# Questions about the Paraxial region for a single spherical surface

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• difalcojr
difalcojr
TL;DR Summary
Is the paraxial equation for a spherical surface an exact equation or an approximate equation? It gives an exact answer, an exact, axial measurement, so it is exact. However, it is only derived using approximations for the very small, paraxial angles and axial distances, to my knowledge. Every text author has used a slightly different derivation with their own chosen, model variables. Does this matter in a logical, mathematical way? Is it a moot point, or just semantics, do you think?

Paraxial rays are an approximation, yes. Consider that in the paraxial approximation you can have an arbitrarily thin lens with an arbitrarily large diameter and an arbitrarily short focal length. That isn't realistic.

Yes, I agree that isn't realistic.

Consider, though, that this equation itself measures only one position of the one ray, the axial ray, at the end point of all the other rays' central axis crossings, the focal point, the point of focus, the brennpunkt.
Why is it then used to approximate the other rays in the paraxial area close to it?

Trigonometry would seem to be the exact method to measure all incident rays, including paraxial rays, right up to, if not exactly at, the focal point.

Solution that you have is exact. Paraxial would assume ##tan( \alpha) = sin( \alpha ) = \alpha##

difalcojr
Yes, thanks, many authors use these approximations. Along with the other approximations they use in the derivation. Incident 'iota' and transmit 'tau' angles in the diagram are often approximated. Sometimes, other lengths, axial, or off-axis, are included as approximations in the derivation. All text authors I've seen derive a little differently, but the equation is the same. All have their own variables, their own 'standard' sign conventions.

Mathematically, though, it is always an approximate derivation of an exact equation. I have not seen an exact derivation, but that may not really matter at all, or I've just missed it.

I have not seen an exact trigonometry program for the non-paraxial region, either, though there may, undoubtedly, be many.

berkeman
Yes, both, I agree. I have lots of questions. I didn't want to ask them all in the OP, just one at a time. Here's another I would like others' opinion on. I think everyone is in agreement with the following statements:

Axial measurements can be exactly determined up to an axial point that is very close to the focal point itself, using the methods of trigonometry. The algebraic paraxial equation is that exact equation needed in addition to the trig methods to determine the focal point of the un-refracted, axial ray.

Also, if r=1 unit, d=1 unit, index1=1, and index2=1.5, the focal point is 3 units left of vertex! You might think it was on the right side of the vertex from the diagram. I always did. Authors don't tell you this that I have seen.
See its full, 30 degree wave diagrammed for rays in the middle plot below.
S for Source point, M for marginal ray, A for axial ray.

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I don't think I understand what's being asked in this thread. Are you asking about exact ray tracing, or the paraxial approximation, or the relationship between them?

difalcojr
Yes, all three.

Well, I started with the paraxial equation because no textbook I've seen gives an exact derivation of the standard algebraic form. Eq. (5) in the OP. I found an easy, exact derivation using only trigonometry, and without any approximations. I thought I might publish that here in a post for the other members to check out, if anyone wanted to see what it looked like.

Also, I think I could show that trigonometric ray tracing is the easiest and best way to measure exact lengths and angles in lens designs.

In textbooks the paraxial equation seems to be the "end" of further explorations of other areas on a spherical surface. I know this is somewhat of an exaggeration. Algebra takes over the job of measurements in the paraxial area. I think I can show that pure, trigonometric measurement is a better method.

If I'm wrong, you can all really have a lot of fun letting me have it.

difalcojr said:
I think I can show that pure, trigonometric measurement is a better method.
Can you quantify this claim? Based on the spectrum of lens examples you've compared, typically by how much does the paraxial approximation differ from your trigonometric approach, expressed, say, in percent?

difalcojr
difalcojr said:
Well, I started with the paraxial equation because no textbook I've seen gives an exact derivation of the standard algebraic form. Eq. (5) in the OP.
I understand equations 1-4 (sine rule twice, angles in a triangle and Snell's Law). I don't see how equation 5 follows, though. Can you explain how you got from your equations 1-4 to equation 5?

Note, for example, that according to equation 5 you can set ##n_1=1##, ##n_2=3/2##, ##b=r## and get ##d=\pm r##. Taking the positive solution yields ##t=0## and ##i\neq 0##, which is not consistent with Snell's Law. So I suspect you have made a mistake somewhere. Furthermore, the diagrams you posted in #8 are consistent with the failure of the paraxial approximation at large angles, which is not consistent with the existence of an approximation-free derivation.

Equation (1) is valid only for i=90 degrees angle. Otherwise it should be ## \sin ( \pi /2 - i)## in the equation.
$$\frac {d+r}{\sin ( \pi /2 - i)} = \frac {r}{\sin ( \alpha_{1})}$$

Gleb1964 said:
Equation (1) is valid only for i=90 degrees angle. Otherwise it should be ## \sin ( \pi /2 - i)## in the equation.
I think it should be ##\sin(\pi-i)## actually, since ##i## is the angle between the outward normal and the incident ray and the interior angle of the triangle (which is what we want) is the angle between the incident ray and the inward notmal. But ##\sin(\pi-i)=\sin(i)##, so it works out.

Yes, right, ## (\pi - i) ##

Ibix
Ibix said:
I understand equations 1-4 (sine rule twice, angles in a triangle and Snell's Law). I don't see how equation 5 follows, though. Can you explain how you got from your equations 1-4 to equation 5?

Note, for example, that according to equation 5 you can set ##n_1=1##, ##n_2=3/2##, ##b=r## and get ##d=\pm r##. Taking the positive solution yields ##t=0## and ##i\neq 0##, which is not consistent with Snell's Law. So I suspect you have made a mistake somewhere. Furthermore, the diagrams you posted in #8 are consistent with the failure of the paraxial approximation at large angles, which is not consistent with the existence of an approximation-free derivation.
Eq. (5) derivations can be found in optics textbooks. I did not want to repeat any here. They are really all a little different from each other. All non-Cartesian in some way. I've looked over a bunch of them.

Not sure why you would set d=r unless the source/sink was the origin of the sphere.

I thought the paraxial equation by itself was the exact location of the axial, focal point, if one solved for b in Eq. (5) in the OP. But only exact for that one point! For the values of b very close to it, the trig equation between (4) and (5) could be used for exact measurement.

You make a point, yes, not sure why it is not consistent, but I can derive Eq. (5) without approximations, I think I did.

difalcojr said:
I can derive Eq. (5) without approximations,
Then show us how you did it. I don't think it's possible for reasons already stated, so I think there's a mistake.

OK, sure, no problem. Give me a day to get it presentable. Just in rough notes now.
Will derive starting from that equation between Eq. (4) and (5) in the OP.
Don't know LaTex from oil-based, either, though.

I'll answer you too, renormalize after that too. Thanks for your feedbacks.

OK, here it is.

So you did make a small angle approximation: that all the cosine terms are approximately one.

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I assume you are referring to the equation just before the paraxial equation, the one with all the cosine terms. At the top of pg.3 above. That is an exact equation that could probably be used to measure paraxial region points around the focal point itself, yes.

The next equation does give the location of the focal point for the axial ray, though, when all angles have reached their limit of zero, and there is no angular refraction. The cosine terms are allowed to reach their limit of one. I don't see any approximations.

In your earlier post, you absolutely analyzed the first diagram to the point! You're right.
Eq.(5) in OP cannot be deduced from those equations as listed. I think this is because it is just the geometric model, without any needed sign conventions. I also tried but couldn't derive to Eq.(5), either. I had just wanted to show a standard model diagram and its geometric equations in the OP. All the text authors use their own sign conventions. Those used in my derivation are different from the others too as seen above.

renormalize said:
Can you quantify this claim? Based on the spectrum of lens examples you've compared, typically by how much does the paraxial approximation differ from your trigonometric approach, expressed, say, in percent?
I can quantify my measurements in a paraxial region, yes. Some shown below.
Don't know the answer to your second question. I do not have any other paraxial measurements from a single surface or single, thin lens to compare.
Here's some measurements for the 30 degree incident wave in the middle diagram below:

[Mentor Note: See post #25 below for a better resolution version of these images]

And the 30 degree numbers and a planar wave's numbers:

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difalcojr said:
I assume you are referring to the equation just before the paraxial equation, the one with all the cosine terms. At the top of pg.3 above. That is an exact equation that could probably be used to measure paraxial region points around the focal point itself, yes.
Assuming you haven't made any arithmetic slips it's an exact equation that could in principle be used for any ray passing through a spherical surface. I don't understand why you think "paraxial" has any place in this, unless you are seeing disregarding the cosine terms as a small angle approximation.

I think the only place the paraxial equation has in this is for the focal point itself. The equation when the 4 angles finally go to zero, and there's no more trig in it. When it's turned into an algebraic equation.

I was also just curious to see how close my trigonometry program would compare to others as an exact, measurement tool for the paraxial region and other areas on a spherical surface.

Don't think I can read those numbers posted above, drat! Too small and I used bold! Can anyone else read them? They show a comparison of axial measurements for points very near to the focal points.

renormalize said:
Can you quantify this claim? Based on the spectrum of lens examples you've compared, typically by how much does the paraxial approximation differ from your trigonometric approach, expressed, say, in percent?
Here's the same as above (post 22) in small but legible numbers, hopefully:

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