Differential computing is of great significance for data analysis and processing, especially in image edge recognition and extraction. Edge enhancement technology is particularly useful in compressing data, checking objects for defects, etc., making it a hot research topic in the past decades. Compared with electric circuits, differentiators that use light as the transport carrier have great advantages in terms of speed, dissipation and stability. However, conventional optical dielectric lenses are still inferior in terms of accurate and rapid response due to some reasons, such as non-negligible losses during reflection, aberrations caused by spherical surfaces, diffraction limits of light and more importantly, their bulky geometries. Using sub-wavelength optical metastructures, i. e., two-dimensional artificial structures, can overcome these inherent constraints and enable the implementation of efficient, responsive and miniaturized optical devices. By rationally utilizing rather than suppressing the interaction between the pixels, non-local metasurfaces can greatly reduce the thickness of the structure and the contrast of refractive indexes between different materials. This allows better tailoring of light behaviors as it propagates in a specially designed structure under specific conditions, thus achieving more complex optical functions. Notably, previous researches tend to focus on particular application scenario but ignore the generality. To remedy this shortcoming, here we combine the optimization algorithm with the inverse designing of the optical metasurface. Different operation logics can be achieved by modifying the objective function of the optimization algorithm and using the design steps of associating the operation logic to be implemented with an objective function, optimizing the structure, and verifying functionality. We propose a one-dimensional second-order differentiator by inversely designing the material distribution, leading to a non-local metasurface. The transfer function of the designed structure agrees well with our objective function, i.e., the relationship between incident angle and transmittance satisfies the quadratic function, especially when the incident angle is smaller than 10 degrees. We also design a Laplace transformer to prove the generality of the method. Then, we assess the robustness of the design by calculating the objective function or transfer function after making appropriate changes to the obtained optimal solution and comparing the results before and after the change. Finally, we verify their functionality in identifying image edges by illuminating light waves perpendicular to the objects, and then calculating their transmitted waves. In this way, we prove that the transmitted waves can reflect the profile of incident waves in the corresponding polarization direction. The optimization effect of a one-dimensional second-order differentiator is remarkable that the error between the transfer function of the optimal and theoretical solutions is significantly decreased. When optimizing the Laplace transformer, there is still some deviation between the transfer functions of the optimal solution and the theoretical solutions, which may be a locally optimal solution due to the fast convergence of the algorithm itself. Through verification, we find that the structure we design still hold feature of differentiation. Specifically, we use a hollowed-out silver layer as the barrier between the incident plane wave and the structures arranged by two thousand unit cells. The barrier layer only allows waves with a spatial distribution consistent with pattern "N" to pass through, and the transmitted wave reflects the profile of the special pattern. Therefore, we prove that this method has a strong generality and error tolerance. In addition, this method can be extended to the design of other spatial operations, such as integration or spatial filter. Predictably, this method has great potential for designing optical computing units considering its high tolerance for the algorithm and manufacturing process. Furthermore, we can map all the material distributions and their transfer functions calculated in the iteration process to form a database, with which the optimization efficiency can be further improved.
