Proof that a spherical lens is stigmatic

In geometric optics, we generally allow that, for example in the case of a convex lens, rays coming from a particular point get refracted towards another particular point on the opposite side of the lens.

How is this proven from Snell-Descartes' law? Do we need to use the paraxial approximation in order for this to be true, or is it actually exactly true?

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I think for two rays to come from one point and converge on another point on the opposite side of the lens you need two convex lenses. physicsclassroom.com/Class/refrn/u14l5b.cfm – AbsoluteƵERØ Mar 31 '13 at 19:47
Check out the diagrams. When light comes from one point it diverges through a convex lens. When light comes from two points it converges to a single point. Your eyes contain lenses. When they are not in focus (non-convergent points) you add another lens to make the points converge. The cones inside of your eyes are multiple points of focus from the single point of light you're trying to focus on (at a distance). – AbsoluteƵERØ Mar 31 '13 at 21:04
A flat plain (such as a pane of glass) allows light to pass from one point to another. A lens bends light. – AbsoluteƵERØ Mar 31 '13 at 21:06

Usual spherical lenses are only approximately stigmatics, that is to say the image of a source-point is itself a point. You need paraxial approximation to ensure approximate stigmatism and avoid spherical aberration.

You can see it by playing around with just a plane interface between two different transparent media like the surface of water. Using paraxial approximation, you can prove the relation $$\frac{n}{HA} = \frac{n'}{HA'}$$ where $n$ is the refraction index of the medium where the object $A$ lies (for example a fish, then $n=1.33$), $H$ is the orthogonal projection of $A$ on the interface, $A'$ is the image and $n'$ is the refraction index of the medium where the observer thinks $A'$ is (for example the air so $n'=1$). If the fish is $40$ cm under the water, you will see it's image only $30$ cm away from the interface (that's why it's difficult to catch a fish in an aquarium on first try).

As you need spherical interfaces to build your lens and it's not even working with planar interfaces (which are a particular case of spherical interfaces), it can not be true for any type of spherical lens.

Here are two animations you can try to see how stigmatism works:

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So let me get this straight. Not only are lenses only approximately stigmatic, they're also only approximately stigmatic under the paraxial approximation? So stigmatism is basically a kind of double approximation. – Jack M Mar 31 '13 at 23:20
No, they are stigmatic under the paraxial approximation. But as this is an approximation, they are only approximately stigmatic as a whole (that is if you use them on the edge, spherical aberration appears quite easily): they will never be perfectly stigmatic. – JJ Fleck Apr 1 '13 at 7:17

According to an article by the Optometric Science Research Group titled Stigmatic optical systems:

"There would appear to be little disagreement on what constitutes an astigmatic system in the case of a thin lens: the cylinder is not zero. A spherical thin lens is stigmatic or not astigmatic. The issue is less clear in the case of a thick system. For example, is an eye stigmatic merely because its refraction is stigmatic (spherical)?"

According to Snell's law, (from Wikipedia) the largest possible angle of incidence which results in a refracted ray is called the critical angle. When something exceeds this angle no there is no refraction point for the object in the image, meaning the result is astigmatic.

Also according to Stigmatic optical systems:

"An eye may be astigmatic despite having a stigmatic refraction."

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