“…The FMM has been used widely for investigating gratings [23][24][25][26][27][28], since the Fourier basis is particularly suitable for periodic structures. Nonetheless, by introducing perfectly matched layers (PMLs) at the boundaries of simulation domains, the FMM has been successfully employed for studying various non-periodic structures such as dielectric waveguides [20,29,30], photonic crystals waveguides [31,32], finite gratings [33], microdisks [34,35], and vertical cavities [36,37]. For the waveguide problems, the PML is typically implemented as a nonlinear coordinate transformation [20], in which the infinite space x is mapped to a finite space x with a width of Λ, i.e.…”