# Numerical Estimation of Bending in Holographic Volume Gratings by Means of RCWA and Deep Learning
## 1 Introduction  
Holographic gratings have emerged as versatile and indispensable tools in optics and photonics, finding applications in fields as diverse as spectroscopy, telecommunications, and laser technology. They occupy a central position in various scientific, industrial, and technological domains because they can disperse, reflect, or diffract light in precise and controlled ways [1; 2; 3; 4]. Often, imperfections in the recording or fabrication process may lead to an unwanted bending of the structure of diffraction gratings, which inevitably impacts its optical properties in uncontrolled ways. A non-uniform deformation of its fringes characterises this phenomenon, and it is mainly associated with a pre-stress in the recording material. On the other hand, many researchers have focused on bending as a crucial element for holographic sensors [5; 6]. [PERSON] et al. [7] considered the bending deformation of a hydrogel actuator to be an essential deformation parameter, and [PERSON] et al. have corroborated the application of holographic sensors as versatile optical sensing platform to characterise the deformation of the polymer due to its narrow diffraction spectrum, high visualisation, and stability [8; 9]  
The phenomenon of bending within holographic gratings has been understudied despite its frequent occurrence. The leading theory previously developed on the subject has been carried out by [PERSON], which is based on [PERSON]'s coupled wave theory with an attenuated grating model [10; 11]. This method focuses solely on the substrate itself and fails to account for air-medium discontinuities, thereby presenting limitations in its applicability. Indeed, over the years, Rigorous Coupled Wave Theory (RCWT) has been proven to be a powerful and versatile numerical method in the context of diffractive optics and photonics. This technique, initially developed by [PERSON] and [PERSON] [12; 13; 14; 15],has significantly advanced our understanding of light diffraction and its interaction with periodic structures. RCWT is founded on the rigorous solution of Maxwell's equations and has been widely adopted in the study of diffraction gratings, photonic crystals, and other periodic optical structures [16]. Over the years, RCWT has evolved and been extended to address complex and practical problems, including incorporating anisotropic and inhomogeneous materials [17]. Its continued development and application have cemented RCWT as a reference of numerical techniques for diffractive optical analysis and design. Accurately determining the bending of holographic grating fringes measured in laboratory settings implies using cost-consuming advanced techniques such as Scanning Electron Microscopy, Ellipsometry, or Interferometry. Due to the complex nature and the non-derivability of the formalism developed for the analysis of bending, an analytical approach to characterise these deformations from diffraction efficiency patterns is not feasible. Therefore, this research aims to set up a novel scheme based on deep learning solutions for estimating the parameters that define the bending in holographic gratings from the diffraction efficiency curves obtained numerically, addressing these limitations and providing a more accessible solution. Neural network-based solutions, particularly due to their nonlinear activation functions, are widely known precisely at regression problems in cases where analytical, linear solutions are not feasible. Thus, the most straightforward and viable solution is developing a computational regression model based on these technologies [18; 19; 20]. For feeding the deep learning engine, we proposed a novel RCWA-SM scheme that considers bending as a phase in dielectric permittivity expansion and uses the shooting method to solve the system of equations. A numerical SF-FDTD scheme was also considered for validating the RCWA-SM implementation. It is worth noting that RCWA-SM was found to be the best candidate for use in the deep learning engine due to its performance compared to the SF-FDTD scheme.