# Physicists solve 2,000-year-old optical problem

A trio of physicists from the National Autonomous University of Mexico and Tec de Monterrey has solved a 2,000-year-old optical problem—the Wasserman-Wolf problem. In their paper published in the journal Applied Optics, Rafael González-Acuña, Héctor Chaparro-Romo, and Julio Gutiérrez-Vega outline the math involved in solving the puzzle, give some examples of possible applications, and describe the efficiency of the results when tested.

Over 2,000 years ago, Greek scientist Diocles recognized a problem with —when looking through devices equipped with them, the edges appeared fuzzier than the center. In his writings, he proposed that the effect occurs because the lenses were spherical—light striking at an angle could not be focused because of differences in refraction. Isaac Newton was reportedly stumped in his efforts to solve the problem (which became known as ), as was Gottfried Leibniz.

In 1949, Wasserman and Wolf devised an analytical means for describing the problem, and gave it an official name—the Wasserman-Wolf problem. They suggested that the to solving the problem would be to use two aspheric adjacent surfaces to correct aberrations. Since that time, researchers and engineers have come up with a variety of ways to fix the problem in specific applications—most particularly cameras and telescopes. Most such efforts have involved creating aspherical lenses to counteract refraction problems. And while they have resulted in improvement, the solutions have generally been expensive and inadequate for some applications.

Now, a means for fixing the problem with any size has been found by González-Acuña, Chaparro-Romo and Gutiérrez-Vega, described in a lengthy math formula. It is based on describing ways in which the shape of a second aspherical surface needs to be given a first surface, along with object-image distance. In essence, it relies on a second surface fixing problems with the first surface. The result is elimination of spherical aberration.

Once the math was established, the researchers tested it by running simulations. They report that their technique can produce lenses that are 99.9999999999 percent accurate. The researchers suggest the formula can be used in applications including eyeglasses, contact lenses, telescopes, binoculars and microscopes.

More information: Rafael G. González-Acuña et al. General formula to design a freeform singlet free of spherical aberration and astigmatism, Applied Optics (2019). DOI: 10.1364/AO.58.001010

Feedback to editors

Aug 09, 2019
Thank you for bringing the article to our attention! One note: Diocles worked with spherical mirrors. Lenses weren't available until much later.

Aug 09, 2019
The cure for spherical aberration in a mirror is to make its surface a paraboloid of revolution rather than a sphere. That's a bit harder to do with a lens, which has two surfaces and refraction; that's why all the complicated math.

Aug 09, 2019
It can only be solved with {FeedBack} that captures the field curvature over time! For humans, that's over 15ms; if ya can't "Morphed that into reality ... mmm; and no repeatable for that user! Or to your computed "reality!

But, what are you guys using to adjust for reality? What see? Or?

Aug 10, 2019
@DMB
Thank you for bringing the article to our attention! One note: Diocles worked with spherical mirrors. Lenses weren't available until much later.
When and where was the first mirror invented? http://www.mirror...mirrors/

Aug 10, 2019
Parabolic mirrors on reflecting telescopes were first used in the nineteenth century: https://history.a...ndex.php

Aug 10, 2019
Was wondering what ever happened to that 'light field' camera called 'Lytro'

Please tell me capitalism didn't kill it...

Aug 10, 2019
It doesn't magnify, @Proto. They might attach it to telescopes that do later.

Aug 10, 2019
It doesn't magnify, @Proto. They might attach it to telescopes that do later.
I recall it was all the rage among the astronomical community, everyone wanted one. There's a reason it doesn't magnify: it has no physical lens. There's a reason it has no physical lens, but that's the subject for this article.

Aug 10, 2019
Please tell me capitalism didn't kill it...

The Lytro camera attempted to bypass the problem of having to focus the camera, but you still have to set up the ISO and shutter/aperture speeds, so you'd still get blurry pictures if the settings were wrong, or if there just wasn't enough light.

Ordinary auto-focus is good enough that the advantage of adjusting the focus after the fact became a moot point. The cameras were awkward boxes that tried to compete with point-and-shoot consumer cameras on a market that was already being lost to cellphone cameras, and the professionals didn't want them either because the light field system afforded them -less- control over the image with far too many compromises elsewhere.

The company tried to push expensive niche products, such as a 360 degree light field camera, but pretty much nobody wanted or needed them, so they folded down and sold the whole business to Google.

Aug 10, 2019
For example, deliberate over-exposure against a white sheet is used in photography to take pictures for ads and catalogs, where the product appears on a white background. Alternatively, an under-exposed black sheet is used for the opposite effect.

The Lytro camera was using computational means to construct the picture from light falling on the entire sensor area, so it could not understand under/over exposure where the sensor is just saturated by light or doesn't actually measure anything but noise.

Professionals couldn't use it to much effect, because the camera was essentially on full-auto mode all the time - no manual controls - like putting a torque converter automatic gearbox into a rally car.

Aug 10, 2019
The Lytro camera was using computational means to construct the picture from light falling on the entire sensor area, so it could not understand under/over exposure where the sensor is just saturated by light or doesn't actually measure anything but noise
Right, the sensor was designed to merely capture the 'light field' including polarization and they opted to keep the processing proprietary. Too bad.

Aug 13, 2019
Or make sure your Analog to Digital from input to output is calibrated!