  We investigate the photon polarization tensor at finite temperature in the
presence of a static and homogeneous external magnetic field. In our scheme,
the Matsubara frequency summation is performed after Poisson summation, which
will be taken easily and convergent quickly in the frame of proper time
representation. Moreover, the dependence of Landau levels is expressed
explicitly. It demonstrates the convergence of summing Landau levels as it has
to be. Consequently, there is no necessary to truncate the Landau level in a
numerical estimation. At zero temperature, the Lowest Landau Level (LLL)
approximation is analytically satisfied for the imaginary parts of the vacuum
photon polarization tensor. Our results examine that, the LLL approximation is
not enough for the thermal photon polarization tensor, it gains the
contribution not only from the lowest Landau level but also up to the
finite-$n$ levels. Such large imaginary ones only show up at finite
temperatures, which is the so called Landau damping. It originates from the
absorption of soft fields by hard plasma constituents, which is a universal
feature of plasma systems. Finally, it was argued that the summation of
Matsubara frequency is not commuted with Landau level ones, such conjecture is
excluded in our calculations.
