Artificial star - what am I missing

[OzEclipse]

This is really stretching the grey matter..... I did a small research project on optical diffraction patterns and resolution when I was a physics student at university in the early 1980s.

My recollection is that you need a function to describe light intensity across the aperture profile. Then you need to apply a fast fourier transform to the aperture profile.

For a typical aperture, and a simple refractor (no obstructions) the resulting diffraction pattern/airy disk is then described by a mutli order Bessel function.

I was investigating apodization masks, masks that apply a modified intensity profile across the aperture to attempt to modify the diffraction pattern to attempt to improve optical resolution. It is now 40 years since I did any of this and I did it at a time when "Physicists were real men who wrote their own fourier transform code." None of the MATLAB / SCILAB nonsense

Seriously though, after 40 years, I am very rusty on this stuff and this is about as much as I can remember of the technique.

Joe

[sdbodin]

I was investigating apodization masks, masks that apply a modified intensity profile across the aperture to attempt to modify the diffraction pattern to attempt to improve optical resolution. It is now 40 years since I did any of this and I did it at a time when "Physicists were real men who wrote their own fourier transform code." None of the MATLAB / SCILAB nonsense

Seriously though, after 40 years, I am very rusty on this stuff and this is about as much as I can remember of the technique.

Joe

Ditto here Joe, about the same time frame, actually wrote a Q-Basic program, I think, to solve the mask contribution to the pattern. It's laying around somewhere on an 8' floppy, probably. Also, cut a couple masks and they worked, but not enough to justify the side effects, rainbow rays and narrow FOV. However, the program did provide a solution to the obstructed aperture condition. Built a 50% obstructed 6" f8 folded Newt. Basically the resolving power improved to 0.5" from 0.8", but the first diffraction ring enlarges. Useful for tight double star splitting, but not planetary observing as the contrast is degraded by the enlarged diffraction.

Fun stuff, if you like juggling Bessel functions. Always a sucker for experimenting.

Steve

[OzEclipse]

Ditto here Joe, about the same time frame, actually wrote a Q-Basic program, I think, to solve the mask contribution to the pattern. It's laying around somewhere on an 8' floppy, probably. Also, cut a couple masks and they worked, but not enough to justify the side effects, rainbow rays and narrow FOV. However, the program did provide a solution to the obstructed aperture condition. Built a 50% obstructed 6" f8 folded Newt. Basically the resolving power improved to 0.5" from 0.8", but the first diffraction ring enlarges. Useful for tight double star splitting, but not planetary observing as the contrast is degraded by the enlarged diffraction.

Fun stuff, if you like juggling Bessel functions. Always a sucker for experimenting.

Steve

Thanks for the confirmation. Interesting we were both working on it at the same time? As I recall, there was an article around that time in either S&T or Astronomy magazines about these apodization masks that inspired the idea for the project.

However, it was so long ago, I wasn't sure if I was remembering the mathematical method correctly and whether or not to post something.

My FT program was also in quick basic and run on an Apple II. I had a mainframe account but it was CPU time limited and preferred to keep it for debugging my comp sci programming assessments. I was never much of a programmer.

I then constructed some 4" masks & used a pinhole artificial star on a 4" f15 Unitron refractor in the lab and photographed to B&W film then scanned on a microdensitometer. But the photographed patterns were so messy I couldn't really experimentally confirm anything. Although the lab was big, it wasn't that big and the star was only located at about 4 x FL. So less than ideal.

I also made some 6" masks for my 6" f7 Newtonian and did some visual field testing. Like you, I found a marginal apparent improvement but it was so small and the radial rainbows really only made it useful for splitting double stars and the difference was marginal. I was using flyscreen mesh for my mask material. An optical density mask was not within budget but might have produced less rainbows and better results.

cheers

Joe

[sdbodin]

I also made some 6" masks for my 6" f7 Newtonian and did some visual field testing. Like you, I found a marginal apparent improvement but it was so small and the radial rainbows really only made it useful for splitting double stars and the difference was marginal. I was using flyscreen mesh for my mask material. An optical density mask was not within budget but might have produced less rainbows and better results.

cheers

Joe

Just amazing, half a world apart, 35 odd years ago, working on the same idea. Same plastic screen material method, I used an old photo light meter to get the approximate light through-put for the beam shading needed. And, like you, the best solution would have been an optical coating that provided the correct function. But too expensive for an amateur.

Well, I will have to dig up that old computer program and see if I actually put in on a 3" floppy, but finding one of those floppy drives around the house might be a problem too.

Steve

[OzEclipse]

I also made some 6" masks for my 6" f7 Newtonian and did some visual field testing. Like you, I found a marginal apparent improvement but it was so small and the radial rainbows really only made it useful for splitting double stars and the difference was marginal. I was using flyscreen mesh for my mask material. An optical density mask was not within budget but might have produced less rainbows and better results.

cheers

Joe

Just amazing, half a world apart, 35 odd years ago, working on the same idea. Same plastic screen material method, I used an old photo light meter to get the approximate light through-put for the beam shading needed. And, like you, the best solution would have been an optical coating that provided the correct function. But too expensive for an amateur.

Well, I will have to dig up that old computer program and see if I actually put in on a 3" floppy, but finding one of those floppy drives around the house might be a problem too.

Steve

Steve,

At my former work, where I still have a 5 year visitor/consultant status, we have a room we call the O.D.D. room. Old disc drives(O.D.D.) are kept with drivers on computers specifically so that old data that wasn't maintained can be read and converted to modern format. It wouldn't surprise me if they even had an 8" floppy drive. If however, it's on a 3.5" , I have one. I am helping to decommission an old mass spectrometer that gave fantastically accurate data but needed such an old processor that we had to write data to a 3.5" FDD then read it into our office computers to process. So I have a 3.5" FDD on my office computer. But then you'll need to find something to run it. ;-)

Joe

[turboscrew]

I attached my calculations about how rays emanating from a point source that is placed at a finite distance from a parabolic mirror, are reflected. It is just geometry based and exact in the sense that no ideal lens formulas are used that assume some approximation.

While I obtained closed formulas for the points of intersections with the optical axis and corresponding path lengths to those points, it does not really say anything about Airy discs because I have no clue how interference of rays in close proximity is described. I found P-V values numerically based on finding the points where the deviation from the optical axis is minimal, for a number of use cases.

All this is probably worth nothing but it resulted in some pretty pictures, I attached one here.



I also attached the Scilab script to execute the math, for who wants to make more pretty pictures.

@SkyHiker, @Gordon, This posting needs to be an article!
viewtopic.php?p=131077#p131077

[SkyHiker]

You know what... from the other thread, Fritz reminded me of a book that I actually have and that I forgot about: "Star testing astronomical telescopes", by Harold Richard Suiter. He indeed describes the minimum distance and maximum aperture of an artificial star.

Regarding the distance, he first quote a rule by Welford that for "normal" F ratio telescopes it should be 20x the focal length. Then he quotes who did some ray tracing by Roger Sinnott (S&T May 1991) who came up with an empirical rule of 336*D/F^3 (D in inches, F=focal ratio) or (1/137)*D/(F^3*lambda) (arbitrary units). Next he comes up with an ellipsoidal shape aligned with the major axis that touches at the origin and the edge, looks at the difference with the parabolic shape, which is (1/128)*D/(F^3*lambda). Let me look into that, it should not be difficult to derive a differential equation that makes the rays converge in one point, then solve it (the hard part I guess) and compare it with the paraboloid, then take 2x the difference.

In other words this problem has been studied well. Of course there's OSLO, which I installed on my computer about 7 years ago that should be able to d some ray tracing I hope, the interface is rather terse. Maybe I should give it another try.

[turboscrew]

You know what... from the other thread, Fritz reminded me of a book that I actually have and that I forgot about: "Star testing astronomical telescopes", by Harold Richard Suiter. He indeed describes the minimum distance and maximum aperture of an artificial star.

Regarding the distance, he first quote a rule by Welford that for "normal" F ratio telescopes it should be 20x the focal length. Then he quotes who did some ray tracing by Roger Sinnott (S&T May 1991) who came up with an empirical rule of 336*D/F^3 (D in inches, F=focal ratio) or (1/137)*D/(F^3*lambda) (arbitrary units). Next he comes up with an ellipsoidal shape aligned with the major axis that touches at the origin and the edge, looks at the difference with the parabolic shape, which is (1/128)*D/(F^3*lambda). Let me look into that, it should not be difficult to derive a differential equation that makes the rays converge in one point, then solve it (the hard part I guess) and compare it with the paraboloid, then take 2x the difference.

In other words this problem has been studied well. Of course there's OSLO, which I installed on my computer about 7 years ago that should be able to d some ray tracing I hope, the interface is rather terse. Maybe I should give it another try.

That's still no excuse not to put the document in the articles. :smile:
People are not always searching an easy way to get by a problem at hand. Some people like to learn too.

[a100171]

Whenever I think that I might be smarter than I am, all I need to do is try to follow some of the math/physics started in this thread.

Or worse yet, read through that book ( Star Testing Astronomical Telescopes by Harold Richard Suiter) mentioned here. I bought this hardcover a while back (fifth edition, 2001), and cannot recall who recommended the book. Yeah, this is quite a book. Makes me wish I still had my old physics/calculus books.

My Evelyn Wood Speed-reading pass through the section on Artificial Stars was not helpful. Definitely not a skimmer of a book. Very comprehensive. But I have got to start getting smarter again.

If I hit the Megabucks on Friday, I will be sure to make sure anyone needing a copy could have one.

[OzEclipse]

Whenever I think that I might be smarter than I am, all I need to do is try to follow some of the math/physics started in this thread.

You have to realise that everybody who posted about math in this thread, was trained in it while doing degrees in Engineering, Physics or Maths. At the time I did my work, I was immersed in these forms of calculus doing my degree in Physics/Maths so it was "easy." As I admitted in one of my first posts in this thread, I am now very rusty in this type of math having barely used this stuff in nearly 40 years. These days, my calculus is very average though my trigonometry which I mostly use in astronomy is still pretty good.

If you have never been trained in this stuff, it will certainly appear very confusing to you and that we are all much smarter than we actually are.

cheers

Joe

[turboscrew]

Whenever I think that I might be smarter than I am, all I need to do is try to follow some of the math/physics started in this thread.

You have to realise that everybody who posted about math in this thread, was trained in it while doing degrees in Engineering, Physics or Maths. At the time I did my work, I was immersed in these forms of calculus doing my degree in Physics/Maths so it was "easy." As I admitted in one of my first posts in this thread, I am now very rusty in this type of math having barely used this stuff in nearly 40 years. These days, my calculus is very average though my trigonometry which I mostly use in astronomy is still pretty good.

If you have never been trained in this stuff, it will certainly appear very confusing to you and that we are all much smarter than we actually are.

cheers

Joe

In this thread, I'm obviously below average in math, but I wasn't born with even that much math, neither did it drop some night from the heaven. Then again, I can't weld and I'm not able to do much more cooking than boil water without burning it.

[SkyHiker]

This is really stretching the grey matter..... I did a small research project on optical diffraction patterns and resolution when I was a physics student at university in the early 1980s.

My recollection is that you need a function to describe light intensity across the aperture profile. Then you need to apply a fast fourier transform to the aperture profile.

For a typical aperture, and a simple refractor (no obstructions) the resulting diffraction pattern/airy disk is then described by a mutli order Bessel function.



I was investigating apodization masks, masks that apply a modified intensity profile across the aperture to attempt to modify the diffraction pattern to attempt to improve optical resolution. It is now 40 years since I did any of this and I did it at a time when "Physicists were real men who wrote their own fourier transform code." None of the MATLAB / SCILAB nonsense

Seriously though, after 40 years, I am very rusty on this stuff and this is about as much as I can remember of the technique.

Joe

You are right that simple geometric reasoning has little to do with diffraction patterns. Huygens' principle describes how wave fronts bend through openings. Fresnel found a way to describe diffraction patterns based on Huygens' principle. But Kirchhoff's wave function based diffraction formula seems to be the only complete solution. Huygens'/Fresnel was still lacking because Huygens only considered forward moving secondary waves. This error was only resolved in 1991 by Miller.

In other words, simple geometric analysis based on exact reflection or refraction of light rays is simply inadequate. The variation in the wave front that I found based on reflection of individual rays was about 10 times larger than the ones in other references. These other references are sometimes based on the lens formula (that I have shown is inexact for parabolic mirrors) and on crossing the Y-axis (which I have shown is largely irrelevant), and sometimes have missing math. In other words, be highly skeptical of such work.

My latest result are based on fitting an ellipse to the quadratic mirror and basing the reflection phase difference at the focal point on twice the distance (not accounting for how far the rays are apart at the point of convergence other than total path length). In that case I get deviations comparable to the other references I have found. Note that the refraction by an elliptic surface is exact if the light source is in a focal point, and that the path traveled is the same for all rays. But the problem still remains that the path length difference is only one aspect, while the other difference (the angle at which the rays are reflected) have a much more severe impact.

While the geometry of ray reflection is definitely interesting, this kind of reasoning can't be trusted to tell anything about diffraction. For that, the wave function must be used with Kirchhoff's formulas, as far as I can tell.

[turboscrew]

This is really stretching the grey matter..... I did a small research project on optical diffraction patterns and resolution when I was a physics student at university in the early 1980s.

My recollection is that you need a function to describe light intensity across the aperture profile. Then you need to apply a fast fourier transform to the aperture profile.

For a typical aperture, and a simple refractor (no obstructions) the resulting diffraction pattern/airy disk is then described by a mutli order Bessel function.



I was investigating apodization masks, masks that apply a modified intensity profile across the aperture to attempt to modify the diffraction pattern to attempt to improve optical resolution. It is now 40 years since I did any of this and I did it at a time when "Physicists were real men who wrote their own fourier transform code." None of the MATLAB / SCILAB nonsense

Seriously though, after 40 years, I am very rusty on this stuff and this is about as much as I can remember of the technique.

Joe

You are right that simple geometric reasoning has little to do with diffraction patterns. Huygens' principle describes how wave fronts bend through openings. Fresnel found a way to describe diffraction patterns based on Huygens' principle. But Kirchhoff's wave function based diffraction formula seems to be the only complete solution. Huygens'/Fresnel was still lacking because Huygens only considered forward moving secondary waves. This error was only resolved in 1991 by Miller.

In other words, simple geometric analysis based on exact reflection or refraction of light rays is simply inadequate. The variation in the wave front that I found based on reflection of individual rays was about 10 times larger than the ones in other references. These other references are sometimes based on the lens formula (that I have shown is inexact for parabolic mirrors) and on crossing the Y-axis (which I have shown is largely irrelevant), and sometimes have missing math. In other words, be highly skeptical of such work.

My latest result are based on fitting an ellipse to the quadratic mirror and basing the reflection phase difference at the focal point on twice the distance (not accounting for how far the rays are apart at the point of convergence other than total path length). In that case I get deviations comparable to the other references I have found. Note that the refraction by an elliptic surface is exact if the light source is in a focal point, and that the path traveled is the same for all rays. But the problem still remains that the path length difference is only one aspect, while the other difference (the angle at which the rays are reflected) have a much more severe impact.

While the geometry of ray reflection is definitely interesting, this kind of reasoning can't be trusted to tell anything about diffraction. For that, the wave function must be used with Kirchhoff's formulas, as far as I can tell.

I wonder if this is of interest...
http://jzbuchwald.caltech.edu/downloads ... action.pdf

[SkyHiker]

I figured out the math by matching an ellipse to the paraboloid and finding the maximum deviation times two to determine the path length difference of a reflected ray. An elliptic mirror is a perfect reflector for a source placed in its farthest focal point, with an equal path length for a ray starting in one focal point and ending in the other. The difference with the earlier geometric approach is that the reflection angle is ignored, which is appropriate for the rays that are involved in diffraction patterns. They come from the edge of the mirror that functions as a light source from the viewpoint of diffraction.

The math was laborious but doable. The outcome is: minimum distance (in m) = 9(D/F)^2 where D is the mirror radius in inches, and F is the focal ratio, for 1/4 wavelength tolerance at 560 nm. The values that I get correspond well with a few references that I found (including an empirical one). Aside from the math I also did the same thing by numerical optimization and put all results in a table. So, this approach is actually valid as opposed to my earlier suspicion. The exact geometric reflection is simply not very relevant.

I attached an updated version of the document just to be complete. I certainly hope it is final, I don't know how many times I had to use Scilab to track down and fix errors in my math! It was fun solving it and the resulting graphics also look cool.

[turboscrew]

GREAT job, SkyHiker!
Now, this NEEDS to go into articles, please!

[OzEclipse]

I figured out the math by matching an ellipse to the paraboloid and finding the maximum deviation times two to determine the path length difference of a reflected ray. An elliptic mirror is a perfect reflector for a source placed in its farthest focal point, with an equal path length for a ray starting in one focal point and ending in the other. The difference with the earlier geometric approach is that the reflection angle is ignored, which is appropriate for the rays that are involved in diffraction patterns. They come from the edge of the mirror that functions as a light source from the viewpoint of diffraction.

The math was laborious but doable. The outcome is: minimum distance (in m) = 9(D/F)^2 where D is the mirror radius in inches, and F is the focal ratio, for 1/4 wavelength tolerance at 560 nm. The values that I get correspond well with a few references that I found (including an empirical one). Aside from the math I also did the same thing by numerical optimization and put all results in a table. So, this approach is actually valid as opposed to my earlier suspicion. The exact geometric reflection is simply not very relevant.

I attached an updated version of the document just to be complete. I certainly hope it is final, I don't know how many times I had to use Scilab to track down and fix errors in my math! It was fun solving it and the resulting graphics also look cool.

Hi Henk,

I only just got around to reading this in detail. Great job BTW.

When I looked at the formula your post and worked out the distance for my 18"f5.5, it came to 24 metres. Intuitively, it didn't seem right, seemed to close. The notation also looked odd. Then I read your paper.

In your post, you wrote, "The outcome is: minimum distance (in m) = 9(D/F)^2 where D is the mirror radius in inches, and F is the focal ratio, for 1/4 wavelength tolerance at 560 nm. "

When I read this, I wondered why you would use D to represent mirror radius instead of R. Using the radius in the formula yields very short minimum distances. When I went through your paper, I noted that D is, in fact, mirror diameter not radius.

The post should read, as per your paper, "The outcome is: minimum distance (in m) = 9(D/F)^2 where D is the mirror diameter in inches, and F is the focal ratio, for 1/4 wavelength tolerance at 560 nm."

Cheers

Joe

[SkyHiker]

I figured out the math by matching an ellipse to the paraboloid and finding the maximum deviation times two to determine the path length difference of a reflected ray. An elliptic mirror is a perfect reflector for a source placed in its farthest focal point, with an equal path length for a ray starting in one focal point and ending in the other. The difference with the earlier geometric approach is that the reflection angle is ignored, which is appropriate for the rays that are involved in diffraction patterns. They come from the edge of the mirror that functions as a light source from the viewpoint of diffraction.

The math was laborious but doable. The outcome is: minimum distance (in m) = 9(D/F)^2 where D is the mirror radius in inches, and F is the focal ratio, for 1/4 wavelength tolerance at 560 nm. The values that I get correspond well with a few references that I found (including an empirical one). Aside from the math I also did the same thing by numerical optimization and put all results in a table. So, this approach is actually valid as opposed to my earlier suspicion. The exact geometric reflection is simply not very relevant.

I attached an updated version of the document just to be complete. I certainly hope it is final, I don't know how many times I had to use Scilab to track down and fix errors in my math! It was fun solving it and the resulting graphics also look cool.

Hi Henk,

I only just got around to reading this in detail. Great job BTW.

When I looked at the formula your post and worked out the distance for my 18"f5.5, it came to 24 metres. Intuitively, it didn't seem right, seemed to close. The notation also looked odd. Then I read your paper.

In your post, you wrote, "The outcome is: minimum distance (in m) = 9(D/F)^2 where D is the mirror radius in inches, and F is the focal ratio, for 1/4 wavelength tolerance at 560 nm. "

When I read this, I wondered why you would use D to represent mirror radius instead of R. Using the radius in the formula yields very short minimum distances. When I went through your paper, I noted that D is, in fact, mirror diameter not radius.

The post should read, as per your paper, "The outcome is: minimum distance (in m) = 9(D/F)^2 where D is the mirror diameter in inches, and F is the focal ratio, for 1/4 wavelength tolerance at 560 nm."

Cheers

Joe

Hi Joe

Thanks for the correction! I had it right in the article but not in the reporting. To repeat it,

The outcome is: minimum distance (in m) = 9(D/F)^2 where D is the mirror diameter in inches, and F is the focal ratio, for 1/4 wavelength tolerance at 560 nm.

Both Carlin and Suiter use the diameter D so for comparison, that's what I aimed for. It evolved over time though. During the math I found that it is more convenient to use the radius in order to avoid powers of 2 everywhere, which I why at some point I defined d (lower case) as the radius (explicitly warning and mentioning that it means the radius not the diameter). In the end, for a user friendly formula that compares with Carlin and Suiter's formulae it is best to express the final result in the diameter D in inches (as much as I favor the metric system). Elsewhere I used D as the path length from source to Y axis crossing - redoing the notation was a bit much at that point especially since the X axis crossings are not all that relevant. My comparison for the test cases was thorough enough for me to have confidence in the results.

Thanks for reading it, the paper should actually be reviewed by others to check the math not just me. For me the fun was in getting a better understanding of mirrors, both parabolic and elliptic. The graphics were great fun too, as was seeing the optimization pinpointing the solutions. Scilab is really a fantastic free tool to do this with, I use it a lot.

[turboscrew]

I think I should have a look into the Scilab.