## 4 Grating Coupler
### Parameter Optimization  
Parameter optimization of the grating coupler can be divided into two steps. In the first step, parameters in the grating part are optimized. Bragg diffraction equation [25] is firstly used to calculate the initial value of each parameter. Then, each parameter is continuously adjusted from its initial value by using the particle swarm optimization method [26], so that coupling efficiency \(\eta_{\text{g1}}\) tends to the maximum. In the second step, width \(b\) and length \(L_{\text{t}}\) are optimized to improve the transmission efficiency \(\eta_{\text{g2}}\). Simulation mode of the grating part is shown in Figure 8a. Since width of the grating part along the \(x\) direction does not affect the coupling efficiency, two-dimensional finite difference time domain (FDTD) simulation is used which also can greatly save the simulation time and computer memory. In Figure 8a, incident angle \(\theta\) denotes the angle between the SMF and normal line, \(z\) denotes the horizontal distance from the left of the grating to the SMF center, \(y\) denotes the vertical distance from the top of the grating to the SMF center, arrows denote propagation directions of the lights, PML denotes the perfectly matched layer boundary condition. Figure 7: Structure of the grating coupler. (**a**) Three-dimensional view; (**b**) Top view. Bragg diffraction equation is given by:  
\[n_{\text{e}}+m\frac{\lambda}{\Lambda}=n_{1}\sin\theta \tag{6}\]  
where \(n_{\text{e}}\) denotes the refractive index of the grating, \(\lambda\) denotes the wavelength of the light, \(\Lambda\) denotes the grating period, \(m\) denotes the diffraction order, \(n_{1}\) denotes the refractive index of the air, \(\theta\) denotes the incident angle of the SMF. Initial value of the incident angle \(\theta\) is chosen to be 20\({}^{\circ}\) to avoid the second-order reflection at the grating. Since initial values of the duty cycle \(d\) and wavelength \(\lambda\) are 0.5 and 0.532 \(\upmu\)m, respectively, refractive index of the grating \(n_{\text{e}}\) can be calculated, and its value is 1.60. According to Equation (6), initial value of the grating period \(\Lambda\) is 0.423 \(\upmu\)m when diffraction order is \(-\)1. Initial value of each parameter is shown in Table 1.  
To get the maximum coupling efficiency, four parameters including the grating period \(\Lambda\), duty cycle \(d\), \(z\), and \(y\) are optimized together using the particle swarm optimization method [26]. After 300 iterations, coupling efficiency \(\eta_{\text{g1}}\) reaches its maximum of 0.35. The final value of each parameter can also be seen in Table 1. It is worth mentioning that coupling efficiency \(\eta_{\text{g1}}\) is polarization sensitive and it gets the maximum when TE mode is in the SMF. Therefore, numerical simulations in the rest of this section use the TE mode. Transmission efficiency \(\eta_{\text{g2}}\) which is determined by the width \(b\) and length \(L_{\text{t}}\) can be calculated as follow:  
\[\eta_{\text{g2}}=\frac{P_{\text{o}}{}^{\prime}}{P_{\text{i}}{}^{\prime}} \tag{7}\]  
where \(P_{\text{i}}{}^{\prime}\) denotes the power on the right of the taper part as shown in Figure 7b, \(P_{\text{o}}{}^{\prime}\) denotes the power on the left. Figure 8b shows the transmission efficiency \(\eta_{\text{g2}}\) with different width \(b\) and length \(L_{\text{t}}\). Firstly, with the increase of the length \(L_{\text{t}}\), transmission efficiency \(\eta_{\text{g2}}\) continues to improve and tends to be 1. Secondly, when width \(b\) increases, length \(L_{\text{t}}\) should increase at the same time to keep the transmission efficiency \(\eta_{\text{g2}}\) constant. Considering the fact that diameter of the mode in the SMF is 3 \(\upmu\)m, grating part whose width \(b\) is 6 \(\upmu\)m, is big enough to accept the whole light spot from the SMF because diameter of