# Numerical Estimation of Bending in Holographic Volume Gratings by Means of RCWA and Deep Learning
## 3 Results
### Suitability of the RCWA-Shooting Method  
This section covers the results of different gratings with different bending degrees. The bending scheme considered is the one defined by Equation (2). The term \(a_{0}\) is not taken into account, since it only implies a displacement in the fringes of the grating, not influencing the diffraction efficiency of the gratings. The results here compare the diffraction efficiency obtained through Kubota's formalism, RCWA-SM (introduced in Section 2.1), and the results from SF-FDTD method. The diffraction grating simulated is a transmission grating \(\phi=90^{\circ}\) with \(\theta_{\text{ref}}=\theta_{\text{obj}}=13.65^{\circ}\). These angles produce a grating with a spatial frequency of \(\Delta=0.823~{}\upmu\)m. The average refractive index was \(n_{0}=1.63\), and the modulation index was \(n_{1}=0.025\). The thickness of the grating wa \(d=12~{}\upmu\)m. This setup defines a volume grating where only one diffracted order will be relevant in efficiency since the parameter \(Q\gg 1\). Regarding FDTD simulations, the grid density was \(\lambda_{0}/40\), and the illumination wavelength wa \(\lambda_{0}=633\) nm. The time resolution was varied as a function of the incidence angle \(\theta_{0}\), ensuring in all cases the Courant-Friederich-Lewy condition defined in Equation (14) in [38], and the number of time steps in the simulation was chosen as 5000, ensuring the steady state of the electromagnetic waves inside the diffractive element. Due to the spatial symmetry, a 2D simulation with 1238 \(\times\) 52 grid cells was considered. Firstly, simulations for a specific value of the \(a_{1},a_{2}\), and \(a_{3}\) were considered. More specifically, Figure 4 shows the results for the grating matching the cases summarised in Table 1. For Figure 4a, the different methods show accurate results when no bending was present in the diffraction grating. The results in Figure 4b-d show how the bending induced a displacement on the curves for the SM-RCWA and Kubota compared to the FDTD. This displacement when bending is present is mainly because both RCWA-SM and Kubota consider a constant grating vector modulus along \(z\) and introduce the bending as a phase term. In contrast, the FDTD computes the diffraction efficiency directly using the interaction of light inside the periodic structure with the bending as they travel along the \(z\) direction. This approach is more accurate, and it translated in the results as a slight displacement of the diffraction efficiency curves compared to the RCWA-SM and Kubota. For the cases where \(a_{2}\) and \(a_{3}\) were the only parameters inducing the bending, SM-RCWA and Kubota's implementations maintained the deviation to some degree (compared to the FDTD). As the modulus of the bending roe (\(|\mathbf{a}|\)) and the amplitude of \(a_{2}\) and \(a_{3}\) were prominent, the SM-RCWA and Kubota significantly reduced its performance compared to the FDTD. The influence of the bending can be summarised as an angular displacement of the diffraction curve; the angular diffraction response was not symmetric around Bragg's angle, and a rise in the curves induced a disappearance of the null diffraction efficiency values. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & \(\mathbf{a_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{a_{3}}\) \\ \hline
1 & 0.5 & 0 & 0 \\
2 & 0 & 0.5 & 0 \\
3 & 0 & 0 & 0.5 \\
4 & 0.1 & 0.1 & 0.1 \\
5 & 0.1 & 0.2 & 0.2 \\
6 & 0.1 & 0.3 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values considered for \(a_{i}\) in Figure 4.  
For analysing more precisely the performance of the different methods, Figure 5 shows the error between the SM-RCWA and Kubota's against the FDTD for the various scenarios here considered (summarised in Table 1). From Figure 5a, it can be seen that the SM-RCWA showed quite good results under the ones obtained by the traditional Kubota implementation. Figure 6 shows the results for a more profound analysis when only varying \(a_{1}\) from \(-0.5\) up to \(0.5\).