## 3 Component Design and Modeling
### Apodized Bidirectional Grating Coupler
If we consider the two diffraction modes to be in phase, the final grating diffraction mode (\(E_{d}(x)\)) can be obtained from the constructive interference of the two excited diffraction fields:  
\[E_{d}(x)=\sqrt{\frac{E_{d1}^{2}(x)+E_{d2}^{2}(x)}{2}} \tag{4}\]  
From Equations (3) and (4), it can be deduced that:  
\[E_{d}(x)=\sqrt{\alpha\Gamma e^{-\alpha L}(e^{-2 ax}+e^{2 ax})} \tag{5}\]  
The fiber mode field \(G(x)\) can be expressed as the following equation:  
\[G(x)\propto\frac{1}{\sqrt{w_{0}}}exp\left(-\frac{(x-\mu)^{2}}{w_{0}^{2}}\right) \tag{6}\]  
As a one-dimensional approximation, the coupling efficiency can be calculated through the overlap integral of the diffraction mode and the single-mode fiber mode:  
\[CE=\left|\int_{\mu-\frac{D}{2}}^{\mu+\frac{D}{2}}E_{d}(x)G(x)dx\right|^{2} \tag{7}\]  
From Equation (5), it is clear that the diffraction field pattern of a bidirectional GC is mainly related to the coupling strength distribution of the grating structure while the amplitude of the E-field is jointly influenced by the grating directionality and the coupling strength distribution. This relationship can be seen from the comparison between the calculated Gaussian field and the bidirectional grating diffraction field shown in Figure 2a. Clearly, there is a big mode mismatch between the grating diffraction mode and fiber mode for a uniform bidirectional grating design. Although a weak grating design (with small \(\alpha\)) can help improve the matching efficiency, it is still impossible to obtain a Gaussian-like diffraction field with uniform grating design. As the coupling strength \(\alpha\) and the grating directionality \(\Gamma\) are both functions of the grating filling factor (FF) and etching depth (d), the grating radiation mode can be reshaped by engineering the grating filling factors and etching depth of each grating cells. To improve the mode matching, apodized grating design is utilized for tailoring the grating coupling strength distribution. As shown in Figure 2b, a carefully designed coupling strength distribution \(\alpha(x)\) can result in a Gaussian-like diffraction field without breaking the grating symmetry. By carefully designing the grating pitches and filling factors, the mode matching can be improved, and the parasitic reflections can be suppressed. Based on the working principle above, we proposed and designed a highly efficient bidirectional grating coupler which combines the technology of grating apodization, overlay technique and subwavelength gratings. Figure 3 shows the structural view of the proposed grating coupler. As can be seen, the grating coupler is designed as central symmetric with 20 grating pitches, of which the inner 10 pitches are standard grating cells while the outer 10 pitches are subwavelength engineered ones. For performance optimization of the optical transmitter, two grating designs were utilized, and the specific design parameters are shown in Table 1.  
Figure 3: The schematic diagram of bidirectional grating coupler design. Figure 2: The comparison between the Gaussian field and the diffraction field of (**a**) uniform bidirectional grating couplers with different coupling strength and grating directionality. (**b**) an apodized bidirectional grating coupler with engineered coupling strength distribution. Due to the bandwidth limitation of the grating couplers, two GCs (ABG\({}_{1}\) and ABG\({}_{2}\)) are designed with different central coupling wavelengths, for decreasing the internal Fabry-Perot reflection [35] and improving the channel uniformity. As can be seen, the GCs are designed with waveguide height (H) of 220 nm, polysilicon overlayer height (h) of 160 nm covering the grating region and etching depth (d) of 230 nm in total.