## 4 Grating Coupler
### 3 dB Fabrication Tolerance of the Grating Coupler  
To quantitatively evaluate the difficulty of fabricating the grating coupler and to compare it with the difficulty of fabricating the Y-branch coupler, 3 dB fabrication tolerance of the grating period \(\Lambda\), duty cycle \(d\) and thickness of the bottom cladding \(h\) are analyzed.  
Effect of the grating period \(\Lambda\) on the coupling efficiency \(\eta_{\mathrm{g1}}\) is shown in Figure 9a. Grating period \(\Lambda\) and incident angle \(\theta\) must satisfy the Bragg diffraction equation (Equation (6)) to get the maximum coupling efficiency \(\eta_{\mathrm{g1}}\). However, the incident angle \(\theta\) remains unchanged in the experiment. This leads to a decrease of the coupling efficiency \(\eta_{\mathrm{g1}}\) regardless of the decrease or increase of the grating period \(\Lambda\). When grating period \(\Lambda\) decreases to 0.40 \(\upmu\)m or increases to 0.44 \(\upmu\)m, coupling efficiency \(\eta_{\mathrm{g1}}\) drops to half of the maximum. Therefore, 3 dB fabrication tolerance of the grating period \(\Lambda\) is \(-\)0.02 \(\upmu\)m\(+\)0.02 \(\upmu\)m.  
Figure 9b shows the effect of duty cycle \(d\) on the coupling efficiency \(\eta_{\mathrm{g1}}\) when grating period \(\Lambda\) is 0.420 \(\upmu\)m. Coupling efficiency \(\eta_{\mathrm{g1}}\) increases first and then decreases with the increase of the duty cycle \(d\). Because too large or too small duty cycle \(d\) enhances the transmission characteristics of the grating and finally leads to the reduction of the coupling efficiency. When duty cycle \(d\) decreases to 0.36 or increases to 0.60, coupling efficiency \(\eta_{\mathrm{g1}}\) drops to half of the maximum. Therefore, 3 dB fabrication tolerance of the duty cycle \(d\) is \(-\)0.12 \(\upmu\)m\(-\)0.12 \(\upmu\)m. In other words, 3 dB fabrication tolerance of the width \(p\) or \(q\) is \(-\)0.05 \(\upmu\)m\(-\)0.05 \(\upmu\)m.  
Thickness of the bottom cladding \(h\) also affects the coupling efficiency \(\eta_{\mathrm{g1}}\). Arrows in Figure 8a clearly show the propagation of the incident light after it enters the grating coupler from the SMF. The light reflected by the substrate (D\({}_{\mathrm{reflect}}\)) interferes with the light propagating to the waveguide (D\({}_{\mathrm{left}}\)), leading to the result that coupling efficiency \(\eta_{\mathrm{g1}}\) changes periodically with the increase of the thickness \(h\). Figure 9c shows this phenomenon clearly. Coupling efficiency \(\eta_{\mathrm{g1}}\) reaches its maximum of 0.35 when thickness \(h\) is 2.52 \(\upmu\)m. Coupling efficiency \(\eta_{\mathrm{g1}}\) drops to half of the maximum once thickness \(h\) decreases to 2.46 \(\upmu\)m or increases to 2.58 \(\upmu\)m. Therefore, 3 dB fabrication tolerance of the thickness \(h\) is \(-\)0.06 \(\upmu\)m\(-\)0.06 \(\upmu\)m.  
Figure 9: (**a**) Influence of grating period \(\Lambda\) on the coupling efficiency \(\eta_{\mathrm{g1}}\) when duty cycle \(d\) is 0.476. (**b**) Influence of duty cycle \(d\) on the coupling efficiency \(\eta_{\mathrm{g1}}\) when grating period \(\Lambda\) is 0.42 \(\upmu\)m. (**c**) Influence of bottom cladding thickness \(h\) on the coupling efficiency \(\eta_{\mathrm{g1}}\).