Defect Solitons in PT-symmetric Optical Lattices with Fractional Diffraction

Optical lattice solitons， as key entities at the intersection of nonlinear optics and photonics， not only provide an ideal model for uncovering fundamental physical principles of nonlinearity and periodicity coupling， but also break through technical bottlenecks in conventional optical transmission， manipulation， and storage. A considerable number of studies have been reported on the existence and stability of defect solitons under both normal and fractional diffraction mechanisms. However， the properties of defect solitons in optical lattices under the combined influence of fractional effects and PT-symmetric potentials have not been thoroughly investigated. This paper focuses on studying the existence， stability， and propagation dynamics of defect solitons in PT-symmetric optical lattices with fractional diffraction， examining the impact of fractional effects and defect types on the bandgap structure， analyzing the existence domain of defect solitons， elucidating their linear stability， and discussing the influence of system parameters on the characteristics of defect solitons.A theoretical model for optical wave propagation in Kerr nonlinear optical lattices with PT-symmetric potentials under fractional-order effects is established based on the fractional nonlinear Schrödinger equation. The plane wave expansion method is used to analyze the effects of defect strength and fractional-order characteristics on the bandgaps of defect solitons. The modified squared-operator iteration method is employed to numerically solve the model and obtain steady-state solutions of the defect solitons. Additionally， perturbations are introduced to the steady-state solutions， and linear stability analysis is performed using the Fourier collocation method. Finally， the propagation properties of the solitons are verified by means of the symmetrized split-step Fourier method.The bandgap range of the optical lattice narrows as the Lévy index decreases. As the defect strength increases from negative to positive， the semi-infinite gap gradually narrows， while the other bandgaps progressively widen. Under negative defect conditions， the power of solitons in both the semi-infinite gap and the first gap generally decreases with increasing propagation constant， except for an initial increase followed by a decrease in the low-power region of the semi-infinite gap and the medium-power region of the first gap. As the Lévy index varies， the stable region of solitons in the semi-infinite gap gradually shrinks， while solitons in the first gap remain unstable. Under zero defect conditions， the soliton power in both the semi-infinite and first gaps decreases monotonically with the propagation constant. The influence of the Lévy index on soliton stability is similar to that under negative defects. Under positive defect conditions， the soliton power in both gaps decreases with increasing propagation constant， but a cutoff point exists where the power reaches zero. Moreover， a smaller α results in a smaller propagation constant at the cutoff point. Similarly， a decrease in α reduces the stable region of solitons in the semi-infinite gap， although stable solitons exist in the first gap. Additionally， for all three defect types， the width of the defect soliton envelope narrows slowly as α decreases， while the peak intensity increases gradually. The power decreases with reduced α， and the peak intensity of the soliton envelope decreases as the defect strength changes from negative to positive.This study investigates the existence， stability， and propagation dynamics of defect solitons in PT-symmetric optical lattices with fractional diffraction. Under fractional diffraction， as the Lévy index increases， the power of defect solitons in the semi-infinite bandgap and the first bandgap gradually increases for all defect lattices， while the stability of solitons improves. In cases of negative defects and zero defects， defect solitons only have stable intervals in the semi-infinite bandgap and are completely unstable in the first bandgap. In cases of positive defects， defect solitons have stable intervals in both the semi-infinite bandgap and the first bandgap. Additionally， within the semi-infinite bandgap， as the Lévy index decreases， the peak of the defect soliton envelope increases while its width narrows； as the propagation constant increases， the peak of the defect soliton envelope decreases rapidly. Within the first bandgap， the peak of the soliton envelope decreases as the Lévy index decreases and decreases as the propagation constant increases under positive defect conditions. In summary， the envelope， power， and stability of defect solitons are significantly influenced by the defect strength， Lévy index， and propagation constant.