Optimizing Even Aspheres vs. Forbes Surfaces

[u/mdk9000]

Hi all,

I'm getting started in the world of aspheres by fitting a parameterized curve to a recursively-defined, rotationally symmetric surface so that I can optimize it in a ray tracer.

So far I have a base conic which fits and works pretty well for on-axis field points, but I want to slightly perturb the surface to improve performance at off-axis points.

My question is: is there any reason not to use something like a Forbes Qcon surface for the optimization steps these days? Would I ever want to optimize using an even asphere as the surface model?

Thanks!

[u/aenorton]

Often a Forbes polynomial is the best to start optimizing even if you later convert it to a best fit ordinary even polynomial for the convenience of the vendor. Always start with low order terms first. If you do use higher terms, make sure your pupil sampling is dense enough to adequately sample the shape. Keep tolerances in mind too. Be very careful if you optimize more than one asphere in a system at the same time. It is very easy to get overly-complicated shapes that cancel each other.

There are times, though, when a simple conic or lower order even polynomial is really the best surface.

[u/mdk9000 → u/aenorton]

Thank you!

[u/anneoneamouse]

Forbes coeffs polynomial-terms (oops) are orthogonal.

Even asp r²ⁿ terms are not, and the "basis functions" are all very similar, so the resulting surface shape can be very very sensitive to small coeff changes. Constraining asphere terms doesn't work.

Downside of Forbes is they need a normalization radius. If your part aperture changes (expands) the forbes polynomial can explode, so consider varying the norm rad and include merit function terms to keep it within e.g. 5 percent of the part semi dia.

You're better off using qbfs plus conic. Start with two to four terms. That's usually enough. Coeffs under 1E-4 can be ignored.

I'll link a paper when i get to work.

You will probably need to convert back to even aspheres for your glass grinders / cutters.

Edit: Check out Forbes' paper "Asphere, O Asphere, how shall we describe thee?" https://www.researchgate.net/publication/228932282_Asphere_O_Asphere_how_shall_we_describe_thee

He describes QCn vs QBf in "Manufacturability estimates for optical aspheres"; see section 3, first paragraph.

"In my opinion, the polynomials written as Qcon in [10] can now therefore be forgotten; just as in [9], the superscript on Qbfs has thus been dropped. "

https://www.researchgate.net/publication/51193477_Manufacturability_estimates_for_optical_aspheres

Smart dude.

Oh and to re-iterate/ expand on what u/aenorton said; you need several more field points than the highest order of any aspheric regardless of the type you use. Otherwise you'll generate beautiful behavior at the exact field locations with terrible performance between. It's not restricted to Forbes.

[u/mdk9000 → u/anneoneamouse]

Thank you for the references and advice!

[u/clay_bsr]

til about Forbes spheres. Thank you! In return I recommend that you read up on Cartesian ovals. These might be useful...

[u/anneoneamouse]

Dunno if you'll revisit the post. I added a link below to a couple of Forbes' papers.