# Optical Realization of Wave-Based Analog Computing with Metamaterials
## 2 Computational Metamaterials
### [PERSON]'s Function Approach
#### 2.2.2 Plasmonic Structure  
By exploiting the critical coupling condition of the surface plasmon polaritons (SPPs), many plasmonic analog computing devices have been realized. In 2017, [PERSON] et al. [135] introduced a classical Kretschmann prism configuration to realize the plasmonic spatial differentiation where the oblique incident wave (transverse profile of \(S_{in}(k_{x})\)) with TM-polarized are used to excite the surface plasmon between glass substrate and silver film, \(k_{x}\) is the transverse component of the wavevector (see Figure 5a). In this case, the amplitude of reflected wave (transverse profile of \(S_{out}(k_{x})\)) is the result of the interference between the direct reflection and the leakage of SPPs. The spatial spectral transfer function around \(k_{x}=0\) can be expressed as:  
\[H(k_{x})=e^{i\varphi}\frac{ik_{x}+A}{ik_{x}+B} \tag{6}\]  
where \(\varphi\) is the phase change related to the process of the direct reflection at the glass-metal interface. \(A=\left(\alpha_{spp}-\alpha_{1}\right)/\cos\theta_{0}\) and \(B=\left(\alpha_{spp}+\alpha_{1}\right)/\cos\theta_{0}\), where \(\alpha_{1}\) and \(\alpha_{spp}\) are the radiative leakage rate of the SPP and the intrinsic material loss rate, respectively. When the critical coupling condition is satisfied (\(\alpha_{1}=\alpha_{spp}\)), Equation (6) can be simplified and approximated as \(H(k_{x})\propto(e^{i\varphi}/B)ik_{x}\), which corresponds to the first-order differentiation.  
Figure 4: (**a**) Schematic and (**b**) photograph of guided-mode resonant diffraction grating for first-order differentiation; adapted with permission from [131] © The Optical Society. (**c**) The schematic and (**d**) spatial spectral transfer function of the optical spatial differentiator based on subwavelength high-contrast gratings. Adapted with permission from [132], AIP Publishing, 2018. (**e**) Schematic of the first-order differentiator based on subwavelength metallic gratings. (**f**) The experimental results for the incident and transmitted images (left) and the normalized field profiles at the position of white dashed line. Adapted with permission from [133] © The Optical Society.  
However, the SPPs is sensitive to the defect of the interface. As such, to further improve its robustness, [PERSON] et al. [137] proposed a unidirectional SPPs-based spatial differentiator which can avoid backscattering by applying an external static magnetic field (shown in Figure 5b). The magnetic field breaks the time-reversal symmetry and makes the system have a nonreciprocal property, which is critical to the unidirectional SPPs leaky mode. As shown in the bottom of Figure 5b, when applying external magnetic field, the SPPs surface mode changes from bidirectional to unidirectional. To verify the practicability when facing real-time image processing, the impact of plasmonic spatial differentiation under time-variant optical signals is analyzed by [PERSON] et al. [136]. By adopting similar setup described in [135], the reflected field profile with central frequency \(\omega_{0}\) in the spatiotemporal coordinate can be expressed as:  
\[s_{out}(x,t)=c_{s}\frac{ds_{in}(x,t)}{dx}+c_{t}\frac{ds_{in}(x,t)}{dt} \tag{7}\]  
where \(c_{s}\) and \(c_{t}\) are two constants where \(c_{s}=e^{\mathrm{i}\phi}\frac{\cos\theta_{0}}{2\mu_{1}}\) and \(c_{t}=e^{\mathrm{i}\phi}(2a_{1})^{-1}\Big{(}\frac{1}{v_{g}}-\frac{\sin\theta_ {0}}{v_{gs}}\Big{)}\), \(\phi\) is phase change of direct reflection without excitation of SPP, \(\theta_{0}\), \(a_{1}\), \(v_{g}\), \(v_{gs}\) are the incident angle, the leakage rate, the group velocity of the SPP, and the speed of light in the prism at central frequency \(\omega_{0}\), respectively. The second term of Equation (7) indicates that the output field profile should consider the change rate of input time-modulated signals. As a conclusion, the authors estimate that the processing speed of plasmonic differentiator is up to the maximum of \(10^{13}\) frame/s which is restricted by the time of establishing the SPP leaky radiation.