Propagation of Two-dimensional Surface Gap Solitons under Fractional Diffraction

Surface waves propagating along the interface of two distinct media have garnered significant attention in recent years， owing to their promising applications in fields such as optical sensing， optical switching， and surface characterization. Extensive research has been conducted on surface solitons based on the standard Schrödinger equation， and many novel characteristics have been found.Recently， an extended version of the standard Schrödinger equation has been introduced into the system， where the diffraction term is described by a fractional-order derivative rather than the second-order derivative in the conventional form. This extended equation is referred to as the fractional Schrödinger equation. Promptly thereafter， an optical realization of the equation was proposed based on the transverse light dynamics in an aspherical optical resonator， alongside a scheme for generating dual-Airy states. Notably， this marked the first introduction of the fractional Schrödinger equation into the field of optics， which opened up a novel research direction in optics. Since then， numerous studies on solitons with fractional diffraction have been reported. However， relevant research on surface solitons supported by the fractional nonlinear Schrödinger equation remains relatively scarce. Focusing on the propagation of two-dimensional surface gap solitons under fractional diffraction， this study investigates the existence， stability， and propagation properties of fundamental and in-phase dipole solitons at the interface between bulk media and an optical lattice in self-defocusing Kerr nonlinear media via numerical methods.The band structure of the optical lattice was calculated via the plane-wave expansion method. Subsequently， the existence region of surface gap solitons was obtained by modified squared operator iteration method， while their stability was analyzed through the Fourier collocation method and further verified via the split-step Fourier method.The results show that　the first bandgap and the semi-infinite bandgap gradually broaden with an increase in the Lévy index， while the other bandgaps progressively narrow as the Lévy index increases. Both fundamental and in-phase dipole solitons can exist within the first gap of the Bloch bands， and both fundamental and in-phase dipole solitons at the interface exhibit an asymmetric morphology. As the Lévy index increases， the peak intensity of the soliton decreases. Similarly， the peak intensity also declines with an increase in the propagation constant. Combined verification through propagation simulations and linear stability analysis reveals that： except for the in-phase dipole solitons away from the interface， which exhibit instability in a small region near the lower edge of the first Bloch band， all other surface gap solitons maintain stability throughout their entire existence range. Additionally， it also indicates that as the Lévy index increases， both the existence interval and stability interval of surface gap solitons gradually broaden. When a phase tilt factor is introduced to the surface gap solitons， applying the phase tilt in a single direction yields distinct behaviors： a small phase tilt causes the soliton center to oscillate periodically only along the direction of the applied phase tilt. A large phase tilt in the y-direction not only induces irregular oscillations of the soliton center along the y-direction but also elicits such oscillations along the x-direction. However， a large phase tilt in the x-direction solely results in irregular oscillations of the soliton center along the x-direction. If the same phase tilt is introduced in both directions， a small phase tilt causes the soliton center to undergo periodic oscillations in both directions， while a large phase tilt leads the soliton center to exhibit irregular oscillations in both directions. The results indicate that a small phase tilt induces slight periodic oscillations in the soliton's mass center， while the soliton shape remains nearly unchanged. In contrast， a large phase tilt leads to irregular oscillations in the mass center， accompanied by gradual distortion of the soliton.