When optimizing a grating coupler, an important consideration to keep in mind is the target bandwidth. Grating couplers are usually optimized at a single frequency or over a relatively narrow bandwidth. There are examples in the literature of grating coupler optimizations that target broadband designs, nevertheless, these tend to employ a sophisticated etching pattern  or undesirable parameter constraints – like a steep fiber angle  – to achieve a broadband response. As demonstrated recently in , it is possible to optimize for different bandwidths using a simple single etch design without unusual parameter constraints by employing inverse design.
The goal of this post is to show that Lumerical’s inverse design parametric adjoint optimization tool can be used to target different design bandwidths and achieve coupling efficiency results comparable to those in . Note that, unlike the topology optimization approach presented in , the parametric adjoint optimization approach employed here leverages the broadband nature of Lumerical’s FDTD solver to avoid frequency sweeps, which are required when using a frequency domain solver.
Grating Coupler Optimization
For this brief demonstration, the target fabrication process is a standard silicon-on-insulator process with a 220 nm Si layer as shown in the diagram below. A TM waveguide mode is excited on the Si layer when the electric field of the source is parallel to the ribs. The source is a Gaussian beam with a 10.4 micron diameter and a tilt angle of 5 degrees.
The goal of the optimization is to tune each individual rib and groove (shown as ai and bi on the diagram) to maximize the coupling efficiency for target design bandwidths of 40, 100 and 120 nm with a center wavelength of 1550 nm. Following reference , four different etch depth values (shown as he in the above diagram) are considered: 40, 60, 80 and 100%.
The plots below show the coupling efficiency (solid line) and back reflection (dashed line) results for each of the target bandwidths and etch depth values are shown below. As expected, larger target bandwidths tend to produce more broadband results, and larger target bandwidths tend to produce lower peak efficiencies.
These results are comparable to those shown in Fig. 2 of reference . Observe that, for some cases, the optimization results presented here tend to have a shorter bandwidth, however, they also tend to have lower back reflections. This is also an inherent tradeoff of broadband designs. Lastly, note that the results presented here were obtained using the same initial structure, number of grates and source position in all cases, so it is possible to further improve the results by varying these quantities as well. If anyone is interested on pursing this analysis further, the simulation files and reproduction steps are attached.
The files grating_coupler_opt.py and grating_coupler_base.fsp perform the optimization for any target bandwidth and etch depth. After running the Python file from the script editor, a simulation file with the optimized structure is created.
Once the optimization has been run for each desired target bandwidth and etch depth, then the simulation files with the optimized structures must be run to extract the coupling efficiency and back reflection. This is can be done by updating the script plot_coupling_efficiency.lsf with the appropriate file names to produce plots like the ones shown above.
 Jason C. C. Mak, Quentin Wilmart, Ségolène Olivier, Sylvie Menezo, and Joyce K. S. Poon, “Silicon nitride-on-silicon bi-layer grating couplers designed by a global optimization method,” Opt. Express 26, 13656-13665, 2018.
 Yun Wang, Wei Shi, Xu Wang, Zeqin Lu, Michael Caverley, Richard Bojko, Lukas Chrostowski, and Nicolas A. F. Jaeger, “Design of broadband subwavelength grating couplers with low back reflection,” Opt. Lett. 40, 4647-4650, 2015.
 Neil V. Sapra, Dries Vercruysse, Logan Su, Ki Youl Yang, Jinhie Skarda, Alexander Y. Piggott, Jelena Vučković, “Inverse design and demonstration of broadband grating couplers,” arXiv:1808.07630 [physics.app-ph].