Phase-gradient metasurfaces exhibit diverse extraordinary diffractive behaviors. These range from reflective or transmissive scattering of an incident wave toward any targeted direction to temporal trapping of waves inside structures. The behaviors are determined by the design parameters, phase gradient and number of discrete unit cells within a supercell. In this study, we propose a higher-level design scheme by integrating a stack of multiple metasurfaces. We focus on a case with negligible separation in which the constituent metasurfaces perfectly interact with each other. A significantly higher degree of diffractive behavior compared with that of a conventional single layer is observed for multilayer metasurfaces specifically in the regime above the critical angle. The scattering angular characteristics can be enriched remarkably in multilayers because, unlike in a single layer, the transformation of the incident tangential wavenumber is not restricted by the number of integer multiples of the phase gradient. The wavenumber transform is available for the entire range of incident angles in multilayers, whereas it is available only for half of this range in a single layer. The temporal duration of internal wave trapping can also be enhanced dramatically in multilayers because the supercell periodicity can be effectively lengthened from those of the constituents by combining different phase gradients. To our knowledge, this is the first report of the distinctive diffractive behaviors of multilayer metasurfaces, particularly for an incident angle above the critical angle. To fully describe the underlying physics, we derive a generalized diffraction law that robustly addresses the multilayers by proposing a novel physical concept to straightforwardly connect repetitive phase accumulations inside metasurface unit cells to diffraction. The reciprocal lattice of a unit cell, rather than that of a supercell, is observed to play an important role in determining dominant scattering. The proposed theories are thoroughly verified by analytical and numerical simulations.

相位梯度超表面表现出多种不同寻常的衍射行为。这些范围从入射波向任何目标方向的反射或透射散射到结构内波的时间俘获。这些行为由设计参数、相位梯度和超级单元内的离散单元的数量决定。在这项研究中，我们通过集成多个超表面的堆栈来提出更高级别的设计方案。我们专注于一个可忽略不计的分离情况，其中组成的超表面完美地相互作用。对于多层超表面，特别是在高于临界角的范围内，观察到与传统单层相比显着更高程度的衍射行为。散射角特性可以在多层中显着丰富，因为与单层不同，入射切向波数的变换不受相位梯度整数倍数的限制。波数变换可用于多层中的整个入射角范围，而它仅适用于单层中该范围的一半。在多层中，内波俘获的时间持续时间也可以显着提高，因为通过组合不同的相位梯度，可以有效地延长超晶胞的周期性。据我们所知，这是关于多层超表面的独特衍射行为的第一份报告，特别是对于高于临界角的入射角。为了充分描述基础物理，我们推导出一个广义的衍射定律，通过提出一个新的物理概念直接将超表面晶胞内的重复相积累与衍射联系起来，从而有力地解决了多层问题。观察到晶胞的倒易晶格，而不是超晶胞的倒易晶格，在确定主要散射中起重要作用。所提出的理论通过分析和数值模拟得到了彻底的验证。观察到在确定主散射中起重要作用。所提出的理论通过分析和数值模拟得到了彻底的验证。观察到在确定主散射中起重要作用。所提出的理论通过分析和数值模拟得到了彻底的验证。