Abstract
For more than 150 years, scientists have advanced aberration theory to describe, analyze and eliminate imperfections that disturb the imaging quality of optical components and systems. Simultaneously, they have developed optical design methods for and manufacturing techniques of imaging systems with ever-increasing complexity and performance up to the point where they are now including optical elements that are unrestricted in their surface shape. These so-called optical freeform elements offer degrees of freedom that can greatly extend the functionalities and further boost the specifications of state-of-the-art imaging systems. However, the drastically increased number of surface coefficients of these freeform surfaces poses severe challenges for the optical design process, such that the deployment of freeform optics remained limited until today. In this paper, we present a deterministic direct optical design method for freeform imaging systems based on differential equations derived from Fermat’s principle and solved using power series. The method allows calculating the optical surface coefficients that ensure minimal image blurring for each individual order of aberrations. We demonstrate the systematic, deterministic, scalable, and holistic character of our method with catoptric and catadioptric design examples. As such we introduce a disruptive methodology to design optical imaging systems from scratch, we largely reduce the “trial-and-error” approach in present-day optical design, and we pave the way to a fast-track uptake of freeform elements to create the next-generation high-end optics. We include a user application that allows users to experience this unique design method hands-on.
Highlights
Optical imaging systems have been playing an essential role in scientific discovery and societal progress for several centuries[1,2]
With the differential equations established and the overall system specifications introduced, two design steps need to be taken: (1) solve the nonlinear first-order case using a standard nonlinear solver or by making use of equivalent first-order optics tools that will provide structurally similar nonlinear equations[42,43]; (2) solve the linear systems of equations in ascending order by setting unwanted aberrations to zero or by minimizing a combination thereof as required by the targeted specifications of the imaging freeform system. These two steps are identical for all freeform optical designs and are implemented as follows: 1. We evaluate Eqs. (8) and (9) for all (i, j, k, l, m) for the first-order case with j +k +l +m = 1
Due to the rotational symmetry of the problem, we can calculate the solution in the x–z-plane
Summary
Optical imaging systems have been playing an essential role in scientific discovery and societal progress for several centuries[1,2]. The last decennia, with the introduction of new ultra-precision manufacturing methods, such as singlepoint diamond turning and multi-axis polishing[4], twophoton polymerization[5,6] or other additive manufacturing technologies[7], it has become possible to fabricate lenses and mirrors that have at least one optical surface that lacks the common translational or rotational symmetry about a plane or an axis Such optical components are called freeform optical elements[8] and they can be used to greatly extend the functionalities, improve performance, and reduce volume and weight of optical imaging systems that are principal parts of spectrometers[9], telescopes[10,11], medical imaging systems[12,13], augmented and virtual reality systems[14], or lithography platforms[15,16]. As such imaging systems including freeform optical elements will be key to tremendously advance science and engineering in a wide range of application domains such as astronomy, material research, chip fabrication, visualization, metrology and inspection, energy production, safety and security, biotechnology and medical imaging and diagnosis[17]
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