# Optical Realization of Wave-Based Analog Computing with Metamaterials
## 2 Computational Metamaterials
### [PERSON]'s Function Approach
#### 2.2.3 Two-Dimensional Dielectric Metasurfaces  
The above discussions are mainly focused on 1D mathematical operations. However, anisotropic edge detection is not sufficient for real-time, high-throughput 2D image processing since multiple measurements are required. In this part, metasurfaces with symmetrically distributed dielectric nano-resonators are briefly reviewed to illustrate their capacity to reveal all information from the boundaries of objects. The basic physical mechanism of these 2D dielectric metasurfaces originates from the modal evolution where a bound state resonance mode with normal incident plane wave will change to a leaky waveguide mode  
Figure 5: **(a)** Top: schematic of the surface-plasmon-based scheme to perform spatial differentiation with the Kretschmann configuration. Bottom: photograph of the three samples with different thicknesses; adapted with permission from [135], Springer Nature, 2017. **(b)** Top: schematic of the backscattering-immune first-order spatial differentiator based on nonreciprocal plasmonic platform. Bottom: the simulated results of real and imaginary parts of the eigenvector with different external magnetic fields. Adapted with permission from [137], APS, 2019.  
at oblique incidence. The transmitted amplitudes encoded with information of spatial variation are sensitive to the transmission when increasing the incidence angle [151].  
A type configuration is illustrated in Figure 6a[143], silicon nitride patch-arrays are embedded in a homogeneous medium of silicon dioxide. By tuning the height of patch-arrays, the spatial bandwidth and resolution of the edge detection can be modulated. The transmission coefficient of \(TE\) incident wave can be expressed as \(T_{TE}(\varphi,k_{r})\approx\alpha(\varphi)k_{r}^{2}\), where \(\varphi\) is the azimuthal angle, \(\alpha(\varphi)\) is the gain of the system, and \(k_{r}=\sqrt{k_{x}^{2}+k_{y}^{2}}\). From Figure 6b, it can be seen that the transmission coefficient on the _xoy_ plane formed the symmetric parabolic distribution due to the geometric symmetric, thereby it can be used to perform the second-order derivative with the quadratic approximation (see Figure 6c). In [142], a similar strategy is adopted to realize a 2D Laplace operator which can be combined with traditional imaging systems with a numerical aperture (NA) up to 0.32. However, since the quasi-guided leaky modes can only couple with \(p\)-polarized incident waves due to the modal symmetry, this scheme can only operate for one polarization. To overcome this limitation, [PERSON] et al. [141] introduce a spatial differentiator can be performed for arbitrary polarized (unpolarized) wave by tailoring the spatial dispersion of electric dipole resonance supported by silicon nanodisks (see Figure 6d). Figure 6e shows that for both \(s\)- or \(p\)-polarized incident waves, the transfer function have similar parabolic shape which indicate the functionality of spatial differentiation for arbitrary polarization. The corresponding experiments verify the performance of this scheme (see Figure 6f).