Sine laws of spherical singlets

[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.

 

[difalcojr]

index2 should equal 2.
Try a marginal ray, maybe. y-axis height is 0.5 for the marginal ray. Everything else the same.

 

[Drakkith]

Ah, I found the issue. The result was actually in engineering notation, but the number of digits after the decimal was so high the e-6 was hidden off to the right inside the text box. So it's very, very close to zero, close enough to agree with your analysis.

 

[difalcojr]

Did it work for both the paraxial and the marginal ray? Ray angles set at zero degrees with the horizontal, and the exit angles the same?

 

[Drakkith]

Just did a ray trace with an initial ray angle of 0.1 and ray height of 0 (hits the first surface in the center). The resulting exit angle was about 0.0999939. The difference from 0.1 is easily attributable to me only using 4 or 5 digits for the input values.

Did it work for both the paraxial and the marginal ray? Ray angles set at zero degrees with the horizontal, and the exit angles the same?

Yes, the first trace was a parallel ray with an angle of 0. The exiting angle was -1.4082972487512724E-06. Or very, very close to zero.

 

[difalcojr]

Terrific! That's good news. You can get a number even closer to zero maybe if you keep lowering the initial ray angle too. So, you have ray trace solfware for thick lenses too. Excellent.
That's two points on the planar wave, exiting with the same angle, the wave reduced in half its size. All the other rays between horizontal axis and marginal ray will do the same if you check. Thanks.

 

[difalcojr]

Here's a last, spider-legged diagram and values. A good test for your software now.
Let me explain how to read this. It is to show ray/wave magnifications and optical path lengths. Under the minimum conditions. At the point when each wave starts to enter the lens, and at the point when each wave finally exits the lens. Same waves as from the other sequences shown. Marginal rays incoming, marginal rays outgoing. Here's some values you can check for the marginal and arbitrary rays. Minimum magnification ratios.

TYPE...............................ANGLE......................MAGNIFICATION.............................................................................................................
converging.......................30.................................10.24:1...........................................................................................................................
converging.......................15....................................3.97:1..........................................................................................................................
planar..................................0...........................................2:1...........................................................................................................................
diverging..........................15..............................................1:0.92.................................................................................................................
diverging..........................30..............................................1:1.71.................................................................................................................
diverging..........................45..............................................1:3.41.................................................................................................................

Hope this chart posts OK to read.
For optical path lengths only the lens itself needs be figured to get the OPL. This formula works at the minimum condition for the lens:

Minimum OPL=(radius1+radius2)(index1). Its value for the lens index of square root of 2 equals 2.4142...
This value is constant for every wave shown.

 

[difalcojr]

Just did a ray trace with an initial ray angle of 0.1 and ray height of 0 (hits the first surface in the center). The resulting exit angle was about 0.0999939. The difference from 0.1 is easily attributable to me only using 4 or 5 digits for the input values.Yes, the first trace was a parallel ray with an angle of 0. The exiting angle was -1.4082972487512724E-06. Or very, very close to zero.

So, I know you are probably super busy with other projects, and I've overloaded everyone with this, and especially you and hutchphd. And it's the weekend now, and it's been 2 weeks with this problem.
When you get a chance, or anyone else please, even teachers needing a student project, maybe, set up this lens model with those easy equations given. Use the diagram for the plane model. Ray trace some different angled rays through the model lens. Convince yourselves that the incoming and outgoing angles are the same for this model's shape and choice of indexes.
I truly expect someone in this forum to be able to confirm or disprove my contention that this model has zero SA. It's only geometrical optics and very easy, right? Thks.

 

[Drakkith]

I managed to slap together a quick design in Optical Ray Tracer (which I've never used before) but either limitations in the software or my own inexperience with it forced me to eyeball things. I was able to get the rays entering the second lens to look close to parallel. You can try it yourself, you just have to fiddle with the thickness of the first lens and the x-position of the 2nd lens to get the distances between the elements correct. Unfortunately it wasn't as easy as just inputting the values we had above, for reasons I don't know.

I basically set up two lenses, the first with asymmetric curvatures, setting the curvature of the 2nd surface (the 'left sphere' in the program) to 0.70710798, the curvature of the 1st surface of the second lens to -0.70710798, and making the 2nd lens very thick. The radius of each lens should be small, something like 0.5 or so. Smaller than the smallest radius of curvature. The second lens should have an x-position that puts it very, very close to the last surface of the first lens with only a tiny gap in between.

 

[Gleb1964]

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?

There is free OSLO EDU raytracing software which is limited up to 10 surfaces for free educational version.

 

[difalcojr]

I managed to slap together a quick design in Optical Ray Tracer (which I've never used before) but either limitations in the software or my own inexperience with it forced me to eyeball things. I was able to get the rays entering the second lens to look close to parallel. You can try it yourself, you just have to fiddle with the thickness of the first lens and the x-position of the 2nd lens to get the distances between the elements correct. Unfortunately it wasn't as easy as just inputting the values we had above, for reasons I don't know.

I basically set up two lenses, the first with asymmetric curvatures, setting the curvature of the 2nd surface (the 'left sphere' in the program) to 0.70710798, the curvature of the 1st surface of the second lens to -0.70710798, and making the 2nd lens very thick. The radius of each lens should be small, something like 0.5 or so. Smaller than the smallest radius of curvature. The second lens should have an x-position that puts it very, very close to the last surface of the first lens with only a tiny gap in between.

Valiant effort! Here's the model diagram again, explicitly, if you want to check that software against this one.