A: How Snell's law (a.k.a. Snell–Descartes law or law of refraction) affects the design of camera lenses?

In order to properly answer to this question, we need to decompose the main question to 2 questions:

• what are the conditions when Snell's law becomes simplified
• how this can be an advantage in lens design

And of course, to give some examples from real life.

Snell's law

Snell's laws is the very basic law of all geometric
optics. Mathematically speaking, it's very simple:

n₁sin(φ₁)=n₂sin(φ₂)

If the angle of light is close to perpendicular to lens - the equation becomes n₁=n₂

n denotes refractive index

In order to avoid non-linear response of sin, there are two solutions: make a small aperture to eliminate unwanted rays, or... Split the element into several and "curve" the ray several times.

Hypergon, Topogon and Metrogon

The quest for lens capable to capture very wide angles was difficult, but it gave birth to incredibly simple, but efficient solution - Hypergon.

Basically, two lenses only.
The advantages included:

• Petzval curvature almost 0
• astigmatism is controlled by the distance between the stop and lens elements
• coma is eliminated due to symmetrical design
• the same goes for distortion
• and for lateral colour

The disadvantage is a huge spherical aberration that can be "fixed" by using high aperture number. Also, the vignetting will be strong.

If you are very, very lucky to find one of Goerz, just buy it, as not complete lenses cost hundreds and mint examples even thousands, like this one from eBay

About 3000 Eur worth Hypergon, from eBay. This strange star-like thing is used to "fix" the vignetting

From Wiki

Next improvement is the addition of the negative flint element to correct spherical and axial chromatic aberrations.

This design is known as Topogon.

If you want to buy a lens with this design, there is a relatively cheap USSR Orion-6 (28mm f/6, about 250 Eur) and Zeiss Topogon that easily exceeds 2000 Eur (designed by Robert Richter).

A very small Zeiss 13mm f/3.5 worth 1500 Eur on eBay

Сделано в СССР, Orion 15, 28mm f/6, 250 Eur on eBay

From Wiki

The last iteration was Metrogon, and as you guess, it included 3 front +2 rear lenses. It's very difficult to find them. Those are mainly made by Bauch and Lomb for aerial recognisance and their price tag is also about 200-300 Eur

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