Josephson Plasma Resonance in Bi₂ Sr₂ Ca Cu₂ O_{8+y} with Spatially Dependent Interlayer-Phase Coherence

We study the Josephson plasma resonance (JPR) in Bi$_2$Sr$_2$CaCu$_2$O$_{8+y}$ (BSCCO) with spatially dependent interlayer-phase coherence (IPC). The half-irradiated BSCCO (HI-BSCCO), in which columnar defects are introduced only in a half of the sample, shows several resonance peaks, which are not simple superposition of the peaks in irradiated- and pristine-parts. JPR in HI-BSCCO changes its character from irradiated- to pristine-type at a crossover frequency ($\omega_{cr}$). We demonstrate that the one-dimensional \LSGE, which takes into account the spatial dependence of IPC, can reproduce most of the experimental findings including the presence of $\omega_{cr}$.