When a lens is designed, typically an optimiser is used to try to find a combination of lens elements that focuses light of all wavelengths to the same spot across the whole image. If all the same type of glass is used, then it's hard for the optimiser to find a solution that corrects for the chromatic aberration across all the spectrum, because if it adjusts (for instance) something to fix the green aberration then it'll mess up the blue and vice-versa. But if a different type of glass is used such as fluorite, which has a different pattern of chromatic dispersion, then that gives the optimiser an extra degree of freedom, so it is more able to independently control the aberration in the different colours and make a lens that performs well.

Over the past ten years, lenses have gotten unbelievably better. Apsherical elements have done some of the lifting, and glass formulas another, but how much have computational tools changed how lenses are designed?

Anecdotally, a lot. Modeling of Aspherical elements has gotten better and it is much easier to make them as well. Mamiya made some of the first APO computer corrected lenses back in 1988 - https://lens-db.com/mamiya-apo-sekor-z-350mm-f56-1988/

What are you basing this observation on? I haven't seen any significant improvement in photography or cinema lens performance over many decades. The principles have been understood for 150 years, and using ray-tracing software is standard since the 70s. Cheap lens systems in phones got a lot better, and digital sensors are much more sensitive than film though.

Not who you're replying to, but it seems this is a classic case of mistaking not noticing change is evidence of no change.

I've been following this newsletter for a while as I have a passing interest in (photographic/cinematic) lens design.

https://www.pencilofrays.com/

The main reason why lenses have become significantly better over the last 10-20 years (certainly at the consumer level) is mainly because camera sensors have become significantly better over the last 10-20 years, and therefore the lenses have had to be engineered to a higher standard to avoid the lens foibles being quite so obvious in the higher resolution images that are now possible. The engineering principles are largely the same with a few smaller advances, but to accomodate the demanded-for improvements consumer lenses are now bigger, heavier, more expensive, and have more interesting optical designs. It's purely demand-driven, not technology-driven.

Exactly. It can't be ignored that these new lenses are much bigger & heavier and much more expensive as a result. That helps a lot. But it makes them worse if a) you need to carry them a long way for your type of photography b) your photography doesn't actually benefit from or require their advantages.

A lot of these benefits seem more important to the measurement focused hobbyists. But if you don't actually care about shooting at f/1.2-f/1.4 a lot of the time you might want to save the money.

All you have to do is compare a recent 50mm f/1.4 lens such as the Sigma Art or Sony GM to an older design before the advent of computer-aided optimization (eg the Canon EF 50/1.4).

The new designs are very contrasty and sharp wide open, almost across the frame. The older designs need to be stopped down to achieve critical sharpness, and suffer in the corners. Just so much progress in the past 15 years.

Possibly a poor analogy, but imagine making a particle accelerator with the goal that objects of different mass put into it are accelerated and leave it going the same velocity.

The current state of technology allows that the best we can do is make one that accelerates lighter objects slightly more than heavier objects.

However, we've found a way to make an accelerator using a different design that accelerates heavier objects more than lighter objects. If we pass objects through the first accelerator, then the second, the second accelerator reverses out some of the non-linearity of the first. Unfortunately this second accelerator is very, very expensive to make.

This is how lenses of different refractive indexes are used: one element partially corrects the dispersion produced by earlier elements. Fluorite has the right refractive index to correct aberrations created in long focal length lenses.

It's a bit buried, but this (to me) is the most interesting sentence :

> Fluorite lenses are also unique in their extraordinary partial dispersion tendencies: the red to green wavelengths are dispersed with the same tendencies as glass, but the green to blue wavelengths are dispersed more than glass.

It has both low dispersion (less overall aberrations) and a unique shape to the dispersion curve (more control). Glasses typically all have similar dispersion curves. The weird shape of fluorite's dispersion curve gives your optimization function an extra lever to play with.

Does this mean it has an index of refraction that depends on wavelength? How can that be? Is the bond length some multiple of blue (or green/red), where there's some quick "change" in what the photon "sees"?

This is, I think, just one of those spots where "all models are wrong, some models are useful" is true. From the wikipedia page on Refractive Index (https://en.wikipedia.org/wiki/Refractive_index):

> The refractive index may vary with wavelength. This causes white light to split into constituent colors when refracted. This is called dispersion. This effect can be observed in prisms and rainbows, and as chromatic aberration in lenses.

--- snip ---

> For most materials the refractive index changes with wavelength by several percent across the visible spectrum. Nevertheless, refractive indices for materials are commonly reported using a single value for n, typically measured at 633 nm.

So we were all lied to in our introductory optics classes. n isn't a constant for a given material but rather n(lambda) but weakly.

It's a consequence of the relationship between absorption and index of refraction in glasses, which follow the Karmers-Kronig relationship [0]. As the wavelength approaches atomic resonance, the index increases or decreases depending on which side of the absorption peak the wavelength is.

0. https://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relatio...

Glass has this too. it's why achromats and apochromats were developed (using rather clever ideas like pairing two different types of glass, which cancel each other's dispersions.

All the physical explanations I've seen invoke the EM equations and quantum mechanics to explain it (and I don't understand it well enough to translate).

3Blue1Brown has a nice animated video about this: https://youtu.be/KTzGBJPuJwM?si=ssBPKL8LVhsQ3wNZ

Amazing question! Yup. Refractive index is a function of wavelength. I bet you already knew that on some level, because that's why prisms do their thing. As an extreme example, imagine X-rays: they barely refract at all, they just pass straight through stuff! Also, the change is actually quite smooth.

https://refractiveindex.info/

https://en.wikipedia.org/wiki/Dispersion_(optics)

Glass has an index of refraction that depends on wavelength; this is how a prism works. This is also why chromatic aberration exists at all.

This article shows a "map" of available glasses, with each glass shown by its refractive index and dispersion constant -- in a particular way. The explanation is that if all you have are glasses on that "glass line," you can make a lens with equal focal length at two wavelengths but not three. You need glasses that are not on the "glass line," and one of them is calcium fluoride. Also, some plastics such as acrylic and polystyrene can be used, but have their own issues such as thermal expansion.

Finding a lens with 2 equal focal lengths is equivalent to a graph of focal length versus wavelength that's roughly a parabola. With 3 equal focal length, the graph looks like a cubic curve.

https://www.opticsforhire.com/blog/apochromatic-lens/

The glasses also have to be economical, able to take a good polish, clear, chemically resistant (to avoid staining), mechanically robust, etc. It's not an easy design problem since the "exotic" glasses with extreme refractive properties also tend to have worse properties overall.

Chromatic aberration happens because of dispersion --- the fact that the refractive index changes with wavelength. By combining materials with different properties (e.g. low dispersion, or high refractive index), you can cancel out the aberration in lens designs such as achromatic doublets, apochromatic triplets, etc.

In general, materials that are high in refractive index have high dispersion (e.g. crown glass) and materials that have low dispersion also have low refractive index. But ideally we want a material that has high refractive index but low dispersion so that it can both bend light without introducing a lot of chromatic aberration.

Fluorite has extraordinarily low dispersion while having a refractive index that's only slightly lower than glass, making it a good material to be used in conjunction with other materials.

I found the article explained exactly why using fluorite helped chromatic aberration. It adds another degree of freedom, due to the partial dispersion.

Isn't that covered in the 7th paragraph?

'the red to green wavelengths are dispersed with the same tendencies as glass, but the green to blue wavelengths are dispersed more than glass. Using a convex fluorite lens element alongside a high-dispersion glass concave lens element therefore eliminates residual chromatic aberration'

I thought the reason underlying that is that fluorite is not a glass, technically. It is a crystal. But I’m not a MatSci person so that doesn’t leave me any bit more informed.

It seemed to, the dispersion is different so you can cancel the dispersion from glass (dispersion is the refractive index variation with wavelength).

Fluorite just has low dispersion and therefore a lens made from it has less chromatic aberration than glass.

> the red to green wavelengths are dispersed with the same tendencies as glass, but the green to blue wavelengths are dispersed more than glass.

I admittedly responded without reading the source, so please accept my apologies.

Despite how it is worded the total dispersion of Fluorite is less than glass.

That quote is referring to the ‘extraordinary partial dispersion’ property. They are using this to better correct the aberration than could be done with two pieces of glass alone. This seems to be illustrated in the diagrams well.

Some further stuff to read up on is the Abbe number that describes the refractive index vs wavelength derived from a set of light sources.

https://en.wikipedia.org/wiki/Abbe_number

The higher the Abbe number the lower the dispersion of a material.

Fluorite is ~95

https://refractiveindex.info/?shelf=main&book=CaF2&page=Mali...

Random choice below but glass tends to be ~25 - 80

https://refractiveindex.info/?shelf=glass&book=HOYA-C&page=E...

> Fluorite just has low dispersion and therefore a lens made from it has less chromatic aberration than glass.

So why not make the entire lens from fluorite? Because pairing an element with one with different refractive index can lower the overall dispersion.

Typically you'll see compound lenses are made of pairs of elements where one is positive (convex) and the other negative (concave): instead of making one lens with power of, say, +4, the group is made from one that's +5 and one that's -1 of different type of glass so the refractive indexes cancel-out dispersion and other aberrations.