# High-Tolerance Grating Couplers for Vertical Backside Coupling
## 3 Results and Discussion  
### Optimization of Grating Coupler  
To reduce optimization time and design complexity of the vertical backside GC, the optimization was conducted using an inverse design approach. Structural parameters were designed in a periodic manner, with the PSO algorithm employed for random search to enhance coupling efficiency (Lumerical, LumOpt, Ansys, Canonsburg, PA, USA). Additionally, simulation of [PERSON]'s equations was performed using a commercial Finite Difference Frequency Domain (Lumerical, FDTD, Ansys, Canonsburg, PA, USA) solver, with simulations executed on a workstation (CPU Intel Xeon Gold 40 cores with 128 G memory, HP, Santa Clara, CA, USA). The optimization of the GC was based on the variation of parameters such as the asymmetric sub-gratings sizes within a grating period, which sets the number of degrees of freedom for the design. The optimization was based on the nine  
Figure 3: Scheme of the transferred PIC including the two GCs, on top of a four-core fiber-tip. Figure 2: Side view of the 2D GC coupled perfectly vertically from the backside. parameters (P1-P4, T1-T4 and L\({}_{I}\)) illustrated in Figure 2. To simplify fabrication, the optimization process constrained the minimum width and spacing to a feature size of 200 nm with only a single layer of shallow etching (70 nm). We first simulated the 2D structure to corroborate the feasibility of the design and the reliability of the PSO algorithm, and then performed 3D FDTD simulations to verify the performances of the final GC design. The starting structure for the 2D optimization is shown in Figure 2, where the PSO algorithm was configured to run with 45 particles for each of the 70 generations, each particle being a set of the nine parameters of interest (four asymmetric sub-gratings and four trenches, as well as the position of fiber's core). Having set the optimization boundaries of those parameters both by the minimum feature size (minimum grating period length of 1.6 \(\upmu\)m) and the target maximum length of the GC itself to make it fit on a fiber tip (core-to-core spacing of 70.71 \(\upmu\)m for an MCF: SM--4C1550), the optimal parameters found in the PSO were as follows: P1 = 333 nm, P2 = 223 nm, P3 = 384 nm, P4 = 224 nm, T1 = 270 nm, T2 = 330 nm, T3 = 205 nm, T4 = 354 nm, and Lf = 5.6 \(\upmu\)m. In Figure 4a, we show the 2D FDTD simulated electric field profile within and around the GC at 1550 nm, when the grating is illuminated with the fundamental z-polarized fiber mode. Figure 4b shows the transmission spectra in each of the four directions, as a function of the wavelength. It clearly shows that the incident light couples efficiently to the left waveguide of interest (output) with a CE -43% at 1550 nm, with merely a small portion of light backpropagating to the right waveguide (<5% at 1550 nm). Due to the requirement that each parameter's minimum feature size must be at least 200 nm, the final grating period after optimization was found to be \(\Lambda\) = 2.323 \(\upmu\)m. According to the PMC in Equation (2), the designed GC generated second-order Bragg diffraction, as expected. With this optimized 2D GC structure considered as an initial starting point for obtaining an efficient 3D GC, the relevance of using a GC coupled perfectly vertically from the backside with an optical fiber was confirmed. Then, the performance of the full 3D structure was investigated by firstly converting the 2D structure into a 3D one. However, due to the limited available area on an optical fiber's facet (diameter = 120 \(\upmu\)m), a focusing grating was employed. Compared to traditional GCs [23], which occupy a large space on the chip, the focusing grating coupler has a smaller footprint [24].