
Mexican physicist’s solution to 2,000-yo problem could lead to improved lenses - elijahparker
https://gizmodo.com/a-mexican-physicist-solved-a-2-000-year-old-problem-tha-1837031984
======
fheld
Relevant thread on the same topic
[https://news.ycombinator.com/item?id=20369960](https://news.ycombinator.com/item?id=20369960)

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gus_massa
Opening sentence in the article:

> _It’s a problem that plagues even the priciest of lenses, manufactured to
> the most exacting specifications: the center of the frame might be razor-
> sharp, but the corners and edges always look a little soft._

The method in the article doesn't solve this. The method only give an analytic
solution for the center and only for one color. All the other points of the
frame and all the other colors look a little soft.

Previous discussion
[https://news.ycombinator.com/item?id=20369960](https://news.ycombinator.com/item?id=20369960)
(845 points, 33 days ago, 210 comments)

I'll repost my previous comment:

It's an interesting mathematical result, but note that gives a solution for
the problem of the spherical aberration, but real lens have also chromatic
aberration. I.E. the speed of light for each color inside the glass is
slightly different, so the value n (refraction index) in the equation is
different, so in the equation you get a different surface for each color.

In the real lens you must pick one surface, so the effect is that one color is
perfectly focused and the other colors are not focused and you get some
rainbow-like effects.

The solution is to use combination of a few lens of different glasses, to
compensate the differences. It's not easy to design these kind of system,
because they must compensante also for other types of aberrations.

More details:

[https://en.wikipedia.org/wiki/Chromatic_aberration](https://en.wikipedia.org/wiki/Chromatic_aberration)

[https://kenrockwell.com/tech/lenstech.htm](https://kenrockwell.com/tech/lenstech.htm)

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m0d0nne11
I note that Roger Cicala over at lensrentals.com (in an interesting article
showing disassembly of a fancy zoom unit) had this to say:

No, no it can't. It was really impressive mathematics (and according to
several mathematicians, the formula was left more complex than necessary to
make it look more impressive). In theory, it could improve ONE of the dozens
of aberrations, but only in the center of the lens, not off-axis.

Not to take anything away from a very impressive intellectual effort, but
basically it solved a problem nobody was particularly trying to solve (the
stuff about '2,000 year old mystery' was so over-the-top it makes me wonder if
the authors were having a bit of fun).

~~~
gus_massa
In the paper and the code with the paper, they have a sane formulation that
use 5 or 6 intermediate variables. (Each variable is calculated using the
parameters and the previous variables, all have closed forms.)

The big formula is only a gimmick for the press article. My guess is that the
journalist asked for it and then they just made Mathematica make all the
replacements and give a nice gif.

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rhn_mk1
Does an analytical solution actually make any difference in practice though? I
would guess that numerical approximations would have reached similar accuracy
as modern manufacturing methods by this point.

------
TheMaskedCoder
Original article (in Spanish):
[https://transferencia.tec.mx/2019/02/21/resuelto-un-
problema...](https://transferencia.tec.mx/2019/02/21/resuelto-un-problema-
optico-con-mas-de-un-siglo-sin-solucion/)

------
Negitivefrags
I hope this can be used to improve eye glasses too. At higher powers the
effect of chromatic aberration as you get away from the center of the lens
really sucks.

I have aspheric lenses which are supposed to help, but I still see the problem
and I’ve been complaining to my optometrist who basically says he can’t make
anything better than what I’ve got.

It’s especially bad in computer interfaces which often have thin high contrast
lines which just splay out into red/blue as you get to the edges of the
monitor.

------
r-bryan
That "mind-melting" equation looks like it could benefit from common
subexpression elimination.

~~~
robinhouston
It’s written much more clearly in the paper.

[https://www.osapublishing.org/ao/fulltext.cfm?uri=ao-57-31-9...](https://www.osapublishing.org/ao/fulltext.cfm?uri=ao-57-31-9341&id=399640)

------
koalaphant
Would love to see a side by side comparison once they make them.

------
robinhouston
Does anyone here know any more details? I’m curious as to how this improves on
the 2014 work by the same author and others:
[https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2014...](https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2014.0608)

I think this is the new paper (from 2018):
[https://www.osapublishing.org/ao/abstract.cfm?uri=ao-57-31-9...](https://www.osapublishing.org/ao/abstract.cfm?uri=ao-57-31-9341)

The introduction to the new paper gives a reasonably clear explanation of how
it relates to previous work:

> The design of optical systems with aspheric surface has the goal to strongly
> reduce spherical aberration. Spherical aberration on lenses has been
> extensively studied by Ref. [1]. Luneberg [2] established a method for
> computing the shape of the second surface from an initial first surface that
> introduces spherical aberration, which he described just for special cases.
> Many authors proposed a lens design with two aspheric surfaces to correct
> spherical aberration [3,4].

> The problem of the design of a singlet free of spherical aberration with two
> aspheric surfaces is also known as the Wasserman and Wolf problem [5]. The
> problem has been solved with a numerical approach by Ref. [6]. Recently,
> Ref. [7] has shown a rigorous analytical solution of a singlet lens free of
> spherical aberration for the special case when the first surface is flat or
> conical. Since its publication, several works inspired by its solution have
> emerged [7–12], all of them free of spherical aberration. The solution has
> six different signs; therefore, it is a set of 2^6 = 64 possible solutions,
> where only one is right. We test the formula provided by those in Ref. [7],
> when the first surface is not flat or conic, and the equation system does
> not give correct answers.

> In this paper, we present a rigorous analytical solution for the design of
> lenses free of spherical aberration. The solution presented here has just
> one sign; therefore, it is a set of just two possible solutions. Our
> solution is robust because the set of solutions is valid for negative and
> positive refraction indices. The model allows use of continuous functions,
> such that the rays inside the lens do not cross each other. This model will
> compute the second surface in order to correct the spherical aberration
> produced by the first surface.

If I understand this correctly, the key difference is that the earlier paper
(referred to as [7] above) only addresses the case where one of the surfaces
of the lens is flat or conical. I’m not sure whether this is a serious
limitation in practice: I would have thought naively that it’s easier to
manufacture a lens with one flat surface.

The (new) paper also includes (in Appendix A) Mathematica code to compute and
plot the lenses produced by this method.

