Design of two-dimensional reflective imaging systems: An approach based on inverse methods

Imaging systems are inherently prone to aberrations. We present an optimization method to design two-dimensional freeform reflectors that minimize aberrations for various parallel ray beams incident on the optical system. We iteratively design reflectors using inverse methods from non-imaging optics and optimize them to obtain a system that produces minimal aberrations. This is done by minimizing a merit function that quantifies aberrations and is dependent on the energy distributions at the source and target of an optical system, which are input parameters essential for inverse freeform design. The proposed method is tested for two configurations: a single-reflector system and a double-reflector system. Classical designs consisting of aspheric elements are well-known for their ability to minimize aberrations. We compare the performance of our freeform optical elements with classical designs. The optimized freeform designs outperform the classical designs in both configurations

An Iterative Least-Squares Method for the Hyperbolic Monge-Amp\`ere Equation with Transport Boundary Condition

A least-squares method for solving the hyperbolic Monge-Amp\`ere equation with transport boundary condition is introduced. The method relies on an iterative procedure for the gradient of the solution, the so-called mapping. By formulating error functionals for the interior domain, the boundary, both separately and as linear combination, three minimization problems are solved iteratively to compute the mapping. After convergence, a fourth minimization problem, to compute the solution of the Monge-Amp\`ere equation, is solved. The approach is based on a least-squares method for the elliptic Monge-Amp\`ere equation by Prins et al., and is improved upon by the addition of analytical solutions for the minimization on the interior domain and by the introduction of two new boundary methods. Lastly, the iterative method is tested on a variety of examples. It is shown that, when the iterative method converges, second-order global convergence as function of the spatial discretization is obtained.Comment: 30 pages, 24 figure

Computation of aberration coefficients for plane-symmetric reflective optical systems using Lie algebraic methods

The Lie algebraic method offers a systematic way to find aberration coefficients of any order for plane-symmetric reflective optical systems. The coefficients derived from the Lie method are in closed form and solely depend on the geometry of the optical system. We investigate and verify the results for a single reflector. The concatenation of multiple mirrors follows from the mathematical framework

Similarities and differences of two exponential schemes for convection-diffusion problems: The FV-CF and ENATE schemes

10 figures, 8 tables.In this paper, we present a comparison of two novel exponential schemes for convection-diffusion problems. An exponential scheme uses in one way or another the analytical solution of the flux of a one-dimensional (1D) transport equation thereby improving the results of the simulation. In a multidimensional problem, the 1D solution is combined with operator splitting. The two approximations to be assessed are the Finite Volume-Complete Flux (FV-CF) and the Enhanced Numerical Approximation of a Transport Equation (ENATE) schemes. They were proposed by the two groups that co-author the current paper. Both schemes share many similarities in 1D but differ, especially in 2D, in some aspects that will be highlighted. In their derivation the algebraic coefficients of the computational stencil are integrals of flow parameters whose calculation is crucial for the accuracy of either method. These factors and their various approximations will be analysed. Some test cases will be used to check the ability of both schemes to provide accurate results.V.J. Llorente and A. Pascau were supported by the European Union through FEDER funding and Diputación General de Aragón “Construyendo Europa desde Aragón” [Government of Aragon “Building Europe from Aragon”].Peer reviewe