Why we use spherical mirrors instead of parabolic mirrors?

[jaumzaum]

Parabolas are the only geometrical shape in which we have a perfect focus (not an approximate one) and does not depend in the angle of incidence being small. So, why do we even build spherical mirrors and not only parabolic mirros?

 

[phyzguy]

It depends. While a parabola brings light to a focus on-axis, it suffers from severe coma off-axis. A Schmidt camera, for example uses a spherical main mirror with a correcting lens. This gives a much wider field of view than a parabolic mirror.

 

[Klystron]

Spherical antennas/reflectors provide advantages over parabolas depending on application such as improved symmetry. This video discusses aspects of spherical reflectors with examples of visible light wavelengths.

 

[Ibix]

Also, spherical surfaces are cheap and easy to grind compared to parabolae.

 

[Baluncore]

So, why do we even build spherical mirrors and not only parabolic mirros?

The traditional way to make a parabolic mirror is to first generate a spherical mirror, then grind it to have a parabolic figure. A spherical mirror is more accurate than a parabolic mirror.

 

[Drakkith]

Parabolas are the only geometrical shape in which we have a perfect focus (not an approximate one) and does not depend in the angle of incidence being small. So, why do we even build spherical mirrors and not only parabolic mirros?

There are many, MANY different designs for telescopes and other optical systems that use mirrors, and they often use both spherical and non-spherical designs to correct all the different aberrations that crop up when designing these systems or to take advantage of certain special properties that the different shapes have. A simple example is a cassegrain telescope, which uses a parabolic primary mirror but a hyperbolic secondary. To quote wiki:

It makes use of the special properties of parabolic and hyperbolic reflectors. A concave parabolic reflector will reflect all incoming light rays parallel to its axis of symmetry to a single point, the focus. A convex hyperbolic reflector has two foci and will reflect all light rays directed at one of its two foci towards its other focus. The mirrors in this type of telescope are designed and positioned so that they share one focus and so that the second focus of the hyperbolic mirror will be at the same point at which the image is to be observed, usually just outside the eyepiece. The parabolic mirror reflects parallel light rays entering the telescope to its focus, which is also the focus of the hyperbolic mirror. The hyperbolic mirror then reflects those light rays to its other focus, where the image is observed.

The cassegrain telescope wouldn't be as effective if it used solely parabolic mirrors.

See the following links for more information.
https://en.wikipedia.org/wiki/Optical_aberration
https://www.telescope-optics.net/index.htm#TABLE_OF_CONTENTS (advanced)
https://www.edmundoptics.com/knowle...tes/optics/comparison-of-optical-aberrations/

 

[sophiecentaur]

This video discusses aspects of spherical reflectors

He refers to "Spherical abbreviations" in that video. SO much for proof reading (=proof listening here). Could really confuse someone at that level.

 

[sophiecentaur]

A convex hyperbolic reflector has two foci and will reflect all light rays directed at one of its two foci towards its other focus.

I think you mean concave?

 

[phyzguy]

I think you mean concave?

No, he means convex. As said above, a Cassegrain telescope uses a concave parabolic primary and a convex hyperbolic secondary to reflect light to a focus. However, again, this is only true on-axis. When using a telescope, we don't just want to focus the light at one point, we want to focus the light at a range of angles so we can build up an image. A parabolic reflector and a Cassegrain telescope perform very poorly in this respect, since the image degrades very quickly as you move away from the center point (off-axis). Many modern telescopes, like the Hubble Space telescope, use the Ritchey-Chrétien design, where both the concave primary and the convex secondary are hyperbolic. This performs just as well as the Cassegrain on-axis, but performs much better off-axis, so you get a larger, sharper image.

 

[sophiecentaur]

A convex hyperbolic reflector has two foci and will reflect all light rays directed at one of its two foci towards its other focus

No, he means convex.

I don't think so. A convex hyperboloid diverges rays and has a virtual focal point (behind the surface) - same as any convex surface. The context is a cassegrain telescope which uses a convex reflector which, in conjunction with the paraboloid main reflector produces a real image but that is not what the bare statement about two foci implies.

 

[Drakkith]

I don't think so. A convex hyperboloid diverges rays and has a virtual focal point (behind the surface) - same as any convex surface. The context is a cassegrain telescope which uses a convex reflector which, in conjunction with the paraboloid main reflector produces a real image but that is not what the bare statement about two foci implies.

As far as I know, the statement is true. Any ray directed at one focus will be reflected towards the other, regardless of how that ray gets directed towards the first focus in the first place.

 

[sophiecentaur]

Any ray directed at one focus

But it's a diverging mirror if it's convex. No?
What goes for a convex lens, goes for a concave mirror.

 

[Drakkith]

But it's a diverging mirror if it's convex. No?

Yes, but in order for the rays to be directed towards the focus, they have to be converging. I guess you could say the convergence of the rays is greater than the divergence introduced by the mirror, thus the rays end up being focused at the other focus.

 

[Drakkith]

@sophiecentaur See the following diagram.

Edit: Imagine the blue surface is bendable. If we start with a sphere and direct the rays towards the focus, it is obvious that they simply reflect back along their original paths. As we bend the surface outwards, away from F₁ , we make it ellipsoidal, and the reflected cone of rays begin to diverge less and less. Once we bend the surface into a parabola, the reflected rays are now parallel to each other, or collimated. Finally, once we bend it further into a hyperbola, the light rays begin to converge after reflection and are focused down to a point.

 

[sophiecentaur]

convergence of the rays is greater than the divergence introduced by the mirror,

Ah yes; it makes sense now. This is not a simple object - image thing. An intermediate image from the prime reflector is formed behind the hyperboloid (but it's never actually there; it's a virtual intermediate object) and the next real image is formed inside the eyepiece.
The focal length of the prime reflector is very long and the image formed needs to be brought to a point just behind the paraboloid. Funny but when this equivalent happens in a Newtonian, the optics are more obvious but the secondary is usually plane.

 

[phyzguy]

I don't think so. A convex hyperboloid diverges rays and has a virtual focal point (behind the surface) - same as any convex surface. The context is a cassegrain telescope which uses a convex reflector which, in conjunction with the paraboloid main reflector produces a real image but that is not what the bare statement about two foci implies.

As others have said, you are misunderstanding. Here is the explanation from Wikipedia, and below is a digram.
"It makes use of the special properties of parabolic and hyperbolic reflectors. A concave parabolic reflector will reflect all incoming light rays parallel to its axis of symmetry to a single point, the focus. A convex hyperbolic reflector has two foci and will reflect all light rays directed at one of its two foci towards its other focus. The mirrors in this type of telescope are designed and positioned so that they share one focus and so that the second focus of the hyperbolic mirror will be at the same point at which the image is to be observed, usually just outside the eyepiece. The parabolic mirror reflects parallel light rays entering the telescope to its focus, which is also the focus of the hyperbolic mirror. The hyperbolic mirror then reflects those light rays to its other focus, where the image is observed."

 

[phyzguy]

@sophiecentaur , perhaps what you are misunderstanding is the following. You are right that with parallel incoming rays, a convex surface will convert these to diverging rays. But in a Cassegrain telescope, the incoming rays to the hyperbolic secondary are already converging, so the hyperbolic secondary converts them to converging rays at a different focus.

 

[Klystron]

He refers to "Spherical abbreviations" in that video. SO much for proof reading (=proof listening here). Could really confuse someone at that level.

Good catch. I searched for a simplistic diagram. Mostly ignored the audio.

 

[sophiecentaur]

Mostly ignored the audio.

There are some truly grim videos around which seem to rely, mostly on just reading the slide captions. I do feel sorry for students for whom that's the only source of learning for them.