This paper introduces a new simulation method that makes it practical to model very long, complex Bragg gratings on silicon chips. Bragg gratings are patterned waveguides that reflect light and are used as filters and delay lines in integrated photonics. When these gratings are millimeters long and their pattern changes along the length, standard full three‑dimensional simulations become too slow to use in design. The authors propose a structure‑aware approach that keeps full 3D physics while cutting computation time from hours or days to minutes.

The method, called hierarchical locally periodic eigenmode expansion (HLP‑EME), turns a smoothly varying grating into a sequence of piecewise‑constant blocks. Each block contains many identical periods, so the solver computes the optical scattering of a single representative period and reuses it many times. The calculation is organized in four hierarchical levels: the slice level (compute modes for thin cross‑section slices), the period level (cascade slice responses to get one period), the block level (raise the single‑period response to the number of periods in the block), and the grating level (cascade block responses to get the whole device). The local modal properties in the slices were obtained with a full‑vectorial eigenmode solver, but the HLP‑EME formulation itself is platform‑independent.

The speed gains are large. The paper reports three example devices with physical lengths of 0.266 mm, 0.516 mm and 1.072 mm. HLP‑EME simulated these devices in 28 seconds, 1.2 minutes and 5.9 minutes, respectively. The same devices would take minutes to hours with a two‑dimensional FDTD (finite‑difference time‑domain) approximation and, according to the authors’ extrapolations, tens to hundreds of hours with full 3D‑FDTD. The authors report speedups exceeding three orders of magnitude compared with 3D‑FDTD, while still performing full 3D calculations rather than a dimensional reduction.