Periodic structures structured as photonic crystals and optical lattices are fascinating for nonlinear waves engineering in the optics and ultracold atoms communities. Moiré photonic and optical lattices — two-dimensional twisted patterns lie somewhere in between perfect periodic structures and aperiodic ones — are a new emerging investigative tool for studying nonlinear localized waves of diverse types. Herein, a theory of two-dimensional spatial localization in nonlinear periodic systems with fractional-order diffraction (linear nonlocality) and moiré optical lattices is investigated. Specifically, the flat-band feature is well preserved in shallow moiré optical lattices which, interact with the defocusing nonlinearity of the media, can support fundamental gap solitons, bound states composed of several fundamental solitons, and topological states (gap vortices) with vortex charge s = 1 and 2, all populated inside the finite gaps of the linear Bloch-wave spectrum. Employing the linear-stability analysis and direct perturbed simulations, the stability and instability properties of all the localized gap modes are surveyed, highlighting a wide stability region within the first gap and a limited one (to the central part) for the third gap. The findings enable insightful studies of highly localized gap modes in linear nonlocality (fractional) physical systems with shallow moiré patterns that exhibit extremely flat bands.

光子晶体和光学晶格结构的周期性结构对于光学和超冷原子领域的非线性波工程非常有吸引力。莫尔光子和光学晶格——介于完美周期性结构和非周期性结构之间的二维扭曲图案——是一种新兴的研究工具，用于研究不同类型的非线性局域波。在此，研究了具有分数阶衍射（线性非局域性）和莫尔光学晶格的非线性周期系统中的二维空间局域理论。具体来说，平带特征在浅莫尔光学晶格中得到了很好的保留，该晶格与介质的散焦非线性相互作用，可以支持基本间隙孤子、由多个基本孤子组成的束缚态以及具有涡旋电荷的拓扑态（间隙涡旋）s = 1 和 2，全部填充在线性布洛赫波谱的有限间隙内。采用线性稳定性分析和直接扰动模拟，调查了所有局部间隙模式的稳定性和不稳定性特性，突出显示了第一个间隙内的宽稳定区域和第三个间隙的有限稳定区域（到中心部分）。这些发现使得能够对线性非局域（分数）物理系统中高度局域化的间隙模式进行深入研究，该系统具有表现出极其平坦带的浅莫尔图案。