Solving a 2,000-Year-Old Mystery

It’s not every day that a 2,000-year-old optics problem is solved. However, Rafael G. González-Acuña, a doctoral student at Tecnológico de Monterrey, set his sights on solving such a problem—spherical aberration in lenses. How can light rays focus on a single point, taking into account differing refraction? It was a problem that, according to Christiaan Huygens back in 1690, even Isaac Newton and Gottfried Leibniz couldn’t sort out, and was formulated two millennia ago in Greek mathematician Diocles’s work, On Burning Mirrors.

But González-Acuña and his colleagues realized that today, they had the use of the Wolfram Language and its computational tools to solve this age-old problem. The result? A breakthrough publication that outlines an analytical solution to why and how lensed images are sharper in the center than at the edges, with 99.999999999% accuracy simulating 500 light beams.

As it happens, González-Acuña was recently at the Wolfram Summer School, and we had the opportunity to ask him a little bit about his work.

How did you become interested in solving problems in optics with the Wolfram Language?

My interest in optics began when I was studying for my bachelor’s in physics; I was very interested in the problem of image formation.

I was working on solving problems symbolically by hand, which is very hard because I had misprints and everything became more complicated. My colleague Héctor A. Chaparro-Romo told me that Mathematica was excellent for symbolic manipulation and computation.

Luckily, at my university, Tecnológico de Monterrey, there is a site license for Mathematica, and I started to use it. I liked Mathematica a lot because of the way it displays the results, the way it evaluates inline and that by default it is symbolic.

What was the problem you were trying to solve?

The problem I wanted to solve was the design of a spherical aberration-free lens. In other words, all the rays that come from an object point that cross through the lens converge in an image point. This old problem had no analytical solution for two millennia, largely because the symbolic expressions are huge—I mean the general equation itself is more than 11 pages!

There were many numerical solutions to this problem, but the difference between a numerical result and an analytical one is immeasurable—the analytical results preserve the physics of the problem. Not only did I find the general solution to the problem, but I also found that the solution is unique, and Mathematica is a great tool for all this.

I published the results in “General Formula for Bi-aspheric Singlet Lens Design Free of Spherical Aberration” and “General Formula to Design a Freeform Singlet Free of Spherical Aberration and Astigmatism.”

Equations solving the problem of spherical aberration in lenses

How did you develop your solution, and what now?

I spent several months working on the problem. It was an obsession. The whole time I was in contact with my coauthor Chaparro-Romo—we were working as team, sharing code, ideas and possible solutions; I had several amazing discussions to get to the generalization of the problem of free-form lenses.

I have read several comments about the problem in relation to Newton. In fact, in his book Treatise on Light, Christiaan Huygens mentions that Newton, Leibniz and Descartes all failed to achieve the solution.

And actually, Huygens got an approximated solution in chapter six of his Treatise:

Excerpts from Christiaan Huygens’s 1690 book Treatise on Light, and an output from Rafael’s Mathematica notebook (far right)

But after all these years, we now have an equation originally sought by Huygens, Descartes, Leibniz and Newton more than 300 years ago, and by Diocles more than 2,000 years ago. Check out our Wolfram Notebook on the Construction of a Spherical Aberration-Free Lens for an overview of Rafael G. González-Acuña and Héctor A. Chaparro-Romo’s solution.