Optical bistability in nonlinear periodical structures with $\mathcal{P}\mathcal{T}$ -symmetric potential

The interaction between forward and backward waves in a periodical structure with -symmetric potential is investigated. The results demonstrate that the -term can change the bandgap of the periodical structure and affect the effective feedback mechanism. The linear solution, reflectivity, dispersion relation, and a generalized analytical solution of this periodical structure are obtained. The influences of the -term and detunings on the characteristic of bistability (or multistability) are also discussed.