Designing a RB67 mirror speedbooster

[Korbel]

The idea is to make an extremely fast lens by focusing a medium format on a tiny film. It is a work in progress I have abandoned around a year ago and maybe it could inspire someone or spark a discussion, or maybe I got it completely wrong.

RB67 has a long flange distance (around 10cm) while for instance micro 4/3 has around 1cm. This gives a quite large distance for a mirror based speedbooster.

In the design I came up with there are two parabolic mirrors (magenta and green). The blue rays are what enters the speedbooster, magenta rays are first reflection and green rays are second.
UW is the film size and DE is the size of image exiting a lens.

I think that the blue rays are ideally parallel when exiting lenses. Such an ideal lens would simplify the design quite a lot, because you can have static elements. Sadly I don't think this is always the case on RB67 system, because I notice quite big focus breathing with my 100-200mm lens. Anyways a measurement of the angle for the lens in a whole focus / zoom range should be done. Worst case you could just focus to infinity and do astral photography with ridiculously fast lens - the area is almost 5x smaller, hence it is 5x brighter, then f3.5 = f0.7.

This project could get even crazier if you take those huge projector lenses!

Mirrors optics shouldn't suffer from aberrations, because the refractive index isn't an issue. There is a slight distortion to the image, UV is around 110% of VW.


Maybe I got some things wrong, I did read a book on optics by Kingslake, Lens Design Fundamentals. On the first few pages he basically said the best one can do is educated trial and error I found that funny because I found it as recommendation for the best optics book.

Uploading the geogebra file too if someone wants to play around.

 

[reddesert]

The generic name for a speed booster is a "focal reducer." This is essentially the converse of a teleconverter (and is sometimes called a telecompressor). A teleconverter is a diverging lens placed behind the primary lens so that the combination has a longer focal length, while a focal reducer is a positive-focal-length lens placed behind the primary to give a shorter focal length. https://en.wikipedia.org/wiki/Telecompressor

The wiki page gives a useful formula for the focal length of the combination, f_new = f_old * (1 - d / f_r),
where d is the distance from the focal-reducing lens to the sensor, and f_r is the focal length of the focal-reducing lens. For the typical kind of speed booster that is commercially available in a lens adapter, they are often 0.67x - 0.7x, so the ratio d / f_r is about 1/3. So the focal length is about 3x the distance of focal reducer to sensor/film (if distance ~ 30mm, focal length is 90mm or so). That means the focal reducer lens doesn't have to be incredibly strong - short focal length lenses are more strongly curved and have more visible aberrations. But to achieve a 5x speed boost like you suggest, the focal reduction is 0.2x, so d / f_r ~ 0.8. For the same distance ~ 30mm, one would need a 37mm focal length. This will be harder to make with good image quality.

It took me a while to figure out what your diagram was trying to show. I think you have parallel rays in blue coming down from the top, bouncing off one off-axis concave paraboloid mirror (magenta), off another convex off-axis paraboloid mirror (green, not concentric with the first), as parallel ray bundles to a new surface (green line). It may be that these parallel rays are each intended to be the chief ray of a bundle imaging some point in the image and the new surface is a new focal plane. The chief rays exiting a real lens are parallel if the lens is "telecentric" - real lenses usually aren't, but that isn't the biggest issue.

Unfortunately, this is not how imaging systems are designed. For each point in the image, there is a converging ray bundle exiting the lens and entering the focal reducer. The focal reducer (whether mirrors or lenses) makes that bundle converge more sharply (shorter focal length) onto a new focal plane. It is necessary to ray-trace the on and off-axis rays to minimize aberrations to get each new bundle to converge more or less on a point. Minimizing the aberrations is complicated even for fairly simple lenses because you quickly get polynomial equations without closed form solutions. That's why Kingslake wrote that the lens designer more or less operated by intuition to develop a design that could then be optimized with mathematics on paper (or a computer).

While mirrors are free of chromatic aberration, they still have all the other aberrations (spherical, astigmatism, coma, field curvature). Also, with off-axis mirrors you lose rotational symmetry, so you have to do the ray tracing not only in the plane of the paper, but out of it. And, if you try to construct such a system, aligning the off-axis mirrors will be darn near impossible. I suggest trying first with simple lenses (a singlet or achromat positive lens) of the appropriate focal length to see what you can get out. It won't be perfect (it will have aberrations and field curvature), but it should make an image, and with enough mechanical adjustment you can get it in focus.

 

[Donald Qualls]

with off-axis mirrors you lose rotational symmetry, so you have to do the ray tracing not only in the plane of the paper, but out of it. And, if you try to construct such a system, aligning the off-axis mirrors will be darn near impossible.

Telescope builders consider off-axis systems some of the hardest to build and produce acceptable image quality -- and they're just looking with an eyepiece and an eyeball behind it. If you want to record an image on film that'll be enlarged for viewing, you need to beat the image quality of a direct viewing telescope by a good bit.

At least one off-axis telescope design (made to have no obstructions in the optical path, unlike Newtonian, Cassegrain, Maksutov, etc.) requires warping the primary mirror to correct the off-axis coma and astigmatism.

The one common design criterion in unobstructed off-axis telescope designs is that they're slow -- because the longer the focal length relative to aperture, the less you have to do to the spherical mirror surface that's easy to make before it produces an acceptable image.

Even common axial reflecting telescopes are usually made at f/4 or slower, because a faster mirror has to be more perfect, optically, to give sharp star images to the edge of the field -- and converting a spherical mirror to a paraboloid is more than half the work in converting a flat glass surface into a telescope mirror (been there, done that; got a 1/6 wave error limit 8" f/6.8 Newton/Dobson in my home office), and even a perfect parabolic mirror has a limited field of view before off-axis coma becomes a problem.

 

[Korbel]

Telescope builders consider off-axis systems some of the hardest to build and produce acceptable image quality -- and they're just looking with an eyepiece and an eyeball behind it. If you want to record an image on film that'll be enlarged for viewing, you need to beat the image quality of a direct viewing telescope by a good bit.

At least one off-axis telescope design (made to have no obstructions in the optical path, unlike Newtonian, Cassegrain, Maksutov, etc.) requires warping the primary mirror to correct the off-axis coma and astigmatism.

The one common design criterion in unobstructed off-axis telescope designs is that they're slow -- because the longer the focal length relative to aperture, the less you have to do to the spherical mirror surface that's easy to make before it produces an acceptable image.

Even common axial reflecting telescopes are usually made at f/4 or slower, because a faster mirror has to be more perfect, optically, to give sharp star images to the edge of the field -- and converting a spherical mirror to a paraboloid is more than half the work in converting a flat glass surface into a telescope mirror (been there, done that; got a 1/6 wave error limit 8" f/6.8 Newton/Dobson in my home office), and even a perfect parabolic mirror has a limited field of view before off-axis coma becomes a problem.

I was trying to avoid the hole in the middle and brought other problems it seems. By "warping" you mean you no longer get a parabolic shape which can be written with some parametric equation? (which simplifies the design I think). And my understanding from what you have written then is, that you either get a great reflective material for the mirrors or you get their perfect shape but cant get both?

One thing I remember for some reason is that a magnifying lens cannot produce higher temperatures than the sun's surface, because of the 2nd law. It seems to me like with this designs some similar boundary is reached on amount of retrievable information from a signal per unit of time.

 

[Korbel]

The generic name for a speed booster is a "focal reducer." This is essentially the converse of a teleconverter (and is sometimes called a telecompressor). A teleconverter is a diverging lens placed behind the primary lens so that the combination has a longer focal length, while a focal reducer is a positive-focal-length lens placed behind the primary to give a shorter focal length. https://en.wikipedia.org/wiki/Telecompressor

The wiki page gives a useful formula for the focal length of the combination, f_new = f_old * (1 - d / f_r),
where d is the distance from the focal-reducing lens to the sensor, and f_r is the focal length of the focal-reducing lens. For the typical kind of speed booster that is commercially available in a lens adapter, they are often 0.67x - 0.7x, so the ratio d / f_r is about 1/3. So the focal length is about 3x the distance of focal reducer to sensor/film (if distance ~ 30mm, focal length is 90mm or so). That means the focal reducer lens doesn't have to be incredibly strong - short focal length lenses are more strongly curved and have more visible aberrations. But to achieve a 5x speed boost like you suggest, the focal reduction is 0.2x, so d / f_r ~ 0.8. For the same distance ~ 30mm, one would need a 37mm focal length. This will be harder to make with good image quality.

It took me a while to figure out what your diagram was trying to show. I think you have parallel rays in blue coming down from the top, bouncing off one off-axis concave paraboloid mirror (magenta), off another convex off-axis paraboloid mirror (green, not concentric with the first), as parallel ray bundles to a new surface (green line). It may be that these parallel rays are each intended to be the chief ray of a bundle imaging some point in the image and the new surface is a new focal plane. The chief rays exiting a real lens are parallel if the lens is "telecentric" - real lenses usually aren't, but that isn't the biggest issue.

Unfortunately, this is not how imaging systems are designed. For each point in the image, there is a converging ray bundle exiting the lens and entering the focal reducer. The focal reducer (whether mirrors or lenses) makes that bundle converge more sharply (shorter focal length) onto a new focal plane. It is necessary to ray-trace the on and off-axis rays to minimize aberrations to get each new bundle to converge more or less on a point. Minimizing the aberrations is complicated even for fairly simple lenses because you quickly get polynomial equations without closed form solutions. That's why Kingslake wrote that the lens designer more or less operated by intuition to develop a design that could then be optimized with mathematics on paper (or a computer).

While mirrors are free of chromatic aberration, they still have all the other aberrations (spherical, astigmatism, coma, field curvature). Also, with off-axis mirrors you lose rotational symmetry, so you have to do the ray tracing not only in the plane of the paper, but out of it. And, if you try to construct such a system, aligning the off-axis mirrors will be darn near impossible. I suggest trying first with simple lenses (a singlet or achromat positive lens) of the appropriate focal length to see what you can get out. It won't be perfect (it will have aberrations and field curvature), but it should make an image, and with enough mechanical adjustment you can get it in focus.

Yes you read the diagram right, the green thick line is the sensor. For telecentricity compensation I have added a movable focus point to the design, my original idea was to somehow measure the angle first and then construct the mirrors. For some reason I though aberrations are a problem only when light passes through mediums of different refractive indexes. Thank you for clarification

The problem of constructing this in real striked me back then, but still I considered it an unresolved project and now I can sleep better knowing it is impossible

 

[Donald Qualls]

I was trying to avoid the hole in the middle and brought other problems it seems. By "warping" you mean you no longer get a parabolic shape which can be written with some parametric equation? (which simplifies the design I think). And my understanding from what you have written then is, that you either get a great reflective material for the mirrors or you get their perfect shape but cant get both?

One thing I remember for some reason is that a magnifying lens cannot produce higher temperatures than the sun's surface, because of the 2nd law. It seems to me like with this designs some similar boundary is reached on amount of retrievable information from a signal per unit of time.

By "warping" I mean actually physically *bending* an initially spherical mirror to make it approximate an off-axis portion of a paraboloid. The alternative is to polish in the off-axis shape, but this is much harder than flexing a spherical mirror.

Any of these methods still use a metal vapor coated surface polished to an accuracy of a fraction of a wavelength of light -- so you get both a highly reflective surface (typically 94-96% for aluminum coating) and, with care the surface shape can be effectively perfect -- but it's much easier to accomplish this with a central obstruction as found in Maksutov mirror lens, where all the surfaces are spherical, parabolic, or at worst elliptical.

 

[Alan Edward Klein]

Metabones makes these adapters. You might Google them.

Metabones®

 

[reddesert]

Metabones makes focal reducers that are largely for mounting full-frame/35mm SLR/DSLR lenses on smaller format, APS-C or micro-4/3, mirrorless cameras. These have a focal reduction of about 0.7x, as I mentioned in post #2. That also results in an equivalent speed boost of 0.7x in f-number, so 1 stop. Metabones says their focal reducers are 5 elements in 4 groups (not simple).

The OP wants to do a much more dramatic focal reduction of about 0.2x of a medium format lens onto a small format. This is much more challenging optically. I mean you might be able to do it with a simple lens or achromat or something, but the images will be highly aberrated and have a lot of field curvature. They might be interesting images, but not general purpose.

Metabones has more technical or marketing-technical information here: https://metabones.com/assets/a/Speed Booster White Paper.pdf
Section 14 on the past of focal reducers and their function in historical lens designs is interesting context.

 

[Korbel]

Btw I mentioned earlier that I believe there might be some theoretical limit to how much information one can retrieve via lens, meaning there might be some theoretical f-stop limit. Same like a magnifier glass cannot make temperatures higher than is the temperature of the source light due to 2nd thermodynamics law. If one would decide to attempt to build such a focal reducer, then it shall not yield any results from some point onward I guess by analogy. Does anynone know more on this subject? There was the f0.7 lens by Zeiss I know about, which nasa and Kubrick used.

 

[Korbel]

Btw I mentioned earlier that I believe there might be some theoretical limit to how much information one can retrieve via lens, meaning there might be some theoretical f-stop limit. Same like a magnifier glass cannot make temperatures higher than is the temperature of the source light due to 2nd thermodynamics law. If one would decide to attempt to build such a focal reducer, then it shall not yield any results from some point onward I guess by analogy. Does anynone know more on this subject? There was the f0.7 lens by Zeiss I know about, which nasa and Kubrick used.

"Numerical aperture (NA) is calculated using the formula:
NA=n×sin(θ), where
n is the refractive index of the medium and θtheta is the half-angle of the maximum cone of light that can enter the lens."

So if we take 180 angle as maximum, air refractive index of 1, then the fastest aperture is f0.5

The Zeiss f0.7 lens is stunningly close to the limit