# Numerical Estimation of Bending in Holographic Volume Gratings by Means of RCWA and Deep Learning
## 4 Conclusions  
This work presents the results of a novel RCWA approach for transmission holographic gratings with bending. These results were compared with those obtained from a set of rigorous SF-FDTD and Kubota's simulations, providing a benchmark for the performance of our approach. The analysis covers a range of different bendings and a parametric analysis of the various parameters that define the bending. The results show that our method is consistent and quite accurate for low and moderate amplitudes of the bending of the fringes (\(\sum|a_{i}|<0.5\)). However, as the amplitude of the bending increased, our approach began to deviate from the numerical results, leading to differences in the angles at which the maximum diffraction efficiency could be found, as well as in the amplitudes of this parameter.  
A deep learning optimum model for the estimation of bending polynomial coefficients was developed, as we compared the performance of two different architecture-based models. The results indicate that while both models achieved good predictive performance, the hybrid model slightly outperformed the FNN-only model in predicting bending coefficients, featuring lower standard deviations and fewer outliers, while the former model excelled in predicting optical depth and index modulation. Despite small bending coefficient losses (0.04), the predicted and true bending phase polynomials revealed high concordance, with the RCWA-SM reconstructions of diffraction patterns being nearly identical. This indicates that the limits of the accuracy of the estimators are due to the insensitivity of the optical system to small variations in bending.  
In future works, the authors plan to extend the current formalism for considering different type of gratings, e.g., slanted and reflection gratings, and to compare the performance of the scheme with experimental results.  
Conceptualisation, J.C.-M., J.J.S.-V. and C.N.; methodology, C.N. and J.F.; software, J.C.-M., C.N., J.S.-V., J.C.B. and J.F.; validation, J.F., J.C.B., A.M.-A. and J.C.-M; formal analysis, J.C.-M. and [PERSON]; investigation, [PERSON] and [PERSON]; resources, A.B.; writing--original draft preparation, [PERSON] and [PERSON]; writing--review and editing, J.F. and C.N.; visualisation, [PERSON] and J.F.; supervision, C.N.; project administration, A.B.; funding acquisition, A.B. All authors have read and agreed to the published version of the manuscript.  
The work was supported by the \"Generalitat Valenciana\" of Spain (projects PROMETEO/2021/006) and the \"Ministerio de Ciencia, Innovacion y Universidades\" of Spain (project PID2021-123124 OB-100). [PERSON] thanks the \"Ministerio de Ciencia, Innovacion y Universidades\" for the grant FPU22/04316.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} RCWA & Rigorous Coupled Wave Analysis \\ FDTD & Finite Difference Time Domain \\ CWT & Coupled Wave Theory \\ FNN & Feedforward Neural Network \\ CNN & Convolutional Neural Network \\ BO & Bending Only \\ RMSProp & Root Mean Square Propagation \\ SGD & Stochastic Gradient Descent \\ RCWA-SM & Rigorous Coupled Wave Analysis--Shooting Method \\ SF-FDTD & Split Field Finite Difference Time Domain \\ \end{tabular}