I Sine laws of spherical singlets

[difalcojr]

I really appreciate the fact that three of you have replied to me in this physics forum over the past week and given me your critical thoughts on a new lens model I wanted to show to all. One more important point of Huygens is useful, I think, to show now, for further clarification.
Here's a copy of his diagram of the aplanatic point of perfect focus within the sphere (without the construction lines). It is from his "Treatise on Light". He was writing about his re-trace of the conics sequence and ovals of Descartes. He says: "But it is worthy of remark, that in one case this oval becomes a perfect circle, namely when the ratio of AD to DB is the same as the ratio of the refractions, here as 3 to 2, as I observed a long time ago."
So, here's another refractive ratio, as seen on the horizontal axis line ratio Huygens found. He used 1.5 for his glass index models. In the above diagram, the line lengths ratio is the square root of 2, the same value as the other ratios of this model.

 

[difalcojr]

This goes with the earlier sequences shown. Might help. Marginal rays of various, incoming waves and their refracted chords inside a sphere are shown. Also, a source point right at the edge of the first surface, and the aplanatic ray with its extension to horizontal axis, these two angles at the limit points of this model. Bisectors to the refracted chords are shown, and you can see the extent of the second radius needed for each wave at its marginal point.

 

[difalcojr]

If that is what the math says then I agree with it. However, note that we really don't have a singlet anymore. That typically implies a single lens inside a single medium such that the refractive index on both sides is the same. You've essentially made a cemented doublet.

Yes, cemented doublets, thanks, though I think I could argue the point for a singlet too a bit further.
Also, I have not really proven anything yet, just shown diagrams that I said I plotted. They could just be AutoSketch pictures. Your caution seems like Hecht's here, reliable and needed, I think. Unless you may accept the fact that the model is independent of incoming angle that showed up in your earlier alarm about the missing sine law relations in the first post.
To my knowledge a perfect lens must also have constant magnification and constant optical path lengths, as per Maxwell's conditions?

 

[Drakkith]

Not sure. But I did find the following snippet from this site:

• Notwithstanding eight propositions about perfect optical systems, Maxwell ended with a ninth proposition that dampened the hopes of would-be instrument designers. An optical system can at best produce a perfect image only if the magnification is equal to the ratio of refractive indices in image and object space. Since in the usual circumstances both object and image are in air then one can’t make a perfect image with any ‘magnification’ other than unity. A truly plain mirror, for example, creates a perfect image but anything designed to magnify won’t. This applies to cameras, microscopes, telescopes and so on.

 

[difalcojr]

Not sure. But I did find the following snippet from this site:

• Notwithstanding eight propositions about perfect optical systems, Maxwell ended with a ninth proposition that dampened the hopes of would-be instrument designers. An optical system can at best produce a perfect image only if the magnification is equal to the ratio of refractive indices in image and object space. Since in the usual circumstances both object and image are in air then one can’t make a perfect image with any ‘magnification’ other than unity. A truly plain mirror, for example, creates a perfect image but anything designed to magnify won’t. This applies to cameras, microscopes, telescopes and so on.

Well, that's an interesting site. I did not know of 9 propositions al all. But I just found what Maxwell also wrote from his source listed on that web page. I'll write it as it is shown in the text with the italics used:

"A perfect instrument must fulfill three conditions:
I. Every ray of the pencil, proceeding from a single point of the object, must, after passing through the instrument, converge to, or diverge from, a single point of the image. The corresponding defect, when the emergent rays have not a common focus, has been appropriately called (by Dr. Whewell) Astigmatism.
II. If the object is a plane surface, perpendicular to the axis of the instrument, the image of any point of it must also lie in a plane perpendicular to the axis. When the points of the image lie in a curved surface, it is said to have the defect of curvature.
III. The image of an object on this plane must be similar to the object, whether its linear dimensions be altered or not; when the image is not similar to the object, it is said to be distorted.
An image free from these three defects is said to be perfect."

 

[difalcojr]

So, this model does not fulfill Maxwell's first condition.
My take on the reason for this, is that the model herein is a projection lens and has no change in the wave angles. What goes in, comes out with same angle, magnified, and the entire wave is magnified.
Also, Maxwell's condition is for a thin lens with an incident planar wave, a lens ideally having parabolic surfaces to get zero SA. Thin lens equations don't fit this model well at all, if you have tried any. For the mechanical drawings, only geometry and trigonometry was needed!

 

[difalcojr]

A diagram may help more, but it's a bit messy, sorry. Planar wave again incoming and its refraction pattern of SA shown, the caustic, believe it's called. Also shown is the normal, second surface output flipped around, so its output is now the input. You can see that its refraction pattern is the exact mirror image of the first surface. Its refractive ratio is the inverse of the other. One pattern is a real image, the other a virtual image. This model matches the caustics of all the waves like this. At that point where the rays begin to cross each other, the bisector point of the marginal, refracted ray the sphere. That bisector length that Huygens found.
The model lens is afocal, no internal focus. Of course, it can project or receive from either direction. It is different from thin lenses in many ways, for sure.

 

[difalcojr]

Here's my favorite picture for relief. 180 degrees diverging, both Huygens' points shown, maximum extension of radius2. No coincidence that it looks like a "Dutch-cut" haircut too? Huygens is the Dutchman.

 

[difalcojr]

View attachment 330275
A diagram may help more, but it's a bit messy, sorry. Planar wave again incoming and its refraction pattern of SA shown, the caustic, believe it's called. Also shown is the normal, second surface output flipped around, so its output is now the input. You can see that its refraction pattern is the exact mirror image of the first surface. Its refractive ratio is the inverse of the other. One pattern is a real image, the other a virtual image. This model matches the caustics of all the waves like this. At that point where the rays begin to cross each other, the bisector point of the marginal, refracted ray the sphere. That bisector length that Huygens found.
The model lens is afocal, no internal focus. Of course, it can project or receive from either direction. It is different from thin lenses in many ways, for sure.

better diagramView attachment 330296

 

[difalcojr]

View attachment 330275
A diagram may help more, but it's a bit messy, sorry. Planar wave again incoming and its refraction pattern of SA shown, the caustic, believe it's called. Also shown is the normal, second surface output flipped around, so its output is now the input. You can see that its refraction pattern is the exact mirror image of the first surface. Its refractive ratio is the inverse of the other. One pattern is a real image, the other a virtual image. This model matches the caustics of all the waves like this. At that point where the rays begin to cross each other, the bisector point of the marginal, refracted ray the sphere. That bisector length that Huygens found.
The model lens is afocal, no internal focus. Of course, it can project or receive from either direction. It is different from thin lenses in many ways, for sure.

better diagram

 

[difalcojr]

Double sided lenses are much more difficult to produce and absent machine tools and other modern techniques only of academic interest.

Yes, agreed, but in molds could be mass produced, probably. Zero SA should be very attractive for a planar wave. What about uses in electronics? Say in signalization devices? Or microscopes/telescopes?
The model's dimensions are simple equations, its shape the same for every wave.
It needs a lot more academic interest, though, I think, now. Due it is something new. It's a lot to digest if you have never seen such a thing.
I only have a few more diagrams to finish up explaining the model. Thank you for your good criticism.

 

[Drakkith]

I'd say do a full raytrace before trying to consider what the system would be good for. SA might be fully corrected for, but what about other aberrations?

 

[difalcojr]

Agreed. Is there existing trace software that can get values for all the positions I've plotted? I used a trigonometric trace and AutoSketch.
What other aberrations? Chromatic? Not sure of that. Have only traced using one index number so far. Might be a big problem for color images, surely. I can think of another lens it absolutely would not be used for. Eyeglasses.

 

[Drakkith]

There are dedicated ray tracing software that will plot all your aberrations, but don't I have a link to one right now. A quick google search should turn up some results.

Edit: You might try Optical Ray Tracer. I haven't used it before, but it's a free program that might be what you need. You can find it here: https://arachnoid.com/OpticalRayTracer/

 

[difalcojr]

Thks. I'll check it out today. Well, there won't be any SA aberrations to plot, I'm contending. Program looks interesting. But to go back to the topic of the initial post, too, here's Huygens' other diagram with XN equaling the second radius. His ray/wave measurements diagram. Not sure what software he was using.

.................................

 

[difalcojr]

Here's a 30 degree incoming wave. A very strange optical system indeed. Hope you may like this plot. Reminds me of the old SF movie, The Day the Earth Stood Still. Do you know that one?

 

[hutchphd]

You may find this treatment interesting (shaping aspheric second surface) .
https://royalsocietypublishing.org/doi/epdf/10.1098/rspa.2014.0608

 

[difalcojr]

You may find this treatment interesting (shaping aspheric second surface) .
https://royalsocietypublishing.org/doi/epdf/10.1098/rspa.2014.0608

Will chk it out and get back to you. Thks.

 

[difalcojr]

Now I know why mathematicians are so special, and why I'm not a mathematician. Wow. Is that it what is takes to find exact position points for a parbolizing, second surface? And they can mass produce molds for those calculated shapes? Stupendous achievement of mathematics.
Looks like the spherical model could do some of the same things except for the real-real examples. Good problem for a cost analysis, maybe, for spheric/aspheric vs. spheric/spheric molds. For someone else.

 

[difalcojr]

There are dedicated ray tracing software that will plot all your aberrations, but don't I have a link to one right now. A quick google search should turn up some results.

Edit: You might try Optical Ray Tracer. I haven't used it before, but it's a free program that might be what you need. You can find it here: https://arachnoid.com/OpticalRayTracer/

I chked it, Optical Ray Tracer. Thks. Could not get it to produce the model shape, but it's probably a good program for thin lenses, I assume.
I don't need a ray trace program, though. My own for this model works fine. It is on an old, discontinued spreadsheet called Improv. Made by Lotus before IBM gobbled them up. I just don't think there's software that can do thick, spherical tracing, anyway.

 

[difalcojr]

Now I know why mathematicians are so special, and why I'm not a mathematician. Wow. Is that it what is takes to find exact position points for a parbolizing, second surface? And they can mass produce molds for those calculated shapes? Stupendous achievement of mathematics.
Looks like the spherical model could do some of the same things except for the real-real examples. Good problem for a cost analysis, maybe, for spheric/aspheric vs. spheric/spheric molds. For someone else.

I take that back, actually, after another look at the article. This model cannot duplicate the aspheric examples shown. They are all either real-real systems or have angle magnifications. This model has wave, but not angle, magnification, in real-virtual or virtual-real systems. Not sure what to make of that.

 

[difalcojr]

You may find this treatment interesting (shaping aspheric second surface) .
https://royalsocietypublishing.org/doi/epdf/10.1098/rspa.2014.0608

In the royal society article, there is not a plane wave input, plane wave output example. Nor online that I can find. I don't think it is possible, even using all that math. So, the question arises: can an aspheric singlet do the same as this model's singlet (doublet)?

Could it do a 16:1 exact reduction like this diagram above?

 

[Drakkith]

No idea. This is a fair bit above my skill level in optics. Which is about two semesters of geometrical optics from five years ago.

 

[difalcojr]

Yes, me too, no idea. Math in the spherics article was way over my head.
Still, if it was possible to reproduce a planar wave magnification without angle change, I think they would have included it in the article. All their models had angle changes.
My thinking is that, if the models shown in this post are unable to do any of the projections that the spheric-aspheric singlets in the royal society article can do, then probably the spheric-aspheric family models cannot do any of this model family's projections, either.
Good problem for all the mathematicians in your audience to solve.

 

[Drakkith]

Hmmm. Does this not violate conservation of etendue?

Per wiki:
The etendue of a given bundle of light is conserved: etendue can be increased, but not decreased in any optical system. This means that any system that concentrates light from some source onto a smaller area must always increase the solid angle of incidence (that is, the area of the sky that the source subtends). For example, a magnifying glass can increase the intensity of sunlight onto a small spot, but does so because, viewed from the spot that the light is concentrated onto, the apparent size of the sun is increased proportional to the concentration.

 

[difalcojr]

No idea. Don't know the term at all.
I wanted this post just to show a lens model family with constant magnifications and zero SA. A singlet in principle, a doublet in most all applications. To show the various wave and ray refractions, and to show the historical math connections to Huygens.
I think giving magnification values that I've calculated, and also showing optical path lengths, will further help to convince everyone that these theoretical lenses are really possible. I'll post that diagram and chart of values tomorrow. Thks for seeing this project along.

 

[Drakkith]

Are the centers of curvature of each surface located at the same place? I was trying to do a paraxial raytrace, but I don't know how far apart the surfaces are.

 

[hutchphd]

No idea. Don't know the term at all.

Probably worth looking at.

 

[difalcojr]

Are the centers of curvature of each surface located at the same place? I was trying to do a paraxial raytrace, but I don't know how far apart the surfaces are.

For the single lens, yes. Monocentric.
Axial distance between surfaces is radius1 plus radius2.
For n0=1, and n2=square root of 2, and radius1=1, radius 2 is then (square root of 2)/2. So, adding radii., you get (1 + .7071...). The answer is 1.7071... for the thickness.

 

[Drakkith]

Hmmm. Using this ynu ray trace calculator, I'm not getting what you're getting. My initially parallel ray diverges after exiting the 2nd surface. Not sure if a paraxial ray trace is inappropriate here, or if I'm doing something wrong.