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Deterministic Approach Reduces Error Potential in Freeform Design Process

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BRUSSELS, May 26, 2021 — Researchers at Brussels Photonics (B-PHOT), Vrije Universiteit Brussel, Belgium, developed a deterministic direct optical design method for freeform imaging systems. The method is based on differential equations derived from Fermat’s principle and solved using power series. The approach, the researchers said, provides a methodology for the design of optical imaging systems from scratch that reduces reliance on a “trial and error” approach.

Fermat’s principle in optics states that light traveling between two points seeks a path where the number of waves, or the optical length between the points, is equal to that in neighboring paths. In other words, the path that a ray of light travels between two points requires either a minimum or a maximum time — meaning that two beams of light diverging from a distant object point and converged by a lens to an image point will have identical optical path lengths.

The method specifically enables the calculation of the optical surface coefficients that ensure minimal image blurring for each individual order of aberrations. Using mirror- and lens-based designs to test the approach, the researchers demonstrated the systematic, deterministic, scalable, and holistic character of the method with catoptric and catadioptric design examples.

The advent of freeform optical systems, the researchers said in a published paper, followed the introduction of ultraprecision manufacturing methods that have enabled fabrication of the types of lenses and mirrors that lack the common translational (or rotational) symmetry about a plane or axis. These freeform optical elements can be used to improve the performance and to reduce the volume and weight of optical imaging systems of which they are components. Because imaging systems prior to the introduction of new ultraprecision manufacturing methods commonly included spherical and aspherical refractive lenses, reflective mirrors, or a combination of both, scientists and engineers were able to use aberration theory to describe and quantify the deviation of light rays from ideal focusing in an imaging system.

Graphical user interface of the developed open-access trial web application that provides readers the opportunity for hands-on freeform design experience. Courtesy of Fabian Duerr and Hugo Thienpont.
Graphical user interface of the developed open-access trial web application that provides readers the opportunity for hands-on freeform design experience. Courtesy of Fabian Duerr and Hugo Thienpont.
Still, the design of many current optical systems relies on raytracing and optimization algorithms. A successful (and widely used) optimization-based optical design strategy consists of a well-known optical system and using it as a starting point from which to achieve incremental improvements throughout the design process. That approach, which the researchers call a “step-and-repeat” approach, requires guesswork and is sometimes referred to as “art and science” — especially in the design of freeform optics.

Beyond the differential equations and solution scheme used in the approach, the scientists summarized the operational principle of their method in two steps: solving the nonlinear first-order case, and solving the linear systems of equations in ascending order.

“Most importantly, these two steps are identical for all (freeform) optical designs,” the researchers said.

“The presented method allows a highly systematic generation and evaluation of directly calculated freeform design solutions that can be readily used as an excellent starting point for further and final optimization,” they added. “As such, it allows the straightforward generation of ‘first time right’ initial designs that enable a rigorous, extensive, and real-time evaluation in solution space when combined with available local or global optimization algorithms.”

The research was published in Light: Science & Applications (www.doi.org/10.1038/s41377-021-00538-1).

Photonics.com
May 2021
opticsEuropeBelgiumfreeform opticsimaging algorithmsmathematicsapplied mathematicsFermat's lawoptical fabricationoptical designResearch & TechnologyeducationB-PHOT Brussels Photonics Team

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