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.

150 多年来，科学家们提出了先进的像差理论来描述、分析和消除干扰光学元件和系统成像质量的缺陷。同时，他们开发了成像系统的光学设计方法和制造技术，其复杂性和性能不断提高，直到现在他们包括表面形状不受限制的光学元件。这些所谓的光学自由曲面元件提供的自由度可以大大扩展功能并进一步提高最先进成像系统的规格。然而，这些自由曲面的表面系数数量急剧增加，给光学设计过程带来了严峻的挑战，以至于直到今天，自由曲面光学器件的部署仍然受到限制。在本文中，我们提出了一种用于自由曲面成像系统的确定性直接光学设计方法，该方法基于源自费马原理并使用幂级数求解的微分方程。该方法允许计算光学表面系数，确保每个像差阶的图像模糊最小。我们通过反射和反射折射设计示例展示了我们方法的系统性、确定性、可扩展性和整体性。因此，我们引入了一种颠覆性的方法来从头开始设计光学成像系统，我们在很大程度上减少了当今光学设计中的“试错”方法，并为快速采用自由形式元素以创建下一代高端光学。我们包含一个用户应用程序，允许用户亲身体验这种独特的设计方法。