Mathematical shape for a lens with no spherical aberration

[jambaugh]

Yet another note... In the quadratic formula the case \pm\to - relied on \rho > 1. If you're working with the \rho < 1 case you must change the sign of the radical of the discriminant.

 

[Redbelly98]

Hi James,

Yes, I realized after my earlier post that for the air→glass case, rays at a large enough angle would simply miss the lens, so the negative discriminant (no solution) makes sense to me now.

Yet another note... In the quadratic formula the case \pm\to - relied on \rho > 1. If you're working with the \rho < 1 case you must change the sign of the radical of the discriminant.

Yup, I noticed this in my spreadsheet.

Interestingly, as A→∞ for air→glass, the surface resembles a hyperbola. I've not confirmed whether it is an actual hyperbola or not, but the asymptotes are clearly evident and make sense physically too. I will post a sample graph shortly.

 

[Redbelly98]

For this plot:

n1 = 1
n2 = 1.5
ρ = n1/n2 = 2/3
Other info as given in graph

 

[jambaugh]

Hi James,
[...] Interestingly, as A→∞ for air→glass, the surface resembles a hyperbola. I've not confirmed whether it is an actual hyperbola or not, but the asymptotes are clearly evident and make sense physically too. I will post a sample graph shortly.

It appears locally to be a hyperbola but note that there is a symmetry in reversing the two foci and refraction indices. You get the same curves as with \rho > 1. The analytic curves continue to enclose the far focus. [See attached plot.]

If you graph both solutions to the quadratic then I think you'll get the other half of the curves.

 

[Redbelly98]

I'm finding a problem with this solution. It seems that Snell's Law is violated, if I have done things correctly.

When I have time I can show my derivation, but for now I'll just outline what I did.

We have, for the lens surface points (z,x):

z = A r cos(θ)
x = A r sin(θ)​

where we use the function r(θ) as shown in Post #22.

Then

dz/dθ = A ( dr/dθ cos(θ) - r sin(θ) )
dx/dθ = A ( dr/dθ sin(θ) + r cos(θ) )​

We can calculate the angle of the surface normal from

θ_norm = atan(-dz/dx)

= atan( -(dz/dr) / (dx/dr) )
​

From the normal angle, we can caculate the angles of incidence and transmission, and use the refractive indices to check whether Snell's Law holds.

I'm seeing a violation of Snell's Law. I'll try to post the derivation later, and perhaps somebody can either confirm it or point out an error.

Mark

 

[Dadface]

Hello James and Mark its me again (Mr diffraction man )I would just like to say that I admire your persistence and I hope you arrive at an answer.If not it will not be for the want of trying.This may be a silly question but have you consulted a textbook on the subject or do you want to get there completely by your own devices?Good luck.

 

[Hobnob]

Hi all

Just come back from my holiday to find that my question has produced quite a lively discussion :) I haven't had a chance to look through it in detail, but just a couple of quick replies:

Regarding lies-to-children: for those who don't recognise the expression, see Ian Stewart, Jack Cohen and Terry Pratchett 'The Science of Discworld' (and other books by Stewart and Cohen too). I use it to mean the simplified examples and explanations we learn as children, which are gradually replaced with more detailed and sophisticated versions. (The negative phrase is only used for humorous effect - of course it's a necessary part of learning) In this case, to my shame I hadn't even heard of spherical aberration until I tried to make a lens model and realized that all my assumptions were wrong.

Regarding diffraction: Of course it's true that diffraction effects are going to be an issue in real life, but my aim here is specifically *not* to model reality but to give kids the picture that teachers expect. They're trying to teach focal length and magnification, and don't need complications to confuse the poor children's minds at this stage! For reality they have actual lenses...

Thanks to all for the input, it's very useful. And just to let you know, I've had some success with my faked solution - I'm using a parabolic lens and tweaking it to give the result I need.

Best
Hob

 

[Dadface]

Cylindrical lenses which stand on the bench are an excellent aid to learning For any educators out there may I suggest that as an introductory lesson you give each group of kids a selection of lenses, mirrors ,glass blocks a large sheet of white paper and a ray box.You need a good black out(I think it was Goethe who said all experiments on light must be done in the dark.Just tell the kids to get on with it, find out what they can and make one or two rough sketches ( the neat diagrams can come later).They will love it but it will tire you out as different groups of kids call you over to see the spectrum or whatever else it is they discover.Try to maintain your own enthusiasm even though you have seen it all many times before.Be prepared because it is likely that you will get many kids asking to borrow a lens to take home . If this is the case the cheap plastic spherical lenses are the best option .Another excellent lesson(probably done best before the lesson above) is for the kids to make their own pinhole cameras but using a box they bring from home The size of the box does not matter (even the lab itself can be a camera the hole being a hole in the blinds )All you will need are some scissors,tape and tracing paper.You should make the holes yourself .You will find several of the kids reporting back to you the next day telling you telling you that their family members were trying out the camera and giving other similar reports.It is strange that in these days of sophisticated devices such as camera phones the humble pinhole camera is still fascinating to both adults an children alike.

 

[jambaugh]

Hello James and Mark its me again (Mr diffraction man )I would just like to say that I admire your persistence and I hope you arrive at an answer.If not it will not be for the want of trying.This may be a silly question but have you consulted a textbook on the subject or do you want to get there completely by your own devices?Good luck.

To answer your last question, No I haven't consulted any texts on the subject. I did note via brief web search that Descartes had worked out that a hyberbolic lens was ideal for focusing a plane wave to a point but I didn't find a good exposition of his derivation.

I may have an error in my derivation or in my original "equal time" assumption. I did do some spot checks of Snells law for the curves I worked out and they appeared pretty close. I however was using numerical calculations built into the the function graphing software and may have missed small scale deviation from Snells law. I haven't been looking at this much recently, (instead I've been working out sign conventions on covariant classical electrodynamics.)

 

[electricsbm]

I'm using a parabolic lens and tweaking it to give the result I need.

Best
Hob

Interestingly, Huygens computed the shape of a perfect lens in terms of geometrical optics by a simple 'recipe'. Idea was: Between two perfectly conjugate points all rays traverse the same optical distance. Therefore, one can assume that the object point is producing a spherically diverging light. The so-called geometric-optics-wise perfect lens (according to its NA) captures a cap of this sphere and converts it into converging spherical cap. This converging spherical cap should be centered at the image point. There is a paper http://www.iop.org/EJ/article/0143-0807/29/3/014/ejp8_3_014.pdf (Fig.19) which illustrates such procedure by assuming one face of the lens to be spherical. If you cannot access this paper, PM me and I shall send you the PDF.

Hope this helps

 

[Redbelly98]

This may be a silly question but have you consulted a textbook on the subject or do you want to get there completely by your own devices?Good luck.

Hmmm, I hadn't looked in a book because James's approach made sense to me when he first proposed it. I have just now checked my copy of Hecht & Zajac's Optics, and it confirms what James said: all rays take the same amount of time to travel from the source point to the image point. However, H&Z do not actually derive the equations for the surfaces.

I may have an error in my derivation or in my original "equal time" assumption. I did do some spot checks of Snells law for the curves I worked out and they appeared pretty close. I however was using numerical calculations built into the the function graphing software and may have missed small scale deviation from Snells law.

Something I didn't mention earlier: I do see agreement with Snell's Law when the image is at infinity. It's the finite-distance case where I am having problems with Snell's Law. I haven't spent much time with it in the last couple of days, but I do want to figure out what's going on.

Interestingly, Huygens computed the shape of a perfect lens in terms of geometrical optics by a simple 'recipe'. Idea was: Between two perfectly conjugate points all rays traverse the same optical distance.

Welcome to PF! Thank you. That should be equivalent to the "equal time" principle:

∑ n×distance = constant​

So I'm confident that James's approach is the right track.

Happy New Year, everybody.

Mark

 

[jambaugh]

That should be equivalent to the "equal time" principle:

∑ n×distance = constant​

Yes, the "optical distance" is simply the time of travel times c, e.g. the time of travel in light-meter units.