We predict the existence of gap solitons in the nonlinear fractional Schrödinger equation (NLFSE) with an imprinted optically harmonic lattice. Symmetric/antisymmetric nonlinear localized modes bifurcate from the lower/upper edge of the first/second band in defocusing/focusing Kerr media. A unique feature we revealed is that, in focusing Kerr media, stable solitons appear in the finite bandgaps with the decrease of the Lévy index, which is in sharp contrast to the standard NLSE with a focusing nonlinearity. Nonlinear bound states composed by in-phase and out-of-phase soliton units supported by the NLFSE are also uncovered. Our work may pave the way for the study of spatial lattice solitons in fractional dimensions.

We introduce a composite optical lattice created by two mutually rotated square patterns and allowing observation of continuous transformation between incommensurate and completely periodic structures upon variation of the rotation angle θ. Such lattices acquire periodicity only for rotation angles cos θ = a/c, sin θ = b/c, set by Pythagorean triples of natural numbers (a, b, c). While linear eigenmodes supported by lattices associated with Pythagorean triples are always extended, composite patterns generated for intermediate rotation angles allow observation of the localization-delocalization transition of eigenmodes upon modification of the relative strength of two sublattices forming the composite pattern. Sharp delocalization of supported modes for certain θ values can be used for visualization of Pythagorean triples. The effects predicted here are general and also take place in composite structures generated by two rotated hexagonal lattices.

We investigate the properties of double-hump solitons supported by the nonlinear Schrödinger equation featuring a combination of parity-time symmetry and fractional-order diffraction effect. Two classes of nonlinear states, i.e., out-of-phase and in-phase solitons are found. Each class contains two families of solitons originating from the same linear mode in both focusing and defocusing nonlinear Kerr media. The critical phase-transition point increases monotonously with increasing Lévy index. For strong gain and loss, out-of-phase solitons in focusing media are stable in a wide parameter window and are almost completely unstable in media with a defocusing nonlinearity. The stability of in-phase solitons is opposite to that of out-of-phase solitons. In-phase solitons in defocusing media are stable in their entire existence domains provided that the gain-loss strength is below a critical value. Meanwhile, the stability region shrinks with the decrease of Lévy index. We, thus, put forward the first example of spatial solitons in fractional dimensions with a parity-time symmetry.

Generalization of Fractional Schrödinger equation (FSE) into optics is fundamentally important, since optics usually provides a fertile ground where FSE-related phenomena can be effectively observed. Beam propagation management is a topic of considerable interest in the field of optics. Here, we put forward a simple scheme for the realization of propagation management of light beams by introducing a double-barrier potential into the FSE. Transmission, partial transmission/reflection, and total reflection of light fields can be controlled by varying the potential depth. Oblique input beams with arbitrary distributions obey the same propagation dynamics. Some unique properties, including strong self-healing ability, high capacity of resisting disturbance, beam reshaping, and Goos-Hänchen-like shift are revealed. Theoretical analysis results are qualitatively in agreements with the numerical findings. This work opens up new possibilities for beam management and can be generalized into other fields involving fractional effects.