Plasma resonance at low magnetic fields as a probe of vortex line meandering in layered superconductors

We consider the magnetic field dependence of the plasma resonance frequency in pristine and in irradiated Bi$_2$Sr$_2$CaCu$_2$O$_8$ crystals near $T_c$. At low magnetic fields applied along the $c$ axis, in the single vortex regime, we calculate linear in field corrections to the plasma frequency due to thermal wiggling of vortex lines by accounting for both Josephson and magnetic interlayer coupling of pancakes. These corrections are similar in solid and liquid phases and they are in agreement with experimental data. This gives evidence that at low fields, near $T_c$, the vortex solid phase melts into a liquid of vortex lines. We found that pinning by columnar defects affects weakly the field dependence of the plasma resonance frequency near $T_c$.