We investigate the photon polarization tensor at finite temperatures in the presence of a static and homogeneous external magnetic field.In our scheme,the summing of the Matsubara frequency is performed after Poisson resummation,which is easily completed and converges quickly.Moreover,the behaviors of finite Landau levels are presented explicitly.It shows a convergence while summing infinite Landau levels.Consequently,there is no necessity to truncate the Landau level in a numerical estimation.At zero temperature,the lowest Landau level (LLL)approximation is analytically satisfied for the vacuum photon polarization tensor.However,we examine that the LLL approximation is not enough for the thermal polarization tensor.The thermal tensor obtains non-trivial contributions from the finite-n Landau levels.And,photon spectra gains a large imaginary contribution in thermal medium,which is the so-called Landau damping.Finally,it is argued that the summation of Matsubara frequency is not commuted with Landau level ones,such conjecture is excluded in our calculations.