This project sought to implement an automatic differentiation scheme around the Rigorous CoupledWave Analysis algorithm to create a tool for parameter retrieval of a device’s structure for 3D-profilometry applications. Rigorous Coupled-Wave Analysis (RCWA) is a semi-analytical Maxwell solver, used to numerically evaluate the properties of three-dimensional structures which are periodic in two dimensions. The three-dimensional Maxwell equations are numerically evaluated in two dimensions using the Plane Wave Expansion Method. This results in a matrix differential equation along the remaining dimension. This differential equation is then solved to transmission and reflection values of incident light using the Transfer Matrix Method. Automatic Differentiation is a method of evaluating the derivative/gradient/Jacobian of a function. By evaluating the gradient of a function, it is possible to perform gradient-descent to find local extrema of a function and the inputs which achieve that extrema. By finding the minimum of a loss function, gradient descent becomes an optimization tool. We sought to apply this to an RCWA algorithm where the loss function is minimized with respect to parameters which define the structure of the device. These optimal parameters then define the structure of a numerical model which most closely replicates measured data, and thereby become ”best guesses” of the device attempted to model. Automatic differentiation was achieved by implementing a conventional RCWA algorithm in TensorFlow. TensorFlow is a software package which has built-in functionality for Automatic Differentiation and gradient descent. For this scheme, we have demonstrated the following: • The ability to replicate the provided validation dataset down to near-rounding error accuracy. • When the parameters are perturbed, the model is able to perform backpropagation and perform parameter retrieval to regain the correct value for every parameter of the device. During the process of implementation, we have invented an alternate formulation for the S-parameter matrix, whose terms serve as analogues to those found in the conventional Scattering parameter matrix method. These are analogous to the W, V, and X matrices. We have demonstrated that using these analogues in lieu of the known values inside the S-parameter matrix algorithm will numerically yield the same values as those found by eigendecomposition. We have demonstrated that these analogues can be calculated using iterative methods. These methods involve the use of matrix multiplications, scalar multiplications, matrix additions, and matrix solves. We have demonstrated that this algorithm can be up to 20 times faster on the HPC hardware provided by TU Delft. The following areas for further research are suggested; • Validation of the algorithm for a wider range of cases; • Exploration of adjustments to the overall RCWA algorithm for improved performance; • Alternate gradient-descent methods which could provide accelerated convergence; • Attempting the use of gradient-descent methods for other Maxwell solvers.

This thesis demonstrates the feasibility of Extreme-ultraviolet (XUV) high-harmonic generation from structured silica, and was performed at the Advanced Research Centre for Nanolithography (ARCNL). The project focuses on High-harmonic generation (HHG) from condensed matter, and further explores the possibility of high-harmonic generation from micro- and nano-structured silica. HHG from solids was discovered less than a decade ago, and it is expected to be a new source for coherent ultrafast pulses, showing potential in many applications such as HHG spectroscopy, imaging and photonic devices. By generating high-harmonics in structured solids, the capability to control HHG properties by engineering the topology of the surface on solids has been demonstrated. Previous research has demonstrated the control of HHG in the visible light regime by generating from structured semiconductors. In this project, we aim to generate high-harmonics in the XUV regime, and control the properties of XUV light. Our work shows the potential of using structured dielectric materials as new XUV optics, and applications on HHG high-resolution lens-less imaging.