Thesis Summary:

The goal of this thesis is to contribute to the development of numerical methods for solving equations arising from modeling light harvesting devices. The optimal design of a photovoltaic solar cell requires efficient solvers that provide accurate approximations of the solution of Maxwell or Helmholtz problems, because these equations are usually solved for a wide range of optical and physical parameters. Our work focuses on two techniques: Asymptotic Models in structures having thin layers and a Perfectly Matched Layer (PML) approach to efficiently handle the radiation conditions. Even though these techniques can be used in different types of structures, in this work we consider one dimensional periodic diffraction gratings. First, we formulate an asymptotic model for implementation in the finite-element method to calculate diffraction from a planar multilayered structure having a shallow surface-relief grating. The thin grating layer containing the shallow grating is replaced by a planar interface with transmission conditions that differ from the standard continuity conditions, thereby eliminating the necessity of representing the grating layer by a fine mesh. The parameters defining the shallow surface-relief grating are thereby removed from the geometry to the transmission conditions. Adoption of the asymptotic model will considerably reduce the computational cost of optimizing the grating shape, since there is no need to re-mesh at every optimization step. In the second part of the thesis we describe a different asymptotic model for the same multilayered structure and provide theoretical error estimates with respect to the thickness of the grating layer. Finally, we introduce a PML approach for finite element calculations of diffraction by metallic surface-relief gratings. We use a non-integrable absorbing function which allows us to use thin absorbing layers, which reduces the computational time when simulating this type of structure. In addition, we numerically determine the best choice of the absorbing layer parameters and show that they are independent of the wavelength.

ISI Publications from the Thesis

Peter MONK, Cinthya RIVAS, Rodolfo RODRíGUEZ, Manuel SOLANO: An asymptotic model based on matching far and near field expansions for thin gratings problems. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 55, pp. S507-S533, (2021).

Cinthya RIVAS, Rodolfo RODRíGUEZ, Manuel SOLANO: A perfectly matched layer for finite-element calculations of diffraction by metallic surface-relief gratings. Wave Motion, vol. 78, pp. 68-82, (2018).

Akhlesh LAKHTAKIA, Peter MONK, Cinthya RIVAS, Rodolfo RODRíGUEZ, Manuel SOLANO: Asymptotic model for finite-element calculations of diffraction by shallow metallic surface-relief gratings. Journal of the Optical Society of America A, vol. 34, 1, pp. 68-79, (2017).