Ray optics of a telescope question

[center o bass]

Its an often prøven fact that a convex lens focused rays parallel with the optical axis into the focal point of the lens, but what about rays that are parallel with each other and with a small angle to the optical axis?

In ray diagrams with telescopes (such like the one below) one often draws that such rays focus into a point straight above (following the perpendicular to the optical axis) the focal point.

How is this proved? A reference to the proof would be apprechiated!
http://www.saburchill.com/physics/images5/211003030.jpg

 

[jtbell]

Use two of the incoming parallel rays, that correspond to principal rays in a typical ray diagram.

1. The ray that passes through the center of the lens is undeviated.

2. The ray that passes through the focal point on the near side of the lens, emerges from the lens going parallel to the optical axis.

Use similar triangles to show that their intersection point is at the same distance from the lens as the focal point on the far side of the lens.

 

[center o bass]

You are right about these two rays, however any other ray seems not to converge at the same point -- a phenomenon called coma.
(http://en.wikipedia.org/wiki/Coma_(optics))

Would you happen to know some calculations on when one can neglect these effects?

Use two of the incoming parallel rays, that correspond to principal rays in a typical ray diagram.

1. The ray that passes through the center of the lens is undeviated.

2. The ray that passes through the focal point on the near side of the lens, emerges from the lens going parallel to the optical axis.

Use similar triangles to show that their intersection point is at the same distance from the lens as the focal point on the far side of the lens.

I agree that this is the case for these two rays, but it seems that other rays that

 

[jtbell]

I was thinking of an ideal thin lens, neglecting aberrations as is normally done in introductory treatments.

The case in which incoming parallel rays that are not parallel to the axis form a point image off the axis at the focal length, is the limiting case of the standard point-object and point-image case that everybody starts out with in ray optics, in the limit as the object distance goes to infinity. (Notice what happens in the thin-lens equation when you let the object distance go to infinity!) The rays all meet at the image point in the first case for the same reason that they all meet at the image point in the second case.

 

[Drakkith]

Use two of the incoming parallel rays, that correspond to principal rays in a typical ray diagram.

1. The ray that passes through the center of the lens is undeviated.

2. The ray that passes through the focal point on the near side of the lens, emerges from the lens going parallel to the optical axis.

Don't all principle rays pass through the center of the lens?

 

[jtbell]

There may be other rays that are called "principal rays" but these are the ones I'm thinking of:

http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/raydiag.html#c1

 

[Drakkith]

There may be other rays that are called "principal rays" but these are the ones I'm thinking of:

http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/raydiag.html#c1

Ah okay.

 

[Drakkith]

Its an often prøven fact that a convex lens focused rays parallel with the optical axis into the focal point of the lens, but what about rays that are parallel with each other and with a small angle to the optical axis?

In ray diagrams with telescopes (such like the one below) one often draws that such rays focus into a point straight above (following the perpendicular to the optical axis) the focal point.

How is this proved? A reference to the proof would be apprechiated!

What do you mean by a proof? A true mathematical proof is going to be pretty in depth. Using approximations such as Snell's law and ray tracing is much simpler, but isn't a "proof".

 

[FactChecker]

You are right about these two rays, however any other ray seems not to converge at the same point -- a phenomenon called coma.
(http://en.wikipedia.org/wiki/Coma_(optics))
Would you happen to know some calculations on when one can neglect these effects?

I don't believe they can be neglected except for non-critical uses. They exist. Even inexpensive photography lenses have several elements to minimize such distortions. I think this is one of the reasons that mirrors are preferred in astronomy. There is an astronomy telescope forum in physicsforums that might know more about that.

 

[Drakkith]

I don't believe they can be neglected except for non-critical uses. They exist. Even inexpensive photography lenses have several elements to minimize such distortions. I think this is one of the reasons that mirrors are preferred in astronomy. There is an astronomy telescope forum in physicsforums that might know more about that.

Mirrors are just as affected by monochromatic aberrations such as coma as lenses are.

 

[jtbell]

I don't believe they can be neglected except for non-critical uses. They exist.

Nevertheless, in introductory textbooks they are neglected in the interest of conveying basic principles, similarly to the way we talk about massless strings and frictionless surfaces, and neglect the effects of air resistance on projectile motion in introductory mechanics.

The OP may want to clarify which level of sophistication we should use.

 

[stargazer193857]

I think hyperbolic lenses might not have comma, but we don't have the ability to make them as smooth as spherical ones, so it would not be an improvement.

As for proofs, set up the equation of your conic section in Excel, do your derivatives, Snell's law, and any starting point you want, and ray trace that system all you want. I tried, and it is a bit involved, but doable.

If you want to avoid aberrations, use slower optics. They will be dimmer. The longer and slower the focal length, and smaller the apparent field of view, the fewer aberrations.

As for neglecting them, that would depend on how sharp your eyes are. If not very, then you don't need as sharp a scope.

 

[Drakkith]

I think hyperbolic lenses might not have comma, but we don't have the ability to make them as smooth as spherical ones, so it would not be an improvement.

It's not that we can't make them as smooth as spherical surfaces, we can, it's that they are much more difficult to make and a single hyperbolic surface, whether a lens or mirror, will not be able to focus the light perfectly without aberrations, even on the optical axis. Only a parabola can do that (though it suffers from severe aberrations off-center, especially coma). However, a multi-surface system can use different shapes and curvatures to achieve near-perfect focus.

 

[stargazer193857]

I thought a parabola was the perfect on axis single mirror, and hyperbola is the best double mirror, and hyperbola is the best lens. Though I see what you mean by single vs double with the lens.

 

[Drakkith]

I thought a parabola was the perfect on axis single mirror

It is.

and hyperbola is the best double mirror

I don't know about being the best, but it is true that hyperbolic surfaces are used in many low-aberration dual-mirror designs.

 

[Andy Resnick]

You are right about these two rays, however any other ray seems not to converge at the same point -- a phenomenon called coma.

I'm not an expert in all-reflective optical systems, but there are several designs that seem to fully correct various aberrations, for example:

Stigmatic (corrected for spherical aberration): a paraboloid single-mirror. Fun fact: Descartes invented analytical geometry for the purpose of solving this problem. Two -mirror designs include Cassegrain and Gregory designs.

Aplanatic (corrected for spherical and coma): two-mirror designs include Ritchey-Chretien telescopes.

Anastigmats (corrected for spherical, coma, and astigmatism): Two-mirror designs include Schmidt (with a refractive corrector plate), Mersenne, and Schwarzschild designs.

Large telescopes have to deal with other issues, for example gravitational sag, thermal gradients, etc. etc.

 

[center o bass]

Nevertheless, in introductory textbooks they are neglected in the interest of conveying basic principles, similarly to the way we talk about massless strings and frictionless surfaces, and neglect the effects of air resistance on projectile motion in introductory mechanics.

The OP may want to clarify which level of sophistication we should use.

The level at which statements can be derived from fermats principle.