Question about reflectivity of mirrors vs transitivity of lenses

[Stargazer19385]

I could not find an optics forum, so I'm posting this here.

I'm trying to learn about the optics of imperfect real mirrors and prisms. I dug out my old physics book, but it just barely touches on this.

1. I read that most aluminum coated telescope mirrors reflect about 90% of the light that hits them. Where does the other 10% go? How much is absorbed? How much is reflected at the wrong angle? Does the 90% only count the light that is within 0.1 degree of accurate? I asked this question on an astronomy forum, but they were not interested.

2. I read that if the surface roughness is less than 1/4 wavelength (1/4 of 500 nm), an aluminized mirror reflects 90% of light, and further smoothing has little effect. I also read online that with surface coatings, 99% of light passes through a lens. Does this mean that the lens has better light gathering for it size than the mirror? (forget about any secondary mirror for now) How accurately does a lens focus monochromatic light, compared to a similarly smooth mirror? Assume both are hyperbolic, but only smooth within 1/4 wavelength.

3. Finally, I read that a 45 degree diagonal inside of a prism reflects about 100% of light that hits it. Is that true? How can it be better than a mirror, if both have surface imperfections?

I have other questions, but that will do for now.

Thank you very much. In order for me to see dim galaxies with binoculars, the binoculars need to be good. Any light that does not make it through accurately can scatter and cause ghost images and haze that make it hard to see the galaxy. The lower brightness is a non-issue compared to the scatter.

 

[Drakkith]

1. I read that most aluminum coated telescope mirrors reflect about 90% of the light that hits them. Where does the other 10% go? How much is absorbed? How much is reflected at the wrong angle? Does the 90% only count the light that is within 0.1 degree of accurate? I asked this question on an astronomy forum, but they were not interested.

The remaining 10% is absorbed. If the mirror is ground accurately practically none of the light is reflected at the wrong angle. I'm not sure if the reflectivity of the mirror depends on the angle of incident.

2. I read that if the surface roughness is less than 1/4 wavelength (1/4 of 500 nm), an aluminized mirror reflects 90% of light, and further smoothing has little effect.

Yep. When surface errors are less than 1/4 of the wavelength of the incident light, they don't affect the direction the light is reflected enough to make further smoothing necessary.

I also read online that with surface coatings, 99% of light passes through a lens. Does this mean that the lens has better light gathering for it size than the mirror? (forget about any secondary mirror for now)

When they are of equal size, yes. But in reality lenses have severe limitations that make mirrors the preferred choice in nearly all telescopes.

How accurately does a lens focus monochromatic light, compared to a similarly smooth mirror? Assume both are hyperbolic, but only smooth within 1/4 wavelength.

I believe they work identically.

3. Finally, I read that a 45 degree diagonal inside of a prism reflects about 100% of light that hits it. Is that true? How can it be better than a mirror, if both have surface imperfections?

The prism uses something called total internal reflection. This is different than the reflection off of a mirror and is able to reflect nearly all of the light.

Thank you very much. In order for me to see dim galaxies with binoculars, the binoculars need to be good. Any light that does not make it through accurately can scatter and cause ghost images and haze that make it hard to see the galaxy. The lower brightness is a non-issue compared to the scatter.

I'm not sure what you're getting at here. The key to seeing dim objects is aperture. There is no difference between a pair of binoculars and a telescope with an equal diameter aperture. (Other than the obvious fact that binoculars are effectively 2 telescopes in one)

If you want to see dim objects you should aim at getting a telescope with the biggest possible aperture without skimping on quality of the optical elements, mount, etc. However binoculars are very easy to use and if you just want to do general stargazing at low magnification they work wonderfully.

 

[Stargazer19385]

Thank you for answering all that. It all makes sense and sounds believable.

My next question is how eyepieces make images. I guess they have to duplicate the angles of all light rays if the object were that many times closer, and work back from there.

I don't understand why refractors would use two convex lenses (Plossl) past the focal point, instead of a concave lens before it, and get the upright image with less glass. Galileo used the concave lens, but with a much reduced apparent field of view. Since a Plossl takes divergent light, and makes it convergent again, I don't know why a concave lens can't take convergent light before the focus, and change it more slightly to give the same desired convergent effect. I wonder if Galileo just did not optimize his eyepiece, and why others have not improved on it.

Placing the concave lens before the focal point would mean slower optics could be used in the same total distance, and no erecting prisms would be needed, combining to mean less chromatic aberration. And less total glass.

Now if the rays coming out the exit pupil are supposed to be divergent, I could see why people would want to use the convex lenses after the focus. I could also see how the concave lens would not be powerful enough to do enough bending.

 

[Drakkith]

My next question is how eyepieces make images. I guess they have to duplicate the angles of all light rays if the object were that many times closer, and work back from there.

See below: (This image shows a refracting telescope, but the principles are identical for a reflecting telescope)

In the image above, the lines represent rays, or light coming in. These rays are on the optical axis and coming from "infinity", which just means that the object that emitted them is so far away that we can approximate it as if it were at infinity and there is no divergence of the rays.

Now, look what happens to the rays when passing through the objective. (Lens in a refracting telescope or mirror in a reflecting.) They are focused down to a spot but then begin to diverge before encountering the eyepiece. The eyepieces job is to take this diverging beam of light and re-focus it back to infinity. It does this because your eye is like a small telescope. Light enters through the pupil and is focused down to a spot on the retina which allows you to see. If the light was NOT re-focused to infinity your eye would not be able to focus the diverging cone very well, nor would you have any magnification.

Now, in this image, the light is coming form an object OFF the optical axis. Notice that the light from off axis exits the eyepiece at a GREATER angle than it entered the telescope objective. THIS is how magnification happens. The eyepiece collimates the light (makes it parallel) and increases the angle. Higher magnification eyepieces increase this angle more than lower magnification eyepieces.

I don't understand why refractors would use two convex lenses (Plossl) past the focal point, instead of a concave lens before it, and get the upright image with less glass. Galileo used the concave lens, but with a much reduced apparent field of view. Since a Plossl takes divergent light, and makes it convergent again, I don't know why a concave lens can't take convergent light before the focus, and change it more slightly to give the same desired convergent effect. I wonder if Galileo just did not optimize his eyepiece, and why others have not improved on it.

This website explains it: http://scitechantiques.com/Galileo-...he_bottom_line:_Looking_through_the_telescope

Now if the rays coming out the exit pupil are supposed to be divergent, I could see why people would want to use the convex lenses after the focus. I could also see how the concave lens would not be powerful enough to do enough bending.

They are not supposed to be divergent. This would make it difficult if not impossible for your eye to bring the image into focus.

 

[Stargazer19385]

Those are the best pictures I've seen yet. Thank you. I see, so in addition to have a shorter focal length than the objective for magnification, the eyepiece has a lower f# too, so the lens is large enough to accommodate off axis rays.

It makes sense that if a galaxy is far away, rays coming from each point would be parallel both before and after magnification, as shown in that picture. And the light from the exit pupil is convergent because the eye reads everything upside down on the large spherical retina, which the brain corrects.

And Galileo picked the upright image telescope despite its extremely small field of view because he was selling them to people for terrestrial use.

The other arrangement I considered was to replace Galileo's concave lens with a convex in the same position, before the objective's focal point, shortening the total system focal length, so that convergent rays send an upright image with the wide angle view. However, I suspect that the eyepiece lens would not be capable of focusing the light in that arrangement, since it is not coming from its focal plane. It would be nice if that did work.

 

[Stargazer19385]

Can a single magnifying glass, held close enough to the eye to capture parallel rays, be used to view the moon? I realize magnifying glasses don't have coatings and are spherical and poorly made. I'm just trying to understand the basic optical principles of one lens vs two.

 

[Drakkith]

Those are the best pictures I've seen yet. Thank you. I see, so in addition to have a shorter focal length than the objective for magnification, the eyepiece has a lower f# too, so the lens is large enough to accommodate off axis rays.

Well, it doesn't have to have a lower f-ratio. It's a trade off between the cost of more glass, size of the eyepiece, aberrations, and a bunch of other stuff.

It makes sense that if a galaxy is far away, rays coming from each point would be parallel both before and after magnification, as shown in that picture. And the light from the exit pupil is convergent because the eye reads everything upside down on the large spherical retina, which the brain corrects.

The rays in each light bundle aren't convergent, the bundles merely intersect each other just like they do coming into your eye normally. Except upside/reversed usually thanks to the telescope.

The other arrangement I considered was to replace Galileo's concave lens with a convex in the same position, before the objective's focal point, shortening the total system focal length, so that convergent rays send an upright image with the wide angle view. However, I suspect that the eyepiece lens would not be capable of focusing the light in that arrangement, since it is not coming from its focal plane. It would be nice if that did work.

A convex lens would not be usable as an eyepiece if you put it before the focal point of the telescope. This is because the light coming into the eyepiece would then be converging, not diverging, and to get it parallel again you would need a concave lens.

Can a single magnifying glass, held close enough to the eye to capture parallel rays, be used to view the moon? I realize magnifying glasses don't have coatings and are spherical and poorly made. I'm just trying to understand the basic optical principles of one lens vs two.

I just tried it. When held close to the eye, the lens wouldn't let me focus on faraway objects. When held out at arms reach, faraway objects were able to be brought into focus, but they lost most/all magnification.

 

[Stargazer19385]

OK. Now I understand. The eye can't focus converging light. It is used to parallel light from far away stuff, and diverging light from close stuff. A magnifying glass aimed at something far, and my convex Gallilean would both give the eye convergent rays. The altered Gallilean would come to a focus, but it would still be no good for long distance viewing.

When I first looked at the second picture, I was puzzled at how it sent parallel rays downwards, without them converging to a focus. Now it makes sense. Because they are after the objective's focus, the steepest one on the objective side is made less steep on the other side, and the least steep one on the objective side is made more steep on the other side. Had they been parallel on the objective side, they would have focused on the other side. But as they are, they are nearly parallel. If they are more divergent, they will make the image appear closer.

And since the eyepiece has such a low f#, it takes two higher f# lenses in a row (Plossl) to bend the light while still staying close enough to hyperbolic in each lens, thus dodging some spherical aberration.

The Gallilean has a wide field of view, but the exit pupil is to large to fit into the human eye. The modern set up has many parts of the field of view criss cross through the small iris opening, so a wider view can be had with a smaller exit pupil.

 

[Stargazer19385]

I suppose two thin lenses have more eye relief than one thick one since the curve of the lens is less and does not stick out into the eye. More lenses necessitates better coatings to avoid ghost image reflections.

There is also the trade off between accurate and smooth spherical lenses or better focusing hyperbolic allowing use of the edge, but more difficult to manufacture accurately and smoothly.

And a larger lens with more FOV will have less accurate edges and aberrations. All of a sudden I don't think I can improve on my current binoculars.

 

[Stargazer19385]

OK, I have one more question, about the difference between a convex convex lens, and a plano convex lens of the same focal distance. Would the plano convex have more chromatic and spherical aberration, if each of the faces of the convex convex were were already at 1/4 wavelength off?

I took the equations for a circle and a parabola that fit together, and found that the circle focus must be twice as far away as the parabola focus to for the curves to fit snuggly. Then I assumed 440nm as the wavelength I wanted to 1/4, and cut the curves at f10, and found that the difference was indeed 1/4 wave length.

I wonder if a spherical lens can have twice the f# of spherical mirror and avoid spherical aberration. However, the rays hitting the second face of the lens are no longer parallel, and are already converging because of the first face.

 

[Drakkith]

OK, I have one more question, about the difference between a convex convex lens, and a plano convex lens of the same focal distance. Would the plano convex have more chromatic and spherical aberration, if each of the faces of the convex convex were were already at 1/4 wavelength off?

Ah, now you're getting into the realm of lens design. This is a complicated area. My book on telescope optics says the curvature of both sides of a lens can be altered while leaving the focal length the same in order to minimize aberrations but it does not go into too much detail, as that is what lens design programs are for. The book simply cannot say what is better because that depends on your setup.

I took the equations for a circle and a parabola that fit together, and found that the circle focus must be twice as far away as the parabola focus to for the curves to fit snuggly. Then I assumed 440nm as the wavelength I wanted to 1/4, and cut the curves at f10, and found that the difference was indeed 1/4 wave length.

The different lenses do not need to fit snugly together, although it is typically a good idea. Some space is allowable, and in some cases, beneficial to correcting aberrations.

I wonder if a spherical lens can have twice the f# of spherical mirror and avoid spherical aberration. However, the rays hitting the second face of the lens are no longer parallel, and are already converging because of the first face.

Spherically curved lenses and mirrors cannot avoid spherical aberration, they can only be reduced by placing additional optical elements that correct for it.