  Recently, much attention has been paid to second-order photonic topological
insulators (SPTIs), because of their support for highly localized corner states
with excellent robustness. SPTIs have been implemented in either transverse
magnetic (TM) or transverse electric (TE) polarizations in two-dimensional (2D)
photonic crystals (PCs), and the resultant topological corner states are
polarization-dependent, which limits their application in
polarization-independent optics. However, to achieve polarization-independent
corner states is not easy, since they are usually in-gap and the exact location
in the topological bandgap is not known in advance. Here, we report on a SPTI
based on a 2D square-lattice PC made of an elliptic metamaterial, and whether
the bandgap is topological or trivial depends on the choice of the unit cell.
It is found that locations of topological bandgaps of TM and TE polarizations
in the frequency spectrum can be independently controlled by the out-of-plane
permittivity $\varepsilon_\perp$ and in-plane permittivity
$\varepsilon_{\varparallel}$, respectively, and more importantly, the location
of in-gap corner states can also be separately manipulated by them. From this,
we achieve topological corner states for both TM and TE polarizations with the
same frequency in the PC by adjusting $\varepsilon_\perp$ and
$\varepsilon_\varparallel$, and their robustness against disorders and defects
are numerically demonstrated. The proposed SPTI provides a potential
application scenario for polarization-independent topological photonic devices.
