Creating large aspheric lens profile for CNC cutting out of Acrylic plastic for huge Camera Obscura 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.
 A: 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)

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:


*

*convex-convex or plano-convex? Modifying just one surface will be significantly easier - only one surface to cut and polish.

*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...)

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.
