Spatial differentiation is important in image-processing applications such as image sharpening and edge-based segmentation. In these applications, of particular importance is the Laplacian, the simplest isotropic derivative operator in two dimensions. Spatial differentiation can be implemented electronically. However, in applications requiring real-time and high-throughput image differentiation, conventional digital computations become challenging. Optical analog computing may overcome this challenge by offering high-throughput low-energy-consumption operations using compact devices. However, previous works on spatial differentiation with nanophotonic structures are restricted to either one-dimensional differentiation or reflection mode, whereas operating in the transmission mode is important because it is directly compatible with standard image processing/recognition systems. Here, we show that the Laplacian can be implemented in the transmission mode by a photonic crystal slab device. We theoretically derive the criteria for realizing the Laplacian using the guided resonances in a photonic crystal slab. Guided by these criteria, we show that the Laplacian can be implemented using a carefully designed photonic crystal slab with a non-trivial isotropic band structure near the $\Gamma$ point. Our work points to new opportunities in optical analog computing as provided by nanophotonic structures.

中文翻译：

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光子晶体平板拉普拉斯算子用于图像区分

空间差异在图像处理应用程序中非常重要，例如图像锐化和基于边缘的分割。在这些应用中，尤为重要的是拉普拉斯算子，它是二维中最简单的各向同性导数算子。空间差异可通过电子方式实现。然而，在需要实时和高通量图像区分的应用中，常规的数字计算变得充满挑战。光学模拟计算可以通过使用紧凑型设备提供高吞吐量，低能耗的操作来克服这一挑战。但是，先前关于纳米光子结构空间分异的研究仅限于一维分度或反射模式，然而，以传输模式操作很重要，因为它与标准图像处理/识别系统直接兼容。在这里，我们表明拉普拉斯算子可以通过光子晶体平板设备以传输模式实现。我们从理论上推导了使用光子晶体平板中的引导共振来实现拉普拉斯算子的标准。在这些标准的指导下，我们表明拉普拉斯算子可以使用精心设计的光子晶体平板来实现，该平板具有不平凡的各向同性能带结构。$\Gamma$观点。我们的工作指出了纳米光子结构在光学模拟计算方面的新机遇。