The application of freeform elements in optical systems increases the number of design variables. In order to use the additional degrees of freedom most efficiently for correcting the system, the optimization process requires a guidance from the lens designer. The knowledge of aberrations generated in the system provides insights for selecting the best starting configuration as well as for choosing the position of the freeform element. In this work we use a new numerical method [Oleszko et al., JOSAA Vol. 34(10), 1856 (2017)] to study surface-by-surface contributions to the total wave aberration of freeform optical systems. Surface contributions are divided due to their origin into intrinsic, induced and transfer components. The study of intrinsic and induced effects assists in finding design solutions corrected for aberrations of orders higher than the fourth in the expansion of the wave aberration function. In contrast to the analytical approach, the method does not incorporate the field dependency into the wave aberrations and the error of the chief ray is studied separately. That allows to visualize the distortion of the image grid at the intermediate image planes.
Analysis of surface-by-surface Seidel aberration contributions is the conventional approach for detecting surfaces sensitive to tolerances in the axially symmetric optical systems. Analogical tool for generalized systems is currently not provided in the optical design programs. Here we present an alternative numerical method to find surface contribution to the total wave aberration without limitation to the certain expansion order and with no constraints on system geometry. Surface contributions are further divided due to their origin into intrinsic, induced and transfer components. Each component is determined from the separate set of rays. In order to specify numerically obtained wavefront errors, the method is combined with Zernike fringe decomposition routine. As an example, sensitivity to tilt errors in a plane symmetric three mirror system consisting of convex mirrors with equal optical powers, was studied. The mirrors in the system are considered with spherical and toroidal basic shapes, with the freeform element placed on different positions, giving in total six configurations. We down select the least and the most sensitive system and present the detailed tolerance analysis.
Freeform surfaces are a new and exciting opportunity in lens design. The technological boundary conditions for manufacturing surfaces with reduced symmetry are complicated. Recently the progress in understanding and controlling this kind of components is ready for use in commercial products. Nearly all procedures of classical design development are changing, if freeform surfaces are used. The mathematical description of the surfaces, the optimization algorithms in lens design and their convergence, the initial design approaches, the evaluation of performance over the field of view, the data transfer in the mechanical design software and in the manufacturing machines, the metrology for characterization of real surfaces and the return of the real surfaces into the simulation are affected. In this contribution, in particular an overview on possible mathematical formulations of the surfaces is given. One of the requirements on the descriptions is a good performance to correct optical aberrations. After fabrication of real surfaces, there are typical deviations seen in the shape. First more localized deformations are observed, which are only poorly described by mode expansions. Therefore a need in describing the surface with localized finite support exists. Secondly the classical diamond turning grinding process typically shows a regular ripple structure. These midfrequency errors are best described by special approaches. For all these cases it would be the best to have simple, robust solutions, that allow for fast calculation in fitting measured surfaces and in raytrace.
Optical systems can benefit strongly from freeform surfaces, however the choice of the right representation isn`t an easy one. Classical representations like X-Y-polynomials, as well as Zernike-polynomials are often used for such systems, but should have some disadvantage regarding their orthogonality, resulting in worse convergence and reduced quality in final results compared to newer representations like the Q-polynomials by Forbes. Additionally the supported aperture is a circle, which can be a huge drawback in case of optical systems with rectangular aperture. In this case other representations like Chebyshev-or Legendre-polynomials come into focus. There are a larger number of possibilities; however the experience with these newer representations is rather limited. Therefore in this work the focus is on investigating the performance of four widely used representations in optimizing two ambitious systems with very different properties: Three-Mirror-Anastigmat and an anamorphic System. The chosen surface descriptions offer support for circular or rectangular aperture, as well as different grades of departure from rotational symmetry. The basic shapes are for example a conic or best-fit-sphere and the polynomial set is non-, spatial or slope-orthogonal. These surface representations were chosen to evaluate the impact of these aspects on the performance optimization of the two example systems. Freeform descriptions investigated here were XY-polynomials, Zernike in Fringe representation, Q-polynomials by Forbes, as well as 2-dimensional Chebyshev-polynomials. As a result recommendations for the right choice of freeform surface representations for practical issues in the optimization of optical systems can be given.