We introduce two-dimensional space plus time (2D+1) structure and numerically investigate it using a developed multilayer simulation framework, for the first time. The new structure is consisting of crossed grating with time-varying permittivity which is inspired from 1D+1. In this regard, we extend Fourier Modal Method (FMM) in a general approach for spacetime multilayer states. Our proposed framework is fast, robust, and powerful compared to various finite difference methods. We use the scattering matrix technique to develop the proposed spacetime simulation method for multilayer structures using a non-uniform stack of layers. The method is perfectly suitable to investigate the spatiotemporal effects of surfaces/metasurfaces which covers both the transverse electric and magnetic (TE & TM) polarizations. The results show more freedom to control the optical outcomes of the multilayer considering two spatial periodicities in addition to the modulation frequency to reach nonreciprocity as one of the main consequences of the proposed structure. Moreover, we investigate the condition and limitation of breaking the Lorentz rule for spacetime structures. 2D+1 structure is more controllable than the 1D+1 due to its greater ability to adjust spatial manipulation in addition to temporal variations to reach nonreciprocity applications, digital coding, beam steering, etc.

中文翻译：

---

使用扩展傅立叶模态方法对多层时空结构进行动力学控制

我们首次引入二维空间加时间 (2D+1) 结构并使用开发的多层模拟框架对其进行数值研究。新结构由具有时变介电常数的交叉光栅组成，其灵感来自 1D+1。在这方面，我们在时空多层状态的一般方法中扩展了傅立叶模态方法（FMM）。与各种有限差分方法相比，我们提出的框架快速、稳健且强大。我们使用散射矩阵技术为使用非均匀层堆叠的多层结构开发建议的时空模拟方法。该方法非常适合研究覆盖横向电和磁（TE＆TM）极化的表面/超表面的时空效应。结果表明，除了调制频率之外，还考虑到两个空间周期性，可以更自由地控制多层的光学结果，以达到非互易性，这是所提出结构的主要结果之一。此外，我们研究了打破时空结构洛伦兹规则的条件和限制。2D+1 结构比 1D+1 结构更可控，因为除了时间变化之外，它还有更大的调整空间操纵的能力，以达到非互易应用、数字编码、波束控制等。我们研究了打破时空结构洛伦兹规则的条件和限制。2D+1 结构比 1D+1 结构更可控，因为除了时间变化之外，它还有更大的调整空间操纵的能力，以达到非互易应用、数字编码、波束控制等。我们研究了打破时空结构洛伦兹规则的条件和限制。2D+1 结构比 1D+1 结构更可控，因为除了时间变化之外，它还有更大的调整空间操纵的能力，以达到非互易应用、数字编码、波束控制等。