doubly degenerate $E'$ mode splits into two non-degenerate modes: one is
$E_{\parallel}'$ mode in which atoms vibrate in parallel to the strain
direction, and the other is $E_\perp'$ mode in which atoms vibrate
perpendicular to the strain direction. The frequency of the $E_{\parallel}'$
mode blue-shifts for a compressive strain, but red-shifts for a tensile strain.
In addition, due to the strain-induced anisotropy in the MoS$_2$ lattice, the
polarized Raman spectra of the $E_{\parallel}'$ and $E_{\perp}'$ modes exhibit
distinct angular dependence for specific laser polarization setups, allowing
for a precise determination of the direction of the uniaxial strain with
respect to the crystallographic orientation. Furthermore, we find that the
polarized Raman intensity of the $A_1'$ mode also shows evident dependence on
the applied strain, providing additional effective clues for determining the
direction of the strain even without knowledge of the crystallographic
orientation. Thus, polarized Raman spectroscopy offers an efficient
non-destructive way to characterize the uniaxial strains in monolayer MoS$_2$.
