Analytical equations for the three diffraction orders of a fused silica grating at the second Bragg incidence

The physical analytical equation is always a pursuit of designing a grating for applications. We presented analytical equations to describe the diffraction process of a deep-etched fused silica grating under the second Bragg incidence by using the simplified modal method, particularly in a special case where the effective refractive index of mode 1 might be equal to mode 2. We found that the joint effect of modes 1 and 2 has no effect on the diffraction efficiency for the 0th order and the -2nd order. The efficiency fluctuates slightly as the grating depth increases. In the context of this situation, we discovered that the modes 0, 1, and 2 influenced the diffraction process together. We derived the analytical diffraction efficiency equations of three orders, with different grating depths h and corresponding diffraction efficiencies to match the coefficient values of efficiency equations. The comparison between these analytical equations and RCWA matches well with the vast majority of the difference between them that are less than 4% for both TE and TM polarizations, which proves the viability of these analytical equations. These analytical equations illustrate a clear physical picture of three modes' contributions under the second Bragg incidence. These analytical equations derived from the simplified modal method could help us understand the physics theory of gratings in general and the design of fused silica gratings for practical applications.