# Optical Realization of Wave-Based Analog Computing with Metamaterials
## 2 Computational Metamaterials
### Metasurface Approach
#### 2.1.2 Transmittive Configurations of MS Approach  
Compared with reflective configurations, the optical computing systems with transmission mode are easily applied to optical devices, however most of them are facing the challenges of low transmission efficiency. In 2015, [PERSON] et al. [32] proposed a concept of \"metalines\" in which three symmetrically stacked graphene-based building blocks were utilized to perform differentiation and integration in the transmitted way (see Figure 3a). By manipulating plasmons surface wave via surface conductivity variation of the graphene, the amplitudes and phases of the transmitted wave could be controlled completely. While it was a proof-of-principle study, it opened up an avenue of the graphene-based structure which are designed to implement the mathematical operators with advantages of dynamic tunable and high-compact characteristics. Then, in 2016, [PERSON] et al. [123] introduced a more practical and flexible computing metamaterial system based on effective medium theory by drilling sub-wavelength hole-arrays with different radiuses, the desired effective permittivity distribution can be obtained to realized functional subblocks (see Figure 3b). To further improve the parallel processing abilities, a multi-way analog computing device combined with additional transformation optical subblock is investigated [124]. Two identical signals on the same input port propagate  
Figure 2: **(a)** The unit cell of plasmonic metasurfaces consist of gold nanobrick atop a dielectric spacer and thick metal ground; adapted with permission from [120], ACS Publications, 2015. **(b)** Schematic representation of dielectric meta-reflect-array for the first-order derivation; adapted with permission from [122] © The Optical Society. **(c)** Top: eleven silver dendritic structure units with different parameters; left: the unit cell of plasmonic metasurfaces with dendritic structure instead of nanobrick; right: comparison of the theoretical and simulated results of the position-dependent reflection coefficient. Adapted with permission from [121] © The Optical Society.  
along opposite directions and perform first- and second-order differentiators simultaneously. Very recently, a one-dimensional high-contrast transmitarray (HCTA) metasurface was experimentally demonstrated by [PERSON] et al. [100] to realize Fourier transform and spatial differentiation with feature size of 140 nm. The on-chip meta-system provides low insertion loss and broad operating bands by tailoring widths and lengths of the rectangular slot arrays (see Figure 3c,d). Inspired by radial Hilbert transform filter, [PERSON] et al. [125] proposed a spin-dependent dielectric metasurface, which can perform a spiral phase filtering operation in which a spiral phase profile is imprinted to the input wave. The tricky part in this proposal is to utilize the \(\pi\) phase difference in opposite azimuth leading to a destructive interference when processing convolution transformation. The authors demonstrate that spiral phase filtering operation is equivalent to the 2D spatial differentiation of incident light field (see Figure 3e).