Fig. 1. (a) Partial schematic diagram of our proposed Si₃N₄/Si grating with a varying duty cycle and width of the Si₃N₄ overlayer. (b) Our proposed grating’s 3D view without SiO₂ cladding.

Fig. 2. (a) Optimized width and duty cycle distribution of the Si₃N₄ overlayer as a function of the grating period order. (b) The corresponding emission profile for our designed grating with a length of 2.247 mm at the wavelength of 1550 nm. (c) The major electric field distribution of the guided mode during the grating periods of 1145–1365. (d) The comparison of the calculated emission profile and the results simulated by FDTD in the same grating region.

Fig. 3. (a) Far-field intensity distribution for our proposed grating at the 1550 nm wavelength. (b) Far-field intensity distribution for beam steering within 1530–1570 nm wavelength range with a 5 nm step.

Fig. 4. Error maps of our proposed grating when σ = 30–60 nm, respectively. The red line indicates the average intensity, and the gray area represents the standard deviation of the intensity error at each point.