  For more than 150 years, scientists have advanced aberration theory to
describe, analyse, 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 'first time right', we eliminate 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.
