How does using more spherical lenses compensate for spherical aberration?

Question

I've heard of three common ways to compensate for spherical aberration:

1. Small aperture (fewer non-paraxial rays).
2. Aspheric lens (compensate with lens shape/material).
3. Corrective spherical lens elements (?).

I have a rough understanding of the first two, but I don't understand the third. How does using more spherical elements correct spherical aberration?

Also, do these extra spherical elements just reduce the circle of least confusion or do they fully correct the aberration? It doesn't seem possible to me that you could fully correct spherical aberration by only using additional spherical lenses.

Answer 1

Playing around in a ray optics simulator I think I figured out how additional spherical lenses can compensate for spherical aberration.

Since spherical aberration is caused by rays further away from the principal axis converging closer to the lens, you can use a spherical concave lens after the first convex lens element to diverge the non-paraxial rays at a steeper angle than the paraxial rays (taking advantage of the same principle that causes spherical aberration, but to diverge non-uniformly rather than converge non-uniformly). Then you can add another convex lens to converge the rays, with the resulting image compensated for spherical aberration as the incoming non-paraxial rays are more diverged than the paraxial rays.

This is what it looks like without the compensation lenses:

This is what it looks like with the compensation lenses:

The above isn't apples-to-apples as the effective focal length of the second system is shorter than the first, but it demonstrates the principle.