I plan to use a CNC mill to cut a plano-convex aspheric lens from a large (300mm dia) thick(30mm) block of Acrylic (PMMA/Plexiglass). I need a focal length on the order of 800-2000mm

The purpose of this large lens will be a lensed Camera Obscura that I hope will have a very bright image.

I have built a lensed camera obscura before with a 150mm dia 2000mm FL glass lens. Documentation of project: https://vimeo.com/94692036

I have also used CNC routers to cut lenses from PMMA before, but they were spheric and made on low-tolerance machines. The lenses I made are 300mm dia 1000mm FL (Plano-Convex PMMA) and are in this video: http://vimeo.com/66783407

How does one go about generating a lens profile? Is there parametric software for the creation of an optimal projection lens as I describe? I am new to optical design, and would love some advice. I had considered buying such a lens, but they are prohibitively expensive or rare.

  • 1
    Have you considered stealing the Fresnel lens from an old rear projection TV? – Chris Mueller Jun 12 '14 at 20:36
  • What awesome clips - the mechanism of your lunar persistence apparatus (are those wooden cogs?) are gorgeous. Since you made a PMMA lens in the past, is the curvature I suggested about right? It would be interesting to know. Light gathering "strength" of a lens goes as diameter / focal length so as you increase $f$ the image will get larger and less bright... You have 300/1000 as a reference (that is an f3.0 lens - considered pretty "fast" already. 300/2000 would be less bright.) – Floris Jun 13 '14 at 11:40
  • Thanks! If the lens I am buying turns out to be unsuitable, I will mill one out of PMMA. I asked a new question about the lens I am buying, to figure out the FoV and image circle. – Robb Godshaw Jun 13 '14 at 17:21
up vote 1 down vote accepted

My first reaction is - don't do it. The image quality of a single element lens is not very good - PMMA has significant chromatic aberration. The right thing to do would be to spend some money on a "real" lens. Here is the refractive index as a function of wavelength (from http://people.csail.mit.edu/jaffer/FreeSnell/pmma.png)

enter image description here

About a 1% difference in refractive index - so blue light (shorter wavelength) will be refracted more. At a distance of 1 m, for an image that is 40 cm diameter, a white point will be smeared out over 6 mm (with blue on the outside and red on the inside).

But assuming this is a "hobby" project and you just want to see what you can do, then you have two choices to make:

  1. convex-convex or plano-convex? Modifying just one surface will be significantly easier - only one surface to cut and polish.
  2. Spherical or aspherical? Aspherical will be "better", but I'm not sure it matters given the chromatic aberration you will get. Spherical is easier to compute.

So - let's assume you are going with plano-convex, spherical. Now we can do the math: we have a refractive index of (say) 1.5, and we want a focal length of f with a lens aperture r. With a source at infinity, you can compute the angle that you need as a function of r with a simple diagram (note - this looks a lot like the beginnings of a Fresnel lens...)

enter image description here

We know from Snell's Law that

$$\frac{n_1}{n_2} = \frac{sin(\theta_2)}{sin(\theta_1)}$$

and that

$$\alpha = \theta_2 - \theta_1$$

We can solve the above and find that the lens surface needs to a a paraboloid - the revolution of a parabola. Now using small angle approximations, where $\alpha = tan(\alpha) = sin(\alpha)$, and putting $n_2=1$ (air), we get

$$\begin{align} \frac{r}{f} &= \theta_2 - \theta_1\\ &= (n_1 - 1) \theta_1\\ \theta_1 = \frac{r}{f(n_1-1)} \end{align}$$

This tells us that the angle of the lens is proportional to the distance off-axis: the equation of a parabola. You can use this equation to compute the shape for a given focal length and refractive index. For example, if you want a 2000 mm focal length, you would get

$$\theta = \frac{r}{2000 (1.5 - 1)}\\ = 0.001 \frac{r}{mm}$$

Translating this to thickness x of the lens:

$$\frac{dx}{dr} = - 0.001 r\\ x = const - 0.0005 (r/mm)^2$$

If we say the lens is zero thickness at 150 mm, then it needs to be 11.25 mm thick in the middle for a 2 m focal length, with a parabolic profile.

That should be enough to get you going... but expect to be making a few iterations before you are happy with the result. Polishing a lens to even a semblance of optical accuracy is hard work - and as I mentioned, chromatic aberration will be pretty severe. Fun project though.

  • Amazing response. Thank you! I have a few new questions— is a parabolic lens spheric or aspheric? I would assume aspheric, but your answer implied otherwise. I found a real lens that seems like a good fit for a camera obscura. 45"in FL and 12" Diam. (Spherical, glass) for $288. I might go for that instead. – Robb Godshaw Jun 13 '14 at 1:15
  • Cool picture on Dropbox! I actually computed it for 150mm radius, not diameter. Your lens will have a focal length of 1 m - and if you start with a 30 cm piece of PMMA there will be a lot of waste... The lens you found online sounds like a reasonable (and quite possibly higher performance) alternative - but wouldn't it be fun to try... Oh - and I started out talking about spherical but ended up parabolic. Honestly with these dimensions it won't make much difference, but parabolic will have slightly lower aberration. And for the CMC it's the same. – Floris Jun 13 '14 at 2:57
  • Ahh- I understand. I might try it for fun. I figure it will be brighter, even if the aberration is high. – Robb Godshaw Jun 13 '14 at 3:52

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