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For more than 150 years, scientists have advanced aberration theory todescribe, analyse, and eliminate imperfections that disturb the imaging qualityof optical components and systems. Simultaneously, they have developed opticaldesign methods for and manufacturing techniques of imaging systems withever-increasing complexity and performance up to the point where they are nowincluding optical elements that are unrestricted in their surface shape. Theseso-called optical freeform elements offer degrees of freedom that can greatlyextend the functionalities and further boost the specifications ofstate-of-the-art imaging systems. However, the drastically increased number ofsurface coefficients of these freeform surfaces poses severe challenges for theoptical design process, such that the deployment of freeform optics remainedlimited until today.  In this paper, we present a deterministic direct optical design method forfreeform imaging systems based on differential equations derived from Fermat'sprinciple and solved using power series. The method allows calculating theoptical surface coefficients that ensure minimal image blurring for eachindividual order of aberrations. We demonstrate the systematic, deterministic,scalable and holistic character of our method with catoptric and catadioptricdesign examples. As such we introduce a disruptive methodology to designoptical imaging systems 'first time right', we eliminate the 'trial and error'approach in present-day optical design, and we pave the way to a fast-trackuptake of freeform elements to create the next-generation high-end optics.

Freeform imaging systems: Fermat's principle unlocks 'first time right' design