The design and manufacturing technology of two-dimensional metrological planar gratings is complex and difficult. Design indicators and processing accuracy of two-dimensional (2D) gratings are important factors determining the performance of planar grating measurement systems. The grating type, grating pitch, groove depth, duty cycle, and characteristics of surface coating materials exert influence on diffraction efficiency and diffraction efficiency equilibrium. There are many parameter combinations and manufacturing methods for gratings. The development and process optimization of gratings feature long cycle time and low efficiency. Our purpose is to propose an accurate design and simulation method for two-dimensional planar gratings to provide sufficient accurate and fast support for grating development, shorten the cycle time of the design and processing of planar gratings, and improve development efficiency.

The design simulation of a silicon based two-dimensional planar grating is studied through the electromagnetic finite-difference time-domain (FDTD) method, and the simulation results are verified by experiments in this study. For two-dimensional grating structures, the FDTD algorithm only needs to simulate the smallest repeatable element with periodic boundary conditions. The calculation speed and accuracy meet the requirements and have been applied and verified in some grating simulations. By approximating and iterating the Maxwell equations in three-dimensional space, we can obtain the spatio-temporal changes in electromagnetic fields. One approximation method is called the central difference method. By linear interpolation approximation of physical quantities and selecting the appropriate truncation error level, high-precision approximation results can be obtained. The grating diffraction simulation in this study employs FDTD Solutions software and the grating unit in the simulation model consists of two parts, and the upper part is a columnar shape compatible with semiconductor technology. In this study, three structures are involved in sequence, including a regular prism, a ladder structure, and a ladder structure with rounded corners. The lower part is the substrate with the coating material of aluminum. Based on the input conditions of the different structure grating model, the actual process development and experimental verification are carried out. The incident light is a plane light pulse with a central wavelength of 780 nm, the incident angle is 27.9° with the normal direction onto the model, and the polarization directions P and S are calculated separately. The longitudinal boundary of the model is a perfect matching layer (PML) with a certain number of layers, and the incident light will pass through the boundary without reflection. The horizontal boundary is the Bloch boundary, which simulates the periodic arrangement of a single computing unit in the horizontal direction to obtain the grating structure. The mesh division of the simulation model is automatically achieved by software preset schemes, with dense grids near the aluminum bumps and loose grids in the air part. The mesh accuracy is generally adjusted between 2 to 4 nm based on convergence requirements.

Firstly, for the regular prism model, the measurement results of the diffraction efficiency of the first round grating sample are 12% for P-polarized light and 64% for S-polarized light. There are some differences between the simulated and experimental results. Combining the actual process results and grating test results, we try to optimize the model structure in the subsequent simulation and build grating simulation models of different structures and parameters, including the ladder structure grating and the ladder structure grating with rounded corner (Figs. 6 and 8). Secondly, the grating model with corner-rounded bump structure is close to the actual process results, and the final diffraction efficiency and polarization equilibrium meet the design requirements. The simulation results are consistent with the actual process results (Table 5). Thirdly, the FDTD simulation accuracy greatly depends on the simulation step size. The smaller step size leads to more accurate characterization of structural details in the simulation, whereas the larger requires computational amount results in longer calculation time. The simulation accuracy is also verified by reducing the simulation mesh size. The accuracy of mesh size 8 nm is acceptable (Table 3). Finally, at 170 mm×170 mm measurement range of grating, the diffraction efficiency, diffraction uniformity, and balance of P-polarized and S-polarized lights satisfy the grating design requirements (Fig. 14).

We study the design and simulation of two-dimensional planar gratings, put forward a simulation method, and conduct process experimental verification. Through the established FDTD simulation method, the variation of polarization diffraction efficiency of two-dimensional gratings under different structures and sizes is explored. The coating layer reflectivity, groove depth, and duty cycle are three main factors affecting the diffraction efficiency of the gratings. At a duty cycle from 45% to 46.7%, the diffraction efficiency of S-polarized light decreases with the increase of groove depth, while the diffraction efficiency of P-polarized light increases with the increase of groove depth. At a groove depth of 260 nm, the diffraction efficiency of different polarizations is nearly constant, and the diffraction efficiency and diffraction efficiency equilibrium meet the design requirements. The simulation results are close to the actual process results, and the simulation accuracy satisfies the requirements. Based on the established FDTD simulation method and manufacturing process, the design efficiency and process development efficiency of planar gratings can be greatly improved.