Edge Sharpness: A Solved Problem?

According to a spate of recent news articles, two Mexican physicists have cracked a problem that stumped Isaac Newton and Diocles, the ancient Greek mathematician who first recognized what is now known as spherical aberration. Two doctoral students at Mexico’s Tecnológico de Monterrey, Rafael G. González-Acuña and Héctor A. Chaparro-Romo, have published a "mind-melting equation" that allegedly solves what is known as the Wasserman-Wolf problem, formulated in 1949, regarding spherical aberration in lenses. The paper, published in the journal Applied Optics (Vol. 57, Issue 31, pp. 9341–9345 [2018]), is titled "General formula for bi-aspheric singlet lens design free of spherical aberration." Supposedly, when lensmakers of all kinds begin using it, spherical aberration will become much simpler to correct and variations in edge-to-edge sharpness in lenses will become a thing of the past.

Sounds good. On the other hand, the story also kinda resembles those "viral" news stories that get going from time to time that the news media just loves to repeat...you know, like redheads are going to go extinct, or alien spaceships resemble flying saucers, or you should drink eight glasses of water a day (a myth that originated in an unresearched article in a women's magazine many years ago and took on a life of its own). What those stories have in common is that there is no scientific basis for any of them, and yet the public, for some reason, just loves them and can't get enough. I'll just note that no journalist has the expertise to check out or evaluate this story.

...Certainly including me. Here; here's the equation. Why don't you check it out and tell me if it'll work? That's farther above my head than a star that's too far away to see with an Orion StarBlast.

González-Acuña (right) was awarded The 2019 Optical Design and Engineering Scholarship from the International Society for Optics and Photonics.

As we speak, huge amounts of research are going into technologies that might be applicable to smartphone camera lenses. That much, at least, is certain.

Mike
(Thanks to Dr. Charlie and several other readers)

Original contents copyright 2019 by Michael C. Johnston and/or the bylined author. All Rights Reserved. Links in this post may be to our affiliates; sales through affiliate links may benefit this site.

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(To see all the comments, click on the "Comments" link below.)
Featured Comments from:

GKFroehlich: "I've worked in the area of electromagnetic propagation for most of the last 45 years—six in graduate school, and 39 on the job. The work involves understanding, characterizing, and describing—mathematically—how electromagnetic waves are affected by the materials they interact with, as the waves travel from one point to another. For example, visible light (which occupies only a tiny portion of the electromagnetic spectrum) might interact with a subject by reflecting off of it, and then by passing through the various glass elements of a lens, before it is 'detected' by an appropriate sensor (e.g., film, our retina, or the sensor in you Fuji). Similarly, a short pulse of microwave energy might interact with a subject by reflecting off of it, and then by passing through the turbulent atmosphere, followed by passage through some kind of antenna, before it is detected by the receiver. The mathematics used to characterize and describe this journey, and the waves' interactions with whatever they encounter, can appear pretty formidable to those not familiar with it. But in fact, the math required to merely characterize and describe is reasonably straightforward.

"Where it starts to become very difficult is when—instead of merely describing the effects of a given path through a given set of materials—you have a very particular outcome in mind, and you wish to discover what materials, geometries, and other properties will provide that outcome. That is a design problem, and those are in general much more difficult problems to formulate and to solve. They are also the most common types of problems encountered. After all, it's usually easy enough to evaluate how an existing system performs; it's a whole different ball game to specify desired performance, and then to find the system that will deliver that performance! And this, of course, is exactly the problem that lens designers, radar designers, and the like must deal with every day.

"In the early days of my career, we had to solve these problems mathematically, and the holy grail was always to find a solution in closed form. That means the desired result could be expressed as an equation—preferably one that could actually be computed when numerical values were substituted for all the symbols. Whenever a closed-form solution eluded us, we would then have to begin making simplifying assumptions and approximations, and applying our experience, to arrive at our best-guess result.

"As computers became more ubiquitous, and much faster, and as memory became less and less expensive, it became feasible to attack these problems iteratively—that is, to make an informed guess, evaluate it (i.e., try it on for size), refine the guess, evaluate this new guess, refine it some more, evaluate it again, and so on—until a satisfactory or acceptable solution was finally obtained. This approach is now widespread and well understood, and it's usually the default method for attacking some of the more difficult design problems in electromagnetic propagation, including lens design.

"Lens designers have long understood that aspherization could be used to minimize spherical aberration, that bending could be used to minimize coma, that proper placement of the field stop could minimize astigmatism, and so on. And for years they have quite effectively applied 'brute force' iterative optimization, with some remarkable results. The contribution of González-Acuña and Chaparro-Romo is to find this particular solution directly in closed form. It is a remarkable achievement in its own right, and should not be minimized. But whether it proves more computationally efficient than current iterative approaches remains to be seen. And one last observation: It appears that their solution is applicable only at a single wavelength, or at least over a very narrow band of wavelengths. If so, practical applicability might still be a long way off."

David Evans: "Diocles proved that replacing a spherical mirror with a paraboloid could eliminate spherical aberration. As far as I know he was interested in burning mirrors and did not consider lenses. He couldn't have got far with lens theory since the law of refraction was not known in his time. The math is way over my head too, but it appears to apply to a single-element lens for which chromatic aberration will still be a problem. I don't know how it applies to complex lenses."

Ernie Van Veen: "I entered the equation into my son's scientific calculator and it seems to work, although, no matter what values are given to the variables, the answer always appears to be 42."

Tim Bradshaw: "I've looked through the paper and it looks at fine to me: I'd have to learn more optical terminology, and sit down with pen, paper and algebra system to actually follow the derivation, but it's immediately clear this isn't cranky in any way, so it's probably correct.

"I think that getting a closed-form solution for this is a remarkable achievement, but how much relevance it has for practical camera lens design, if any, is not clear. This is for two reasons.

• One of the assumptions they make is that the refractive index of the lens material is constant. This is probably a good assumption in the sense that the refractive index should be the same throughout the body of the lens, but it's horribly wrong in that it will vary with the frequency of the light: with colour in other words, unless the material of the lens has no dispersion. Well, of course, real materials do have dispersion which is why no-one uses singlet lenses, and this solution does not clearly help there.
• Computers exist. If you want to design a lens you can either seek a closed-form solution like this, or you can attack the problem numerically using a computer to iteratively find a good solution. Attacking it numerically is repugnant to people like me who like to do our maths with pen and paper, but if what you want is a good lens, especially a good zoom lens (there's a reason that there were essentially no zoom lenses before computer lens design became possible) then attacking it numerically is the way to go. So this solution is practically interesting for camera lenses if it makes the numerical simulation of lenses easier: if it doesn't, well, it's not.

"None of this is meant to detract from this result: I have a background in mathematical physics and I'm very interested in closed-form solutions. But I suspect it has limited, and perhaps no, relevance for camera lens design. I think it may very well be important in some other areas though: in particular if you are interested in light only at a single frequency then this may well be very interesting. An example of that is lasers and laser optical systems are used quite pervasively, for instance in the fibre-optics which will now send this comment from my computer to yours. There are probably more lenses for laser-based systems than there are for cameras."

Comments

[Craig]

For what it's worth, it looks like Applied Optics is a legit peer-reviewed scientific publication. It's been around since 1962 and is published by The Optical Society, a century-old professional organization with a membership of over 22,000 worldwide. The current president is a professor of physics at the Norwegian University of Science and Technology, and the immediate past president is the Chair of Experimental Physics at Imperial College London and a Fellow of the Royal Society. It doesn't sound like the kind of publication that you'd have to worry about printing stuff like "redheads are going extinct".

[James]

Mike,
I have just worked the equation thru and can report that unfortunately, it will only work on "full-frame" lenses.

[Richard Tugwell]

At last!!!! I can sleep easy...

[John Krumm]

That equation was a little blurry on my high res monitor, so I can't verify its effectiveness.

[David Saxe]

I checked out the equation. Seems about right.

[Ernest Zarate]

Oh, yeah. I saw that formula a few days ago. But it was simplified into something mere mortals can deal with:

8 ÷ 2(2+2) = ?

Have fun with that!

[Tony Rowlett]

Leica will be the first to introduce this finding in a new line of lenses called Equalux, and they will cost $43,000.

[Robert Roaldi]

I know the equation looks complicated (I am not making any sort of claim of being an optical designer) but there are many areas of science where equations like that are more or less routine. There are theses and articles being published every day with stuff like that in them. This one just happened to catch the attention of the nonscientific media.

The "easy" problems have all been solved. :)

[Trevor Johnson]

I looked at the formula and can understand how it is constructed, but not what it means! However, it is a formula rather than an equation.

[Lothar Adler]

Imagine f.e. all the wonderful pictures of Alvin Langdon Coburn, the early Edward Steichen, Alfred Stieglitz and all the Holga and Diana artists out there without spherical aberration ... .
Wonderful that this new formula was developed so late in the history of photography.

[Speed]

Spherical aberration? Doesn’t Photoshop have a slider for that?

Stay hydrated.

[c.d.embrey]

Merriam-Webster says: peer review noun
: a process by which something proposed (as for research or publication) is evaluated by a group of experts in the appropriate field

From what I read several days ago, this has been peer reviewed. Unlike several articles mentioned by Mike.

Day-by-day we are getting closer to the demise of traditional cameras. Computational Photography and multi lenses will change everything.

Some photographer can't-walk-and-chew-bubblegum. Fickle-photographers carry several brands of bubblegum. While other photographers don't need bubblegum to [email protected] The same goes for shooting with a cell-phone-camera—which type are you 8-)

[Jim Kofron]

I think they forgot to carry the 10's digit...

OMG. That's a heck of an equation!!!

[Mike Plews]

I hope this works although I am content with the lenses I now have. It would be sad if all that math just added up to something like cold fusion, trickle down economics or 10,000 steps a day (actual doctors say 4,000 is plenty).
As for your telescope choice you may have undershot. I don't think you could hit that star with this.

https://www.obsessiontelescopes.com/telescopes/25/index.php

Obsession Telescopes wonderfully lives up to its name.

[Yonatan K]

On the other hand, the story also kinda resembles those "viral" news stories that get going from time to time that the news media just loves to repeat...

An article based on mathematics that has been vetted by the appropriate scientific community is not the same as a rumor started in an unsubstantiated article written by non-experts.

Just because the average person (or government official) doesn't understand it, doesn't make the science wrong.

(Sorry for the persnickety tone)

[The science might be fine, probably is, but the popular articles about it might still be drawing unwarranted conclusions and generalizing its applications. In the 31 years I've been covering photography I've probably read "clusters" of articles a dozen or more times about scientific "breakthroughs" that are going to revolutionize this or that in optics. Then that's the last you hear of it. This one could be different...all I'm saying is that the journalists repeating ech other's conclusions don't actually know one way or the other. --Mike]

[Patrick]

I saw this a few days ago and wondered if someone was pulling our leg. Every now and then academic journals do receive papers that are scams. Sounds like others with some expertise have looked this over, but I have a wait and see attitude.

[JohnW]

I recognize the () and the √; the rest is .....

[hugh crawford]

I wonder, has anyone has plugged that equation into Matlab yet?

From what I understand talking to people smarter than I am, recent computer aided optical design has consisted of trying out a bunch of random design variations then testing them in the computer, throwing out the bad ones and tweaking the better ones and repeating over and over until the design is optimized. Think of it as optimization by automated trial and error. The reason being that the technology to test the designs is far more advanced and cheaper than the technology to originate the design.

From what I am reading, this is a big advance in getting a good simple design in the first iteration without the complexity of many elements having a bunch of errors correcting each other. It also looks like it will be particularly applicable to fast wide angle lenses, and it may involve some really complicated moulded lens surfaces that aren’t lenses you can grind in the traditional way.

Overall pretty cool and practical since it doesn’t depend on exotic materials like GRIN optics which had so much promise but seem to be resistant to actual manufacturing.

[Dan Khong]

That equation looks about right. It has 22 elements in 8 groups and shows various levels of multi-coatings that further refine the formula.

The ways to judge this optical scientist are:

1. He looks intelligent and sincere
2. He does not need glasses

So we can fully trust him at his word.

[David Mackenzie]

We'll need the variables identified before we can verify the equation is correct!

[PDLanum]

Why all the concern about edge sharpness? Most people these days are applying heavy handed vignettes to all of their images.

It's as if the entire world is suffering from reverse macular degeneration. Or most people just like looking at the world through a toilet paper tube.

[RLC]

The best comment I've read on this one is here: https://www.reddit.com/r/technology/comments/cn9ckb/a_mexican_physicist_solved_a_2000year_old_problem/ew9ocav/

The short of that is that this is a good mathematical solution, but unlikely to make big changes in real-world lens manufacture.

[Yonatan K]

Mea culpa - I haven't actually read the photography-site articles on this, and I don't doubt that speculation about practical implementation may be getting ahead of itself. (I read an article in a scientific journal that was just talking about the math).

Happily, I apologized in advance! :)

[Earl Dunbar]

As I may have stated previously (perhaps to you, Mike, directly) the state of mobile phonography is quite remarkable. I really don’t look for spherical aberrations or other such nonsense - all I know is that the iPhone XS shot I made of Peter Turnley’s beloved Brasserie de l’Isle Saint Louis astonished both me and the man who makes my prints. While I still look for the Fujinon 240A at a bargain price for my 4x5 work, I sometimes wonder ...

[A. Dias]

That seems to be a detailed analytical formulation of general aspherical surfaces for a singlet lens design. That formula can indeed be used to grind complex lens surfaces, free of conventional spherical aberrations. The lens is no longer a spherical surface but a complex multi-radii 3D surface.

[Gary]

Bravo to GKFroehlich. I (a layperson)actually understood most of what this scientist/engineer wrote. That's good expository writing.

[Bill Cowan]

I wonder what happens when you turn your complex multi-radii 3D surface lens and camera from landscape to portrait format.

[Mark Roberts]

News flash — Scientist discovers way to beat spherical aberration: Use an aspheric lens.

[A. Dias]

@Bill_Cowan - re landscape or portrait. The proposed fully aspheric lens design still is circularly symmetric, this camera orientation makes no difference.

[Steve Deutsch]

42

[Dan Matiesanu]

Or Plant based diets...